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International Conference on Information Technology and Nanotechnology (ITNT)

Stochastic Non-Markovian Schroedinger equation for a three-level quantum system

Vitalii Semin, Andrei Pavelev

Published Jan 1, 2017• 1 citation
Publisher

Abstract

Non-Markovian dynamics of a three-level system is studied with the help of stochastic Schrodinger equation (SSE). We derive a new form of SSE for a three-level system driven by four independent Ornstein-Uhlenbeck stochastic noises. The main advantage of the suggested SSE is the ensuring of the complete positivity of the reduced density operator. We demonstrate significant influence of the non-Markovian noises on the dynamics of the three-level quantum system.

DOI
10.18287/1613-0073-2017-1904-263-265

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Non-Markovian dynamics of a three-level system is studied with the help of stochastic Schrodinger equation (SSE). We derive a new form of SSE for a three-level system driven by four independent Ornstein-Uhlenbeck stochastic noises. The main advantage of the suggested SSE is the ensuring of the complete positivity of the reduced density operator. We demonstrate significant influence of the non-Markovian noises on the dynamics of the three-level quantum system.
Firstly, the Markovian stochastic Schrödinger equations are presented, together with their connections with the theory of measurements in continuous time. Moreover, the stochastic evolution equations are translated into a simulation algorithm, which is illustrated by two concrete examples — the damped harmonic oscillator and a two-level atom with homodyne photodetection. We then consider how to introduce memory effects in the stochastic Schrödinger equation via coloured noise. Specifically, the approach by using the Ornstein-Uhlenbeck process is illustrated and a simulation for the non-Markovian process proposed. Finally, an analytical approximation technique is tested with the help of the stochastic simulation in a model of a dissipative qubit.
Positivity preservation is naturally guaranteed in exact non-Markovian master equations for open quantum system dynamics. However, in many approximated non-Markovian master equations, the positivity of the reduced density matrix is not guaranteed. In this paper we provide a general class of time-local, perturbative, and positivity-preserving non-Markovian master equations generated from stochastic Schr\"odinger equations, particularly quantum-state-diffusion equations. Our method has an expanded range of applicability for accommodating a variety of non-Markovian environments. We show the positivity-preserving master equation for a three-level system coupled to a dissipative bosonic environment as a particular example to exemplify our general approach. We illustrate the numerical simulations with an analysis explaining why the previous approximated non-Markovian master equations cannot guarantee positivity. Our work provides a consistent master equation for studying the non-Markovian dynamics in ultrafast quantum processes and strong-coupling systems.
In this paper, dynamics of a qubit in an external laser field is investigated for the non-Markov environment described by an Ornstein-Uhlenbeck process. A non-linear non-Markovian stochastic Schrödinger equation is derived for the quantum system. The stochastic equation is preserved for a positive semi-definite of the density matrix and its trace. By the numerical solution of the equation, the probability of finding the system in the excited state and the influence of non-Markov noise parameters on this probability are studie.