Quantum trajectory approach to stochastically‐induced quantum interference

Fortschr. Phys. 51, No. 2–3, 179 – 185 (2003) / DOI 10.1002/prop.200310022

Quantum trajectory approachto stochastically-induced quantum interference

A. Karpati1,∗, P. Adam1, W. Gawlik2, B. Lobodzinski2,3, and J. Janszky1

1 Department of Nonlinear and Quantum Optics, Research Institute for Solid State Physics and Optics,Hungarian Academy of Sciences, P.O. Box 49, 1525 Budapest, Hungary

2 Jagellonian University, Reymonta 4, 30-059 Krakow, Poland3 Present address: DESY Zeuthen, Platanenallee 6, 15738 Zeuthen, Germany

Received 10 June 2002, accepted 19 June 2002Published online 25 February 2003

PACS 42.50.Gy, 42.50.Lc

Stochastic perturbation of two-level atoms strongly driven by a coherent light field is analyzed by the quantumtrajectory method. Narrow resonances as hole burning and dispersive Fano-like profile occur in the spectrain the regime where the noise dominates the Rabi oscillations. It is shown that in this case the stochasticnoise can unexpectedly create phase correlation between the neighboring atomic dressed states. This phasecorrelation is responsible for quantum interference between the related transitions resulting in anomalousmodifications of the resonance fluorescence spectra.

1 Introduction

Quantum interference is one of the most intriguing phenomena of quantum mechanics. Over the pastdecade several effects in atom-light interaction which have their origin in quantum interference have beenpredicted and demonstrated experimentally [1]. Some characteristic examples are reduction and cancellationof absorption [2–6] and spontaneous emission [7–10], and narrow resonances in fluorescence [11, 12]. Aprerequisite of quantum interference between the transition channels is the existence of some stable timecorrelation of the atomic system under consideration. A possible way of achieving such correlation is theapplication of coherent coupling in a multi-level atomic system. Although some interference effects havealso been found in two-level systems interacting with two light beams [13], so far quantum interference hasbeen observed exclusively in at least three-level systems.

Generally, various incoherent perturbations destroy the phase correlation between the states involved inthe interfering transition pathways and the coherently induced quantum interference disappears. However,under special circumstances, even incoherent perturbation can be responsible for quantum interference.

Recently, in an experiment with coherently driven two-level atoms, anomalous resonance fluorescencespectra were found when the collisional relaxation rate exceeded the Rabi frequency [14]. The spectra wereof the form of a pressure-broadened line with a narrow, not collisionally broadened dip. These results,unexpected in a collisionally perturbed two-level system, were interpreted as a consequence of quantuminterference between different dressed-state transition channels. These effects can also occur in the case ofa non-monochromatic, e.g. phase-diffusing, laser field [14, 15]. What in both cases appears essential forobservation of quantum interference and anomalous spectra is that the incoherent perturbation (collisionsor phase diffusion of the light field) dominates over the Rabi oscillations.

∗ Corresponding author E-mail: [email protected]

c© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 0015-8208/03/2–302-0179 $ 17.50+.50/0

180 A. Karpati et al.: Quantum trajectory approach

These two examples raise an intriguing question how can a stochastic noise lead to stable time correlationresulting in quantum interference in two-level systems.

In this paper we analyze in detail the system of stochastically perturbed two-level atoms applying thequantum trajectory method to explain quantum interference effects and the underlying physical processes.

2 Quantum interference effects

The system of incoherently perturbed and coherently driven two-level atoms can be modeled in several ways.Here we make a rather general assumption that the stochastic perturbation is responsible for fluctuations ofthe atomic resonance frequency which obeys the Gaussian statistics. In particular, such fluctuations mayresult from e.g. elastic, dephasing collisions.

In our model the Hamiltonian of the strongly, coherently driven atom subjected to stochastic perturbationhas the form

HAL = (ωa(t) − ωL)Sz + 12Ω(S− + S+) (1)

in the interaction picture, where ωa(t) = ωa + δωa(t) is the fluctuating atomic transition frequency, ωL isthe frequency of the laser, Ω the Rabi frequency, and Sz , S+, S− are the atomic operators defined in theexcited state (|e〉) – ground state (|g〉) basis:

Sz =12

(1 0

0 −1

), S+ =

(0 10 0

), S− =

(0 01 0

). (2)

We assume that the noise in the transition frequency satisfies

〈δωa(t)δωa(t′)〉 = 2Γδ(t − t′), (3)

where Γ stands for the magnitude of the stochastic noise. If the noise is due to collisions, this quantity isthe collision rate between the atoms. This model can also describe the system of two level atoms driven bya laser field with fluctuating phase, if the phase drift is neglected [15]. In such case, Γ represents the laserlinewidth.

The time-evolution of the system defined by the Hamiltonian in Eq. (1) is described by the followingmaster equation, taking into account also the spontaneous emission processes:

ρ =1i

[〈HAL〉, ρ] + Lρ, (4)

where

Lρ = (Lρ)sp + (Lρ)st, (5)

(Lρ)sp = γ(−12(S+S−ρ + ρS+S−) + S−ρS+), (6)

(Lρ)st = 4Γ(−12(SzSzρ + ρSzSz) + SzρSz), (7)

and 〈HAL〉 is the mean atomic Hamiltonian obtained by averaging over the stochastic noise of Eq. (3), andγ is the natural linewidth of the atom.

It is natural to introduce the dressed-state basis in which the atomic Hamiltonian HAL is diagonal:

|1〉 = cos Θ|g〉 + sin Θ|e〉, (8)

|2〉 = − sin Θ|g〉 + cos Θ|e〉, (9)

where

Fortschr. Phys. 51, No. 2–3 (2003) 181

Θ = − 12 arctan

( Ω∆

),

and ∆ = ωa − ωL is the laser detuning.The mean atomic Hamiltonian 〈HAL〉 in the dressed-state basis |1〉, |2〉 can be written as

〈HAL〉 = E1|1〉〈1| + E2|2〉〈2|, (10)

where

E1,2 = ∓ 12

√Ω2 + ∆2.

The effect of incoherent perturbation on the pure states can be determined from the Lindblad form (7)of the master equation in Eq. (4). The action of the operator 2

√ΓSz corresponds to an event generated by

the stochastic noise. Without detuning, this operator generates transitions between the dressed states |1〉and |2〉:

2√

ΓSz|1〉 =√

Γ|2〉, (11)

2√

ΓSz|2〉 =√

Γ|1〉. (12)

In the system of coherently driven stochastically perturbed two-level atoms, quantum interference effectscan be seen in the resonance fluorescence spectra [14, 15]. The resonance fluorescence can be describedby transitions between appropriate dressed states of the atom. If spectral modifications are due to quantuminterference, some time correlation should exist between the dressed states of the atom involved in theinterfering transition channels.

For analyzing time correlations in a quantum system, quantum trajectory methods are particularly appro-priate. These methods are based on the simulation of quantum trajectories, that are individual realizationsof the evolution of the system conditioned on particular sequence of observed events. By tracking the timeevolution of a single quantum trajectory, the time correlations can be revealed.

We apply the quantum trajectory method of [16] for simulating the time evolution of the coherently driven,stochastically perturbed two-level atom. In this system, a single quantum trajectory evolves coherentlyaccording to the Hamiltonian of Eq. (10), interrupted by incoherent gedanken measurements due to noiseevents and spontaneous emission. The evolution of the density operator of the system is obtained byaveraging the density operators of the individual quantum trajectories. The resulting density operator isalso the solution of the master equation of the system.

Within dipole approximation the resonance fluorescence spectrum S(ω) can be calculated as the realpart of the two-time correlation function

ΓN1 (ω) = lim

t→∞

∫ ∞

0exp(−iωτ)〈S+(t + τ)S−(t)〉 dτ, (13)

for an arbitrary initial condition:

S(ω) = ReΓN1 (ω), (14)

where ω is the detuning of the emitted light from ωL. The method of [17] was used for calculating theabove resonance fluorescence spectra from the numerical results. Fig. 1 presents the resonance fluorescencespectrum of the atom irradiated by a resonant (∆ = 0), strong (Ω γ) laser field with low noise (Ω > Γ).The spectrum exhibits a three-peak structure, but with a suppressed and broadened central peak comparedto the standard Mollow triplet. As the noise increases, the central peak disappears and for Γ nearly equal toΩ we get a two-peak structure with a relatively broad dip, as depicted in Fig. 2. When the noise magnitudeis much larger than the Rabi frequency, the dip becomes very narrow, as shown in Fig. 3. For large detuning(∆ Ω) and low noise (Γ Ω), a two-peak spectrum is obtained with an asymmetric Fano-like structure

182 A. Karpati et al.: Quantum trajectory approach

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-3 -2 -1 0 1 2 3

S(ω

) [a

rb. u

nits

]

ω/Ω

0

0.05

0.1

0.15

0.2

0.25

-10 -5 0 5 10

S(ω

) [a

rb. u

nits

]

ω/ΩFig. 1 The resonance fluorescence spectrum for lownoise magnitude and strong laser field (Γ/Ω = 0.2,γ/Ω = 0.05) in the case of no detuning (∆ = 0).

Fig. 2 The resonance fluorescence spectrum for noisemagnitude comparable to Rabi frequency and strong laserfield (Γ/Ω = 1.1, γ/Ω = 0.05) in the case of no detuning(∆ = 0).

0.010.020.030.040.050.060.070.080.090.1

-15 -10 -5 0 5 10 15

S(ω

) [a

rb. u

nits

]

ω/Ω

0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

-6 -4 -2 0 2 4 6

S(ω

) [a

rb. u

nits

]

ω/ΩFig. 3 The resonance fluorescence spectrum for highnoise magnitude and strong laser field (Γ/Ω = 6, γ/Ω =0.05) in the case of no detuning (∆ = 0), showing anarrow dip at the center of the spectrum.

Fig. 4 The resonance fluorescence spectrum for lownoise magnitude, strong laser field and large detuning(Γ/Ω = 0.2, γ/Ω = 0.05, ∆/Ω = 3), showing a Fano-like structure at the driving frequency.

at the center, as depicted in Fig. 4. Increasing the noise magnitude the Fano-like peak transforms to anasymmetric Fano profile, a narrow dip on the side of the broadened part of the spectrum due to stochasticnoise, and a narrow peak on the other side next to the dip (Fig. 5).

The narrow dip in the spectrum (in the case of high noise magnitude, resonant excitation) and theasymmetric Fano profile (in the case of large detuning and high noise magnitude) are signatures of quan-tum interference in the stochastically perturbed system. The quantum interference emerges if long-timephase correlation exists between quantum states connected by different transition channels. To reveal thephenomena let us investigate the behavior of phase difference ∆φ between the dressed states, defined asfollows:

|Φ〉 = a1eiφ1 |1〉 + a2eiφ2 |2〉, ∆φ = φ2 − φ1, (15)

where |Φ〉 is a pure state of the atom, while |1〉 and |2〉 are the dressed states defined in Eqs. (8) and (9).The phase difference can be calculated straightforwardly from a single quantum trajectory. It is found that

Fortschr. Phys. 51, No. 2–3 (2003) 183

0.010.020.030.040.050.060.070.080.090.1

-15 -10 -5 0 5 10

S(ω

) [a

rb. u

nits

]

ω/Ω

Fig. 5 The resonance fluorescence spectrum for highnoise magnitude, strong laser field and large detuning(Γ/Ω = 3, γ/Ω = 0.05, ∆/Ω = 3), showing an asym-metric Fano profile at the driving frequency.

a b

0

π/2

π

3π/2

2π

0 200 400 600 800 1000

∆φ [r

ad]

Ωt

0

π/2

π

3π/2

2π

0 200 400 600 800 1000

∆φ [r

ad]

Ωtc

0

π/2

π

3π/2

2π

0 200 400 600 800 1000

∆φ [r

ad]

Ωt

Fig. 6 The phase difference between the dressed statesin the case of resonant excitation (∆ = 0), a) for alow noise magnitude (Γ/Ω = 0.2, γ/Ω = 0.05); b)for a noise magnitude comparable to the Rabi frequency,(Γ/Ω = 1.1, γ/Ω = 0.05); c) for a high noise magni-tude, (Γ/Ω = 5, γ/Ω = 0.05). The initial state was theexcited state |e〉 in these simulations.

the phase difference behaves differently in the low and high noise magnitude regimes. In Fig. 6a the noisemagnitude is much less than the Rabi frequency, Rabi oscillations are rarely disrupted by noise events, hencethe phase difference is essentially linearly dependent on time: ∆φ(t) = 2Ωt. Consequently, the shape ofthe phase difference as the function of time shows no structure. When the noise magnitude increases, asdepicted in Fig. 6b, the uniform shape changes to a picture showing some structure of gaps appearing fromtime to time between 0 and π values of the phase difference. For high noise magnitude (Fig. 6c), the phasedifference tends to stabilize around values 0 and π for some time intervals.

In order to characterize the observed phenomena quantitatively, we introduce the correlation function ofcos ∆φ by the definition

Ccos(τ) = c

∫ T

t=0(cos ∆φ(t + τ) − cos ∆φ)

×(cos ∆φ(t) − cos ∆φ) dt, (16)

184 A. Karpati et al.: Quantum trajectory approach

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-20 -15 -10 -5 0 5 10 15 20

Cco

s(τ)

Ωτ

Γ/Ω=0.2Γ/Ω=1.1

Γ/Ω=5

Fig. 7 Correlation function of cos ∆φ(t) for low, mediumand high noise magnitude, in the case of no detuning andstrong laser field (Γ/Ω ∈ 0.2, 1.1, 5, ∆ = 0, γ/Ω =0.05).

|1,n

|1,n-1

|2,n

|2,n-1 Fig. 8 Different dressed-state transition channels for acoherently driven and stochastically perturbed two-levelatom.

where cos ∆φ is the mean value of the cosine of the phase difference for the simulated time interval and cis a normalization constant fixed by the condition Ccos(0) = 1. The correlation function of sin ∆φ(t) isdefined similarly.

The Ccos(τ) function is shown in Fig. 7 for the same parameter values as those used in Figs. 6a–c. Thequalitative picture of emerging correlations as the noise magnitude increases is now backed up by thewidening of the correlation functions. On the other hand, the correlation of the sine of the phase difference,Csin(τ) (defined similarly as Ccos(τ)) tends towards a δ-like shape when the noise increases, so when Γstrongly exceeds Ω, sin ∆φ(t) remains uncorrelated. This means that the phase difference is locked tovalues 0 and π for some time intervals, though it spans a phase interval no less than π

2 around these phasevalues.

The stabilization of the phase difference between the dressed states of the stochastically perturbed andcoherently driven two-level atom is the underlying physical process which makes the quantum interferencepossible. This stabilization supports the following interpretation first suggested for collisional and phasenoise-induced quantum interference effects in resonance fluorescence spectrum in [14]. Resonance fluo-rescence of a strongly-driven two-level atom is emitted in cascade transitions downward the ladder of thedressed-state doublets. Fig. 8 shows two adjacent doublets and all possible spontaneous and noise-inducedtransitions between the dressed-atom states. According to Eq. (2), noise events generate transitions betweenthe dressed states |1〉 and |2〉 and couple them as indicated by double arrows in Fig. 8. As we have seen inthe previous section, in the noise-dominated regime, i.e. when Γ > Ω, the phase difference between dresseddoublets tends to stabilize for some time intervals due to frequent noise events. Moreover, the resonancefrequencies of all fluorescence contributions are the same in this regime. Among several possible emissionchannels there are two pairs: |1, n〉 → |2, n − 1〉 → |1, n − 1〉 and |1, n〉 → |2, n〉 → |1, n − 1〉 (or|2, n〉 → |1, n〉 → |2, n − 1〉 and |2, n〉 → |1, n − 1〉 → |2, n − 1〉) that differ exclusively by time orderingbetween collisional mixing and photon emissions. Photons emitted along these channels are indistinguish-able, so their interference is possible. Due to opposite signs of the relevant matrix elements this interferenceis destructive and creates a dip in the line center. On the other hand, other emission channels are not thatequivalent, hence the corresponding photons cannot interfere and contribute to non-zero intensity at ω = 0.

Fortschr. Phys. 51, No. 2–3 (2003) 185

3 Conclusion

We have applied the quantum trajectory method to the system of two-level atoms strongly driven by acoherent light field and perturbed by stochastic noise. The simulation of a single quantum trajectoryrevealed that for high noise magnitude the phase difference between the dressed states tends to stabilizearound fixed values. When calculating the resonance fluorescence spectra, narrow resonances as centraldip and dispersive Fano-like profile occurred in the regime where the noise dominated the Rabi oscillations.These modifications of the resonance fluorescence spectra are associated with the stabilization of the dressed-state phases and stochastically-induced quantum interference between various emission channels.

Acknowledgements This work was supported by the Research Fund of Hungary under contract No. T034484 andby the Polish Committee for Scientific Research (grant 2PO3B 015 16). It was also a part of a general program of theNational Laboratory of AMO Physics in Torun, Poland (PBZ/KBN/032/P03/2001).

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