The Journal of Chemical Physics

Second-quantized molecular time scale generalized Langevin equation theory and the quantum oscillator

H. Keith McDowell

Published Oct 15, 1987• 10 citations

Abstract

The short-time Gaussian approximation to the molecular time scale generalized Langevin equation (MTGLE) friction kernel is introduced and used to compute the time dependence of a fluctuation time correlation function. The fluctuations are shown to have two time scales, namely, the dissipation time scale and a quantum time scale given by exp[−2πkTt/ℏ]. Absorption and emission spectral functions for a quantum oscillator coupled to a bath are derived with no approximations made. The second-quantized MTGLE approach is applied to the problem of a quantum oscillator coupled linearly to a bath of quantum oscillators. The method is shown to be consistent with previous work and to provide a systematic methodology to examine more general many-body boson problems.

DOI: 10.1063/1.452849

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The Journal of Chemical Physics

Second-quantized molecular time scale generalized Langevin equation theory and the quantum oscillator

H. Keith McDowell

Published Oct 15, 1987• 10 citations

Publisher

Abstract

DOI: 10.1063/1.452849

Related papers

Second-quantized molecular time scale generalized Langevin equation theory: Boson equivalent chain

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The Journal of Chemical PhysicsFeb 1, 198715 citations

Second-quantized molecular time scale generalized Langevin equation theory: Fermion equivalent chain

H. Keith McDowell

The Journal of Chemical PhysicsMay 15, 198710 citations

Quantum non-Markovian Langevin equations and transport coefficients

Vazgen Valerikovich Sargsyan, Z. Kanokov, G. G. Adamian +1 more

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quant-pharXiv:2602.10421Feb 12, 20260 citations

We derive a

c

-number generalized Langevin equation (GLE) describing the evolution of the expectation values ${\ensuremath{\langle}{x}_{i}\ensuremath{\rangle}}_{t}$ of the atomic position operators

{x}_{i}

of an open system. The latter is coupled linearly to a harmonic bath kept at a fixed temperature. The equations of motion contain a non-Markovian friction term with the classical kernel [L. Kantorovich, Phys. Rev. B 78, 094304 (2008)] and a zero mean non-Gaussian random force with correlation functions that depend on the initial preparation of the open system. We used a density operator formalism without assuming that initially the combined system was decoupled. The only approximation made in deriving quantum GLE consists of assuming that the Hamiltonian of the open system at time

t

can be expanded up to the second order with respect to operators of atomic displacements ${u}_{i}={x}_{i}\ensuremath{-}{\ensuremath{\langle}{x}_{i}\ensuremath{\rangle}}_{t}$ in the open system around their exact atomic positions ${\ensuremath{\langle}{x}_{i}\ensuremath{\rangle}}_{t}$ (the ``harmonization'' approximation). The noise is introduced to ensure that sampling many quantum GLE trajectories yields exactly the average one. An explicit expression for the pair correlation function of the noise, consistent with the classical limit, is also proposed. Unlike the usually considered quantum operator GLE, the proposed

c

-number quantum GLE can be used in direct molecular dynamic simulations of open systems under general equilibrium or nonequilibrium conditions.

Building upon the work of Hu, Paz, and Zhang [1,2] on open quantum systems we consider the quantum Brownian motion (QBM) model with one oscillator (position variable

x

) as the system, {\it nonlinearly} coupled to an environment of

N

harmonic oscillators (with mass

m_n

, natural frequency

ω_n

, position

q_n

and momentum

p_n

variables) in the form

\sum_{n}\left(v_{n1}(x)q_{n}^{k}+v_{n2}(x)p_{n}^{l}\right)

where

k, l

are integers (the present work only considers the

k=l=2

cases). The vertex functions $v_{n1}, v_{n2} $ are of the form

v_{n1}=λC_{n1} f(x), v_{n2}(x)=-λ\,C_{n2}m_{n}^{-2}ω_{n}^{-2}f(x)

where

C_{n1,2}

are the coupling constants with the

n

th oscillator,

f(x)

is any arbitrary function of

x

, and

λ

is a dimensionless constant. Employing the closed-time-path formalism the influence action

S_{IF}

is calculated using a perturbative expansion in

λ

. It is possible to identify the terms in

S_{IF}

quadratic or higher in

Δ(s)\equiv f(x_{+}(s))-f(x_{-}(s))

to constitute the noise kernel, while terms linear in

Δ

to that of the dissipation kernel. The non-Gaussian noise kernel gives rise to non-zero three-point correlation function of the corresponding stochastic force. The pathway presented here should be useful for the exploration of \textit{non-Gaussian properties of systems nonlinearly coupled with their environments}; examples in early universe cosmology and in quantum optomechanics (QOM) are mentioned. A modified fluctuation-dissipation relation (FDR) is also established, which ensures the consistency of the model and the accuracy of results even at higher perturbative orders. Another result of significance is the derivation of a nonlinear Langevin equation which is expected to be useful for many open quantum system applications.