rare quantum fluctuations in the driven dissipative jaynes-cummings...

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.


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:

ρ = −(i/~)[HJC , ρ] + κ[n(ωc) + 1]L{a, ρ}+ κn(ωc)L{a†, ρ}++ (γ/2)[n(ωq) + 1]L{σ−, ρ}+ (γ/2)n(ωq)L{σ+, ρ}++ (γφ/2)L{σz , ρ},

(4)

where L{B, ρ} = 2BρB† − B†Bρ− ρB†B.

Characteristic functions

χA(z , z∗) = tr(ρe izae iz∗a†) and χS(z , z∗) = tr(ρe iza+iz∗a†). (5)

Their Fourier transforms

Q,W (α, α∗) =

∫χA,S(z , z∗)e−iz∗α∗e−izα d2z . (6)

are quasi-probability density functions.


The resonant case (energy spectrum)

At (total) resonance: ωc = ωq = ωd :

For weak driving fields (2 |εd | /g � 1) we expect theJaynes-Cummings spectrum plus a perturbative correction of order(2 |εd | /g)2.

For strong driving fields (2 |εd | /g � 1) the roles of the interactionsare reversed: the JC interaction becomes the perturbation.

The second order phase transition is expected at 2 |εd | = g .

With drive-cavity detuning the large mean photon number is

n =|εd |2

κ2 + [(ωd − ωc)∓ g/(2√

n)]2. (7)



Quasi-energies are associated to a time-independent Scrodingerequation with Hamiltonian:

H = i~g(a†σ− − aσ+

)+ ~

(εda† + ε∗da

)(8)

For strong driving fields (2 |εd | /g � 1) the roles of the interactionsare reversed: the JC interaction becomes the perturbation.

Potential energy ~εd(a† + a) proportional to the position of aharmonic oscillator. We anticipate a continuous spectrum.

The transition is anticipated at 2 |εd | = g .



Define a squeeze parameter (think of the parametric oscillator):

e−2r ≡√

1− (2 |εd | /g)2. (9)

Quasi-energies En,U(L) = ±√

n ~ge−3r .

Spectrum (m − 1/2)ωA + En,±/~, m = 0, 1, 2, ... below threshold.

At threshold the discrete states merge into a continuum.

For arbitrary detunings the JC Hamiltonian acquires the term12~∆ωqσz + ~∆ωca†a.


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)


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)


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.


Complex amplitude bistablity

Along the “wall” of a first-order dissipative phase transition:


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


Perturbation series for the cavity bistability

Effective dressed Duffing oscillator (weak excitations):

W (α, α∗) =2

πe−2|α|2 |0F1 (c , 2εd α

∗)|2

0F2(c , c∗, 2|εd |2)

=(2/π)e−2|α|2

0F2(c , c∗, 2|εd |2)

∣∣∣∣1 +z

D1+

z2

2!D2+ · · ·

∣∣∣∣2 .(17)

with c = (κ− i∆ω′c)/(iχ) and εd = εd/(iχ) with χ = (g4/δ3)σz , whilez =√−8εd α∗.


From amplitude to phase bistability

Following the crossover for fixed (left) and varying (right)|δ| = |ωq − ωc |. The scale parameter is nsc = δ2/4g2 and (for γ = 0):

α = −iεd

{κ− i

[∆ωc −

g2

δ

(1 +

4g2

δ2|α|2

)−1/2]}−1

.


Fast scan of the detuning


Variation of the scale parameter

Appearance and breakdown of the photon blockade with diminishingδ/g captured in the entanglement entropy Sq = −tr(ρq ln ρq)(ρq = trcρ):


Second-order transition at resonance

Critical point (white spot) of a second-order quantum dissipative phasetransition in zero dimensions: γ2/8g2 → 0. Amplitude and phasebimodality:

44H. J. Carmichael, Breakdown of Photon Blockade: A Dissipative Quantum Phase

Transition in Zero Dimensions, PRX, 2015 (Fig. 2)Themis Mavrogordatos, UCL Rare Quantum Fluctuations in the Driven Dissipative Jaynes-Cummings Oscillator 18 / 28

Spontaneous dressed-state polarization

Second-order quantum dissipative phase transition in zero dimensions:γ2/8g2 → 0. Critical point: |εd | = g/2, ωc = ωq = ωd .


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.


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


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)


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)


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:


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.


Spontaneous dressed-state polarization

For slowly-varying operators: 〈A〉 = e iωAt 〈A〉, we define 7: Cavity field:

x + iy ≡ N−1/2[ie−iarg(εd ) 〈a〉]. (25)

Collective Bloch vector:

m ≡ N−1{

2Re[ie−iarg(εd ) 〈J−〉]x + 2Im[ie−iarg(εd ) 〈J−〉]y + 〈Jz〉 z}(26)

Equations:

x = −κx +1

2

√N gmx + |εd |/

√N,

y = −κy +1

2

√N gmy ,

(27)

giving:m = B×m, m ·m = 1, (28)

with B ≡ 2√

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:


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!


rare quantum fluctuations in the driven dissipative jaynes-cummings...

Documents