Solution of quantum Langevin equation: Approximations, theoretical and numerical aspects

Banerjee, D.; Bag, Bidhan Chandra; Banik, Suman Kumar; Ray, Deb Shankar

doi:10.1063/1.1711593

Cited by 62 publications

(41 citation statements)

References 30 publications

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“…Although questionable [64], a common approximation [65,66,67,68,69,70,71] is to abandon its operator character and to replace the non-commutating q-number noise by a c-number noise while taking the same power spectrum. One then obtains a quasiclassical Langevin equation which leads to a reasonable description for systems which are nearly harmonic and to possible violations of the Heisenberg relations.…”

Section: The Possible Noises For the Stochastic Termmentioning

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: The Possible Noises For the Stochastic Termmentioning

confidence: 99%

The Schrödinger–Langevin equation with and without thermal fluctuations

2016

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“…The general advantage of the stochastic GLE is that dissipation and the statistical properties of the noise are entirely described by the so-called memory kernel being simply a function of time. If such a memory kernel can be obtained for a real system then the full quantum-mechanical treatment of the bath can be performed analytically, leading to a quantum version of the GLE [28][29][30][31][32]. Alternatively, a density matrix theory either via the Feynman-Vernon influence functional approach [33,34] or hierarchy type EOMs [35,36] can be employed.…”

Section: Introductionmentioning

confidence: 99%

Parametrizing linear generalized Langevin dynamics from explicit molecular dynamics simulations

Gottwald

Karsten

Ivanov

et al. 2015

The Journal of Chemical Physics

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Fundamental understanding of complex dynamics in many-particle systems on the atomistic level is of utmost importance. Often the systems of interest are of macroscopic size but can be partitioned into few important degrees of freedom which are treated most accurately and others which constitute a thermal bath. Particular attention in this respect attracts the linear generalized Langevin equation (GLE), which can be rigorously derived by means of a linear projection (LP) technique. Within this framework a complicated interaction with the bath can be reduced to a single memory kernel. This memory kernel in turn is parametrized for a particular system studied, usually by means of time-domain methods based on explicit molecular dynamics data. Here we discuss that this task is most naturally achieved in frequency domain and develop a Fourier-based parametrization method that outperforms its time-domain analogues. Very surprisingly, the widely used rigid bond method turns out to be inappropriate in general. Importantly, we show that the rigid bond approach leads to a systematic underestimation of relaxation times, unless the system under study consists of a harmonic bath bi-linearly coupled to the relevant degrees of freedom. INTRODUCTIONStudying complex dynamics of many-particle systems has become one of the main goals in modern molecular physics. The fundamental understanding of the underlying microscopical processes requires the interplay of elaborate experimental techniques and sophisticated theoretical approaches. Experimentally, (non-)linear spectroscopy revealed itself as a powerful tool for probing the dynamics and for determining the characteristic timescales, such as dephasing/relaxation times and reaction rates to name but two. For interpreting the experimental spectra theoretical models are needed which can give insight into the atomistic dynamics. Often, a reduction of the description to few variables is convenient in many cases since this can not only ease the interpretation, but enable the identification of key properties [1]. Such a reduced description can formally be obtained from the so-called system-bath partitioning, where only a small subset of degrees of freedom (DOFs), referred to as system, is considered as important for describing a physical process under study. All the other DOFs, referred to as bath, are regarded as irrelevant in the sense that they might influence the time evolution of the system but do not explicitly enter any dynamical variable of interest. Practically, such a separation is often natural, for instance, when studying a reaction with a clearly defined reaction center or solute dynamics in a solvent environment. Further, reduced equations of motion (EOMs) for the system DOFs can be derived in which the influence of the bath is limited to dissipation and fluctuations.The most simple formulation of this idea is provided by the Markovian Langevin equation, where dissipation and fluctuations take the form of static friction and stochastic white noise, respectively [2][3][4]. Sit...

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“…The simple quasiclassical approach considered in this article ignores both the quantum corrections to the force field [31,32,36,60] and the proper treatment of the quantum fluctuations by means of the path integral formalism [40,48,61,62]. However, it is surprisingly accurate in the limit of strong system-bath interaction, at a computational cost essentially equal to that of its classical counterpart.…”

Section: Discussionmentioning

confidence: 99%

“…In fact, the problem of dissipative tunnelling has been solved theoretically by using path-integral techniques [23][24][25][26]. Alternative approaches which make use of a c-number 1 quantum GLE [28][29][30][31][32][33][34] are in principle better suited for numerical simulations since they are based on real-time equations of motion. However, the existing approaches are either more computationally demanding than the classical GLE or their applicability to the strong system-bath coupling regime has not been fully demonstrated, yet.…”

Section: Introductionmentioning

confidence: 99%

Modelling a Bistable System Strongly Coupled to a Debye Bath: A Quasiclassical Approach Based on the Generalised Langevin Equation

Stella¹,

Ness²,

Lorenz³

et al. 2017

CMST

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Bistable systems present two degenerate metastable configurations separated by an energy barrier. Thermal or quantum fluctuations can promote the transition between the configurations at a rate which depends on the dynamical properties of the local environment (i.e., a thermal bath). In the case of classical systems, strong system-bath interaction has been successfully modelled by the Generalised Langevin Equation (GLE) formalism. Here we show that the efficient GLE algorithm introduced in Phys. Rev. B 89, 134303 (2014) can be extended to include some crucial aspect of the quantum fluctuations. In particular, the expected isotopic effect is observed along with the convergence of the quantum and classical transition rates in the strong coupling limit. Saturation of the transition rates at low temperature is also retrieved, in qualitative, yet not quantitative, agreement with the analytic predictions. The discrepancies in the tunnelling regime are due to an incorrect sampling close to the barrier top. The domain of applicability of the quasiclassical GLE is also discussed.

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Solution of quantum Langevin equation: Approximations, theoretical and numerical aspects

Cited by 62 publications

References 30 publications

The Schrödinger–Langevin equation with and without thermal fluctuations

The Schrödinger–Langevin equation with and without thermal fluctuations

Parametrizing linear generalized Langevin dynamics from explicit molecular dynamics simulations

Modelling a Bistable System Strongly Coupled to a Debye Bath: A Quasiclassical Approach Based on the Generalised Langevin Equation

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