The Schrödinger–Langevin equation with and without thermal fluctuations

Abstract

The Schrödinger-Langevin equation (SLE) is considered as an effective open quantum system formalism suitable for phenomenological applications involving a quantum subsystem interacting with a thermal bath. We focus on two open issues relative to its solutions: the stationarity of the excited states of the non-interacting subsystem when one considers the dissipation only and the thermal relaxation toward asymptotic distributions with the additional stochastic term. We first show that a proper application of the Madelung/polar transformation of the wave function leads to a non zero damping of the excited states of the quantum subsystem. We then study analytically and numerically the SLE ability to bring a quantum subsystem to the thermal equilibrium of statistical mechanics. To do so, concepts about statistical mixed states and quantum noises are discussed and a detailed analysis is carried with two kinds of noise and potential. We show that within our assumptions the use of the SLE as an effective open quantum system formalism is possible and discuss some of its limitations.

Heavy quarkonium suppression in a fireball

“…While this work derives its original inspiration from [25], it is quite different from all previous studies on the subject [48][49][50][51][52][53][54][55][56], as the derived evolution equations fulfill three essential conditions: they conserve the total number of heavy quarks (i.e., Tr{ρ s } + Tr{ρ o } is preserved by the evolution equations); they account for the non-Abelian nature of QCD (through gluon exchanges color-singlet quarkonia may dissociate into quark-antiquark color-octet states, and vice versa quark-antiquark color-octet states may generate quarkonia); and, finally, they do not rely on classical approximations but rather follow from the closed-time-path formalism applied to quantum field theory. The work substantially extends, updates, and completes a previous strongly-coupled analysis done in [10].…”

“…The real part of −i times (B8) gives (54), and the imaginary part of −i times (B9) gives (55). The function Σ s that appears in the evolution equation (38) has been defined in (29).…”

Heavy-flavor production and medium properties in high-energy nuclear collisions --What next?

“…At the same time, a new Schrödinger-Langevin approach has been put forward [107], which, with the inclusion of a non-linear contribution in the stochastic evolution of the quarkonium wave function, allows the heavyquarkonium state to thermalize at late times. It has been thoroughly explored in 1-dimensional settings and its formulation in terms of a Schrödinger equation bodes well for relating its parameters to quantities on the EFT side in the future.…”

“…So, it is clearly not applicable to ground state bottomonium. In that case, a full quantum treatment is required [107]. A possible way out is to treat the tightly bound Υ (1S) as a distinct particle whose number is described by a rate equation, while other bottomonium states are treated using Langevin dynamics [193].…”

A Langevin Canonical Approach to the Study of Quantum Stochastic Resonance in Chiral Molecules

“…One of them it is the well known logarithmic nonlinear Schrödinger equation proposed for an open dynamics due to Kostin [5]; in particular, for the Brownian motion (linear dissipation). This equation is known as the Schrödinger-Langevin, or Kostin, equation. Recently, an application of this equation to the harmonic oscillator under the presence of thermal fluctuations (white and colored) has been reported [6]. A generalized equation for nonlinear dissipation has also been proposed [7,8].…”