O ct 1 99 5 Exact semiclassical wave equation for stochastic quantum optics ∗

Semiclassical (stochastic) wave equations are proposed for the coupled dynamics of atomic quantum states and semiclassical radiation field. All relevant predictions of standard unitary quantum dynamics are exactly reproducible in the framework of stochastic wave equation model. We stress in such a way that the concept of stochastic wave equations is not to be restricted to the widely used Markovian approximation. submitted to: Stochastic Quantum Optics, ed.: H. Carmichael (special issue of Quantum and Semiclassical Optics) E-mail: [email protected] 0 1) Introduction. Since many years it has been understood that one can not reproduce all statistical predictions of quantum theory from pure classical statistics [1]. One can do it for certain predictions after all. Reduced dynamics of quantum subsystems (i.e. effective dynamics of open quantum systems) can always be interpreted as hybrids of quantum and classical mechanisms. The corresponding semiclassical dynamics (which may on equal footing be called semiquantized) is described by stochastic wave equations (SWEs). Since ten years they have been attracting permanent interest in research of foundations [2] and, since the 90’s [3], in quantum optics, too. Consider a system of atoms interacting with radiation field. Basically, all attainable results are approximations: SWEs have been derived in weak coupling Markovian limit [4]. In the present work, however, we investigate the exact unitary dynamics of the atom+radiation system and we ask the following question. Can we describe the radiation field in terms of classical (stochastic) variables instead of quantum ones in such a manner that all relevant predictions remain exactly identical with those of the unitary quantum theory? The answer will be affirmative in almost all respects. Forthcoming results were anticipated once [5] by detailed relativistic calculations. Here we intend to give a short account on exact SWEs. Readers interested in practical rather than conceptual aspects may, after reading Section 2, undertand SWEs directly from Section 5. 2) The quantum model. For simplicity’s sake, we consider single mode (cavity) quantum-electrodynamics. The Hamiltonian for the composite system of the atom plus radiation field has the following general form Ĥ = Ĥ + ωââ+ Ĵ â + Ĵâ (1)