Wave packet dynamics for a non-linear Schrödinger equation describing continuous position measurements

Zander, Claudia; Plastino, A.; Díaz-Alonso, J.

doi:10.1016/j.aop.2015.07.019

Cited by 17 publications

(14 citation statements)

References 41 publications

(101 reference statements)

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“…The study of its solutions without stochastic term has been carried out in many specific cases, either analytically [18,31,32,41,42,43,44,45,46] or numerically [24,33,41,47,48,49,50,51]. Along these analysis, it has been advocated that the stationary eigenstates of H 0 are also stationary states of the equation [18,43].…”

Section: Introductionmentioning

confidence: 99%

The Schrödinger–Langevin equation with and without thermal fluctuations

2016

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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.

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Section: Introductionmentioning

confidence: 99%

The Schrödinger–Langevin equation with and without thermal fluctuations

2016

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“…Indeed, in this latter reference, the frequencies where under the form n 4 + 2n 2 λ 0 ρ 2 where λ 0 was fixed and ρ the varying parameter. The condition (32) is thus a consequence of this result by replacing λ 0 ρ 2 by the varying parameter λ in the proof of [18,Lemma 2.2]. Note that the arguments are similar to the one used in [3,2,31].…”

Section: 4mentioning

confidence: 85%

“…On the other hand, the Schrödinger-Langevin equation (1) first appears in [29] as a possible way to give a stochastic interpretation of quantum mechanics in the context of Bohmian mechanics. It had a recent renewed interest in the physics community, in particular in quantum mechanics in order to describe the continuous measurement of the position of a quantum particle (see for example [30], [32] or [28]) and in cosmology and statistical mechanics (see [13], [14] or [15]).…”

Section: Introductionmentioning

confidence: 99%

Around plane waves solutions of the Schr{ö}dinger-Langevin equation

Chauleur¹,

Faou²

2021

Preprint

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We consider the logarithmic Schrödinger equations with damping, also called Schrödinger-Langevin equation. On a periodic domain, this equation possesses plane wave solutions that are explicit. We prove that these solutions are asymptotically stable in Sobolev regularity. In the case without damping, we prove that for almost all value of the nonlinear parameter, these solutions are stable in high Sobolev regularity for arbitrary long times when the solution is close to a plane wave. We also show and discuss numerical experiments illustrating our results.

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“…The solution of this nonlinear differential equation was given by Pinney for the conservative case [41] (η=0) when  is replaced by an arbitrary constant. Zander et al [42] have also used the same ansatz to solve the Kostin equation under the presence of a continuous measurement. This procedure can also be seen as the 'wave packet approximation' due to Gutzwiller [43] where it is supposed that within the spatial range where the wave function is appreciably different from zero, the interaction potential V changes slowly enough so that it can be approximated to second order.…”

Section: Bohmian Stochastic Trajectoriesmentioning

confidence: 99%

Quantum surface diffusion in Bohmian mechanics

Miret‐Artés

2018

J. Phys. Commun.

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Surface diffusion of small adsorbates is analyzed in terms of the so-called intermediate scattering function and dynamic structure factor, observables in experiments using the well-known quasielastic Helium atom scattering and Helium spin echo techniques. The linear theory applied is an extension of the neutron scattering due to van Hove and considers the time evolution of the position of the adsorbates in the surface. This approach allows us to use a stochastic trajectory description following the classical, quantum and Bohmian frameworks. Three different regimes of motion are clearly identified in the diffusion process: ballistic, Brownian and intermediate which are well characterized, for the first two regimes, through the mean square displacements and Einstein relation for the diffusion constant. The Langevin formalism is used by considering Ohmic friction, moderate surface temperatures and small coverages. In the Bohmian framework, analyzed here, the starting point is the so-called Schrödinger-Langevin equation which is a nonlinear, logarithmic differential equation. By assuming a Gaussian function for the probability density, the corresponding quantum stochastic trajectories are given by a dressing scheme consisting of a classical stochastic trajectory followed by the center of the Gaussian wave packet, and issued from solving the Langevin equation (particle property), plus the time evolution of its width governed by the damped Pinney differential equation (wave property). The Bohmian velocity autocorrelation function is the same as the classical one when the initial spread rate is assumed to be zero. If not, in the diffusion regime, the Brownian-Bohmian motion shows a weak anomalous diffusion.

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Wave packet dynamics for a non-linear Schrödinger equation describing continuous position measurements

Cited by 17 publications

References 41 publications

The Schrödinger–Langevin equation with and without thermal fluctuations

The Schrödinger–Langevin equation with and without thermal fluctuations

Around plane waves solutions of the Schr{ö}dinger-Langevin equation

Quantum surface diffusion in Bohmian mechanics

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