Funding:
IQC 2011-10-17
Lev S Bishop
Joint Quantum Institute and
Condensed Matter Theory Center, University of Maryland
Collaborators:
Theory:
Eran Ginossar (Surrey)
Erkki Thuneberg (Oulu)
Jens Koch (Northwestern)
Steve Girvin (Yale)
Experiment:
Jerry Chow (IBM)
Andrew Houck (Princeton)
Matt Reed (Yale)
Leo DiCarlo (Delft)
Dave Schuster (Chicago)
Rob Schoelkopf (Yale)
…

• 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

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 2qubit density matrix
• Measured (not theory)
• 85% algorithm fidelity

•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…

• 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

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
A. Wallraff et al., Nature 431 , 162 (2004)

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?

Two main results:
‘Supersplitting’ and  n peaks
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 a y a as in photon counting)
Steady state solution of Bloch equations:
(T
1
, T
2 get renormalized)

Full model
• Extended Jaynes-Cummings Hamiltonian with drive:
• Include dissipation via Master equation
• Measure heterodyne transmission amplitude, not

• 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)

2
D I Schuster et al Nature 445 , 515

!
c
MD Reed et al.
PRL 105 , 173601 (2010)
!
c
+ Â
Four transmons
Very strong driving (10,000 photons if linear response)
Strong-dispersive bad-cavity regime

~ !
c

JC Hamiltonian
HUGE simplification: seems unlikely to be useful but let’s try anyway
Exact Diagonalization detuning total excitations critical photon number

Expand in
Dispersive approximation
Plus Kerr term…
Can continue the expansion, but only converges for
For typical cQED parameters, the dispersive approximation breaks down before N=N crit
: anharmonicity ® =2g 4 / ±
3 is approx. linewidth g=200MHz, ± =1 GHz, ® =3.2MHz

!
c
¡ Â
0.010
n
1000 g/ ± =0.1
800
|0 i
600
400
|1 i
Kerr nonlinearity :
H = ! a y a + ´ (a y a) 2
200
0.005
!
c
!
ij
Transition frequency
0.005
0.010
!
c
+ Â

10 g/ ± =0.1
8
6
4
2
0
0 20 40 60 80 n
Matrix elements of a do not change, O(n -1/2 )*O(g/ ± )
100
Elements of ¾ z
, 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

•Heterodyne amplitude: | h a i |
• 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

t=2.5/ ·

t= 1

• 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
800
JC 600
400
200
Rabi
6.96
6.98
7.00
7.02
7.04
7.06

• 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)

10
8
6
4
2
0
10 5 5 10
X
0

2
2
2

Region of bistability
Like a phase diagram with 2 critical points
(careful, no Maxwell construction, etc)
Dip is in classically bistable region
Readout protocol operates close to upper critical point

• 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!

2
Log scale
Linear scale

30
!
d
= !
c
20
|0 i
10
|1 i
Neglect for large N
0
10
15 10
Input power/dB
5
Not for one-qubit case, because of symmetry
0
|0 i
1000
800
600
400
200
|1 i
0.010
0.005
0.005
(there is still information in the phase)
0.010

Pure 2-level qubit has (almost) symmetry
Two qubits, one ‘active’ one ‘spectator’
One transmon

• Uses nonlinearity of qubit, not additional element
• Non-latching mode of operation
– JBA could do this also: similar gain at C
1
• C
2 easy to find, brighter
• Frequency of C
2
‘independent’ of qubit state
• Chaos?
and C
2 cf Mallet, F. Ong, et al Nature Physics 5 (2009) 791

• 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
’

• 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 C
2
• Special kind of symmetry breaking ( »
2 on transmon state(s), but not

2
– Is very helpful for readout
) depends

• 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 ·
Neither small Hilbert space nor point in classical phase-space
Total frequency shift from “end-toend” due to anharmonicity should be smaller than the linewidth.
find quasi-harmonic states, coexisting with photon-blockaded states (for same parameters and drive).
 Quantum states coexisting with semiclassical states (bistability)
Photon blockade
0.04
0.03
0.02
0.01
0
0
0.08
0.07
0.06
0.05
10 20
¹n ¡ 2 p
¹n
30 n
¹n + 2 p
40
¹n
<n>=25
50 60

Quantum trajectory simulations of quasi-coherent states several time slices (

=0) from before post-selection
0.07
0.06
0.05
0.04
0.03
initial state
0.02
0.01
0
0 20 after 9
 -1
40 60 80 100 after
 -1
120

Coexistence of blockaded and long lived quasi-coherent states
Probability for decay after
· ¡ 1
Quasi-coherent states lifetimes
Cavity drive strength [GHz]
• Lifetime is large on the scale of the cavity lifetime
• Should be obtainable experimentally for typical circuit QED parameters

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
L [!
d
(t); » d
(t); !
q
(t); · ; ° ] jbr i ghti jn ¼ 0i

High fidelity readout : Coherent control
Optimization of a linear chirp readout protocol in the bistable regime
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.

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.

• 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!