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

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

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

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

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

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)

…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…

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

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

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?

‘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)

Extended Jaynes-Cummings Ladder

J-C Hamiltonian extended to include higher transmon levels:

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)

Full model

• Extended Jaynes-Cummings Hamiltonian with drive:

• Include dissipation via Master equation

• Measure heterodyne transmission amplitude, not

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)

Strong-dispersive regime

• Cavity-pull Â=g2/± many linewidths, though g/±À 1

D I Schuster et al Nature 445, 515

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)

Essential mechanism• Diminishing anharmonicity of the

Hamiltonian

~!c

Undriven HamiltonianJC Hamiltonian

Exact Diagonalization

detuning total excitations critical photon number

HUGE simplification: seems unlikely to be usefulbut let’s try anyway

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

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

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

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

Transient (via trajectories)

t=2.5/·

Steady state (via solution of M.E.)

t=1

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

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)

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)

Semiclassical equation• Self-consistent equations for the amplitude

A2=X2+P2

• Treat A as constant (ignore harmonic generation, chaos)

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

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!

Switching

• Slow timescale À cavity lifetime• Initialize in g.s., takes a long time for dip to

move to the left

Lots of gain near C2

Log scale

Linear scale

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

Symmetry breaking

Pure 2-level qubithas (almost) symmetry

Two qubits, one ‘active’ one ‘spectator’

One transmon

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

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’

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

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)

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

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

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]

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); · ;°]

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

Initial chirp: achieving selectivity via coherent oscillation

• Chirping in the quasi-dispersive regime can be thought of as oscillator ringing

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

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.

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!