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Physical Review A

Quantum-classical correspondences for atomic operators: A doubled atomic-space approach

Li You, J. Cooper

Published Dec 1, 1994• 0 citations
Publisher

Abstract

An operational method is presented that can avoid some of the difficulties associated with numerically simulating the Langevin operator equations for a dissipative quantum optical system. The approach is based on a set of quantum-classical correspondence rules that relate atomic operators in the Heisenberg picture to their corresponding double-dimensioned classical atomic variables. With this approach, the Langevin operator equations can be transformed into a doubled set of classical stochastic differential equations (SDE's). The relationship to the quantum regression theory up to the two-time correlation function and similarities to the positive P representation are discussed. As an illustrative example, we show that both the resonance fluorescence spectrum (Mollow spectrum) as well as the absorption spectrum for a two-level atom interacting with a near-resonance field can be simulated from the doubled set of classical SDE's without explicit reference to the quantum regression theorem.

DOI
10.1103/physreva.50.5264

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An operational method is presented that can avoid some of the difficulties associated with numerically simulating the Langevin operator equations for a dissipative quantum optical system. The approach is based on a set of quantum-classical correspondence rules that relate atomic operators in the Heisenberg picture to their corresponding double-dimensioned classical atomic variables. With this approach, the Langevin operator equations can be transformed into a doubled set of classical stochastic differential equations (SDE's). The relationship to the quantum regression theory up to the two-time correlation function and similarities to the positive P representation are discussed. As an illustrative example, we show that both the resonance fluorescence spectrum (Mollow spectrum) as well as the absorption spectrum for a two-level atom interacting with a near-resonance field can be simulated from the doubled set of classical SDE's without explicit reference to the quantum regression theorem.
In the previous chapter, we developed the equation of motion for a system as it evolved under the influence of an unobserved (reservoir) system. We used the density matrix approach and worked in the interaction picture. In this chapter, we consider the same problem of the system-reservoir interaction using a quantum operator approach. We again eliminate the reservoir variables. The resulting equations for the system operators include, in addition to the damping terms, the noise operators which produce fluctuations. These equations have the form of classical Langevin equations, which describe, for example, the Brownian motion of a particle suspended in a liquid. The Heisenberg–Langevin approach discussed in this chapter is particularly suitable for the calculation of two-time correlation functions of the system operator as is, for example, required for the determination of the natural linewidth of a laser. We first consider the damping of the harmonic oscillator by an interaction with a reservoir consisting of many other simple harmonic oscillators. This system describes, for example, the damping of a single-mode field inside a cavity with lossy mirrors. The reservoir, in this case, consists of a large number of phonon-like modes in the mirrors. We also consider the decay of the field due to its interaction with an atomic reservoir. An interesting application of the theory of the system–reservoir interaction is the evolution of an atom inside a damped cavity. It is shown that the spontaneous transition rate of the atom can be substantially enhanced if it is placed in a resonant cavity.
We discuss a procedure for the calculation of operator correlation functions that arise frequently in quantum-optics problems. The starting point of our method is the tradiational quantum regression theorem, and its implementation requires only the solution of the master equation for the density operator of the system of interest. An attractive feature of this scheme of calculation is that it offers some distinctive advantages over the conventional Langevin-equations approach or other procedures that rely on the mapping of the density operator into appropriate phase-space distribution functions. Although the practical implementation of this method is likely to require numerical computations for virtually every model of physical interest, the procedure is sufficiently general to allow consideration of both field and atomic correlation functions, as well as multitime expectation values of mixed products of atomic and field operators. In this paper we provide a fairly detailed derivation of the formulas of interest, and illustrate our results with specific examples and tests of the procedure.
A number of physically intuitive results for the calculation of multi-time correlations in phase-space representations of quantum mechanics are obtained. They relate time-dependent stochastic samples to multi-time observables, and rely on the presence of derivative-free operator identities. In particular, expressions for time-ordered normal-ordered observables in the positive-P distribution are derived which replace Heisenberg operators with the bare time-dependent stochastic variables, confirming extension of earlier such results for the Glauber-Sudarshan P. Analogous expressions are found for the anti-normal-ordered case of the doubled phase-space Q representation, along with conversion rules among doubled phase-space s-ordered representations. The latter are then shown to be readily exploited to further calculate anti-normal and mixed-ordered multi-time observables in the positive-P, Wigner, and doubled-Wigner representations. Which mixed-order observables are amenable and which are not is indicated, and explicit tallies are given up to 4th order. Overall, the theory of quantum multi-time observables in phase-space representations is extended, allowing non-perturbative treatment of many cases. The accuracy, usability, and scalability of the results to large systems is demonstrated using stochastic simulations of the unconventional photon blockade system and a related Bose-Hubbard chain. In addition, a robust but simple algorithm for integration of stochastic equations for phase-space samples is provided.
Quantum stochastic differential equations have been used to describe the dynamics of an atom interacting with the electromagnetic field via absorption/emission processes. Here, by using the full quantum stochastic Schrödinger equation proposed by Hudson and Parthasarathy, we show that such models can be generalized to include other processes into the interaction. In the case of a two-level atom we construct a model on the basis of some physical requirements, the main being a balance equation on the fluxes of the ingoing and outgoing photons; in this model the atom-field interaction turns out to be due either to absorption/emission processes either to direct scattering processes, which simulate the interaction due to virtual transitions to the levels which have been eliminated from the description. To see the effects of the new terms, we consider both direct and heterodyne detection of the fluorescence light emitted by an atom stimulated by a monochromatic coherent laser and we deduce from these two detection schemes the expressions of the total, elastic and inelastic electromagnetic cross sections and the spectral distribution of the fluorescence light. The total cross section, as a function of the frequency of the stimulating laser, can present not only a Lorentzian shape, but the full variety of Fano profiles; intensity dependent widths and shifts are obtained. The fluorescence spectrum can present complicated shapes, according to the values of the various parameters; when the direct scattering is not important the usual symmetric triplet structure of the Mollow spectrum appears (for high intensity of the stimulating laser), while a strong contribution of the direct scattering process can distort such a triplet structure or can even make it to disappear.