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Physical Review A

Stochastic incoherences of optical Bloch equations

K. Wódkiewicz

Published Apr 1, 1979• 116 citations
Publisher

Abstract

The theory of optical Bloch equations with stochastic coefficients is discussed in detail. It is shown that Bloch equations with stochastic fields lead to multiplicative stochastic processes for which exact solutions can be obtained. The steady state and the transient regimes of the phase-diffusion model of the laser bandwidth are treated, and exact solutions of Bloch equations are obtained. The case of fluctuation due to collisions is also discussed. A simple stochastic model of radiation-induced collisions is presented. For each stochastic model of incoherences formulas for the longitudinal and transverse lifetimes are derived exactly. It is shown that the finite bandwidths of light or the collision incoherences do not, in general, have a simple additive effect in atomic equations of motion.

DOI
10.1103/physreva.19.1686

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•Jan 1, 1985•0 citations
The theory of optical Bloch equations with stochastic coefficients is discussed in detail. It is shown that Bloch equations with stochastic fields lead to multiplicative stochastic processes for which exact solutions can be obtained. The steady state and the transient regimes of the phase-diffusion model of the laser bandwidth are treated, and exact solutions of Bloch equations are obtained. The case of fluctuation due to collisions is also discussed. A simple stochastic model of radiation-induced collisions is presented. For each stochastic model of incoherences formulas for the longitudinal and transverse lifetimes are derived exactly. It is shown that the finite bandwidths of light or the collision incoherences do not, in general, have a simple additive effect in atomic equations of motion.
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A system of N identical superimposed two-level atoms (the Dicke model) interacting with one another and driven by a fluctuating laser field of finite phase and amplitude bandwidths is considered. An exact master equation for the phase-averaged reduced atomic density operator is derived using the theory of multiplicative stochastic processes. It is shown that in the absence of amplitude fluctuations, this equation admits an exact steady-state solution. The exact solution is further used to obtain and analyse the behaviour of the phase-averaged atomic observables and atomic correlation functions in the steady state. The principal results of our analysis is that the phase fluctuations tend to smooth the discontinuities in the observables and the correlation functions found in the non-fluctuating case. Further, including the amplitude fluctuations as well we derive in the high-field limit a Markovian master equation for the density operator averaged over the ensembles of both phase and amplitude fluctuations. The quantum regression theorem is used to derive and discuss exact analytical expressions for the fluorescent spectrum and the second-order intensity correlation function for a system of two interacting atoms.
Recently1 a detailed analysis was given for the role played by laser phase fluctuations in NL optics. In that work, the existence of new extra resonances induced by stochastic fluctuations was predicted, based on a derivation starting from the phase diffusion model. In this work we present the results of numerical simulations of laser phase fluctuations. The optical Bloch equations are solved numerically for a phase fluctuating laser. A model of a piecewise constant phase jumping at random between all values is treated, and other models are discussed. Molecular collisions are simulated and compared to phase fluctuations. The collisions give rise to the pressure-induced extra resonance (PIER4), while the phase jumps give rise to stochastic fluctuations-induced extra resonance (SFIER). These simulations show clearly that phase fluctuations may be analogous to real collisions and may cause extra resonances, and indeed, the SFIER4 resonance has recently been observed experimentally.2
The effect of field fluctuations on the motion of a two-level atom in resonant laser radiation is investigated theoretically. The internal motion of the atom is treated quantum mechanically (optical Bloch equations); the translational motion, which is coupled to the internal motion, is treated classically (Ehrenfest's theorem); and the fluctuating radiation is treated as a classical electromagnetic wave with stochastic phase (phase-diffusion model). An expression for the radiation force and equations of motion for the ensemble-averaged Bloch vector are derived and used to calculate (i) the radiation force in a plane running wave, (ii) the steady-state radiation force in a general standing wave, and (iii) the dissipative force in a plane standing wave in the limit of strong phase fluctuations. In case (iii) it is found that the atomic velocity is damped when the radiation is tuned above resonance, which is opposite to the case of a weak coherent standing wave, in which damping occurs for radiation tuned below resonance. It is shown that fluctuations have a substantial effect on the radiation force only when the spectral width of the radiation approaches or exceeds the spontaneous emission rate.
The effect of stochastic laser phase fluctuations on nonlinear optical phenomena are investigated within the phase-diffusion model. Averaged equations of motion are derived in the limit of infinitely short correlation times and the results are applied to multiphoton absorption and to four-wave mixing. The effect of the stochastic phase fluctuations is shown to be equivalent to a T2{T}_{2}T2​ dephasing process, and a procedure is described for the inclusion of this equivalence in many nonlinear-optical problems. When a given mode contributes n photons to a multiphoton absorption process, its contribution to the width of the multiphoton transition is n2{n}^{2}n2 times its own stochastic width. When correlated lasers are used as the excitation sources in four-wave-mixing experiments, (new) stochastic-fluctuation-induced extra resonances in four-wave mixing (SFIER4) will be observed, in complete analogy to the extra resonances induced by collisions [pressure-induced extra resonances in form-wave mixing (PIER4)].