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arXiv:1006.0640 · Izvestiya Mathematics

Randomized Hamiltonian Feynman integrals and Schrödinger-Itô stochastic equations

John Edward Gough, OLeg O Obrezkov, O G Smolyanov

Published Dec 31, 2005• 6 citations
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Abstract

In this paper, we consider stochastic Schroedinger equations with two-dimensional white noise. Such equations are used to describe the evolution of an open quantum system undergoing a process of continuous measurement. Representations are obtained for solutions of such equations using a generalization to the stochastic case of the classical construction of Feynman path integrals over trajectories in the phase space.

DOI
10.1070/im2005v069n06abeh002290

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In this paper, we consider stochastic Schroedinger equations with two-dimensional white noise. Such equations are used to describe the evolution of an open quantum system undergoing a process of continuous measurement. Representations are obtained for solutions of such equations using a generalization to the stochastic case of the classical construction of Feynman path integrals over trajectories in the phase space.
Abstract Quantum stochastic differential equations are shown to naturally arise because of the mathematical background of the correctness of the Schrödinger equation, which describes the interaction of an open quantum system with broadband quantized fields in the Markov approximation. The main classical and quantum stochastic processes appear in the notation in the Markov approximation of the operator of the electric broadband quantized field strength and the effective Hamiltonian of the interaction of a quantum system with it obtained by the methods of algebraic resonance perturbation theory. All basic stochastic quantum processes, including the counting and two-quantum Lévy processes, are constructed based on quantum white noise, which differs significantly from the classical case. At the same time, the apparatus of stochastic differential equations demonstrates consistency and parallelism.
The principle of "indirect continuous measurement" in "open quantum system theory" is usually described by non-usual types of stochastic differential equations. These equations are called "stochastic Schrödinger equations" and their solutions are called "quantum trajectories". Physically, they describe the random evolution of the state of a quantum system undergoing indirect quantum measurement (such models are widely used in quantum optics, quantum computing and quantum information theory). In this chapter, we consider a physically realistic discrete-time setup for two-level quantum systems and we present the theory of "discrete quantum trajectories". These discrete trajectories are Markov chains which can be expressed as solutions of "discrete-time" stochastic differential equations". In particular, these equations appear as time discretization of "stochastic Schrödinger equations". Going to the continuous-time limit, we justify the stochastic Schrödinger equations associated to the two-level systems. Within this approach, we obtain two different types of behaviors described either by jump-type or diffusive-type stochastic differential equations. Finally we investigate the large time behavior of the solutions and we prove return to equilibrium properties for the associated physical models.
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]:
A class of functions ℋnF was chosen, which are boundary values of holomorphic functions. It was shown that if the potential V∈ℋnF and the initial wave function ψ∈ℋnF then the solution of the Schrödinger equation is ψt∈ℋnF. The unitary evolution ψ → ψt can be expressed by an integral with respect to the Wiener measure. To each ψt∈ℋnF is associated a complex Markov process qt fulfilling stochastic Hamilton equations. The time evolution of ψ as well as multitime correlation functions of arbitrary observables in quantum mechanics take the form of expectation values with respect to qt. Some statistical and ergodic aspects of quantum mechanics resulting from the stochastic interpretation of Feynman’s sum over trajectories are discussed.