Stochastic Bohmian mechanics within the Schrödinger-Langevin framework: A trajectory analysis of wave-packet dynamics in a fluctuative-dissipative medium

Abstract

A Bohmian analysis of the so-called Schrödinger-Langevin or Kostin nonlinear differential equation is provided to study how thermal fluctuations of the environment affects the dynamics of the wave packet from a quantum hydrodynamical point of view. In this way, after obtaining the Schrödinger-Langevin-Bohm equation from the Kostin equation its application to simple but physically insightful systems such as the Brownian-Bohmian motion, motion in a gravity field and transmission through a parabolic repeller is studied. If a time-dependent Gaussian ansatz for the probability density is assumed, the effect of thermal fluctuations together with thermal wave packets leads to Bohmian stochastic trajectories. From this trajectory based analysis, quantum and classical diffusion coefficients for free particles, thermal arrival times for a linear potential and transmission probabilities and characteristic times such as arrival and dwell times for a parabolic repeller are then presented and discussed. I. INTRODUCTIONIt seems that the first attempt for generalizing Chandrasekhar's phenomenological theory [1] of Brownian motion to the quantum realm was made by introducing a stochastic term in the Caldirola-Kanai framework [2]. It was shown that for the Boltzmann statistics, quantum stochastic dynamics lead to a diffusion constant which is the same as that obtained from the classical Brownian motion. The effect of noise on the Brownian motion of a quantum oscillator via the corresponding Caldirola-Kanai equation with a noise term was also studied [3].On the other hand, Kostin [4,5] derived heuristically the so-called non-linear Schrödinger-Langevin equation (SL) from the quantum Langevin equation through Heisenberg position and momentum operators. This equation fulfills the unitarity condition, satisfies the uncertainty relation but violates the superposition principle due to its non-linear nature [6]. Recently, a different generalized Schrödinger equation has been proposed to describe dissipation in quantum systems [7] where the Kostin equation without the random potential is a special case of this generalized equation. Thanks to its straightforward formulation and its numerical simplicity, the Kostin equation "can be considered as a solid candidate for effective description of open quantum systems hardly accessible to quantum master equations" [8]. Although this equation has been widely used in the literature to study pure dissipative effects [8][9][10][11], only a few works are found where the presence of a stochastic or random force is introduced [8]. A generalized Schrödinger-Langevin equation has been proposed in the literature for nonlinear interaction providing a state-dependent dissipation process exhibiting multiplicative noise [12]. Afterwards, this equation has been extended to a non-Markovian problem [13].Bohmian mechanics [14][15][16] as an alternative interpretation of quantum mechanics has the advantage to give a clear picture of quantum phenomena in terms of trajectories in configuration space. This theory has ...

Dissipative quantum backflow

“…In both CK and CL approaches, the equation of motion for the quantum state is linear. The other approaches that one could choose are the logarithmic non-linear Schrödinger equations [16,22,23] and the stochastic Schrödinger equation [24,25]. In these approaches the linearity is not fulfilled and the form of the initial superposed state is not preserved under such dynamics.…”

The quantum mean square displacement of thermalized CO on Cu(100) in the short time approximation

“…50 In some of these works, potential energy functions were obtained from ab initio calculations. Truly time dependent quantum mechanical methods used to investigate diffusion are based on Bohmian dynamics, 51,52 wave packet dynamics using a stochastic Schrödinger equation, 53 or quantized forms of the generalized Langevin equation. 54,55 None of these time dependent methods are ab initio , however, and many rely on coupling models with adjustable parameters.…”

Justifying Born’s Rule Pα = |Ψα|2 Using Deterministic Chaos, Decoherence, and the de Broglie–Bohm Quantum Theory

“…However, in later works, especially in the work conducted with Vigier [27] and then subsequently Hiley [14], Bohm modified the original de Broglie-Bohm dynamics by introducing stochastic and fluctuating elements associated with a subquantum medium forcing the relaxation towards quantum equilibrium ρ(X) → |Ψ(X)| 2 . In this context, we mention that very important works have been done in recent years concerning "Stochastic Bohmian mechanics" based on the Schrödinger-Langevin framework, the Kostin equation and involving nonlinearities [28][29][30]. While this second semi-stochastic approach was motivated by general philosophical considerations [31], proponents of BBQT have felt divided concerning the need for such a modification of the original framework.…”