iqc 2011-10-17 lev s bishop strong driving in circuit qed texpoint fonts used in emf. read the...
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IQC 2011-10-17
Lev S Bishop
Strong driving in Circuit QED
Collaborators:
Theory:
Eran Ginossar (Surrey)
Erkki Thuneberg (Oulu)
Jens Koch (Northwestern)
Steve Girvin (Yale)
Funding:
Experiment:
Jerry Chow (IBM)
Andrew Houck (Princeton)
Matt Reed (Yale)
Leo DiCarlo (Delft)
Dave Schuster (Chicago)
Rob Schoelkopf (Yale)
…
Joint Quantum Institute and
Condensed Matter Theory Center, University of Maryland
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Outline
• Background– Circuit QED, approximations, Jaynes-Cummings
• Resonant strong coupling regime (quantum oscillator)– Photon Blockade, multiphoton transitions, supersplitting
• Strong-dispersive regime (semiclassical oscillator)– Special kind of bifurcation with 2 critical points– readout
• Intermediate regime– Quantum control and readout
• Conclusions and future directions
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Jaynes-Cummings Physics
Qubit=atom=transmon Cavity=resonator coupling
(two-level approx.: Rabi)
(RWA: Jaynes-Cummings)
Open-system (drive & dissipation) is where it gets interesting
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These circuits are designed for quantum computing
• DiCarlo et al., Nature 460, 240-244, (2009)
• Real part of 2-qubit density matrix
• Measured (not theory)
• 85% algorithm fidelity
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From cavity QED to circuit QED
•Strong coupling, strongly dispersive regimes: easy with circuit QED
•Atom spatially fixed, no field inhomogeneity effects, etc
•Drive strength easily tunable over a wide power range
•Atom frequency can be tuned quickly
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Quantum optics with circuits…
Probing photon states via ‘number splitting’ effect
! Transmon as a detector for photon states
J. Gambetta et al., PRA 74, 042318 (2006)
D. Schuster et al., Nature 445, 515 (2007)
Single microwave photons ‘on demand’
! Transmon as a microwave photon emitter
A. A. Houck et al., Nature 449, 328 (2007)
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…More quantum optics with circuits
Generation of Fock states and measurement of subsequent decay
! Phase qubit used to climb the Fock state ladder one rung at a time
H. Wang et al., PRL 101, 240401 (2008)
Generation of arbitrary states of a resonator
M. Hofheinz et al. Nature 454, 310 (2008)
And more…
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Outline
• Background– Circuit QED, approximations, Jaynes-Cummings
• Resonant strong coupling regime (quantum oscillator)– Photon Blockade, multiphoton transitions, supersplitting
• Strong-dispersive regime (semiclassical oscillator)– Special kind of bifurcation with 2 critical points– readout
• Intermediate regime– Quantum control and readout
• Conclusions and future directions
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A. Wallraff et al., Nature 431, 162 (2004)
Strong coupling: Vacuum Rabi Splitting
•Signature for strong coupling
Placing a single resonant atom inside the cavity leads to a splitting of the cavity transmission peak
Vacuum Rabi Splitting
Observed in:
Cavity QED:R. J. Thompson et al, Phys. Rev. Lett 68, 1132 (1992)
Circuit QED:A. Wallraff et al., Nature 431, 162 (2004)
Quantum dot systems:J.P. Reithmaier et al., Nature 432, 197 (2004)T. Yoshie et al., Nature 432, 200 (2004)
2008
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Vacuum Rabi splitting: Linear Response
• Jaynes-Cummings model
• Lorentzian lineshape• Separation: • Linewidth:
Circuit QED is ideally suited to go beyond linear response
Increase of microwave power is simple
Atom is spatially fixed
Question: heterodyne transmission beyond linear response?
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‘Supersplitting’ and n peaks
Two main results:
1) Supersplitting of each vacuum Rabi peak
Simple 2-level model based on ‘dressing of dressed states’
(H. J. Carmichael)
2) Emergence of n peaks
Probing higher levels in the Jaynes-Cummings ladder (n anharmonicity)
Here: up to n=6
Related work on n anharmonicity:I. Schuster et al., Nature Physics 4, 382 (2008)J. M. Fink et al., Nature 454, 315 (2008)M. Hofheinz et al., Nature 459, 546 (2009)
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Extended Jaynes-Cummings Ladder
J-C Hamiltonian extended to include higher transmon levels:
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Supersplitting: 2-level model
Restriction to 2-level subspace:
‘Dressing of dressed states’
Measure heterodyne amplitude:(Not aya as in photon counting)
Steady state solution of Bloch equations:
(T1, T2 get renormalized)
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Full model
• Extended Jaynes-Cummings Hamiltonian with drive:
• Include dissipation via Master equation
• Measure heterodyne transmission amplitude, not
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Outline
• Background– Circuit QED, approximations, Jaynes-Cummings
• Resonant strong coupling regime (quantum oscillator)– Photon Blockade, multiphoton transitions, supersplitting
• Strong-dispersive regime (semiclassical oscillator)– Special kind of bifurcation with 2 critical points– readout
• Intermediate regime– Quantum control and readout
• Conclusions and future directions
LSB, Ginossar, Girvin PRL 105, 100505 (2010)Boissonneault, Gambetta, Blais PRL 105, 100504 (2010) Reed et al PRL 105, 173601 (2010)
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Strong-dispersive regime
• Cavity-pull Â=g2/± many linewidths, though g/±À 1
D I Schuster et al Nature 445, 515
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A strange dataset
! c + Â
! c
Four transmonsVery strong driving (10,000 photons if linear response)Strong-dispersive bad-cavity regime
MD Reed et al. PRL 105, 173601 (2010)
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Essential mechanism• Diminishing anharmonicity of the
Hamiltonian
~!c
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Undriven HamiltonianJC Hamiltonian
Exact Diagonalization
detuning total excitations critical photon number
HUGE simplification: seems unlikely to be usefulbut let’s try anyway
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Perturbative expansion
Dispersive approximation
Plus Kerr term…
Can continue the expansion, but only converges for
Expand in
For typical cQED parameters, the dispersive approximation breaks downbefore N=Ncrit: anharmonicity ®=2g4/±3 is approx. linewidth
g=200MHz, ±=1 GHz, ®=3.2MHz
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Transition frequencies
0 .010 0 .005 0 .005 0 .010
200
400
600
800
1000
! c + Â! c ¡ Â
!c-!ij
g/±=0.1
|1i|0i
Transition frequency
n
Kerr nonlinearity :H = ! aya+ ´(aya)2
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Transformed drive & dissipation
Matrix elements of a do not change, O(n-1/2)*O(g/±)
Elements of ¾z , ¾§ do change, cf “dressed dephasing” Boissonneault et al, PRA 79, 013819 (2009), PRA 77, 060305(R) (2008)
Take ‘bad cavity limit’ ·À°, look at timescales short compared to the qubit relaxation t¿1/° (‘freeze the qubit’)
Remaining degree of freedom is the JC oscillator
0 20 40 60 80 100
0
2
4
6
8
10
n
g/±=0.1
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Master equation
•Heterodyne amplitude: |hai|•Effective parameters are chosen to be representative, not fitted•Integrate to t=2.5/· using quantum trajectories
-RWA in the drive-Truncate at 10,000 Fock states (up to ~1 cpu week/pixel)
-Inefficient, can be improved-(NB Transient: Steady-state quantitatively different)
experiment theory
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Transient (via trajectories)
t=2.5/·
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Steady state (via solution of M.E.)
t=1
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Why does JC model work?• Several reasons to be surprised!• Multiple transmons• Higher transmon levels (>10 occupied)• Breakdown of RWA going from Rabi to JC
Hamiltonian • Answer: Still exhibits return to bare frequency
6 .96 6 .98 7 .00 7 .02 7 .04 7 .06
200
400
600
800
JCRabi
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Semiclassical JC Oscillator• Quantum model works nicely, but want to simplify further
• In limit of anharmonicity ¿ linewidth. – final part of my talk is about opposite limit
• Rewrite Hamiltonian in terms of canonical variables
gives
cf Peano & Thorwart, EPL 89,17008 (2010)
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Semiclassical potential
• Perturbation to quadratic potential looks like |X| for large X
10 5 0 5 100
2
4
6
8
10
X
Sqr
t(1+
N/N
crit)
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Semiclassical equation• Self-consistent equations for the amplitude
A2=X2+P2
• Treat A as constant (ignore harmonic generation, chaos)
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Semiclassical results
Region of bistability
Like a phase diagram with 2 critical points
(careful, no Maxwell construction, etc)
Dip is in classically bistable regionReadout protocol operates close to upper critical point
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Frequency response
• Dip is from noise-driven switching between semiclassically allowed states• Analytic solution (hypergeometric functions) for the case of a Kerr oscillator
- Including dip and even multiphoton peaks!
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Switching
• Slow timescale À cavity lifetime• Initialize in g.s., takes a long time for dip to
move to the left
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Lots of gain near C2
Log scale
Linear scale
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How to use this for qubit readout?
Not for one-qubit case, because of symmetry
|1i
|0i
0 .010 0 .005 0 .005 0 .010
200
400
600
800
1000
15 10 5 010
0
10
20
30
|1i|0i
Input power/dB
Tra
ns.
pow
er/d
B
(there is still information in the phase)
! d = ! c
Neglect for large N
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Symmetry breaking
Pure 2-level qubithas (almost) symmetry
Two qubits, one ‘active’ one ‘spectator’
One transmon
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Comparison to JBA/Kerr Oscillator
• Uses nonlinearity of qubit, not additional element• Non-latching mode of operation
– JBA could do this also: similar gain at C1 and C2
• C2 easy to find, brighter• Frequency of C2 ‘independent’ of qubit state• Chaos? cf Mallet, F. Ong, et al Nature Physics 5 (2009) 791
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Other single-atom bistabilities
• Absorptive bistability– V. different regime: weak coupling, good cavity– Maxwell-Bloch (keeps qubit dynamics)
• Spontaneous dressed-state polarization/single-atom phase stability– Strong coupling, bad cavity– But: qubit & cavity on resonance– Drive above ‘»2’
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Conclusions
• JC oscillator is appropriate qualitative model for the readout– Surprising: return to bare frequency is the important
thing
• Beyond dispersive approximation• Beyond Kerr nonlinearity• Beyond perturbation expansion• A new kind of nonlinear oscillator(?)• Lots of gain at C2
• Special kind of symmetry breaking (»2 depends on transmon state(s), but not )– Is very helpful for readout
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Outline
• Background– Circuit QED, approximations, Jaynes-Cummings
• Resonant strong coupling regime (quantum oscillator)– Photon Blockade, multiphoton transitions, supersplitting
• Strong-dispersive regime (semiclassical oscillator)– Special kind of bifurcation with 2 critical points– readout
• Intermediate regime– Quantum control and readout
• Conclusions and future directions
Ginossar, LSB, Schuster, Girvin. Phys. Rev. A 82, 022335 (2010)
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Quasi-harmonic long lived states
Coherent state with average occupation <n> obeying approximately
! ¹n+2¾¡ ! ¹n¡ 2¾ ¼·
0 10 20 30 40 50 600
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
n
P(n
)
<n>=25
Total frequency shift from “end-to-end” due to anharmonicity should be smaller than the linewidth.
¹n + 2p
¹n¹n ¡ 2p
¹n
find quasi-harmonic states, co-existing with photon-blockaded states (for same parameters and drive).
Quantum states coexisting with semiclassical states (bistability)
Photon blockade
Neither small Hilbert space nor point in classical phase-space
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0 20 40 60 80 100 1200
0.01
0.02
0.03
0.04
0.05
0.06
0.07several time slices (=0) from before post-selection
initial state
after 9-1after -1
Quantum trajectory simulations of quasi-coherent states
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Coexistence of blockaded and long lived
quasi-coherent states
• Lifetime is large on the scale of the cavity lifetime
• Should be obtainable experimentally for typical circuit QED parameters
Probability for decay after · ¡ 1 Quasi-coherent states lifetimes
Cavity drive strength [GHz]
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High fidelity readout : a dynamical mapping
• Idea: use co-existence of bright (quasi-harmonic) and dim (photon blockade) states to readout qubit.
• Selective state transfer problem in quantum coherent control
jn = 0i jn ¼0i
jbrighti
L [! d(t);»d(t);! q(t); · ;°]
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High fidelity readout : Coherent control
1) An initial strong pulse excites the cavity-qubit system selectively (quasi-dispersive regime)
2) A weak long pulse displaces the quasi-coherent state and does not affect the blockaded state, thus generating the readout contrast.
Optimization of a linear chirp readout protocol in the bistable regime
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Initial chirp: achieving selectivity via coherent oscillation
• Chirping in the quasi-dispersive regime can be thought of as oscillator ringing
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Cumulative probability distributions (s-curves)
• Very high fidelities for a low photon threshold, trades off with contrast
• Very Robust against variations of the system and control parameters
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Summary and outlook
• New type of bistability in the JC ladder between photon blockaded states and quasi-coherent metastable states.
•See also DiVincenzo and Smolin arXiv:1109.2490 (2011).
• We demonstrated an efficient coherent control protocol for high fidelity (98%) readout, with full quantum mechanical simulation including the decay processes.
• A simple architecture: apply a different readout protocol -No additional parts necessary on the circuit except the qubit and cavity.
Open questions:
• Theory for the timescales for switching between the bistable states?
• Apply optimal control
• Consider multi-qubit readout?
• Effect of additional levels of realistic (e.g. Transmon) systems.
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Overall conclusions• Extreme parameters of circuit QED (compared to other cavity QED
implementations) allow observation of interesting quantum optics effects in different regimes
• These can be useful for qubit readout• Some other strong driving effects (many others):
– Autler-Townes, Mollow triplet (Baur et al Phys. Rev. Lett. 102, 243602 (2009)), (Li et al Phys. Rev. B 84, 104527 (2011))
– Photon blockade (Hoffman et al Phys. Rev. Lett. 107, 053602 (2011))• Quantum control
– For gates, eg DRAG and GRAPE (Motzoi et al Phys. Rev. Lett. 103, 110501 (2009))
– For readout, eg chirped driving/autoresonance (Naaman et al PRL 101,117005 (2008)
• Better qubits, fancier architectures (multiple cavities), additional nonlinear elements, etc, etc
• Some inspiration from other cavity QED implementations, some unique to circuits.
See forthcoming “Fluctuating nonlinear oscillators” M. Dykman (ed), OUP (2011).
Thank you!