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Rob Schoelkopf, Applied Physics, Yale University
PI’s:RSMichel DevoretLuigi Frunzio Steven Girvin Leonid Glazman
Quantum Optics in Circuit QED: From Single Photons to Schrodinger Cats….
Postdocs & grad students
wanted!
Thanks to cQED Team Thru the Years!
Theory
Experiment (past)
Steve Girvin, Michel Devoret, Luigi Frunzio, Leonid Glazman
Alexandre BlaisLev BishopJay GambettaJens KochEran GinossarA. NunnenkampG. CatelaniLars TornbergTerri YuSimon NiggDong ZhouMazyar MirrahimiZaki Leghtas
Andreas WallraffDave SchusterAndrew HouckLeo DiCarloJohannes MajerBlake JohnsonJerry ChowJoe Schreier
Experiment (present)
Hanhee PaikLuyan SunGerhard KirchmairMatt ReedAdam SearsBrian VlastakisEric HollandMatt ReagorAndy FragnerAndrei PetrenkoJacob BlumhoffTeresa Brecht
Outline
• Cavity QED vs. Circuit QED
• How coherent is a Josephson junction?
• Scaling the 3D architecture
• A bit of nonlinear quantum optics
• Deterministic Schrödinger cat creation
Cavity Quantum Electrodynamics (cQED)
2g = vacuum Rabi freq.
k = cavity decay rate
g = “transverse” decay rate
† †12
ˆ ( )( ) ˆ2a
zr a a a agH H H
Quantized Field Electric dipole Interaction
2-level system
Jaynes-Cummings Hamiltonian
Strong Coupling = g > k , g
Dissipation
2012: Year of Quantum Measurement"for ground-breaking experimental methods that enable
measuring and manipulation of individual quantum systems"
Serge Haroche (ENS/Paris)
Cavity QEDw/ Rydberg atoms
Dave Wineland (NIST-Boulder)
Quantum jumpsw/ trapped ions
Josephson-junctionqubits7 GHz in
outtransmissionline “cavity”
Thy: Blais et al., Phys. Rev. A (2004)
Qubits Coupled with a Quantum Bus
“Circuit QED”
use microwave photons guided on wires!
Superconducting Qubits
nonlinearity from Josephson junction
(dissipationless)electromagnetic oscillator 01 ~ 5 10 GHz
See reviews: Devoret and Martinis, 2004; Wilhelm and Clarke, 2008
Ener
gy
0
101
1201 12 Superconductor
Superconductor (Al)
Insulating barrier1 nm
• Engineerable spectrum• Lithographically produced features• Each qubit is an “individual”• Decoherence mechanisms?
CCj
Lj
Transmon
5 m m
Vacuum fields: mode volumezero-point energy density enhanced by
Transition dipole:0 /g d E
0~ 40,000d ea
L = l ~ 2.5 cm
coaxial cable
R
6 310
610
Supports a TEM modelike a coax:
Advantages of 1d Cavity and Artificial Atom
x 10 larger than Rydberg atom
Vacuum fields: mode volumezero-point energy density enhanced by
Transition dipole:0 /g d E
0~ 40,000d ea6 310
610
Advantages of 1d Cavity and Artificial Atom
x 10 larger than Rydberg atom
1 100 MHz~ ~ 0.02
5 GHzg
6~ 10g
Circuit QED
compare Rydberg atomor optical cQED:
much easier to reach strong interaction regimes!
The Chip for Circuit QED
Qubittrapping
easy:it’s
“soldered”down!
Nb
Nb
Si Al
Expt: Wallraff et al., Nature (2004)
Cavity QED: Resonant Case
r a
vacuumRabi
oscillations
“dressed state ladders”
g e
# ofphotons
qubit state
+ ,0 ,1e g
- ,0 ,1e g
(see e.g. “Exploring the Quantum…,” S. Haroche & J.-M. Raimond)
“phobit”
“quton”
Strong Resonant Coupling: Vacuum Rabi Splitting
Review: RS and S.M. Girvin, Nature 451, 664 (2008).
Nonlinear behavior: Bishop et al., Nature Physics (2009).
2g ~ 350 MHz
Can achieve “Fine-Structure Limit”
6.75 6.85 6.95 7.05
25 9~ ~ 10 10
g
Cooperativity:
g >> [k, g]
200 MHz~ ~ ~ 0.04
5 GHz r
g
ra
But does it “compute”?
Algorithms: DiCarlo et al., Nature 460, 240 (2009).
1 ns resolution
cavity: “entanglement bus,” driver, & detector
transmon qubits
DC - 2 GHz
A Two-Qubit Processor
T = 10 mK
General Features of a Quantum Algorithm
Qubitregister
Workingqubits
M
createsuperposition
encode functionin a unitary
processinitialize
measure
will involve entanglementbetween qubits
Maintain quantum coherence
1) Start in superposition: all values at once!2) Build complex transformation out of one-qubit and two-qubit “gates”3) Somehow* make the answer we want result in a definite state at end!
*use interference: the magic of the properly designed algorithm
Total pulse sequence:104 nanoseconds
Coherence time ~ 1 ms
The correct answer is found
>80% of the time!
ideal 10 Grover Algorithm Step-by-Step
Previously implemented in NMR: Chuang et al., 1998Ion traps: Brickman et al., 2003Linear optics: Kwiat et al., 2000
Will it ever scale?
or,
“Come on, how coherent could this squalid-state thing ever really get?”
(H. Paik et al., PRL, 2011)
Progress in Superconducting Charge Qubits
Similar plots can probably be made for phase, flux qubits
Schoelkopf’s Law:Coherence increases 10x every 3 years!
Materials: Dirt Happens!Qubit: two 200 x 300 nm junctions
Rn~ 3.5 kOhmsIc ~ 40 nA
Current Density ~ 30- 40 A/cm2
Dolan Bridge TechniquePMMA/MAA bilayerAl/AlOx/Al
Why Surfaces Matter…
Increase spacings decreases energy on surfaces
increases Q Gao et al. 2008 (Caltech)O’Connell et al. 2008 (UCSB)Wang et al. 2009 (UCSB)
as shown in:
+ + --
E d
a-Al2O3
Nb
“participation ratio” = fraction of energy stored in material
even a thin (few nanometer) surface layer will store ~ 1/1000 of the energy
5 mm
If surface loss tangent is poor ( tand ~ 10-2) would limit Q ~ 105
One Way to Be Insensitive to Surfaces…3-D waveguide cavity
machined from aluminum(6061-T6, Tc ~ 1.2 K)
TE101 fundamental mode
50 mm
Observed Q’s to 5 MIncreased mode volume
decreases surface effects!
cav 100T sm
Transmon Qubit in 3D Cavity
50 mm
~ mm
g 100 MHz
Still has same net coupling!
Smaller fields compensated by larger dipole
Vacuum capacitor
-50
0
50
Sig
nal (
a.u.
)
20151050Delay Time (ms)
-50
0
50
Sig
nal (
a.u.
)
20151050Delay Time (ms)
100
50
0
T1
Sig
nal (
a.u.
)
4003002001000
Delay Time (ms)
T1 = 60 ms
T2 = 14 msmeas.
/2p /2pDt /2p
Coherence Dramatically ImprovedDt
p
Dt /2p /2/2p /2pp
Dt /2p /2Techo = 25 ms
61 2 10Q T
Ramsey Experiment/Hahn Echo
T2echo = 145 m s
Remarkable Frequency Stability
f01 = 6 808 737 605 (608) Hz
No change in Hamiltonian parameters > 80 ppb in 12 hours!?
608 Hzf
Overall precision after 12 hours: ~ 19 Hz or 3 ppb
Charge Qubit Coherence, Revised
QEC limit?
Schoelkopf’s law 10x every 3 years!
Ringdown of TE011
Fit (Black):τ = 3.7msQL=ωτ=265M
Milliseconds and Beyond?
M. Reagor et. al. to be published
Now this is aQuantum Memory for qubits!!
0.6 Billion
E
best qubits
Building Blocks for ScalingOne AtomOne Cavity
Two AtomsOne Cavity
One AtomTwo Cavities
Many AtomsMany Cavities
Two-Cavity Design
45mm
1.2mm
900μm Al2O3
500nm
Strong dispersive limit:QND measurement of single photons
Algorithms: DiCarlo et al., Nature 460, 240 (2009).
Dispersive Limit of cQED
cavity
qubit
2 242
rE n
g n
Diagonalizing J-C Hamilt.:
r a
Dispersive (>>g): 24 3
, ,0
2( / )rg n g
gE E n O g
2g
,0 ,1e g
“phobit”
,0 ,1e g “quton”
reff† †
2 a z za a a aH
Strong Dispersive Hamiltonian:
n=0
n=1
n=0
n=1
n=2
n=2
Photon Numbersplitting
2~ 0n
~ 0.5n
~ 1n
0n1n2n
Qubit Frequency (GHz)
2
~ ,g
qubi
t ab
sorp
tion
“doubly-QND” interaction
QND Measurement of Photon Number
nX
“Got any ‘s?1n
Quantum “go-fish”
gg e
cavity
qubit2) then measure qubit state using second cavity
1) perform n-dependent flip of qubit
Repeated QND of n=0 or n=1:B. Johnson, Nature Phys., 2010
“Click!”
Coherent Displacementscreate
Coherent Displacements
Using a cavity as a memory:Schrodinger cats on demandexperiment theoryG. Kirchmair M. MirrahimiB. Vlastakis Z. Leghtas
“No, no mini-Me, we don’t freeze our kitty!”
Driving a Quantum Harmonic OscillatorGiving a classical ‘drive’ to a quantum system:
Where:
with
Our state is described by two continuous variables, an amplitude and phase.A ‘coherent’ state.
Phase-space portrait of oscillator state:
x̂
p̂
ˆ cos( )
ˆ sin( )
x
p
What’s a Coherent State?
E
x
x
2 / 2x m Glauber (coherent) state
2E
0t
maxx x
What’s a Coherent State?
E
x
x
2Glauber (coherent) state
2E
2t
What’s a Coherent State?
E
x
x
2Glauber (coherent) state
2E
t
Measured Q functions of a Coherent State
21( , )Q e
D D
nX gg e
Deterministic Cat Creation
cavity
qubit
•
•
•
Deterministic Cat Creation
cavity
qubit
•
•
• •
Deterministic Cat Creation
cavity
qubit
•
•
•
cavity transmission
5nspulse
•
Deterministic Cat Creation
cavity
qubit
•
•
• •
Deterministic Cat Creation
cavity
qubit
•
•
• •
Deterministic Cat Creation
cavity
qubit
•
•
• •
Deterministic Cat Creation
cavity
qubit
•
•
• •
Deterministic Cat Creation
cavity
qubit
•
•
• •
So, What’s a Cat State?
E
x
x
2Schrödinger cat state
2E
0t
/ 2x m
1
2
2D x Superposition with distinguishability, D
So, What’s a Cat State?
E
x
x
2Schrödinger cat state
2E
/ 2x m
1
2
2t
What happens now, when packets collide?
Seeing the Interference: Wigner Function
Parity
Thy:
Negative fringes =“whiskers”
Expt’l. Wigner tomography: Leibfried et al., 1996 ion traps (NIST)Haroche/Raimond , 2008 Rydberg (ENS)Hofheinz et al., 2009 in circuits (UCSB)
Seeing the Interference: Wigner FunctionDcavity 1
cavity 2
qubit q wait X/2
mapX/2
measurement
Wigner Function: Interpretation
x
2 / 2x m
Wigner Function: Interpretation
x
2 / 2x m
2t
Fringes for different cat sizes
Creating Curious Cats
State used for a
protected memory
multicomponent interference fringesZ. Leghtas ,M. Mirrahimi et.al. arXiv. 1207.0679
(2012)
“Bulldog State?” Y
Curiouser and Curiouser…
So Now What?
“Age of Coherence”
“Age of Entanglement”
“Age of Measurement.”
“Age of Qu. Feedback.”
“Age of Qu. Error Correction.”
So Now What?• Coherence won’t be the reason it doesn’t work…
• In next few years, we will be building non-trivial (i.e. non-calculable) quantum systems from the “bottom-up”
• Beginning era of “active” quantum devices – incorporating: – quantum feedback – quantum error-correction – engineered dissipation
• Advent of analog quantum simulations and artificial many-body physics?
• What (if any?) are the medium-term applications of quantum information technology?
Error Correction with Minimal HardwareLeghtas, Mirrahimi, et al., arXiv 1207:0679
also known as a Zurek “compass state”
Then photon loss can be monitored/corrected by repeated photon parity measurement using qubit
0 ( )L C N
1 ( )L i i i C N10e eg Lg Lc cc c
• Correction for a single bit / phase flip: at least 5 qubits• A single cavity mode: infinite dimensional Hilbert Space• Minimal QEC hardware: a single cavity mode coupled to a qubit
Idea: encode a qubit in a 4 component parity state
Numerical simulations
2
40 MHz, T1,qubit T2,qubit 100ms, Tcav 2 ms, 2 4.
Circuit QED Team Members 2012
KevinChou
HanheePaik
BrianVlastakis
JacobBlumhoff
LuyanSun
LuigiFrunzio
MattReed
SteveGirvin
AndreiPetrenko
Funding:
AdamSears
Eric Holland
TeresaBrecht
NissimOfek
AndreasFragner
MichelDevoret
MattReagor
GerhardKirchmair
LeonidGlazman
Z. Leghtas M. Mirrahimi
Summary
• We won’t be able to use coherence as an excuse anymore!Qubits: T2 ~ 2*T1 ~ 0.0001 secCavities: T1 ~ 0.01 sec
• 3D approach has led to 2+ orders of magnitude improvements!
Paik et al., PRL 107, 240501 (2011).
• New physics: single-photon Kerr and dispersive revivals
• New approaches: cats in cavities as logical qubits!
Kirchmair, Vlastakis et al., in preparation.
Leghtas, Mirrahimi et al.,
ArXiv:1205.2401 and ArXiv:1207.0679