Quantum information processing with trapped ionsmsafrono/650/Lecture16.pdf · shine the light pulse...
Transcript of Quantum information processing with trapped ionsmsafrono/650/Lecture16.pdf · shine the light pulse...
1
Quantum information processingwith trapped ions
Courtesy of Timo Koerber Institut für Experimentalphysik
Universität Innsbruck
1. Basic experimental techniques
2. Two-particle entanglement
3. Multi-particle entanglement
4. Implementation of a CNOT gate
5. Teleportation
6. Outlook Lectures 15 - 16
The requirements for quantum information processing
D. P. DiVincenzo, Quant. Inf. Comp. 1 (Special), 1 (2001)
I. Scalable physical system, well characterized qubits
II. Ability to initialize the state of the qubits
III. Long relevant coherence times, much longer than gate operation time
IV. “Universal” set of quantum gates
V. Qubit-specific measurement capability
2
S1/2
P1/2
D5/2
„quantumbit“
Experimental Setup
Important energy levels
• The important energy levels are shown on the next slides; a fast transition is used to detect ion fluorescence and for Doppler cooling, while the narrow D5/2 quadrupole transition has a lifetime of 1 second and is used for coherent manipulation and represents out quantum bit. Of course a specific set of Zeeman states is used to actually implement our qubit. The presence of other sublevels give us additional possibilities for doing coherent operations.
3
P1/2
S1/2
= 7 ns
397 nm
D5/2
S1/2 – D5/2 : quadrupole transition
729 nm
= 1 s
Ca+: Important energy levels
P1/2
S1/2
D5/2
„qubit“
P1/2
= 7 ns
„quoctet“ (sp?)
Ca+: Important energy levels
4
qubit
qubit
Encoding of quantum information requires long-lived atomic states:
microwave transitions
9Be+, 25Mg+, 43Ca+, 87Sr+, 137Ba+, 111Cd+, 171Yb+
optical transitions
Ca+, Sr+, Ba+, Ra+, Yb+, Hg+ etc.
S1/2
P1/2
D5/2
S1/2
P3/2
Qubits with trapped ions
Linear RF Paul trap
RF electrode
High dc potential
control electrodeLow dc voltage
control electrode
Positive ion
• Drive freq ~ 100-150 MHz• RF amp ~ 200-400 V• Secular freq
– Radial ~ 15 MHz– Axial ~ 4 MHz
Slides from :R. Ozeri
5
Multi-zone ion trap
control
controlrf
rf
view along axis:
rf filter board• Gold on alumina
construction• RF quadrupole
realized in two layers
• Six trapping zones• Both loading and
experimental zones
• One narrow separation zone
• Closest electrode ~140 m from ion
segmented
linear trapping
region
Ion transport
100 m
6-zone alumina/gold trap(Murray Barrett, Tobias Schaetz et al.)
200 mseparation zone• Ions can be moved
between traps.– Electrode potentials
varied with time
• Ions can be separated efficiently in sep. zone– Small electrode’s
potential raised
• Motion (relatively) fast– Shuttling: several 10 s– Separating: few 100 s
6
Innsbruck segmented trap (2004)
row of qubits in a linear Paul trap forms a quantum register
String of Ca+ ions in Paul trap
7
50 µm
row of qubits in a linear Paul trap forms a quantum register
MHz 20.7zMHz45.1, yx
String of Ca+ ions in linear Paul trap
50 µm
row of qubits in a linear Paul trap forms a quantum register
MHz 20.7zMHz45.1, yx
String of Ca+ ions in linear Paul trap
8
Addressing of individual ions
CCD
Paul trap
Fluorescencedetection
electrooptic deflector
coherentmanipulation of qubits
dichroicbeamsplitter
inter ion distance: ~ 4 µm
addressing waist: ~ 2.5 µm
< 0.1% intensity on neighbouring ions
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Exc
itatio
n
Deflector Voltage (V)
Ion addressing
The ions can be addressed individually on the qubit transition with an EO deflector which can quickly move the focus of the 729 light from one ion to another, using the same optical path as the fluorescence detection via the CCD camera.
How well the addressing works is shown on the previous slide: The graph shows the excitation of the indiviual ions as the deflector is scanned across the crystal.
9
External degree of freedom: ion motion
Notes for next slides:
Now let's have a look at the qubit transition in the presence of the motional degrees of freedom. If we focus on just one motional mode , we just get a ladder of harmonic oscillator levels.
The joint (motion + electronic energy level) system shows a double ladder structure. With the narrow laser we can selectively excite the carrier transition, where the motional state remains unchanged...
Or use the blue sideband and red sideband transitions, where we can change the motional state.
We can walk down the double ladder by exciting the red sideband and returning the ion dissipatively to the grounsstate. With this we can prepare the ions in the motional ground state with high probability, thereby initializing our quantum register.
harmonic trap
…
External degree of freedom: ion motion
10
harmonic trap
…
2-level-atom joint energy levels
External degree of freedom: ion motion
harmonic trap
…
2-level-atom joint energy levels
Laser cooling to the motional ground state:
Cooling time: 5-10 ms
> 99% in motional ground state
External degree of freedom: ion motion
11
Interaction with a resonant laser beam :
: Rabi frequency
: phase of laser field
: rotation angle
Laser beam switched on for duration :
Coherent manipulation
harmonic trap2-level-atom joint energy levels
…
If we resonantly shine in light pulse at the carrier transition, the system evolves for a time tau with this Hamiltonian,
where the coupling strength Omega depends on the sqroot of the intensity, and phi is the phase of the laser
field with respect to the atomic polarization.
Coherent manipulation
Let's now begin to look at the coherent state manipulation. If we resonantly shine the light pulse at the carrier transition, the system evolves for a time with this Hamiltonian, where the coupling strength depends on the square root of the intensity, and is the phase of the laser field with respect to the atomic polarization.
The effect of such a pulse is a rotation of the state vector on the Bloch sphere, where the poles represent the two states and the equator represents superposition states with different relative phases. The roation axis is determined by the laser frequency and phase. The important message is here that we can position the state vector anywhere on the Bloch sphere, which is a way of saying that we can create arbitrary superposition states.
The same game works for sideband pulses. With a /2 pulse, for example, we entangle the internal and the motional state! Since the motional state is shared by all ions, we can use the motional state as a kind of bus to mediate entanglement between different qubits in the ion chain.
12
D state population
„Carrier“ pulses: Bloch sphererepresentation
Coherent excitation: Rabi oscillations
coupled system„Blue sideband“ pulses:
D state population
Entanglement between internal and motional state !
Coherent excitation on the sideband
13
P1/2 D5/2
=1s
S1/2
40Ca+
Experimental procedure
1. Initialization in a pure quantum state:laser cooling,optical pumping
3. Quantum state measurementby fluorescence detection
2. Quantum state manipulation onS1/2 – D5/2 qubit transition
P1/2
S1/2
D5/2
Dopplercooling Sideband
cooling
P1/2
S1/2
D5/2
Quantum statemanipulation
P1/2
S1/2
D5/2
Fluorescencedetection
50 experiments / s
Repeat experiments100-200 times
One ion : Fluorescence histogram
counts per 2 ms0 20 40 60 80 100 120
0
1
2
3
4
5
6
7
8S1/2 stateD5/2 state
P1/2 D5/2
=1s
S1/2
40Ca+
Experimental procedure
1. Initialization in a pure quantum state:Laser sideband cooling
3. Quantum state measurementby fluorescence detection
2. Quantum state manipulation onS1/2 – D5/2 transition
P1/2
S1/2
D5/2
Dopplercooling Sideband
cooling
P1/2
S1/2
D5/2
Quantum statemanipulation
P1/2
S1/2
D5/2
Fluorescencedetection
50 experiments / s
Repeat experiments100-200 times
Spatially resolveddetection withCCD camera:
Multiple ions:
14
1. Basic experimental techniques
2. Two-particle entanglement
3. Multi-particle entanglement
4. Implementation of a CNOT gate
5. Teleportation
6. Outlook
Pulse sequence:
…
… …
…
Creation of Bell state
15
Generation of Bell states
Ion 1: /2 , blue sideband
Pulse sequence:
…
… …
…
Creation of Bell states
Ion 1: /2 , blue sideband
Ion 2: , carrier
Pulse sequence:
…
… …
…
Creation of Bell states
16
Ion 1: /2 , blue sideband
Ion 2: , carrier
Ion 2: , blue sideband
Pulse sequence:
…
… …
…
Creation of Bell states
Fluorescencedetection withCCD camera:
Coherent superposition or incoherent mixture ?
What is the relative phase of the superposition ?
SSSDDS
DD SSSDDSDD
Measurement of the density matrix:
Analysis of Bell states
17
Reconstruction of a density matrix
Representation of as a sum of orthogonal observables Ai :
is completely detemined by the expectation values <Ai> :
For a two-ion system :
Joint measurements of all spin components
Finally: maximum likelihood estimation (Hradil ’97, Banaszek ’99)
Preparation and tomography of Bell states
SSSD
DSDD SSSDDSDD
SSSD
DSDD SSSDDSDD
SSSD
DSDD SSSDDSDD
Fidelity:
Entanglementof formation:
Violation of Bell inequality:
F = 0.91F = 0.91
E(exp) = 0.79
S(exp) = 2.52(6)> 2
SSSD
DSDD SSSDDSDD
C. Roos et al., Phys. Rev. Lett. 92, 220402 (2004)
18
sensitive to:
laser frequency magnetic field exc. state lifetime
Different decoherence porperties
1. Basic experimental techniques
2. Two-particle entanglement
3. Multi-particle entanglement
4. Implementation of a CNOT gate
5. Teleportation
6. Outlook
19
Ion 1: 1 , blue sideband
Ion 2,3: , carrier
Ion 2: 2 , blue sideband
Pulse sequence:
…… …
…Ion 3: 3 , blue sideband
1. 2.3.
Generation of W-states
Density matrix of W – state
experimental result theoretical expectation
Fidelity: 85 %
20
DDDDDDDS
SSSSDDDD
SSSS
14.4.2005
Four-ion W-states
DDDDDDDDDS
SSSSS
SSSSS
DDDDD15.4.2005
Five-ion W-states
21
Ion detectionon a CCD camera
(detection time:4ms)
5µm
5 10 15 20 25
1
2
3
5 10 15 20 25
1
2
3
5 10 15 20 25
1
2
3
5 10 15 20 25
1
2
3
5 10 15 20 25
1
2
3
5 10 15 20 25
1
2
3
5 10 15 20 25
1
2
3
5 10 15 20 25
1
2
3
all ions in |S>
ion 1 in |S>
ion 6 in |S>
ion 4 in |S>
ion 5 in |S>
ions 1 and 5 in |S>
ions 1,2,3, and 5 in |S>
ions 1,3 and 4 in |S>
123456
Detection of six individual ions
22.4.2005729 settings, measurement time >30 min.
F=73%preliminary
result
Is there 6-particle entanglement present?
• 6-particle W-statecan be distilled from thestate (O. Gühne)
• 6-particle entanglementpresent, unresolved issues with error bars
Six-ion W-state
22
control
target
1. Basic experimental techniques
2. Two-particle entanglement
3. Multi-particle entanglement
4. Implementation of a CNOT gate
5. Teleportation
6. Outlook
other gate proposals include: • Cirac & Zoller • Mølmer & Sørensen, Milburn• Jonathan & Plenio & Knight• Geometric phases• Leibfried & Wineland
controlcontrol targettarget
...allows the realization of a universal quantum computer !
control
target
Cirac-Zoller two-ion controlled-NOT operation
23
ion 1
motion
ion 2
control qubit
target qubit
SWAP
Cirac-Zoller two-ion controlled-NOT operation
1 2
First we swap the quantum states of the control qubit and the motional qubit…
ion 1
motion
ion 2
control qubit
target qubit
Cirac-Zoller two-ion controlled-NOT operation
1 2
Next we perform a CNOT gate between the motional qubit and ion 2 and …
24
ion 1
motion
ion 2
SWAP-1control qubit
target qubit
Cirac - Zoller two-ion controlled-NOT operation
1 2
F. Schmidt-Kaler et al., Nature 422, 408 (2003)
Finally we reverse the SWAP operation,
we have used the motional qubit as a
quantum messenger between the two ions.
ion 1
motion
ion 2
SWAP-1SWAP
Ion 1Ion 1
Ion 2Ion 2
pulse sequence:pulse sequence:
Cirac - Zoller two-ion controlled-NOT operation
control qubitcontrol qubit
target qubittarget qubit
laser frequencypulse lengthoptical phase
Phase gate
25
Experimental fidelity of Cirac-Zoller CNOT operation
input
output
F. Schmidt-Kaler et al.,Nature 422, 408 (2003)
Protecting qubits (from being measured)
26
Threshold
ion #1 in |D>
Tomography after the measurement result is available!
ion #1 in |S>
Selective read-out of an atom in a W-state
1. Basic experimental techniques
2. Two-particle entanglement
3. Multi-particle entanglement
4. Implementation of a CNOT gate
5. Teleportation
6. Outlook
27
No : • Infinite amount of information needed to specify • Measurement on yields just one bit of information
Phys. Rev. Lett. 70, 1895 (1993)
„Alice“ „Bob“
A Bclassical communication
Is it possible to transfer an unknown quantum state
from „Alice“ to „Bob“ by classical communication ?
unknown state
Two bits
Yes, if Alice and Bob share a pair of entangled particles !
EPR pair
Quantum state teleportation
Ion 3
Ion 2
Ion 1
Bell
state
initialize #1, #2, #3
classical communication
conditional rotations
CNOT --Bell basis
Alice
Bob
Selectiveread out
recovered on ion #3
Teleportation protocol
28
Ion 3
Ion 2
Ion 1
conditional rotations using electronic logic, triggered by PM signal
conditional rotations using electronic logic, triggered by PM signalP
U
U
P
C C C
B
B B B B C
CU P
B
C
C P
spin echo sequencespin echo sequence
full sequence:26 pulses + 2 measurements
B
C
blue sideband pulsesblue sideband pulses
carrier pulsescarrier pulses
P
C
B
B
Teleportation protocol, details
29
Teleportation procedure, analysis
Input state Output state
TP
Initial Final
U U-1
Fidelities
Quantum teleportation with atoms: results
83 %
class.: 67 %
no cond. op. 50 %
30
How to build a large-scale quantum computer with trapped ions
Linear crystal of 20 confined atomic 171Yb+ ions laser cooled to be nearly at rest
C. Monroe, and J. Kim Science 2013;339:1164-1169
31
Harnessing ion-trap qubits
David P. DiVincenzo Science 2011;334:50-51
Jonathan P. Home et al. Science 2009;325:1227-1230
Schematic of the sequence of operations implemented in a single processing region for building up a computation in the architecture
32
Separating memory and processor zones
Scientific American (August 2008), 299, 64-71
Ion trap structure for the shuttling of ions through a junction
C. Monroe, and J. Kim Science 2013;339:1164-1169
33
Concept of a quantum CCD trap
Image credit: National Institute of Standards and TechnologyC. Monroe, and J. Kim Science 2013;339:1164-1169
Version 2: Photonics coupling of trapped ions qubits
C. Monroe, and J. Kim Science 2013;339:1164-1169
Energy levels of trapped ion excited with a fast laser pulse (blue upward arrow) that produces single photon whose color, represented by the state or , is entangled with the resultant qubit state.
34
Version 2: Photonics coupling of trapped ions qubits
Two "communication qubit" ions, immersed in separate crystals of other ions, each produce single photons when driven by laser pulses (blue). With some probability, the photons arrive at the 50/50 beamsplitter and then interfere. If the photons are indistinguishable (in polarization and color), then they always leave the beamsplitter along the same path. The simultaneous detection of photons at the two output detectors means that the photons were different colors, this coincidence detection heralds the entanglement of the trapped ion qubits.
Scientific American (2008), 299, 64-71
35
Scientific American (2008), 299, 64-71
Elementary logic unit (ELU)
Phys. Rev. A 89, 022317 (2013)
36
Modular distributed quantum computer
Several elementary logic units (ELU)s are connected through a photonic network by using an optical crossconnect switch, inline fiber beamsplitters, and a photon-counting imager.
C. Monroe, and J. Kim Science 2013;339:1164-1169
Application to quantum communication
Trapped ion quantum repeater node made up of communication qubit ions (such as Ba+) and memory qubit ions (such as Yb+), with two optical interfaces per node. Multiple communication qubits are used per optical interface to inject photons into the optical channel, while the results for successful entanglement generation at the detectors are reported back to this node. Only qubits corresponding to successful events will be transported to the memory qubit region for use in quantum repeater protocol.
C. Monroe, and J. Kim Science 2013;339:1164-1169
37
Long-distance quantum communication
C. Monroe, and J. Kim Science 2013;339:1164-1169
A chain of quantum repeater nodes can distribute quantum entanglement over macroscopic distances. The photons generated by the ions must be converted to telecommunication wavelengths for long-distance transport, which can be achieved by nonlinear optical processes.