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.
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.
A generalization of the stochastic wave-function method to open quantum systems under the influence of strong laser fields is derived. To this end, quantum-statistical ensembles of reduced state vectors are represented in terms of probability distributions on the projective Hilbert space of the open system. By employing the Floquet expansion of the quantum time evolution, this representation is shown to lead in the Markov approximation to a piecewise deterministic stochastic process for the reduced wave function. The realizations of the stochastic process consist of smooth deterministic parts that are interrupted by sudden jumps. The jump operators are the eigenoperators of the Floquet Hamiltonian and describe instantaneous transitions between Floquet states. Two examples are given that serve to illustrate the method and the stochastic simulation technique of the process.
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.
This article presents functional integral representations for the heat semigroups with the infinitesimal generators given by self-adjoint Hamiltonians describing an interaction of a non-relativistic charged particle and a quantized radiation field in the Coulomb gauge without the dipole approximation.Special attention is paid to definition of the "time-ordered Hilbert space-valued stochastic integrals associated with a family of isometries from a Hilbert space t,o another one" and semigroup techniques.Some inequalities are derived, which are infinite degree versions of those known for finite dimensional Schr\"odinger operators with classical vector potentials.
In this paper, we develop an action formulation of stochastic dynamics in the Hilbert space. In this formulation, the quantum theory of random unitary evolution is easily reconciled with special relativity. We generalize the Wiener process into
1+3-dimensional spacetime, and then define a scalar random field which keeps invariant under Lorentz transformations. By adding to the action of quantum field theory a coupling term between random and quantum fields, we obtain a random-number action which has the statistical spacetime translation and Lorentz symmetries. The canonical quantization of the theory results in a Lorentz-invariant equation of motion for the state vector or density matrix. We derive the path integral formula of
S matrix, based on which we develop a diagrammatic technique for doing the calculation. We find the diagrammatic rules for both the stochastic free field theory and stochastic ${\ensuremath{\phi}}^{4}$ theory. The Lorentz invariance of the random
S matrix is strictly proved by using the diagrammatic technique. We then develop a diagrammatic technique for calculating the density matrix of final quantum states after scattering. In the absence of interaction, we obtain the exact expressions of both
S matrix and density matrix. In the presence of interaction, we prove a simple relation between the density matrices of stochastic and conventional ${\ensuremath{\phi}}^{4}$ theory. Our formalism leads to an ultraviolet divergence that has a similar origin as that in quantum field theory. The divergence can be canceled by renormalizing the coupling strength to random field. We prove that the stochastic quantum field theory is renormalizable even in the presence of interaction. In the models with a linear coupling between random and quantum fields, the random field excites particles out of the vacuum, driving the Universe towards an infinite-temperature state. The number of excited particles follows the Poisson distribution. The collision between particles is not affected by the random field. But the signals of colliding particles are gradually covered by the background excitations caused by random field.