Strong driving in circuit quod erat demonstrandum (QED)

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Strong driving in Circuit QED
IQC 2011-10-17
Lev S Bishop
Joint Quantum Institute and
Condensed Matter Theory Center, University of Maryland
Collaborators:
Experiment:
Theory:
Jerry Chow (IBM)
Eran Ginossar (Surrey)
Andrew Houck (Princeton)
Erkki Thuneberg (Oulu)
Matt Reed (Yale)
Jens Koch (Northwestern)
Leo DiCarlo (Delft)
Steve Girvin (Yale)
Dave Schuster (Chicago)
Rob Schoelkopf (Yale)
Funding:
…
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 2qubit 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
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)
2008
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)
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?
‘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 JaynesCummings 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
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
Essential mechanism
• Diminishing anharmonicity of the
Hamiltonian
~!c
Undriven Hamiltonian
JC Hamiltonian
HUGE simplification: seems unlikely to be useful
but let’s try anyway
Exact Diagonalization
detuning
total excitations
critical photon number
Perturbative expansion
Expand in
Dispersive approximation
Plus Kerr term…
Can continue the expansion, but only converges for
For typical cQED parameters, the dispersive approximation breaks down
3
before N=Ncrit: anharmonicity ®=2g4/± is approx. linewidth
g=200MHz, ±=1 GHz, ®=3.2MHz
Transition frequencies
n
1000
g/±=0.1
800
600
|0i
400
|1i
Kerr nonlinearity :
H = ! ay a + ´ (ay a) 2
200
0.010
!c¡ Â
0.005
!c-!ij
Transition frequency
0.005
0.010
!c+ Â
Transformed drive & dissipation
10
g/±=0.1
8
6
4
2
0
0
20
40
60
80
100
n
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
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
800
JC
600
Rabi
400
200
6.96
6.98
7.00
7.02
7.04
7.06
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
Sqrt(1+N/Ncrit)
10
8
6
4
2
0
10
5
0
5
10
X
• Perturbation to quadratic potential looks
like |X| for large X
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 region
Readout 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
Trans. power/dB
How to use this for qubit readout?
30
!
d
= !
c
20
|0i
10
|1i
Neglect for large N
0
10
15
10
5
0
Input power/dB
Not for one-qubit case, because of symmetry
|0i
1000
800
600
400
200
0.010
0.005
0.005
(there is still information in the phase)
0.010
|1i
Symmetry breaking
Pure 2-level qubit
has (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 2)
– 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
!
n
¹ + 2¾ ¡
!
Neither small Hilbert space nor
point in classical phase-space
n
¹ ¡ 2¾
¼·
0.08
0.07
<n>=25
find quasi-harmonic states, coexisting with photon-blockaded
states (for same parameters and
drive).
 Quantum states coexisting with
semiclassical states (bistability)
0.06
0.05
P(n)
Total frequency shift from “end-toend” due to anharmonicity should
be smaller than the linewidth.
0.04
0.03
0.02
0.01
0
0
Photon
blockade
10
20
p
n
¹ ¡ 2 n
¹
30
n
40
p
n
¹ +2 n
¹
50
60
Quantum trajectory simulations of quasi-coherent states
several time slices (=0) from before post-selection
0.07
0.06
after 9-1
after -1
0.05
0.04
0.03
initial state
0.02
0.01
0
0
20
40
60
80
100
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
jbr i ghti
jn = 0i
jn ¼ 0i
L [! d (t); »d (t); ! q(t); · ; ° ]
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.
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!
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