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ORTHOGONAL JUMPS OF THE WAVEFUNCTION IN WHITE-NOISE POTENTIALS

Lajos Diósi

Published Jan 1, 1985• 26 citations

Abstract

The master equation of the quantum noise theory is derived by means of Feynman's path integral method. We propose an equivalent s ochastic process where the wavefunction satisfies a nonlinear SchrOdinger equation except for random moments at which it shows orthogonal jumps. Since the early works [ 1-3] on the theory of quantum noise, great interest has been shown in quantum-me-chanical systems affected by random external forces and also new aspects have appeared [4-6]. Usually, the quan-tum state of such systems i described by a density operator obeying a deterministic differential equation (master equation). If we attribute a state vector (instead of the density operator) to the given system, then the state vector should be considered as a stochastic variable governed by a certain stochastic process. In the first part of our work we derive the master equation for the density operator by a seemingly new applica-tion of the path integral technique. Then, we construct a stochastic process for the wavefunction of the system, while retaining its equivalence with the density operator formalism. In the final part we shortly discuss favoured features of the above stochastic process proposed for the wavefunction. This work is an anticipation of a more general and lengthy explanation of the stochastic properties of the wave-function in the presence of random forces. Now, we are going to introduce the notion of stochastic potential of white-noise type. Here we shall investigate the effect of gaussian white-noise potentials. For brevity, we restrict ourselves to the case of a single point.like par-ticle moving in one dimension. Let us assume that the potential V(x, t) acting on the particle is a stochastic variable (x, t stand for the coordi-nate and time). We define the probability distribution of V by the following generator functional G [h]:

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The master equation of the quantum noise theory is derived by means of Feynman's path integral method. We propose an equivalent s ochastic process where the wavefunction satisfies a nonlinear SchrOdinger equation except for random moments at which it shows orthogonal jumps. Since the early works [ 1-3] on the theory of quantum noise, great interest has been shown in quantum-me-chanical systems affected by random external forces and also new aspects have appeared [4-6]. Usually, the quan-tum state of such systems i described by a density operator obeying a deterministic differential equation (master equation). If we attribute a state vector (instead of the density operator) to the given system, then the state vector should be considered as a stochastic variable governed by a certain stochastic process. In the first part of our work we derive the master equation for the density operator by a seemingly new applica-tion of the path integral technique. Then, we construct a stochastic process for the wavefunction of the system, while retaining its equivalence with the density operator formalism. In the final part we shortly discuss favoured features of the above stochastic process proposed for the wavefunction. This work is an anticipation of a more general and lengthy explanation of the stochastic properties of the wave-function in the presence of random forces. Now, we are going to introduce the notion of stochastic potential of white-noise type. Here we shall investigate the effect of gaussian white-noise potentials. For brevity, we restrict ourselves to the case of a single point.like par-ticle moving in one dimension. Let us assume that the potential V(x, t) acting on the particle is a stochastic variable (x, t stand for the coordi-nate and time). We define the probability distribution of V by the following generator functional G [h]:

The dynamics of an open quantum system coupled to an external reservoir is studied on the basis of a recently proposed formulation of quantum statistical ensembles in terms of probability distributions on projective Hilbert space. The previous result is generalized to include interaction Hamiltonians of the form

{\mathit{tsum}}_{\mathit{i}}

{\mathit{A}}_{\mathit{i}}

\ensuremath{\bigotimes}

{\mathit{B}}_{\mathit{i}}

, where

{\mathit{A}}_{\mathit{i}}

and

{\mathit{B}}_{\mathit{i}}

are operators acting on the Hilbert space of the reduced system and of the reservoir, respectively. The differential Chapman-Kolmogorov equation governing the dynamics of the conditional transition probability of the reduced system is derived from the underlying microscopic theory based on the Schr\"odinger equation for the total system. The stochastic process turns out to be a piecewise deterministic Markovian jump process in the projective Hilbert space of the reduced system. The sample paths are derived and shown to be similar to those of the Monte Carlo wave function simulation methods proposed in the literature. Finally, a diffusion-noise expansion of the Liouville master equation is performed and demonstrated to yield a stochastic differential equation for the state vector of the open system.

A formulation of quantum statistical ensembles in terms of probability distributions on a projective Hilbert space is developed. The combination of statistically independent systems and the reduction of a system to one of its subsystems are described by means of a tensor product of probability distributions and a reduction formula for the reduced probability distribution. Within this framework the dynamics of open quantum systems is investigated starting form a microscopic system-plus-reservoir model. Employing the Markov approximation of the classical theory of stochastic processes the short time behavior of the conditional transition probability is derived and shown to yield a differential Chapman-Kolmogorov equation. The latter has the form of a Liouville master equation for the reduced probability distribution of the open quantum system. Thus it is shown that the time-dependent wave function of an open quantum system represents a well-defined and unique stochastic process in the space of rays of the underlying Hilbert space. This stochastic process consists of a continuous time evolution generated by a nonlinear Schr\"odinger equation and a discontinuous jump process; the realizations of the processes correspond to those of the Monte Carlo wave function simulation method and to those of the jump-type evolution of the quantum trajectory method. The quantum master equation for the statistical operator is derived as the equation of motion for the two-point correlation function of the stochastic process.

We investigate the quantum dynamics of a particle in a one-dimensional parabolic potential subject to both timeperiodic driving and stochastic perturbations modeled as white noise. Using the non-stationary Schrödinger equation and the Green’s function formalism, we derive analytical expressions for the time evolution of the wave function and its statistical moments. It is shown that, due to the quadratic nature of the potential, the wave function retains a Gaussian form, allowing a reduced description in terms of its first and second moments. Under resonant periodic driving, the mean energy exhibits quadratic growth in time. In the presence of white noise, the wave packet undergoes diffusive spreading and loss of coherence. For combined deterministic and stochastic driving, we identify a characteristic decoherence (delocalization) time, which scales inversely with the noise intensity. The results are relevant to quantum transport, driven systems, and tunneling processes in fluctuating environments.

We study the quantum oscillator perturbed by external random noise which imparts a series of kicks to the system: the kicks are Poisson distributed in time and the probability measure associated with the jump size is assumed to correspond to symmetric stable L\'evy distributions. The Feynman-Vernon theory of influence functionals yields a master equation for the density operator with a term related to the characteristic function of the measure associated with the jump distribution. A scaling argument shows that the constant associated with the external-noise-induced diffusion carries fractal space dimension 0<\ensuremath{\alpha}\ensuremath{\le}2. In the noise-dominated regime the system exhibits singularities for 0\ensuremath{\alpha}2 in contrast to the behavior for \ensuremath{\alpha}=2 corresponding to the Gaussian case.