New Approach for Stochastic Quantum Processes, their Manipulation and Control

TL;DR

This paper introduces a new stochastic differential equation approach to model open quantum systems, enabling exact analysis of thermodynamic potentials and uncertainties in quantum harmonic oscillators interacting with environments.

Contribution

It develops a Langevin-Schroedinger type SDE for open quantum systems and introduces the stochastic density matrix method for exact thermodynamic and uncertainty analysis.

Findings

Derived Langevin-Schroedinger SDE for quantum systems with environment

Developed stochastic density matrix method for relaxation processes

Calculated thermodynamic potentials and uncertainty relations

Abstract

The dissipation and decoherence (for example, the effects of noise in quantum computations), interaction with thermostat or in general with physical vacuum, measurement and many other complicated problems of open quantum systems are a consequence of interaction of quantum system with the environment. These problems are described mathematically in terms of complex probabilistic process (CPP). Particularly, treating the environment as a Markovian process we derive an Langevin-Schroedinger type stochastic differential equation (SDE) for describing the quantum system interacting with environment. For the 1D randomly quantum harmonic oscillator (QHO) model L-Sh SDE is a solution in the form of orthogonal CPP. On the basis of orthogonal CPP the stochastic density matrix (SDM) method is developed and in its framework relaxation processes in the uncountable dimension closed system of "QHO+environment" is investigated. With the help of SDM method the thermodynamical potentials, like nonequilibrium entropy and the energy of ground state are exactly constructed. The dispersion for different operators are calculated. In particular, the expression for uncertain relations depending on parameter of interaction with environment is obtained. The Weyl transformation for stochastic operators is specified. Ground state Winger function is developed in detail.