APPLICATIONS OF SUPERINDUCTORS IN
SUPERCONDUCTING QUANTUM CIRCUITS
by
WENYUAN ZHANG
A dissertation submitted to the
School of Graduate Studies
Rutgers, The State University of New Jersey
In partial fulllment of the requirements
For the degree of
Doctor of Philosophy
Graduate Program in Physics and Astronomy
Written under the direction of
Michael Gershenson
And approved by
New Brunswick, New Jersey
May, 2019
ABSTRACT OF THE DISSERTATION
Applications of Superinductors in Superconducting Quantum
Circuits
By WENYUAN ZHANG
Dissertation Director:
Michael Gershenson
Superinductors are inductors whose microwave characteristic impedances are greater than
the resistance quantum, RQ = h/(2e)2 ≈ 6.5kΩ. They can be implemented using Josephson
junction chains and high kinetic inductance nanowires. In this dissertation, we explore ap-
plications of superinductors in both implementations in superconducting quantum circuits.
The dissertation consists of three parts. In the rst part, we discuss the uxon-parity-
protected qubit consisting of a Cooper-pair box (CPB) shunted by a superinductor made of a
chain of coupled asymmetric Superconducting Quantum Interference Devices (CASQUIDs).
The spectroscopic measurement of a prototype of the uxon-parity-protected qubit was per-
formed. We observed almost complete suppression of the single uxon tunneling across the
CPB due to the destructive Aharonov-Casher interference when the oset charge on the
CPB island was set to e mod(2e). A uxon-parity-protected qubit with a higher superin-
ductance can potentially be used to perform fault-tolerant Cliord gates. In the second part,
we studied the microwave losses in high-kinetic-inductance granular Aluminum lms using
superconducting coplanar-waveguide (CPW) resonators made of the lms. We observed that
the intrinsic losses in these resonators at low temperatures were limited by resonator cou-
pling to the two-level systems (TLS) in the environment. The demonstrated internal quality
ii
factors are comparable with those for CPW resonators made of conventional superconduc-
tors. The characterized granular Aluminum lms can be used to fabricate superinductors
for a wide range of applications in quantum metrology and quantum information processing.
In the third part, we discuss the one-dimensional Josephson metamaterial made of a similar
structure as the superinductor used in the uxon-parity-protected qubit. The metamaterial
demonstrated strong Kerr nonlinearity with the Kerr constant tunable over a wide range
from positive to negative values by the magnetic eld. The metamaterial is promising for use
as an active medium for quantum-limited Josephson traveling-wave parametric ampliers.
iii
Acknowledgments
I would like to thank my advisor Michael Gershenson for his mentorship throughout this
work. I cannot express my gratitude enough for his motivation and support. I would also
like to thank Professor Lev Ioe, whose theoretical proposal of the uxon-parity protected
qubit is the motivation of this experimental work.
Dr. Matthew Bell was the rst to introduce me to fabrication and measurement in the
lab. Though he moved to UMass Boston to start his tenure-track position a year after I joined
the lab, his guidance and support continued. He has oered great help in the fabrication of
the uxon-parity protected qubit discussed in Chapter 4. Later on, we collaborated on the
work discussed in Chapter 6 about the nonlinear Kerr eect in a metamaterial transmission
line.
Wen-Sen had worked in the semiconductor industry before coming to Rutgers for his
PhD. He has helped me better understand and appreciate the art of sample fabrication. He
is also full of cool ideas of DIY lab gadgets, which made the fabrication process much easier
for us. I can't be more grateful to have a lab mate like him.
Dr. Konstantin Kalashnikov joined the lab as a postdoc when I was about to nish my
graduate study. I have beneted from discussions with him as he brought up questions to
concepts I took for granted and helped further my understanding of the work I have done.
We also had a productive collaboration in analyzing the data about the low temperature
losses in the lms of granular Aluminum discussed in Chapter 5.
I would like to thank my committee members, Professors Karin Rabe, Weida Wu,
Amitabh Lath and Vladimir Manucharyan, for their valuable time and suggestions which
helped me in completion of this work.
I would like to thank the graduate students Plamen Kamenov and Tom DiNapoli for
their help with fabrications of the samples discussed in Chapter 5. I would also like to
iv
thank Hypres, Inc. for fabrication of the samples discussed in Chapter 6.
I thank my friends for their support and encouragement. Zhuohui, Rene, Lin-Ing, Wen-
Qing, Qin Xiao and many others.
Last, I thank my family especially my maternal grandma. She had generously supported
my education nancially as I grew up. Without her support, this journey would not be
possible. She is also an inspiration to me as an accomplished hematologist who has saved
thousands of lives with her knowledge and wisdom.
v
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Superconducting qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2. Superinductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3. A chain of coupled asymmetric SQUIDs (CASQUIDs) . . . . . . . . . . . . . 6
1.4. Fluxon-parity-protected qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5. Josephson traveling-wave parametric amplier based on CASQUIDs . . . . . 9
2. Fluxon-parity-protected qubit: theoretical background . . . . . . . . . . 13
2.1. Single-qubit state and coherence . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2. Parity-protected qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3. Fluxon-parity-protected qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4. Inductive coupling of uxon-parity-protected qubit and readout resonator . . 21
3. Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1. Sample fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.1. Josephson junction fabrication technique used for uxon-parity-protected
qubit fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.2. Deposition of disordered granular Aluminum lms . . . . . . . . . . . 27
vi
3.2. Measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3. Sample holders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4. Fluxon-parity-protected qubit : a prototype device . . . . . . . . . . . . . 32
4.1. Sample design and measurement . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2. First-tone measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3. Two-tone measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5. Microresonators fabricated from high-kinetic-inductance Aluminum lms 39
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2. Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.3. Microwave characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.3.1. The resonance frequency analysis . . . . . . . . . . . . . . . . . . . . . 42
5.3.2. The quality factor analysis . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3.3. The two-tone and time-domain measurements . . . . . . . . . . . . . . 46
5.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6. Josephson metamaterial with a widely tunable positive or negative Kerr
constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.2. Metamaterial Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.3. Microwave Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7. Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.1. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.2. Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
vii
Appendix A. Design of hybrid superinductor . . . . . . . . . . . . . . . . . 71
Appendix B. Increase in transmission power through metamaterial tranmis-
sion line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
viii
List of Tables
4.1. Parameters of Josephson junctions in the representative device. . . . . . . . . 34
5.1. Summary of the measured parameters of AlOx resonators. . . . . . . . . . . . 42
5.2. Summary of the tting parameters. . . . . . . . . . . . . . . . . . . . . . . . . 44
6.1. Parameters of two Josephson metamaterial devices. . . . . . . . . . . . . . . . 54
ix
List of Figures
1.1. (a) A Josephson junction consisting of two electrodes (in white and grey)
separated by the tunnel barrier; (b) Schematic representation of a Josephson
junction with zero losses; (c) Schematic of a superconducting qubit. . . . . . 4
1.2. Schematic of the tunable superinductor consisting of two large (blue) and two
small (yellow) Josephson junctions per unit cell. α and α′ correspond to the
superconducting phases across the large junctions, and β1 and β2 correspond
to those across the small junctions. . . . . . . . . . . . . . . . . . . . . . . . 6
2.1. Bloch sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2. Schematic of a uxon-parity-protected qubit. . . . . . . . . . . . . . . . . . . 16
2.3. Schematic of the potential wells and the wave functions of the uxon-parity
protected qubit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4. Circuit diagram of a uxon-parity-protected qubit coupled inductively to a
readout resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5. Schematic of the magnitudes of the transmissions through the microwave
transmission line corresponding to |0〉 (red) and |1〉 (blue) states of the qubit
and the bare LC resonator (black). |0〉 and |1〉 states shift the resonant
frequency dispersively by χ0 and χ1 respectively. . . . . . . . . . . . . . . . . 24
3.1. Schematic representation of Josephson junction fabrication using the Man-
hattan pattern technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2. SEM images for junctions with in-plane dimensions (a) 100 × 100nm2 and
(b) 300×300nm2. (c) Variations of normal state resistance among nominally
identical junctions with in-plane dimension 100× 100nm2. . . . . . . . . . . 27
3.3. Schematics of the microwave measurement setup. . . . . . . . . . . . . . . . 29
x
3.4. Photographs of the sample holders. (a) Sample holder for launching from
SMA to microwave stripline and the sample holder (b) Sample holder for
launching from SMA to coplanar waveguide (CPW). . . . . . . . . . . . . . . 31
4.1. Sample design. (a) The schematics of the circuit containing the device and
the read-out lumped-element resonator. The CPB Josephson junctions are
shown as crosses. (b) The layout of the device, the read-out resonator, and
the MW transmission line. The superinductor consists of 36 coupled cells,
each cell represents a small superconducting loop interrupted by three larger
and one smaller Josephson junctions . . . . . . . . . . . . . . . . . . . . . . . 33
4.2. Spectroscopy of the readout resonator around full-frustration of the superin-
ductor loop. At ΦL/Φ0 = 0.5, the superinductor reaches maximum induc-
tance, resulting in a minimum of the readout resonant frequency. . . . . . . . 34
4.3. Panel (a): The transmitted microwave power |S21|2 at the rst-tone frequency
f1 as a function of the second-tone frequency f2 and the gate voltage Vg
measured at a xed value of ΦL = 0.5Φ0. The power maxima correspond to
the resonance excitations of the device (f2 = f01), the superinductor (fL), and
the read-out resonator (fR). Note that the resonance measurements could not
be extended below ∼ 1 GHz because of a high-pass lter in the second-tone
feedline. Panel (b): The frequency dependence of the transmitted microwave
power measured at Vg = 0V and ΦSL = 0.5Φ0. . . . . . . . . . . . . . . . . . 36
xi
4.4. Panel (a): The ux dependence of the device energy levels obtained by nu-
merical diagonalization of the Hamiltonian. The solid curves correspond to
ng = 0.5, the dashed curves - to ng = 0 (the blue curves correspond to the
ground state |0〉, the yellow curves - to the state |1〉, and the green curves
- to the state |2〉). For comparison we also plotted the dotted curves that
correspond to the fully suppressed uxon tunneling; in this case there are
no avoided crossings between the parabolas that represent the superinduc-
tor energies EL(m,Φ) = 12EL(m − Φ
Φ0)2 plotted for dierent m. Panel (b):
The dependences of the resonance frequencies f01 (red dots - ng = 0, red
squares - ng = 0.5) and f02 (blue down-triangles - ng = 0, blue up-triangles -
ng = 0.5 ) on the ux in the device loop. The theoretical ts (solid curves -
ng = 0.5, dashed curves - ng = 0) were calculated with the following parame-
ters: EJ = 6.25 GHz, the asymmetry between the CPB junctions 4EJ = 0.5
GHz, EC = 6.7 GHz, EL = 0.4 GHz (L = (Φ02π )2/EL h 0.4µH), ECL = 5 GHz. 37
5.1. (a) Microphotograph of a portion of the half-wavelength resonator capaci-
tively coupled to the coplanar waveguide transmission line. Light green - Al
ground plane and the central conductor of the transmission line, green - silicon
substrate, black - the central strip of the resonator made of strongly disor-
dered Al. (b) Several resonators with dierent resonance frequencies coupled
to the transmission line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2. The temperature dependences of resonance frequency shift δfTLSr (T )/fr0. . . 43
5.3. The dependencesQi(n) at T ≈ 25 mK for the resonators with dierent widths.
Solid curves represent the theoretical ts of the quality factor governed by
TLS losses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.4. a) The dependences of Qi for resonator #1 on the pump tone power Pp for
several values of detuning ∆f between resonance and pump frequencies. (b)
The values of Qi measured versus detuning ∆f at a xed number of the pump
tone photons in the resonator np ≈ 1000. The error bars are derived from
the covariance matrix obtained from nonlinear tting of the measurement of
S21(f). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
xii
5.5. The time dependence of Re[S21] measured at T = 25 mK at a xed frequency
on the slope of a resonance dip. The microwave power corresponds to 〈n〉 ∼
1000. Each point corresponds to the data averaging over 1 sec. . . . . . . . . 48
5.6. (a) The pulse sequence. (b) The time dependence of |S21| measured at f0 =
2.4258 GHz. The pump pulse at fp = f0 + 1 MHz was applied between
t = 0 s and t = 0.5 s. The pump tone power corresponds to np ≈ 1000. Each
data point was averaged over 4000 cycles with the same readout delay time.
The inset shows CW measurement of S21 versus f with (red) and without
(blue) the pump signal and indicates the position of f0 used in the relaxation
time measurement. The readout power was at the single photon level for all
measurements on this plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.1. Josephson metamaterial based on a chain of coupled asymmetric SQUIDs.
(a) Circuit schematic of the metamaterial. Each unit-cell of the metamaterial
consists of two asymmetric SQUIDs coupled with a shared junction and is of
length a. Each SQUID in the unit-cell is threaded with a magnetic ux Φ and
has a capacitance to ground Cgnd. (b) Illustration of the three-metal-layer
layout of the device. Metal layer M0 (gray) represents the ground plane, M1
and M2 are the two metal layers which form the electrodes of coupled asym-
metric SQUIDs, red and green vias between M1 and M2 represent Josephson
junctions and M1-to-M2 vias respectively. The purpose of the ngers on M0
in gray and M1 in green which extend into the foreground is to increase the
capacitance of the SQUID array to ground (M0). (c) Optical image of the
measured Josephson metamaterial. . . . . . . . . . . . . . . . . . . . . . . . . 55
6.2. Low-power transmission measurements of the phase shift across the Josephson
metamaterial as a function of the magnetic ux for device 1 (lower panel) and
device 2 (upper panel) at dierent measurement frequencies. . . . . . . . . . 56
6.3. Wavenumber as a function of frequency for devices 1 (blue circles) and 2 (red
squares). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
xiii
6.4. Measurements of the microwave phase shift as a function of signal power where
= -70 dBm, at dierent values of the magnetic ux in the metamaterial unit
cells for device 1 (upper panel) and device 2 (middle panel). . . . . . . . . . 57
A.1. Design of hybrid superinductor. . . . . . . . . . . . . . . . . . . . . . . . . . 71
B.1. Increase in transmission power through the metamaterial transmission line. . 72
xiv
1
Chapter 1
Introduction
Quantum computers have the capacity to outperform classical computers in solving hard
problems such as quantum simulations [1, 2], factoring large numbers [3] and indexing un-
structured database [4]. A quantum computer utilizes coherent quantum systems called
quantum bit (qubit) for computation. A qubit provides a quantum superposition of |0〉 and
|1〉 states in contrast to a classical bit which is in either 0 or 1 state. The power of quantum
computing relies on the ability to control the qubits while preserving their coherence. Fault-
tolerant quantum computing can be realized by encoding a logical qubit using many physical
qubits as long as the error rate per physical qubit is below a threshold value [58]. The error
rate is dened as the ratio between the single gate operation time and the qubit coherence
time. The estimates of the threshold for dierent error correction codes very between 10−6
and 10−2. Lower error rates require lower circuit redundancy for a given correction code.
Physical implementations of qubits include trapped ions [9], quantum dots [10, 11],
nuclear [1214] and electron spins [15], photons [16] and superconducting circuits [17] among
others. Trapped ions and superconducting qubits have demonstrated gates with high delity
necessary for the surface error-correction code [18]. Though the delity of single qubit gates
of trapped ions is higher than that of the superconducting qubits, the gate operation time
of superconducting qubits is ∼ 103 times less [19, 20]. Thus the superconducting qubits
compare favorably with the trapped ion qubits for fault-tolerant quantum computing. In
addition, the fabrication and operation of superconducting circuits use technologies already
available in the semiconductor and telecommunication industries.
The circuit redundancy required for universal quantum computing with superconducting
qubits so far remains very high. It is estimated that it would require 103− 104 the state-of-
the-art physical superconducting qubits to realize a single logical qubit. One way to reduce
2
the circuit redundancy is to implement the so-called parity-protected qubits, which have
degenerate ground states corresponding to even and odd parities of Cooper-pairs or ux-
ons [21]. Parity-protected qubits have built-in circuit symmetry to suppress the transition
between dierent parities. So it requires lower circuit redundancy to realize fault-tolerant
Cliord gates using parity-protected qubits than using non-protected qubits.
One of the essential elements for the realization of the uxon-parity-protected qubits is
the so-called superinductor - an inductor with the impedance greater than the resistance
quantum RQ = h/(2e)2 ≈ 6.5kΩ. Superinductors oer a abroad range of applications for
novel quantum circuits. For example, they enable realization of high-impedance environment
and are an important resource for amplication of the amplitude of quantum uctuations
in phase. In this dissertation, we explore two types of superinductors. The rst type of
superinductor is based on Josephson inductance. It consists of a chain of coupled asym-
metric Superconducting Quantum Interference Devices (CASQUIDs). The advantage of
this superinductor is the tunability of its inductance and its nonlinearity by the external
magnetic ux. These superinductors can be used in the circuits where linearity of the in-
ductance is desirable. We studied the prototype uxon-parity-protected qubit consisting of
a Cooper-pair box (CPB) shunted by a CASQUIDs-based superinductor. The spectroscopy
of a prototype device of the qubit demonstrated that the parity of uxons in the loop was
preserved when the oset charge on CPB island was set to e mod (2e). A qubit with similar
design and a even larger superinductance can potentially be used to implement fault-tolerant
Cliord gates. The second type of superinductors is made of the high-kinetic-inductance
superconducting nanowires. We studied superinductors based on high-kinetic-inductance
disordered Aluminum lms. To study the dissipation processes in these lms, we fabri-
cated the superconducting coplanar waveguide (CPW) resonators and measured microwave
losses at ultra-low temperatures. We observed that the intrinsic losses in these resonators
were limited by the resonator coupling to two-level systems (TLS) in the environment. The
demonstrated internal quality factors were comparable to those for CPW resonators made
of conventional superconductors. The disordered Aluminum nanowires are promising for
applications in quantum metrology and quantum information processing.
In addition to fault-tolerant gate operations, fault-tolerant state measurements are needed
3
when performing quantum error corrections to avoid the propagation of errors detrimental
to the quantum information stored in the logical qubit. Fault-tolerant measurements re-
quire amplication of signals with quantum-limited noise level. Since fault-tolerant quan-
tum computing involves measurements of multiple qubits, the amplication also needs to
be broadband. Kerr nonlinearity in metamaterial transmission lines based on Josephson
junctions have been previously utilized to realize Josephson traveling-wave parametric am-
pliers (JTWPA) with nearly quantum-limited noise [22]. In this dissertation, we present
a novel matematerial on the basis of CASQUIDs with strong Kerr nonlinearity and Kerr
constant tunable by the magnetic eld over a wide range from positive to negative values.
The metamaterial is promising as an active medium for quantum-limited JTWPA.
1.1 Superconducting qubit
A Josephson junction consists of two superconducting electrodes separated by an insulating
barrier as illustrated in Fig. 1.1(a). The Al/AlOx/Al Josephson junctions are the most
widely used for superconducting qubits. Cooper pairs can tunnel without dissipation across
the junction. The supercurrent depends nonlinearly on the phase φ dierence between the
electrodes
I = Icsin(φ), (1.1)
where Ic is the critical current. The change in φ with time is associated with a voltage V
across the junction
dφ
dt=
2eV
~. (1.2)
From the two Josephson equations above, we can obtain the inductance associated with the
Josephson junction
LJ(φ) =Φ0
2π(dI
dφ)−1
=Φ0
2πIc cos(φ)
, (1.3)
4
Figure 1.1: (a) A Josephson junction consisting of two electrodes (in white and grey) sepa-rated by the tunnel barrier; (b) Schematic representation of a Josephson junction with zerolosses; (c) Schematic of a superconducting qubit.
where Φ0 is the ux quantum. Thus, a Josephson junction can be viewed as a non-dissipative
element with nonlinear inductance.
A Josephson junction is characterized by two energies. The rst one is the Josephson
energy EJ = ∆RQ/8RN , where ∆ = 1.76kBTc is the superconducting gap, and RN is
the normal state resistance of the junction [23]. The second one is the charging energy
EC = 4e2/2C corresponding to the transfer of one Cooper pair across the junction with
capacitance C. The Hamiltonian of the Josephson junction is
HJ = 4EC n2 − EJcosφ, (1.4)
where n and φ are a conjugate pair of coordinates corresponding to the number of Cooper
pairs stored in the capacitor and the phase across the junction respectively. Figure. 1(b)
shows the schematic representation of a Josephson junction with zero losses.
Figure 1.1(c) shows the schematic of a superconducting qubit consisting of a Josephson
junction shunted by an inductor. The Hamiltonion of the qubit is
H = 4EC n2 − EJcosφ+ ELφ
2, (1.5)
where EL = (Φ0/2π)2/L corresponds to the inductive energy of the inductor. The energy
levels of the system are non-equidistant. The two lowest energy states correspond to the
states |0〉 and |1〉 of the qubit. Various qubit designs corresponding to dierent EJ/EC and
EL/EJ have been explored in the process of improving the qubit coherence time during the
past two decades. A Mendeleev-like table of superconducting qubits can be found in Ref.
5
[24].
Superconducting qubits operate at temperatures < 20mK. At those temperatures, the
numbers of thermally excited quasiparticle and phonon are suppressed, which increases the
coherence time of qubits. Typically the energy dierence between the |0〉 and |1〉 states of
the superconducting qubit are designed to correspond to microwave frequencies < 10GHz.
1.2 Superinductor
If a wire's inductance is geometrical, its impedance Z would be limited by the ne structure
constant (Z ≤ 1/137RQ) since the geometric inductance is associated with energy stored in
electromagnetic elds. The kinetic inductance is not limited the same way for it is associated
with the kinetic energy of the Cooper pairs . Though kinetic inductance exists in normal
metals, it becomes signicant only at very high (terahertz) frequencies since electrons are
scattered on a short time scale. Cooper pairs in the superconducting condensate ow fric-
tionlessly at microwave frequencies [25], thus superinductors realized using superconductors
can operate at microwave frequencies with low losses, making it suitable for applications in
superconducting qubits.
Quantum uctuations of charge and ux uctuations in a LC resonator are related to
the ratio between its characteristic impedance Z0 =√L/C and the resistance quantum RQ
as [26]
δΦ/Φ0 =
√1
4πZ0/RQ (1.6)
δQ/2e =
√1
4πRQ/Z0. (1.7)
Therefore, superinductors allow quantum uctuations in ux larger than Φ0 and quantum
uctuations in charge less than 2e.
The kinetic inductance per unit length of a superconducting nanowire is
6
LK =m
2e2Ans, (1.8)
where A is the area of the cross section of wire, ns is the density of Cooper pairs and m is
the mass of an electron. Conventional superconductors have large kinetic inductance when
the current approaches the critical value because the density of electrons decreases. But
they have large losses for the density of normal electrons increases at the same time.
Superinductors can be realized using nanowires of disordered superconducting materi-
als, such as TiNx, NbN, and granular Aluminum, and using Josephson junction arrays (see
Eq. 1.3 for Josephson inductance of a single junction). Each approach has its advantages
and disadvantages. Superinductors made of Josephson arrays typically have larger in-plane
dimensions than that of nanowire-based superinductors. This results in a greater stray ca-
pacitance of Josephson arrays and lower frequencies of the self-resonance modes. On the
other hand, fabrication of compact superinductors based on nanowires requires enhanc-
ing the strength of disorder by approaching the disorder-driven superconducting-insulating
transition (SIT). Soft modes that might appear near the SIT may aect detrimentally the
performance of these superinductors by increasing losses [27]. Also, it is challenging to
reproducibly fabricate strongly disordered lms near the SIT.
1.3 A chain of coupled asymmetric SQUIDs (CASQUIDs)
Figure 1.2: Schematic of the tunable superinductor consisting of two large (blue) and twosmall (yellow) Josephson junctions per unit cell. α and α′ correspond to the superconduct-ing phases across the large junctions, and β1 and β2 correspond to those across the smalljunctions.
7
Figure 1.2 shows the schematics of a chain of coupled asymmetric Superconducting Quan-
tum Interference Devices (CASQUIDs). Each unit cell consists of two large junctions with
Josephson energy EJL (blue) and one small junctions with Josephson energy EJS (yellow),
where EJL > EJS . Two adjacent unit cells are coupled through a common large junction.
The large junctions form the meandered backbone for the chain of junctions.
The CASQUIDs is translationally invariant. The structure stays the same when shifted
horizontally by one unit cell and then mirror-reected with respect to the horizontal axis. In
Figure. 1.2, α and α′ correspond to the phase across the two large junctions, and β1 and β2
correspond to that across the two small junctions. Due to the transnational symmetry, the
phase dierence across the two large junctions within one unit cell are the same (α = α′).
Assuming the positive current ows clockwise, the phase across the junctions satises the
following constraints
α+ α′ + β1 =2π
Φ0Φ, and (1.9)
α+ α′ + β2 = −2π
Φ0Φ, (1.10)
where Φ corresponds to the external ux through the loop. By dening α+α′ ≡ ϕ, we have
β1 =2π
Φ0Φ− ϕ, and (1.11)
β2 = −2π
Φ0Φ− ϕ. (1.12)
In the classical limit where the charging energy is ignored 1, the total energy of the CASQUIDs
is
EJ(α) = −2EJLcos(ϕ
2)− EJScos(ϕ−
2π
Φ0Φ)− EJScos(ϕ+
2π
Φ0Φ). (1.13)
1The discussion in this section does not include the quantum eect by taking into consideration thecharging energy. Interested readers can refer to Ref. [28] for the discussion with quantum eects considered.
8
The inductance per unit cell for this CASQUIDs is
LJ(ϕ,Φ) =
(∂E2
J(ϕ)
∂2ϕ
)−1
(1.14)
=Φ0
2πIJS
([r
2+ 2 cos
(2π
Φ
Φ0
)]−[r
16+ cos
(2π
Φ
Φ0
)]ϕ2
)−1
. (1.15)
where r = EJL/EJS and IJS is the critical current corresponding to the large junction. The
linear and nonlinear part of inductance depend on r and are tunable by the external ux Φ
through the unit cell. For a certain r, the inductance increases and reaches the maximum
as the external ux increases from zero to full frustration (Φ = Φ0/2). For an innitely long
chain of CASQUIDs, the maximal inductance of the chain at full frustration correspond
to r = 4. The tunable inductor made of CASQUIDs have demonstrated an increase in
inductance by 1-2 orders of magnitude with the ux increases from zero to full frustration
[28].
In the superinductors realized using the CASQUIDs, the junctions of the backbone is
designed phase-slip-free with Josephson energy much greater than the charging energy as
the phase slip rate across a Josephson junction is ∼ exp(−√
8EJ/EC). In the experiment
shown in Ref. [28], EJS/EJC ∼ 100. The backbone of CASQUIDs can be replaced by
a nanowire made of disordered superconducting materials. In this case, the rst term in
Eq. 1.13 is replaced by the inductive energy ELϕ2/2 associated with the inductance of the
segment of the nanowire within one unit cell. When EL = 2EJS , the inductance at full
frustration reaches the maximal value.
1.4 Fluxon-parity-protected qubit
A uxon-parity-protected qubit consists of a Cooper pair box shunted by a superinductor
made of a chain of CASQUIDs. The uxon-parity-protected qubit is an example of physical
systems with built-in error correction. The Josephson energy EJ and the charging energy
EC of the junctions in the CPB satisfy the condition EJ > EC . The dynamics of the low
energy states of the qubit correspond to uxons tunneling across the Josephson junctions
in the CPB in and out of the superconducting loop. Each uxon is associated with phase
9
slips that change the superconducting phase across the CPB by an integer number of 2π.
The uxon tunneling process is dual to the Josephson eect that corresponds to Cooper
pairs tunnel cross the insulating barrier separating the two electrodes. The uxons that
change the phase by even and odd numbers of 2π are of even and odd parities respectively.
When the oset charge on the island of the CPB is emod(2e), the phase slip process that
change the phase across the CPB by 2π is suppressed due to Aharonon-Casher eect. The
lowest two energy states |0〉 and |1〉 of the qubit are superpositions of states associated with
uxons of even and odd parities respectively. Since the 2π phase slips are suppressed, so
is the transition between the states of even and odd parities. The qubit is thus protected
from energy decay. Meanwhile, the double phase slips corresponding to simultaneous 2π
phase slips across each junction in the CPB are allowed. When the double phase slip rate
is much greater than the energy dierence between the uxons separated by 4π, each qubit
state corresponds to a superposition of multiple uxon states of the same parity. When
the condition is met, the energy dierence between the two qubit states becomes negligibly
small and insensitive to charge and ux noises in the environment to the rst order. The
qubit is thus protected from pure dephasing.
Fault-tolerant Cliord gates can be performed by using certain types of nonlinear cou-
pling between the uxon-parity-protected qubit and the control channels. For example,
the fault-tolerant π/4 phase gate can be realized through quadratic coupling by tuning the
magnetic eld through the loops in the superinductor based on CASQUIDs [21]. Because of
the inherent properties of the Hamiltonian of the protected-qubit, it requires less eort to
actively check and correct for errors than that involved in fault tolerant operations realized
using non-protected qubit [29].
1.5 Josephson traveling-wave parametric amplier based on CASQUIDs
A qubit state is usually probed by the dispersive measurement using a high-Q readout
resonator. The qubit coupling to the resonator introduces a shift in the resonant frequency.
The state of the qubit can be derived by measuring the transmission of the readout resonator
at a xed frequency close to its resonance. To minimize the perturbation of qubit due to
readout signal, the readout resonator is usually measured with power corresponding to
10
the single photon level. For a critically coupled resonator of frequency fr = 6 GHz and
Qi = 10, 000 , this corresponds to microwave power ∼ −135 dBm = 4πhf2r /Qi.
Multiple stages of amplication are required for processing such a low-power signal. It
is necessary to amplify the signals at low temperatures before it reaches the room tempera-
ture pre-amps. Fault-tolerant operations require low temperature ampliers with quantum-
limited noise level to perform operations such as high delity measurement of supercon-
ducting qubit state and qubit entanglement [30]. The commercially available semiconductor
based high electron mobility transistor (HEMT) ampliers are able to amplify the signals at
4K, but would introduce noise corresponding to 10-20 photons due to thermal noise in the
transistor. Parametric ampliers based on Josephson junctions, rst developed by Yurke et
al. at Bell labs [31], are able to meet the requirement for fault-tolerant operations.
In a parametric amplier, a weak input signal is amplied through interacting with a
strong pump signal that modulates certain parameters of the nonlinear medium, such as the
index of refraction in nonlinear optical systems. There are mainly two types of parametric
ampliers based on Josephson junctions: the Josephson parametric amplier (JPA) and the
Josephson traveling-wave parametric amplier (JTWPA). The JPA relies on high-quality
resonators to prolong the interaction time between signals in the nonlinear medium [32
35]. As a result it has a limited bandwidth and dynamic range. The narrow bandwidth is
a limitation for multiplexing when a large number of qubits are measured simultaneously.
The JTWPA overcomes the limit by allowing interacting signals to propagate for a distance
much longer than their wavelength.
In a JTWPA, parametric amplication is realized in a transmission line that couples a
strong pump tone (ωp) to a weak signal (ωi) and an idler (ωi) via a degenerate four-wave
mixing process such that 2ωp = ωi + ωs. When the phase matching condition between the
three waves
∆k = 2kp − ks − ki = 0 (1.16)
is met, where kp, ks, and ki are the wave vectors of the pump, signal and idler respectively,
the gain of the signal ωi grows exponentially with the length of the transmission line.
11
The wave vector k of any signal transmitted through Josephson-junction-based transmis-
sion lines consists of a power independent part kL and a power dependent part kNL. kNL
is proportional to the intensity of the electromagnetic eld ∝ |E|2 due to Kerr eect similar
to that in optic bers [36]. For example, in the case of two dierent signals E1e−iω1t and
E2e−iω2t traveling simultaneously in the nonlinear transmission line, the power dependent
part of the wave vector of the signal at ω1 is modulated by the intensities of signals at ω1
and ω2
kNL(ω1) ∝ K(|E1|2 + 2|E2|2), (1.17)
where K is the Kerr constant. The two modulations are called self-phase modulation (SPM)
and cross-phase modulation (XPM).
In a chain of Josephson junctions, ∆kL = 2kLp − kLs − kLi > 0 and ∆kNL = 2kNLp −
kNLs − kNLi > 0 [37, 38]. At low pump power, both terms are close to zero. As the pump
power increases, ∆k = ∆kL + ∆kNL < 0. One method to achieve the phase matching
condition called resonant phase matching (RPM) is to modify the dispersion of the pump
tone by periodically loading the chain with resonators with the resonant frequencies close
to the pump frequency [22, 38]. The nonlinear dispersion of the pump close to the resonant
frequency compensates for the phase mismatch. For the JTWPA based on RPM, the pump
frequency can only be varied over a small range (∼ 100MHz ) around the resonant frequency
of the loaded resonators, and the signals at frequencies within the range can not be amplied.
In the transmission line made of a chain of CASQUIDs, the Kerr constant is tunable with
magnetic eld and even changes its sign. ∆kNL, which is proportional to the Kerr constant,
can be tuned to positive to compensate ∆kL. The pump frequency can be varied over a
broad range in the JTWPA based on the transmission line. Numerical studies have shown
that amplication can be realized for dierent pump frequencies in the range of 1-9GHz in
the CASQUIDs based JTWPA [39]. In addition, the nonlinearity in a chain of CASQUIDs
facilitates stronger interaction between a pump (ωp) and a signal (ωi) than that in a chain
of Josephson junctions. A JTWPA based on CASQUIDs is able to achieve comparable gain
as that based on RPM at shorter length. It is estimated to achieve 20dB gain with a length
12
∼ 650µm [39] , while it requires ∼ 3.3cm using RPM for similar gain [22].
13
Chapter 2
Fluxon-parity-protected qubit: theoretical background
2.1 Single-qubit state and coherence
A qubit state can be expressed as
|ψ〉 = cos(θ
2)|0〉+ eiφ sin(
θ
2)|1〉. (2.1)
with θ ∈ [0, π] and φ ∈ [0, 2π]. It can also be represented using the density matrix ρ = |ψ〉〈ψ|
ρ =1
2(I + axσx + ayσy + azσz), (2.2)
where σx, σy and σz are the Pauli matrices and ax = sin θ cosϕ, ay = sin θ sinϕ and az =
cos θ are their expectation values respectively. The qubit state can be represented by the
vector (ax, ay, az) called the Bloch vector within the sphere of unit radius called the Bloch
sphere. Figure. 2.1 shows the schematic representation of a Bloch sphere (black) and a
Bloch vector (blue).
For a qubit initialized at the excited state |1〉, its Bloch vector is of unit length and
pointing vertically downward. Due to the coupling between the qubit and the noise in
the environment, the Bloch vector decay to the position corresponding to the qubit state
at thermal equilibrium. The longitudinal and transversal decays are characterized by T1
and T ∗2 , called energy relaxation time and pure dephasing time respectively. The energy
relaxation is related to the noise at qubit frequency that is able to induce the transition
between |0〉 and |1〉 state. And the pure dephasing is due to the low frequency noise that
induces random change in the energy dierence between |0〉 and |1〉 state. The overall decay
14
Figure 2.1: Bloch sphere.
process is characterized by the coherece time T2
1
T2=
1
2T1+
1
T ∗2. (2.3)
2.2 Parity-protected qubit 1
In the quantum error-correction code (QECC) [58], a single logical qubit is encoded in
multiple physical ones. The logical qubit reside in the subspace corresponds to mutually
commuting operators Oα. Oα have eigenvalues +1 and −1. Eigenvalues of Oα cor-
responding to the code space are +1. The operators Oα are chosen such that any error
would project the logical qubit from the code space to its orthogonal space where at least one
of the eigenvalues of Oα corresponding to the logical qubit becomes −1. Oα are called
error syndromes for its capability of diagnosing the errors. By performing error syndrome
measurement, we are able to check for errors before applying corrections.
The process of implementing the QECC can be simplifed by designing a physical qubit
corresponds to the Hamiltonian H = −∑
α ∆αOα with degenerate ground states. The
degenerate ground states correspond to the code space. In order for the ground state to be
degenerate, additional symmetries Pβ need to be satised by H, [H,Pβ] = 0, and Pβ
does not commute with Oα. In addition, the energy required to excite the qubit from
1This section summarizes the discussions in Ref. [21] relevant to the purpose of this work.
15
the degenerate ground states to the excited states is larger than the thermal noise in the
environment (∆α kBT ) so that the error rate is suppressed.
In the phase space corresponding to a pair of canonical coordinates φ and q,
[cos(2πq), cos(mφ)] = 0, (2.4)
where m is an integer. We can use the two operators as error syndromes. We detect any
errors in φ and q by measuring cos(mφ) and cos(2πq) respectively. For the Hamiltonian
H = −Eqcos(2πq)− Eφcos(mφ), (2.5)
the lowest energy states are m-fold degenerate. The wave functions presented in the coor-
dinate representation are
ψs(φ) =∑k
δ[φ− 2π(k +s
m)], (2.6)
where s are integers between 0 andm−1 and k is summed over all integers. In the momentum
representation, the wave functions are
ψs(p) = exp(−2πiqs/m)∑l
δ(q − l). (2.7)
By embedding the system in a harmonic oscillator, we are able to create a energy gap
that separates the degenerate ground states from all higher excitations. Such a system
correspond to the Hamiltonian
H = −Eq cos(2πq)− Eφ cos(mφ) + 1/2V (t)φ2 +1
2m∗q2. (2.8)
When V (t) 1 and Eφm∗ 1, the coordinate φ are conned to the discrete values
2πk/m, where k is integer, corresponding to the maximum of cos(mφ). The ground states
are separated from excited states by√Eφ/m∗. Denote the discrete states as |k〉. The low
16
energy states of the system expressed in this basis becomes
H = −1/2Eφ (|k +m〉〈k|+ |k〉〈k +m|) +1
2V (t)
(2πk
m
)2
. (2.9)
When m = 2, the ground states are two fold degenerate corresponding to k at even and
odd numbers. They form the code space for the parity-protected qubit. When V (t) 1, the
qubit is decoupled from noise as well as external control and becomes a quantum memory.
But certain fault tolerant Cliord gates, such as the π/4 gate, can be performed by varying
V (t) [21].
2.3 Fluxon-parity-protected qubit
Figure 2.2: Schematic of a uxon-parity-protected qubit.
Figure 2.2 shows the schematic of the uxon-parity-protected qubit. The qubit consists
of a Cooper-pair Box (CPB) shunted by a superinductor. The CPB consists of two nomially
identical Josephson junctions separated by a superconducting island. EJ and EC correspond
to their Josephson and charging energies of the junctions respectively. The phase φ across
the CPB and the oset charge q associated with the superinductor are a conjugate pair of
coordinates [q, φ] ∼ i. We choose the gauge so that the phase at the two ends of the CPB are
17
±φ/2. The number of the charge n and the phase ϕ on the CPB island are also a conjugate
pair of coordinates [n, ϕ] ∼ i. ng is the oset charge on the CPB island controlled by the
external DC voltage bias. In the following discussion, φ and ϕ are the phases normalized by
Φ0/2π and q, n and ng are the charges normalized by 2e.
The CPB Hamiltonian is expressed as
HCPB = −EJcos(ϕ+ φ/2)− EJcos(ϕ− φ/2) + 4Ec(n− ng)2, (2.10)
= −2EJcos(ϕ)cos(φ/2) + 4Ec(n− ng)2. (2.11)
In the presence of the external magnetic eld Φ, the phase cross the superinductor cor-
responds to φ − 2πm0, where m0 = Φ/Φ0. The Hamiltonian of the superinductor with
inductive energy EL and charging energy ECL is expressed as
HSI =1
2EL(φ− 2πm0)2 + 4ECLq
2. (2.12)
EL is dened as (Φ0/2π)2(1/LSI), where LSI is the inductance of the superinductor. Its
charging energy ECL is associated with its parasitic capacitance to the ground. In the basis
of charging states |n〉, the CPB Hamiltonian is expressed in matrix form as
HCPB =
. . . −EJcos(φ/2) · · · · · · · · ·
−EJcos(φ/2) 4EC(N − 1− ng)2 −EJcos(φ/2) · · · · · ·... −EJcos(φ/2) 4EC(N − ng)2 −EJcos(φ/2) · · ·...
... −EJcos(φ/2) 4EC(N + 1− ng)2 · · ·...
......
.... . .
,
(2.13)
where N is an integer. When ng = N + δN , where δN ∈ [0, 1), in the limit of EJ < EC , the
eect of the transition between |n〉 and |n + 1〉 is signicant when n ' N . The low energy
18
subspace of the CPB is spanned by |N〉 and |N + 1〉 and the corresponding Hamiltonian is
HCPB =
−4EC(0.5− δN)2 −EJcos(φ/2)
−EJcos(φ/2) 4EC(0.5− δN)2
. (2.14)
When δN = 0.5, the Hamiltonian of the CPB when represented in the basis of 1/√
2(|N〉+
|N + 1〉), 1/√
2 (|N〉 − |N + 1〉) becomes
HCPB =
−EJcos(φ/2) 0
0 +EJcos(φ/2)
. (2.15)
The total Hamiltonian H = HCPB +HSI becomes
H =
H− 0
0 H+
(2.16)
where
H− = 4ECLq2 − EJcos(φ/2) +
1
2EL(φ− 2πm0)2, and (2.17)
H+ = 4ECLq2 + EJcos(φ/2) +
1
2EL(φ− 2πm0)2. (2.18)
H± can be considered as a system where a ctitious particle of mass 1/4ECL is moving in
the potential wells
V±(φ) = ±EJ cos(φ/2) +1
2EL(φ− 2πm0)2. (2.19)
The top plot in Figure 2.3 shows the schematic for V±(φ) with EJ EL. When EJ ?
ECL EL, the ground states of H± are localized around the minimums of ±EJ cos(φ/2).
Phenomenological, the localized states correspond the instantons called uxons. A uxon
state |m〉 corresponds to the state with phase dierence φ = 2πm (m ∈ Z) across the CPB.
Even and odd m correspond to uxon states of even and odd parities respectively. The |0〉
and |1〉 states of the qubit correspond to superpositions of uxons of even and odd parities.
19
Figure 2.3: Schematic of the potential wells and the wave functions of the uxon-parityprotected qubit. (a) Schematic of the potential wells V (φ)± with with EJ EL. Theinset shows the condition required for a uxon-parity-protected qubit. The energies of theuxons are separated from the energies corresponding to higher excitations by the plasmafrequency ωp. The energy dierence between the uxon states conned in two adjacentpotential minimums is ∼ 8π2EL. The two uxon states are coupled through the doublephase-slip process at the rate Edps. The uxon-parity-protected qubit correspond to thecondition when ωp Edps 8π2EL. (b) The wave functions for |0〉 and |1〉 states of theuxon-parity-protected qubit at m0 = 0 and ng = emod(e). The wave functions of |0〉and |1〉 states are superpositions of wave packages localized at even and odd numbers of 2πrespectively along φ, corresponding to states of even and odd parities.
20
The energy of uxons is separated by the plasma frequency ωp =√
2EJECL from the higher
excitations, the plasmons. The transition between two uxons |m〉and |m′〉 corresponds to
the phase-slip that change φ by 2π(m−m′). The 2π phase slip across the CPB corresponding
to transition from |m〉 and |m± 1〉 is suppressed when the oset charge on the CPB island
is emod(2e) due to the Aharonov-Casher eect [40]. The double phase-slips that change the
phase across the CPB by 4π become the dominant transition process. We will see later that
the description using uxons is only valid when we shunt the CPB with a superinductor.
The dynamics of the uxon states correspond to even or odd parities can be expressed
using the Hamiltonian 2
H = −Edps|m〉〈m+ 2| − Edps|m+ 2〉〈m|+ 2π2EL(m−m0)2, (2.20)
where the double phase-slip rate expressed in terms of g = 4√
2EJ/ECL and ωp =√
2EJECL
is
Edps ∼ g1/2exp(−g)ωp. (2.21)
The double phase slip rate is equivalent to the energy splitting corresponding to the particle
with kinetic energy 4ECLq2 tunneling between two minimums of the potential well V (φ). We
see from Eq. 2.20 that in order for the description using uxons to be valid, EL needs to be
less than the resonant frequency of the inductor that shunts the CPB. This is equivalent to
that the impedance of the inductor is greater than resistance quantum. So the superinductor
is necessary for uxon states to exist in the qubit.
The dependence of the energy dierence E01 between even and odd parity on m0 is
E01(m0) ∝
(√8Edpsπ2EL
)1/2
exp
(−
√8Edpsπ2EL
)√2EdpsEL| cos(4πm0)|. (2.22)
At full frustrations, |0〉 and |1〉 qubit states are degenerate and their energy dierence is
2The description of the system based on Josephson junction chains by instantons is similar to that inRef. [41].
21
rst order sensitive to the ux noise. At zero frustration, the energy dierence is non-zero
but rst order insensitive to the ux noise. When the amplitude of the double phase slips is
much greater than the energy between two uxons separated by δm = 2, Edps/8π2EL 1,
the two qubit states become almost degenerate. The qubit is thus protected from pure
dephasing at zero frustration.
When the oset charge on the CPB island is emod(2e), the charge noise induces the
nite coupling between |0〉 and |1〉 at the rate
t01 ∼ g1/4EC(δN − 0.5)√Edps/ωp. (2.23)
The eect is small when Edps ωp. Under this condition, the qubit is protected from
energy relaxation.
The condition to protect the qubit from decoherene, which includes the energy relaxation
and the pure dephasing, is ωp Edps 8π2EL [schematically shown in the inset in Figure
2.3(a)]. Under this condition, the double-phase slip rate is high enough for the mixing of
uxons of the same parity, such as |m − 2〉 and |m〉. Meanwhile it is much lower than
the plasma frequency to avoid mixing uxons with plasmons. The bottom plot in Figure.
2.3 schematically shows the wave functions corresponding to the protected qubit states
represented in the basis of φ. The wave functions are superpositions of wave packages
localized at positions corresponding to even and odd numbers of 2π. They share the same
Gaussian envelope corresponding to the ground state of the harmonic oscillator with respect
to the superinductor. The fault tolerant π/4 gate can be realized with the protected qubit
by varying the inductance of the superinductor adiabatically over a certain time while the
qubit still satises the aforementioned condition [21].
2.4 Inductive coupling of uxon-parity-protected qubit and readout res-
onator
The state of the uxon-parity-protected qubit can be determined dispersively through the
inductive coupling to the readout LC resonator, which is coupled inductively to a microstrip
22
Figure 2.4: Circuit diagram of a uxon-parity-protected qubit coupled inductively to areadout resonator.
transmission line. Figure 2.4 shows the circuit diagram corresponding to the uxon-parity-
protected qubit coupled to the resonator through the inductor Lc. The Hamiltonian of the
system can be divided into three parts: the resonator, the qubit and their coupling
H = Hres +Hqubit +Hcouple. (2.24)
We express Hres in terms of the creation a† and annihilation a operators of photons as
Hres =1
2~ωra†a(n+ 1), (2.25)
where n is the number of photons and ωr is the angular frequency of the resonator. The
Hamiltonian of the qubit with transition energy ω01 between |0〉 and |1〉 can be expressed
as
Hqubit = ~ω01σz. (2.26)
Let's dene the current Iq ows counter-clockwise around the qubit loop and the cur-
rent Ir ows clockwise around the resonator loop as shown in Fig. 2.4. The Hamiltonian
corresponding to the coupling between the qubit and the readout resonator is
Hcouple = LcIrIq. (2.27)
The phase around the qubit loop satisfy the constraint imposed by the magnetic ux Φext
23
through the loop
Φ0
2πφ+ LcIr + (Lc + LSI)Iq = Φext, (2.28)
where Lc and LSI are the inductance of the coupler and superinductor respectively. Re-
placing Iq in Eq. 2.27 using Eq. 2.28 and keeping terms containing parameters of both the
qubit and the LC resonator, we obtain
Hcoupler = −Φ0
2π· LcLc + LSI
φIr. (2.29)
The current Ir can be expressed in terms of a† and a as
Ir = −√
~ωr2L
(a− a†). (2.30)
Combine Eq. 2.30 and 2.29, we have
Hcoupler = ~ωrLc
Lc + LSI
√RQ
4πZ0φ(a− a†), (2.31)
where Z0 =√L/C is the characteristic impedance of the resonator. Dene the coupling
constant
g = ~ωrLc
Lc + LSI
√RQ
4πZ0. (2.32)
Hcoupler is expressed as
Hcoupler = gφ(a− a†). (2.33)
Next we discuss the dispersive shift of the resonant frequency for dierent qubit states.
Dene |i, n〉 as the eigenstate for Hres +Hqubit with energy E0i,n = Ei + 1/2~ωr(n+ 1) when
the qubit is in state i with energy Ei and there is n number of photons in the resonator.
24
Figure 2.5: Schematic of the magnitudes of the transmissions through the microwave trans-mission line corresponding to |0〉 (red) and |1〉 (blue) states of the qubit and the bare LCresonator (black). |0〉 and |1〉 states shift the resonant frequency dispersively by χ0 and χ1
respectively.
When the coupling is weak (g |ωr−ω01|), Hcoupler perturbs E0i,n by a small amount δEi,n
resulting in the dispersive shift χi = Ei,n+1 − Ei,n (Ei,n = E0i,n + δEi,n) of the resonant
frequency of the LC resonator. Fig. 2.5 shows a schematic example of the transmission
magnitude through the transmission line corresponding to |0〉 (red) and |1〉 (blue) states of
the qubit and the bare LC resonator (black). The qubit states |0〉 and |1〉 shifts resonant
frequency by χ0 and χ1 respectively. We are able to monitor the qubit state by measuring
the transmission at a xed frequency that corresponds to dierent transmission magnitudes
when the qubit is at |0〉 and |1〉. Note that the ideally protected states of the uxon-
parity-protected qubit is decoupled from linear coupling (〈0|φ|1〉 = 0). Nonlinear coupling
is required to operate and monitor the qubit at the protected state. For the experiment
presented in Chapter 4, since the superinductance is small compared to that in a ideally
protected qubit, we are still able to detect the state through the coupling method discussed
here.
25
Chapter 3
Experimental techniques
3.1 Sample fabrication
3.1.1 Josephson junction fabrication technique used for uxon-parity-protected
qubit fabrication
Al/AlOx/Al Josephson junctions were used for the sample of the uxon-parity-protected
qubit discussed in Chapter 4. The junctions were prepared using the Manhattan pattern
technique. In this technique, the metal deposition mask is formed from a bilayer e-beam re-
sist with an undercut created during the lithography. It allows bridge-free e-gun depositions
of top and bottom electrodes from two directions separated by 90 degrees.
The layout of the junction was designed using computer-aided design (CAD) as two in-
tersecting lines [see Fig. 3.1(a)]. E-beam lithography was used to pattern the the lift-o
mask for metal deposition according to the designed layout on top of the substrate. The
lift-o mask is made of a bilayer e-beam resist consisting of a 350-nm-thick poly(methyl
methacrylate) (PMMA) layer on top of a 100-nm-thick copolymer Methyl methacrylate
(MMA) layer [see Fig 3.1 (b) ]. The lithography was performed using a scanning electron
microscope (SEM). In the SEM, electrons scissor molecular chains by interacting with poly-
meric molecules in the e-beam resist. The blue region in Fig. 3.1(c) indicates the exposed
volume in the resist. The copolymer MMA has shorter molecular chains than does the
PMMA and is thus more sensitive to the exposure of electrons. As a result, an undercut was
formed at the bottom of the trench after the development [see Fig. 3.1(d)]. Following the
development, the sample was put into an Ozone asher for 5 minutes to remove the e-beam
resist residue on the substrate surface before metal deposition. Afterwards, the sample was
loaded into an oil-free high vacuum chamber with a bass pressure 4e-8 mTorr for metal
26
Figure 3.1: Schematic representation of Josephson junction fabrication using the Manhattanpattern technique. (a) The layout of the Josephson junction. (b-d) The e-beam lithographyprocess using a bilayer e-beam resist. (e-g) Metal deposition. (h) The cross section of aJosephson junction after metal lift-o.
deposition. The Aluminum electrodes were deposited at 45 degrees with respect to the sur-
face at rate 1A/s. Figure 3.1(e) shows the schematic representation of the deposition for
horizontal and vertical trenches labeled A and B respectively. The two black arrows indicate
the deposition directions. When the Aluminum was deposited along trench A [see Fig. 3.1
(f)], an electrode would form on the substrate only in trench A as long as the trench width
is less than the PMMA thickness [see Fig. 3.1 (f) and (g)]. The deposition along trench
B works similarly. We can start the deposition along either trenches. After depositing the
bottom Aluminum electrode, it was oxidized with dry oxygen without breaking vacuum be-
fore depositing the top electrode. The oxygen pressure and oxidation time was set to meet
the requirement for the normal state resistance of the Josephson junction. The thicknesses
for the top and bottom electrodes were 20nm and 60nm respectively. After deposition, the
device was passivated with 100 torr dry oxygen for 5 minutes before exposed to atmosphere.
27
The e-beam resist was then lift-o by by soaking the sample in the solution of Microchem
PG remover for 20 minutes at 80C. Figure 3.1 (h) shows the cross section of the junction
after lift-o.
By using the Manhattan pattern technique, we are able to fabricate Josephson junctions
with in-plane areas of 100×100nm2−300×300nm2 [See SEM images for Josephson junctions
in Fig. 3.2(a-b)]. Standard deviation of normal state resistance is ∼ 2.4% for junctions of
area 100 × 100nm2 [See Fig. 3.2(c)]. The variations of the in-plane areas for nominally
identical junctions are < 20%. The small spreading of the normal state resistance and of
the in-plane areas in Josephson junctions results in small variations in the Josephson energy
EJ and the charging energy EC among the nominally identical junctions. This is important
for realizing the uxon-parity-protected state since the CASQUIDs based superinductor
achieves the maximal superinductance when the unit cells are simultaneously frustrated and
the complete suppression of the single uxon tunneling requires identical phase slip rate
∼ exp(−√
8EJ/EC
)across each junction in the Cooper-pair box (CPB).
Figure 3.2: SEM images for junctions with in-plane dimensions (a) 100 × 100nm2 and (b)300×300nm2. (c) Variations of normal state resistance among nominally identical junctionswith in-plane dimension 100× 100nm2.
3.1.2 Deposition of disordered granular Aluminum lms 1
The standard method for the fabrication of disordered Al lms is the deposition of Al at a
reduced oxygen pressure [42, 43]. Such lms consist of nanoscale grains (3−4 nm in diameter)
partially covered by AlOx. The disordered granular Aluminum lms in the sample discussed
1This section is based on the supplemental material published in W. Zhang et al., Phys. Rev. Applied11, 011003 (2019).
28
in Chapter 5 were fabricated by DC magnetron sputtering of a 6N-purity Al target in the
atmosphere of Ar and O2. The base pressure for the sputtering system is < 1× 10−6mbar.
Typically, the partial pressures of Ar and O2 were 5× 10−3 mbar and (3÷ 7)× 10−5 mbar,
respectively. In order to improve reproducibility, prior to the disordered Al deposition the
target was pre-cleaned in a pure Ar-plasma by sputtering at a rate of 0.6 nm/s for 5 minutes.
The reactive DC sputtering of disordered Al was then initiated by introducing 1 sccm O2
and 115 sccm Ar gas mixture from two independent feedback-controlled mass ow meters
(MicroTrakTM and SmartTrakTM). By controlling the deposition rate and O2 pressure, the
resistivity of the lms can be tuned between 10−4 Ω·cm and 10−1 Ω·cm.
3.2 Measurement setup2
The experiments discussed in this work were performed using a BlueFors dilution refrig-
erator (DR) rated 200 µW at 100mK with a base temperature of ∼25 mK. The DR was
outtted with microwave coaxial cables from room to base temperature to perform mi-
crowave transmission measurement. A µ − metal shield was installed outside the cryostat
to reduce background magnetic elds. Fig. 3.2(a) shows the measurement setup for the
experiment on the uxon-parity-protected qubit discussed in Chapter 4. Inside the cryo-
stat, coaxial cables with stainless steel inner and outer conductor were used for wiring
inside the DR to transmit signals between the stages with dierent temperatures. Atten-
uators and low-pass lters were installed in the microwave input line to prevent leakage
of thermal radiation into the sample. The signal was amplied by a high-electron mo-
bility transistor (HEMT) amplier (Caltech CITCRYO 1-12, 35 dB gain between 1 and
12 GHz) at 4K before reaching room temperature and further amplied by two ampli-
ers with a total of 60 dB gain. Two Pamtech isolators (each provides 18dB isolation
between 3 and 12 GHz) were anchored at base temperature immediately after the device
to reduce the noise from ampliers. At room temperature, the probe signal at f and the
pump signal at fp, generated by two microwave synthesizers, were coupled to the input
of the cryostat through a directional coupler. At the output of the cryostat, the probe
2This section is based on the supplemental material published in W. Zhang et al., Phys. Rev. Applied11, 011003 (2019).
29
Figure 3.3: Schematics of the microwave measurement setup.
30
signal was downconverted to the intermediate frequency (IF) fIF = |f − fLO| ≈ 30MHz
using mixer M1 with the local oscillator signal fLO. The IF signal was acquired using
AlazarTech ATS 9870 at 1GS/s sampling rate. The magnitude and phase of the signal S21
was obtained by digital demodulation as a =√
(〈a2(t) sin2(2πft)〉+ 〈a2(t) cos2(2πft)〉) and
φ = arctan(〈a2(t) sin2(2πft)〉/〈a2(t) cos2(2πft)〉) − φ0 (here 〈...〉 stands for the time aver-
aging over integer number of periods, typically 106). The reference phase φ0 was provided
by mixer M2. The setup enables two-tone (pump-probe) spectroscopy and time-domain
measurements with microwave powers corresponding to 1− 1000 photon levels. A DC-gate
line was used for measuring the uxon-parity-protected qubit to tune the oset charge on
the CPB island. The DC-gate line was ltered with RC lters at the room temperature
and with a stainless-steel powder lter at the base temperature. A Niobium wire was used
to connect between the 4 K and the 700 mK anges. The DC and microwave signals are
combined using a bias-Tee at base temperature. The experiments on the granular aluminum
lms discussed in Chapter 5 and on the one-dimensional metamaterial discussed in Chapter
6 used the same setup except for the removal of DC-gate line [see Fig. 3.2(b)].
3.3 Sample holders
The samples were connected using wirebonds to RF-tight sample holders made of oxygen-free
high thermal conductivity (OFHC) copper. The sample holders contain SMA feedthroughs
and printed circuit boards (PCB) to route the signals from SMA connectors to the chips.
Two sample holders were designed to house samples with dierent types of input and output
ports. The sample holder shown in Fig. 3.3(a) was used for the sample with ports made of
microstrip lines. A 7 × 7mm2 square recess (left) was used to house the chip so that the
surface of the chip ush with the surface of the CPB. A plate with a rectangular window
(middle) was screwed on top of it to press the chip against the ground. The plate was
designed to be thick enough to prevent the bonding wires from touching the lid (right) but
not too thick that would induce parasitic modes within the frequency range of interest.
The sample holder shown in Fig. 3.3(b) was used for sample with ports made of coplanar
waveguide. Similar to the sample holder described above, the left piece was used to house
the sample. The middle piece containing the PCB routes the excitation signals to the
31
chip. The ground plane of the sample was connected to the ground plane of the PCB using
as many wirebonds as possible with lengths as short as possible so that the frequencies
corresponding to parasitic modes due to the inductance of wirebonds will be beyond the
measurement bandwidth. Silver paste was used for the electrical connections between the
PCB ground to the sample holder and between the central conductors of the SMA connector
and the PCB to avoid non-uniform magnetic eld in the device loops due to magnetic uxes
trapped in the solder.
The sample holder was mounted to the mixing chamber stage through cold ngers also
made of OFHC. In the experiment on the uxon-parity-protected qubit (Chapter 4) and
one-dimensional metamaterial transmission line (Chapter 6), the sample holder was housed
in another RF-tight cylindrical copper box. A superconducting coil providing magnetic elds
was slided on to the cylindrical box. The sample holder and the superconducting coil were
housed in a µ−metal shield to avoid stray eld due to magnetic parts within the cryostat.
Figure 3.4: Photographs of the sample holders. (a) Sample holder for launching from SMA
to microwave stripline and the sample holder (b) Sample holder for launching from SMA to
coplanar waveguide (CPW).
32
Chapter 4
Fluxon-parity-protected qubit : a prototype device 1
In this chapter, we present the microwave spectroscopy experiments of the prototype device
for uxon-parity-protected qubit consisting of a Cooper pair box (CPB) shunted by a tunable
superinductor. We observed the eect of the Aharonov-Casher (AC) interference on the
spectrum. The device spectrum varies as we change the oset charge ng on the CPB island
with the period ∆ng = 2e. The periodic variations are attributed to the charge modulated
AC interference between the uxon tunneling processes in the CPB Josephson junctions.
The |0〉 → |1〉 transition energy depends linearly on external ux Φext as it varies between
integers and half integers of ux quantum. The linear dependence corresponds to the energy
dierence between two uxons that dier by a single ux quantum Φ0. The measured phase
and charge dependences of the frequencies of the |0〉 → |1〉 and |1〉 → |2〉 transitions are in
good agreement with our numerical simulations. Almost complete suppression of the single
uxon tunneling due to destructive interference is observed for the charge ng = emod(2e).
As a result, the |0〉 and |1〉 states correspond to even and odd uxon parity. By further
increasing of the superinductance, the uxon-parity-protected qubit can be used for fault-
tolerant π/4 gate.
4.1 Sample design and measurement
The studied device (Fig. 4.1) consists of a superconducting loop that includes a CPB and a
36 unit cell superinductor[28]. We refer to this loop as the device loop. The magnetic ux Φ
in this loop controls the phase dierence across the CPB. Each unit cell of the superinductor
in this work consists of three large and one small Josephson junctions. The inductance L
1This chapter is based on the work published in M. T. Bell, W. Zhang et al., Phys. Rev. Lett. 116,107002 (2016).
33
Figure 4.1: Sample design. (a) The schematics of the circuit containing the device and theread-out lumped-element resonator. The CPB Josephson junctions are shown as crosses.(b) The layout of the device, the read-out resonator, and the MW transmission line. Thesuperinductor consists of 36 coupled cells, each cell represents a small superconducting loopinterrupted by three larger and one smaller Josephson junctions [44].
reaches its maximum when the unit cell is threaded by half a ux quantum, ΦL = Φ0/2. In
this regime of full frustration, L exceeds the Josephson inductance of the CPB junctions by
2 orders of magnitude. The device loop area ( 1850µm2) was designed much greater than
the superinductor unit cell area (15µm2) .
For dispersive measurement of the device resonances, a narrow portion of the device
loop with the kinetic inductance Lsh was coupled to the read-out lumped element resonator
as shown in Fig.4.1(b). The LC resonator is then coupled to a transmission line with
50Ohm characteristic impedance. The global magnetic eld, which determines the device
loop Φ and the unit cell of the superinductor ΦL, has been generated by a superconducting
solenoid. The oset charge on the CPB island was varied by a gate voltage Vg applied to
the microstrip transmission line [Fig. 4.1(b)]. The device was measured using the setup
described in Chapter 3.
The sample were fabricated using multiangle electron-beam deposition of aluminum
through a PMMA/MMA bilayer lift-o mask as described in Chapter 3. Six devices were
fabricated on the same chip; they can be addressed individually due to dierent resonant fre-
quencies of the read-out resonators. The parameters of the CPB junctions were nominally
the same for all six devices, whereas the maximum inductance of the superinductor was
systematically varied across six devices by changing the in-plane dimensions of the small
34
Table 4.1: Parameters of Josephson junctions in the representative device. Parameters ofthe CPB junctions correspond to the tting parameters; parameters of the superinductorjunctions were estimated using the Ambegaokar-Barato relationship and the resistance ofthe test junctions fabricated on the same chip [44].
Junctions In-plane areas, µm2 EJ ,GHz EC ,GHz
CPB 0.11× 0.11 6 6.4Superinductor large 0.30× 0.30 94 3.3Superinductor small 0.16× 0.16 25 11
junctions in the superinductor. Below we discuss the data for one representative device;
Table I summarizes the parameters of junctions in the CPB junctions and superinductor
(throughout the chapter, all energies are given in the frequency units, 1K ≈ 20.8GHz). All
measurements were performed at the base temperature of a dilutional fridge ∼ 20mK.
Figure 4.2: Spectroscopy of the readout resonator around full-frustration of the superinduc-tor loop. At ΦL/Φ0 = 0.5, the superinductor reaches maximum inductance, resulting in aminimum of the readout resonant frequency.
4.2 First-tone measurement
We perform rst-tone spectroscopy by measuring the transmission for the readout resoanator
as we vary the global magnetic eld while keeping the external DC bias for CPB island xed.
At full frustration, the superinductance reaches the maximum corresponding to the minimum
resonant frequency. Fig. 4.2 shows the rst-tone spectroscopy when ΦL ∼ 0.5Φ0. Because
35
the area of the unit cell is much smaller than the device loop, the phase across the CPB can
change by many periods within the region corresponding to the maximal superinductance.
The periodic change in the phase across the CPB corresponds to the wiggles around the
minimum in Fig. 4.2.
4.3 Two-tone measurement
In the two-tone measurements, the microwaves at the second-tone frequency f2 excited the
transitions between the |0〉 and |1〉 quantum states of the device, which resulted in a change
of its impedance [45]. This change was registered as a shift of the resonance of the readout
resonator probed with microwaves at the frequency f1. The microwave set-up used for these
measurements has been described in Refs. [28, 46, 47]. The resonance frequency f01 of the
transition between the |0〉 and |1〉 states was measured as a function of the charge ng and
the ux in the device loop. The f01 measurements could not be extended below ∼ 1 GHz
because of a high-pass lter in the second-tone feedline.
4.4 Results
The resonances corresponding to the |0〉 → |1〉 transition are shown in Fig. 4.3a as a
function of the gate voltage Vg at a xed value of the magnetic eld that is close to full
frustration of the superinductor unit cells (ΦL ' 0.5Φ0). The dependence f01 (Vg) is periodic
in the charge on the CPB island, ng, with the period ∆ng = 1 (here and below the charge
is measured in units 2e (mod 2e)). The increase of temperature above 0.3K resulted in
reducing the period in half due to the thermally generated quasiparticles population. Figure
4.3 also shows the resonance of the read-out resonator at fR = 6.45 GHz and the self-
resonance of the superinductor fL ≈ 5.5GHz. All three resonances are shown in Fig. 2b for
ng ≈ 0.47(Vg = 0) and ΦL ≈ 0.5Φ0. Weaker resonances observed at f2 ≈ 3 GHz and 4.8
GHz at Vg = −30mV correspond to the multi-photon excitations of the higher modes of the
superinductor.
Note that no disruption of periodicity neither by the quasiparticle poisoning [48] nor
by long-term shifts of the oset charge was observed in the data in Fig. 4.3(a) that were
36
Figure 4.3: Panel (a): The transmitted microwave power |S21|2 at the rst-tone frequencyf1 as a function of the second-tone frequency f2 and the gate voltage Vg measured at axed value of ΦL = 0.5Φ0. The power maxima correspond to the resonance excitations ofthe device (f2 = f01), the superinductor (fL), and the read-out resonator (fR). Note thatthe resonance measurements could not be extended below ∼ 1 GHz because of a high-passlter in the second-tone feedline. Panel (b): The frequency dependence of the transmittedmicrowave power measured at Vg = 0V and ΦSL = 0.5Φ0 [44].
measured over 80 min. With respect to the quasiparticle poisoning, this suggests that on
average, the parity of quasiparticles on the CPB island remains the same on this time scale.
In the opposite case, the so-called eye patterns would be observed on the dependences of
the resonance frequency on the gate voltage [49]. Signicant suppression of quasiparticle
poisoning was achieved due to the gap engineering [48] (the superconducting gap in the thin
CPB island exceeded that of the thicker leads by ∼ 0.2K), as well as shielding of the device
from infrared photons [50].
The expected ux dependence of the energy levels of the device is shown in Fig.4.4. This
ux dependence can be understood by noting that in the absence of uxon tunneling (the
dotted curves in Fig. 4.4a corresponding to ng = 0.5 and identical CPB junctions) dierent
states are characterized by a dierent number m of uxons in the device loop. At EJ EL
the energies of these states are represented by crossing parabolas EL(m,Φ) = 12EL(m− Φ
Φ0)2.
The phase slip processes mix the states with dierent numbers of uxons and lead to the
37
Figure 4.4: Panel (a): The ux dependence of the device energy levels obtained by numericaldiagonalization of the Hamiltonian (see the Supplemental Material of Ref. [44] for details,the tting parameters are listed below). The solid curves correspond to ng = 0.5, thedashed curves - to ng = 0 (the blue curves correspond to the ground state |0〉, the yellowcurves - to the state |1〉, and the green curves - to the state |2〉). For comparison we alsoplotted the dotted curves that correspond to the fully suppressed uxon tunneling; in thiscase there are no avoided crossings between the parabolas that represent the superinductorenergies EL(m,Φ) = 1
2EL(m − ΦΦ0
)2 plotted for dierent m. Panel (b): The dependencesof the resonance frequencies f01 (red dots - ng = 0, red squares - ng = 0.5) and f02 (bluedown-triangles - ng = 0, blue up-triangles - ng = 0.5 ) on the ux in the device loop.The theoretical ts (solid curves - ng = 0.5, dashed curves - ng = 0) were calculated withthe following parameters: EJ = 6.25 GHz, the asymmetry between the CPB junctions4EJ = 0.5 GHz, EC = 6.7 GHz, EL = 0.4 GHz (L = (Φ0
2π )2/EL h 0.4µH), ECL = 5 GHz[44].
level repulsion. The qualitative picture of uxon tunneling and AC interference is in good
agreement with the observed level structure shown in Fig. 4.4b.
Figure 4.4(b) shows the dependences of the resonance frequencies of the |0〉 → |1〉 and
|0〉 → |2〉 transitions (f01 and f02, respectively) on the ux in the device loop for the charges
ng = 0 and 0.5. In line with the level modeling, at ng = 0 the frequency f01 periodically
varies as a function of phase, but never approaches zero. On the other hand, when ng =
0.5, the amplitudes of uxon tunneling across the CPB junctions acquire the Aharonov-
Casher phase dierence π. Provided that the CPB junctions are identical, the destructive
interference should completely suppress uxon tunneling, which results in vanishing coupling
between the states |m〉 and |m ± 1〉 and disappearance of the avoided crossing. Since the
38
dierence EL(m,Φ)−EL(m± 1,Φ) is linear in Φ, the spectrum at ng = 0.5 should acquire
the sawtooth shape. This is precisely what has been observed in our experiment. To better
t the experimental data, we have assumed that the Josephson energies are slightly dierent
for the CPB junctions (4EJ < 0.5 GHz); for this reason, the minima of the theoretical
sawtooth-shaped dependence f01(Φ) are slightly rounded. Fitting allowed us to extract all
relevant energies (see the caption to Fig. 4.4). The amplitude of the single phase slips does
not exceed 0.2 GHz, the amplitude of the double phase slips Edps is 0.4 GHz, which is ∼ EL.
A protected qubit suitable for fault-tolerant operation requires Edps EL. This condition
can be satised by decreasing EL and meanwhile increasing ECL. It requires superinductor
with higher inductance and smaller in-plane dimension per unit length. A superinductor
with the backbone made of high-kinetic inductance disordered superconducting material will
be able to meet the requirement.
4.5 Conclusion
We have observed the eect of the Aharonov-Casher interference on the spectrum of the
Cooper pair box (CPB) shunted by a tuanble superinductor. Large values of L (EL EJ)
are essential for the observation of the AC eect with the Cooper pair box. We have
demonstrated that the amplitudes of the uxon tunneling through each of the CPB junctions
acquire the relative phase that depends on the CPB island charge ng. In particular, the phase
is equal to 0 (mod 2π) at ng = 2ne and π (mod 2π) at ng = e (2n+ 1). The interference
between these tunneling processes results in periodic variations of the energy dierence
between the ground and rst excited states of the studied quantum circuit; the period of the
oscillations corresponds to ∆q = 2e. The phase slip approximation provides quantitative
description of the data and the observed interference pattern evidences the quantum coherent
dynamics of our large circuit. By further increase of the superinductance, the qubit is able
to perform fault-tolerant Cliord gate.
39
Chapter 5
Microresonators fabricated from high-kinetic-inductance
Aluminum lms1
5.1 Introduction
The development of novel quantum circuits for information processing requires the imple-
mentation of ultra-low-loss microwave resonators with small dimensions [24]. Superconduct-
ing resonators have become ubiquitous parts of high-performance superconducting qubits
[51, 52] and kinetic-inductance photon detectors [53]. An important resource for resonator
miniaturization is the kinetic inductance of superconductors, LK , which can exceed the
magnetic (geometrical") inductance by orders of magnitude in narrow and thin supercon-
ducting lms [25]. High kinetic inductance translates into a high characteristic impedance
Z of the microwave (MW) elements, slow propagation of electromagnetic waves, and small
dimensions of the MW resonators. Ultra-narrow wires and thin lms of Nb and NbN [53, 54],
TiN [55], InOx[56, 57], and granular Al [58] were studied recently as candidates for high-LK
applications.
Research in high-LK elements also has an important fundamental aspect. According
to the Mattis-Bardeen (MB) theory [59], the kinetic inductance of a thin superconducting
lm LK(T = 0) is proportional to the resistance of the lm in the normal state, RN , and
thus increases with disorder. This theory, however, cannot be directly applied to strongly
disordered superconductors near the disorder-driven superconductor-to-insulator transition
(SIT). Recent theories predict a rapid decrease of the superuid density near the SIT and
the emergence of sub-gap delocalized modes that would result in enhanced dissipation at
microwave frequencies [27, 60]. Thus, the study of the electrodynamics of strongly disordered
superconductors may also contribute to a better understanding of the disorder-driven SIT.
1This chapter is based on the work published in W. Zhang et al., Phys. Rev. Applied 11, 011003 (2019).
40
In this chapter, we present a detailed characterization of the half-wavelength microwave
resonators fabricated from disordered Aluminum lms. Our interest in high-LK lms was
stimulated by the possibility of fabrication of superinductors (dissipationless elements with
microwave impedance greatly exceeding the resistance quantum RQ = h/(2e)2 [28, 61, 62]),
and the development of uxon-parity-protected qubit [44]. We have fabricated resonators
with an impedance Z as high as 5 kΩ, ultra-small dimensions and relatively low losses. The
study of the temperature dependences of the resonance frequency fr and intrinsic quality
factor Qi at dierent MW excitation levels allowed us to identify resonator coupling to
TLS in the environment as the primary dissipation mechanism at T . 250 mK; at higher
temperatures the losses can be attributed to thermally excited quasiparticles.
5.2 Experimental details
All microwave (MW) resonators studied in this chapter consisted of two parts. First, the
50-Ohm coplanar MW transmission line (TL) was formed on an intrinsic Si substrate by
electron beam deposition of a 140-nm-thick lm of pure Al through a lift-o mask, which
comprised of a 300-nm-thick e-beam resist (the top layer) and a 150-nm-thick copolymer
(the bottom layer). After the deposition of the bilayer resist and its patterning with e-beam
lithography, the sample was placed in a reactive ion etching system and etched with 75
mbar O2 plasma at a power of 30 watts for 30 seconds to remove any resist residue from
the substrate surface. The use of this pure Al transmission line facilitated the impedance
matching with the MW tract and eliminated spurious resonances (a large number of these
resonances is observed if high-Lk lms are used for both the TL and resonator fabrication).
After the second e-beam lithography with alignment precision better than 0.5 µm, several
half-wavelength disordered Al resonators were fabricated on the same substrate by reactive
DC magnetron sputtering as described in Chapter 3 (Fig. 5.1). The parameters of several
representative samples are listed in Table 5.1.
For the resonator characterization at ultra-low temperatures, we used a microwave setup
developed for the study of superconducting qubits described in Chapter 3. The resonators
were designed with the resonance frequencies fr ≈ 2 − 4 GHz, which allowed us to probe
the rst three harmonics of the resonators within the setup frequency range (2÷ 12) GHz.
41
Figure 5.1: (a) Microphotograph of a portion of the half-wavelength resonator capacitivelycoupled to the coplanar waveguide transmission line. Light green - Al ground plane andthe central conductor of the transmission line, green - silicon substrate, black - the centralstrip of the resonator made of strongly disordered Al. (b) Several resonators with dierentresonance frequencies coupled to the transmission line [63].
Dierent resonance frequencies of the resonators enabled multiplexing in the transmission
measurements. In order to ensure accurate extraction of the internal quality factor Qi, the
resonators were designed with a coupling quality factor Qc of the same order of magnitude
as Qi.
5.3 Microwave characterization
The resonators were characterized using a wide range of MW power PMW , two-tone (pump-
probe) measurements, and time domain measurements. The resonator parameters fr, Qi,
and Qc were found from the simultaneous measurements of the amplitude and the phase of
the transmitted signal S21(f) using the procedure described in Refs. [64, 65]. The kinetic
inductance LK of the central conductor of the resonators, which exceeded the magnetic
inductance by several orders of magnitude, was calculated as LK = 1/4f2rC (the capacitance
C between the resonator strip and the ground was obtained in the Sonnet simulations). The
parameters of several representative resonators are listed in Table 5.1.
The measured sheet kinetic inductance LK2 ≈ 2 nH/2 is similar to that reported for
granular Al lms in Ref. [66] and TiN in Ref. [67], and exceeds by a factor-of-2 LK2
42
realized for ultra-thin disordered lms of InOx [56, 68]. For the disordered Al lms with
ρ < 10 mΩ·cm, LK2 is in good agreement with the result of the MB theory [59], LK2(T =
0) = ~R2/π∆(0), where ∆(0) is the BCS energy gap at T = 0 K. Very large values
of LK2 allowed us to realize the characteristic impedance Z =√LK/C as high as 5 kΩ
for the resonators with narrow (w = 0.7 µm) central strips. The speed of propagation of
the electromagnetic waves in such resonators does not exceed 1% of the speed of light in
free space; accordingly, their length is two orders of magnitude smaller than that for the
conventional CPW resonators with the impedance Z = 50 Ω.
To identify the physical mechanisms of losses in the resonators, we measured the depen-
dences of fr and Qi on the temperature (T = 25÷450 mK) and the microwave power PMW .
Below we show that in the case of moderately disordered lms (resonators #2 − 4), both
the dissipation and dispersion at T < 0.25 K can be attributed to the resonator coupling
to the TLS [69] in the environment, whereas at higher temperatures they are controlled by
the T dependence of the complex conductivity of superconductors, σ(T ) = σ1(T )− iσ2(T )
[59].
Table 5.1: Summary of the measured parameters of AlOx resonators [63].
#w, l, fr, ρ, Tc, LK , Z,
µm µm GHz mΩ·cm K nH/
2 kΩ
1 11.0 1090 2.42 19.2 1.4 2.0 0.6
2 7.4 765 4.05 4.2 1.7 1.2 1.1
3 1.4 445 3.69 4.2 1.7 1.2 2.9
4 0.7 265 3.88 9.9 1.75 2.0 5.0
5.3.1 The resonance frequency analysis
We start the data analysis with the non-monotonic temperature dependence of the rela-
tive shift of the resonance frequency δfr(T )/fr0 ≡ [fr(T ) − fr(25mK)]/fr(25mK). Figure
5.2(a) shows the dependences δfr(T )/fr0 measured for three resonators (#2 − 4) with dif-
ferent width w. The non-monotonic character of these dependences is due to competing
eects of TLS [70] and thermally-induced quasiparticles on fr. The low-temperature part of
δfr(T )/fr0 is governed by the T -dependent TLS contribution to the imaginary part of the
complex dielectric permittivity ε(T ) = ε1(T )+iε2(T ). It should be noted that, in contrast to
43
Figure 5.2: The temperature dependences of resonance frequency shift δfTLSr (T )/fr0 (a)and the internal quality factor Qi (b) for the resonators #2− 4 measured at n ≈ 1(5) andn 1(4). The tting curves correspond to Eq. (5.2) and Eq. (5.7), respectively [63].
the TLS-related losses, the frequency shift δfTLSr is expected to be weakly power-dependent
[71]. Indeed, the temperature dependences measured for the dierent values of PMW almost
coincide; this simplies the analysis and reduces the number of tting parameters. The
low-temperature part of δfTLSr (T ) is well described by the following equation [53]:
δfTLSr (T )
fr0=Vfδ0
π
[Ψ<
(1
2+
1
2πi
hfrkBT
)− ln
(hfrkBT
)]. (5.1)
Here Ψ<(x) is the real part of the complex digamma function, the TLS participation
ratio Vf is the energy stored in the TLS-occupied volume normalized by the total energy
in the resonator, and the loss tangent δ0 characterizes the TLS-induced microwave loss in
weak electric elds at low temperatures kBT hfr. The product Vfδ0 is the only tting
parameter, its values are listed in Table 5.2. The obtained values of Vfδ0 are close to
that found for Al-based [71] and AlOx-based resonators [67, 72]. Note that resonator #4
demonstrates the most pronounced increase of fr(T ) with temperature due to the stronger
electric elds and a larger participation ratio characteristic of the high-Z resonators [73].
At T > 0.25 K, fr rapidly drops due to the decrease of the superuid density. The
44
Table 5.2: Summary of the tting parameters [63].
# ∆(0)/kBTc β Vfδ0 · 10−4 nc(0) · 10−3
2 1.96 0.60 1.4 50
3 1.98 0.55 4.8 1.6
4 1.88 0.38 6.7 0.23
dependences δfr(T ) over the whole studied T range can be described as
δfr(T )/fr0 = δfTLSr (T )/fr0 + δfMBr (T )/fr0 (5.2)
where
δfMBr (T )
fr0=
1
2
[σ2(T )− σ2(25mK)
σ2(25mK)
](5.3)
is the resonance shift due to the T -induced break of Cooper pairs and subsequent increase
of the kinetic inductance, calculated in the thin lm limit [71]. The only free parameter in
δfMBr (T )/fr0 is the gap energy ∆(0), which can be found by tting of the high-T portion
of δfr(T )/fr0 [Eq. (5.2)]; the measured ratio ∆(0)/Tc is about 10% greater than the BCS
value of 1.76kB, which is consistent with previously reported data [74].
5.3.2 The quality factor analysis
We now proceed with the analyses of losses. We observed the enhancement of the internal
quality factor Qi with increasing the average number of photons in the resonators, n =
2PMWQ2l /(Qchf
2r ) [75], where Ql = (1/Qi + 1/QC)−1 is the loaded quality factor. The
dependences Qi(n) for three resonators with dierent w measured at the base temperature
≈ 25 mK are shown in Fig. 5.3. Similar behavior of Qi(n) have been observed for many types
of CPW superconducting resonators (see, e.g. [53, 76] and references therein), including the
resonators based on disordered Al lms [58, 66]. Note that the increase of Qi with the input
MW power PMW is limited by the resonance distortion by bifurcation at PMW > P∗. For
the resonators with Ql & 104 the onset of bifurcation is observed for the microwave currents
I∗ =√
2P∗/Z which scale approximately as Idp/√Ql [77], where Idp is the Ginzburg-Landau
depairing current in the central strip [25].
45
The power-dependent intrinsic losses can be attributed to the resonator coupling to the
TLS with the Lorentzian-shaped distribution
g(ETLS) ∼ 1
(ETLS − hfr)2 + (~/τ2)2, (5.4)
where ETLS is the energy of TLS and τ2 is its dephasing time [78]. Once the MW power PMW
reaches some characteristic level Pc and the Rabi frequency of the driven TLS ΩR ∼√PMW
exceeds the relaxation rate 1/√τ1τ2, the population of the excited TLS increases, and the
amount of energy that the TLS with fTLS ≈ fr can absorb from the resonator decreases.
Thus, the high PMW burns the hole" in the density of states (DoS) of dissipative TLS. The
width of the hole is κ/2πτ2, the power-dependent factor can be written as
κ =
√1 +
(n
nc
)β, (5.5)
where n and nc correspond to PMW and Pc, respectively. Note that the exponent β is known
to be dependent on the electric eld distribution in a resonator [79], and the characteristic
power nc increases with temperature by orders of magnitude due to a strong T -dependence
of τ1 and τ2 [80, 81]. Taking into account the TLS saturation at high temperature, the power
dependence of the TLS-related part of the loss tangent can be expressed as follows [73]:
δTLS(n, T ) =Vfδ0
κtanh
(hfr
2kBT
). (5.6)
By tting the experimental data with Eq. (5.6) we found β and nc, the obtained param-
eters are listed in Table . We found that larger values of β correspond to wide strips, and
the extracted nc(0) scales as the square of the electric eld on the surface of the resonator.
The experimental dependences Qi(T ) measured for resonators #2−4 at n w 1 and n 1
[Fig. 5.2(b)] are well described by the sum of the TLS contribution [Eq. (5.6)] and the MB
term δMB = σ1(T )/σ2(T ) [71]:
Qi(T ) = δTLS(T, β, nc, Vfδ0) + δMB[T,∆(0)]−1. (5.7)
46
Figure 5.3: The dependences Qi(n) at T ≈ 25 mK for the resonators with dierent widths.Solid curves represent the theoretical ts of the quality factor governed by TLS losses [Eq.(5.5), see the text for details] [63].
The agreement of measured Qi with the prediction of Eq. (5.7) over the whole measured
temperature range proves that the losses in the developed resonators are limited by the sum
of TLS and MB terms.
5.3.3 The two-tone and time-domain measurements
We obtained an additional information on the TLS-related dissipation by performing the
pump-probe experiments in which Qi was measured at a low-power (n w 1) probe signal
while the power Pp of the pump signal at the frequency fp was varied over a wide range.
Figure 5.4(a) shows the dependences Qi(Pp) measured at dierent detuning values ∆f =
fp−fr = 0, ±1 MHz, and ±10 MHz. Note that we have not observed any changes in Qi when
the pump signal was applied at the second and third harmonics of the resonator. Also, Qi
was Pp-independent when we monitored the second harmonic and applied the pump signal
at the rst harmonic.
47
Figure 5.4: (a) The dependences of Qi for resonator #1 on the pump tone power Pp forseveral values of detuning ∆f between resonance and pump frequencies. (b) The valuesof Qi measured versus detuning ∆f at a xed number of the pump tone photons in theresonator np ≈ 1000. The error bars are derived from the covariance matrix obtained fromnonlinear tting of the measurement of S21(f) [63].
Since the resonator coupling to the pump signal varies by several orders of magnitude
within the detuning range 0 ÷ 10 MHz, it is more informative to analyze Qi as a function
of the average number of the pump" photons in the resonator, np = Pp(1 − |S21(fp)|2 −
|S11(fp)|2)/hf2p , where S21 and S11 = 1−S21 are the transmission and reection amplitudes
at the pump frequency, respectively. The dependence Qi on the detuning ∆f for a xed
np ≈ 1000 is depicted in Fig. 5.4(b). The resonance behavior of Qi(∆f) is expected since
only a narrow TLS band [Eq. (5.4)] contributes to dissipation: the hole" extension in the
DoS is limited by ∼ κ/τ2 around the pump frequency. Indeed, using the approach developed
in [82], one can obtain the following expression:
Qi(∆) = Q0
[1 +
(κ/2πτ2)2
∆f2 + κ(1/2πτ2)2
], (5.8)
where Q0 is the o-resonance quality factor, and introduced by Eq. (5.5) factor κ might
be calculated as κ = Qmax/Q0. The dephasing time is the only tting parameter and it is
found to be τ2 ≈ 60 ns. This result agrees with the measurements of the dephasing time for
individual TLS in amorphous Al2O3 tunnel barrier in Josephson junctions [83].
Interactions between the high-frequency (coherent, E > kBT ) TLS with the low-frequency
(thermal, E < kBT ) uctuators result in the TLS spectral diusion as well as the icker
48
noise. The telegraph noise in the resonance frequency fr is expected if some of the TLS
with f ≈ fr are strongly coupled to a resonator. Typical TLS densities for Al/AlOx junc-
tions are ∼1 (GHz·µm2)−1 [69]. Interestingly, the number of strongly coupled TLS for our
resonators (assuming that the strongly coupled TLS are in the oxide layer of the resonator)
is of the order of unity [1 (GHz · µm2)−1 × 0.1MHz × 104µm2]. To study the telegraphy
noise, we repetitively measured S21 at a xed frequency on a slope of the resonance dip for
a few minutes. Figure 5.5 shows an example of the measured telegraph noise in Re[S21].
The characteristic time scale of random switching between two Re[S21] levels is 1-10 sec-
onds. This noise can be attributed to interactions of the resonators with a small number of
strongly coupled TLS.
Figure 5.5: The time dependence of Re[S21] measured at T = 25 mK at a xed frequencyon the slope of a resonance dip. The microwave power corresponds to 〈n〉 ∼ 1000. Eachpoint corresponds to the data averaging over 1 sec [63].
We have performed the time domain measurements of the TLS relaxation time for res-
onator #1 using the pulse sequence shown in Fig. 5.6(a). A 0.5 s-long pump pulse was
applied to the resonator at the beginning of each duty cycle. A readout pulse at the single-
photon power level lasting for 36 ms followed the pump pulse and was digitized to obtain
S21. The readout delay time was varied between 0 s and 1 s. Figure 5.6(b) shows the re-
sult of the experiment at the readout frequency f0 = 2.4258 GHz and the pump frequency
fp = f0 + 1 MHz. The change in |S21(f0)| at t = 0.5 s is consistent with CW measurements
49
at the same readout frequency and power level when a pump tone was turned on and o.
This indicates that an upper limit of the TLS relaxation time for our sample is much less
than 36 ms.
Figure 5.6: (a) The pulse sequence. (b) The time dependence of |S21| measured at f0 =2.4258 GHz. The pump pulse at fp = f0 +1 MHz was applied between t = 0 s and t = 0.5 s.The pump tone power corresponds to np ≈ 1000. Each data point was averaged over 4000cycles with the same readout delay time. The inset shows CW measurement of S21 versusf with (red) and without (blue) the pump signal and indicates the position of f0 used inthe relaxation time measurement. The readout power was at the single photon level for allmeasurements on this plot [63].
5.4 Summary
In conclusion, we have fabricated CPW half-wavelength resonators made of strongly disor-
dered Al lms. Because of the very high kinetic inductance of these lms, we were able to
signicantly reduce the length of these resonators, down to ∼ 1% of that of conventional
CPW resonators with a 50 Ω impedance. Due to ultra-small dimensions and relatively
low losses at mK temperatures, these resonators are promising for the use in quantum su-
perconducting circuits operating at ultra-low temperatures, especially for the applications
50
that require numerous resonators, such as multi-pixel MKIDs [53, 77]. The high impedance
Z =√LK/C of the narrow resonators can be used for eective coupling of spin qubits
[84, 85]. The high resonator impedance imposes limitations on the strength of resonator
coupling to the transmission line. For the studied CPW resonators with Z ∼ 5 kΩ, the
strongest realized coupling (when half of the resonator length was used as the element of
capacitive coupling to the transmission line) corresponded to Qc ∼ 104. On the other hand,
for many applications, such as large MKID arrays that require a high loaded Q factor, this
should not be a limitation.
We have shown that the main source of losses in these resonators at T Tc is the
coupling to the resonant TLS. A comparison of our results with those of the other groups
shows that the obtained Qi values, increasing from (1÷2)×104 in the single-photon regime
to 3×105 at high microwave power, are typical for the CPW superconducting resonators
with similar TLS participation ratios. This implies that the disorder in Al lms does not
introduce any additional, anomalous losses. Most likely, the relevant TLS are located near
the edges of the central resonator strip either in the native oxide on the Si substrate surface
or in the amorphous oxide covering the lms. Further increase of Qi can be achieved by the
methods aimed at the reduction of surface participation, such as substrate trenching (see
[86] and references within) and increasing the gap between the center conductor and the
ground plane [79]. The evidence for that was provided by the results of Ref. [66] obtained
for the modied three-dimensional microstrip structures based on disordered Al lms. It is
also worth mentioning that the losses can be reduced using TLS saturation by the microwave
signal outside of the resonator bandwidth but within the TLS spectral diusion range. A
fundamental issue pertinent to all strongly disordered superconductors is the development of
a better understanding of the impedance of superconductors near the disorder-driven SIT.
This issue requires further research, and the microwave experiments with the resonators
made of disordered Al and other disordered materials demonstrating the SIT may shed light
on the nature of this quantum phase transition.
51
Chapter 6
Josephson metamaterial with a widely tunable positive or
negative Kerr constant1
6.1 Introduction
In conventional optics, a material whose refractive index n is aected by the intensity of
an electric eld n ∝ |E|2 is known as a Kerr medium [87]. Analogous to nonlinear optics,
microwave superconducting circuits exhibit the Kerr eect due to a nonlinear response of
their kinetic Josephson inductance that determines the circuit impedance. In supercon-
ducting circuits based on Josephson junctions the Kerr eect originates from the φ2 term
in the expansion of the Josephson inductance L(φ) = Φ0/(2πIc cosφ), where φ is the su-
perconducting phase across the junction, Ic is the junction critical current, and Φ0 is the
magnetic ux quantum. The Kerr eect in superconducting circuits has been used to gen-
erate squeezed states of light [88], traveling-wave parametric ampliers [22, 39, 89, 90], and
superconducting quantum bits [91].
In this chapter, we demonstrate a novel Josephson metamaterial with a Kerr constant
tunable over a wide range that includes both positive and negative values. Such a nonlinear
medium can nd applications in wave-packet rectication [92], analogues of nematic optical
materials [93], superinductances [28], and in Josephson traveling-wave parametric ampliers
(JTWPA) [39], which was the motivation behind the present work. The metamaterial is
composed of a one-dimensional chain of asymmetric superconducting quantum interference
devices (SQUIDs) with nearest-neighbor coupling through common Josephson junctions Fig.
6.1(a). The same magnetic ux threads all SQUIDs to allow for tunability of the chromatic
and nonlinear dispersion. Tunable superconducting metamaterials [94] composed of passive
[95], and electrically active meta-atoms such as SQUIDs [9699] or qubits [100102] have
1This chapter is based on the work published in W. Zhang et al., Phys. Rev. Appl. 8, 051001 (2017).
52
been investigated. Below we discuss a novel topology with direct coupling between meta-
atoms in a structure with a tunable Kerr constant which can change sign. This design
oers signicant advantages for several applications, including parametric amplication in
a JTWPA. As the magnetic ux on the metamaterial is varied, we observe a monotonic
dependence of the chromatic dispersion and a Kerr constant which varies over a wide range
from positive to negative. This novel metamaterial compares favorably with the Josephson
circuits previously used for parametric amplication [103] in two important aspects. First,
the Kerr eect is much stronger and the magnitude of a Kerr constant can be easily tuned
by the magnetic ux Φ in the SQUID loops. Second, the sign of the Kerr constant is also
ux-dependent, which is an important resource for the development of quantum-limited
parametric ampliers and other superconducting circuits.
6.2 Metamaterial Design
The design of the proposed metamaterial is shown in Fig. 6.1(a); it resembles the design
of the Josephson superinductor introduced by us in Ref. [28] . Each unit-cell of length a
is composed of two coupled asymmetric SQUIDs with a single smaller Josephson junction
with critical current Ijs0 and capacitance Cjs in one arm and two larger Josephson junctions
with critical current Ijl0 = rIjs0 and capacitance Cjl = rCjs in the other arm. Here r is the
ratio between the areas of the larger and smaller junctions. The eld dependent Josephson
inductance of the metamaterial is
L(φ,Φ) = L0
([r2
+ 2 cos(
2πΦ
Φ0
)]−[ r
16+ cos
(2π
Φ
Φ0
)]φ2)−1
,
(6.1)
where L0 = ϕ0/Ijs0, ϕ0 = Φ0/(2π), and φ is the phase dierence across a unit-cell. At
a critical value r0 = 4 the rst term in L(φ) vanishes at Φ/Φ0 = 0.5 and the quadratic
term dominates [28, 39]. Propagation of electromagnetic waves with wavelengths λ a in
this metamaterial in the absence of dissipation is described by the following nonlinear wave
equation for superconducting phases on the nodes between unit-cells ϕ(z, t) [28, 37]
53
a2
L0
[r2
+ 2 cos(
2πΦ
Φ0
)]∂2ϕ
∂z2+ a2Cjs
(r2
+ 2) ∂4ϕ
∂t2∂z2−
Cgnd∂2ϕ
∂t2− γ ∂
∂z
[(∂ϕ∂z
)3]= 0,
(6.2)
where γ = a4/(3ϕ20L0)[r/16 + cos(2πΦ/Φ0)]. Cgnd is the distributed capacitance between
the metamaterial and the ground plane. The linear (low-power) dispersion relation and
solution to Eq. (6.2) is
k =ω√L0Cgnd
a
√[r2 + 2 cos
(2π Φ
Φ0
)]− ω2L0Cjs
(r2 + 2
) (6.3)
and A(z) = A0e−i(k+α)z respectively, where
α =3γk5|A0|2
8ω2Cgnd, (6.4)
and A0 is the superconducting phase amplitude, see Ref. [39]. Electromagnetic waves
which propagate in this metamaterial acquire a phase shift −αz, where z is the direction of
propagation along the metamaterial which depends on the intensity |A0|2 analogous to light
traveling in a Kerr medium [36, 87]. The Kerr constant γ and thus the intensity dependent
phase shift can vary over a wide range with magnetic ux tuning, and can even change sign
from positive to negative.
6.3 Microwave Characterization
To demonstrate the tunable properties of the Josephson metamaterial, several devices were
fabricated at Hypres Inc. using the standard Nb/AlOx/Nb trilayer process with a nominal
critical current density of 30 A/cm2. The devices are shown schematically in Fig. 6.1(b).
The perforated bottom metal layer M0 (gray) acted as the ground plane; it was separated
from the metamaterial structure by 150 nm of SiO2. Metal layers M1 (green) and M2
54
Table 6.1: Parameters of two Josephson metamaterial devices [104].
Device r Cjs(fF)
Cjl(fF)
Cgnd(fF)
Ijs0(µA)
Ijl0(µA)
RNs(kΩ)
RNl(kΩ)
1 5.9 50 300 75 0.25 1.5 8.44 1.42
2 7 50 350 75 0.25 1.75 8.43
(blue) form the coupled asymmetric SQUID structure of the metamaterial. The Josephson
junctions are shown in red, and the vias between M1 and M2. Fig. 6.1(c) shows an optical
image of the device. The design parameters of two representative devices are listed in Table
1. The junction critical currents were determined from the Ambegaokar-Barato formula
[25] using the normal state resistance RN(s,l) of the on-chip smaller and larger test junctions,
respectively, measured at room temperature. The variations in the normal state resistance
within the same batch of devices did not exceed 1%. Each SQUID in the unit-cell has a
loop area of 13 × 7µm2 and the unit-cell which is composed of two SQUIDs has a length
a = 14µm. Each device measured contains 125 unit-cells and have a physical length l = 125a
(1.75 mm).
Investigation of the chromatic and nonlinear dispersion in the Josephson metamaterial
was performed in the cryostat described in Chapter 3. The Josephson chain was included in
the microwave transmission line, and transmission measurements were performed with an
Anritsu 37369A vector network analyzer. A superconducting solenoid was used to provide
a uniform magnetic ux bias to all SQUID loops in the metamaterial.
The dispersion of the Josephson metamaterial in the linear (low-power) regime was in-
vestigated with the transmission measurements of the phase shift as a function of magnetic
ux (Fig. 6.2). The linear transmission measurements were performed at a signal power
of (−130 dBm) (−100 dBm), where S21 was independent of the signal intensity. Figure
6.2 illustrates the low-power phase shift across the metamaterial −lk(Φ, f) measured for
devices 1 and 2. Solid lines are ts to Eq. (6.3) utilizing the design parameters listed in
Table 6.1 and Ijs0 as the only tting parameter (for brevity only one measurement frequency
f = 4 GHz is shown for device 1). The values of Ijs0 = 0.21 ± 0.01µA, the same for both
devices, were slightly lower than Ijs0 = 0.25µA estimated using the Ambegaokar-Barato
formula (Table 6.1). The chromatic dispersion of the metamaterial at Φ/Φ0 = 0.5 is shown
55
Figure 6.1: Josephson metamaterial based on a chain of coupled asymmetric SQUIDs. (a)Circuit schematic of the metamaterial. Each unit-cell of the metamaterial consists of twoasymmetric SQUIDs coupled with a shared junction and is of length a. Each SQUID inthe unit-cell is threaded with a magnetic ux Φ and has a capacitance to ground Cgnd. (b)Illustration of the three-metal-layer layout of the device. Metal layer M0 (gray) representsthe ground plane, M1 and M2 are the two metal layers which form the electrodes of coupledasymmetric SQUIDs, red and green vias between M1 and M2 represent Josephson junctionsand M1-to-M2 vias respectively. The purpose of the ngers on M0 in gray and M1 in greenwhich extend into the foreground is to increase the capacitance of the SQUID array toground (M0). (c) Optical image of the measured Josephson metamaterial [104].
for both devices in Fig. 6.3. Solid lines are the expected k(f) dependence calculated with
the initial design parameters in Table 6.1. The eective plasma frequency at Φ/Φ0 = 0.5 of
the elementary unit-cell is fp = [L(Φ = 0.5Φ0)(r/2 + 2)Cjs]−1/2/2π which corresponds to 8
GHz and 12 GHz for device 1 and 2, respectively. The phase velocity υ = a/√L(Φ)Cgnd
for both devices varied between 3 × 106 m/s and 1.5 × 106 m/s for Φ = 0 and Φ = 0.5Φ0
, respectively. The characteristic impedance Z =√L(Φ)/Cgnd of the metamaterial varied
between 60 Ω and 145 Ω over the magnetic ux range from 0 to 0.5Φ0.
Figure 6.4 shows the main result of this work: the dependence of the microwave phase
shift at a signal frequency of 4 GHz as a function of signal power P . An estimated signal
56
Figure 6.2: Low-power transmission measurements of the phase shift across the Josephsonmetamaterial as a function of the magnetic ux for device 1 (lower panel) and device 2(upper panel) at dierent measurement frequencies. Solid lines are ts to Eq. (6.3) [104].
Figure 6.3: Wavenumber as a function of frequency for devices 1 (blue circles) and 2 (redsquares) extracted from the tting procedure of the data at Φ/Φ0 = 0.5 in Fig. 6.2. Solidlines are a plot of Eq. (6.3) with the design parameters listed in Table 6.1 [104].
57
Figure 6.4: Measurements of the microwave phase shift as a function of signal power Pwhere P0 = -70 dBm, at dierent values of the magnetic ux in the metamaterial unit cellsfor device 1 (upper panel) and device 2 (middle panel). Transmission measurements wereperformed at a signal frequency of 4 GHz. Solid lines are ts to Eq. (6.4). (lower panel)shows the Kerr constant of the metamaterial unit cell derived from the ts of the powerdependent transmission phase data in the upper two panels normalized to the Kerr constantof the small Josephson junction γ/γJJ as a function of magnetic ux. As magnetic ux istuned, γ/γJJ varies over a wide range and changes sign from positive to negative [104].
.
58
power at the mixing chamber of P0 = -70 dBm was attenuated at room temperature with
a programmable attenuator (Aeroex 8311) to vary the signal power P to port 1 of the
metamaterial. Microwave transmission measurements were performed over several xed
values of the magnetic ux 0 ≤ Φ ≤ Φ0/2. For each magnetic ux, the phase across the
metamaterial depends on the input power. Near zero eld, the phase across the metamaterial
decreases with signal power (i.e. a positive Kerr constant), similar to a linear chain of
Josephson junctions which would exhibit a non-tunable Kerr constant γJJ = a4/(6ϕ20L0)
[37] which is magnetic ux independent. Fig. 6.4 (lower panel) shows the Kerr constant of
the metamaterial unit-cell derived from the ts of the power dependent transmission phase
data in the upper two panels of Fig. 6.4 normalized to γJJ of the small junctions in the
metamaterial. In contrary to a linear chain of junctions, as the magnetic ux increases, the
magnitude and sign of the phase shift changes: γ becomes negative at 0.3 ≤ Φ/Φ0 ≤ 0.5 as
shown in Fig. 6.4 (lower panel). The magnitude of the Kerr constant in the metamaterial is
similar to γJJ for a linear chain of junctions with critical currents equal to that of the smaller
junctions in the metamaterial γ/γJJ = 2[r/16 + cos(2πΦ/Φ0)]. However, the metamaterial
can be driven with higher microwave power since the majority of the current ows through
the backbone formed by larger critical current junctions. This feature allows for an increase
in the dynamic range of JTWPAs composed of this metamaterial in comparison to linear
chains of junctions. According to Eq. (6.4) it was expected that the Kerr constant γ whould
change sign at a magnetic ux of Φ/Φ0 = cos−1(−r/16)/(2π), which is Φ/Φ0 = 0.3 and
Φ/Φ0 = 0.33 for device 1 and 2 respectively. Indeed, for both devices the sign change of the
Kerr constant was observed in the ux range 0.3 − 0.33(Φ/Φ0) as shown in the γ/γJJ vs.
Φ/Φ0 data in Fig. 6.4 (lower panel). In Fig. 6.4 the solid lines are ts to Eq. (6.4) calculated
with the design parameters in Table 6.1 and Ijs0 = 0.21µA. Best ts were obtained with
P0 = −70 dBm ±1.5 dB as a tting parameter which takes into account the uncertainty
of actual signal power level at Port 1 of the metameterial for dierent tunings of magnetic
ux. The nonlinear wave equation (Eq. (6.2)) which describes the behavior of the Josephson
metamaterial is in good agreement with the phase measurement data.
59
A nonlinear medium with a tunable Kerr constant, which can change sign, is very promis-
ing for the development of Josephson traveling-wave parametric ampliers [39]. State-of-
the-art JTWPAs rely on a four-wave-mixing process which require perfect phase matching
between signal, idler and pump waves propagating along a nonlinear transmission line. In
recent works [22, 38, 89, 105] the required relations between chromatic dispersion and self-
(SPM) and cross-phase (XPM) modulations were realized by tuning the pump frequency
near a pole or band-gap introduced into the nonlinear transmission line via sophisticated
dispersion engineering techniques. The unique feature of the studied metamaterial enables
phase matching due to compensation of the positive chromatic dispersion between the signal,
idler, and pump waves by the negative SPM and XPM . The theory of operation of such a
novel JTWPA was described in Ref. [39].
6.4 Summary
In conclusion, we have developed a unique one-dimensional Josephson metamaterial whose
Kerr constant is tunable over a wide range, and can change sign from positive to negative.
The metamaterial is composed of a chain of coupled asymmetric SQUIDs. The dispersion
properties of the metamaterial are varied with an external magnetic ux threading each
SQUID loop in the array. The transmission measurements of the phase of microwaves prop-
agating along the metamaterial at low and high signal powers veried predictions of a non-
linear wave equation governing the microwave response of the medium. Such a metamaterial
can be used as the nonlinear medium for parametric amplication and phase-matching in a
four-wave-mixing process in Josephson traveling-wave parametric ampliers, its use elimi-
nates the need for complex dispersion engineering techniques.
60
Chapter 7
Conclusions and future work
7.1 Conclusions
In this dissertation, we explored (a) superinductor-based parity-protected qubits, (b) novel
disordered granular Aluminum lms for the superinductor fabrications, and (c) superinductor-
inspired one-dimensional superconducting circuits for parametric amplication of microwave
signals at ultra-low temperatures.
In Chapter 2 and 4, we discussed the operation of the parity-protected qubits and our ex-
periments with a prototype uxon-parity-protected qubit formed by a superconducting loop
consisting of a Cooper-pair box (CPB) with EJ < EC and a superinductor. The superin-
ductor in the qubit was realized as a chain of coupled asymmetric SQUIDs (CAQUIDs)
frustrated by the external magnetic eld. The backbone of the superinductor consists of
junctions with EJ EC . The dynamics of the low energy states of the qubit correspond
to uxons tunneling across the Josephson junctions in the CPB in and out of the supercon-
ducting loop. The uxon tunneling process is dual to the Josephson eect where Cooper
pairs tunnel between two electrodes separated by an insulating barrier. The phase dierence
of the uxon tunneling amplitudes across the two junctions of a CPB depends on the oset
charge on the CPB island due to Aharonov-Casher eect. By measuring the qubit spec-
trum when biasing the oset charge on the CPB at emod(2e), we observed almost complete
suppression of the tunneling of a uxon that change the phase across the CPB by 2π. The
lowest two energy states are composed of uxons of even and odd parities carrying even and
odd numbers of ux quantum. Symmetry between EJ and EC of the two junctions forming
the CPB is essential as it allows the 2π-phase-slip process to happen across either one of
the CPB junctions with equal probability. We reduced the spread of the Josephson and
61
charging energies in the CPB junctions down to ~5% by using the Manhattan pattern fab-
rication technique. Further increase of the inductance of the superinductor will enhance the
quantum ucuations in phase across the CPB in order to perform fault tolerant operations
based on uxon-parity-protected qubits which require each qubit states to be superpositions
of multiple states corresponding to dierent uxons of the same parity.
In Chapter 5, we studied superconducting coplanar-waveguide (CPW) resonators fab-
ricated from disordered (granular) lms of Aluminum. The studied disordered supercon-
ducting lms have high kinetic inductance and low microwave losses. The intrinsic quality
factors of the resonators are limited at ultra-low temperatures by the resonator coupling to
two-level systems in the environment and are comparable with those of resonators made of
conventional superconductors. Nanowires of width 1µm made of the high kinetic inductance
material have characteristic impedance greater than resistance quantum. The wavelength for
microwaves traveling in the transmission lines made of the nanowires are extremely short
(∼ 200µm). The high kinetic inductance nanowires are easier to design and have more
compact in-plane dimensions and higher self-resonant frequencies compared to superinduc-
tors made of a chain of Josephson junctions of the same inductance. In addition to high
kinetic inductance and small dimension, the well-understood losses make these disordered
Aluminum resonators promising for a wide range of microwave applications which include
kinetic inductance photon detectors and superconducting quantum circuits.
In Chapter 6, we studied a noval superinductor-inspired metamaterial transmission lines
based on CAQUIDs. In a transmission line based on Josephson junctions, because of the
nonlinear Josephson inductance, the wave vectors for a transmitted signal composed of
a part linear with respect to its frequency and a nonlinear part due to the power of its
own or that of other signals. The nonlinear eect is called Kerr eect similar to that in
optic bers. The ratio between the nonlinear part versus the power is proportional to a
constant called Kerr constant. We observed that the Kerr constant in the transmission
line based on CASUIQDs is tunable over a wide range from positive to negative values
with magnetic elds. The tunable Kerr nonlinearity of these CAQUID-based transmission
lines facilitate the phase matching of the four-wave mixing process that couples a strong
pump tone to a weak microwave signal for the quantum-limited parametric amplication at
62
ultra-low temperatures.
7.2 Future work
One of the directions of future work on uxon-parity-protected qubits would be optimization
of CASQUIDs-based superinductors. In particular, replacement of Josephson junctions in
the superinductor backbone with a nanowire made of granular Aluminum could signi-
cantly increase the self-resonance frequencies of the superinductor. The optimal working
point for uxon-parity-protected qubits, as discussed in Chapter 2, is when the magnetic
eld through the qubit loop is at zero-frustration and the oset charge on the CPB island
is emod(2e). In the prototype qubit, we observed that the qubit frequency under the two
conditions are close to the resonant frequency of the readout resonator and the frequency
corresponding to self-resonant mode of the superinductor. In order to bring down the qubit
frequency and to raise the self-resonant frequency of the superinductor at the same time,
we need to increase the superinductance while reducing its parasitic capacitance. For the
superinductor used in the prototype qubit, the Josephson inductance corresponding to the
backbone per unit cell is 3nH. The same inductance can be realized using the a 1.5 μm long
and 0.1μm wide wire made of disordered granular Aluminum lms. The in-plane area of the
superinductor can thus be shrunk by at least ve times (see Appendix A for the design of
the "hybrid" superinductor made of granular-Aluminum and Al/AlOx/Al Josephson junc-
tions). The superinductor with smaller in-plane dimensions would have smaller parasitic
capacitance and thus higher self-resonant frequency.
The studied metamaterial transmission line have shown an average 10dB increase in
signal when the pump is applied (see Appendix B). Future studies on the transmission line
include characterization of the gain by comparing the transmission powers with respect to
the signals passing and bypassing the transmission line and characterization of the noise
associated with the amplication.
63
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Appendix A
Design of hybrid superinductor
Figure A.1: Design of hybrid superinductor. The backbone of the superinductor (black)consists of nanowires made lms of granular Aluminum. Josephson junctions are formed atthe crossing of two Aluminum nanowires (blue).
72
Appendix B
Increase in transmission power through metamaterial
tranmission line
Figure B.1: Increase in transmission power through the metamaterial transmission linewith a strong pump applied at 3.5 GHz and Φ/Φ0 ≈ 0.5 . The ripples are due to impedancemismatch between the transmission line and the embedding environment.
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