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Physical Review A

Excited states in stochastic electrodynamics

H. M. França, Trevor W. Marshall

Published Oct 1, 1988• 25 citations
Publisher

Abstract

We show that the set of Wigner functions associated with the excited states of the harmonic oscillator constitutes a complete set of functions over the phase space. An arbitrary probability distribution can be expanded in terms of these Wigner functions. By studying the time evolution, according to stochastic electrodynamics, of the expansion coefficients, we are able to separate explicitly the contributions of the radiative reaction and the vacuum field to the Einstein A coefficients for this system. We also supply a simple semiclassical explanation of the Weisskopf-Heitler phenomenon in resonance fluorescence.

DOI
10.1103/physreva.38.3258

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We show that the set of Wigner functions associated with the excited states of the harmonic oscillator constitutes a complete set of functions over the phase space. An arbitrary probability distribution can be expanded in terms of these Wigner functions. By studying the time evolution, according to stochastic electrodynamics, of the expansion coefficients, we are able to separate explicitly the contributions of the radiative reaction and the vacuum field to the Einstein A coefficients for this system. We also supply a simple semiclassical explanation of the Weisskopf-Heitler phenomenon in resonance fluorescence.
We apply the theory of stochastic electrodynamics to the study of the (nonrelativistic) harmonic oscillator by using the Fokker–Planck method. It is demonstrated that the equilibrium distribution in phase space is exactly equal to that given by quantum statistical mechanics, i.e., the corresponding Wigner distribution, and that analysis of this distribution by means of a decomposition in terms of canonical densities leads automatically to the usual description of quantum mechanics in terms of excited states. All fundamental equations of quantum mechanics are recovered as aprproximations to zero order in the radiation terms; the first-order terms lead to the radiative corrections predicted by quantum electrodynamics, namely, the decay of states and the Lamb shift of the energy levels. The necessary differences between both treatments of the oscillator and their implications are briefly discussed.
We derive a classical Schrodinger type equation from the classical Liouville equation in phase space. The derivation is based on a Wigner type Fourier transform of the classical phase space probability distribution, which depends on an arbitrary constant ααα with dimension of action. In order to achieve this goal two requirements are necessary: 1) It is assumed that the classical probability amplitude Ψ(x,t)Ψ(x,t)Ψ(x,t) can be expanded in a complete set of functions Φn(x)Φ_n(x)Φn​(x) defined in the configuration space; 2) the classical phase space distribution W(x,p,t)W(x,p,t)W(x,p,t) obeys the Liouville equation and is a real function of the position, the momentum and the time. We show that the constant ααα appearing in the Fourier transform of the classical phase space distribution, and also in the classical Schrodinger type equation, has its origin in the spectral distribution of the vacuum zero-point radiation, and is identified with the Planck's constant ℏ\hbarℏ.
Abstract The Wigner function was introduced as an attempt to describe quantum-mechanical fields with the tools inherited from classical statistical mechanics. In particular, it is widely used to describe the properties of radiation fields. In fact, a useful way to distinguish between classical and nonclassical states of light is to ask whether their Wigner function has a Gaussian profile or not, respectively. In this paper, we use the basis of Fock states to provide the closed-form expression for the Wigner function of an arbitrary quantum state. Thus, we provide the general expression for the Wigner function of a squeezed Fock, coherent and thermal states, with an arbitrary squeezing parameter. Then, we consider the most fundamental quantum system, Resonance Fluorescence, and obtain closed-form expressions for its Wigner function under various excitation regimes. With them, we discuss the conditions for obtaining a negative-valued Wigner function and the relation it has with population inversion. Finally, we address the problem of the observation of the radiation field, introducing physical detectors into the description of the emission. Notably, we show how to expose the quantumness of a radiation field that has been observed with a detector with finite spectral resolution, even if the observed Wigner function is completely positive.
The relation between quantum electrodynamics and (classical) stochastic electrodynamics is elucidated by means of a general construction which associates with every Gaussian quantum field (for example, vacuum-free fields or coherent states at zero or nonzero temperature) a classical random field, which in the case of quantum electrodynamics yields stochastic electrodynamics. The associated classical and quantum theories yield the same results for all nondisturbing (causally separated) field measurements. This is an additional classical characteristic of coherent quantum states which implies that Bell's inequality is satisfied by quantum field measurements. Furthermore, stochastic electrodynamics is identified as the Wigner distribution of quantum electrodynamics. We point out that in these Gaussian states stochastic electrodynamics is more suitable as a classical model of the quantum field than is the Euclidean field, which results from applying Nelson's stochastic mechanics to each field mode.