The strongly driven dissipative Jaynes-Cummingsmodel

Themis MavrogordatosSupervisor: Dr. Marzena Szymanska

Co-supervisor: Dr. Eran Ginossar

Quantum Optics in Coherent Artificial Systems, 9 December 2015

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Brief Outline

The Piecewise Deterministic Process.

The Diffusive approximation and the Fokker-Planck Equation.

The Jaynes-Cummings Model and the√

n oscillator:

I. At resonanceII. Dispersive regime

Bimodality and Quantum Trajectories.

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The Piecewise Deterministic Process I

The density matrix ρS(t) of an open quantum system satisfies theMarkovian master equation (in Lindblad form) 1

d

dtρS(t) = −i [H, ρS(t)]+

∑i

γi

(AiρS(t)A†i −

1

2A†i AiρS(t)− 1

2ρS(t)A†i Ai

).

(1)Liouville master equation for a PDP in Hilbert space:

∂

∂tP[ψ, t] = i

∫dx

{δ

δψ(x)G (ψ)(x)− δ

δψ∗(x)G (ψ)∗(x)

}P[ψ, t]

+

∫DψDψ∗{W [ψ|ψ]P[ψ, t]−W [ψ|ψ]P[ψ, t]},

(2)The first integral corresponds to a deterministic evolution under theaction of the operator:

G (ψ) = Hψ +i

2

∑i

γi‖Aiψ‖2ψ. (3)

1Breuer and Petruccione, The Theory of Open Quantum Systems, Oxford 20063 / 27

The Piecewise Deterministic Process II

The linear operator H is defined as:

H = H − i

2

∑i

γiA†i Ai . (4)

The last term is added in order to preserve the norm of thewavefunction, leading to the ultimate unravelling of the masterequation.

The operator H is non-Hermitian. It reflects the decaying norm ofthe wavefunction corresponding to no photon count.

The transition rate is

W [ψ|ψ] =∑

i

γi‖Ai ψ‖2 δ

[Ai ψ

‖Ai ψ‖− ψ

]. (5)

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The Piecewise Deterministic Process III

Assume one jump operator A with a corresponding rate γ0. Adiffusion expansion of a given master equation can be realized if

the size of the transitions among the states becomes arbitrarily smallandthe number of transitions in any finite time interval becomesarbitrarily large.

If the diffusion limit of the Liouville master equation exists it yields aFokker-Planck equation for the probability density functional, whichis in turn equivalent to a stochastic Schroedinger equation.

To formulate these conditions, we write the Lindblad operator interms of a small dimensionless parameter, ε, as below:

A = I + εC , (6)

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Diffusive approximation and the Fokker-Planck Equation I

We keep terms to second order in ε. For the deterministic evolutionwe have:

G (ψ) =Hψ − i

2γ0{I + ε(C † + C ) + ε2C †C}ψ+

i

2γ0{1 + ε

⟨C † + C

⟩ψ

+ ε2⟨C †C

⟩ψ}ψ

(7)

and for the stochastic evolution:

W [ψ|ψ] = γ0

(1 + ε

⟨C † + C

⟩ψ

+ ε2⟨C †C

⟩ψ

)δ

(I + εC )ψ√1 + ε 〈C † + C 〉ψ + ε2 〈C †C 〉ψ

− ψ

= γ0

(1 + ε

⟨C † + C

⟩ψ

+ ε2⟨C †C

⟩ψ

)δ[ψ − ψ + εM(ψ) + ε2N(ψ)

],

(8)

The Liouville Master Equation is reduced to a Fokker-Planckequation for the probability density functional.

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Diffusive approximation and the Fokker-Planck Equation II

The non-linear drift operator K (ψ) becomes

K (ψ) = Hψ +i

2γ0ε(C − C †)ψ+

+ iγ0ε2

{1

2

⟨C + C †

⟩ψ

C − 1

8

⟨C + C †

⟩2ψ− 1

2C †C

}ψ.

(9)

Non-vanishing and finite diffusive contribution requires γ0 = ε−2γ,where γ is independent of ε. On the other hand, the drift operatorK (ψ) contains a term which is proportional to γ0ε = ε−1γ. To avoiddivergence we impose that the operator C is self-adjoint.

K (ψ) = Hψ + iγ

{〈C 〉ψ C − 1

2〈C 〉2ψ −

1

2C 2

}ψ, (10)

The operator M assumes the form:

M(ψ) = (C − 〈C 〉ψ)ψ. (11)

The conditions leading to the diffusion limit depend on the numberand the nature of the jump operators.

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Diffusive approximation and the Fokker-Planck Equation III

With these expressions, the Fokker-Planck equation is equivalent tothe following stochastic Schroedinger equation in Ito form:

dψ(t) = −iK (ψ(t))dt +√γM(ψ(t))dW (t), (12)

dW (t) is an increment of a real Wiener process.

Numerical solution with various algorithms:

The Euler scheme. A scheme of order 1.

Combination of the 4th order Runge-Kutta scheme for the drift andthe Euler scheme for the diffusion. A scheme of order 1.

The second-order weak scheme (Platen, 1992). A scheme of order 2.

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The Jaynes Cummings Model

H =1

2~ωAσz + ~ωca

†a + i~g(a†σ− − aσ+

)+ ~

(E0e−iω0ta† + E∗0e iω0ta

).

(13)

Two competing interactions: the JC interaction between the atomand the cavity mode, and the interaction of the cavity mode withthe external driving field. 2

In resonance fluorescence the bare atomic levels split as a result ofthe atom-field interaction:

HS =1

2~ωAσz + ~ωAa†a + ~

(κaσ+ + κ∗a†σ−

)(14)

The new energies of the dressed states are

En,± =

(n +

1

2

)~ωA ±

√n + 1 ~ |κ| . (15)

We anticipate ’dressing’ of the ’dressed states’. The threshold forspontaneous dressed-state polarization occurs at 2

∣∣E0

∣∣ = g .2Carmichael, Statistical Methods in Quantum Optics, 2, Springer 2008

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The resonant case

At resonance ω0 = ωA = ωc :

For weak driving fields (2∣∣E0

∣∣ /g � 1) we expect theJaynes-Cummings spectrum plus a perturbative correction of order(2∣∣E0

∣∣ /g)2.

For strong driving fields (2∣∣E0

∣∣ /g � 1) the roles of the interactionsare reversed: the JC interaction becomes the perturbation.Assuming a real excitation field the relevant term, in the interactionpicture is ~E0(a† + a): potential energy proportional to the positionof a harmonic oscillator, hence we anticipate a continuous spectrum.

The transition is anticipated at 2∣∣E0

∣∣ = g .

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The resonant case

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

H = i~g(a†σ− − aσ+

)+ ~

(E0a† + E∗0a

)(16)

For strong driving fields (2∣∣E0

∣∣ /g � 1) the roles of the interactionsare reversed: the JC interaction becomes the perturbation.

Assuming a real excitation field the relevant term, in the interactionpicture is ~E0(a† + a): potential energy proportional to the positionof a harmonic oscillator, hence we anticipate a continuous spectrum.

The transition is anticipated at 2∣∣E0

∣∣ = g .

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The resonant case (Energy spectrum)

Quasi-energies En,± = ±√

n ~g [1− (2∣∣E0

∣∣ /g)2]3/4. This yields thespectrum (m − 1/2)ωA + E/~, m = 0, 1, 2, ... below threshold.

Define a squeeze parameter:

e−2r ≡√

1−(2∣∣E0

∣∣ /g)2. (17)

For arbitrary detunings the JC Hamiltonian acquires the term12~∆ωAσz + ~∆ωca

†a.

For non-zero cavity detuning the operator is quartic and cannot bediagonalized by a suitable choice of displacement and squeezeoperators.

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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:

HS +1

2~ωA = 0 |G 〉 〈G |+

(~ωAU†U + ~g

√U†U

)+

+(~ωAL†L− ~g

√L†L)

+ ~(E0a† + E∗0a

) (18)

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

)(19)

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The resonant case (Two regimes)

Weak excitation limit:

⟨a†a⟩ss≈∣∣∣∣ E0

κ+ ig

∣∣∣∣2 ≈(∣∣E0

∣∣g

)2

. (20)

Strong excitation-quasi resonant, with detuning g/(2√

n):

⟨a†a⟩ss≈

∣∣∣∣∣ E0

κ+ ig/(2√〈a†a〉ss)

∣∣∣∣∣2

≈

(∣∣E0

∣∣κ

)2

−( g

2κ

)2

. (21)

Mean-field equations for zero system-size (scale ofnsat = γ2/(8g2)→ 0) predict above threshold:

A2 =

(∣∣E0

∣∣κ

)2 [1−

(g

2 |E0|

)2]

(22)

Interference between pathways (quantum fluctuations) is hereignored.

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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.

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The resonant case (Quantum Fluctuations-I)

Quantum fluctuations perform the symmetry breaking.

The mean photon number behaves similarly to the√

n oscillator, butthe fluctuations are markedly different.

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The resonant case (Quantum Fluctuations-II)

Above threshold the internal coupling is not sufficiently strong toproduce energy splitting and hence levels in excess of the levelwidths.

The treatment of semi-classical dynamics plus “small fluctuations” isinapplicable.

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The resonant case (Quantum Fluctuations-III)

Single Trajectory for γ = 0. The state follows the contour specifiedby the Wigner function, marking the fluctuations.

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The resonant case (Quantum Fluctuations-IV)

Single Trajectory for γ = 2κ. The “skirt” of the Wigner function isunravelled.

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The resonant case (Quantum Fluctuations-V)

The state of the cavity field obeys the stochastic equation:

dα

dt= −[κ+ iεg/(2|α|)]α + iE0, (23)

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

ρAα = −ig/(2√

n)1

2

(dz [a†a, ρ] + [a†a, ρ]dz

). (24)

Performing the secular transformation yields the “switching terms”

...+ γ/4 (d−ρd+ + d+ρd−) + ... (25)

with d+ = |u〉 〈l | and d− = |l〉 〈u|These terms couple the U and L paths.

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The resonant Case (Forming a Quantum Trajectory)

Emission times: t1, t2, ...tN . 3

S : collapse operator and e(L−S)(tj−tj−1): propagator

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 .

(26)

with S O = (γ/4) (d−Od+ + d+Od−) and

(L− S)O: determining evolution between switching events(deterministic time evolution).

3Alsing and Carmichael, Quantum Opt., 199121 / 27

The dispersive regime I (Yet another√

N oscillator!)

Following a transformation the decouples the qubit from the cavity,we obtain (keeping the same form for the transformed drive): 4

H = ωca†a + (ωc −∆)

1

2σz + ~

(E0a† + E∗0a

)(27)

∆ =√δ2 + 4g2N, with δ = ωA − ωc the detuning and

N = a†a + 12σz + 1

2 : total number of excitations.

‘Decoupled’ quantum master equation:

ρ = −i [H, ρ] +κ

2

([aρ, a†] + [a, ρa†]

). (28)

To avoid photon blockade, N � g4/(κδ3). The semiclassical modelis invalidated by the non convergence of the root expansion inpowers of N/Ncrit, where Ncrit = δ2/(4g2).

4Bishop, Ginossar and Girvin, PRL, 201022 / 27

The dispersive regime II

Viewed as a saturable extension of the Kerr model 5:

α = −iE0

{κ− i

[(ω0 − ωc)− g2

δ

(1 +

4g2

δ2|α|2

)−1/2]}−1

(29)

In the dispersive regime g2/δ2 � 1, hence the nonlinearity manifestsin the high photon number regime.

This abides by the mean-field absorptive optical bistability where thenonlinearity is brought into play for photon numbers:

|α|2 ∼ nsat = γ2/(8g2). (30)

Markedly different from the resonant case where the magnitude ofthe Bloch vector is preserved and the nonlinearity diverges for smallphoton numbers.

5Carmichael, 201523 / 27

The dispersive regime (Single Quantum Trajectory I)

Bimodality in the photon number < a†a > and the spin operator< σz > in accordance to the mean field results.

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The dispersive regime (Single Quantum Trajectory II)

Poissonian distribution corresponding to the two coherent states.

During switching there is a ‘spiral rotation’ of the two states in thephase diagram.

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The dispersive regime (Single Quantum Trajectory III)

The two states coexist during the transition —1st order transition.

During switching there is a ‘spiral rotation’ of the two states in thephase diagram.

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Thank you for your attention!

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