Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator Themis Mavrogordatos * Supervisor: Dr. Marzena Szyma´ nska Co-supervisor: Dr. Eran Ginossar * University College London (UCL), UK Tuesday 12 th of September 2017, UK Meeting on Superconducting Quantum Devices (SQD), Lancaster University, UK Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 1 / 28
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Rare Quantum Fluctuations in the DrivenDissipative Jaynes-Cummings Oscillator
Themis Mavrogordatos ∗
Supervisor: Dr. Marzena SzymanskaCo-supervisor: Dr. Eran Ginossar
∗University College London (UCL), UK
Tuesday 12th of September 2017, UK Meeting on SuperconductingQuantum Devices (SQD),Lancaster University, UK
Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 1 / 28
Brief Outline of Concepts and Tools
The Jaynes-Cummings Model and the√
n oscillator:
I. At resonanceII. In the dispersive regime
Mean-field results for amplitude and phase bistability.
Dissipative quantum phase transitions and the “thermodynamiclimit”.
Quantum fluctuations: Master Equation and single trajectories.
Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 2 / 28
The driven Jaynes-Cummings model
HJC =1
2~ωqσz + ~ωca†a + i~g
(a†σ− − aσ+
)+ ~
(εde−iω0ta† + ε∗de iω0ta
).
(1)
Two competing interactions: the JC interaction between the atomand the cavity mode, and the interaction of the cavity mode withthe external driving field. 1
In resonance fluorescence the bare atomic levels split as a result ofthe atom-field interaction:
HRF =1
2~ωqσz + ~ωca†a + ~
(λaσ+ + λ∗a†σ−
)(2)
The new energies of the dressed states are
En,± =
(n +
1
2
)~ωq ±
√n + 1 ~ |κ| . (3)
With driving we expect ’dressing’ of the ’dressed states’. Thethreshold for spontaneous polarization occurs at 2 |εd | = g .
1Carmichael, Statistical Methods in Quantum Optics, 2, Springer 2008Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 3 / 28
Master Equation, Wigner and Q representations
The Master Equation with dissipation (at rates 2κ for the cavityphotons and γ, γφ for the atom) is:
At threshold the discrete states merge into a continuum.
For arbitrary detunings the JC Hamiltonian acquires the term12~∆ωqσz + ~∆ωca†a.
Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 7 / 28
The resonant case (two ladders)
Two quasi-annihilation operators U and L for two ladders beginningfrom the same ground state.
The JC Hamiltonian can be written as (ωA = ωc = ωq = ωd):
HS +1
2~ωA = 0 |G 〉 〈G |+
(~ωAU†U + ~g
√U†U
)+
+(~ωAL†L− ~g
√L†L)
+ ~(εda† + ε∗da
) (10)
Two√
n anharmonic oscillators driven away from resonance.
For a small cavity damping we form the Master Equation(U)-oscillator
ρ =1
i~
[H+√
n, ρ]
+ κ(2U† ρU − U†Uρ− ρU†U
)(11)
Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 8 / 28
The two ladders of eigenstates
2
2H. J. Carmichael, Statistical Methods in Quantum Optics 2, 2008Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 9 / 28
The resonant case (two regimes)
Weak excitation limit (√
n oscillator):
⟨a†a⟩ss≈∣∣∣∣ εdκ+ ig
∣∣∣∣2 ≈ ( |εd |g
)2
. (12)
Strong excitation-quasi resonant, with detuning g/(2√
n) (for the√n oscillator):
⟨a†a⟩ss≈
∣∣∣∣∣ εd
κ+ ig/(2√〈a†a〉ss)
∣∣∣∣∣2
≈(|εd |κ
)2
−( g
2κ
)2
. (13)
Mean-field equations for zero system-size (scale number ofnsc = γ2/(8g2)→ 0) predict above threshold (JC):
A2 =
(|εd |κ
)2[
1−(
g
2 |εd |
)2]
(14)
Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 10 / 28
The resonant case (stability of the stationary states)
Zero-system size: for one atom and strong coupling the semiclassicalstationary states are “attractors”.
This is corroborated by the Wigner function (two Gaussians centredat αsemiclassical).
Well above threshold, in the presence of spontaneous emission(γ 6= 0) we find an amalgamation of (a) spontaneous dressed statepolarization and (b) absorptive optical bimodality.
A spontaneous emission event places the atom in a superposition of|U〉 and |L〉 states. The system is not localized on either branch. Ifthe polarization switches, then the phase of the intracavity field willalso switch. A “skirt” connecting the peaks appears.
Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 11 / 28
Complex amplitude bistablity
Along the “wall” of a first-order dissipative phase transition:
Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 12 / 28
The dispersive regime (again the√
n oscillator)
Following a transformation the decouples the qubit from the cavity,we obtain (keeping the same form for the transformed drive): 3
H = ~ωca†a + ~(ωc −∆)1
2σz + ~
(εda† + ε∗da
)(15)
∆ =√δ2 + 4g2N, with δ = |ωq − ωc | the detuning and
N = a†a + 12σz + 1
2 : total number of excitations.
‘Decoupled’ quantum master equation:
ρ = −(i/~)[H, ρ] + κ([aρ, a†] + [a, ρa†]
). (16)
To avoid photon blockade, n� g4/(2κδ3). The semiclassical modelis invalidated by the non convergence of the root expansion inpowers of n/nsc, where nsc = δ2/(4g2).
3L. S. Bishop, E. Ginossar and S. M. Girvin, Response of the Strongly-DrivenJaynes-Cummings Oscillator, PRL, 2010
Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 13 / 28
Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 19 / 28
Resonance (forming a quantum trajectory from jumps)
The state of the cavity field obeys the stochastic equation:
dα
dt= −[κ+ iεg/(2|α|)]α + iεd , (18)
ε = ±1 representing the random switching events.
In the strong coupling the JC interaction term gives rise to theoperator dz = i(σ− − σ+). This in turn generates the coupling term
ρqα = −ig/(2√
n)1
2
(dz [a†a, ρ] + [a†a, ρ]dz
). (19)
Performing the secular transformation yields the “switching terms”
...+ γ/4 (d−ρd+ + d+ρd−) + ... (20)
with d+ = |u〉 〈l | and d− = |l〉 〈u|These terms couple the U and L paths.
Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 20 / 28
Resonance (forming a quantum trajectory)-cont.
Emission times: t1, t2, ...tN between which there is coherentevolution with a non-Hermitian Hamiltonian.
S : collapse operator and e(L−S)(tj−tj−1): the propagator 5
ρc =
e(L−S)(t−tj−1)ρc(tj−1)
tr[e(L−S)(t−tj−1)ρc(tj−1)], tj−1 ≤ t < tj
Se(L−S)(t−tj−1)ρc(tj−1)
tr[Se(L−S)(t−tj−1)ρc(tj−1)], t = tj .
(21)
with S O = (γ/4) (d−Od+ + d+Od−) and
(L− S)O: determining evolution between switching events(deterministic time evolution).
5P. Alsing and H. J. Carmichael, Spontaneous dressed-state polarization of acoupled atom and cavity mode, Quantum Opt., 1991
Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 21 / 28
The effect of decoherence (ladder switching)
a∣∣En, (U,L)
⟩=
√n +√
n + 1
2
∣∣En−1, (U,L)
⟩+
√n −√
n + 1
2
∣∣En−1, (L,U)
⟩(22)
σ−∣∣En, (U,L)
⟩=
1
2|En,U〉+
1
2|En, L〉 . (23)
(i.e. 50% probability of ladder switch). For γ/κ = 0.1, 1, 10, 100 6.
6H. J. Carmichael, Breakdown of Photon Blockade: A Dissipative Quantum PhaseTransition in Zero Dimensions, PRX, 2015 (Fig. 7)
Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 22 / 28
Qubit trajectory and the dark state
The Maxwell-Bloch equations (mean-field) predict that the qubitinversion is always in the southern hemisphere:
zss = −[
1 +8g2n
γ2 + 4∆ω2q
]−1
, n = |α|2ss (24)
Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 23 / 28
Qubit and cavity trajectory; the dark state
The equatorial plane of the Bloch sphere evidences the coherentcancellation and the dark state as an analogue of the quasi-distributionfunction of the field in the phase space. Unstable periodic orbits:
Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 24 / 28
Dark state: Master Equation and single trajectory
Probability transfer between the dim and dark states. The dim state(re)appears in the switching from the dark to the bright state.
The dark state is substantiated by quantum fluctuations with a lifetimecomparable to the states of mean-field bistability.
Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 25 / 28
Spontaneous dressed-state polarization
For slowly-varying operators: 〈A〉 = e iωAt 〈A〉, we define 7: Cavity field:
N g(−y , x , 0): dynamically changing magnetic field.7P. Alsing and H. J. Carmichael, Spontaneous dressed-state polarization of a
coupled atom and cavity mode, Quantum Opt., 1991Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 26 / 28
The associated strong-coupling limit
|α|2
nsc
[|α|2
nsc+ 1−
(2εdg
)2]
= 0. (29)
If the drive is tuned to the n-photon resonance:
n~ωd = n~ωc ∓√
n ~g (30)
then
En+1,(U,L) − En,(U,L) − ~ωd ≈√
nsc
n~κ. (31)
nsc = [g/(2κ)]2 = 250000, 2500, 25:
Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 27 / 28
Concluding remarks
The√
n nonlinearity is the distinct feature of the Jaynes-Cummingsmodel (dispersive regime and resonance) at the core of quantumoptics and electrodynamics.
The critical point associated with the development of bistability inthe dispersive regime is subject to a perturbation expansion withnsc = [δ/(2g)]2.
In contrast, at resonance there is photon blockade breaking down bymeans of a dissipative first order quantum phase transition.
The limit of “zero system size” (nsc = [γ/(2√
2g)]2 → 0) atresonance accounts for new (semiclassical) stationary statespredicting a threshold related to a second-order dissipativequantum phase transition.
In the dispersive regime the limit γ/(2κ)→ 0 reveals a long-lived‘dark state’ corresponding to a non-classical attractor.
Thank you for your attention!
Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 28 / 28