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Physical Review A

Stochastic description of vacuum fluctuations

K. Wódkiewicz

Published Sep 1, 1988• 5 citations
Publisher

Abstract

It is shown that vacuum fluctuations can be incorporated in the atomic equations of motion as an external multiplicative stochastic process. From the quantum-electrodynamics (QED) vacuum-vacuum transition amplitude it is shown that under the standard Wigner-Weisskopf approximation, this external noise is a Gaussian white noise with an operator-valued diffusion. This property reflects the fundamental difference between QED and a purely stochastic theory in which atomic fluctuations are neglected. It is shown that spontaneous emission results from a stochastic average over this external noise. The role of vacuum and atomic fluctuations in such a process is exhibited.

DOI
10.1103/physreva.38.2932

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It is shown that vacuum fluctuations can be incorporated in the atomic equations of motion as an external multiplicative stochastic process. From the quantum-electrodynamics (QED) vacuum-vacuum transition amplitude it is shown that under the standard Wigner-Weisskopf approximation, this external noise is a Gaussian white noise with an operator-valued diffusion. This property reflects the fundamental difference between QED and a purely stochastic theory in which atomic fluctuations are neglected. It is shown that spontaneous emission results from a stochastic average over this external noise. The role of vacuum and atomic fluctuations in such a process is exhibited.
The atomic Bloch equations driven by a quantum noise are investigated in the framework of c-number multiplicative stochastic Rabi amplitudes. It is shown that the stochastic broad-band vacuum fluctuations can be incorporated in such a way that spontaneous absorption is removed. It is shown that the multiplicative stochastic noise with no spontaneous absorption is equivalent to the quantum vacuum noise. A stochastic model of the broad-band squeezed vacuum is introduced and investigated. From such a stochastic description the fast and the slow damping rates of the Bloch polarizations are obtained. The strong-field resonance fluorescence spectrum and the steady-state atomic inversion for the stochastic model of squeezed vacuum fluctuations are derived.
The role of vacuum fluctuations and radiation reaction in the process of spontaneous emission is examined with particular reference to the Heisenberg picture of quantum electrodynamics. In this picture the radiation reaction is found to be responsible for spontaneous emission if the equations are ordered normally. However, in the case of nonnormally ordered equations there appears to be an explicit contribution to the observable effects from vacuum fluctuations as well. We point out that this contribution from the vacuum field is an artifact of the approximation methods employed for eliminating the free-field variables from the nonnormally ordered equations These approximations are the same as those used in the master-equation approach, in which spontaneous emission appears to be due solely to the vacuum fluctuations. We show that the vacuum field does not contribute explicitly in any ordering if it is eliminated nonperturbatively and the approximation methods are invoked only after its elimination from the equations of motion.
A study of the non-dissipative Brownian motion in vacuum is presented. The noise source associated to the stochastic process assumed in this work is vacuum fluctuations of some quantum field capable of interact with a massive particle. For the associated Fokker-Planck equation we do a perturbation theory which has terms presenting dynamics similar to that satisfied by the probability amplitude in Quantum Theory even though the physical interpretation be classical. In particular classical observables present expected values that coincide with those calculated using Quantum Mechanics.
We propose a stochastic Schrödinger equation in which the momentum is coupled to a white Gaussian noise. In the stochastic representation, the kinetic energy representing the self-interaction of momentum is reduced to a linear term of momentum. As such, the quantum evolution operator factorizes into two contributions due to the momentum and the potential, respectively. The exact quantum propagator thereby becomes an expectation of the stochastic one in which the amplitude results from the potential with a fluctuating position: The particle moves with a constant velocity, subjected to a complex Brownian motion. We demonstrate that the stochastic Schrödinger equation can be feasibly used to derive the quantum propagators for the linear potential and the harmonic oscillator system. Novel semiclassical and other approximations may be developed from the new representation of quantum mechanics.
It is shown that the vacuum state of weakly interacting quantum field theories can be described, in the Heisenberg picture, as a linear combination of randomly distributed incoherent paths that obey classical equations of motion with constrained initial conditions. We call such paths ``pseudoclassical'' paths and use them to define the dynamics of quantum fluctuations. Every physical observable is assigned a time-dependent value on each path in a way that respects the uncertainty principle, but in consequence, some of the standard algebraic relations between quantum observables are not necessarily satisfied by their time-dependent values on paths. We define ``collective observables'' which depend on a large number of independent degrees of freedom and show that the dynamics of their quantum fluctuations can be described in terms of unconstrained classical stochastic processes without reference to any additional external system or to an environment. Our analysis can be generalized to states other than the vacuum. Finally, we compare our formalism to the formalism of coherent states and highlight their differences.
Vacuum fluctuations play a central role in spontaneous emission. Recently, it has been suggested that these fluctuations could also be fundamental in the absorption dynamics, breaking the superposition inherent to the linear quantum evolution. We analyze the consistency of that proposal with previous results in double spontaneous emission. Moreover, for the case of single absorption by two atoms, we present a test based on the time dependence of the subsequent spontaneous emission patterns, which can experimentally settle the question. This test is more viable than the original proposal, built on the Casimir effect. Our approach also allows for the comparison between the time scales of vacuum fluctuations as a disentangling mechanism and an emission trigger.