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Fortschritte der Physik/Progress of Physics

Stochastic Dynamics of Reduced Wave Functions and Continuous Measurement in Quantum Optics

Heinz‐Peter Breuer, Francesco Petruccione

Published Jan 1, 1997• 15 citations
Publisher

Abstract

The stochastic dynamics of open quantum systems interacting with a zero temperature environment is investigated by employing a formulation of quantum statistical ensembles in terms of probability distributions on projective Hilbert space. It is demonstrated that the open system dynamics can consistently be described by a stochastic process on the reduced state space. The physical meaning of reduced probability distributions on projective Hilbert space is derived from a complete, orthogonal measurement of the environment. The elimination of the variables of the environment is shown to lead to a piecewise deterministic process in Hilbert space defined by a differential Chapman-Kolmogorov equation. A Hilbert space path integral representation of the stochastic process is constructed. The general theory is illustrated by means of three examples from quantum optics. For these examples the microscopic derivation of the stochastic process is given and the general solution of the differential Chapman-Kolmogorov equation is constructed by means of the path integral representation.

DOI
10.1002/prop.2190450104

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The stochastic dynamics of open quantum systems interacting with a zero temperature environment is investigated by employing a formulation of quantum statistical ensembles in terms of probability distributions on projective Hilbert space. It is demonstrated that the open system dynamics can consistently be described by a stochastic process on the reduced state space. The physical meaning of reduced probability distributions on projective Hilbert space is derived from a complete, orthogonal measurement of the environment. The elimination of the variables of the environment is shown to lead to a piecewise deterministic process in Hilbert space defined by a differential Chapman-Kolmogorov equation. A Hilbert space path integral representation of the stochastic process is constructed. The general theory is illustrated by means of three examples from quantum optics. For these examples the microscopic derivation of the stochastic process is given and the general solution of the differential Chapman-Kolmogorov equation is constructed by means of the path integral representation.
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.
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 tsumi{\mathit{tsum}}_{\mathit{i}}tsumi​Ai{\mathit{A}}_{\mathit{i}}Ai​\ensuremath{\bigotimes}Bi{\mathit{B}}_{\mathit{i}}Bi​, where Ai{\mathit{A}}_{\mathit{i}}Ai​ and Bi{\mathit{B}}_{\mathit{i}}Bi​ 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.
Employing a formulation of quantum statistical ensembles in terms of classical probability distributions on projective Hilbert space, it is shown that the wave function of an open quantum system represents a stochastic process. The stochastic dynamics of the state vector is obtained from a microscopic system-plus-reservoir model by deriving within the Markov approximation the differential Chapman-Kolmogorov equation for the classical distribution of the reduced system. The realizations of the stochastic process are found to be similar to those of the Monte Carlo wave function simulation method proposed, in general form, by Zoller et al. [Phys. Rev. A 46, 4363 (1992)].
Abstract This chapter contains the foundations of the stochastic formulation of continuous measurements carried out on an open quantum system. It shows that the time evolution of the state vector, conditioned on the measurement record, is given by a piecewise deterministic process in Hilbert space involving continuous time-evolution periods which are broken by randomly occurring, sudden quantum jumps. This so-called unravelling of the quantum master equation in the form of a stochastic process is based on a close relation between quantum dynamical semigroups and piecewise deterministic processes. The general theory is illustrated by means of a number of examples, such as direct, homodyne, and heterodyne photodetection. The chapter also treats the corresponding description of continuous measurements through stochastic Schr ö dinger equations and stochastic differential equations for the density matrix, as well as the stochastic representation of multitime correlations functions.
Abstract This chapter characterizes the statistical properties of a quantum mechanical ensemble in terms of a density matrix. However, if selective measurements of one or several observables are carried out on the ensemble, it will split into a number of sub-ensembles, each sub-ensemble being conditioned on a particular outcome of the measurements. The mathematical description of the collection of sub-ensembles thus created leads to probability distributions on projective Hilbert space. The chapter develops an appropriate mathematical framework which enables the general formulation of such a distribution, and leads to the concepts of stochastic state vectors and stochastic density matrices. These concepts are required in later chapters to construct appropriate stochastic differential equations describing the continuous monitoring of open quantum systems.
A Hilbert space path integral for the dissipative dynamics of matter in thermal radiation fields is derived from the Hamiltonian of quantum electrodynamics. This path integral represents the conditional transition probability of a stochastic Markov process as a sum over sample trajectories in Hilbert space. The realizations of the process are piecewise deterministic paths broken by instantaneous quantum jumps. It is shown that the operators which define the possible quantum jumps form a continuous family parametrized by the polarization vector of the emitted or absorbed photons. The stochastic process is shown to be representation-independent and to be invariant with respect to space rotations. The precise physical interpretation of the stochastic process is given. In particular, the expansion of the density matrix in terms of quantum jumps is derived for finite temperatures from the Hilbert space path integral.