Minimising the Decoherence of Rare Earth Ion Solid State ......Minimising the Decoherence of Rare...
Transcript of Minimising the Decoherence of Rare Earth Ion Solid State ......Minimising the Decoherence of Rare...
Minimising the Decoherence
of Rare Earth Ion
Solid State Spin Qubits
Elliot Fraval
A thesissubmitted for the degree
ofDoctor of Philosophy
of theAustralian National University
The Australian National UniversityJuly 2005
Statement of authorship
This thesis contains no material which has been accepted for the award
of any other degree or diploma in any university. To the best of the author’s
knowledge and belief, it contains no material previously published or written
by another person, except where due reference is made in the text.
Elliot Fraval
August 22, 2006
iv
“In the beginning the Universe was created. Thismade a lot of people very angry and has been widelyregarded as a bad move.”
– Douglas Adams
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Acknowledgements
Firstly, I would like to thank my primary supervisor Matt Sellars. I havethoroughly enjoyed working with Matt and have come to respect his talentfor problem solving and enthusiasm for this area of physics. I’d also like tothank Neil Manson who as my supervisor and the head of our group hasbeen fundamental in creating the excellent working environment within theSolid State Spectroscopy group. Their interest in not just physics but life ingeneral has contributed greatly to the experience of my study.
Particular thanks go to the Laser Physics Centre technicians, withoutwhom the project would simply not have happened. Many thanks to IanMcRae, the god of all cryostats, John Bottega for his award winning singing,comedy, liquid helium and, like Craig McLeod, his invaluable hands. Thanksfor helping me out in the workshop to get my head around the vast array oftools.
It has been a pleasure to share an office with Jevon Longdell and JoHarrison. I thank both of them for putting up with my atonal singing,extreme musical taste and slapping out the occasional funk bass line to keepme sane in the lab. Big thanks to the lunch time crew for their laughs andbizarre conversational tangents pursued well beyond their reasonable ends.
Big thanks and group hugs to all of my muso friends that have helpedme keep the correct balance of sanity and lack of it with the free psychiatrythat playing original music is. In particular Nic, Marky, Dan and Candiethat make up the rest of eyTis as well as the indominable Red Rocko andThe Whalebone from The Milk. Youse guys rock! Thanks for keepin it realin the arts end of town :)
Last but by no means least thanks to all of my family and friends thathave had to put up with me over the last few years. Definitely thanks toIrma for her editorial prowess and Amelie for her abstract corrections. Tothose friends I haven’t had the spare time for I will call.... no really, I will!
Elliot FravalJuly 2005
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Abstract
This work has demonstrated that hyperfine decoherence times sufficiently
long for QIP and quantum optics applications are achievable in rare earth ion
centres. Prior to this work there were several QIP proposals using rare earth
hyperfine states for long term coherent storage of optical interactions [1, 2, 3].
The very long T1 (∼weeks [4]) observed for rare-earth hyperfine transitions
appears promising but hyperfine T2s were only a few ms, comparable to rare-
earth optical transitions and therefore the usefulness of such proposals was
doubtful.
This work demonstrated an increase in hyperfine T2 by a factor of ∼7 × 104 compared to the previously reported hyperfine T2 for Pr3+:Y2SiO5
through the application of static and dynamic magnetic field techniques.
This increase in T2 makes previous QIP proposals useful and provides the
first solid state optically active Λ system with very long hyperfine T2 for
quantum optics applications.
The first technique employed the conventional wisdom of applying a small
static magnetic field to minimise the superhyperfine interaction [5, 6, 7], as
studied in chapter 4. This resulted in hyperfine transition T2 an order of
magnitude larger than the T2 of optical transitions, ranging fro 5 to 10 ms.
The increase in T2 was not sufficient and consequently other approaches were
required.
Development of the critical point technique during this work was crucial
to achieving further gains in T2. The critical point technique is the applica-
tion of a static magnetic field such that the Zeeman shift of the hyperfine
transition of interest has no first order component, thereby nulling decoher-
ing magnetic interactions to first order. This technique also represents a
global minimum for back action of the Y spin bath due to a change in the
Pr spin state, allowing the assumption that the Pr ion is surrounded by a
thermal bath. The critical point technique resulted in a dramatic increase of
the hyperfine transition T2 from ∼10 ms to 860 ms.
Satisfied that the optimal static magnetic field configuration for increas-
ing T2 had been achieved, dynamic magnetic field techniques, driving ei-
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ther the system of interest or spin bath were investigated. These tech-
niques are broadly classed as Dynamic Decoherence Control (DDC) in the
QIP community. The first DDC technique investigated was driving the Pr
ion using a CPMG or Bang Bang decoupling pulse sequence. This sig-
nificantly extended T2 from 0.86 s to 70 s. This decoupling strategy has
been extensively discussed for correcting phase errors in quantum computers
[8, 9, 10, 11, 12, 13, 14, 15], with this work being the first application to solid
state systems.
Magic Angle Line Narrowing was used to investigate driving the spin
bath to increase T2. This experiment resulted in T2 increasing from 0.84 s to
1.12 s. Both dynamic techniques introduce a periodic condition on when QIP
operation can be performed without the qubits participating in the operation
accumulating phase errors relative to the qubits not involved in the operation.
Without using the critical point technique Dynamic Decoherence Control
techniques such as the Bang Bang decoupling sequence and MALN are not
useful due to the sensitivity of the Pr ion to magnetic field fluctuations.
Critical point and DDC techniques are mutually beneficial since the critical
point is most effective at removing high frequency perturbations while DDC
techniques remove the low frequency perturbations. A further benefit of
using the critical point technique is it allows changing the coupling to the
spin bath without changing the spin bath dynamics. This was useful for
discerning whether the limits are inherent to the DDC technique or are due
to experimental limitations.
Solid state systems exhibiting long T2 are typically very specialised sys-
tems, such as 29Si dopants in an isotopically pure 28Si and therefore spin free
host lattice [16]. These systems rely on on the purity of their environment
to achieve long T2. Despite possessing a long T2, the spin system remain
inherently sensitive to magnetic field fluctuations. In contrast, this work has
demonstrated that decoherence times, sufficiently long to rival any solid state
system [16], are achievable when the spin of interest is surrounded by a con-
centrated spin bath. Using the critical point technique results in a hyperfine
state that is inherently insensitive to small magnetic field perturbations and
therefore more robust for QIP applications.
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Contents
Acknowledgements vii
Abstract ix
1 Introduction: From Classical to Quantum Information 1
1.1 Classical Information Processing . . . . . . . . . . . . . . . . . 2
1.1.1 Theoretical Developments . . . . . . . . . . . . . . . . 2
1.1.2 Towards Quantum Hardware . . . . . . . . . . . . . . . 4
1.1.3 Information goes Quantum . . . . . . . . . . . . . . . . 5
1.1.4 The Power of Hilbert Space . . . . . . . . . . . . . . . 7
1.2 Quantum Computing Requirements . . . . . . . . . . . . . . . 8
1.3 The Two Level Atom . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 The Density Matrix . . . . . . . . . . . . . . . . . . . . 12
1.3.2 Dynamics on the Bloch Sphere . . . . . . . . . . . . . 14
1.3.3 Quantum Process Tomography . . . . . . . . . . . . . 16
1.4 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.1 System Bath Interactions . . . . . . . . . . . . . . . . 20
1.4.2 Decoherence on the Bloch Sphere . . . . . . . . . . . . 22
1.5 Error Correction . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5.1 Quantum Error Correction Codes . . . . . . . . . . . . 24
1.5.2 Decoherence Free Subspaces . . . . . . . . . . . . . . . 25
1.5.3 Dynamic Decoherence Control . . . . . . . . . . . . . . 25
1.5.4 A Quiet Corner of Hilbert Space . . . . . . . . . . . . . 26
1.6 Why the Rush? . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.7 Other application and fundamental interest . . . . . . . . . . . 27
2 Rare Earth Ion Spectroscopy 29
2.1 Introducing The Lanthanides . . . . . . . . . . . . . . . . . . 30
2.2 4f Energy Levels . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Hyperfine Interaction . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.1 Yttrium Orthosilicate: The Gracious Host . . . . . . . 36
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2.3.2 Hyperfine Interaction in Praseodymium Doped Y2SiO5 37
2.3.3 M and Q . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 Homogeneous and Inhomogeneous Broadening . . . . . . . . . 41
2.5 Optically Detected NMR and Coherent Transients . . . . . . . 42
2.5.1 Raman Heterodyne . . . . . . . . . . . . . . . . . . . . 42
2.5.2 Spin Echos . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5.3 Spin Echo Decays . . . . . . . . . . . . . . . . . . . . . 46
3 QC Benchmarks and Benefits of Rare Earth QC 50
3.1 Rare Earth Ion ODNMR Quantum Computing . . . . . . . . . 51
3.2 Rare Earth Quantum Computing Architecture . . . . . . . . . 52
3.3 Rationale for System Comparison . . . . . . . . . . . . . . . . 57
3.4 Liquid Phase NMR . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.1 Limitations of Liquid State NMR . . . . . . . . . . . . 61
3.5 The Case for Solids . . . . . . . . . . . . . . . . . . . . . . . . 62
3.6 Limitations of Rare Earth QC due to Hyperfine Decoherence . 64
4 Hyperfine Decoherence with Small Applied Mangetic Field 66
4.1 Pr3+:Y2SiO5 Hyperfine Decoherence . . . . . . . . . . . . . . 66
4.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 Maximising Hyperfine T2 using Static Magnetic Fields 79
5.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2.1 Finding a Critical Point . . . . . . . . . . . . . . . . . 86
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.4.1 Future Improvements . . . . . . . . . . . . . . . . . . . 97
6 Dynamic Decoherence Control 99
6.1 Application to Pr3+:Y2SiO5 . . . . . . . . . . . . . . . . . . . 100
6.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . 104
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.4.1 Bang Bang Process Tomography . . . . . . . . . . . . 110
7 Extending T2 Through Driving the Environment 116
7.1 Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
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8 Future Decoherence Challenges 124
8.1 Exchanging Praseodymium for Europium . . . . . . . . . . . . 124
8.2 Stoichiometric Materials . . . . . . . . . . . . . . . . . . . . . 125
8.3 Considerations for QIP in Stoichiometric Materials . . . . . . 126
8.4 Minimising Decoherence in Stoichiometric Defect QIP Systems 127
9 Conclusions and Future Work 129
9.1 Strategies for Further Increases in Decoherence Time . . . . . 132
9.1.1 Improved RF Control . . . . . . . . . . . . . . . . . . . 132
9.1.2 Rabi Frequency and Inhomogeneous Broadening . . . . 132
9.1.3 Eulerian Decoupling . . . . . . . . . . . . . . . . . . . 133
9.2 Other Applications For Long T2 Optically Active Solids . . . . 133
9.2.1 Slow and Stopped Light . . . . . . . . . . . . . . . . . 133
9.2.2 Stark Echo Quantum Memory . . . . . . . . . . . . . . 134
Appendices 135
A Y2SiO5 site position calculation 136
B Full Critical Point List 150
C Published Papers 153
Bibliography 175
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xiv
Chapter 1
Introduction: From Classical to
Quantum Information
“I think there is a world market for maybe five computers.”
- Thomas Watson (1874-1956), Chairman of IBM, 1943
“There is no reason for any individual to have a computer in his home.”
–Ken Olsen, President, Digital Equipment, 1977
The time we live in is often referred to as the ‘information age’ since the
ability to gather and process information is crucial for the decision making
process of any entity. Modern computing enables information processing
tasks that were previously impossible as a result of the dramatic advances
made by computing technology. The rate at which information processing
technology has advanced is unprecedented in human experience, prompting
the question: what are the fundamental limits to processing information in
our Universe? Our knowledge of physics tells us this limit is defined by
quantum physics. Exploring these limits will result in advances to both
information science and quantum physics.
The first generation of quantum communication systems that exploit the
measurement principals of quantum mechanics to secure the transmitted in-
formation, known as quantum cryptography, have already been deployed.
Current research is trying to create the quantum analogue of other required
building blocks of information processing, being quantum memories and
quantum computers.
2 Introduction: From Classical to Quantum Information
Fundamental to the new information processing applications of quantum
mechanics is the ability to preserve superposition states. Superposition states
can be observed in a vast range of systems, such as polarisation states of light,
electronic states of atoms, nuclear spin orientation in a magnetic field and
vibrational state of trapped ions. The loss of superposition information of
large quantum systems is what gives rise to classical physics [17, 18]. This
work focuses on methods to preserve superposition states of hyperfine nuclear
spin states such that they are useful for implementing quantum information
processing technology.
1.1 Classical Information Processing
Algorithmic processes and their implementation have been important for
many cultures throughout human history for tasks as diverse as calculat-
ing the seasons to economics and construction. Physical systems were often
sought to implement algorithms that were difficult to perform mentally, par-
ticularly if some measurement was required. In this context, many ancient
structures, from Stonehenge to the pyramids form in part a physical imple-
mentation to aid the calculation of seasonal variables. Such structures are
clearly not reprogrammable and implementation of the algorithm was still
performed by the user. For the development of engineering and economics it
quickly became important for numeric representations where several parties
could agree on the outcome. Flexible structures such as the abacus were
developed, however the algorithm was always implemented by the user.
1.1.1 Theoretical Developments
While there were several attempts to create early mechanical computers de-
signed for specific calculations, the basis of modern computer science was
created by Alan Turing in 1936 [19]. Turing developed the concept of a
Turing Machine, a machine that implements a calculation by following an
algorithmic process. Theoretical investigations centre on the Universal Tur-
ing Machine, one that could simulate any other Turing machine.
Turing was not alone in considering this problem with significant contri-
butions being made by Alonso Church, Kurt Godel and Emil Post. Despite
each description initially appearing different, due to the different perspec-
tives on algorithmic computing, they were shown to be equivalent models
[20]. This reinforced the universality of the models, and since the Turing
Machine provided a common reference for both hardware and software it
1.1 Classical Information Processing 3
became the standard conceptual reference.
Turing considered that any physical system that performed an algorithmic
process was in some sense a Turing machine. Therefore a universal Turing
machine could simulate any physical algorithmic process. This strengthened
assertion is known as the Church-Turing Principle, stated as:
Every ‘function which could be naturally computable’ can be computed by
the universal Turing machine.
Computer science changed very quickly from a mathematical curiosity to
a fundamentally important area of technology in the years following 1936.
It became important to understand whether a computer could provide an
answer in a reasonable amount of time, how that time changed as the com-
plexity of the problem increased and consequently how efficient the algorithm
was.
An algorithm’s efficiency is the scaling of the resources required to achieve
a solution as the size of the problem is increased. Resources in this context
are the number of steps required in the computation and the amount of in-
formation required to be stored by the algorithm. The definition that arose
was that if the resources required to implement the algorithm were bounded
by a polynomial in the complexity of the problem then it is ‘efficient’. Many
algorithms require an exponential increase in resources and are consequently
considered ‘inefficient’. Algorithms are typically compared by the resources
they consume as the problem tends to large size. In this limit efficient al-
gorithms will always outperform inefficient algorithms. Discussions of this
nature resulted in a strengthening of the Church-Turing assertion that any
algorithm could be implemented on a universal Turing machine efficiently.
Shortly after the strengthened Church-Turing principal was presented,
stochastic algorithms were developed that solved problems more efficiently
than deterministic algorithms [21]. Although each iteration of the algorithm
has a finite probability of failing, stochastic methods will typically converge
to a solution much faster than analogous deterministic algorithms. There-
fore the definition of the conceptual efficient universal computing machine
required modification to a probabilistic Turing machine.
The universality of the Turing machine approach taken by computer sci-
entists is verified by the vast array of computing applications. There are,
however, many processes that do not have distinct algorithmic states. Con-
sequently measuring and processing information in a discreet state space such
that it mimics a continuous process presents many limitations and necessi-
tates approximations.
4 Introduction: From Classical to Quantum Information
1970 1975 1980 1985 1990 1995 2000 200510
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Figure 1.1: Moores law has been a remarkably simple description of the increase incomplexity of computer chips. Data courtesy of the Intel Website, www.intel.com
1.1.2 Towards Quantum Hardware
In unison with the theoretical developments the physical systems used to
implement Turing machines underwent rapid change. Mechanical systems
quickly became electro-mechanical and the all electrical, initially using valves
and then transistors. In 1965 Gordon Moore [22] noted that the number of
transistors contained in a integrated circuit had doubled roughly every 18
months for the previous three years despite occupying the same physical
space. If this progression were to continue Moore predicted that by 1975
integrated circuits would contain in excess of 50,000 elements. This was
an accurate prediction and what has become known as ‘Moore’s Law’ has
continued to be a good estimate to this day, as shown in figure 1.1.
There are many areas of research hidden under the exponentially increas-
ing complexity of Moores Law. Miniaturisation has been a major driver of
Moores Law. By reducing the size of a transistor the benefit is twofold: the
time required for a signal to propagate through the device is reduced and the
amount of material required to change state is reduced. Therefore the device
can operate faster and consume less energy. In the context of information
storage and manipulation, this results in less atoms being required to store
the same amount of information. Given our current knowledge of physics this
1.1 Classical Information Processing 5
progression is limited by information being represented on a single quantum
object.
Quantum effects will, however, dominate the behaviour of a computing
structure before miniaturisation attains single atom feature size. Electrons
tunnelling between conductors is one of the many quantum effects that will
cause the flow of signals to deviate from classical descriptions in a counter
productive manner. At this stage the classical computing concepts are invalid
and the design philosophy must change to one that begins with considering
the quantum nature of how the desired signals will propagate and interact.
Some devices already exploit these effects such as tunnelling diodes, however
the signals external to the device are still classical.
In conjunction with miniaturisation the materials used to fabricate the
devices has been an area of unrelenting research. This is important because
while the power requirements are somewhat reduced by miniaturisation, in-
creasing the density of computing elements would eventually result in power
consumption that would melt the device as Moores Law continued. From the
initial 12V transistor logic we are now below 1V with a similar reduction in
required current resulting in massive reduction in power consumption. With-
out this progress any portable information processing device would not be
possible in it’s current form. If the automotive industry had made equivalent
efficiency gain in combustion engine technology as the electronics industry
did from 1975 to 1995 you could drive a car around Australia on one tank
of fuel. Eventually this process also leads to a fundamental limit of what is
the minimum amount of energy that can be used to store and manipulate
information. This again leads to quantum mechanics.
1.1.3 Information goes Quantum
Computation on a Turing machine is typically irreversible. Given a compu-
tational result and knowing the algorithm that processed it is not generally
sufficient to reconstruct the input states. Therefore information has been lost
during the calculation. Irreversibility was considered to be intrinsic to com-
puting until, in 1973, Bennett, a researcher at IBM, proved this conjecture
false [23]. Reversibility is a fundamental feature of quantum computing.
Despite rigorous definitions of efficiency and extensive investigation of
numerical methods the perceived limits of algorithmic performance did not
rely on physical arguments. This was unsatisfying since it did not permit any
argument that a Turing machine was the optimal framework permitted by
physics. Furthermore, it was becoming apparent that a Turing machine could
not efficiently simulate a quantum system as highlighted by Feynmann [24].
6 Introduction: From Classical to Quantum Information
During the mid-1980s David Deutsch began the search for a derivation based
on physical laws of something analogous to the Church-Turing Principle.
The first phase of this was to rephrase the Church-Turing principle such
that the physical implications were clearer [25].
Every finitely realisable physical system can be perfectly simulated by a
universal model computing machine operating by finite means.
Deutsch was able to show that while Quantum mechanics obeyed this
principal, classical physics does not. Classical physics, due to its continuity
will always require a continuum of input and output states, however there
is only a finite mapping of input to output states in the computer and con-
sequently fails the ‘perfectly simulated’ part of the Church-Turing principle.
Therefore, while numerical techniques allow good approximations to physical
systems they cannot perfectly simulate them. Deutsch showed the Universal
Quantum Computer could simulate any Turing machine and finite physical
system that has equal or lower complexity than the computer.
A quantum computer as proposed by Deutsch is a register of two state
quantum systems initialised into some known state, typically |0〉, the ground
state of the system (|1〉 being the excited state). The system than has some
time reversible operations, termed unitary operations, performed on one or
many qubits at a time to implement the quantum algorithm. On completion
of the algorithm the state of the register is measured. Quantum algorithms
are structured such that despite qubits being in a superposition of basis states
during the algorithm, once the algorithm has finished the state of the qubits
are pure basis states. This avoids the problems associated with measuring an
arbitrary state and the associated quantum tomography procedures required.
Unitarity of the quantum information processing requires that no information
is lost during the computation, an essential feature of quantum computing.
Unitarity allows the input state to be recovered by applying the conjugate
of the algorithm to the output state.
Perhaps the most obvious situation that a quantum computer would out-
perform a classical computer is in simulating a quantum system. Accurately
simulating quantum physics is what led Feynmann to consider problems as-
sociated with using a classical computer and propose a quantum computer
[24]. Already an algorithms to simulate many-body fermionic systems [26]
have been developed, answering Feynmann’s uncertainty as to whether his
description of a quantum computer could simulate fermionic systems. An
algorithm for finding eigenvalues and eigenvectors of atomic Hamiltonians
[27] have also been developed.
1.1 Classical Information Processing 7
Conceptually, the simplest advantage is that a quantum computer can
generate truly random numbers efficiently. A classical computer cannot gen-
erate true random numbers since classical physics is deterministic. Therefore
only pseudo random numbers can be generated and this is done so ineffi-
ciently [24]. In a quantum computer this is achieved by measuring a super-
position state, a very simple quantum algorithm. The same qubit cannot be
used to generate a series of random numbers without correlations, however
Deutsch also showed this can be used to advantage to build up more complex
stochastic systems. Bell’s theorem states that classical systems, even if given
perfect random number generator hardware, cannot reproduce the statistics
of consecutive measurements of a quantum state.
Deutsch also introduced the notion of Quantum Parallelism whereby in
the many worlds interpretation of quantum mechanics quantum computer
performs a computation in many near universes and combines the answers
at the output. Only one result is accessible in each universe, however this
result can be the product of many interfering universes.
1.1.4 The Power of Hilbert Space
The clearest method to visualise the benefit of quantum over classical com-
puting is to consider the state space in which it operates. To completely
describe the state space S of the quantum computer we need to describe
both the population of the states and the coherence between them. Using a
density matrix representation the population of the states are the diagonal
elements of a matrix (αnn) while the coherence terms are the off diagonal
elements (βmn). In classical computation knowledge of the value of each el-
ement is sufficient to completely describe the system. Therefore the state
space of the classical computer is described by the diagonal elements of S.
S =
α11 β12 · · · β1n
β21 α22 · · · β2n
......
. . .
βn1 βn2 αnn
(1.1)
It is clear that the state space is larger for a quantum system and as such
a quantum computer is able to represent more information given the same
number of elements. Considering the increase in system complexity due to
adding another element to the computer we can see that in the classical case
adding a bit to the system results in a linear increase in complexity. Adding
a qubit to a quantum computer results in an exponential increase in system
8 Introduction: From Classical to Quantum Information
complexity since this new qubit can have a coherent relationship with any
other qubit. Therefore the computational space grows exponentially, leading
to an exponential ability to process information.
1.2 Quantum Computing Requirements
While the work by Deutsch showed that quantum computing had enormous
potential it did not specify any rigorous criterion for candidate systems. Any
system with coherent interactions was a potential physics system. David
DiVincenzo [28] proposed five requirements for that the physical system must
meet if it is to be useful for quantum computing.
1. A scalable physical system with well characterised qubits.
2. The ability to initialise the state of the qubits to a simple starting state,
such as |000...〉.
3. Very long coherence times relative to the time required for quantum
gates.
4. A ‘universal’ set of quantum gates.
5. A qubit-specific measurement capability.
All of these requirements are inextricably linked with each other. Scala-
bility is very important to satisfy since a quantum computer is only worth
building if it outperforms a classical computer. Since we can already build
complex classical computers a useful quantum computer must span a large
state space.
Scaling up a quantum computing system is challenging. Each qubit must
be only weakly coupled to it’s environment, to allow long decoherence times.
Qubits, however, must also be strongly coupled to each other to allow quan-
tum logic gates to be performed well within the decoherence time of the
system. Therefore when adding qubits to make our quantum computer more
complex we are adding noise sources that, while weakly coupled to their
environment, are strongly coupled to all other qubits in their locality.
Although methods to correct for some decohereing interactions have been
developed (reviewed in section 1.5) many require more qubits to implement
and therefore must introduce more decoherence to the system as a whole.
1.2 Quantum Computing Requirements 9
Unfortunately we cannot settle for a simple trade-off between the number
of elements and their decoherence since both the number of individually
controllable quantum elements and the decoherence time are required to be
pushed significantly further than currently available.
The ability to manipulate the qubit state fast compared to the decoher-
ence rate necessitates that there is a strong interaction with a driving field,
and therefore limits the isolation from the rest of the universe, providing po-
tential decoherence mechanisms. Decoherence is also directly introduced by
imperfect driving fields, an unavoidable consequence of amplitude and phase
noise in the driving electronics.
While manipulating large ensembles of identical quantum systems is rou-
tinely performed in spectroscopy, atom optics and many other areas of physics,
achieving individual control of each qubit is a more complex challenge. The
approaches are necessarily different for each candidate system, however the
general trend is to use spatial confinement, inhomogeneities in transition fre-
quencies or specifically placed nanostructures to address individual qubits.
All of these have limitations: spatial selectivity of the driving field is lim-
ited by the wavelength; inhomogeneous broadening implies disorder; nanos-
tructues distort the local environment and cannot be totally decoupled from
their assigned qubit. Again there are decoherence implications to all ap-
proaches.
It should also be noted that detection of single quantum systems is a
current area of intense research. Single molecule or single site detection
is of interest to the biotechnology [29] and quantum computing fraternities
[30, 31]. Very coherent systems must be very isolated and consequently they
are some of the hardest systems to achieve reliable measurement of a single
quantum system. Again we are faced with a problem for which a simple
compromise does not exist. The more interactive our system is, the faster it
will decohere and therefore it is less useful for QIP.
The physical requirements of isolation and interaction are paradoxical and
will be hard to satisfy. Consequently the predictions of a useful quantum
computer requiring ∼106 qubits [32] should be taken with a grain of salt.
Having complete control over a Hilbert space millions of dimensions large is
an extremely ambitious task.
Assuming we can eventually meet the physical requirements the universal-
ity criterion of quantum logic operations is comparatively simple. In classical
computing any boolean logic function can be realised with only NAND gates
and consequently NAND gates form a universal set of gates. A “universal”
set of quantum gates is a set of unitary quantum operations that can be
10 Introduction: From Classical to Quantum Information
concatenated to produce an arbitrary unitary transformation of the input
state. DiVincenzo [33] was able to show that single qubit operations and
the conditional-not (CNOT) operation between two qubits was sufficient to
form a universal set. The CNOT gate inverts the “target” qubit conditional
on the state of a “control” qubit. Using the basis |00〉 , |01〉 , |10〉 , |11〉 this
has the matrix representation:
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
(1.2)
The CNOT and single qubit rotations are typically demonstrated by
quantum computing proposals to show that the system is a strong candi-
date. In fact it has been shown that almost any quantum gate forms a
universal set [34]. Consequently this approach is somewhat an artefact of
the CNOT and single qubit being the first demonstrated universal set and
having a direct analogy with Boolean logic gates.
One of the hardest requirements is that decoherence time of the system
must be very long compared to the time required to perform gate operations.
This necessitates that the system is well isolated. However, to implement a
gate you require strong interaction between the qubits of interest and some
driving field such that it can be performed quickly. Therefore we have a
paradox since the system cannot be well isolated and strongly interacting.
This is only compounded by the last requirement of being able to measure
the final state of the system, since this also must be done on the time scale
of the gate operations [35]. In order to discuss this point in detail some basic
two level atom theory is required.
1.3 The Two Level Atom
Quantum computing is generally considered using an array of interacting two
level quantum systems. While quantum systems with more than two states
per qubit are considered by some researchers [36] it significantly complicates
the discussion. This treatment of two level atom theory is focused at giving
the reader sufficient tools to consider the state manipulations used in this
thesis.
The Bloch equations provide the best tool to visualise the evolution of a
two level system interacting with a coherent driving field. These equations
were originally derived to describe NMR experiments in spin 1/2 systems
1.3 The Two Level Atom 11
but is formally equivalent to any closed two level systems. Visualisation
of processes described by these equations is achieved via the Bloch Sphere,
shown in figure 1.2, which is the primary conceptual tool for considering
single qubit rotations. The “poles” of the sphere represent the ground and
excited state with the rest of the surface representing possible phases of a
superposition state. Discussion of State representation and evolution on the
Bloch Sphere is restricted spin 1/2 systems for simplicity, with no loss of
generality.
If we consider our fictitious spin 1/2 particle to have |0〉 as the z = −1/2
state and |1〉 as the z = +1/2,
|0〉 =
[
0
1
]
, |1〉 =
[
1
0
]
(1.3)
then a vector describing an arbitrary state of this system can be mapped into
the visualisation space via projection onto the Pauli spin operators:
rψ = (〈ψ|X|ψ〉 , 〈ψ|Y |ψ〉 , 〈ψ|Z|ψ〉) (1.4)
where X, Y and Z are the Pauli spin operators:
X =
[
0 1
1 0
]
, Y =
[
0 −ii 0
]
, Z =
[
1 0
0 −1
]
(1.5)
Due to normalisation of the states |ψ〉 the projection rψ will always be a unit
vector. To demonstrate this let us consider the mapping of a arbitrary state
to the Bloch sphere with the state given by:
|ψ〉 = sin θ |0〉 + eiφ cos θ |1〉 (1.6)
which, as shown in figure 1.2, gets mapped to:
r = (cos φ sin θ, sin φ sin θ, cos θ) (1.7)
This mapping is clearly one to one with the Z axis poles of the sphere
representing the pure states and the angle φ represents the phase of the
superposition. It is clear that this allows the visualisation of an arbitrary,
and therefore in general, mixed state. For both mathematical and conceptual
clarity the rotating wave approximation is applied to the Bloch equations.
This results in the Bloch sphere effectively being spin around the Z axis
at the resonant frequency of the transition. Therefore a Bloch vector has a
12 Introduction: From Classical to Quantum Information
| >0
| >1
| >0 | >1+i
| >0 | >1+
| >0 | >1i-
Z
X
Y
| >Ψ
φ
θ
Figure 1.2: Projection of an arbitrary state, given by equation 1.6 on the Bloch sphere
stationary projection on the sphere if it is on resonance. Detuning and driven
evolution is discussed in section 1.3.2
1.3.1 The Density Matrix
The density matrix is a useful tool to describe mixed states. A pure state can
be described by a single wavefunction and when measured will always yield
the same result. Superposition states are a linear combination of pure states
which when measured, collapse into a pure state in order to interact with
the measuring apparatus. Mixed states are ensemble states and therefore
can be a mixture of different superposition states, as well as pure states.
Consequently when measuring a mixed state some members of the ensemble
will have opposite projections on the measurement basis. Therefore the total
measured projection over all basis states will not necessarily be normalised.
The density matrix, traditionally denoted as ρ is used to describe mixed
states. If the system could be in any linear combination of states |ψi〉, with
the probability of being in a particular state being given by pi the density
matrix would be given by:
ρ =∑
i
pi |ψi〉 〈ψi| (1.8)
Some important properties of the density matrix are that it has a trace of
1.3 The Two Level Atom 13
one and the eigen values are real, resulting in det(p) ≥ 1. The expectation
value for a quantity associated with an operator M is
〈M〉 = Tr(ρM) (1.9)
where Tr denotes the trace. The evolution of the density matrix can be
derived from the Schrodinger equation as:
ρ(t) =−i~
[H, ρ(t)] (1.10)
The density matrix can be directly related to a vector on the Bloch sphere
by tracing the projection on the Pauli operators:
rρ = (Tr(ρX),Tr(ρY ),Tr(ρZ)) (1.11)
The one to one mapping of the density matrix to a Bloch vector is confirmed
by the following identity, which is true for all 2 × 2 Hermitian matrices.
ρ =1
2(Tr(ρ)I + Tr(ρX)X + Tr(ρY )Y + Tr(ρZ)) (1.12)
This identity is also useful to determine the density matrix from a series of
measurements on a quantum system and will be used in the process tomogra-
phy in section 1.3.3. The magnitude of the vector is, however, now bounded
by the Bloch sphere and need not strictly exist on the surface of the sphere.
This can be seen from the determinant:
det ρ = 1 −X2 − Y 2 − Z2 ≥ 0 (1.13)
⇒ X2 + Y 2 + Z2 ≤ 1 (1.14)
An incoherent state is mapped to a point at the origin of the Bloch sphere
and therefore any deviation from the surface of the sphere represents decay
of some form as will be discussed in section 1.4.
In circumstances where the Hamiltonian is also time dependant he equa-
tions of motion of the density matrix are described by the Louiville-Von
Neumann equation.d
dtρ(t) = −i [H(t), ρ(t)] (1.15)
This equation is required for evaluating the result of manipulating quantum
states by the application of pulsed driving fields.
14 Introduction: From Classical to Quantum Information
Z
X
Y
Figure 1.3: Schematic representation of an inhomogeneously broadened line evolving intime showing the precession of the Bloch vectors around the Z axis at a rate proportionalto the detuning.
1.3.2 Dynamics on the Bloch Sphere
The Bloch sphere was designed as a visualisation tool for the Bloch equations
which describe a closed two level quantum system [37]. The real power of
the Bloch sphere lies in it’s ability to visualise dynamic processes. The Bloch
equations are given extensive treatment in a number of texts [38, 39, 40,
41, 42, 35] and consequently the following discussion is intended to provide
sufficient background to properly visualise the system dynamics described in
later chapters.
If we consider a fictitious spin 1/2 particle experiencing a strong, static
magnetic field aligned with the Z axis, B0, the degeneracy of the spin states
|0〉 and |1〉 is lifted transition frequency between them is:
ω0 =2π
~µB0Z (1.16)
In rare-earth doped crystals there will always be some static inhomoge-
neous broadening and the detuning from line centre will be represented as
∆ = (ω − ω0). The rotating wave approximation results in a Bloch vector
on resonance having a stationary projection. However if the detuning is
nonzero the Bloch vector will precess around the Z axis with a frequency of
∆ as shown in figure 1.3. The consequences for the projection of an inho-
mogeneously broadened ensemble on the Bloch sphere will be discussed in
section 2.5.2.
1.3 The Two Level Atom 15
Z
XY
θ
Z
X
Y
a
bc
Figure 1.4: Shows the action of a driving field on a series of Bloch vectors. All Blochvectors are rotated around the rotation axis, defined by the phase of the driving field. TheBloch sphere on the right describes the action of resonant driving a) in phase, b) phaseshifted by 90 and c) phase shifted by 90 and detuned.
When the system is driven on resonance the resulting action is a rotation
of the Bloch vector as shown in figure 1.4. The axis of rotation is in the X,Y
plane and by convention a driving field with no phase shift rotates about the
X axis. Introducing a phase shift of θ into the driving field results in a shift
of the rotation axis in the X,Y plane by an angle of θ relative to the X axis.
This is described by the equations of motion for Bloch vectors:
X = −∆Y
Y = ∆X + ΩZ
Z = −ΩY
(1.17)
where Ω is the Rabi frequency, which describes rotation on the Bloch sphere
due to the action of the driving Hamiltonian, Hd. The Rabi frequency is
given by:
Ω =
√
Tr (HdX)2 + Tr (HdY )2 + Tr (HdZ)2 (1.18)
with the rotation axis being,
n(θ, φ) = (Tr (HdX) + Tr (HdY ) + Tr (HdZ)) (1.19)
In the case of our spin 1/2 particle in the magnetic field, B0 the rotation is
caused by driving the system with an RF field at the transition frequency
ω0, perpendicular to B0. Assuming the driving field is in phase this results
in a Hamiltonian of the form:
Hd =µB1
2cos (ωt)X (1.20)
16 Introduction: From Classical to Quantum Information
If the driving field is close to the transition resonance (ω ≈ ω0) then the
contribution from cos (ωt)X has a non-zero average. This can be rewritten
in the usual rotating frame as:
H =∆
2Z +
Ω
2X (1.21)
This results in the rotation axis, n being the X axis as depicted in figure
1.4, path a. If the driving field is rotated by 90 it can be seen from equation
1.19 that the rotation axis n is also shifted by 90 as shown in figure 1.4.
Detuning from resonance results in n leaving the X,Y plane as per equation
1.19. In this case the Rabi frequency should be replacesd with the generalised
Rabi frequency, which includes detuning effects and for nuclear spins has the
form:
Ω =
√
∆2 +
(
µB1
~
)2
(1.22)
1.3.3 Quantum Process Tomography
When driving a quantum system the state manipulation will never be per-
fect. Consequently we require a mathematical tool to asses what the actual
state manipulation was performed by the experiment. This is of particular
importance to QIP applications since we require high fidelity manipulation
of an arbitrary state and therefore any non-ideal state manipulations in-
troduce errors. Quantum mechanics tells us it is impossible to completely
determine the state of the system given only one opportunity to measure the
system. Quantum Process Tomography is a method by which the process
superoperator O can be determined over a series of experiments by preparing
a number of known input states ρini and measuring the output state, ρfin.
This method was derived by Jevon Longdell [43] and is equivelent to methods
derived by Chuang and Nielsen [44, 43]. Using density matricies the action
of the process superoperator O is expressed as:
ρfin = O(
ρini)
(1.23)
where ρini and ρfin are the initial and final state of the system and O is the
process superoperator. The process operator is linear and can therefore be
decomposed as:
ρfink
=∑
j
λjkρini
j(1.24)
By preparing a set of known input states that span the input state space,
ρini, and measuring the projection on σi we can create a set of measurements
1.3 The Two Level Atom 17
si.
si = Tr(
σiO(
ρinii))
= vec (σi) · vec(
O(
ρinii))
= σiTλρini
i(1.25)
where vec is the operation of flattening a matrix into a vector, defined as
implemented in fortran libraries:
A =
[
a b
c d
]
vec(A) =
a
b
c
d
(1.26)
If we flatten both sides this produces
si =(
ρi
T)
⊗(
σiT)
vec (λ) (1.27)
where ⊗ is the kronecker product, which maps two arbitrarily dimensioned
matrices into a larger matrix with a block structure, defined as
A =
a1,1 · · · a1,n
.... . .
...
am,1 · · · am, n
B =
b1,1 · · · b1,k...
. . ....
aj,1 · · · aj, k
A⊗ B =
a1,1B · · · a1,nB...
. . ....
am,1B · · · am,nB
(1.28)
In order to determine what λ is, and therefore O we need to solve the inverse
problem such that we are confident of a physically reasonable result. The
18 Introduction: From Classical to Quantum Information
general linear inverse problem is characterised by the following equation [45]:
y = Cx+ n (1.29)
where x ∈ RM is the measured initial state, y ∈ R
N is the measured final
state, n N dimensional variable representing the noise present in the mea-
surements and C is the process that maps the input state to the output state.
The noise, n, is assumed to be a Gaussian distributed random variable with
zero mean and a covarience matrix Γ. The noise present in the measurement
can lead to unphysical results and therefrore we draw on Bayes’ theorem
ensure a physically reasonable result. Bayes’ theorem relates joint and con-
ditional probabilities between two events, which in theis case are the initial
and final states of the system. Given two events, x and y, Bayes’ theorem
states that:
p(x, y) = p(x|y)p(y) = p(y|x)p(x) (1.30)
and therefore
p(x|y) =1
p(y)p(y|x)p(x) (1.31)
where p(x) represent what we know about x before making the observation
y, known as the prior probability, p(x|y) represents what we know about x
after making the observation y, or posterior probability. Equation 1.31 states
that a prior probability is turned into a posterior probability by multiplying
by the forward probability, p(y|x) and a normalisation factor 1/p(y). In
this context p(y|x) is considered a likelyhood function, i.e. how likely is it
that the observation y was dependant on the prior state x. This equation is
interpreted as telling us how we should change our state of knowledge of x
as a result of making an observation which yields the result y.
When applied to linear inverse problems, as stated in equation 1.29, the
likelihood function becomes [45]:
p(y|x) = p(n = y − Cx) =1
(2π)N/2√detΓ
exp
[
−E (x; y)
2
]
(1.32)
where the misfit function, E is
E (x; y) = (y − Cx)TΓ( − 1)(y − Cx) (1.33)
When E is minimised, the mapping of observations x to y has a ’maximum
likleihood’ of being described by process C. The maximum likleihood solu-
tion is obtained when
‖y − Cx‖ (1.34)
1.4 Decoherence 19
which, restated in the context of the quantum process tomography is
‖s−Mλ‖ (1.35)
where,
M =(
ρi
T)
⊗(
σiT)
(1.36)
which is satisfied by
λ = pinv(M)s (1.37)
where pinv is the Morse-Penrose pseudoinverse. We can therefore determine
λ and therefore construct the processes superoperator O using equation 1.24.
In order for the mapping of λ to s to be completely specified we require that
Rank(M) ≥ dim(λ) (1.38)
This is simply that we require at least as many measurements as there are
parameters of the system. For a single qubit, or any two state quantum
system, this can be achieved by preparing the input states |0〉, (|0〉+ i |1〉)/2,
(|0〉 − i |1〉)/2 and |1〉 projecting on the σi basis states, which are the Pauli
spin matrices (σx, σy, σz, I). This results in 12 independent measurements
on the system with the remainder of the 16, or 2n2 measurements (where n is
the number of Hilbert space dimensions) being known due to the properties
of the density matrix, defined in equation 1.12.
1.4 Decoherence
Since the quantum computer is a finite physical system it will have inter-
actions with its surrounding environment. Therefore, the finite dimension
Hilbert space of the quantum computer HQ is a subset of the total semi
infinite dimension hilbert space of the quantum computer and rest of the
universe, HE . Performing unitary operations on the finite Hilbert space HQ
results in non-unitary dynamics of HQ due to its’ projection on the total
Hilbert space, HQ ×HE . Undesired interactions between the quantum com-
puter and it’s environs represent a loss of information from the quantum
computer to the environment, referred to as Decoherence. This loss of in-
formation from the system is exactly analogous to the effect of entropy on
the quantum computer [25]. Consequently Deutsch was able to show that
using the non-unitary dynamics of the quantum computer interacting with
it’s environment and the physical interpretation of the Church-Turing prin-
ciple the third law of thermodynamics can be derived. For the first time real
20 Introduction: From Classical to Quantum Information
physical significance could be derived from computational theories. This also
enabled new perspectives on entropy and it’s relation to the computational
accumulation of knowledge.
Decoherence is the border between the quantum and classical worlds. It
is the loss of phase coherence between quantum objects that leads to classical
physics emerging from quantum mechanics [17, 18, 46]. Irrespective of the
final application reducing decoherence of quantum systems is important for
developing technology that utilises quantum mechanics.
Population decay and Decoherence, or longitudinal and transverse relax-
ation in NMR terminology has been an major area of study since the advent
of NMR. Reducing the rate at which a system relaxes allows greater spec-
tral resolution thereby allowing subtle interactions to be discerned. While a
detailed discussion of these developments is beyond the scope of this thesis,
several results of this work are fundamentally important for this work. There
are a number of models that have emerged that treat the interaction between
the quantum system of interest and the surrounding environment in different
levels of detail. These models can be put in three groups according to the
assumptions about the system-environment interaction.
1.4.1 System Bath Interactions
All decoherence that is not purely due to population decay results from in-
teractions between the quantum system of interest and the surrounding en-
vironment. In the context of the systems investigated in this work we are
interested in a nuclear spin system IS surrounded by a nuclear spin bath, IB.
The most general Hamiltonian to describe this is as follows:
H0 = HS ⊗ IB + IS ⊗HB + HSB
HSB =∑
γ
Sγ ⊗ Bγ (1.39)
where HS describes the quantum system of interest, HB describes the sur-
rounding “bath” and HSB is the interaction between them with Sγ and Bγ
being operators on the system and bath respectively. The Sγ operators are
responsible for decoherence of the system due to interaction with the bath.
The result of the system-environment interaction, HSB, is to induce a
time varying perturbation in the state energies[38, 39, 47, 41]. When con-
sidering the effect of HSB on a transition of interest in HS both the time
scale of the perturbation, τB, and the frequency shifts, ∆ω, that they in-
duce are important. The system-environment interaction is a homogeneous
1.4 Decoherence 21
process as the environment is considered to be equivalent between different
sites. The effect of the ∆ω is generally discussed as either spectral diffusion
or decoherence depending on the ratio of ∆ω to τB. Spectral diffusion refers
to large, slow shifts in transition frequency or τB∆ω ≫ 1, resulting in a slow
random walk. While these processes certainly contributes to dephasing by
accumulating a phase relative to the reference oscillator generally a faster,
lower energy process will have decohered the transition before a significant
amount of configurations of the spectral diffusion processes are progressed
through. This makes the homogeneous linewidth dependant on the period of
measurement. The short time dephasing is of most interest to this work and
will be examined in more detail.
In the regime where where the shifts are small and rapidly change, τB∆ω ≪1, the effect of the perturbations on a transition of HS acts like a homoge-
neous broadening mechanism. In this case a phenomenological description
of an ensemble of HS systems interacting with a bath can be constructed
using decay constants, T1 and T2, describing the population and coherence
decay respectively. In the basis of in phase superpositions, quadrature su-
perpositions and population (+,−, Z) the relaxation matrix can be written
as:
R =
1/T2 0 0
0 1/T2 0
0 0 1/T1
(1.40)
These constants can be applied to the Bloch equations as will be shown in
section 1.4.2.
This approach is convenient and conceptually simple, becoming standard
terminology in NMR and many other areas of spectroscopy. The ability to
apply it to the Bloch equations allows visualisation of the effect of dephasing
and it provides a simple parameter to describe homogeneous decay processes
and compare lifetimes of different quantum systems.
If, however, τB∆ω ≈ 1 or we want to modify the effect of the interaction
between the system and it’s environment then we need to understand the
perturbations in greater detail. For this we can use a semi-classical approach
where the effect of the environment is modelled as a time varying field due
to an array of particles in the lattice. The interaction of the particles in the
environment with each other is treated by statistical models with a charac-
teristic time for reconfiguration of the environment. This will be investigated
for the specific case of nuclear spins interacting with a nuclear spin ‘bath’ in
section 2.5.3.
Knowledge of the range of possible fields due to environmental configura-
22 Introduction: From Classical to Quantum Information
tions and the time scale on which the field changes allows an assessment of
how the system or environment can be driven to minimise the effect of their
interaction. These techniques known as Dynamic Decoupling techniques can
drastically increase the observed T2 of the system [41, 39, 42, 38, 13, 48, 49,
50, 51, 52, 53, 54, 55, 56, 9, 8, 57, 58], often by several orders of magnitude.
This illustrates that T2 is a convenient conceptual tool, but the underlying
dynamics that it represents should always be carefully considered.
There are also cases where the system of interest is sufficiently strongly
coupled to the environment that driving the quantum system of interest also
drives the local environment, resulting in back action on the system of interest
[59]. In this case a full quantum mechanical description of the coupling to
the environment is required to adequately describe the response to a driving
field. In this circumstance T2 is far too simple a description to encapsulate
the dynamics of the system. Such a system is also not suitable for quantum
computing because the system cannot be approximated as a closed two level
system.
1.4.2 Decoherence on the Bloch Sphere
The effect of population decay and decoherence, if able to be approximated
by a T2 decay can be incorporated into the Bloch equations as follows:
X = −∆Y − X
T2
Y = ∆X − Y
T2+ ΩZ
Z =Z − Z0
T1− ΩY
(1.41)
where Z0 is the equilibrium state population, ∆ is the detuning (ω − ω0), Ω
is the Rabi frequency, T1 and T2 are the population lifetime and decoherence
time respectively and dotted variables denote the time derivative.
The action of T2 on the Bloch vector causes it to decay toward the Z axis.
The effect of T1 on a Bloch Vector is that the population component will decay
toward the mixture of states defined by the Boltzmann distribution for the
given temperature. The bahaviour is clearest when considering a system at
zero temperature, in which case the Bloch vector decays to the |0〉 state at the
“south pole”. The remainder of the discussion will consider zero temperature
systems for conceptual clarity. The effects of T1 and T2 on the evolution of
an ensemble spin vector in a 50-50 superposition state on the Bloch sphere
are shown in figure 1.5. First considering a spin ensemble on resonance as
1.5 Error Correction 23
Z
X
Y
aZ
X
Y
b
Z
X
Y
cZ
X
Y
d
Figure 1.5: Schematic representation of an inhomogeneously broadened line evolving intime showing the precession of the Bloch vectors around the Z axis at a rate proportionalto the detuning.
shown in figure 1.5a we see that if T2 < T1 the spin vector undergoes a T2
decay toward the origin before the T1 process becomes significant. If the only
coherence loss if from population decay, or T2 = T1 then the decay proceeds
toward |0〉 and the Z axis at an equal rate. If we follow a detuned ensemble,
T2 decay causes the spin vector to spiral toward the Z axis as shown for
T2 ≪ T1 and T2 < T1 in figure 1.5c and d.
1.5 Error Correction
As previously mentioned in section 1.4 Deutsch was able to show that using
the non-unitary dynamics of the quantum computer interacting with it’s
environment and the physical interpretation of the Church-Turing principle
the third law of thermodynamics can be derived [25]. Therefore entropy
will cause information to be lost to the environment via T1 and T2 processes,
with a corresponding accumulation of errors. The precision or fidelity of qubit
manipulations also contributes to the error accumulation during calculations.
This requires the computation to proceed with a finite probability of an
error occurring with strategies to correct the error broadly known as “Fault
Tolerant” quantum computing [60, 32, 61, 62, 63]. As with analog computers,
the possible errors form a continuum and accumulate over the course of a
calculation. Small errors of this kind are difficult to correct and, due to
entropy are impossible to completely remove.
Estimates of the tolerable error rate are of the order of 10−6 [32], requiring
24 Introduction: From Classical to Quantum Information
extremely high fidelity (F > 0.999999) from all operations performed on the
quantum computer. If these fidelities are to be reached, the decoherence
time of the system needs to be significantly larger than the gate operation
time. Estimates of the required ratio of the gate operation time to the
decoherence time of the system are of the order of 105 [35]. These benchmarks
are extremely hard to reach and as yet no scalable system has achieved both
gate fidelity and coherence time to gate operation time ratio. Consequently
investigating the decoherence present in the candidate quantum system is
of paramount importance to understanding the potential of the system to
function as a quantum computer.
The emerging strategies for correcting errors in quantum computers draw
extensively from NMR decoupling methods and classical error correction
techniques. These strategies can be broadly grouped as Quantum Error Cor-
rection Codes (QECC) [60, 64, 65, 66], Decoherence Free Subspaces (DFS)
[67, 68, 69, 70] and Dynamic Decoherence Control [8, 10, 11, 12, 13, 71].
1.5.1 Quantum Error Correction Codes
Quantum Error Correction Codes adapt classical coding concepts to deal with
the more challenging problem of preserving phase and population informa-
tion. All codes draw from classical techniques of redundancy of information
by encoding information using several physical qubits to represent one log-
ical qubit. Active schemes can then be to correct the error. This can have
several forms either measure one physical qubit or partially measure several
physical qubits [72] to check for errors without collapsing the logical qubit.
The measurement scheme must be carefully constructed such that it only
measures the error and does not gain any information about the state of the
qubit before the error occurred [72]. Alternatively the correction can take
place as part of the decoding process of a logical qubit [66].
The encoding is important so that these quantum codes, like their classi-
cal counterparts, can discern between valid computational states and errors.
This requires that any superposition of code words is also a code word and
that there is a volume around each code word that is only occupied due to
an error. If an error is detected, a correcting operation can be performed
on the logical qubit before the calculation proceeds. These codes are typi-
cally limited to correcting a single error in population and phase of the state
[60, 64, 56]. Only correcting for single errors allows a minimum “distance”
between code words, thereby making efficient use of the code space. It is
important to note that the rate at which error correction must be performed
is minimally the rate at which errors occur, and therefore directly related to
1.5 Error Correction 25
the decoherence experienced by the system.
QECC codes have major implications for scalability as codes that can
correct for all errors generally require at 5 to 9 physical qubits [60, 64, 65, 73].
1.5.2 Decoherence Free Subspaces
Decoherence Free Subspaces (DFS) are a set of techniques that encode a
logical qubit using a number of physical qubits to protect against specific
decoherence mechanisms [67, 74, 70, 64, 68, 69]. Consequently they are
sometimes referred to as “noiseless” or “error avoiding schemes”. Like QECC
a single qubit is encoded using several physical qubits to create a DFS. This
encoding is designed to create multi qubit states that have special symmetries
with respect to the dephasing mechanisms. This results in a the definition
that a Hilbert space H is said to possess a decoherence free subspace H if
the evolution inside H ⊂ H is purely unitary.
Encoding information in this way is only effective at preventing decoher-
ence if the perturbing field is correlated for all physical qubits that constitute
the logical qubit. In certain liquid phase NMR systems and ion traps this has
been shown to be effective at reducing decoherence [73, 75]. In general, how-
ever, one cannot assume correlated, equivalent perturbations at two or more
different qubit sites. In solid state systems there may be several elements of
the dephasing bath between nearest neighbour qubits. To correct for asym-
metric perturbations or uncorrelated perturbations, other techniques such as
QECCs or Dynamic Decoherence Control need to be incorporated [55].
DFS codes can be effectively implemented with less physical qubits than
QECCs, the simplest requiring only 2 qubits [67]. Asymmetric or uncorre-
lated perturbations acting on the physical qubits still result in dephasing.
1.5.3 Dynamic Decoherence Control
Dynamic Decoherence Control (DDC) techniques, or phase cycling in NMR
terminology, were developed by NMR spectroscopists remove unwanted con-
tributions to the spin Hamiltonian [48, 57, 51, 52, 50]. This is achieved by
driving the system of interest or the bath such that they are decoupled and
the effect of system - bath interactions is minimised. These techniques have
been particularly useful in increasing the resolution of liquid phase NMR ex-
periments [76, 77, 78, 79, 41, 42, 80, 81, 48, 51, 82, 53, 83, 84]. Of this large
range of potential DDC techniques many are designed to decouple a partic-
ular interaction and often the technique is specifically designed to protect a
particular state and as such do not leave an arbitrary state unchanged. In
26 Introduction: From Classical to Quantum Information
order to perform QIP we need to use arbitrary states which limits our choice
of DDC techniques.
There are two DDC techniques investigated in this work. The technique
given most attention by the quantum computing community for increasing
decoherence times is a variant of the Carr-Purcell Mieboom-Gill (CPMG)
pulse sequence [81], well known to NMR spectroscopists and renamed as
“Bang Bang” decoupling [85, 13, 62, 8, 9] due to analogy with the classical
error correction protocol [8]. This scheme ideally drives the system of interest
faster than the reconfiguration time of the bath such that the action of the
bath averages to zero. Application of this technique is studied in chapter 6.
While the Bang Bang decoupling sequence acts on the quantum system
of interest we can also act on the bath to increase the decoherence time. If
the bath is reconfiguring slowly, resulting in a random walk of the transition
frequency the transition linewidth can be reduced by driving the bath suf-
ficiently hard such that it’s contribution to transition frequency is averaged
out. This process, called Magic Angle Line Narrowing (MALN) [53, 54, 41]
is routinely used in NMR to decouple a spin system from a surrounding spin
bath and is investigated in chapter 7.
1.5.4 A Quiet Corner of Hilbert Space
All of these previously mentioned decoherence control methods are still lim-
ited by the underlying dynamics of their local environs and the sensitivity of
the transition used to those dynamics. QECCs must check for errors before
there is a significant probability of an error occurring. Decohering of DFS
codes is limited by asymmetry of the System-Bath interactions and Dynamic
Decoherence Control techniques must be applied faster than the characteris-
tic time for the bath to reconfigure to be effective [13]. Therefore all of these
techniques benefit from reducing the effect of system bath interactions. Fur-
ther, any resources used for error correction cannot be used for computation.
Therefore a useful quantum computer needs to be built in a quiet corner of
Hilbert space such that it does not spend most of it’s time error correcting
itself.
The increase in the number of qubits required for error correction is a
major obstacle to achieving Fault Tolerant quantum computation. Qubits
are a precious commodity and the difficulty of increasing the number of
available qubits by a factor of ∼5 is very challenging.
Consequently the approach of our group has been to find quantum sys-
tems that have are inherently low decoherence and determine how far de-
coherence can be reduced. Rare Earth ions have the narrowest optical
1.6 Why the Rush? 27
linewidths observed in solids [5] as well as displaying moderately long co-
herence times in hyperfine ground states. This is due to Rare Earth ions
being well isolated from their surrounding environment. In this work we in-
vestigate the limits of decoherence as experienced by Praseodymium ions in a
Yttrium Orthosilicate host to create a low noise environment for the storage
and processing of quantum information using nuclear spins.
1.6 Why the Rush?
While the theoretical development of quantum computing was fundamentally
interesting for physics and computer science quantum computation did not
gain widespread attention until in 1994 Peter Shor showed that a quantum
computer could efficiently solve a problem of great interest. This was finding
two prime factors given only their product [86]. No known classical algorithm
can perform this calculation efficiently and as a consequence it provides the
security of what is known as Public key cryptography.
Public key cryptography allows two parties to exchange secure messages
with all of their correspondence public and no prior exchange of code books.
The lack of need for exchanging code books fundamentally changed the
manner in which secure communications took place. Consequently it is
an enabling technology for secure personal or business communications, e-
commerce and heavily utilised in defence applications. Therefore the interest
in breaking this encryption has been intense since the advent of public key
cryptography.
Therefore a quantum computer, comparatively simple to the current su-
percomputers could unlock a significant amount of protected information
with obvious application in military and industrial espoiange. In combina-
tion with this is the desire by intelligence agencies and finance sector to
protect their information from hacking by a quantum computer. This has
resulted in frantic research to find a suitable system for scalable quantum
computing to take advantage of this opportunity. It also poses some serious
questions as the what will be the spoils for the victor.
1.7 Other application and fundamental inter-
est
The required parameters for a scalable quantum computer are still unsat-
isfied by any system and some would argue that the requirements are too
28 Introduction: From Classical to Quantum Information
paradoxical to be satisfied. Despite this there are several areas that benefit
from the study of decoherence times and the extension thereof.
The most closely related is that of quantum memories. Quantum commu-
nication links are already in place, so irrespective of the future of quantum
computing, the ability to store and recall a quantum state is important. If
these networks are to extend beyond point to point links there will be a
need to cache and route quantum information. The ability to store optical
information using nuclear spins is clearly beneficial for integration into op-
tical networks. This memory could be based on either Electromagnetically
Induced Transparency (EIT) or stark echos as described by Alexander et al.
[87].
Most EIT experiments are performed in dilute gasses and as such the
coherence time is limited by diffusion and collisions. The atomic motion is
a problem, even at the limit of trapped dilute gas techniques since in a Bose
Einstein Condensate (BEC). For a Rubidium 87 BEC at 100nK atoms move
of the order of 5mm/s and consequently will move half an optical wavelength
in approximately 6×10−6s, moving through a 1mm laser beam in only 0.2s.
This creates some hard limits as to maximum time scales that can be expected
from EIT experiments, even after pushing dilute gas technology to it’s limit
with a BEC.
In a rare-earth doped crystal since the ions are not moving the upper limit
is the population lifetime of the hyperfine ground states, which can be several
weeks [4]. If techniques can be developed to achieve a significant fraction
of the population lifetime rare-earth doped crystals have great promise to
provide a very coherent Λ system for quantum optics experiments and non-
linear optics. EIT is one application that could achieve slower light and
stopping light for longer periods, thereby extending prior work in dilute gases.
In extending the decoherence time the goal is to remove the effect of
the dominant dephasing mechanism. In doing so more subtle interactions
can be discerned. The techniques presented herein allow the observation
of strain broadening in the hyperfine ground state levels of Pr3+:Y2SiO5 ,
which is usually dominated by magnetic dephasing interactions. This allows
for a future study of the correlation of strain broadening between optical and
hyperfine inhomogeneous linewidths.
Chapter 2
Rare Earth Ion Spectroscopy
“We all agree that your theory is crazy, but is it crazy enough?”
- Niels Bohr (1885-1962)
Rare-Earth ions exhibit rich spectroscopy making them both interesting
scientifically and an indispensable part of our modern technology. Spec-
troscopists were initially fascinated by the observation of optical spectral
features that are unusually narrow for solid state systems, which prior to
the invention of the laser required massive spectrometers to obtain data that
was not instrument limited [47, 88]. The vast array of transition energies
available from rare-earth compounds revealed by this work proved ideal for
application to phosphors for TV screens, X-ray intensifying screens and tri-
phosphor fluorescent lamps.
Spectroscopy of rare-earth ions, like many other areas of spectroscopy un-
derwent rapid development in the 1960s with the invention of the laser. This
initial phase focused on finding and studying the dynamics of lasing systems,
for which the narrow optical transitions of rare-earth ions were particularly
attractive. Materials containing Neodymium, such as Nd:YAG and Nd:YLF
are of particular note, becoming the de facto standard for many lasing appli-
cations due to the high output power and narrow linewidth. More recently
Erbium has become an indispensable part of long haul fibre optic commu-
nications with the development of in-line Erbium Doped Fibre Amplifiers
(EDFAs).
The development of tunable single frequency dye lasers in the 1970s dra-
matically increased the systems that could be studied. This development
provided researchers with sufficient resolution to study inhomogeneous broad-
ening effects. With the use of holeburning and time-domain techniques the
30 Rare Earth Ion Spectroscopy
homogeneous broadening of these transitions could also be probed. Homoge-
neous linewidths can be up to 107 times narrower than the inhomogeneous
linewidths with linewidths as small as 122Hz for optical transitions have been
reported [5].
These studies led to new applications using holograms which exploited the
inhomogeneous broadening to store [89, 90], process [91, 92, 93] and route
[94] classical information. Although not yet reaching commercial application
this demonstrates diverse range of applications that a detailed knowledge of
rare-earth spectroscopy allows.
This chapter provides a description of the interactions that create the
rich spectroscopy of rare-earth ions. The discussion focuses on details that
relate to hyperfine transitions of Pr3+:Y2SiO5 and as such is not a complete
discussion for all rare-earth ions.
2.1 Introducing The Lanthanides
The rare-earth elements are formally known as the Lanthanide group, con-
sisting of the 15 elements from Lanthanum (Z = 57) to Lutetium (Z = 71)
during which the 4f electron shell is filled. The term rare-earth was ascribed
to this group by Johann Gadolin in 1794. Rare due to initially only being
found in abundance in Ytterby mine in Sweden, in “earthy” coloured oxide
mixtures. The only member of the group that actually is rare is promethium
due to being radioactive and is often found in uranium ores since it is part
of the uranium decay process [95]. All stable rare-earth ions have similar
abundance to arsenic and mercury, neither of which are considered rare.
The rare-earth elements share the same bonding electrons, consisting of
the 6s, 5s and 5p shells (figure 2.1) and consequently share many chemical
properties. The elements all exhibit strong electropositvity and their bond
is often well approximated as ionic. The most stable oxidation state (M3+)
dominates chemistry of the rare-earth elements in a manner not seen in any
other group in the periodic table. Therefore the distinguishing feature of rare-
earth ions is their size, which uniformly reduces across the group. This is
know as the Lanthanide contraction. Yttrium, due to sharing this oxidation
state (M3+) and having a similar size is often included in a discussion of
rare-earth ions.
2.1 Introducing The Lanthanides 31
Figure 2.1: Radial distribution functions of the 4f, 5s, 5p and 6s states calculated forGd+[96]. The 4f electrons are partially shielded from perturbations by the 5s, 5p and 6sorbitals
32 Rare Earth Ion Spectroscopy
2.2 4f Energy Levels
In high symmetry hosts the 4fn → 5dn transitions are forbidden by parity
conservation. It is only when rare-earth ions are placed in low symmetry hosts
that the orbital angular momentum is no longer a good quantum number and
optical 4fn → 5dn transitions are allowed.
The rich nature rare-earth spectroscopy stems from the purely electronic
4fn → 5dn transitions. The partial shielding due to the 5s and 5p electrons
results in narrow optical transitions and reduces the influence of crystal strain
and of lattice phonons.
There are a number of contributions to the energy levels of the 4fn → 5dn
transitions that result in the rich spectra observed in rare-earth spectroscopy.
In a purely coulombic potential the 4f states are degenerate. Rare Earth ions
have strong spin orbit coupling, typical of heavy atoms, which splits the 4f
states into manifolds. These manifolds consist of states with the same total
angular momentum, J . Since J is conserved by spin orbit coupling it is a
good quantum number to label the manifolds. The spin orbit coupling is the
largest contribution and therefore primarily defines the spectrum. The spec-
tra produced by the spin orbit coupling as the rare-earth group is progressed
through is best shown by the seminal work of Dieke and Carnall [97, 98] us-
ing a lanthanum trichloride host, shown in figure 2.2. Interactions between
the rare-earth ion and the host lattice can be treated as perturbations to the
spin orbit coupling. At liquid helium temperatures each J manifold is then
split into a maximum of 2J + 1 levels by interaction with the crystal field.
When considering their magnetic properties, rare-earth ions can be di-
vided into groups according to having an even or odd number of f electrons.
Those with an odd number of electrons have a large magnetic moment due
to the unpaired spin, which splits the energy levels into a doublet known as
a Kramers Doublet. For rare-earth ions with an even number of electrons
or non-Kramers ions, such as Pr3+ used in this work, placed in sites with
axial or lower symmetry the crystal field levels are singlets due to “quench-
ing” of the angular momentum. Quenching refers to all states having zero
angular momentum due to the Hamiltonian not commuting with an angular
momentum operator in any direction.
All of the work presented herein was performed using trivalent praseodymium
doped yttrium orthosilicate (Pr3+:Y2SiO5). Pr is a non-Kramers ions and
Y2SiO5 is a low symmetry host belonging to the C62h space group. Pr can sub-
stitute for Y at both Y sites, each of which has C1 symmetry. Consequently
to investigate the remaining perturbations the discussion will be restricted
to rare-earth non-Kramers ions in hosts with lower than axial symmetry.
2.2 4f Energy Levels 33
Figure 2.2: Energy levels of triply ionised rare earth ions in LaCl3. The semicirclesindicate fluorescing transitions
34 Rare Earth Ion Spectroscopy
2.3 Hyperfine Interaction
Pr has nuclear spin of 5/2 with each crystal field split level further split into
three levels levels by the hyperfine interaction. The hyperfine splittings are of
the order of 10MHz with each level being doubly degenerate in zero applied
magnetic field. Manipulation of these hyperfine states is the basis of this
thesis. For the experiments performed in this thesis, rare-earth ions can be
described using the following Hamiltonian:
H = [HFI + HCF ] + [HHF + HQ + HZ + Hz] (2.1)
As described previously the spin orbit coupling and crystal field couplings
dominate the spectrum, represented as HFI (Free Ion) and HCF respectively.
The remaining terms in order of appearance are the hyperfine coupling be-
tween the 4f electrons and the nuclear spin, nuclear electric quadrupole
interaction, electronic Zeeman and nuclear Zeeman.
As previously mentioned angular momentum of non-Kramers ions in low
symmetry hosts is “quenched”. This results in the degeneracy within the J
manifolds being completely lifted and all states are electronic singlets with
only second order contributions from the last four terms in equation 2.1.
The last four terms result in approximately equal contributions to state en-
ergies, hence the grouping of terms in equation 2.1. Hyperfine and electronic
magnetic effects are therefore described by applying second order perturba-
tion theory. The effective Hamiltonian including the nuclear Zeeman and
quadrupole interactions is [47]:
Heff = g2Jµ
2BB ·Λ ·B − (2AJgJµBB ·Λ · I −Hz)− (A2
JI ·Λ · I −HQ) (2.2)
where µB is the Bohr magneton, g is the Landau g value, B is the magnetic
field vector, I is the nuclear spin state, AJ is given by
AJ = 2µB~γN〈r−3〉〈J‖N‖J〉 (2.3)
which is the hyperfine interaction for a particular LSJ state given the mean
4f electron-nuclear distance 〈r−3〉 and nuclear gyromagnetic ratio γN . The
tensor Λ is given by
Λ =
2J+1∑
n=1
〈0| Jα |n〉 〈0| Jβ |n〉∆En,0
(2.4)
where 〈0| is the level described by Heff , |n〉 is the other crystal field level of
2.3 Hyperfine Interaction 35
the LSJ state involved and ∆En,0 is the energy difference between the states
En − E0.
Equation 2.2 can be rewritten as
Heff =≡ g2Jµ2BB · Λ · B + H′
z + H′Q (2.5)
where the first term is the quadratic Zeeman shift, H′z is the enhanced nu-
clear Zeeman and H′Q is the effective quadrupole, incorporating both the
quadrupole and pseudoquadrupole interaction. H′z has the form [47]
H′z = −~ [γxBxIx + γyByIy + γzBzIz] (2.6)
where x, y and z are the axes of the Λ tensor with the effective gyromagnetic
ratios given by
γα = γN + 2gJµBAJΛαα/~ (2.7)
The quadrupole interaction is described by
HQ = P[(
I2z′ − I(I + 1)/3
)
+ (η/3)(
I2x′ − I2
y′
)]
(2.8)
where P is the quadrupole coupling constant, η is the electric field gradient
assymetry parameter and x′, y′, z′ are the principal axes of the electric field
gradient tensor.
The second order magnetic hyperfine or pseudo-quadrupole interaction,
described by the term A2JI ·Λ ·I in equation 2.2 can be rewritten in the same
form as the quadrupole contribution HQ, i.e.,
Hpq = Dpq
[
I2z − I (I + 1) /3
]
+ Epq[
I2x − I2
y
]
(2.9)
The zero field pseudoquadrupole parameters can be determined from the
original Λ description as follows:
Dpq = A2J ·
[
1
2(Λxx + Λyy) − Λzz
]
(2.10)
Epq = A2J ·
1
2(Λxx + Λyy) (2.11)
When combining the pseudo-quadrupole and pure quadrupole terms to cre-
ate a single effective quadrupole Hamiltonian it should be noted that in low
symmetry sites the axes of Λ and HQ (x′, y′, z′) are in general different. When
the sum of these terms is diagonalised, a third set of principal axes is found
result, (x′′, y′′, z′′). In low symmetry sights the crystal field and effective
36 Rare Earth Ion Spectroscopy
quadrupole principal axes will only coincide when the enhanced nuclear Zee-
man and pseudo-quadrupole interactions dominate the contribution of the
nuclear Zeeman and electric quadrupole.
H′Q = D
[
I2z′′ − I (I + 1) /3
]
+ E[
I2x′′ − I2
y′′
]
(2.12)
In order to accurately calculate the enhanced Zeeman and pseudo-quadrupole
interactions the crystal field wavefunctions that define Λ need to be known
precisely. Precise determination of Λ requires fitting these theoretical re-
sults to experimental data. For low symmetry sights fitting all the required
parameters is difficult due the principal axes being free parameters.
2.3.1 Yttrium Orthosilicate: The Gracious Host
Yttrium Orthosilicate (Y2SiO5) is a low symmetry (C62h) insulator host, the
crystallographic properties of which are summarised in table 2.1. The trans-
lational unit cell consists of with two formula units with the resulting crystal
structure shown in figure 2.4. This creates two crystallographically inequiv-
alent sites at which the Pr can substitute for Y, labelled ‘site 1’ and ‘site 2’
[6]. While the crystal has C62h symmetry the Y sites have C1 or no symmetry.
Each crystallographically identical site in fact consists of a pair of magneti-
cally inequivalent sites, related by the C2 axis, labelled ‘a’ and ‘b’. Only site
1 ions are used in this work.
While Y is a good substitutional ion for all rare-earth ions, Pr and Y
are a particularly good match with the ionic radius for the 3+ oxidation
state being 1.03 and 0.97 respectively. As previously mentioned both Pr and
Y have a stable 3+ oxidation state and consequently substitution does not
require any charge compensation in the crystal.
Despite the structure of Yttrium Orthosilicate being known since 1969
[99] the ability to grow Y2SiO5 crystals did not eventuate until the late
1980s. This was partially due to being a high temperature crystal which
poses significant fabrication challenges. Consequently Y2SiO5 is a relatively
new host for rare-earth ions [100, 88]. Due to the small magnetic moment
of Yttrium both Y2SiO5 and Y2O3 when doped with rare-earth ions result
in the some of the narrowest homogeneous optical transitions observed in a
solid [5, 100].
2.3 Hyperfine Interaction 37
Space group C62h
Y site symmetry C1
Lattice constants
a (A) 14.371b (A) 6.71c (A) 10.388β 122.17
Table 2.1: Y2SiO5 Crystal parameters
2.3.2 Hyperfine Interaction in Praseodymium Doped
Y2SiO5
The complete description of the state energy within one LSJ state of equation
2.2 can be rewritten using an effective Zeeman M tensor and an effective
quadrupole tensor Q as [101]
H = B · M · I + I · Q · I (2.13)
This description is experimentally usefull since we can measure the zero field
aplittings produced by the I ·Q ·I interaction and the response to a magnetic
field and therefore the B ·M ·I interaction. The zero field splittings produced
by the quadrupole splittings are 10.2 and 17.3MHz [102] as shown in figure
2.3.
2.3.3 M and Q
In order to determine M we need to measure the spectrum of the hyperfine
ground state for a number of magnetic field configurations. The applied
magnetic field must be rotated such that it spans the space. Peaks in the
spectrum are then assigned to a particular state and a minimisation routine,
such as simulated annealing method [104, 105], is used to fit the M and Q
tensors. Knowledge about the site symmetry can be used to constrain the
axes of M and Q if the symmetry is sufficiently high and thereby reduce the
free parameters.
In axial or higher symmetry Λ and HQ share the same principle axes, re-
ducing the free parameters. Consequently the Y site being C1 or no symmetry
is the least constrained situation, resulting in 12 free parameters [105, 107].
This is further complicated since each crystallographic site consists of a pair
of magnetically inequivalent sites, related via the Y2SiO5 C2 axis. Therefore,
any y component of the applied field experienced by site ‘b’ ions will appear
rotated by 180 with respect to site ‘a’ ions. Consequently there are two sets
38 Rare Earth Ion Spectroscopy
32/+-
12/
+-
52/
+-
32/+-
12/
+-
52/
+-
4.6 MHz
4.8 MHz
17.3 MHz
10.2 MHz
H
D1
2
34
605.7 nm
Figure 2.3: The energy levels of the Pr ground and excited state in Pr3+:Y2SiO5 labelledaccording to Ham et al [103, 102]
of spectral lines whenever the applied field has a projection on the y axis.
By performing a detailed field rotation study the M and Q tensors for site
1 were recently determined [105, 107] for Pr3+:Y2SiO5 to be:
Q = R(αQ, βQ, γQ)
−E 0 0
0 E 0
0 0 D
RT (αQ, βQ, γQ)
M = R(αM , βM , γM)
gx 0 0
0 gy 0
0 0 gz
RT (αM , βM , γM)
(2.14)
where E = 0.5624Mhz, D = 4.4450Mhz, g = (2.86, 3.05, 11.56)kHz/G, and
the Euler angles are (αQ, βQ, γQ) = (−94, 58.1,−20.7) and (αM , βM , γM) =
(−99.7, 55.7,−40). These values are for the crystal aligned with the C2 axis
in the y direction, and the z axis is the direction of linear polarization of
the Pr optical transitions. These tensors are highly anisotropic due the low
symmetry of the site. The agreement between experiment and the fitted M
and Q tensors is very good, as shown in figure 2.5.
2.3 Hyperfine Interaction 39
a
c
b
o
YSi
O
Figure 2.4: Yttrium Orthosilicate structure. Figure originally printed in Ching et.al.[106]
40 Rare Earth Ion Spectroscopy
0 500 1000 1500 20000
10
20
30
Magnetic Field (Gauss)
Freq
(M
Hz)
32/ 52/+- +-
12/
32/+- +-
52/12/+- +-
Figure 2.5: Shows part of the data used to fit M and Q with the spectrum predicted bythe fitted data
2.4 Homogeneous and Inhomogeneous Broadening 41
Figure 2.6: An inhomogeneously broadened transition consists of a continuum of ho-mogeneously broadened packets offset in frequency. The height represents the number ofions in each packet. This diagram is not to scale and in a typical rare-earth system theratio of inhomogeneous to homogeneous linewidth is of the order of107
2.4 Homogeneous and Inhomogeneous Broad-
ening
The broadening of transitions can be separated into two categories of Homo-
geneous and Inhomogeneous. Homogeneous broadening refers to broadening
that is common to all ions, while inhomogeneous broadening is experienced
differently for each member of the ensemble. The homogeneous linewidth
would be be the linewidth measured if a single ion was probed. The lower
limit of the homogeneous linewidth is determined by the excited state life-
time of the transition studied. In general, dynamic interactions with the
host broaden the transition significantly beyond this limit. At higher tem-
peratures phonon interactions dominate the homogeneous linewidth contri-
butions, while at liquid helium temperatures the exchange of energy between
spins tends to dominate.
Every crystal has some degree of strain due to solidifying at a finite tem-
perature. Combined with the strain are point defects, chemical impurity and
other effects which cause a spatially varying local potential. This results in
a different resonant frequency for each ion in the ensemble and is referred to
as inhomogeneous broadening. This detuning is static for the crystals used
42 Rare Earth Ion Spectroscopy
in this work. For the systems considered in this work and rare-earth systems
in general, the inhomogeneous broadening is significantly greater than the
homogeneous broadening and consequently the transitions are referred to as
inhomogeneously broadened.
2.5 Optically Detected NMR and Coherent
Transients
All experiments performed in this work fall under the banner of Optically
Detected Nuclear Magnetic Resonance (ODNMR). Optical detection has sig-
nificant sensitivity advantages compared with direct detection of the RF
photons traditionally used in NMR due to the higher efficiency of optical de-
tectors. ODNMR therefore allows probing significantly more dilute samples
than traditional methods with a large Signal to Noise Ratio (SNR).
2.5.1 Raman Heterodyne
While Raman heterodyne is a useful technique for many areas of spectroscopy
the following discussion is limited to optical transitions split by the hyperfine
interaction. Raman heterodyne signals are only observed in systems which
have a common excited state for both hyperfine ground states, often referred
to as a Λ system. This only occurs in systems with sufficiently low symmetry
that spin is no longer a good quantum number and all transitions are allowed.
For the purpose of this discussion we will label the hyperfine ground states
as |1〉 and |2〉 with the common excited state being |3〉 as shown in figure
2.7. Driving the system at ω13 and ω23 a coherent relationship is set up
between all of the levels. If there is a population difference between |1〉 and
|2〉 then the system will emit coherent radiation at the frequency ω32. The
hyperfine splitting for Pr3+:Y2SiO5 are on the order of 10 MHz and as such
the associated wavelength is large compared to typical sample dimensions
of several mm. Therefore the phase matching is effectively satisfied over
the sample and as such the coherent emission at ω32 is collinear with ω23.
This results in a beat note on the transmitted light at the frequency of the
hyperfine transition ω12, the amplitude of which is defined by the population
difference between the levels.
The benefit of using Raman heterodyne detection of nuclear spin states is
the detection efficiency and flexibility. Optical detectors such as photodiodes
are commonly available with high quantum efficiency (> 80%). Direct RF
detection of spin resonances typically requires resonant detection coils placed
2.5 Optically Detected NMR and Coherent Transients 43
23
12
31
| >1
| >3
| >2
Figure 2.7: The Raman heterodyne interaction: By driving the system at ω13 and ω23
a coherent relationship is set up between all 3 levels, resulting in coherent emission at ω32
proportional lto the population difference between the ground states |1〉 and |2〉
close to the sample, limiting the experimental configurations, whereas optical
detection only requires a beam path through the sample. A further benefit
of using optical detection is that significantly lower concentration samples
can be used which is of particular importance for maximising T1, as will be
discussed in chapter 4.
2.5.2 Spin Echos
As previously discussed in section 1.3.2, detuning from resonance causes the
spin vector to rotate round the Bloch sphere at a rate equal to the detuning.
The different rate of precession around the Bloch sphere of each component of
an ensemble causes the total projection of an ensemble superposition state to
decay, termed a Free Induction Decay (FID) as shown in figure 2.8. An FID
is typically observed by driving the transition and observing the coherence
decay as the driving field is switched off and as such is maximised if the
transition is driven to a 50-50 superposition state. The FID decay rate is
proportional to the inhomogeneous linewidth of the transition, given by the
following relation.
ωinh =2π
T ∗2
(2.15)
where T ∗2 is the time taken for the ensemble projection to reach 1/e of the
value when the driving field is turned off and ωinh is the inhomogeneous
linewidth of the transition. If the inhomogeneous broadening is static then
despite the projection of the ensemble on the Bloch sphere quickly tending
to zero the coherence each individual component of the ensemble can persist
significantly longer. The measured decay, with the constant T ∗2 , is therefore
due to our inability to follow each member of the ensemble.
44 Rare Earth Ion Spectroscopy
Z
X
Y
Z
X
Y
Z
X
Y
a cb
Time
Ense
mb
le C
oh
ere
nce a
c
b
Figure 2.8: Bloch Sphere representation of an ensemble placed in a 50-50 superpositionstate undergoing a Free Induction Decay (FID)
If we consider an experiment as described in figure 2.9 where we apply
a π/2 pulse to an ensemble initially in the ground state it will exhibit an
FID signal. Applying a π pulse after a period τ1 results the ensemble being
rotated by π radians around the X axis, thereby effectively time reversing the
evolution of the ensemble, causing the spin vectors to return to a macroscopic
coherence after a further time τ1. This macroscopic rephasing of the ensemble
is referred to as a “Spin Echo” and the delays before and after the π pulse
are referred to as the dephasing and rephasing periods respectively.
Any difference in transition frequency between dephasing and rephasing
period will result in the spin vector accumulating a phase shift. Changes in
transition frequency considered in this work are due to reconfiguration of the
local magnetic field resulting in a time varying Zeeman shift. For an ensem-
ble of ions, each interacting with it’s own local field there will therefore be
a distribution of accumulated phase resulting in attenuation of the echo am-
plitude. Changing the delay τ allows the phase shift due to the environment
to accumulate over different periods, building up a decay sequence which
describes the decoherence time of the ensemble, T2.
It is assumed in the previous discussion that the evolution of the state
during a pulse can be described by only considering the action of the pulse
on the state. Pulses that satisfy this assertion are referred to as “hard”
2.5 Optically Detected NMR and Coherent Transients 45
Z
X
Y
π
Z
X
Y
π/2
ππ/2
Z
X
Y
Z
X
Y
Z
X
Y
1
2
3
4
5
1
23
4
5
Time
Time
En
sem
ble
C
oh
ere
nce
Ap
plie
d R
f
τ τ
Figure 2.9: Bloch Sphere representation of a 2 pulse Spin Echo.
46 Rare Earth Ion Spectroscopy
pulses in NMR terminology. For this approximation to be valid the Rabi
frequency must be significantly larger than the linewidth of the transition
such that for all ions the action of the driving field is the same. If the
detuning approaches the Rabi frequency then off resonant effects become
important and each member of the ensemble will respond differently to the
driving field.
Γ ≪ γB1 (2.16)
where B1 is the ampltide of the RF magnetic field, Γ is the linewidth of the
transition.
2.5.3 Spin Echo Decays
The decay of a spin echo is due to uncontrolled interaction of the spin of in-
terest with it’s local environment. An exact calculation of each ion’s possible
local environment configurations is prohibitively complex and often unknown.
Consequently statistical models are used to approximate the effect of local
environment reconfiguration. For solid state NMR experiments these models
were rigorously investigated theoretically using a semi classical approach by
Klauder et al. [108] and Herzog [109] and experimentally by Mims [110].
The complete derivations are lengthy and for this work we are primarily
interested in what different decay behaviours indicate about the dephasing
environment. These derivations assume certain dynamics of the bath and
then calculate the expected decay functions. In these discussions the spin
echo amplitude of the spins of interest, or “A ” spins, is studied in the
presence of a random magnetic field due to an array of “B ” spins within the
host. The analysis requires that the experiment is performed using “hard”
pulses (see equation 2.16). The concentration of A spins is assumed to be
sufficiently low that interactions between A spins can be neglected. Under
these conditions the dominant perturbation of the A frequencies is due to
dipolar interactions with the array of B spins. The total interaction between
an A spin, at point r, with a magnetic moment of mj = ~γAIj and a number
of other B spins, each with a moment of mk = ~γBSk is given by:
HD = γAγB~2Ij ·
∑
k
Sk
rjk3− 3rjk [Sk · rjk]
rjk5(2.17)
where rjk = rj − rk is the distance between the spin A and the jth B spin
and mj is it’s magnetic moment. Reconfiguration of the B spins produce a
time dependant magnetic field fluctuation which results in a frequency shift
2.5 Optically Detected NMR and Coherent Transients 47
of the A spin, ∆ω(t), given by
∆ω(t) = γA∑
k
mk(t)(
1 − 3 cos2 θjk)
/r3jk (2.18)
The spin echo amplitude for a given pulse separation τ can therefore be
described by
E(2τ) =
⟨⟨
exp
i∑
k
αjξk(2τ)
⟩
time
⟩
lattice
(2.19)
where the angled brackets represent averages in time and over all A spin
environments in the ensemble, ξj(t) are the magnetic field fluctuations, αj
includes the A spin gyromagnetic ratio and the geometric factor for the dipole
fields
αj = γA(
1 − 3 cos2 θjk)
/r3jk (2.20)
The magnetic field fluctuations, ξj(t) are given by
ξk(t) =
∫ t
0
s(t′) (mk(t′) − m(0)) dt′ (2.21)
where s(t′) is the π rotation around the y axis at time τ , represented by
s(t′) = 1, t′ < τ
= −1, t′ > τ (2.22)
Statistical models of how the B spin bath reconfigures are required to find
suitable averages for equation 2.19. All of the models involve a Markoffian
evolution of the B spin configurations such that the next configuration only
depends on the current configuration and the bath therefore has no memory.
Consequently there is a bath reconfiguration rate parameter, R, in all of
the models that is directly related to T2. R is equivalent to the inverse of
population lifetime, 1/T1 of the bath.
Herzog et al. [109] assume that the the average over all bath configu-
rations results in a Gaussian lineshape independent of time. They further
consider the instantaneous distribution of detunings to also be Gaussian and
derive a probability for a particular detuning at time t of
P (∆ω, t,∆ω0) =1
γA [2π (1 − e−2Rt)]1
2
exp
[
−(
∆ω − ∆ω0e−Rt
)2
2γA2 (1 − e−2Rt)
]
(2.23)
48 Rare Earth Ion Spectroscopy
It is assumed that all B spin neighbourhoods are the same and therefore can
be averaged over using the same probability distribution. This results in the
spin echo amplitude being described by
E(2τ) = E0exp
(
−(γAR
)2[
Rτ − 1 + (1 +Rτ)e−2Rτ]
)
(2.24)
where E0 is the maximum echo amplitude. If the bath is rapidly flipping
with respect to τ , the time scale of the echo, such that Rτ ≫ 1 equation 2.24
approximates to
E(2τ) ≈ E0exp
(
−(
γA2τ
R
))
(2.25)
If the B spin bath is flipping slowly, such that Rτ ≪ 1, then over the time
scale of the echo only a small number of possible bath configurations are
experienced by the A. It is not possible to argue that a small number of
randomly selected elements of a Gaussian distribution also form a Gaussian
distribution [110]. If we average over a number of A spins, each experiencing
an independent small number of B configurations the resulting distribution is
not Gaussian but Lorentzian [110]. Therefore if we examine the broadening
of a group of spins, all starting the echo sequence with an initial transition
frequency ω0 at time t0 it will broaden into a distribution [108]
K(ω − ω0,∆t) =
(
2R∆ω1/2∆t)
/π
(ω − ω0)2 +(
2R∆ω1/2∆t)2 (2.26)
where ∆t = t − t0 and ∆ω1/2 is the half width half maximum defined by
the distribution of all possible bath configurations and ∆t. Klauder and
Anderson [108] show that this results in the echo decay function
E(2τ) = E0e−(2τ/TM )2 (2.27)
Where TM is the phase memory, rather than coherence time T2 is used when
the decay time dependence is non-linear, reflecting the non-Bloch behaviour
of the decay [110]. In this work we will however refer to the 1/e point of all
echo decays as T2 for clarity.
In between these two extremes when Rτ ≈ 1 there is no simple statistical
approximation [110] and consequently no decay functions have been derived.
From equations 2.25 and 2.27 it is clear that two distinctly different spin
echo decays will be observed depending on the relative time scale of the echo
and B spin bath reconfigurations. In the laboratory observing the decay
characteristics allows the time scale of the bath perturbation to be estimated.
2.5 Optically Detected NMR and Coherent Transients 49
If the echo decay fits well to equation 2.25 only a lower limit can be placed
on R. When the echo decay is described by equation 2.27 noting the point
at which the gradient starts to increase allows estimation of R. Having these
indicators of the dynamic processes that cause decoherence is important for
choosing the right approach to minimise decoherence as will be seen in this
work.
Chapter 3
QC Benchmarks and Benefits
of Rare Earth QC
“It is not worth an intelligent man’s time to be in the majority. By definition,
there are already enough people to do that.”
- G. H. Hardy (1877 - 1947)
As discussed in section 1.2 Quantum computing places conflicting re-
quirements on quantum systems. Long coherence times are characteristic of
systems that are well isolated from their environment while the ability to
perform fast state manipulation and readout requires that the system can be
strongly coupled to the driving field. Generally systems that can be strongly
coupled to driving fields interact strongly with their environment. A diverse
range of systems has been proposed as quantum computing candidates. The
main emerging approaches focus on NMR [20, 111, 35, 112], trapped atoms
[113, 114], quantized magnetic fields [115], charge fluctuation [116] and lin-
ear optics [117, 118]. DiVincenzo’s requirements are an extremely useful tool
to assess proposals and all, if the advocates are honest, are either seriously
challenged by at least one requirement (section 1.2) or currently fail outright
with no change in sight.
The comparison of candidate quantum computing systems is however an
extremely useful process. It has provided a motivation and method to com-
pare our ability to initialise, manipulate and measure quantum information
in a vast range of systems that are otherwise rarely contrasted due to very
different mechanisms of interaction and areas of application. Each proposal
has strengths that typically stem form long term research investigating a
specific quantum interaction in particular class of materials. Since the re-
3.1 Rare Earth Ion ODNMR Quantum Computing 51
quirements for quantum computing require a system that is a “Master of all
trades” it is not sufficient just to focus exclusively on the traditional line of
research of quantum interactions in the particular system. With the require-
ments of a scalable quantum computer being so challenging to achieve it is
also important to consider as many systems as possible as they all contribute
to our knowledge of both desirable and undesirable system properties.
3.1 Rare Earth Ion ODNMR Quantum Com-
puting
The ability to optically manipulate both the electronic state and nuclear spin
state of rare-earth ions results in an interesting solution to DiVincenzo’s re-
quirement, desiring an isolated system that also strongly interacts (section
1.2). Superpositions of nuclear spin states are typically long lived and are
therefore a good place to store information, reflecting their isolation from
their environs. Their isolation is of course a hinderence to implementing
control Hamiltonians and interacting information on different nuclear spins.
Optical transitions in contrast typically have short lifetimes and higher os-
cillator strengths providing us a mechanism to strongly interact with the
nuclear spin state.
Using Raman techniques, coherence and population of a nuclear spin tran-
sition can be transferred to an electronic transition [119, 120]. For rare-earth
ions such as Praseodymium (Pr) or Europium (Eu) a change in electronic
state is accompanied by a change in static electric dipole moment. This
change stark shifts optical transitions of nearby ions by up to several GHz
which is significantly larger than both the laser linewidth and hyperfine split-
tings. This effect, termed instantaneous spectral diffusion, is well known to
rare-earth spectroscopists and has been observed in numerous rare-earth sys-
tems [121, 122, 123, 124, 125, 126].
We can therefore use a electronic ground state hyperfine transition as
a “non-interacting” qubit and an electronic transition as an “interacting”
qubit. This gives strong switchable coupling between qubits. This use of the
electronic ground state hyperfine transitions as a “non-interacting” qubit
and an electronic transition as an “interacting” qubit are common features
of all rare-earth quantum computing proposals [1, 2, 3].
To consider how this enables quantum computation we will examine the
operation of a CNOT gate, discussed in section 1.2, with pure input states.
A CNOT, or Controlled NOT operation involves two qubits: the “control”
and “target”. For the purpose of this discussion the electronic ground state
52 QC Benchmarks and Benefits of Rare Earth QC
hyperfine levels are labelled |0〉 and |1〉 with a common electronic excited
state |e〉 as in figure 3.1. Since the states that form the control qubit change
during the gate operation these have been labelled |Q0〉 and |Q1〉.The first stage of the CNOT is to change the “non-interacting” qubit
into an “interacting” qubit by transferring everything in |1〉 to |e〉. If the
control qubit is initially in |1〉 it will be promoted to |e〉 and stark shift the
transition of the neighbouring target qubit as shown in figure 3.1. The Stark
shift is large enough such that it no longer interacts with the driving fields,
leaving the state of the target qubit unchanged. The control qubit is then
returned to “non-interacting” state by transferring population |e〉 in to |1〉.Conversely if the control qubit is initially in |0〉 it will not be promoted to
|e〉 and will not Stark shift the target qubit. Therefore the driving fields will
remain resonant with the target qubit and invert it’s state as shown in figure
3.1.
3.2 Rare Earth Quantum Computing Archi-
tecture
The simplest architecture using the interactions described in the previous
section was proposed by Ohlsson et al. [2]. This scheme relies on finding
pairs of rare-earth dopant ions that have sufficiently strong interactions in a
randomly doped sample. Due to the difficulty of detecting emission from a
single rare-earth ion this architecture was proposed as an ensemble approach.
Using holeburning techniques two narrow features, or spikes can be pre-
pared in the inhomogeneous line that will form the two physical qubits we
desire to interact as shown in figure 3.2. The spikes now need to have the
ions removed which do not appreciably stark shift a corresponding ion in
the other spike. To achieve this Ohlssen et al. proposed optically exciting
the spike which produces a stark shift on interacting ions in the other spike,
thereby broadening the spike. The unshifted ions can be holeburned away
leaving two spikes of interacting ions. The ratio of homogeneous linewidth to
nearest neighbour Stark shift for rare-earth ions can exceed 107 [127]. Given
that hyperfine splittings are on the order of ∼ 100MHz and a laser linewidth
of < 1kHz ions need not be nearest neighbours to be sufficiently shifted and
therefore many ions can interact with each other via this mechanism.
The problem with the scheme arises when the number of remaining ions
are considered. Ohlssen et al. calculated that optically exciting a spike
resulted in a frequency shift that was ∼4% of the width of the spike. The
random direction of the frequency shift results is > 99% of the ions in each
3.2 Rare Earth Quantum Computing Architecture 53
1| >|0>
| >e
1| >|0>
| >e
1
Initial
TargetControl
|Q >|Q >0
Control
Final
Target
|Q >1
|Q >0
Intermediate
TargetControl
|Q >1
|Q >0
Initial
TargetControl
|Q >1
|Q >0
Final
TargetControl
|Q >1
|Q >0
Intermediate
TargetControl
|Q >1
|Q >0
Figure 3.1: CNOT implementation using rare-earth ions showing the two extreme casesof the control in either a pure |0〉 or |1〉 state. Transitions to the excited state |e〉 areoptical. The detuning ∆ω is much larger than the difference in energy between states |0〉and |1〉
54 QC Benchmarks and Benefits of Rare Earth QC
spike being thrown away due to insufficient interaction strength. Selecting
for 3 or more interacting ions results in removing an impractically large
proportion of the initial ensemble. Consequently this approach does not
scale to complex quantum computing demonstrations.
The only previous gate demonstration using rare-earth ions used a modi-
fied form of this approach [127]. This modified approach was able to resolve
shifts within the width of the spike and consequently allowed more ions to
be kept. The scaling problems still remained however and the distribution
of interaction strengths was shown to limit the operation of the gate [127].
A different approach, recently developed by Sellars [128], is to introduce
point defects into a stoichiometric crystal. A magnetic defect is which has a
magnetic dipole moment that significantly differs from that of the stoichio-
metric ion will be the the focus of the discussion. Charge defects can also
be considered and it is also important to note that with a dopant ion will
typically differ in both electric and magnetic dipole moments. A system that
would realise a predominantly magnetic defect this is a stoichiometric Eu
crystal into which we lightly dope Er such as Er3+ : Eu2SiO5. Due to the
unpaired electron spin of Er (γEr ≈ 103γEu) many Eu ions will be Zeeman
shifted out of resonance with the bulk Eu, thereby creating a large region of
ions with unique detunings.
Such a system would exhibit rich spectroscopy, behaving like an artificial
molecule. The Zeeman splitting can be investigated to map the distance
between a particular rare-earth ion and the magnetic defect. Studying the
stark shift due to selective excitation of a particular rare-earth ion detuning
can then be used to determine the spatial relationship of each unique rare-
earth ion spectral feature. As the bulk rare-earth ions will have a different
hyperfine splitting to the rare-earth ions of interest near the defect, the rare-
earth ions in the bulk can be optically pumped into a single spin state. This
stops any cross relaxation within the bulk, thereby suppressing the associated
magnetic field fluctuations and increasing the hyperfine T2 of the frozen core
rare-earth ions.
Therefore such a system can create a small region of uniquely addressable
qubits, enabling simple fabrication of many qubit solid state quantum com-
puting. It is expected that a > 10 qubit device would be realisable with this
approach [128]. Due to the low oscillator strength of the optical transitions
many magnetic defect features would be detected as an ensemble to achieve
sufficient SNR. It would therefore perform as an ensemble of small quantum
computers with the final result representing an ensemble average.
Each computational step of an algorithm generally involves interacting
3.2 Rare Earth Quantum Computing Architecture 55A
bso
pb
tio
n
Frequency
π
π
π
π
1
2
3
4
5
Figure 3.2: Schematic representation of the selection process for obtaining two ensemblesof interacting ions from a randomly doped sample. 1) two narrow spikes are prepared inlarge holes burnt into the inhomogeneously broadened optical transition. 2) one of thespikes is optically excited using a π pulse while the unshifted ions of the other spike areholeburned away. 3) the initial spike is returned to the ground state. 4) Unshifted ions inthe first peak are holeburned away. 5) Two less intense spikes containing interaction ionsremain.
56 QC Benchmarks and Benefits of Rare Earth QC
different qubits. In quantum computing systems that are limited to nearest
neighbour interactions information needs to be moved around the computer
such that the required qubits can interact. This requires a number of swap
operations [129], in which the interacting qubits “swap” their state. Further
to this each swap operation must be fault tolerant and therefore error cor-
rected, taking more time, resulting in more decoherence and increasing the
number of qubits required. Many schemes are limited to, or favour nearest
neighbour interaction, such as Liquid Phase NMR and most solid state NMR
proposals [130, 131].
The ability to perform operations not restricted to nearest neighbour in-
teractions, as afforded by Sellars’ rare-earth Stark shift scheme is a significant
advantage. By allowing each qubit to interact with many others algorithms
can be implemented with less concern form the specific physical structure
of the quantum computer. When information is required to be moved, or
swapped, it can be moved many qubits in the desired direction. This pro-
vides flexibility as to how the information is routed within the computer
and the number of acceptable nanostructures for implementing a quantum
computer.
More complex structures are of course possible. Network analysis has
shown that with only a few interconnecting nodes between highly connected
local networks the resulting global network is still well connected [132].
Therefore an effective, scalable architecture could in principal be created by
introducing another rare-earth ion species inbetween these magnetic defect
features to act as an interconnect between local computation nodes. There-
fore limits to the number of qubits per magnetic defect is not a fundamental
limitation to the number of possible interacting qubits.
This approach is not a complete solution to scaling of Rare-Earth quan-
tum computing architectures up to the desired 106 qubits [32] it demonstrates
the flexibility inherent in this approach. When compared to other solid state
systems the fabrication required to reach a 10 qubit device is negligible since
a random positioning of the magnetic defect is adequate and therefore no
nanostructuring is required. It should also be noted that an infinitely scal-
able quantum computing system, as routinely discussed to demonstrate the
power of quantum computing is not physically reasonable. No natural sys-
tem continues exponential growth indefinitely. At some point the system
will not be able to sustain coherent interactions between the most distant
qubits and as such the system will reach a more classical regime where the
computational phase space does not increase exponentially with the addition
of qubits.
3.3 Rationale for System Comparison 57
While the final quantum computer will involve some degree of nanostruc-
turing it is not required for initial investigations. Crystals randomly doped
with rare-earth ions are sufficient to investigate some of the physical issues
with controlling optical transitions [105], the Stark shift interaction [127] and
decoherence of hyperfine transitions investigated in this thesis. Therefore ini-
tial quantum computing investigations of rare-earth systems can use existing
crystals, many of which have been examined in detail by spectroscopists and
are therefore well understood. This circumvents many of the problems in-
volved with proposals that require exotic nanostructures, many of which have
not been built before and require fabrication research and characterisation
before the QIP applications can be fully assessed.
3.3 Rationale for System Comparison
Quantum computing demonstrations using Liquid state NMR systems have
to date set the benchmark for candidate quantum computing systems. Liq-
uid phase NMR exhibits very long coherence times [133] and was the first
to implement an algorithm [134]. It has demonstrated the largest number
of interacting qubits [135], 7 as well as the most sophisticated algorithm im-
plementations [111, 112, 135]. It has therefore played a fundamental role in
verifying the theory underlying quantum computing and become a bench-
mark for quantum computing. There are however well recognised limitations
to the approach which our approach does not suffer from. Consequently I will
compare the system we have chosen to investigate with Liquid phase NMR
quantum computing. Furthermore, the approach we are persuing can be
classed as Optically Detected Nuclear Magnetic Resonance (ODNMR) and
therefore has many similarities, allowing for direct comparison of important
parameters.
It should be noted that trapped ions are also a particularly attractive
physical system [114, 113]. Like NMR quantum computing investigations in
the area have benefited substantially from prior work maturing the experi-
mental apparatus and techniques. The atom-optics community has developed
numerous trapping structures and methods of manipulating trapped atoms
that have served to fast track quantum computing experiments.
3.4 Liquid Phase NMR
Liquid Phase NMR experiments typically consist of a sample placed in a
large, homogeneous static magnetic field applied by a solenoid such that
58 QC Benchmarks and Benefits of Rare Earth QC
the strength of the applied field can be precisely varied. Great attention
is paid to the homogeneity of the applied magnetic field with the sample
space, typically of the order of 1cm3 having a variation of less than one part
in 109. Both driving and detecting nuclear spin transitions is accomplished
using a coil pair perpendicular to the applied static field. By convention the
coordinate system is chosen such that the static magnetic field lies along the
z axis. This results in a Hamiltonian of the form:
H = µ (B0Z +BRF (t)X) (3.1)
where X and Z are the Pauli spin operators while B0 and BRF are the static
and RF magnetic fields respectively.
Changes in the nuclear spin state are detected via a change in the magne-
tization of the sample. This change is very small due to the correspondingly
small nuclear magnetic moment and consequently would be difficult to di-
rectly observe over the static magnetic field. Magnetization perpendicular to
the z axis precesses at the Lamor frequency around the z axis at a rate of
µB/~. Detecting the magnetization perpendicular to the z axis removes the
large offsets and any associated fluctuations of B0 from the measurement.
Another benefit is that the desired signal is moved away from DC with the
benefits of removing 1/f noise. The detection coil pair is part of a high Q
resonant circuit at a fixed frequency. This frequency is chosen such that the
range of static magnetic fields that can be generated is sufficient to shift the
spins of interest into resonance with the detection coils.
Ideally, B0 should be as large as possible for a number of reasons. Signal
amplitude is determined by the magnetization or population difference within
the sample. Therefore to maintain the population difference the transition
energy should be much greater than the thermal energy kBT . Even for
the largest realistic fields the energy difference between spin states, ≈ µNB
is smaller than kBT due to the small magnetic moment of nuclei and the
high temperatures that working with a fluid requires. Increasing the applied
field also increases the resolution of the system since changes in the nuclear
magnetic moment result in larger energy differences.
Nuclear spin interactions are typically dominated by magnetic dipole -
dipole interactions [41], given by the following Hamiltonian.
HD1,2 =
γ1γ2~
4r3[−→σ1 · −→σ2 − 3 (−→σ1 · −→n ) (−→σ2 · −→n )] (3.2)
where γ1 and γ2 are the gyromagnetic ratios of the two spins of interest, −→nis the vector between the spins and σ is the Pauli spin matrix.
3.4 Liquid Phase NMR 59
In a fluid the molecules are reorienting with no particular directional-
ity and consequently the dipole- dipole interaction is described using a time
average. For low viscosity fluids the molecules are reorienting significantly
faster than the nuclear spin transition energy, resulting in the time aver-
age of the dipole - dipole interaction being vanishingly small [41]. This is
clearly demonstrated by observing the homogeneous linewidth of nuclear spin
transitions as the liquid is frozen. In a frozen liquid the mean position of
molecules is fixed, resulting in −→n ≈ n(t0) and therefore resulting in a non-
zero average to the inter molecular dipolar coupling. As the liquid is cooled
dramatic increases in the homogeneous linewidth are observed [41]. There-
fore, surprisingly, the chaotic nature of molecules in a liquid is a benefit
for achieving long nuclear spin coherence times by removing inter molecular
magnetic dipole interactions due to the vastly different time scales.
With the use of a suitable solvent the molecule of interest can be diluted
such that the spectrum of the NMR signal is dominated by intra molecu-
lar interactions. Nuclear spin transitions are shifted by the specifics of the
local molecular bonding due to coupling between the electron and nuclear
spins. Such shifts are termed chemical shifts since the shift determined by
the chemical bonds. This shift is of the order of tens to several hundred
kilohertz, which is much larger the typical transition linewidth. A change in
nuclear spin state will change the energy of other nuclear spin states within
the molecule. The energy change is mainly due to J-coupling whereby a
change in a nuclear spin state is coupled to other nuclei in the molecule by
overlap in bonding electron wavefunctions. This interaction is given by
HJ1,2 =
~J
4−→σ1 · −→σ2
=~J
4Z1Z2 +
~J
8[σ−σ+ + σ+σ−] (3.3)
where J is typically scalar (most generally it is a tensor), Zi are the Pauli Z op-
erator for the ith spin, σ+ and σ− are in phase and quadrature superposition
states. Using the rotating wave approximation the terms ~J8
[σ−σ+ + σ+σ−]
can be ignored. If the coupling, J, is weak or the resonant frequencies of the
two nuclei considered are very different this can be approximated as
HJ1,2 ≈
~
4JZ1Z2 (3.4)
This is a very good approximation for molecules with different nuclear species,
large chemical shifts or the same nuclear spins separated by a number of
bonds. It is clear that this effect will not be averaged out when a large
60 QC Benchmarks and Benefits of Rare Earth QC
number of liquid molecules is considered.
The shift due to the J-coupling is typically larger than the direct dipole-
dipole coupling and consequently is very useful for determining the relative
position of nuclear spins in a molecule as well as the bonding between them.
This application has made liquid phase NMR indispensable in the fields of
chemistry and biology and remains the only method to extract detailed struc-
tural information about complex molecules. An impressive library of spec-
tral “fingerprints” have been accumulated so that simple molecules, whose
structure is simple and known can be instructive in discerning more complex
molecular structures. Combined with this is a formidable array of techniques
to drive the system to manipulate the spin states and enhance or suppress
particular interactions between spins. Of particular interest to this work are
those techniques that are designed to extend T2 beyond the undriven value.
These techniques are referred to as Phase Cycling or decoupling sequences
by the NMR community and Dynamic Decoherence Control (DDC) by the
QIP community. Two of these techniques are investigated experimentally in
chapters 6 and 7.
To investigate the general concept consider the Hamiltonian that de-
scribes a multiple spin system interacting with an RF driving field:
H =∑
k
ωkZk +∑
j,k
HJj,kZk + HRF +
∑
j,k
HDj,kZk + Henv (3.5)
where ωk are the resonant frequencies for each nuclei due to the applied
magnetic field and the chemical shifts, Zk is the Pauli Z operator for the ith
spin and α2Z1Z2 is the coupling between the nuclei. HRF describes the action
of the RF field on the nuclear spins and Henv is the uncontrollable interaction
of the environment with the system that results in decoherence.
To extend T2 by driving the system the first step is always understanding
which interaction currently limits T2. Once this is established generally one
of two approaches are taken. Firstly the system can be driven such that
the time scales of the dephasing process and transition of interest differ suf-
ficiently that they become decoupled. This is directly analogous with the
rapid reorientation of molecules in a liquid averaging out inter molecular
dipole coupling as previously discussed. Decoherence time extension based
on this principal is investigated in chapter 7. The second approach is to ap-
ply a cyclic driving sequence such that the effect of the dephasing interaction
is equal and opposite during successive periods, thereby removing the con-
tribution of dephasing at the end of the cycle. This is investigated in detail
in chapter 6.
3.4 Liquid Phase NMR 61
3.4.1 Limitations of Liquid State NMR
The steady state population of nuclear spin states, in contact with an envi-
ronmental bath of temperature T is dominated by the first term in equation
3.5∑
k ωkZk. Since kT ≫ ~ωk the system is in a highly mixed state. The
population excess in the ground state that is utilised to perform NMR exper-
iments is typically very small. The Overhauser effect can be used to induce
larger population differences [41] however they require the ability to couple
the spins of interest to a highly polarised spin reservoir. These techniques
are not applicable to all nuclear spins and total spin polarisation is rarely
achievable.
Observables in NMR are traceless since it is the superposition states that
radiate perpendicular to the static magnetic field direction. Therefore any
excess diagonal terms due to thermal states do not contribute to the mea-
surement. It can therefore be argued that if the experiment is properly
constructed the excess diagonal thermal population is not detected and the
result of an experiment performed on a highly mixed state with a small pop-
ulation excess is identical to an experiment which used a pure initial state
[35]. Hence the term pseudopure state is often used when referring to NMR
experiments. There are two methods used to achieve this termed temporal
labelling and logical labelling [35].
Despite the result of a properly constructed experiment achieving math-
ematically the same result using a pseudopure state as using a pure state
the systems fails DiVincenzo’s second requirement: that the system must
be able to be initialised into a specific initial state. While this may seem
to be splitting hairs, it is contentious if manipulating the small population
excess used as the ensemble is categorically a quantum computation. This is
because there are an infinite number of expansions possible of a pseudopure
state and therefore the interpretation of the evolution and interactions of
such a state is not explicitly defined.
The most fundamental limitation to liquid state NMR providing a prac-
tically useful quantum computer is the number of interacting qubits that can
be achieved. While using molecular bonds as the interconnections between
qubits has obvious advantages for creating ready made quantum computing
circuits it introduces two principle limitations.
Firstly, the coupling between spins at opposite ends of a molecule becomes
very weak as the size of the molecule increases. This requires that very small
energy differences be resolvable, which in turn means long detection periods.
This poses problems for manipulating and observing the system well within
the decoherence time and severely limits the speed at which the computer
62 QC Benchmarks and Benefits of Rare Earth QC
can operate. For the current state of the art 7 qubit implementation this
resulted in an energy difference of 0.25Hz being resolvable, resulting in a
detection period approaching 1 second. The typical signal to noise ratio
requires averaging a number of experiments making the total computation
period very long. Clearly this pushes detection limits and requires excessive
coherence times if the number of qubits is to be increased. To compound this
both logical and temporal labelling result in an exponential decrease in signal
strength as the number of bits is increased. This is because as the number of
molecules in any particular state decreases as the number of possible states
is increased if the available thermal energy exceeds the energy required to
change the molecular state.
Secondly, the coupling between qubits is not switchable since it is defined
by the molecular bonds. Many NMR Phase Cycling pulse techniques are
specifically designed to decouple spins within the same molecule, however
they are often optimised to preserve a specific state and therefore do not
necessarily preserve an arbitrary state [41]. Incorporating such a decoupling
scheme significantly increases the complexity of the required pulse sequences
to manipulate desired qubits. Consequently this increases the required deco-
herence time and fidelity with which control sequences are implemented. As
previously discussed, both of which are already challenging existing technol-
ogy.
Direct detection of RF photons makes detecting a conventional NMR
signal challenging. Detection takes place in the near field using a High Q
resonant circuit with the detection coil placed as close as possible to the
sample. Due to the low energy of RF photons current technology does not
allow efficient detection approaching the single photon limit. Consequently
conventional NMR experiments require a high concentration of spins, often
in combination with signal averaging to achieve a sufficient SNR.
3.5 The Case for Solids
Solid state quantum systems suitable for QIP are difficult to find. Each
quantum object that makes up a solid is typically strongly coupled to every
other quantum object in it’s locality due to bonding and close proximity.
There are many microstructures in solids that exhibit coherent oscillations,
such as quantum dots [116] or superconducting structures [115]. While these
structures are attractive since their properties can be tailored by fabrication
parameters, their decoherence time is typically sub nanosecond [115] due to
uncontrollable interactions with the environment and within the microstruc-
3.5 The Case for Solids 63
ture.
Using dopant ions as the quantum system to implement qubits has po-
tential to remove much of the fabrication and allows exploitation of known
spectroscopy of dopant ions. The maximum utilisation of available quantum
systems is for each dopant ion to be a physical qubit. This however comes
at the expense of easily addressing the qubits. The wavelength of the driv-
ing field is typically significantly longer than the inter qubit distance, which
removes the ability to utilise spatial selectivity to address qubits. Therefore
large field gradients [130] or exploitation of inhomogeneous broadening or
specifically introducing defects to produce known, unique detunings, as dis-
cussed in section 3.2, is required to individually address the qubits spectrally.
Optically detecting nuclear spins has several benefits when compared to
conventional NMR. Through the use of Raman heterodyne spin concentra-
tions orders of magnitude lower than conventional NMR can be used while
retaining high SNR. This allows the use of sufficiently dilute samples that
resonant relaxation between the spins of interest is negligible, often the lim-
iting factor on coherence in conventional solid state NMR [39]. Given the
efficiency of photodetectors and the ability to use interferometric techniques
single photon detection can be readily achieved [136]. It is therefore possi-
ble, in principal, to detect the nuclear spin state of a single optically active
nuclear spin system. Detecting RF photons, as in conventional NMR is a
significantly more difficult problem since detection takes place in the near
field regime and the energy of a single RF photon is significantly below the
thermal noise of detectors. Consequently conventional NMR is limited to
detecting ensembles for fundamental reasons.
There are two distinct benefits to ODNMR when compared to conven-
tional NMR. Directly detecting the RF photon, as in conventional NMR, is
difficult due to the low quantum efficiency, circumvented by detecting Ra-
man heterodyne beats on optical detectors. Conventional NMR achieves
population differences between nuclear spin states by a combination of ma-
nipulating the Boltzmann distribution and pulsed techniques. By applying
large magnetic fields (∼ 15T ) higher energy spin states become energetically
unfavourable and some spin polarisation due to the Boltzman distribution
arises. A variety of Overhauser based pulse sequences exist [41] to exchange
spin polarisation between different spin populations, typically used if a par-
ticular spin species of interest is difficult to polarise. The ability to prepare
nuclear spin population differences via optical excited states is a significant
advantage. Complete polarisation of a nuclear spin state can be achieved
using holeburning techniques [47].
64 QC Benchmarks and Benefits of Rare Earth QC
Optical state manipulation also has a significant benefit for satisfying
Divincenzo’s initialisation requirement (section 1.2). The initialisation re-
quirement can be achieved using spontaneous emission from an optically
excited state to spin polarise the ions interacting with the laser beam [127].
Holeburning spectroscopy has exploited this ability extensively [47]. This is
in stark contrast to liquid phase NMR which is forced to utilise pseudopure
states afforded by applying large magnetic fields (∼ 15T ) such that some de-
gree of spin polarisation is induced by the Boltzmann distribution. A variety
of Overhauser based pulse sequences exist [41] to exchange spin polarisation
between different spin populations, typically used if a particular spin species
of interest is difficult to polarise. Complete polarisation is rarely achievable
through any method.
The rare-earth proposal like all others does however has a fly in it’s oint-
ment. This takes the form of low oscillator strength of the optical transitions,
which is due to the transitions being forbidden in high symmetry situations
and only weakly allowed for low symmetry hosts [47]. This limits the number
of photons able to be detected before spontaneous emission changes the spin
state by decaying to a different spin ground state. At present this will re-
strict us to an ensemble approach, whereby small clusters of qubits perform
identical computation, the results of which are read simultaneously. There
is also potential to couple the final answer to another colour centre or high
Q nanostructure can probed more easily. One such colour centre is the N-V
centre in diamond [137]. Nanostructuring a metallic film on the surface would
allow the formation of high Q microcavities, based on plasmon resonances,
which the ions to be read out could by stark shifting the ion to be read out
into resonance microcavity.
The density of qubits achievable with solid state systems is encouraging
for scaling systems toward a useful quantum computer. Scalability of solid
state systems has been amply demonstrated by conventional electronics.
3.6 Limitations of Rare Earth Quantum Com-
putation due to Hyperfine Decoherence
Rare- earth quantum computing schemes such as those discussed in the sec-
tion 3.2 as well as other proposals [3, 1, 138] use the hyperfine states as long
term storage for information contained on optical transitions. On commenc-
ing this work the T2 of the optical transitions was approximately equal to
that of hyperfine transitions [5, 102] at a few ms. While this is very close
to being lifetime limited for the optical transitions the hyperfine transition
3.6 Limitations of Rare Earth QC due to Hyperfine Decoherence 65
T1 can be several weeks [4]. This significantly longer upper limit to T2 has
led to the use of hyperfine states in the QIP proposals without any actual
experimental evidence that T2 can approach T1 for hyperfine transitions.
If the T2 of the hyperfine transitions are approximately the same as the
optical then hyperfine transitions are not useful for QIP. Transferring infor-
mation from an optical to a hyperfine transition will always introduce some
noise due to non-unit fidelity. Consequently, the decoherence of the hyperfine
transition and the decoherence induced by the swap operation must be less
than the optical transition over the same period to be beneficial. However for
the proposed QIP applications to be successful we require hyperfine transi-
tion decoherence times to significantly exceed that of optical transitions such
that decoherence of the hyperfine transition is negligible.
Research involving rare-earth ions has typically focused on the optical
properties of the ions for application to new lasing materials or holographic
memory and signal processing. Consequently development of rare-earth ma-
terials has focused on obtaining narrow homogeneous optical transitions.
This has resulted in rare-earth doped insulators possessing the narrowest
optical transitions observed in a solid [5]. Hyperfine transitions have typi-
cally only been used for their ability to store population differences created
by driving optical transitions for use in holographic signal processing. As
such hyperfine decoherence has not been a limiting factor in applications
and has not received the same attention as the optical transitions.
In this work we seek to increase the hyperfine T2 to such a point that rare-
earth ions are useful for QIP applications as described in current proposals
[3, 2, 1, 138]. To achieve this we investigate the dependence of hyperfine
transition T2 to both static and dynamic magnetic field techniques.
Chapter 4
Decoherence of Pr3+:Y2SiO5
Hyperfine Transitions with
Small Applied Magnetic Fields
“When we try to pick out anything by itself, we find it hitched to everything
else in the Universe.”
– John Muir
Understanding decoherence mechanisms is paramount to formulating strate-
gies to remove or minimise the effect of decoherence. As previously mentioned
in section 3.6 decoherence studies of rare-earth ions have focused on optical
transitions [139, 53, 54, 140, 141, 142, 7, 122, 5, 6, 143, 144, 88, 145, 4].
These studies showed that T2 for optical transitions are limited by magnetic
interactions [53, 54, 7, 5]. In this chapter the understanding gained from de-
coherence studies in rare-earth systems will be used to build up a description
of decoherence in Pr3+:Y2SiO5, culminating with experiments investigating
the described interactions.
4.1 Pr3+:Y2SiO5 Hyperfine Decoherence
The use of Optically Detected NMR (ODNMR) allows sufficiently dilute
dopant concentrations that dopant-host interactions dominate dopant-dopant
interactions. As discussed in section 3.5, this regime is not achievable in con-
ventional solid state NMR due to detection SNR limitations. Therefore, the
properties of the host are the most important for determining the decoher-
ence properties of the dopant ion.
High quality insulator hosts have no free charges and therefore time vary-
ing electric field perturbations typically contribute less to dephasing than
4.1 Pr3+:Y2SiO5 Hyperfine Decoherence 67
magnetic perturbations for both optical and hyperfine transitions [5, 7, 142].
Consequently the materials developed to achieve long T2 optical transitions
also help achieving long hyperfine transition T2. Magnetic fluctuations in in-
sulating hosts of interest originate from changes in the state of host nuclear
spins. These host nuclear spins exchange spin via cross relaxation as well
as spin flipping due to spin-lattice relaxation. Therefore the magnetic field
due to the array of host nuclear spin magnetic dipoles as seen by the Pr ion
fluctuates in time. This magnetic field Zeeman shifts the Pr ion’s hyperfine
states, causing the transition frequency to change. This results in the transi-
tion phase relative to a reference oscillator becoming uncertain, resulting in
decoherence.
During the early 1990s “low noise” Yttrium oxide based hosts became
available. While Yttrium had been used as a substitutional ion for some time
in common lasing materials such as YAG and YLF these materials contained
Fluorine or Aluminium which have magnetic moments orders of magnitude
larger than Yttrium. Consequently hosts such as Yttrium oxide (Y2O5) and
Yttrium orthosilicate (Y2SiO5), in which Yttrium is the only ion possessing
spin (assuming isotopically puse 28Si) have a significantly smaller fluctuating
magnetic field, hence their description as “low noise” host. This results in
significantly longer decoherence times with the optical transition linewidths
of 122Hz observed in Eu3+ : Y2SiO5 [5], the narrowest ever observed in a
solid. Approximately 20Hz of this linewidth was attributed to magnetic field
fluctuations due to host nuclear spins.
For a Pr3+:Y2SiO5 sample with a 0.05% concentration of Pr dopants
the resonant cross relaxation rate, given by equation 2.17, for two Pr ions
separated by the mean distance (|r| ≈ 30A) is maximally ∼ 60 Hz if both are
in the ±5/2 state and minimally ∼ 2 Hz if both are in the ±1/2 state. The
inhomogeneous broadening of hyperfine transitions in similar samples result
in linewidths of up to 30kHz [6]. Due to the inhomogeneous linewidth being
significantly larger than the dipole-dipole interaction the cross relaxation will
typically be non-resonant, detuned by several homogeneous linewidths and
therefore proceeds at a significantly reduced rate. Since T2 is 500µs [102],
decohering interactions with the host dominate and we can consider each Pr
ion as being surrounded by a continuous host which does not contain any
other Pr ion.
The Pr-Y interaction is significantly more complex than an ion of inter-
ested surrounded by a thermal spin bath. The magnetic moment of Pr ions is
over an order of magnitude larger than that of Y. The presence of the Pr ion
Zeeman shifts the nearby host nuclear spins and with no applied magnetic
68 Hyperfine Decoherence with Small Applied Mangetic Field
field also locally defines the quantization axis of Y spins. By Zeeman shifting
the nearby Y spins it lifts their hyperfine degeneracy and detunes them from
other nuclear spins in the host, hereafter referred to as the “bulk”. This
results in a drastically reduced cross relaxation rate for the spins that are
Zeeman shifted. This is known as a “frozen core” [53], since the spin state
of the ions within it are almost static compared to the bulk. Frozen cores
have been observed in many analogous systems [146, 147, 53, 88]. This effect
tends to increase the decoherence time of the Pr ion since the nearest Y spins
which induce the largest field at the Pr site spin flip at a reduced rate.
Yttrium ions for which the Pr-Y magnetic dipole-dipole interaction (HPr−YDD )
exceeds the Y-Y magnetic dipole-dipole interaction (HY−YDD ) are considered
as being part of the frozen core. In order to define this we first require to
know what the mean magnetic dipole-dipole interaction is for the bulk Y.
The distribution of magnetic fields due to the bulk Y was calculated
by using the Y2SiO5 crystal structure [99, 106] and summing the magnetic
dipole contributions after the spins had been randomly oriented. The code
used to calculate the Y positions, written by Jevon Longdell, is shown in
appendix A. After numerous randomisations of the spins had been performed
a histogram of magnetic field values at the Pr site was generated. It was found
that including more than 50 Y spins did not significantly alter the resulting
distribution of fields. The result of these calculations, shown in figure 4.1,
indicate that the most probable magnetic field experienced by the Pr ion
due to randomly oriented Y dipoles is ∼0.035 G. This results in a bulk Y
linewidth of 7.6 Hz. Calculating the Pr-Y magnetic dipole-dipole interaction
using equation 2.17 results in the nearest 81 Y ions satisfying HPr−YDD > HY−Y
DD
and are therefore considered as being part of the frozen core.
As previously mentioned the magnetic field due to the Pr defines the
quantisation axis of the nearby Y nuclear spins. Due to the Pr ion defining
the local Y quantisation axis a change in the Pr ion spin state will change the
quantisation axis of nearby Y spins. The change in quantisation axis mixes
the Y spin states and results in a high probability of a spin flip. This coupling
introduces the possibility that as a direct result of driving a Pr hyperfine
transition several Y spins also change state, known as the superhyperfine
interaction [7, 143, 148].
When transitions involving both Pr and Y spin flips are excited, it results
in many quantum pathways between the initial and final states that interfere.
The frequency of the quantum pathways will differ by integer multiples of
the Y Zeeman splitting. The echo will therefore be modulated by all quan-
tum pathways with the modulation intensity proportional to the transition
4.1 Pr3+:Y2SiO5 Hyperfine Decoherence 69
0 0.02 0.04 0.06 0.08 0.1 0.120
500
1000
1500
2000
2500
Magnetic Field (G)
Nu
mb
er
of
con
fig
ura
tio
ns
wit
hin
win
do
w
Figure 4.1: Histogram of the magnetic field magnitude at the Pr site due to randomlyoriented Y spins
70 Hyperfine Decoherence with Small Applied Mangetic Field
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
0= =
23= 2=
23=
OnResonance
Detuned
Resultant
Figure 4.2: Bloch sphere representation of a spin echo at the rephasing time 2τ whentwo ensembles, one detuned from resonance result in a modulated echo.
strength. This can be visualised by considering a Bloch sphere with several
sub ensembles, detuned by integer multiples of the Y Zeeman splitting as
shown in figure 4.2. As the echo delay is increased these sub ensembles will
rotate around the Bloch sphere according to their detuning, being initially
in phase with the sub ensemble on resonance, 180 out of phase at time
τ = nπ/∆ω and in phase at time τ = 2nπ/∆ω where n = 1, 2, ..k. It should
be noted that T2 cannot strictly be used to describe the loss of coherence
from the system since it is no longer a two level system and therefore does
not obey the Bloch equations.
The expected magnetic field strengths can be calculated created at the
Pr site due to the nearest neighbour Y ion (BY nn) and visa versa (BPrnn).
The magnetic field due to a dipole is given by the following equation:
B (r) = ~γ
[
I
|r|3 +3I · r|r|5 r
]
(4.1)
where γ is the gyromagnetic ratio, which for Pr is the effective Zeeman tensor,
M , I is the nuclear spin state and r is the distance from the dipole. For
Y2SiO5 the nearest neighbor Y sites are seperated by |r| = 3.4 A, γY = 209
Hz/G and the Pr effective Zeeman tensor is described in section 2.3.2. This
calculation results in a BY nn = 0.011 G and BPrnn = 1.4 G.
Since the magnetic field defines the spin quantisation axis the superhyper-
fine transition probability can be calculated by determining the projection of
4.1 Pr3+:Y2SiO5 Hyperfine Decoherence 71
the initial magnetic field onto the final magnetic field as described in equation
4.2.
PSHF =Bi · Bf
max (Bi · Bi,Bf · Bf)(4.2)
where Bi is the initial magnetic field, Bf is the final magnetic field and
normalised by dividing by the maximum projection of each field onto itself.
As an applied magnetic field is increased the spin quantisation axis is
increasingly defined by the applied field rather than the magnetic field due
to the Pr ion. Therefore mixing of Y spin states due to changes in the spin
state of the Pr ion is decreased and consequently the superhyperfine transi-
tion probability is also decreased. Increasing the applied magnetic field will
also increase the Y Zeeman splitting, and therefore the echo modulation fre-
quencies. The echo modulation frequencies will increase as integer multiples
of γY and since Y is a spin 1/2 system the Zeeman splitting will be linear
within the range of fields used. as the applied magnetic field is increased we
should therefore observe the superhyperfine transition frequency increase as
the transition strength decreases.
The superhyperfine transition probability as an external magnetic field is
applied was calculated using equation 4.2 and is shown in figure 4.3. This
predicts that the superhyperfine transitions will be significantly attenuated
for applied magnetic fields of the order of 10 G. The complex behaviour of the
superhyperfine transition probability when the applied field has a magnitude
of ∼1 G is due to the applied magnetic field and the magnetic field due
to the Pr ion being approximately equal. It is unlikely that the complex
behaviour can be observed experimentally due to residual magnetisation in
the equipment, such as the optical table, being of the order of a few Gauss
making it difficult to achieve true zero applied magnetic field.
Suppressing the multiple superhyperfine transitions results in an increase
in decoherence time. This has been known for some time and is often used
to reduce the magnetic linewidth contribution for optical transitions [5, 142].
These studies inferred the suppression of superhyperfine interactions from
increases in the homogeneous optical lifetime but did not directly observe
the beat frequency. More detailed studies were carried out on Pr3+:LaF3
by Pryde [148]. Superhyperfine interaction however has not been studied for
hyperfine transitions of Pr3+:Y2SiO5.
The previous discussion of superhyperfine interaction focuses on Y spin
flip process that create a different magnetic field configuration at the site
of the Pr ion. Changes in the Pr spin state can also induce two Y ions to
mutually spin flip, or cross relax, resulting in near degenerate initial and
final magnetic field configurations as investigated by Pryde in Pr3+:LaF3
72 Hyperfine Decoherence with Small Applied Mangetic Field
0 5 10 15 20 25 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Applied field (G)
Tra
nsi
tio
n p
rob
ab
ility
Figure 4.3: Calculation of the superhyperfine transition probability for changes in thePr spin state of mI = +1/2to + 3/2, mI = +1/2to − 3/2, mI = −1/2to + 3/2, mI =−1/2to− 3/2. The applied magnetic field is aligned to the z axis.
[148]. Pryde’s work showed that these interactions do not display the same
magnetic field dependence as the non-degenerate superhyperfine interactions
since they require a significantly smaller change in the spin quantisation axis.
Therefore, near degenerate superhyperfine interactions are unlikely to be sup-
pressed by the application of moderate magnetic fields. Pryde demonstrated
that this interaction resulted in transition frequency shifts peaked around 4
kHz with a transition probability of 0.0035. Comparing the gyromagnetic
ratios of Flourine (γF ) and Yttrium (γY ) we find that γF ≈ 20γY and conse-
quently expect only a ∼ 10 Hz contribution for Pr3+:Y2SiO5. The timescales
of coherent transients in Pr3+:Y2SiO5 with low applied magnetic field are of
the order of 500 µs [102] and as such the frequency shifts due to the degen-
erate Y spin flips will not be possible to measure experimentally. Precise
calculation of these parameters was therefore not performed. It is important
to note that these near degenerate superhyperfine interactions are errors in
state manipulation when performing quantum computing operations. The
estimated transition probability calculated above, while small is sufficient to
induce errors of the order of 10−5, the expected threshold for fault tolerant
quantum computing [35].
As previously mentioned we need to understand the decoherence mech-
anisms in order to develop strategies to suppress interactions or the effect
4.2 Experimental Setup 73
of interactions that cause decoherence. From previous work on rare-earth
doped insulators the dominant decohering interaction with low applied mag-
netic fields is the superhyperfine interaction [5, 6, 7] and consequently it is
the first interaction investigated. Studies at low field also provide a bench-
mark to assess whether our attempts to increase T2 beyond the original value
are beneficial.
4.2 Experimental Setup
Coherence is studied using two pulse spin echos detected via Raman het-
erodyne, as discussed in section 2.5.1, the pulse sequence of which is shown
in figure 4.6. The experiments, shown schematically in figure 4.5 were per-
formed using a Coherent 699 frequency stabilised tunable dye laser tuned to
the 3H4 − 1D2 transition at 605.977nm. This laser can be operated in “high”
resolution mode with a FWHM ∼1MHz or in “ultra-high” resolution mode
in which the linewidth is reduced to < 100 Hz. For these experiments it
is operated in high resolution mode. The laser power incident on the crys-
tal was 40mW, focused to ∼100 µm and could be gated using a 100MHz
acousto-optic modulator (AOM). The RF used to drive the AOM was sup-
plied by an 80 MHz oscillator switched by a Mini Circuits ZASW RF switch
before being amplified using an in house built amplifier to 1W maximum.
The ZASW 2 input 1 output bidirectional RF switch provides 100dB isola-
tion with a switching time of 10 ns [149]. This was the only RF switch used
in the experiments, hereafter simply referred to as an RF switch. The RF
used to drive the hyperfine transitions was supplied by the Pulse Blaster Tx
output amplified by an Amplifier Research 10W1000 RF amplifier. This was
applied to the sample using a non-resonant 6 turn 5 mm diameter RF coil
which resulted in a RF Rabi frequency of ΩRF = 50 kHz. The magnetic
field was applied using a solenoid in the z direction, defined by the crystal
properties as shown in figure 4.4.
The Pr3+:Y2SiO5 crystal was placed in a Cryo Industries gas exchange
cryostat with the temperature held at ∼ 4.2K for the duration of the ex-
periment. The laser prepared a population difference in the excited ions
for 1s before the pulse sequence and was scanned over 1.2GHz of the opti-
cal transition to avoid hole burning effects. The laser is off during the RF
pulse sequence to minimize coherence loss from optical pumping as shown
in figure 4.6. The laser and RF pulses were controlled using a Pulse Blaster
pattern generator with a timing resolution of 10ns. The Pulse Blaster is
computer controlled, allowing automation of the experiment. The RF pulses
74 Hyperfine Decoherence with Small Applied Mangetic Field
x
y
z
3mm
5mm
4mmC2 axis
OpticalPolarisation
Figure 4.4: Relationship of laboratory axes to crystallographic axes: y is the C2 axis, zis the direction of polarisartion of the optical transition for site 1 and x is perpendicularto both.
applied were sufficiently hard (ΩRF = 50 kHz) that all transitions of interest
are within the Fourier width of the pulses. The required magnetic field are
supplied by a solenoid aligned with the z axis of the crystal.
The transmitted laser beam is incident on a photodetector with a 125MHz
bandwidth, the output of which is fed into a spectrum analyser. The spec-
trum analyser is operated in a non scanning mode such that when tuned
to the transition it detects the intensity of the Raman heterodyne beat fre-
quency. The input is band pass filtered using the internal resolution band-
width filter, set to 300 kHz. The RF applied to the hyperfine transition is
generated by the spectrum analyser’s tracking generator, thereby ensuring
the detection and excitation frequencies are locked. The intensity measured
by the spectrum analyser was recorded by a digital oscilloscope and down-
loaded onto a computer.
4.3 Results
The modulation of the echo intensity is clearly visible when a large number of
echo delays are studied as shown in figure 4.7. If this decay is to be fit to an
exponential decay the maximum peak heights should be chosen and aliasing
due to an inadequate number of samples will yield an incorrect measurement
4.3 Results 75
Co
he
ren
t6
99
RFAmp
GPIB
Trig
RFSwitch
Pulse TTLout
Tx Rx
BlasterRF out
Cryostat
RF CoilsDC CoilsAOM
DC CoilsSample
Detector
Computer
CRO
Spectrum
Analyzer
Tracking
Generator
RF in I out
Figure 4.5: Schematic diagram of the experimet steup. The laser is gated via an AOMbefore entering the cryostat and being incident on the sample. A static magnetic field canbe applied to the sample using pairs of coils, with the RF radiation applied via the Rfcoils. The transmitted laser beam is incident on a photodetector, the output of which isfed into a spectrum.
Laser
RF2
Figure 4.6: Two pulse Raman heterodyne spin echo pulse sequence.
76 Hyperfine Decoherence with Small Applied Mangetic Field
0 1 2 30
0.2
0.4
1
0.6
0.8
Delay Time (ms)
15 shot running aveln
(E/E
)0
Figure 4.7: Two pulse echo series with 1000 single shot points with an applied field of7.5G. Solid line is a 15 shot running average used for the Fourier analysis. τ is the delaybetween the π/2 and π pulses, as shown in figure 4.6.
of the coherence in the system.
When the echo decay is viewed in the frequency domain it is clear that
there are a number of components of the ensemble with different detunings
contributing to the echo (figure 4.8). Given the π pulse width of 10µs all
frequency components are well within the Fourier width of the pulse for all
field values and therefore excitation bandwidth can be ignored. As an exter-
nal magnetic field is applied, the frequency of each superhyperfine transition
increases by integer multiples of the Y Zeeman splitting.
Figure 4.8 shows that at low applied magnetic fields superhyperfine tran-
sitions involving up to three Y spin flips have significant contribution to the
echo amplitude. The DC component of the FFT is proportional to the un-
modulated echo and consequently each FFT spectrum is normalised to the
DC component. The superhyperfine transitions are attenuated by the ap-
plied field at a rate proportional to how many Y spin flips are involved. This
is consistent with observation in analogous systems [150]. Fitting an expo-
nential decay to the Signal to Carrier Ratio for the spectral component due
to single Y spin flips yields a decay constant of 7G. This is consistent with
the calculations presented in figure 4.3. It is possible that the superhyperfine
transitions are mediated by an exchange interaction involving a shared bond-
ing electron, not simply magnetic dipole-dipole coupling. This would cause
the interaction to persist long after direct magnetic dipole-dipole coupling
4.3 Results 77
−10 -8 -6 -4 -2 0 2 4 6 8 100
1
2
3
4
5
Frequency (kHz)
Am
plit
ud
e (a
.u)
Figure 4.8: Fourier transform of averaged echo sequences at different field values aslabelled on the trace. The FFT of each field value is normalised to the DC componentand offset for clarity.
78 Hyperfine Decoherence with Small Applied Mangetic Field
was quenched by the external field. Since the superhyperfine transitions are
significantly attenuated when the external field is of the order of the Pr-Y
magnetic dipole-dipole coupling, an exchange interaction can be ruled out.
The Zeeman shift due to the Pr ion on local Y ions is the origin of the
frozen core since it detunes the local Y ions from the bulk Y ions. However, as
shown by this experiment, with low applied magnetic field it is premature to
consider the local Y ions as constituting a frozen core for coherent transient
techniques due to the superhyperfine interaction. The local Y ions can be
considered a frozen core when the applied field is much larger than the Pr-Y
dipole-dipole coupling, when changes in the Pr spin state only weakly perturb
the local Y quantization axis.
Although a simple exponential coherence decay is not a complete descrip-
tion of coherence decay at low field the stated value in the literature [144] of
T2 = 500µs is a good fit to the decay envelope. If the peak heights of the
echo are used to determine T2 it is clear that with an applied field of 7.5 G
as shown in figure 4.7 T2 is approximately 3ms.
The Pr-Y magnetic dipole-dipole coupling strength allows us to approxi-
mate the field strengths required for dynamic coherence time extension tech-
niques such as magic angle line narrowing, investigated in chapter 7. This
technique flips the dephasing spins fast, at the magic angle such that their
average contribution is zero. For it to be effective the RF field must exceed
the Pr-Y magnetic dipole-dipole coupling. Therefore an RF field of > 10 G
would be required.
The study by Pryde [148] of near degenerate superhyperfine transitions in
Pr3+:LaF3 indicates that near degenerate Pr-Y will persist. Therefore larger
applied magnetic fields than those used in this experiment should be used to
suppress the superhyperfine interactions such that the hyperfine transitions
behave more like a closed two level atom. Any deviation from closed two
level atom behaviour results in information being lost to the environment
directly due to driving the transition. The majority of previous studies of
Pr3+:Y2SiO5 were done at zero or low field [6, 141, 103, 151, 102]. This is gen-
erally because of the simpler energy level structure at zero field, of particular
benefit if the experiment requires an optical repump scheme [102]. This work
however demonstrates that at low applied magnetic field the system cannot
be considered to be a two level atom and as such QIP investigations should
be performed with larger applied magnetic fields. Larger applied magnetic
fields are investigated in the following chapter.
Chapter 5
Maximising Hyperfine T2 using
Moderate Static Magnetic
Fields
“You must be the change you want to see in the world.”
– Mahatma Gandhi
In the previous chapter it was demonstrated that the decoherence time
of Pr hyperfine transitions in Pr3+:Y2SiO5 is improved by the application of
a small field. It was established that the applied magnetic field minimises
the change in the local Y spin bath due to a change in the Pr spin state. In
this low field regime the sensitivity of the Pr hyperfine transitions to fluctu-
ating fields magnetic fields is largely unchanged. In this chapter I investigate
minimising the sensitivity of the Pr hyperfine transitions to fluctuating mag-
netic fields by application of a static magnetic field with precise direction
and magnitude.
In a spin 5/2 system with zero field splittings when the Zeeman split-
ting approaches the zero field splitting there are numerous level anticross-
ings which result in non-linear Zeeman shifts. The sensitivity of transition
frequency to magnetic field perturbations is given by the gradient of the
Zeeman shift. The non-linearities demonstrate that the sensitivity of the
transition frequency to magnetic field is dependant on the applied magnetic
field. Therefore it is possible that a particular magnetic field magnitude
and direction can be found for which the hyperfine transition sensitivity is
minimised and therefore T2 is maximised.
80 Maximising Hyperfine T2 using Static Magnetic Fields
5.1 Theory
Searching the magnetic field space experimentally for the global minimum of
transition sensitivity is impractical given 15 transitions and the complexity
of the Zeeman shift due to the low site symmetry. As such, knowledge of
the parameters of the reduced Hamiltonian for hyperfine ground state in-
teractions (equation 2.13) is required to enable computational searches of
the magnetic field space. The field rotation study undertaken by Longdell
et al. [105] determined the M and Q tensors for hyperfine interactions in
Pr3+:Y2SiO5, as discussed in section 2.3.3.
H = B · M · I + I · Q · I
The predicted hyperfine spectrum as the magnetic field is applied along
the x, y and z axis is shown in figure 5.1. The complexity of the Zeeman
shift non-linearities make finding a global minimum of transition sensitivity
prohibitively complex. While numerous turning points in hyperfine transi-
tion frequency are observed in one dimensional magnetic field sweeps (figure
5.1), representing one dimensional sensitivity minima, the minimum three
dimensional sensitivity was not known.
Transition sensitivity minima with zero first order Zeeman shifts in any
direction were searched for using a Nelder-Mead simplex (direct search)
method, implemented as part of Matlab Function Functions and ODE Solvers
toolbox. For each magnetic field value tried by the minimisation routine the
frequency shift due to a perturbing magnetic field along each axis was calcu-
lated as follows:
〈∆ω〉 =(
δωx2 + δωy
2 + δωz2)1/2
(5.1)
where 〈∆ω〉 is the geometric mean frequency perturbation and δωi is the
change in transition frequency due to the perturbing magnetic field aligned
with the i axis. The perturbing magnetic field strength was chosen to be
of the order expected due to the array of Y ions. This was calculated in
section 4.1 to have a mean value of 0.035G with the distribution previously
shown in figure 4.1. The minimisation routine was initiated from 125 points
within the ±2 kG region for all hyperfine transitions, extensively searching
the anticrossing region.
The minimisation routine found 61 magnetic field values that satisfied
the criterion zero first order Zeeman shift in any direction for a particular
hyperfine transitions. These points will hereafter be referred to as a critical
point with reference to the definition of a critical point on a curve.
5.1 Theory 81
0 1 2
Bz (kG)
0
10
20
30
40
Fre
qu
en
cy (M
Hz)
0
10
20
30
40
Bx (kG)0 1 2
Fre
qu
en
cy (M
Hz)
0 1 2
By (kG)
0
10
20
30
40
Fre
qu
en
cy (M
Hz)
Figure 5.1: Ground State hyperfine spectrum predicted by theory as the applied mag-netic field is varied from zero to 2kG in the x, y and z directions.
82 Maximising Hyperfine T2 using Static Magnetic Fields
mI Bx By Bz f (MHz) δfδB2
+12↔ −3
2732 173 −219 8.6 102
-309 -1124 -425 3.5 119-451 361 -678 5.6 193
+12↔ −5
2-465 372 -698 7.7 192
−32↔ +3
2438 -351 659 2.2 386237 -1094 382 1.7 486
−52↔ −3
2-683 -357 -1000 7.5 241664 373 971 5.8 2441567 454 155 10.2 320-1086 93 489 10.2 345-890 487 -174 11.5 828212 55 -844 11.5 828
-1093 -253 -453 11.5 829-418 -793 561 11.5 830
+52↔ −3
2-518 566 494 8.5 132-256 1109 -411 5.8 243-1062 -857 602 10.2 320-1562 -457 -150 10.2 321-1289 -646 209 10.2 3451079 -87 -499 10.2 345-214 -54 841 11.5 829
Table 5.1: List of all critical points which have less than 1kHz/G2 second order frequencydependance.
Those critical points that had a second order sensitivity < 1kHz/G2 are
listed in table 5.1 with the full list presented in Appendix B. The critical
point with the lowest seconder order sensitivity was investigated on the mI =
−1/2 ↔ +3/2 transition with BCP = 732, 173,−219G and a transition
frequency of 8.6MHz.
In order to understand the dependence of the transition sensitivity on
the applied magnetic field in the vicinity of the critical point used in the
experiment the first order Zeeman shift was calculated for a large range of
field values around the optimal critical point, shown in figure 5.2. As figure
5.2 shows, there is a continuous reduction in first order transition sensitivity
as the critical point is approached.
The optimum critical point the magnetic field is required to be accurate
to within the magnetic field fluctuations inerrant to the material, which for
Y2SiO5 is of the order of 0.035G. Therefore, given the critical point field
magnitude is ∼780G, this requires an accuracy of one part in ∼2×104. Since
the fitting of the hyperfine parameters to the field rotation data was only
accurate to ∼4% [105] the predicted transition frequencies are not accurate
5.2 Experimental Setup 83
Bx (G)
∆ω
∆Β
Hz/
G
Figure 5.2: Theoretical calculation of the first order Zeeman shift due to a 0.035 Gperturbation in each spatial direction calculated as per equation 5.1.
enough. Further, the inhomogeneous linewidth of hyperfine transitions not
at the critical point is of the order of 10kHz [6]. Therefore, measuring the
spectrum alone is not sufficient to ensure an optimal field alignment has been
obtained. We can, however, predict the expected critical point magnetic
field with sufficient accuracy to optimise the field alignment experimentally.
Given the topology near the critical point, in particular the lack of nearby
local minima, we can be confident of achieving an optimised critical point
when T2 is optimised from an approximate initial magnetic field alignment.
5.2 Experimental Setup
The experiment configuration shown in figure 5.3 has many schematic sim-
ilarities with the previous configuration described in section 4.2. However,
in this experiment the Pr3+:Y2SiO5 crystal maintained at temperature of
∼1.5K in an Oxford Instruments MD10 liquid helium bath cryostat [152].
The applied magnetic fields to achieve the critical point field configuration
were supplied by two orthogonal superconducting magnets supplying a z
field, and x, y field. The axis used to describe the magnetic fields are related
to the crystal properties as shown in figure 4.4 and was discussed in section
2.3.3. The sample was rotated about the z axis to provide the correct ratio
84 Maximising Hyperfine T2 using Static Magnetic Fields
Co
he
ren
t6
99
GPIB
Trig
Pulse TTLout
Tx Rx
BlasterRF out
Cryostat
RF CoilsDC Coils
DC CoilsSample
Detector
Computer
CRO
RF out
J850 DDS
RFAmp
RFAmp
USB
USB1 2 3 4
AOM
RFSwitch
RFSwitch
Figure 5.3: Schematic diagram of the experiment setup. The laser is gated via an AOMbefore entering the cryostat and being incident on the sample. A static magnetic field canbe applied to the sample using pairs of coils, with the RF radiation applied via the RFcoils. The transmitted laser beam is incident on a photodetector, the output of which isfed into a spectrum.
of fields along the x and y axes for the critical point in magnetic field space.
The x, y coil is a split pair magnet, incorporated into the cryostat, specif-
ically designed to produce very homogeneous fields for NMR experiments
and therefore used to provide a majority (∼71%) of the magnetic field. The
field in the x, y plane could also be adjusted using a small correction coil
mounted orthogonal to the MD10 split pair magnet. The inhomogeneity in
magnetic field across the sample was measured using a hall probe to be <2G,
dominated by inhomomgeneities in the z coil.
The Coherent 699 dye laser was operated in “ultra-high” resolution mode,
resulting in a sub kilohertz linewidth. The laser power incident on the crystal
was 40mW , focused to ∼100µm and could be gated using a 100MHz acousto-
optic modulator. The RF driving the AOM was upgraded to a purpose built
Direct Digital Synthesis (DDS) agile RF source combined with an array of
RF switches and amplifiers capable of producing 1W . This unit was built
to our specification by the RSPhysSE Electronics Unit, referred to by it’s
project number as the J850. This enabled implementation of the complex
burn back sequence described later.
The hyperfine transition was excited using a six turn coil with a diameter
of 5mm, driven by a 10W RF amplifier resulting in a Rabi frequency ΩRF =
5.2 Experimental Setup 85
π2π
ττ x10
x500
x10
ω 3
ω CP
ωp
ωr
ω r scan
ω1 ω2,
State Preperation
Figure 5.4: Pulse sequence used in the experiment, showing the repump scheme, with thenumber iterations for wach section indicated below, followed by the 2 pulse echo sequence
91kHz. The RF pulse and digital control sequences were generated using
a direct digital synthesis system referenced to an oven controlled crystal
oscillator. The pulse sequences used in the experiment are illustrated in
figure 5.4 The Raman heterodyne signal, seen as a beat on the optical beam,
was detected by a 125MHz photodiode. This signal was analysed using a
mixer and a phase controlled local oscillator referenced to the RF driving
field.
Prior to applying each Raman heterodyne pulse sequence the sample was
prepared with the optical/RF repump scheme as shown in figure 5.5. The re-
pump frequencies were ωr−ωp = 18.2MHz, ω1 = 12.2, ω2 = 15.35MHz and
ω3 = 16.3MHz. Optical inhomogeneous broadening is significantly larger
than the excited state Hyperfine splittings. Therefore, since the repump
scheme is only designed to select for ground state hyperfine levels there will
be 6 optical subgroups for which the optical transition frequency is satis-
fied. Therefore the common excited state is not labelled. The repump RF
was pulsed with a duty cycle of 10% to reduced sample heating, while the
readout laser frequency ωr was scanned 200kHz to hole burn a trench in the
inhomogeneous optical line where detection would take place. This was im-
plemented by applying ω1 and ω2 pulses alternately while ωr was scanned,
86 Maximising Hyperfine T2 using Static Magnetic Fields
- 1 2/
- 3 2/
+ 3 2/
- 52/
+ 52/
+ 1 2/
3 2/+-5 2/+-
+- 1 2/
r
1
23
p
Figure 5.5: All ground state hyperfine levels other than mI = −1/2 interact via RFfrequencies ω1−3 with laser radiation either at the read frequency ωr or pump frequencyωp. Through spontaneous emission from the electronic excited state the group of ionsinteracting with both ωr and ωp via any common excited state will be holeburned into themI = −1/2 state. Bold states are the critical point transition.
then applying ω3 pulses while ωp was applied and iterating this 500 times as
shown in figure 5.4. This repump scheme ensures that all Pr ions interacting
with the laser radiation are forced into the mI = −1/2 state, creating a pure
state ensemble. It also ensures there is no initial population near the laser
frequency used for Raman heterodyne detection. The use of a sub kilohertz
linewidth laser and the repump scheme resulted in a significant improvement
in the signal to noise compared to work performed earlier [152].
5.2.1 Finding a Critical Point
Sweeping the magnetic field from 0G to 2BCP results in transition frequen-
cies changing as shown in figure 5.6. As can be seen, the transition of interest
goes through a very gradual critical point. Site 1 consists of two magnetically
inequivalent sites as discussed in section 2.3.1. Consequently the y compo-
nent of the magnetic field experienced by “site 1a” ions is the opposite of
“site 1b” ions, resulting in the more complex level structure seen in figure 5.6.
As the sites are crystallographically equivalent it is not important which site
is used and consequently we will refer to “a” as the site that is used. However
since both “a” and “b” ions will interact with the laser radiation they will
both contribute to Raman heterodyne signals and are therefore important
experimentally.
While there are many distinct spectral features, such as other turning
points and points where transition frequencies cross, most are not close
enough to the critical point to be of use. The main spectral feature of use is
5.2 Experimental Setup 87
200 400 600 800 1000 1200 1400
2
4
6
8
10
12
14
16
18
20
B field (G)
Fre
qu
en
cy (
MH
z)
Figure 5.6: Ground State hyperfine transition frequencies as the applied magnetic fieldis varied from zero to twice the Critical Point Field. Site 1a ions are shown as a solid linewith site 1b ions shown as dashed lines. The critical point transition is bold.
88 Maximising Hyperfine T2 using Static Magnetic Fields
that the critical point transition is crossed near the optimum magnetic field
value by the mI = −1/2 ↔ +3/2 transition of site 1b as shown in figure
5.6. As discussed in section 5.1, measuring the spectrum is not sufficient to
achieve an optimal critical point magnetic field configuration.
Coarse magnetic field alignment focused initially on the x, y field by ap-
plying the predicted field magnitude and rotating the sample rod until the
hyperfine spectrum matched the predicted spectrum shown in figure 5.7.
Sample rod rotation has the least resolution of any magnetic field adjustment
due to seals producing hysterisis in the adjustment and not having a microm-
eter actuated rotation mechanism. Consequently this was only adjusted if
there was insufficient field strength from the perturbing coil to achieve an
optimised critical point field alignment. In practice the sample rod needed
to be adjusted several times due to the inhomogeneous linewidth masking the
exact transition crossing points and a lack of other distinct spectral features.
The z axis magnetic field is then applied and the magnetic field is swept
by a small amount to verfy that the frequency of the critical point transi-
tion changes very little compared to surrounding transitions. The predicted
response to a small field sweep on each axis is shown in figure 5.8.
Once the approximate position of the critical point in magnetic field space
has been obtained the decoherence time of the transition is the best guide
for improving the magnetic field alignment. Fitting an entire echo sequence
is time consuming whereas maximising the amplitude of an echo with a long
delay is sufficient. The echo delay should be adjusted so that the echo is eas-
ily distinguishable from the background but there is still plenty of dynamic
range left in the detection system. The magnetic field can then be iteratively
adjusted until the amplitude of the echo is optimised using the longest delay
possible. The lack of further gains will be due to one of three things: the
magnetic field is more accurate than the magnetic field fluctuations in the
material; the magnetic field alignment is limited by inhomogeneity of the ap-
plied field or there is insufficient resolution in the adjustment of the magnetic
field.
5.3 Results
Initial experiments to verify the concept resulted in a significant increase
in decoherence time as shown in figure 5.9. The typical gains due to the
application of a magnetic field were observed, increasing the T2 of transitions
not at the critical point from 0.5ms to 9.9ms for the site 1b mI = −1/2 ↔+3/2 transition and 5.9ms for the site 1a mI = −3/2 ↔ +3/2 transition.
5.3 Results 89
0 2 4 6 8 10 12 14 16 18 200
5
10
15
20
25
30
35
40
Rotation angle (degrees)
Fre
qu
en
cy (
MH
z)
Figure 5.7: Hyperfine spectrum as the sample rod is rotated with the x, y field magnitude(752 G) applied. Site 1 ’b’ ions are dotted lines and the critical point transition is bold
90 Maximising Hyperfine T2 using Static Magnetic Fields
−10 −8 −6 −4 −2 0 2 4 6 8 108.642
8.643
8.644
8.645
8.646
8.647
8.64810G field sweep around Critical Point
Detuning from optimal Critical Point field (G)
Tra
nsiti
on F
requ
ency
(M
Hz)
Figure 5.8: Frequency change of the mI = +1/2 ↔ +3/2 transition due to a change of10G in the x (solid),y (dashed) and z (dotted)
5.3 Results 91
0 0.02 0.04 0.06 0.08 0.1−4
−3
−2
−1
0
delay (s)
ln(E(2 )/E ) 0
τ
Figure 5.9: A comparison of two pulse echos at zero field (×) and three transitions withthe for the mI = +1/2 ↔ +3/2 at site 1a (). The mI = +1/2 ↔ +3/2 site 1b transition(∗); the mI = −3/2 ↔ +3/2 transition (+) and the low field.
The application of the critical point field has a further significant increase
on these values to yield T2 = 82ms. These experiments were limited by only
having control of the applied magnetic field in two dimensions, requiring the
ratio of the x and y field being adjusted via rotation of the sample rod. Given
the low resolution and hysteresis of such an adjustment we were only able to
attain a magnetic field alignment within ∼5G of the ideal critical point field.
Despite these limitations the critical point transition displayed an order of
magnitude longer T2 than transitions not near a critical point [152].
With the addition of the y axis perturbation coil, allowing 3D control of
the applied magnetic field, greater timing resolution and improved RF control
the experiment was significantly improved. This is clearly demonstrated in
the two pulse echo data presented in figure 5.11, in which T2 was increased to
860 ms for an optimised critical point. A summary of the T2 for the different
magnetic field configurations is shown in table 5.3.
The initial critical point results were fit to the exp[−2τ/TM ] model de-
rived by Mims [110] which describes the echo decay due to a dephasing pro-
cess long compared to the echo time scale. Besides being longer the decay
92 Maximising Hyperfine T2 using Static Magnetic Fields
ln(E(2
)/
E ) 0
τ
τ
0
-0.04
0 20 40 60
(ms)
Figure 5.10: Two pulse echo decays with improved experimental apparatus showing theshort time decay of the optimised critical point field alignment.
Mangetic Field Configuration (G) T2 (ms)zero field ∼0.5
7 ∼312 ∼5780 5.9780 9.9
Initial CP 82Optimised CP 860
Detuned 2G (z) from CP 320Detuned 5G (z) from CP 100
Table 5.2: List of the observed T2 for the static magnetic field values experimentedwith. CP is an abbreviation of critical point and all ∼ are due to the superhyperfineinteraction resulting in T2 not being a good description of the system for the magneticfield configuration. 780G is the field magnitude of the critical point field alignment withechos performed on transitions not at a critical point.
5.3 Results 93
0 0.2 0.4 0.6 0.8 1 1.2
0
delay (s)
ln(E
(2
)/E
) 0τ
Figure 5.11: Two pulse echo decays with improved experimental apparatus for anoptimised critical point (⋄), ∼2G detuned from the optimised critical point (·) and ∼5Gdetuned in the z direction to yield a T2 = 100ms decay ().
94 Maximising Hyperfine T2 using Static Magnetic Fields
can no longer be described by standard echo decay function with a single
time constant [110]. There are three distinct regions for the optimised criti-
cal point field, as seen in figures 5.10 and 5.11. As shown in figure 5.10, for
pulse separations less than 20 ms the decay rate is less than 1/4s−1. When
the pulse seperation is of the order of 30 ms there is a distinct shoulder with
the decay rate increasing to 1/0.4s−1 as the pulse separation reaches 60 ms.
From 150 ms onwards the decay rate asymptotically approaches a value of
1/0.86s−1. This asymptotic decrease in the decay rate was only observed
for magnetic fields within 0.5 G of the optimal field. When the field was
more than 0.5 G away from the critical point a simple exponential decay was
observed for delays longer than 50 ms.
5.4 Discussion
During the initial stage of the decay (τ = 0 → 10ms) there is almost no
decay, indicating that there is very little reconfiguration of the local Yttrium
ions on this time scale. The shoulder in the decay at τ = 30 ms is interpreted
as indicating that the majority of the dephasing is due to perturbations that
occur on time scales between 10 and 100ms. This is consistent with an
exp [−(τ/TM )2] decay when the dephasing time scale is longer than the echo
time scale as discussed in section 2.5.3.
For delays longer than 100mswe would normally expect to see an exp [−τ/T2]
decay emerge, corresponding to a dephasing process acting faster than the
echo time scale (see section 2.5.3). The asymptotic behaviour of the decay
is attributed to a variation in the T2 within the ensemble resulting primar-
ily from inhomogeneity in the applied magnetic field. Ions experiencing a
field closer to the critical point condition will have a longer T2 and conse-
quently their contribution to the echo intensity will dominate for long pulse
separations. Quadrupole and pseudoquadrupole inhomogeneities will also
contribute to producing a range of ideal critical point field values within an
ensemble. However, given the measured magnetic field inhomogeneity of 2G
it is expected to be the current limit rather than the quadrupole interactions.
Therefore, the asymptotic decay rate gives an upper limit for the contribu-
tion to the decoherence due to second order magnetic interactions of 1.16
Hz.
The decay affords insight as to which Y ions are contributing to the
decoherence. The kinds of magnetic field perturbations experienced by the Pr
ion can be divided into two groups. Y ions near the Pr ion are part of a frozen
core and therefore spin flip slowly, causing large but infrequent magnetic
5.4 Discussion 95
field perturbations. Nearest neighbour Y ions resonantly exchange spin at
approximately 10 Hz, however in the bulk given the vast number of spins that
can resonantly exchange spin the actual cross relaxation rate is significantly
faster. This results in rapid but small magnetic field perturbations.
The critical point technique results in only second and higher order con-
tributions to the Zeeman shift. Consequently the rapid, small magnetic field
perturbations originating from the bulk Y ions have a vanishingly small ef-
fect on the hyperfine transition frequency. This leaves only the large, slow
magnetic field perturbations of the frozen core Y able to significantly change
the hyperfine transition frequency.
Removing transition frequency perturbations due to bulk Y is fundamen-
tally important for QIP applications. Any change in transition frequency
that is not controlled within a quantum computer results in a phase error
accumulating. The vast number of Y ions in the bulk means that on any time
scale it would be reasonable to expect some change in the bulk Y spin con-
figuration. This in turn means that Dynamic Decoherence Control (DDC,
or error correction protocols) could not be applied fast enough to correct
all errors. Further, DDC assumes that the transition frequency is constant
during the application of the pulses required to implement the particular
DDC scheme. This assumption is not valid if the bulk Y can perturb the
transition frequency. Therefore, despite the optimisation of the critical point
only showing significant gains at long delay times it is still fundamentally
important to error correction of the system.
QIP operations have to be performed in a time short compared to decay
of coherence from the system. Therefore the most interesting timescale for
QIP is within the first ∼ 10ms, shown in figure 5.10 during which decoher-
ence is negligable. On this timescale decoherence for each member of the
interacting Pr ensemble is different due to different local Y environments
with individual perturbation magnitudes and dynamics. For long delays, as
previously discussed, each Pr ion has experienced a represantitive set of local
Y configurations and only the ions experiencing an optimised critical point
significantly contribute. Therefore the T2 at long delays remains the most
usefull measure of the Pr ion sensitivity to magnetic perturbations for QIP
applications despite operating on a significantly shorter timescale.
A subtle point is that if there is no change in Pr hyperfine transition
frequency to first order as the magnetic field is perturbed it follows than the
magnetic moment dipole of each hyperfine state must be the same to first
order. Since the magnetic dipole moment of each state is the same to first
order there is no change to the spin quantisation axis to first order. This
96 Maximising Hyperfine T2 using Static Magnetic Fields
property of a critical point magnetic field alignment makes it the optimal
field alignment for suppressing near degenerate superhyperfine interactions.
Consequently, changes in the Pr spin state when the transition being driven
is at a critical point no longer result in back action from the bath on the Pr
ion. We have therefore created a situation in which it is accurate to consider
the Pr ion be surrounded by a decohering thermal spin bath. As there will
always be second order contributions to the Zeeman shift it is considered
that the critical point with the lowest second order Zeeman shift represents
the global minimum of hyperfine transition sensitivity and therefore is the
global maximum of T2, excluding spin polarisation of the bulk Y.
Minimising the back action of the bath on the Pr ion is fundamentally
important for QIP applications. If there is any back action from the bath in
response to changes in the state of a qubit it cannot be considered as a two
level atom and therefore driving the system increases the probability of an
error in QIP applications. Dynamic Decoherence Control (error correction
schemes), a requirement of fault tolerant quantum computing, are derived
under the universal assumption is that the interactions used to perform error
correction do not induce errors. This is assumption is not valid if there is
any back action from the bath. It is an open question as to wether any error
correction can be succesfully implemented if there is appreciable back action
from the bath.
The main requirements for the critical point technique are the existence
of a zero field splitting in the spin states and the ability to apply a sufficiently
strong magnetic field so that the Zeeman splitting is comparable to the zero
field splittings. Zero field splitting are observed for sites with axial or lower
symmetry. Axial symmetry sites, due to the higher symmetry than investi-
gated in this work will result in a circle in magnetic field space of minimised
decoherence instead of a single point. Anticrossings are also required and
hence the system should have a nuclear spin I > 1.
An interesting aside is that the critical point technique is unlikely to have
been discovered using conventional NMR. Limits imposed by resonant dipole-
dipole cross relaxation between the spins of interest is removed by using dilute
samples in ODNMR (sections 2.5.1 and 3.1), compared to molecular tumbling
or Magic Angle Spinning for liquid and solid state NMR respectively. Both
molecular tumbling and sample spinning render the critical point concept
inapplicable due to the continuous reorientation of molecules or crystal with
respect to the applied field. Further, the applied fields used to tune the
nuclear spin transition frequency into resonance with the detection coil are
typically many Tesla and is significantly beyond the anticrossing region of
5.4 Discussion 97
most nuclear spin systems. Therefore, ODNMR experiments allow studying
spin systems which have a common, static reference frame for all spins of
interest, with the ability to apply an arbitrary magnetic field magnitude and
direction, required for the critical point technique.
5.4.1 Future Improvements
Improvements to the magnetic field homogeneity over the sample can be
made by either using a thinner sample or improving the homogeneity of the
applied field. The solenoid in the Oxford MD10 bath cryostat is specifically
designed to provide a homogeneous field along the optical axis with the field
inhomogeneity contributing less than 0.1G variation accross the laser beam
volume, determined from the calibration data supplied with the cryostat by
Oxford. Therefore it was used to provide a majority of the applied magnetic
field (x and y components), the ratio of which was adjusted by sample rod
rotation. The MD10 cryostat used in these experiments has a small sample
space and as such limits the coil designs in both the z and y axis. With
modifications to the sample mount the z field could also be produced using
the Oxford solenoid, thereby making both the z and y coils perturbation
coils. A superconducting switch is implemented on the Oxford solenoid and
should also be implemented on the y perturbation and z coils to increase
current stability. Using a thinner sample reduces the volume over which the
field is required to be homogeneous. The sample used in these experiments
is 3 × 4 × 5 mm and the group has recently acquired some 1 × 4 × 5 mm
samples. Signal strength is currently sufficiently large to allow single shot
measurements. When using thinner samples if SNR becomes a problem the
measurements can be averaged, the sample can be placed in a cavity, or
interferometric detection can be used.
With no applied field the hyperfine transition linewidths in Pr3+:Y2SiO5
are ∼70kHz, dominated by magnetic broadening. For an optimised critical
point field alignment the linewidth was measured as 4kHz, which did not
change as the magnetic field was detuned to the point where T2 = 100ms.
Since there is almost an order of magnitude change in the magnetic field sensi-
tivity, indicated by T2, with no corresponding change in the inhomogeneous
linewidth of the transition it was concluded the 4kHz of inhomogeneous
broadening of the transition at the critical point field is not due to magnetic
interactions.
The inhomogeneous broadening at the critical point field is most likely
dominated by strain broadening within the crystal, which is not intrinsic to
the site and can be reduced by refining standard crystal growing techniques.
98 Maximising Hyperfine T2 using Static Magnetic Fields
The strain broadening couples to the hyperfine transition via the crystal field
interacting the quadrupole and pseudo quadrupole moment [47]. Techniques
for reducing strain broadening are discussed in section 6.4.1
It is possible, in principal, to spin polarise the bath and thereby stop bath
spin flips using large magnetic fields and low temperatures. This is achieved
by making one of the spin states energetically unfavourable due to the Boltz-
mann distribution, such that kT ≪ µB. This has been used to increase T2
in electron spin systems [153] and T2 increases in nuclear spin systems have
been attributed to this mechanism despite lack of complete spin polarisation
[154]. For Y this would require extreme experimental conditions of B > 10T
and T ≈ 1mK. While technically possible, this regime is very challenging
and, in general, is not feasible for nuclear spin systems. The hyperfine transi-
tions would remain sensitive, with potential enhancement of their sensitivity
due to non linearity of Zeeman shifts at high fields and therefore magnetic
field fluctuations generated by experimental equipment will strongly deco-
here the transitions. Due to the very low temperatures it would also be a
fragile thermal state and performing frequent state manipulations is likely
to heat the sample, if only the spin temperature, beyond the required tem-
perature. Therefore this regime is considered impractical. Consequently we
consider the critical point to provide the optimum practical magnetic field
configuration for achieving a long hyperfine T2 in Pr3+:Y2SiO5.
Chapter 6
Dynamic Decoherence Control
“If everything seems under control, you’re just not going fast enough.”
– Mario Andretti
In this chapter we investigate the possibility of increasing T2 using time
varying magnetic fields to periodically drive transitions of the quantum sys-
tem of interest. We can describe a quantum system of interest, HS , inter-
acting with a driving field HRF and an environment HE using the following
Hamiltonian:
H′ = HS + HSS + HE + HSE + HRF (6.1)
where HSE is the system - environment interaction and HSS in the interaction
within the quantum system of interest. As discussed in section 4.1, the first
step is to understand what the dominant decohering interaction is. Once this
has been determined strategies can be formulated to use HRF to minimise
the effect of this interaction on HS . The strategy used in this chapter is to
apply a cyclic driving sequence to HS such that the effect of the dephasing
interaction is equal and opposite during successive periods, thereby removing
the contribution of the dephasing interaction at the end of the cycle.
Dynamic Decoherence Control is a label quantum information theorists
give to decoupling techniques suitable for QIP, many of which are existing
techniques which NMR spectroscopists broadly refer to as Phase Cycling or
Coherent Averaging techniques. Only a subset of existing NMR techniques
are useful for QIP because many decoupling techniques are designed to pro-
tect a specific state from decohering and consequently do not protect an
arbitrary state well.
The simplest DDC technique used in NMR is the Carr-Purcell Mieboom-
Gill (CPMG) pulse sequence [80, 81]. This sequence is also the most com-
monly discussed DDC method for QIP applications [8, 9, 10, 11, 12, 13, 14,
100 Dynamic Decoherence Control
15]. In QIP discussions the CPMG technique was renamed Bang Bang de-
coupling [8], in reference to an analogous classical error correction protocol.
The Bang Bang decoupling method is a direct extension of a spin echo
and consequently has a long history in NMR. As discussed in section 2.5.2
spin echos “refocus” the inhomogeneous broadening of an ensemble. If the
reconfiguration of the dephasing bath is longer than the time scale which
pulses can be applied to the system then the frequency perturbations due to
the dephasing bath remain constant during each echo period and are therefore
also rephased. Consequently if π pulses can be applied sufficiently quickly
the driven T2 will exceed the undriven T2.
This was noted very early in the development of NMR techniques by Carr
and Purcell [80]. It was also noted by Mieboom and Gill [81] that to max-
imise the measured T2 the applied RF power had to be rigorously optimised.
They deduced that this was due to systematic errors in Rabi frequency ac-
cumulating as shown in figure 6.1. Mieboom et. al. then modified the pulse
sequence with the second π pulse having a 180 phase shift, to cancel any
systematic error in Rabi frequency. This modified sequence is commonly
referred to as the CPMG sequence and is also shown in figure 6.1.
Detailed analysis of the evolution of an ensemble interacting with a pulsed
driving field have been conducted initially by liquid NMR spectroscopists
[80, 41, 42, 82, 48] and more recently by quantum information theorists [8,
10, 11, 13]. All of these analyses lead to an equivalent statement of the simple
time scale argument [13] that:
ωbτc . 1 (6.2)
where ωb is the cut-off frequency of the dephasing bath modes and τc is the
delay between the pulses, or cycling time as shown in figure 6.1. Therefore if
the decoupling sequence is to be effective we need the ability to apply pulses
faster than the rate at which the bath reconfigures. This provides a simple
test for whether application CPMG like sequences are expected to increase
T2 of the system based directly on experimental measurements.
6.1 Application to Pr3+:Y2SiO5
In the previous chapter it was discussed that the magnetic field perturbations
due to the Y ions can be divided into two groups: frozen core Y ions near
the Pr ion producing large but infrequent magnetic field perturbations; and
Y ions in the bulk which result in small, rapid magnetic field perturbations.
6.1 Application to Pr3+:Y2SiO5 101
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
π π
Z
X
Y
−π
π
Z
X
Y
π
Z
X
Y
π/2
ππ/2
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
1
Ensemble Coherence
Carr Purcell RF τ2τ τ4 τ6 τ8 τ10 τ12 τ14 τ16 τ18
ππ/2
Time
Ensemble Coherence
Mieboom - Gill RF
τ2τ τ4 τ6 τ8 τ10 τ12 τ14 τ16 τ18
τ
3τ 5τ
Time t0
CarrPurcell
Common
MieboomGill
Figure 6.1: Bloch sphere visualisation of the Carr Purcell and Carr-Purcell Mieboom-Gill(CPMG) pulse sequence following a subgroup of an inhomogeneously broadened ensemble.This illustrates the effect of the π phase shift in cancelling systematic Rabi frequency errors
102 Dynamic Decoherence Control
Z
X
Y
π/2Z
X
Y
Z
X
Y
π/2
Z
X
Y
π
π
Z
X
Y
Z
X
Y
Z
X
Y
−π
−π
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
τ
2τ 3τ 4τ
Time t0
Figure 6.2: Bloch sphere visualisation of the action of the Bang Bang pulse sequenceon an inhomogeneously broadened ensemble with two different initial coherent states: inphase and shifted by π/2.
6.1 Application to Pr3+:Y2SiO5 103
Perturbation Timescale
Fre
qu
en
cy S
hif
t
1/τ c
Figure 6.3: conceptual representation of the effect of DDC and critical point techniqueson the expected decohering magnetic field perturbations. The critical point techniqueminimises all transition frequency perturbations at all frequencies and is particularly ef-fective for the small perturbations. This is indicated by the difference between the zerofield distribution (solid line) and the critical point (dashed line). DDC techniques are onlyeffective at removing the perturbations occurring slower than τc, indicated by the blueshading.
DDC techniques and the critical point technique are complimentary tech-
niques. While the critical point technique minimises the decohering effect of
magnetic field perturbations at any frequency, it is best at rephasing small
perturbations since the second order contribution remains negligible as shown
in figure 6.3. Therefore it is most effective at removing the decoherence due
to the bulk Y. Larger magnetic field perturbations due to nearby Y ions
have a larger second order contribution and therefore still contribute to de-
coherence. We can consider the effect of the DDC as a decoherence filter,
in which low frequency decoherence within the bandwidth of the “filter” is
corrected. Consequently DDC will be effective at removing the decoherence
due to nearby Y ions but poor at negating the effect of the bulk Y. Without
application of the critical point technique the required τc will be significantly
smaller and therefore the schemes are complementary and DDC will be in-
vestigated in addition to the critical point technique.
Successful application of DDC techniques rests upon the whether the
required τc can be achieved and if so, can sufficiently hard pulses be achieved.
We therefore need to estimate the characteristic reconfiguration time of the
dephasing environment. For this we revisit the two pulse spin echo decay
for short delays, initially presented in figure 5.10, reproduced in figure 6.4.
As discussed in section 5.4 for echo delays less than 10ms there is almost
no decoherence due to reconfiguration of the spin bath. The duration of a
104 Dynamic Decoherence Control
delay (ms)0 20 40 60
0
-0.2
-0.4
ln(E
(2
)/E
) 0τ
Figure 6.4: Two pulse spin echo amplitude as a function of delay for short delays. Thevery low decoherence during the first 10ms indicates that dynamic decoupling should beable to significantly extend T2, ideally as a tangent to the echo decay.
π pulse in the current experimental configuration is ∼10µs controlled with
10ns timing resolution by the PulseBlaster. Therefore inequality 6.2 can be
satisfied and Bang Bang DDC techniques should be effective at decoupling
the a Pr hyperfine transition at a critical point. This should result in the
hyperfine coherence evolving at a tangent to the decay curve, as shown in
figure 6.4, dramatically extending T2.
6.2 Experimental Setup
The experimental configuration remained largely unchanged from the critical
point experiment configuration. All of the DDC experiments use either an op-
timised critical point magnetic field configuration or a with the field detuned
by 5G in the z direction. By detuning the magnetic field the sensitivity of
the Pr hyperfine transition to magnetic field fluctuations is increased thereby
requiring the decoupling sequence to rephase more decoherence. The critical
point was optimised using long delay echos, as outlined in section 5.2.1 and
from this optimised configuration the field was detuned if desired.
The major change to the experimental apparatus was the addition of
phase sensitive detection of the Raman heterodyne signal as shown in figure
6.2 Experimental Setup 105
Coherent
699
GPIB
Trig
Pulse TTLout
Tx Rx
BlasterRF out
Cryostat
RF CoilsDC Coils
DC CoilsSample
Detector
Computer
CRO
RF out
J850 DDS
RFAmp
RFAmp
USB
USB1 2 3 4
AOM
RFSwitch
RFSwitch
Figure 6.5: Schematic diagram of the experiment setup. The laser is gated via an AOMbefore entering the cryostat and being incident on the sample. A static magnetic field canbe applied to the sample using pairs of coils, with the RF radiation applied via the RFcoils. The transmitted laser beam is incident on a photodetector, the output of which isfed into a spectrum.
6.5. Phase sensitive detection is required for performing quantum process
tomography [35] which, as described in section 1.3.3, is used to determine
the process operator O of the DDC technique. This involved down converting
the RF Raman Heterodyne beat note using two mixers with a 90 phase shift
on the Local Oscillator input such that both the in phase and quadrature
components were detected. In order to perform phase sensitive measurements
over T1 time scale of ∼ 100s at 8.65MHz requires a reference oscillator
stability of ∼10−9. All of the RF equipment was referenced to an Oven
Controlled Crystal Oscillator (OCXO) contained within a Stanford Research
Systems DS345 which has an Allen variance of < 5× 10−11 over a 1s period.
Therefore, our measurements should not be limited by the phase stability of
the reference oscillator.
The pulse sequences used in this experiment are shown in figure 6.6. As
before, two pulse spin echos were used to measure the undriven T2. The
inversion recovery sequence is used to measure T1, the upper limit that could
be expected from DDC techniques. The decoupling pulse sequence is also
shown in figure 6.6, with τ1 denoting the period between the initial pulse and
the start of the decoupling sequence and τc is the cycling time, the period
between the pulses used in the decoupling sequence.
106 Dynamic Decoherence Control
π x10
x500
x10
ω 3
ω CP
ωp
ωr
ω r scan
ω1 ω2,
State Preperation
1τ1τ
π
2π
1τ
a)
2τ2τ2π/
−π
π π1τ cτcτ 1τ2π/
b)
c) N
Figure 6.6: Pulse sequence used in the experiment, showing the repump scheme, withthe number iterations for each section indicated below, followed by the a) 2 pulse echosequence b) Inversion recovery pulse sequence and c) decoupling pulse sequence. Thedecoupling pulse sequence is enclosed by the grey box and is iterated N times.
In the current investigation τ1 = 1.2ms and the cycling time τc was varied
from 0.05ms to 20ms. The choice of τ1 was made such that it could remain
constant for all choices of τc without being a simple integer multiple of τc.
This avoids echo artefacts that result from the DDC sequence rather than
the initial coherence by temporally separating the signals. Therefore we can
be certain that the measured echo amplitude is due to coherence generated
by the initial pulse.
6.3 Results
The decoupling pulse sequence was investigated varying the cycling time τc
from 0.05ms to 20ms, with several echo sequences shown in figure 6.7. In
order to differentiate the decoherence times for the driven and undriven cases
in the following discussion they will be referred to as T2D and T2 respectively.
Also shown in figure 6.7 is the result of an inversion recovery measurement
used to determine the T1 of the transition. The inversion recovery measure-
ments were performed using the pulse sequence described in figure 6.6(b).
As can be seen in figure 6.7 the decoupling sequence significantly increases
the Pr hyperfine coherence time though T2D/T1 < 1/2, even for the longest
value of τc.
6.4 Discussion 107
0 10 20 30 40 50−4
−3
−2
−1
0
Total cycling time (s)
ln(E(2 )/E ) 0
τ
Figure 6.7: Decoupled echo decays, varying τc 7.5ms(×), 10ms(∇), 15ms(⋄), 20ms(⋆)corresponding to T2D = 27.9, 21.1, 15.2, 10.9s. Inversion recovery measurements () yieldT1 = 145s.
T2D was studied as a function of τc for both an optimised critical point
(T2 = 860ms) and with the magnetic field detuned by ∼5G in the z direction
such that the two pulse echo decay was reduced to T2 = 100ms. As shown in
figure 6.8, there are three regions: for τc > 5ms, there are significant gains
in T2D made by reducing τc. T2D then plateaus at 33s for 0.2 < τc < 5ms
before again rising to a maximum of 70s for an optimised critical point.
It was also noted during the experimental work that there was a significant
drop in the echo decay amplitude during the first iterations of the decoupling
pulse sequence. As seen in figure 6.9 this initial fast decay was short lived
and all T2D measurements were fit to the latter portion of the decay.
6.4 Discussion
For τc > 5ms there is a significant reduction in the T2D observed using
the decoupling sequence for the measurements made with the detuned field.
As τc is decreased throughout this region T2D increases exponentially at a
similar rate for both data sets. The offset between the optimal critical point
and the detuned data sets is due to the increased sensitivity to magnetic
field fluctuations. Away from the critical point smaller, faster bulk processes
108 Dynamic Decoherence Control
10-2
10-1
100
101
102
10
20
30
40
50
60
70
80T 2
(s)
Bang Bang cycling time τc (ms)
Figure 6.8: Dependence of decoherence time on the Bang Bang cycling time τc both atthe critical point () and with the magnetic field misaligned to give a coherence time ofT2D = 100ms (×). Trend lines do not represent a physical model.
0 5 10 15 20 25 300.5
0.6
0.7
0.8
0.9
1
Iterations
ln(E
(2
)/E
) 0τ
Figure 6.9: Echo decay during the initial iterations of a Bang Bang decoupling sequenceshowing an initial fast decay.
6.4 Discussion 109
contribute an equivalent amount of dephasing and consequently τc must be
shorter to leave the same residual dephasing contribution.
During the plateau region of 0.2 < τc < 5ms both data sets average
to ∼30s. This implies that T2D in this region is not limited by the ion’s
sensitivity to magnetic field fluctuations. As τc is reduced further (0.05 <
τc < 0.2ms) there is another dramatic increase in T2D. The most likely source
of this limitation was determined to be the driving RF waveform. Changes in
RF amplitude and phase that differed between the π and −π pulses are not
cancelled by the pulse sequence and hence accumulate, causing decoherence
relative to the reference oscillator. This second increase in T2D is due to the
driving pulses being applied to the ensemble faster than phase errors occur in
the driving field. Therefore, the phase of the ensemble tracks the phase of the
driving field, observed in numerous spin locking experiments [41]. Reducing
τc further was not possible due to the RF coil heating the liquid helium bath
past the superfluid lambda point. An interesting feature of the data set
for a detuned critical point is that T2D is actually shorter for τc = 0.05ms
compared to τc = 0.1ms. This was not due to noise in the experimental data
and was verified on different days that T2D decreased from ∼60s to ∼53s
for these delays. We consider this to be due to the combination of phase
noise and increased sensitivity to magnetic field fluctuations. While this is
somewhat unresolved it is considered to be an issue dominated by noise of
the driving RF and consequently should be revisited one the RF control is
improved.
The phase noise of the Pulse Blaster was characterised using a mixer and
the Stanford Research Systems DS345, reference oscillator. A continuous
stream of pulses, differing by 180 was mixed down to DC using the reference
oscillator. Over long time scales (¿1s) the phase noise was 1.15. As the time
scale was reduced to the range of 100ms → 0.5ms, transients in the phase
error became resolvable. It was not until the pulse width was reduced to
∼100µs that the number of phase noise glitches per pulse were noticeably
reduced by the time scale of the measurement. Pulse widths of 20µs typically
had no phase difference between the pulses. This indicated that the phase
noise was primarily glitches in the Pulse Blaster DDS circuitry, unrelated
to specifics of the pulse duration. This observation was consistent with the
decoupling sequence data, which demonstrated an increase in T2D in the
region which phase noise glitches were minimised. If the T2D increased at
the same rate as prior to the plateau the for a cycling time of τc = 1ms a
T2D of the order of 100s is expected. This could allow significant progress
toward the T1 = T2 limit.
110 Dynamic Decoherence Control
-1
0
1
|0 0|
|0 1|
|1 0|
|1 1| |0 0|
|0 1|
|1 0|
|1 1|
-1
0
1
|0 0|
|0 1|
|1 0|
|1 1| |0 0|
|0 1|
|1 0|
|1 1|
-1
0
1
|0 0|
|0 1|
|1 0|
|1 1| |0 0|
|0 1|
|1 0|
|1 1|
Real Imaginary
-1
0
1
|0 0|
|0 1|
|1 0|
|1 1| |0 0|
|0 1|
|1 0|
|1 1|
Ideal
WorstCase
Figure 6.10: The λ matrix for any DDC scheme is to leave an arbitrary state unchangedand therefore is the identity matrix. The worst case is that the DDC process takes anarbitrary state and maps it to an incoherent state loosing all quantum information
6.4.1 Bang Bang Process Tomography
In order to assess how well the decoupling sequence preserved an arbitrary
quantum state process tomography was performed to determine the process
superoperator O, as described in section 1.3.3. Using the quamtum process
tomography method described in section 1.3.3 we determine λ, which can be
used to construct O using equation 1.24. Process tomography was performed
on the input state and 1, 10, 100 and 1000 iterations of the decoupling
sequence. The initial delay was τ1 = 1.2ms, with a cycling time of τc =
2ms. The total period over which the tomography was performed was 4ms,
40ms, 400ms and 4s respectively. Figure 6.10 shows the best and worst
case scenarios for a decoupling sequence. Ideally the decoupling sequence
preserves the input state without manipulating it, resulting in λ being the
identity matrix. The representation of an incoherent state, also shown in
figure 6.10 is the worst case, since any input state will be turned into an
incoherent state by the application of the pulse sequence.
The imaginary component of the process tomography, as shown in figure
6.11 is only shown for the experimental data since the modelling results were
always zero. Ideally the decoupling sequence process operator is the identity
matrix, leaving the state unchanged. It was observed that the component
6.4 Discussion 111
−1
0
1
−1
0
1
−1
0
1
− 1
0
1
− 1
0
1
− 1
0
1
− 1
0
1
− 1
0
1
− 1
0
1
− 1
0
1
|0 0|
|0 1|
|1 0|
|1 1| |0 0|
|0 1|
|1 0|
|1 1|
|0 0|
|0 1|
|1 0|
|1 1| |0 0|
|0 1|
|1 0|
|1 1|
|0 0|
|0 1|
|1 0|
|1 1| |0 0|
|0 1|
|1 0|
|1 1|
0
1
100
1000
10
ImaginaryReal Realπ,−πpairs
−1
0
1
−1
0
1
−1
0
1
−1
0
1
−1
0
1
Experiment Theory
Figure 6.11: Plots of λ constructed from process tomography of the Bang Bang pulsesequence for 1, 10, 100 and 1000 iterations for experiment and theoretical modelling.Imaginary components of theory are omitted for clarity.
112 Dynamic Decoherence Control
of the Bloch vector in the coherence plane for a given state was preserved
well, while the population component of the Bloch vector rapidly decayed.
The fidelity of the decoupling sequence for 1, 10, 100 and 1000 iterations was
99%, 65%, 54% and 43% respectively.
The evolution of the ensemble was modelled using the Bloch equations
assuming an infinite T1 and T2, with an inhomogeneous linewidth of 4kHz
(FWHM) and a Rabi frequency of 100kHz. The results from this mod-
elling are shown in figure 6.11, along side the experimental data. Despite the
model not including any homogeneous dephasing the rapid decay of popula-
tion terms compared to coherence terms of the simulated process tomography
(figure 6.11) match the experimental data. Simulations indicate that the de-
cay rate of the population terms can be reduced by increasing the ratio of the
Rabi frequency to the linewidth. A suitable criteria for when the application
of the decoupling sequence is useful for preserving arbitrary quantum states
is when the decay rate of the population terms in the presence of the pulse
sequence is slower than that of the coherence terms in the absence of the
decoupling sequence. For the present case, where T2 = 0.86s the simulation
indicates that to meet this criteria it will be necessary to achieve a ratio of
Rabi frequency to linewidth of ΩRF /ωinh ≈ 100. There is limited capacity to
increase the Rabi frequency of the driving field without the possible excitation
of off-resonant transitions. Therefore for the application of the decoupling
sequence to the investigated critical point on the mI = −1/2 ↔ +3/2 hyper-
fine transition to be useful it will be necessary to reduce the inhomogeneous
broadening of the transition by a factor of ∼10.
The simulations also exhibit the fast decay during the initial few iterations
of the decoupling sequence. Shown in figure 6.12 is the predicted projection
on the Bloch sphere axes as a function of decoupling sequence iteration for
the experimentally measured parameters. While the magnitude of the effect
differs from the experiment the general behaviour is the same.
Further simulations indicated that the decay rate of the population terms
can be reduced by increasing the ratio of the Rabi frequency to the linewidth.
A suitable criteria for when the application of the decoupling pulse sequence
is useful for preserving arbitrary quantum states is when the decay rate of
the population terms in the presence of the pulse sequence is slower than that
of the coherence terms in the absence of the sequence. For the present case
where T2 = 0.86s the simulation indicates that to meet this criteria it will be
necessary to achieve a ratio of Rabi frequency to linewidth of ΩRF/ωinh≈100.
In order to achieve a high fidelity (1 −mathcaF < 10−5) coherence time
extension using the decoupling pulse sequence the modelling predicted that
6.4 Discussion 113
0 100 200
0
1
Ech
o a
mp
litu
de
Decoupling Sequence iterations
0.5
Figure 6.12: Modelling prediction of the Bloch sphere projection during the initialiterations of a Bang Bang decoupling sequence for a coherent state (dash) and a populationstate (solid) with the experimentally measured parameters.
ΩRF /ωinh≥103. This can be addressed in part by using a resonant RF coil
such that all RF power is applied to the sample rather than a significant frac-
tion being dumped into the 50Ω load. The maximum transition specific Rabi
frequency is ∼1MHz which would result in ∆ωinh/ΩRF = 250. Therefore
inhomogeneous broadening of the hyperfine levels should ideally be reduced
by a factor of ∼10 for Bang Bang decoupling sequences to be effective.
It was argued in section 5.4 that the remaining inhomogeneous broad-
ening at the critical point field was due to strain broadening. The strain
broadening couples to the hyperfine transition via the crystal field interact-
ing the quadrupole and pseudo quadrupole moment [47]. A reduction in
strain broadening by over an order of magnitude has been achieved in anal-
ogous materials [143] through reducing the dopant concentration and using
isotopically pure materials. In the available Pr3+:Y2SiO5 crystals Si29, a spin
1/2 isotope of silicon, exists in natural abundance of ∼4%. The magnetic
moment of Si29 is a factor of 4 larger than that of Y, thereby contributing to
dephasing via magnetic field fluctuations and increasing disorder in the crys-
tal lattice. Repeated annealing of the crystal has demonstrated a reduction
in optical linewidth from 2.4 GHz to 520 MHz for Eu3+:Y2SiO5 [88]. Both
of these inhomogeneous linewidth observations were for optical transitions
114 Dynamic Decoherence Control
and the hyperfine linewidth was not measured during these studies. If the
assertion in section 5.4 that the remaining inhomogeneous broadening on the
hyperfine at the critical point is due to inhomogeneities in the quadrupole
and pseudo quadrupole is correct then it is the same broadening mechanism
for both the optical and hyperfine transitions. Therefore we could expect a
similar reduction in hyperfine linewidth at the critical point as observed in
the optical transitions.
Concluding the discussion above, by using isotopically pure materials with
very low dopant concentrations and repeated annealing of the crystal we ex-
pect a reduction of inhomogeneous linewidth by a factor of approximately 50.
Other materials also warrant investigation and will be discussed in chapter
8.
Holeburning techniques can also be used to modify the hyperfine inhomo-
geneous linewidth. Sinc pulses have a “top hat” frequency response, thereby
allowing a narrow window of detunings to be resonant with the pulse. By
applying 2π sinc pulses to the hyperfine transition only those within the top
hat bandwidth of the sinc pulse will be returned to the initial state. Us-
ing holeburning techniques the remainder of the inhomogeneous line can be
transferred to other spin states that will not be involved in the experiment.
Therefore we can narrow the inhomogeneous linewidth of the ensemble used
to the required ratio even if refining the crystal growth techniques does not
provide sufficient inhomogeneous line narrowing.
Using a combination if Rabi frequency increase and inhomogeneous linewidth
reduction there is potential to increase ∆ωinh/ΩRF such that it is no longer a
limitation to the performance of the decoupling pulse sequence. In combina-
tion with actively modifying the inhomogeneous linewidth using holeburning
techniques this creates a real possibility of performing high fidelity DDC
experiments using Pr3+:Y2SiO5 in the near future.
Composite pulses are often used in NMR to minimise undesired effects due
to detuning and Rabi frequency errors [41]. By applying several consecutive
pulses the resulting operation, or composite rotation, can be made less depen-
dent on Rabi frequency and detuning errors [42, 41, 58, 155, 156, 157, 158].
Again we were restricted to composite pulses that can act on an arbitrary
state, such as those designed by Levitt [158], Wimpres [58] and Tycko [157].
Application of the decoupling sequence utilising the Wimpres BB1, BB2 [58]
and Levitt composite pulses resulted in a shorter T2D. Modelling confirmed
that the ratio ΩRF/ωinh was too low for the composite pulses to be effec-
tive. Composite pulses become effective when ΩRF /ωinh ≈ 103, the same
condition for the decoupling sequence itself. This is not surprising since this
6.4 Discussion 115
represents the regime where the pulses can be considered hard. While the
simulations typically showed only minor improvement due to using compos-
ite pulses there was no modelling of noise in the driving field and hence, in
practice, the improvement should be more significant.
To be certain that the detected spin echo was only due to the initial pulse
and not an artefact of the decoupling sequence, τ1 and τc were chosen such
that τc would not be an integer multiple of τ1. The optimal condition is
actually τc = 2τ1 [81] such that the inhomogeneous broadening is refocused
in between the π pulses. In the τc = 2τ1 case coherence generated by the
decoupling sequence will contribute to a spin echo at the same time as the
echo due to the input pulse. As this is the first test of Bang Bang DDC in
a solid the the experiment was designed such that the interpretation was as
clear as possible. While T2 can be expected to be longer for this optimised
condition more detailed investigation would be required to determine the
artefact contribution.
While T2 has been dramatically extended by the application of DDC, it
has also imposed a periodic condition for QIP operations. If we consider
the optimised case of τc = 2τ1 the inhomogeneous broadening is rephased at
the points half way between the π pulses. Consequently it is only at these
times that QIP operations can be performed without introducing phase shifts
relative to ions not involved in the QIP operation. To predict the phase shifts
incurred performing QIP operations not at τc/2 the detuning of the ions
involved must be precisely known. Since DDC techniques are employed to
remove the effect of time dependant inhomogeneities in transition frequency
this is not information that is possible to determine prior to executing the
operation. It therefore appears that as the cycling time is reduced, T2 is
increased and there are more opportunities per unit time to operate on the
system. This of course comes at the expense of applying more DDC pulses,
therefore causing errors due to pulse imperfections accumulate faster.
Careful attention to the reference oscillator and the phase distortions
inherent in RF synthesizing equipment will become increasingly important
in future experiments. In current experiments phase sensitive measurements
over a period of 100s at 8.65MHz requires a reference oscillator stability of
∼10−9. The phase stability of the driving field should ideally be an order
of magnitude greater than the measurement requirements such that it is not
the limiting noise source. State of the art OCXOs that provide stabilities of
∼10−12 over a 10s period are available from oscillator manufacturers [159].
Chapter 7
Extending T2 Through Driving
the Environment
“In theory, there is no difference between theory and practice. But, in prac-
tice, there is.”
– Jan L.A. van de Snepscheut
In the previous chapter we investigated a DDC technique that drove the
quantum system of interest, HS , to minimise the decoherence of that system.
In this chapter we investigate driving the environment, HE to decrease the
decoherence of HS . Magic Angle Line Narrowing is an established NMR
technique to drive HE to decouple the environment from the system, thereby
minimise the effect of system-environment interaction HSE [41] and increase
T2.
Magic Angle Line Narrowing (MALN) is used to decouple our spin of
interest, A, from the B spin bath undergoing dipole-dipole interactions by
driving the bath such that the B spins are rapidly flipped. Rapidly flipping
the B spins averages out dipolar interactions, resulting in a similar effect
to the tumbling of molecules in liquid NMR [41]. In order to be effective
the interaction of B spins with the driving field must dominate the dipolar
interaction between the A and B spins. When the driving field is applied at a
particular detuning relative to the transition linewidth to provide the “Magic
Angle” all orientations experience the same Rabi frequency. Therefore, all of
the B spins are decoupled effectively, resulting in optimal decoupling when
applied to complex systems or powdered samples.
The primary criterion for successful application of MALN is achieving the
following ratio of magnetic fields [54]:
B0 ≫ B1 > BAB (7.1)
7.1 Experiment Setup 117
where B0 is the static magnetic field, B1 is the magnetic field of the RF
driving the spin transition and BAB is the magnetic field due to the A spin
as seen by the nearest neighbour B spin. This can be interpreted as B0 must
significantly exceed B1 in order for there to be a defined quantisation axis
for spins and B1 > BAB such that the interaction with the driving field is
stronger than the interaction between the spins. The Magic Angle condition
for the detuning is given by [54]:
θM =ωd∆ω
= 3cos2β (7.2)
where β = 54.75 is the Magic Angle, ωd is the driving frequency and ∆ω is
the detuning from line centre.
MALN has been applied to many solid state systems with the first demon-
stration in solid state ODNMR spectroscopy performed on Pr3+:LaF3 by
Rand et. al. [53] and MacFarlane et. al. [54]. In systems that have large
nuclear spins in close proximity this can often result in impractically large
RF fields. This often occurs in solid state systems limiting the applicability
of this technique.
7.1 Experiment Setup
The experimental setup underwent minor modifications to the magnetic field
control from the previous experiments, described in section 6.2. Given the
magnetic field applied to reach the critical point the Y nuclear spin transi-
tion frequency, ωY , is 162khz. In order to get a sufficiently large RF field,
perpendicular to the dominant field direction ωY was applied to the sample
using the z coil as shown in figure 7.1. This required an RF choke to isolate
the RF from the DC power supplies. Since ωY is very low frequency in terms
of RF electronics, the choke could not provide very high isolation with prac-
tical inductances. Therefore we limited the RF power applied to minimise
the risk of damaging the DC power supply. The RF power supply delivered
1.4W into the z coil, resulting in a field of B1 = 6.4G, which produced a
Rabi frequency of ΩY = 1.3kHz.
T2 was measured using Raman heterodyne two pulse echos using the same
repump scheme and pulse sequence as the critical point experiments shown
in figure 7.1. The only addition to this was the RF driving field ωY was
applied during both the state preparation and echo sequence.
To determine how well we expect MALN to work we need to examine these
experimental conditions in light of the inequality 7.1 and the Pr-Y interaction
118 Extending T2 Through Driving the Environment
Coherent
699
GPIB
Trig
Pulse TTLout
Tx Rx
BlasterRF out
Detector
Computer
CRO
RF out
J850 DDS
RFAmp
RFAmp
USB
USB1 2 3 4
AOM
RFSwitch
RFSwitch
DC Coils
DC CoilsRF Coil
Sample
Figure 7.1: Experimental setup for the MALN experiments showning the DC and RFdrive to the z coil.
strength measured in chapter 4 as 9G. The experimental configuration fails
to provide an RF field B1 larger than BAB. We were unable to generate a
larger RF field with the available equipment due to the low RF frequency
limiting our ability to isolate the DC power supply.
Although the experiment did not satisfy inequality 7.1 it was close and
consequently we persisted. Despite not being an optimal configuration if
significant gains in T2 are achievable through optimised MALN we should
be able to achieve minimal gains with the current configuration. Since we
have not achieved B1 > BAB, the “Magic Angle” is not meaningful and
therefore we will refer to the current experiments as Driven Environment
Line Narrowing (DELN).
7.2 Results
The frequency of the DELN driving field was optimised using a Raman het-
erodyne two pulse spin echo with a long delay as per critical point optimi-
sation (section 5.2). When the DELN driving field was applied at a fixed
frequency in the range 140kHz < ωY < 180kHz the increase of the echo
amplitude not significantly above the small shot to shot variation.
It became clear that due to not satisfying the inequality 7.1 only a sub-
7.3 Discussion 119
group of the Y ions were interacting with the DELN field. The requirement
in inequality 7.1 that B1 > BAB can be interpreted that the Rabi frequency,
and therefore power broadening, must exceed the inhomogeneous broadening
of the B spins due magnetic field of the A spin. Therefore, if the inequality
7.1 is satisfied all Y ions interact with driving field.
Scanning the driving field was tried such that all of the Y ions would
interact with the driving field. This was the only experimental configuration
that resulted in a gain in the long delay spin echo amplitude. The spin echo
amplitude was optimised when ωY was centred at 162kHz and scanned over
600Hz with a 10Hz scan rate. The decoupling field was applied during both
the state preparation and echo sequence. It should be noted that given the
field of 9G magnetic field due to the Pr as seen by the nearest neighbour Y
and γY = 209Hz/G we expect the Y inhomogeneous linewidth to be 1.8kHz.
The Rabi frequency of 1.3kHz combined with scanning 600Hz matches the
Y inhomogeneous broadening.
The two pulse echo decay with and without the DELN decoupling field
are shown in figure 7.2. It is clear that the decay is significantly longer
when the DELN decoupling field is applied. In this experimental run the
critical point T2 was optimised to 820ms which DELN increased to 1.12s,
an increase of 36%. The overall characteristics of the decay are unchanged
with the three decay regions evident, as discussed in section 5.4. The two
pulse echo decay becomes linear at different times due to the DELN field
(0.7s and 1s ) but at the same echo amplitude, i.e. ln(E(2τ)/E0) ≈ −3.4.
Therefore, the proportion of ions experiencing an optimised critical point is
not changed by DELN. At short time there is no discernible difference caused
by the DELN driving field as shown in figure 7.3.
7.3 Discussion
Magic Angle Line Narrowing represents the optimal experimental conditions
for removing decoherence due to dipole-dipole interactions by driving the
bath spins. Since the MALN conditions were not satisfied these experiments
should be regarded as a feasibility of increasing T2 through driving the bath.
This experiment has demonstrated a 36% increase in T2 and as such an
optimised MALN experiment should perform significantly better.
The increase in T2 due to the DELN was most prominent in the second re-
gion of the two pulse echo decay (see section 5.4) where ions not experiencing
an optimised critical point create non-linear time response in the decay. The
increase in coherence in this region of the echo decay indicates that driving
120 Extending T2 Through Driving the Environment
0
0
0.4 0.8 1.2 1.6
Delay (s)
ln(E
(2
)/E
) 0τ
Figure 7.2: Spin echo decay amplitude as a function of pulse delay with (·) and without() the DELN decoupling field.
7.3 Discussion 121
0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
0
Delay (s)
ln(E(2 )/E ) 0
τ
Figure 7.3: Spin echo decay amplitude as a function of pulse delay with no Y drivingfield(×), fixed frequency DELN () and the DELN frequency scanned over 600Hz (·).
122 Extending T2 Through Driving the Environment
the bath is effective at extending the T2 of ions both at and near the critical
point.
Inhomogeneous broadening of Y hyperfine transitions is dominated by
magnetic interactions with the Pr ion. Since spin 1/2 nuclei, such as Y
have no electric dipole moment and are therefore not broadened by crystal
field inhomogeneities. Therefore, the Y hyperfine transition frequency is
proportional to the proximity of Pr ions, resulting in the frozen core and
bulk delineation of the Y spin bath, previously discussed in section 4.1. By
not scanning ωY either the bulk Y and the outer frozen core are driven or
the inner frozen core is driven. This results in many bulk Y or a few inner
frozen core Y still contributing as normal to dephasing the Pr ions. As such
either the bulk contributes many weak dephasing sources or the inner frozen
core provides a few strong dephasing sources. Some Y ions will be strongly
coupled to the driven subset despite not being driven themselves, such as the
ions on the edge of the frozen core when either the bulk Y or inner frozen
core Y are driven. The ions strongly coupled to the driven Y ions will have
an increased spin flip rate due to the rapid flipping of nearby Y ions with
a small detuning. This will tend to increase the decoherence contribution
from the undriven coupled ions as their spin flip rate is increased, but not
sufficiently increased to decouple the dipole - dipole interactions. This results
in the minimal gain in T2 due to the fixed frequency DELN driving field.
An increase in RF power will allow the MALN magnetic field condition
(inequality 7.1) to be satisfied. Higher RF powers were not available without
risk to equipment and hence reserved for future investigations. Further gains
in T2 are expected and, in the opinion of the author, achieving T2 ≈ 2s is
reasonable for an optimised MALN experiment.
In the previous chapter problems associated with imperfect driving fields
acting on HS became apparent. While much of this can be avoided through
the improvement of the RF circuitry the fidelity will always be less than
unity. By driving HE rather than HS to extend T2, errors in the driving field
are coupled to HS non-resonantly and as a result accumulate differently.
The MALN driving field introduces a time dependant Zeeman shift on the
hyperfine transitions of the Pr ion at the driving frequency. This is often
considered to be a phase modulation by NMR spectroscopists due to the
appearance of sidebands on the transition frequency, separated by integer
multiples of the driving frequency. Pulsed techniques have been developed
to counter the phase modulation in a similar manner to the refocusing in spin
echo experiments. The most commonly used technique is TOtal Sideband
Suppression (TOSS) [83] which involves a minimum of 4 π pulses, precisely
7.3 Discussion 123
timed relative to the period of the MALN driving frequency. It should be
noted that TOSS is identical to a CPMG or Bang Bang decoupling scheme
which rephases time dependant frequency shifts of the transition due to the
MALN field.
Consequently, as with the Bang Bang DDC we have introduced a periodic
condition for performing operations on the system if phase shifts between
qubits due to operations are to be avoided. While TOSS can be implemented
with a minimum of 4 π pulses, significantly more can be used such that QIP
operations can be performed more often. This however highlights that there
is no consequence free DDC method to extend T2 for QIP applications.
Implementing a TOSS scheme is very similar to the CPMG or Bang Bang
DDC schemes investigated in the previous chapter, with the addition of the
MALN field. The main importance of the DELN experiment is to show
that the technique is compatible with the critical point technique, allowing
for concatenation of QIP error correction schemes. Concatenation of these
schemes could be usefull in situations, such as removing residual first order
Zeeman contributions if it is not possible to achieve an optimised critical
point for all qubits.
There are a series of experiments will only be worthwhile if the Magic
Angle conditions are met, in particular studying the effectiveness of TOSS
at and near the critical point. These experiments require an optimised MALN
driving field, thereby removing the requirement to scan the frequency of the
driving field, and is therefore left to future work.
Chapter 8
Future Decoherence Challenges
“A bend in the road is not the end of the road... unless you fail to make the
turn.”
–Anon
The investigation of the decoherence mechanisms and suppression thereof
in Pr3+:Y2SiO5 has been very successful, however other materials are of
greater interest for long term rare-earth QIP development. Currently ma-
terials using Europium appear most attractive and stoichiometric crystals,
rather than lightly doped materials also appear promising. Decoherence ben-
efits and challenges of these future directions are discussed in this chapter.
8.1 Exchanging Praseodymium for Europium
In examining possible replacement ions for Pr we have strict criteria. The
ion must be an optically active non-Kramers ion, ie. it has a nuclear spin,
but not an electron spin. Rare-earth ions that are non-Kramers ions in the
3+ oxidation state are Pr, Pm, Eu, Tb, Ho and Tm, however Pm can be
excluded as being impractical due to having a short nuclear decay half life.
We require the ability to use the critical point technique, and as such the
nuclear spin to be I > 1 5.4, such that in a low symmetry host there is a
zero field splitting and an anticrossing region. This allows us to exclude Tm
since it is a spin 1/2 nucleus. Holmium is a spin 7/2 system with a much
larger dipole moment than Pr, reflected in the significantly larger linewidths
when used in the same hosts as Pr [88]. This indicates that inhomogeneous
broadening issues that arose in this work would be exacerbated by using Ho.
Terbium is still a candidate system, exhibiting equivalent linewidths to Pr
8.2 Stoichiometric Materials 125
[88]. However, the lack of interest in Tb for both holographic research and
achieving narrow optical transitions suggests there are experimental issues
with the ion. The most promising replacement ion appears to be Europium,
as will be discussed in the remainder of this section.
The greater promise of materials containing Eu is due to differences in
hyperfine transition T1, which for Pr3+:Y2SiO5 is 145s compared to 23 days
for Eu3+:Y2SiO5 [4]. Europium therefore provides a significantly longer up-
per bound on T2. This is due to negligible cross relaxation rates in low
concentration samples and in general cross relaxation is only significant in
stoichiometric samples. The smaller magnetic moment of Eu (γ ≈ 3kHz/G,
measured in Eu3+:Y2SiO5 [160]) implicitly reduces the hyperfine transition
sensitivity to magnetic field fluctuations.
The larger nuclear quadrupole of Eu results in zero field hyperfine split-
tings of the order of 100MHz in Eu3+:Y2SiO5 [140, 7, 6]. This larger splitting
allows higher Rabi frequencies to be used and remain transition specific. The
smaller γ of Eu when compared to Pr does however have two minor draw-
backs. Higher power driving fields are, however, required to obtain the same
Rabi frequency and larger magnetic fields are required to reach the hyperfine
anticrossing region to achieve critical point field configurations. Both are
technical issues.
Europium materials were not used in these initial investigations primar-
ily due to the difficulty of obtaining good Raman heterodyne signals in non
stoichiometric samples and the lack of field rotation data for Eu:Y2SiO5. Re-
cent developments within our research group [160, 128] have demonstrated
that using a suitable optical pumping scheme high SNR Raman heterodyne
signals can be obtained. Therefore investigating critical point field configura-
tions and dynamic decoupling methods in lightly doped Eu materials warrant
investigation after field rotation studies are performed.
8.2 Stoichiometric Materials
Stochiometric materials containing Eu are also of interest. Stoichiomet-
ric samples produce the lowest possible strain and therefore results in the
narrowest inhomogeneous linewidth. Reducing inhomogeneous linewidth is
beneficial for DDC techniques, as discussed in section 6.4.1. Investigat-
ing crystals that have inherently narrow inhomogeneous linewidths such as
EuCl6H2O will be interesting for further Dynamic Decoherence Control ex-
periments. Optical linewidths of the order of 100MHz have been observed
for EuCl6H2O [128], significantly narrower than the several GHz observed
126 Future Decoherence Challenges
in Eu3+:Y2SiO5.
The drawback with these materials is that resonant cross relaxation be-
tween the spins of interest dramatically increases the relaxation rate, with
the resulting decrease of T1 and T2. Stoichiometric materials lightly doped
with magnetic defects, such as Er3+:Eu2SiO5, discussed in detail in section
3.2, will hopefully provide a solution to this by only using the Eu ions within
the “frozen core”, and therefore allow low strain crystals which also have long
T2. Performing QIP in such a situation has potential complications above
and beyond simple T2 considerations and will be discussed in section 8.3.
8.3 Considerations for Implementing QIP in
Stoichiometric Materials
Performing QIP in stoichiometric samples as described in quantum computer
architecture (section 3.2) creates some problems due to the close proximity
of the ions used as qubits. The following discussion considers these issues
in relation to the Er3+:Eu2SiO5 system as proposed in section 3.2. It is
assumed that the bulk Eu ion can be optically spin polarised via holeburn-
ing techniques and therefore decohereing magnetic perturbations originating
from the bulk are vanishingly small. The following discussion also assumes
that a critical point for Eu can be found with similar properties to the critical
point investigated in this work.
Both the electric and magnetic dipoles change when the ion is optically
excited. Consequently there is a change in magnetic field on the order of a
Gauss seen by nearest neighbour ions. The use of the critical point technique
will be helpful to minimise the effect of the field perturbation, however there
are two primary differences between this situation and the Pr-Y critical point
investigated in this work. Firstly the Eu ions near the Er ion have a range
of hyperfine detunings due to the frozen core generated by the Er ion, and
therefore result in a range of critical point field configurations. Secondly,
the magnetic field change due to optically exciting Eu is significantly larger
than for a Y spin flip and therefore second order Zeeman shifts will be more
important. This is another benefit of the smaller magnetic moment of Eu
with respect to Pr.
The back action of the Y bath due to a change in the Pr spin state, as
investigated in chapter 4, also needs to be reconsidered in the context of
Er3+:Eu2SiO5. Changing the magnetic dipole moment of an Eu ion will
change the local spin quantisation axis and therefore can induce superhy-
perfine and near degenerate mutual spin flip transitions amongst nearby Eu
8.4 Minimising Decoherence in Stoichiometric Defect QIP Systems 127
ions. Superhyperfine transitions should be well suppressed by the applica-
tion of the critical point field assuming the magnetic field magnitude for a
critical point in Eu is, like that for Pr, many orders of magnitude larger than
the magnetic dipole moment of the Eu ion. Therefore, the near degener-
ate superhyperfine transitions that represent the biggest problem. The near
degenerate superhyperfine spin flips will most likely be other qubits rather
than members of the bath due to their proximity. This represents a very se-
rious potential for errors to occur as a direct consequence of QIP operations.
Quantum Error Correction Codes can be employed to correct these kinds of
errors at the expense of using several physical qubits for one logical qubit
[72, 65, 63, 161, 74, 68, 162, 62, 70, 66].
There is also potential for the change in electric dipole moment of an ex-
cited ion coupling to the hyperfine state of a neighbour through the quadrupole
and pseudo quadrupole moments. This needs to be carefully investigated to
determine the limits it creates for particular applications. The Er3+:Eu2SiO5
system discussed in the previous section provides a very good system to in-
vestigate these effects and how they scale with distance.
In light of these considerations the previous discussion in section 3.2 of the
optical qubit as being interacting and the hyperfine qubit as being isolated is
not completely true. Assessing the limits these interactions impose on QIP
is a particularly important direction for future research in the area. Char-
acterisation of these interactions will be required to formulate appropriate
quantum error correction strategies.
8.4 Minimising Decoherence in Stoichiomet-
ric Defect QIP Systems
The quantum computing architecture discussed in section 3.2 creates uniquely
addressable qubits by introducing defects. All QIP therefore occurs inside
the frozen core of the defect site. While it was previously stated that devices
possessing greater than 10 qubits could be produced, the frozen cores can be
very extensive. If we consider the Er3+:Eu2SiO5 system a rough calculation
shows that γEr/γEu ≈ 25γPr/γY and using the size of the Pr-Y frozen core
therefore we can expect the Er-Eu frozen core to contain of the order of 104
Eu ions. This clearly has the potential to create far more than 10 individ-
ually addressable qubits. The size of the frozen core also has the distinct
advantage for decoherence since the bulk is a long way from the inner frozen
core used for QIP. Therefore, the fluctuating magnetic field due to the bulk
as seen by the inner frozen core will be small and tend to reflect the mean
128 Future Decoherence Challenges
field value.
This work has demonstrated the dramatic increase in T2 afforded by the
critical point technique and that DDC techniques can be used to further
increase T2. The introduction of defects to the system provide two primary
challenges the the methods followed in this work to increase T2. Firstly the
state of the defect must be controlled. In the case previously discussed of
Er3+:Eu2SiO5 we require spin polarisation of the Er since any change of state
of the Er will introduce catastrophic errors via the superhyperfine interaction.
Therefore magnetic field configurations will be dictated by achieving this
result with the benefits of a critical point configuration being secondary.
The second challenge is that we introduced the defect to create repeat-
able inhomogeneities, however inhomogeneities limit the effectiveness of the
critical point technique. The magnetic field configuration required to spin po-
larise the defects may preclude the use of the critical point technique. There
is, however, also the possibility of satisfying a critical point field configura-
tion for a particular set of detunings such that many of the other rare-earth
ions experience partial benefit. Operation of the quantum computer would
need to reflect the difference in T2 and response to field perturbations among
the available qubits in this case.
The defect related inhomogeneities will also have implications for DDC
techniques. As was shown in section 6.4.1 the inhomogeneities were the
reason that an arbitrary state was not well preserved. Consequently this will
require harder RF pulses or a qubit specific Raman optical implementations
of DDC techniques.
Chapter 9
Conclusions and Future Work
“Any sufficiently advanced technology is indistinguishable from magic.”
–Arthur C. Clarke
This work has demonstrated that hyperfine decoherence times sufficiently
long for QIP and quantum optics applications are achievable in rare-earth
doped insulators. Prior to this work there were several QIP proposals using
rare earth hyperfine states for long term coherent storage of optical inter-
actions [1, 2, 3]. The very long T1 (∼ 23 days [4]) observed for rare-earth
hyperfine transitions appears promising but the hyperfine transition T2 was
typically only a few ms [144], comparable to rare-earth optical transitions.
Therefore transferring information from an optical to a hyperfine transition,
with the associated time and fidelity considerations for only a minimal in-
crease in T2 made the usefulness of such proposals dubious. This work demon-
strated an increase in hyperfine T2 by a factor of ∼ 7× 104 compared to the
previously reported hyperfine T2 for Pr3+:Y2SiO5 through the application of
static and dynamic magnetic field techniques. This increase in T2 makes pre-
vious QIP proposals useful and provides the first solid state optically active
Λ system with very long hyperfine T2 for quantum optics applications.
The first technique employed the conventional wisdom of applying a small
static magnetic field to minimise superhyperfine interaction [5, 6, 7], as stud-
ied in chapter 4. This resulted in hyperfine transition T2 an order of mag-
nitude larger than the T2 of optical transitions, ranging from 3 to 10 ms.
Estimates of the required T2 to the time taken to perform a quantum oper-
ation, τop, such that the system is useful for complex QIP yield T2/τop > 105
[35]. In rare-earth systems the operation time is expected to be ∼10 µs given
Rabi frequencies of approximately 1 MHz [127] and swapping the information
130 Conclusions and Future Work
to and from the optical transition. Therefore minimising the superhyperfine
interaction resulted in T2/τop ≈ 103, which falls too short of the 105 estimate.
In addition to this requirement, most of the proposals involving rare-earth
ions used the hyperfine states as the long term storage [1, 2, 3]. In analogy
to classical computing the optical transitions are used as a RAM/processor
combination, while the hyperfine transitions are used as a hard drive. This
implies that hyperfine T2 is required to be longer, such that the decoherence
is negligible over the desired storage times.
Development of the critical point technique during this work was crucial
to achieving further gains in T2. The critical point technique is the applica-
tion of a static magnetic field such that the Zeeman shift of the hyperfine
transition of interest has no first order component, thereby nulling decohering
magnetic interactions to first order. This technique also represents a global
minimum for back action of the Y spin bath due to a change in the Pr spin
state, as discussed in section 5.4, allowing the assumption that the Pr ion
is surrounded by a thermal bath. The critical point technique resulted in a
dramatic increase of the hyperfine transition T2 from ∼10 ms to 860 ms. The
critical point method should be applicable to any spin system with I ≥ 1 and
zero field splitting, as discussed in section 5.4. This technique is anticipated
to be of general interest. Using the critical point technique allowed us to
realise a system with T2/τop ≈ 105, thereby experimentally demonstrating
the suitability of rare-earth doped systems for QIP applications.
Satisfied that the optimal static magnetic field configuration for increas-
ing T2 had been achieved, dynamic magnetic field techniques, driving ei-
ther the system of interest or spin bath were investigated. These tech-
niques are broadly classed as Dynamic Decoherence Control (DDC) in the
QIP community. The first DDC technique investigated was driving the
Pr ion using CPMG or Bang Bang decoupling pulse sequence. This sig-
nificantly extended T2 from 0.86 s to 70 s. This decoupling strategy has
been extensively discussed for correcting phase errors in quantum computers
[8, 9, 10, 11, 12, 13, 14, 15], with this work being the first application to
solid state systems. The time scales for pulsed decoupling are easily satisfied
for the Pr3+:Y2SiO5 system unlike expectations of other solid state quantum
computing candidate systems [13].
Magic Angle Line Narrowing was used to investigate driving the spin
bath to increase T2. The experiment demonstrated the applicability of this
technique despite not reaching the desired field regime, as discussed in section
7.3. This experiment resulted in T2 increasing from 0.84 s to 1.12 s. Both
DDC techniques introduce a periodic condition on when QIP operation can
Conclusions and Future Work 131
be performed without the qubits participating in the operation accumulating
phase errors relative to the qubits not involved in the operation.
Without using the critical point technique Dynamic Decoherence Control
techniques such as the Bang Bang decoupling sequence and MALN are not
useful due to the sensitivity to magnetic field fluctuations. Critical point
and DDC techniques are mutually beneficial since the critical point is most
effective at removing high frequency perturbations while DDC techniques
remove the low frequency perturbations as discussed in section 6.1. A further
benefit of using the critical point technique is it allows changing the coupling
to the spin bath without changing the spin bath dynamics. This was useful
for discerning whether the limits are inherent to the DDC technique or are
due to experimental limitations.
Solid state systems exhibiting long T2 are typically very specialised sys-
tems, such as 29Si dopants in an isotopically pure 28Si and therefore spin free
host lattice [16]. These systems rely on on the purity of their environment
to achieve long T2. Despite possessing a long T2, the spin system remain
inherently sensitive to local magnetic field fluctuations. Sources of magnetic
field fluctuations, from local defects in the crystal, noise in the controlling
fields and noise from lab equipment can all strongly couple to the spin of in-
terest and therefore place practical limits on achievable T2. Nanostructuring
always introduces some level of disorder at the interface between different
materials, creating excess charge or trapped spins. Therefore problems asso-
ciated with the qubit’s sensitivity scale proportionally to the complexity of
their environment. Complex, nanostructured systems for realising quantum
computing architectures in such a system therefore face significant scaling
challenges.
In contrast, this work has demonstrated that decoherence times, suffi-
ciently long to rival any solid state system [16], are achievable when the spin
of interest is surrounded by a concentrated spin bath. Using the critical
point method results in a hyperfine state that is inherently insensitive to
small magnetic field perturbations and therefore more robust for QIP appli-
cations.
In the present work we have achieved a hyperfine transition with a res-
onator quality factor, Q of 6 × 108, which is expected to increase with up-
grades to RF equipment, more sophisticated DDC techniques and investigat-
ing other promising materials. Previously discussed approaches to achieve
these goals are summarised in the remainder of this chapter as well as inter-
esting quantum optics applications.
132 Conclusions and Future Work
9.1 Strategies for Further Increases in Deco-
herence Time
While the 70s T2 achieved dwarfs the previously achieved few ms T2 limit
for optically active nuclear spin system there is great promise to increase
this further. The discussion will initially focus on improvements that would
benefit experiments on any system before addressing material specific issues.
9.1.1 Improved RF Control
The current limit on the T2 achievable using DDC techniques, such as Bang
Bang decoupling, appears to be due to the driving RF, as discussed in section
6.4. The PulseBlaster [163] DDS circuitry was identified as the dominant RF
noise source in section 6.4 and consequently using higher purity RF source is
critical to removing the plateau observed in the Bang Bang dynamic decou-
pling experiments. If T2 increased at the same rate as prior to the plateau
the for a cycling time of τc = 1ms a T2 of the order of 100s is expected. This
could allow us to approach the T1 = T2 limit.
9.1.2 Rabi Frequency and Inhomogeneous Broadening
As discussed in section 6.4.1 a combination of increasing in Rabi frequency
and a decrease in linewidth are required if Bang Bang decoupling sequences
can preserve an arbitrary state. If work is continued in Pr3+:Y2SiO5 repeat-
edly annealed isotopically pure crystals are desirable due to the expected re-
duction in hyperfine inhomogeneous linewidths, as discussed in section 6.4.1.
If such crystals are not available then modifying the inhomogeneous linewidth
via holeburning techniques is feasible for further DDC investigations.
Europium is the other rare-earth ion with particular application to this
work. Doping Eu into Y2SiO5 or analogous crystals such as Y2O5 allows
directly analogous systems to be investigated with potential for higher Rabi
frequencies due to the increased hyperfine splittings, as discussed in section
8.1. Other materials such as Eu6H2O have shown optical inhomogeneous
linewidths of the order of 10MHz [128] and consequently are expected to
have narrow hyperfine inhomogeneous widths. As such DDC investigations
could benefit from this or analogous materials.
9.2 Other Applications For Long T2 Optically Active Solids 133
9.1.3 Eulerian Decoupling
While in Pr3+:Y2SiO5 it is possible to achieve the required cycling time,
τc, for the Bang Bang pulse sequence to be effective the underlying theory
still requires unbounded Rabi frequencies during the applied pulses. This is
clearly not physically reasonable and as such the more sophisticated method
of Eulerian decoupling has been proposed to relax the pulse amplitude re-
quirement [11, 13]. Eularian decoupling is gaining favour with the QIP theory
community since it brings theory and experiment a step closer. I believe that
given how successful the application of the Bang Bang decoupling scheme was
rare earth ion systems are the best solid state system to investigate these new
dynamic decoherence control schemes.
9.2 Other Applications For Long T2 Optically
Active Solids
Quantum optics experiments are typically performed using dilute gas sys-
tems due to the availability of optically active Λ systems with long coherence
times. In atomic vapour systems the atoms will always have a non-zero ve-
locity resulting in a finite time for an atom to drift such that it experiences
a different phase or amplitude of the optical driving field and eventually
completely leave the interaction region. Utilising solid state systems removes
these problems, at the cost of introducing inhomogeneous broadening via the
crystal field, which as discussed earlier, can be engineered. The static rela-
tionship of ions in solids has been exploited for classical optics experiments,
using holograms for high bandwidth RF signal processing [91, 92, 93] and
optical routing [94]. Utilising the static, ultra coherent optically active Λ
systems afforded by this work has direct application to two experiments of
particular interest in quantum optics.
9.2.1 Slow and Stopped Light
Electromagnetically Induced Transparency (EIT), commonly referred to as
slow or stopped light, has been proposed as a potential quantum memory
[164, 165]. While there have been a number of EIT demonstrations in dilute
gas systems there has only been one demonstrations in a solid state system
prior to this work [102]. Recently the critical point technique, in combina-
tion with the Bang Bang DDC technique has been used to stop light in a
solid for 6s [166], dwarfing the longest previous storage time of 1ms [167].
134 Conclusions and Future Work
The efficiency of the EIT feature was also an order of magnitude larger that
that of dilute gas experiments [43] and can be increased further by modifying
the dopant concentration or crystal dimensions to increase the optical depth.
Utilising a solid also allows a larger number of beam geometries to be consid-
ered, in particular the probe and coupling beams can counter propagate and
as such simplify the detection of a weak probe in the presence of a strong
coupling beam [43].
9.2.2 Stark Echo Quantum Memory
EIT memories have inherently small bandwidth due to using a narrow spec-
tral feature. A higher bandwidth quantum memory scheme was recently
developed in our group by Sellars and Alexander [128, 160]. This scheme
uses electric field gradients to create a controlled inhomogeneous broadening
on an optical transition that can be reversed, resulting in a photon echo [87].
Using Raman techniques to store the information on the optical transitions
will allow a both critical point and DDC techniques to extend the storage
time.
Appendices
Appendix A
Y2SiO5 site position calculation
This appendix lists the code used to calculate the Y site positions for Y2SiO5The
files lsted are pairs.cpp, rot.c, rot.h, assign.h, clapack.h, parameters.h and
makeJ.h
----- pairs.cpp -----
#include <iostream.h>
#include <complex>
#include <math.h>
#include <unistd.h>
#include <tnt/tnt.h>
#include <tnt/vec.h>
#include <tnt/fmat.h>
#include <tnt/cmat.h>
using namespace std;
using namespace TNT;
double a = 10.410,b=6.726,c=12.495; //unit cell lengths in Angstroms
double beta = (102.65)/180.0*M_PI; //beta = 102.65deg
Y2SiO5 site position calculation 137
Fortran_Matrix<double> xtal_to_cart(3,3);
double jmod(double x,double y)
double temp = fmod(x,y);
if (temp<0) temp+=y;
return temp;
double norm(Vector<double> v)
double norm=0;
for(int k=1;k<=v.size();k++)
norm+=v(k)*v(k);
norm=sqrt(norm);
return norm;
Vector<Vector<double> > Y_positions(Vector<double>atom_coords)
//return array of yttrium positions within unit cell
// in crystalographic coordinates
// space groups I 2/a origin same as int tables
Vector<Vector<double> > Y;
int k,j;
Y.newsize(8);
for(k=1;k<=8;k++)
Y(k).newsize(3);
Y(1) = atom_coords;
// if(n==1) //return site one positions
138 Y2SiO5 site position calculation
// Y(1)(1) = 0.30657;
// Y(1)(2) = 0.37701;
// Y(1)(3) = 0.14154;
// else
// Y(1)(1) = 0.42839;
// Y(1)(2) = 1.0-0.25506;
// Y(1)(3) = 1.0-0.03701;
//
// if(n==-99)
// Y(1)(1) = 0.1;
// Y(1)(2) = 0.2;
// Y(1)(3) = 0.3;
//
//do inversion operation
for(k=1;k<=3;k++)
Y(2)(k) = jmod(-Y(1)(k),1);
//do rotation operator
for(j=1;j<=2;j++)
Y(j+2)(1) = jmod(0.5-Y(j)(1),1);
Y(j+2)(2) = jmod(Y(j)(2),1);
Y(j+2)(3) = jmod(-Y(j)(3),1);
//do translation by 1/2 1/2 1/2
for(j=1;j<=4;j++)
for(k=1;k<=3;k++)
Y(j+4)(k) = jmod(Y(j)(k)+0.5,1);
return Y;
Vector <Vector<double> > Y_dipoles(Vector<double> dipole_of_Y1)
Y2SiO5 site position calculation 139
//return array of yttrium dipoles within unit cell corresponding to
//
// cartesian coords (see notebook for axes)
// space groups I 2/a origin same as int tables
Vector<Vector<double> > Y;
int k,j;
assert(dipole_of_Y1.dim() == 3);
Y.newsize(8);
for(k=1;k<=8;k++)
Y(k).newsize(3);
Y(1) = dipole_of_Y1;
//do inversion operation
for(k=1;k<=3;k++)
Y(2)(k) = -Y(1)(k);
//do rotation operator
for(j=1;j<=2;j++)
Y(j+2)(1) = -Y(j)(1);
Y(j+2)(2) = Y(j)(2);
Y(j+2)(3) = -Y(j)(3);
//do translation by 1/2 1/2 1/2
for(j=1;j<=4;j++)
for(k=1;k<=3;k++)
Y(j+4)(k) = Y(j)(k);
return Y;
Vector<Vector<Vector<double> > > all_within(Vector<double> r,
140 Y2SiO5 site position calculation
double distance,
Vector<double> atom_coords,
Vector<double> dipole)
//returns a vector length 2, element one is a vector of position vectors
// for all the atoms with coordinates atom_coords
// within distance angstroms of the position r
// element two are the corresponding dipole moments
// dipole = the dipole moment of the atom with position vector
// equal to the atomic coords
// r, atom_coords,output(1) in xtalographic units
// dipole,output(2) in cartesian
//
// r is assumed to be with the 0-1,0-1,0-1 unit cell
Vector<Vector<Vector<double> > > output_before_thining;
Vector<Vector<Vector<double> > > output;
Vector<Vector<double> > one_unit_cell_posns;
Vector<Vector<double> > one_unit_cell_dipoles;
int h,k,l,j,count;
int n = (int)(distance/c)+1;
one_unit_cell_posns = Y_positions(atom_coords);
one_unit_cell_dipoles = Y_dipoles(dipole);
output_before_thining.newsize(2);
output.newsize(2);
for(k=1;k<=2;k++)
output_before_thining(k).newsize((2*n+1)*(2*n+1)*(2*n+1)*8);
output(k).newsize((2*n+1)*(2*n+1)*(2*n+1)*8);
for(k=0;k<output_before_thining(1).dim();k++)
output_before_thining(1)[k].newsize(3);
Y2SiO5 site position calculation 141
count=1;
for(h=-n;h<=n;h++)
for(k=-n;k<=n;k++)
for(l=-n;l<=n;l++)
for(j=1;j<=8;j++)
output_before_thining(1)(count)(1)=one_unit_cell_posns(j)(1)+h;
output_before_thining(1)(count)(2)=one_unit_cell_posns(j)(2)+k;
output_before_thining(1)(count)(3)=one_unit_cell_posns(j)(3)+l;
output_before_thining(2)(count)=one_unit_cell_dipoles(j);
count++;
count=0;
for(k=1;k<=output_before_thining(1).dim();k++)
if (norm(xtal_to_cart*(output_before_thining(1)(k)-r))<distance)
count++;
output(1)(count) = output_before_thining(1)(k);
output(2)(count) = output_before_thining(2)(k);
output(1).newsize(count);
output(2).newsize(count);
count=0;
for(k=1;k<=output_before_thining(1).dim();k++)
if (norm(xtal_to_cart*(output_before_thining(1)(k)-r))<distance)
count++;
output(1)(count) = output_before_thining(1)(k);
output(2)(count) = output_before_thining(2)(k);
return output;
142 Y2SiO5 site position calculation
main()
int k,l,j;
double temp;
double re;
int nj;
xtal_to_cart(1,1) = a;
xtal_to_cart(1,3) = c*cos(beta);
xtal_to_cart(2,2) = b;
xtal_to_cart(3,3) = c*sin(beta);
Vector<double> site_one_coords(3,"0.30657 0.37701 0.14154");
Vector<double> site_two_coords(3,"0.42839 0.74494 0.96299");
Vector<double> site_Si_coords(3,"0.6270 0.9070 0.1810");
Vector<Vector<Vector<double> > > posns_and_dipoles1 ;
Vector<Vector<Vector<double> > > posns_and_dipoles2 ;
Vector<Vector<Vector<double> > > posns_and_dipoles3 ;
Vector<Vector<Vector<double> > > posns_and_dipoles4 ;
Vector<Vector<Vector<double> > > posns_and_dipolesSi ;
Vector<Vector<double> > posns,dipoles;
Vector<double> r(1501);
Vector<double> n1(1501);
Vector<double> n2(1501);
Vector<double> n3(1501);
Vector<double> n4(1501);
re= 200;//angstroms
/* cout<<"###site ones\n";
posns_and_dipoles1 = all_within(site_one_coords,re,site_one_coords,site_one
Y2SiO5 site position calculation 143
nj = posns_and_dipoles1(1).dim();
for(k=1;k<=nj;k++)
posns_and_dipoles1(1)(k) = xtal_to_cart*posns_and_dipoles1(1)(k);
for(j=1;j<=3;j++)
cout<<posns_and_dipoles1(1)(k)(j)<<" ";
cout<<endl;
/*posns_and_dipoles1(2)(k) = xtal_to_cart*posns_and_dipoles1(2)(k);
for(j=1;j<=3;j++)
cout<<posns_and_dipoles1(2)(k)(j)<<" ";
cout<<endl;* /
;
cout<<"###site twos\n";
posns_and_dipoles2 = all_within(site_one_coords,re,site_two_coords,site_one_coord
nj = posns_and_dipoles2(1).dim();
for(k=1;k<=nj;k++)
posns_and_dipoles2(1)(k) = xtal_to_cart*posns_and_dipoles2(1)(k);
// cout<<"len="<<posns_and_dipoles2(1).size()<<endl;
for(j=1;j<=3;j++)
cout<<posns_and_dipoles2(1)(k)(j)<<" ";
cout<<endl;
/* posns_and_dipoles2(2)(k) = xtal_to_cart*posns_and_dipoles2(2)(k);
for(j=1;j<=3;j++)
cout<<posns_and_dipoles2(2)(k)(j)<<" ";
cout<<endl; * /
;*/
cout<<"###site Si\n";
posns_and_dipolesSi = all_within(site_Si_coords,re,site_Si_coords,site_Si_coords);
nj = posns_and_dipolesSi(1).dim();
for(k=1;k<=nj;k++)
posns_and_dipolesSi(1)(k) = xtal_to_cart*posns_and_dipolesSi(1)(k);
for(j=1;j<=3;j++)
cout<<posns_and_dipolesSi(1)(k)(j)<<" ";
cout<<endl;
;
144 Y2SiO5 site position calculation
// temp=0.0000001;
// for(k=0;k<1501;k++)
// r[k]=temp;
// temp+=0.005;
// posns_and_dipoles1 = all_within(site_one_coords, r[k],site_one_coords,site_
// posns_and_dipoles2 = all_within(site_one_coords, r[k],site_two_coords,site_
// posns_and_dipoles3 = all_within(site_two_coords, r[k],site_one_coords,site_
// posns_and_dipoles4 = all_within(site_two_coords, r[k],site_two_coords,site_
// n1[k] = posns_and_dipoles1(1).dim()-1;
// n2[k] = posns_and_dipoles2(1).dim();
// n3[k] = posns_and_dipoles3(1).dim();
// n4[k] = posns_and_dipoles4(1).dim()-1;
// cout<<r[k]<<"\t"<<n1[k]<<"\t"<<n2[k]<<"\t"<<n3[k]<<"\t"<<n4[k]<<endl;
//
// posns = posns_and_dipoles(1);
// dipoles = posns_and_dipoles(2);
// for(k=0;k<posns.dim();k++)
// cout<<k<<"\t"<<posns[k]<<endl;
//
---- end pairs.cpp ----
---- rot.c ----
#include <math.h>
Y2SiO5 site position calculation 145
//#include <f2c.h>
/* rotation matrix describing a rotation size |axis| around the vector axis */
#define EPS(i,j,k) ((int)((i<j)&&(j<k))-(int)((i>j)&&(j>k))+(int)((j<k)&&(k<i))-(int)
void axis2so3(double * axis,double * R)
double theta;
double unit_vec[3];
double col[3];
int k,l,m;
for(k=0;k<9;k++)
R[k]=0;
/* normalise */
theta = sqrt(axis[0]*axis[0]+axis[1]*axis[1]+axis[2]*axis[2]);
for(k=0;k<3;k++)
unit_vec[k]=axis[k]/theta;
//x’ = x*cos(t) + sin(t) (unit_vec X x) + (1-cos(t))u.(u.x)
for(k=0;k<3;k++)
R[4*k] = cos(theta);
for(k=0;k<3;k++)
for(l=0;l<3;l++)
for(m=0;m<3;m++)
R[k+3*l]+=EPS(k,m,l)*unit_vec[m]*sin(theta);
R[k+3*l]+=unit_vec[k]*unit_vec[l]*(1-cos(theta));
146 Y2SiO5 site position calculation
void euler2so3(double alpha, double beta, double gamma, double * R)
int k;
char side = ’l’;
int three = 3;
char trans = ’n’;
double naada = 0;
double unity = 1;
double sa = sin(alpha);
double ca = cos(alpha);
double sb = sin(beta);
double cb = cos(beta);
double sg = sin(gamma);
double cg = cos(gamma);
double rx[9] = ca,sa,0,-sa,ca,0,0,0,1;
double ry[9] = cb,0,-sb,0,1,0,sb,0,cb;
double rz[9] = cg,sg,0,-sg,cg,0,0,0,1;
double hold[9] = 0,0,0,0,0,0,0,0,0;
/* hold = ry*rz */
dgemm_(&trans,&trans,&three,&three,&three,&unity,ry,&three,rz,&three,
&naada,hold,&three);
dgemm_(&trans,&trans,&three,&three,&three,&unity,rx,&three,hold,&three,
&naada,R,&three);
/* for(k=0;k<9;k++) */
/* R[k] = hold[k]; */
/* */
Y2SiO5 site position calculation 147
---- end rot.c ----
---- rot.h ----
#include <math.h>
/* rotation matrix describing a rotation size |axis| around the vector axis */
void axis2so3(double * axis,double * R);
void euler2so3(double alpha, double beta, double gamma, double * R);
void apply_rot(double *R, double *A, double *RARt);
---- end rot.h ----
---- parameters.h ----
typedef struct
double alf;
double bet;
double gam;
double gx;
double gy;
double gz;
double az;
double el;
double alf2;
double bet2;
double gam2;
double E;
double D;
double B_0x;
double B_0y;
148 Y2SiO5 site position calculation
double B_0z;
Parameters;
---- end parameters.h ----
---- clapack.h ----
#ifndef __CLAPACK_H
#define __CLAPACK_H
typedef struct double r, i; doublecomplex;
/* Subroutine */ int zheev_(char *jobz, char *uplo, int *n, doublecomplex
*a, int *lda, double *w, doublecomplex *work, int *lwork,
double *rwork, int *info);
void zgemm_(char*,char*,int*,int*,int*,doublecomplex*,doublecomplex*,
int*,doublecomplex*,int*,doublecomplex*,doublecomplex*,int*);
#endif /* __CLAPACK_H */
---- end clapack.h ----
---- makeJ.h ----
void makeJ(doublecomplex** J, doublecomplex** JJ,const int hsdim);
void makeHeff(const double* const M,
const double* const Q,
doublecomplex** J,
doublecomplex** JJ,
doublecomplex* Heff,
const int nsq,
const double *B,
int t,
double *temp);
void revert_params(Parameters *params,double old_value,int kj);
Y2SiO5 site position calculation 149
void choose_new_params(Parameters *params,
Parameters *del,
double alpha,
int kj,
double *old_value);
---- end makeJ.h ----
Full Critical Point List 151
Appendix B
Full Critical Point List
Transition Bx By Bz δfδB2 f(MHz)
−12↔ +1
2-101 -369 -139 1769 18.27
−12↔ +3
2-101 -369 -139 1790 18.74
+12↔ −3
2-732 -172 219 102 8.64
-309 -1123 -424 118 3.54
-451 361 -677 193 5.57
+12↔ −5
2-464 371 -697 191 7.72
-38 -9 11 2207 10.18
+12↔ +5
2-492 391 -734 16028 23.88
496 -399 750 16119 23.88
-384 -89 112 17584 27.05
−32↔ +3
2-103 -376 -142 86 4.77
438 -350 658 385 2.16
236 -1093 382 486 1.73
−52↔ +3
294 713 312 1002 9.86
-312 -765 -247 1009 9.86
-102 -369 -139 1819 16.24
-203 -738 -279 277156 9.90
−52↔ −3
2-683 -356 -999 240 7.55
664 372 971 243 5.82
1566 453 155 320 10.21
-1085 92 488 345 10.20
-889 486 -173 827.95 11.55
211 55 -843 828 11.55
-1093 -252 -453 828 11.55
-417 -793 561 829 11.55
402 -252 -287 1049 11.39
99 370 138 1823 11.01
576 -464 871 18102 16.03
152 Full Critical Point List
Transition Bx By Bz δfδB2 f(MHz)
−52↔ −3
2317 -704 489 417327 12.49
-203 -738 -279 419271 9.90
-520 -35 -767 488098 12.49
+52↔ −3
2-517 566 493 131 8.47
-256 1109 -411 243 5.82
-1061 -857 601 320 10.21
-1562 -456 -149 320 10.21
-1289 -646 209 345 10.20
1079 -87 -499 345 10.20
-213 -53 840 828 11.55
-95 -713 -312 1005 9.86
-103 -368 -141 1823 11.01
+52↔ +3
2311 765 247 1013 9.86
101 369 139 1819 16.24
-408 316 -594 16350 18.40
373 -309 579 16532 18.40
203 738 279 250783 9.90
-317 705 -490 330466 12.49
-520 -33 -769 330550 12.49
Appendix C
Published Papers
This appendix contains papers published during the course of my PhD work.
Not all of the work is specifically discussed in this thesis, in particular the
paper on spectral features of Europium pair sites.
154 Published Papers
Published Papers 155
156 Published Papers
Published Papers 157
158 Published Papers
Published Papers 159
160 Published Papers
Published Papers 161
162 Published Papers
Published Papers 163
164 Published Papers
Published Papers 165
166 Published Papers
Published Papers 167
168 Published Papers
Published Papers 169
170 Published Papers
Published Papers 171
172 Published Papers
Published Papers 173
174 Published Papers
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