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Departments of Physics and Applied Physics, Yale University
Introduction to Circuit QEDLecture 2
TheorySMGLiang JiangLeonid GlazmanM. Mirrahimi
Marios MichaelVictor AlbertRichard BrierleyClaudia De GrandiZaki LeghtasJuha SalmilehtoMatti SilveriUri VoolHuaixui ZhengYaxing Zhang+…..
ExperimentMichel DevoretLuigi FrunzioRob Schoelkopf
Andrei Petrenko Nissim OfekReinier HeeresPhilip ReinholdYehan LiuZaki LeghtasBrian Vlastakis+…..
http://quantuminstitute.yale.edu/
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- use quantized light shift of qubit frequency
†q 2
2za a
Reminder from Lecture 1: Measuring Photon Number Parity
ˆ ˆ22 2ez z
i nt i ne
ˆ 0,2,4,...n ˆ 1,3,5,...n x
z
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Lecture 2: Quantum State Manipulation
and Measurement in Circuit QED
The ability to measure photon number parity withoutmeasuring photon number is an incredibly powerful tool.
• Quantum Optics at the Single Photon Level
• Measuring Wigner Functions
• Creating and Verifying Schrödinger Cat States
• Cat in Two Boxes
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Quantum optics at the single photon level
0 1 2 30 1 2 3a a a a
• Photon state engineering
Goal: arbitrary photon Fock state superpositions
Use the coupling between the cavity (harmonic oscillator)and the two-level qubit (anharmonic oscillator) to achieve this goal.
Dispersively coupled cavity-qubit system is fully controllable.
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Previous State of the Art for Complex Oscillator States
Haroche/Raimond, 2008 Rydberg (ENS)
Expt’l. Wigner tomography: Leibfried et al., 1996 ion traps (NIST – Wineland group)
Hofheinz et al., 2009 (UCSB – Martinis/Cleland)
Rydberg atom cavity QED Phase qubit circuit QED
~ 10 photons ~ 10 photons
Q5
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What concepts do we need to know to understand a
Schrödinger Cat State?
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Photons in First Quantization
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Coherent state is closest thing to a classicalsinusoidal RF signal
0( ) ( )
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9(normalization is only approximate)
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1even2
1odd2
(normalization approx. only)
How cats die:
even odd
odd even
a
a
2
2
2 | | (4 )2
n
n
a
a
Novel property:
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How do we create a cat?
‘Classical’ signal generators only displace the vacuum and create coherent states.
We need some non-linear coupling to the cavity via a qubit.
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Strong Dispersive Hamiltonian
q† †r damping2
z zH a a a a H
resonator qubit dispersivecoupling
rcavity frequency z
eg
r r
‘strong-dispersive’ limit
32 ~ 2 10
,
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Strong‐Dispersive Limit yields a powerful toolbox
eg
r r
Microwave pulse at this frequency excites cavityonly if qubit is in ground state
Microwave pulse at this frequency excites cavityonly if qubit is in excited state
gD Conditional displacement of cavityEngineer’s tool #1:
Cavity frequency depends onqubit state
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n Conditional flip of qubit if exactly n photonsEngineer’s tool #2:
q† †r damping2
z zH a a a a H
resonator qubit dispersivecoupling
Reinterpret dispersive term:- quantized light shift of qubit frequency
†q 2
2za a
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3 124…
- quantized light shift of qubit frequency(coherent microwave state)
†q 2
2za a
N.B. power broadened100X
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†DISPERSIVE
zV a a
strong dispersive coupling I
2
n 0n 1n 2
Qubit Spectroscopy
Coherent state in the cavity
Conditional bit flip n
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Strong Dispersive Coupling Gives Powerful Tool Set
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Cavity conditioned bit flip
Qubit-conditioned cavity displacement gD
n
• multi-qubit geometric entangling phase gates (Paik et al.)• Schrödinger cats are now ‘easy’ (Kirchmair et al.)
experiment theoryG. Kirchmair M. MirrahimiB. Vlastakis Z. LeghtasA. Petrenko
Photon Schrödinger cats on demand
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Deterministic Cat State Production
Will skip over details of cat state production;Focus on proving the cat is not an incoherent mixture:
- measure photon number parity in the cat
- measure the Wigner function (phase space distribution of cat)
Vlastakis et al. Science 342, 607 (2013)
12
g
18
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Photon number
Rea
dout
sig
nal
0246810
Spectroscopy frequency (GHz)
Coherent state:
Mean photon number: 4
Even parity cat state:
Odd parity cat state:
Only photon numbers: 0, 2, 4, …
Only photon numbers: 1, 3, 5, …
Proving phase coherence via photon number distribution
P̂
P̂
2
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Qubit Spectrum
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1.0 1.0
2.0 2.0 ODD CAT
ODD CAT EVEN CAT
EVEN CAT
Number of parity jumps Number of parity jumps
Prob
abilit
y (%
)Pr
obab
ility
(%)
Prob
abilit
y (%
)Pr
obab
ility
(%)
0 2 4 6 8 100 2 4 6 8 10
0 2 4 6 8 10Number of parity jumps
0 2 4 6 8 10Number of parity jumps
12 12
1212
20
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We have proven our states have the correct parity and photon number distribution.
We have not (strictly) verified all the phases are correct.
Need full state tomography via measurement of the Wigner Function.
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Wigner Function MeasurementVlastakis, Kirchmair, et al., Science (2013)
Density Matrix:*( , ) ( ) )(
Wigner Function (definition):
, )2
( ( , )2
iQr r rQ dr eW
22
, r2 2
Define center of mass and relative coordinates:
Combines position and momentum information by Fourier transforming relative coordinate
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Wigner Function = “Displaced Parity”Vlastakis, Kirchmair, et al., Science (2013)
(( ˆ) ) ( )D DW P
Handy identity (Luterbach and Davidovitch):
ˆˆ ( 1) parityNP
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Full state tomography on large dimensional Hilbert space can be done very simply over a single input-output wire.
Simple Recipe: 1. Apply microwave tone to displace
oscillator in phase space.2. Measure mean parity.
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Wigner Function of a Coherent State
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‐4
Re
Im Q
(( ˆ) ) ( )D DW P ˆˆ ( 1) parityNP
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‐4
Re
Im Q
(( ˆ) ) ( )D DW P ˆˆ ( 1) parityNP
Wigner Function of a Coherent State
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0
0
4
4
‐4
‐4
Wigner Function of a Cat StateVlastakis, Kirchmair, et al., Science (2013)
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Re
Im Q
Rapid parity oscillationsWith small displacements
Interference fringes prove cat is coherent:
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0
0
44
‐4
‐4
Deterministic Cat State Production
Data!
Expt’l Wigner function
Vlastakis, Kirchmair, et al., Science (2013)
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0
0
44
‐4
‐4
Deterministic Cat State Production
0.8
0.4
0.0
‐0.4
‐0.8‐2 0 2 ‐2 0 2 ‐2 0 2 ‐2 0 2
18.7 photons 32.0 photons 38.5 photons 111 photons
determined by fringe frequency
Data!
Expt’l Wigner function
111 photons
Most macroscopic superposition ever created?
Vlastakis, Kirchmair, et al., Science (2013)
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0
0
44
‐4
‐4
Deterministic Photon Cat Production
0.8
0.4
0.0
‐0.4
‐0.8‐2 0 2 ‐2 0 2 ‐2 0 2 ‐2 0 2
18.7 photons 32.0 photons 38.5 photons 111 photons
determined by fringe frequency
Three-component cat: Four-component cat:
111 photons
Vlastakis, Kirchmair, et al., Science (2013)
Zurek ‘compass’ state for sub-Heisenberg metrology
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Non-Deterministic Cat State Production
Using Parity Measurement
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Cat State = Coherent State Projected onto Parity
L. Sun et al., Nature (July 2014)
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even odd2 2 2 2
time evolve to entangle spin with cat states:
even odd
2 2
x x x
x x
ˆ ˆ22 2ez z
i nt i ne
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Wigner Tomography of cats entangled with qubitL. Sun et al., Nature (July 2014)
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even odd2 2
x x
Wigner function of cavity (tracing out qubit) yields an incoherent MIXTURE of two coherent states and not a cat. (no fringes)
Equivalently: mixture of even and odd cats.
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“qubit is in |+x>”
“qubit is in |-x>”
Fidelity of produced cats:
Wigner Tomography Conditioned on Qubit StateL. Sun et al., Nature (July 2014)
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Cat In Two Boxes
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Cat in Two Boxes Qubit measures joint parity!1 2ˆ ˆ( )
12 1 2i n nP P eP
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Theoretical proposal by Paris group:Eur. Phys. J. D 32, 233–239 (2005)
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Cat in Two Boxes Qubit measures joint parity!1 2ˆ ˆ( )
12 1 2i n nP P eP
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Experiment by Yale group:Science 352, 1087 (2016)
- Universal controllability- 3-level qubit can measure
1 2 12, , and P PP
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Cat in Two Boxes
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Two-cavities:4-dimensional phase space and Wigner functions.
Theory
Experiment
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Entanglement of Two Logical Cat-Qubits
CHSH Bell: 2 2 2B
CHSH: (evaluate Wigner at 4 points in 4D phase space)
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The ability to measure photon number parity without measuring photon number is an incredibly powerful tool.
Lecture 2: Using parity measurements for:
• Wigner Function Measurements• Creation and verification of photon cat
states
Lecture 3: Using parity measurements for:
• Continuous variable quantum error correction
Summary of Lecture 2:
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For separate discussion offline:
Detailed Recipe to Make a
1. Schrödinger Cat2. Schrödinger Cat State
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Strong Dispersive Coupling Gives Powerful Tool Set
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Cavity conditioned bit flip
Qubit-conditioned cavity displacement gD
n
• multi-qubit geometric entangling phase gates (Paik et al.)• Schrödinger cats are now ‘easy’ (Kirchmair et al.)
experiment theoryG. Kirchmair M. MirrahimiB. Vlastakis Z. LeghtasA. Petrenko
Photon Schrödinger cats on demand
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cavity
qubitP
M
Making a cat: the experiment
Q
(*fine print for the experts: this is the Husimi Q function not Wigner)43
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cavity
qubitP
M
Making a cat: the experiment
Q
44
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cavity
qubitP
M
Making a cat: the experiment
Q
45
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cavity
qubitP
M
Making a cat: the experiment
Q
46
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cavity
qubitP
M
Making a cat:
qubit acquires phase per photon…
t after time:
Q
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cavity
qubitP
M
Making a cat:
qubit acquires phase per photon…
t after time:
Qubit fully entangled with cavity‘cat is dead; poison bottle open’‘cat is alive; poison bottle closed’
Q
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12
g
Qubit in ground state; cavity in photon cat state
12
g e
We have a ‘cat’
We want a ‘cat state’
How do we disentangle the qubit from the cavity?
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Combining conditional cavity displacements with conditionalqubit flips, one can disentangle the qubit from the photons
12
g e ‘cat’
D 2 012
eg
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Combining conditional cavity displacements with conditionalqubit flips, one can disentangle the qubit from the photons
D
12
g e ‘cat’
2 012
g g 0
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Combining conditional cavity displacements with conditionalqubit flips, one can disentangle the qubit from the photons
D
12
g e ‘cat’
12
g g 0D
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Combining conditional cavity displacements with conditionalqubit flips, one can disentangle the qubit from the photons
12
g D0gD
12
g e ‘cat’
‘cat state’