Journal of Physics Conference Series

Application of stochastic calculus for certain classes of quantum models

Andrei Pavelev, Vitalii Semin

Published Feb 1, 2021• 0 citations

Abstract

Abstract In this work, stochastic calculus is applied for describing open quantum systems. The suggested approach is based on a modified stochastic Schroedinger equation. We apply the approach to investigate a qubit relaxation in Markovian and non-Markovian regimes.

DOI: 10.1088/1742-6596/1745/1/012056

Related papers

Stochastic Schrödinger equation derivation of non-Markovian two-time correlation functions

Rafael Carballeira, David Dolgitzer, Peng Zhao +2 more

We derive the evolution equations for two-time correlation functions of a generalized non-Markovian open quantum system based on a modified stochastic Schrödinger equation approach. We find that the two-time reduced propagator, an object that used to be characterized by two independent stochastic processes in the Hilbert space of the system, can be simplified and obtained by taking ensemble average over one single noise. This discovery can save the cost of computation, and accelerate the converging process when taking the average over noisy trajectories. As a result, our method can be widely applied to many open quantum models, especially large-scale systems and extend the quantum regression theory to the non-Markovian case. In the short-time simulations, it is observed a significant difference between Markovian and non-Markovian cases, which can be applied to realize the environmental spectrum detection and enhance the measurement sensitivity in varying open quantum systems.

Scientific ReportsJun 4, 202114 citations

The non-Markovian stochastic Schrödinger equation for open systems

Lajos Diósi, Walter T. Strunz

We present the non-Markovian generalization of the widely used stochastic Schrodinger equation. Our result allows to describe open quantum systems in terms of stochastic state vectors rather than density operators, without approximation. Moreover, it unifies two recent independent attempts towards a stochastic description of non-Markovian open systems, based on path integrals on the one hand and coherent states on the other. The latter approach utilizes the analytical properties of coherent states and enables a microscopic interpretation of the stochastic states. The alternative first approach is based on the general description of open systems using path integrals as originated by Feynman and Vernon.

Jan 1, 1997143 citations

Stochastic simulation of dissipation and non-Markovian effects in open quantum systems

Denis Lacroix A

The exact dynamics of a system coupled to an environment can be described by an integro-differential stochastic equation for the reduced density. The influence of the environment is incorporated through a mean field which is both stochastic and nonlocal in time and where the standard two-time correlation functions of the environment appear naturally. Since no approximation is made, the presented theory incorporates exactly dissipative and non-Markovian effects. Applications to the spin-boson model coupled to a heat bath with various coupling regimes and temperature show that the presented stochastic theory can be a valuable tool to simulate exactly the dynamics of open quantum systems. Links with the stochastic Schrödinger equation method and possible extensions to "imaginary time" propagation are discussed.

arXiv:0802.1981Physical Review EApr 29, 200822 citations

Convolutionless Non-Markovian master equations and quantum trajectories: Brownian motion

Walter T. Strunz, Ting Yu

Stochastic Schr\"odinger equations for quantum trajectories offer an alternative and sometimes superior approach to the study of open quantum system dynamics. Here we show that recently established convolutionless non-Markovian stochastic Schr\"odinger equations may serve as a powerful tool for the derivation of convolutionless master equations for non-Markovian open quantum systems. The most interesting example is quantum Brownian motion (QBM) of a harmonic oscillator coupled to a heat bath of oscillators, one of the most employed exactly soluble models of open system dynamics. We show explicitly how to establish the direct connection between the exact convolutionless master equation of QBM and the corresponding convolutionless exact stochastic Schr\"odinger equation.

arXiv:quant-ph/0312103Physical Review AMay 24, 2004165 citations

Dynamics of non-Markovian open quantum systems

Inés de Vega, Daniel Alonso

This review gives a summary of the many techniques that are used in the analysis of open quantum systems. Emphasis is on those cases where it is unsuitable to use a memoryless or Markovian point of view, generally because there is no large separation of time scales between system and environment dynamics. The approaches reviewed include master equations, Heisenberg equations of motion, chain mapping representations, and various stochastic methods such as path integral Monte Carlo and stochastic equations. Guidance is given on how to evaluate the suitability of each of these methods for application in different physical problems.

arXiv:1511.06994Reviews of Modern PhysicsJan 20, 20171,230 citations

Stochastic Schrodinger equations and applications to Ehrenfest-type theorems

Franco Fagnola, Carlos M. Mora

We study stochastic evolution equations describing the dynamics of open quantum systems. First, using resolvent approximations, we obtain a suf- ficient condition for regularity of solutions to linear stochastic Schrodinger equa- tions driven by cylindrical Brownian motions applying to many physical systems. Then, we establish well-posedness and norm conservation property of a wide class of open quantum systems described in position representation. Moreover, we prove Ehrenfest-type theorems that describe the evolution of the mean value of quantum observables in open systems. Finally, we give a new criterion for the existence and uniqueness of weak solutions to non-linear stochastic Schrequations. We apply our results to physical systems such as fluctuating ion traps and quantum measurement processes of position.

Jan 1, 20139 citations