A stochastic approach to open quantum systems

Abstract

Stochastic methods are ubiquitous to a variety of fields, ranging from physics to economics and mathematics. In many cases, in the investigation of natural processes, stochasticity arises every time one considers the dynamics of a system in contact with a somewhat bigger system, an environment with which it is considered in thermal equilibrium. Any small fluctuation of the environment has some random effect on the system. In physics, stochastic methods have been applied to the investigation of phase transitions, thermal and electrical noise, thermal relaxation, quantum information, Brownian motion and so on. In this review, we will focus on the so-called stochastic Schrödinger equation. This is useful as a starting point to investigate the dynamics of open quantum systems capable of exchanging energy and momentum with an external environment. We discuss in some detail the general derivation of a stochastic Schrödinger equation and some of its recent applications to spin thermal transport, thermal relaxation, and Bose-Einstein condensation. We thoroughly discuss the advantages of this formalism with respect to the more common approach in terms of the reduced density matrix. The applications discussed here constitute only a few examples of a much wider range of applicability.

Towards a dynamical approach to the calculation of the figure of merit of thermoelectric nanoscale devices

“…54 It can be proven that this is the most general memoryless master equation valid up to second order in λ that preserves the positivity of the density matrix. 59 The equations (18) and (19) can be easily generalised to the case of the system being in contact with more than one bath operator B.…”

“…The stochastic Schrödinger equation (SSE) (14) is able to reproduce the dynamics of the density matrix for the system, 54,55,57,58 …”

A stochastic reorganizational bath model for electronic energy transfer

“…Within the stochastic Schrödinger equations, the non-Markovian terms and the detailed balance condition are important for describing the thermal relaxation process. 44 The lack of reorganization is due to the neglect of the H (t) term, which plays an essential role in the reorganization process that follows photoexcitation. The parameters m and 2 m for the dynamical equations can be obtained from MD simulations and excited-state calculations.…”

“…39 Different methodologies for simulating energy transfer have been widely reviewed. [40][41][42][43][44][45][46][47] The most expensive and accurate methods are unified models that combine electronic structure directly with bath models. 48 The details of exciton-phonon coupling are crucial in light-harvesting systems because they strongly modify the excited-state dynamics.…”

Exact classical stochastic representations of the many-body quantum dynamics

“…The field of research, which is the subject of such an activity, is truly interdisciplinary: quantum optics [4], utracold atoms in traps [5][6][7], quantum phase transitions [8], open quantum systems [9], and the measurement theory [10]. The motivation behind such studies range from purely practical (to provide exact data in order to verify a theory, or in order to interpret an experiment) to fundamental (to characterize the relationship between the classical and quantum computational complexity).…”