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Physical Review Letters

Quantum Langevin Equation

Rafael D. Benguria, Mark Kac

Published Jan 5, 1981• 305 citations
Publisher

Abstract

It is shown (by means of a perturbation series) that for a class of potentials $V(x)$ the stationary distribution of the solution $x(t)$ of the quantum Langevin equation approaches in the weak-coupling limit ($f\ensuremath{\rightarrow}0$) the quantum mechanical canonical distribution of the displacement of the oscillator, subject to the potential $V(x)$, if and only if $E(t)$ is the operator version of the purely random Gaussian process so that, in particular, higher symmetrized averages ${〈E({t}_{1})\ensuremath{\cdots}E({t}_{n})〉}_{s}$ are expressible in terms of pair correlations, in the usual way.

DOI
10.1103/physrevlett.46.1

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It is shown (by means of a perturbation series) that for a class of potentials V(x)V(x)V(x) the stationary distribution of the solution x(t)x(t)x(t) of the quantum Langevin equation approaches in the weak-coupling limit (f\ensuremath→0f\ensuremath{\rightarrow}0f\ensuremath→0) the quantum mechanical canonical distribution of the displacement of the oscillator, subject to the potential V(x)V(x)V(x), if and only if E(t)E(t)E(t) is the operator version of the purely random Gaussian process so that, in particular, higher symmetrized averages 〈E(t1)\ensuremath⋯E(tn)〉s{〈E({t}_{1})\ensuremath{\cdots}E({t}_{n})〉}_{s}〈E(t1​)\ensuremath⋯E(tn​)〉s​ are expressible in terms of pair correlations, in the usual way.
The macroscopic description of a quantum particle with passive dissipation and moving in an arbitrary external potential is formulated in terms of the generalized Langevin equation. The coupling with the heat bath corresponds to two terms: a mean force characterized by a memory function \ensuremath{\mu}(t) and an operator-valued random force. Explicit expressions are given for the correlation and commutator of the random force. The random force is never Markovian. It is shown that \ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(z), the Fourier transform of the memory function, must be a positive real function, analytic in the upper half-plane and with Re[\ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(\ensuremath{\omega}+i0+{0}^{+}0+)] a positive distribution on the real axis. This form is then derived for the independent-oscillator model of a heat bath. It is shown that the most general quantum Langevin equation can be realized by this simple model. A critical comparison is made with a number of other models that have appeared in the literature.
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 xxx) as the system, {\it nonlinearly} coupled to an environment of NNN harmonic oscillators (with mass mnm_nmn​, natural frequency ωnω_nωn​, position qnq_nqn​ and momentum pnp_npn​ variables) in the form ∑n(vn1(x)qnk+vn2(x)pnl)\sum_{n}\left(v_{n1}(x)q_{n}^{k}+v_{n2}(x)p_{n}^{l}\right)∑n​(vn1​(x)qnk​+vn2​(x)pnl​) where k,lk, lk,l are integers (the present work only considers the k=l=2k=l=2k=l=2 cases). The vertex functions $v_{n1}, v_{n2} $ are of the form vn1=λCn1f(x),vn2(x)=−λ Cn2mn−2ωn−2f(x)v_{n1}=λC_{n1} f(x), v_{n2}(x)=-λ\,C_{n2}m_{n}^{-2}ω_{n}^{-2}f(x)vn1​=λCn1​f(x),vn2​(x)=−λCn2​mn−2​ωn−2​f(x) where Cn1,2C_{n1,2}Cn1,2​ are the coupling constants with the nnnth oscillator, f(x)f(x)f(x) is any arbitrary function of xxx, and λλλ is a dimensionless constant. Employing the closed-time-path formalism the influence action SIFS_{IF}SIF​ is calculated using a perturbative expansion in λλλ. It is possible to identify the terms in SIFS_{IF}SIF​ quadratic or higher in Δ(s)≡f(x+(s))−f(x−(s))Δ(s)\equiv f(x_{+}(s))-f(x_{-}(s))Δ(s)≡f(x+​(s))−f(x−​(s)) to constitute the noise kernel, while terms linear in ΔΔΔ 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.
We first point out that nonrelativistic quantum mechanics formulated at imaginary time s = it can formally be recognized as the Fokker-Planck description of a classical Brownian motion, which occurs (in general) in a medium where particles are absorbed and emitted. We next offer a new formulation of quantum dynamics, which is basically the Langevin treatment of that Brownian motion. More explicitly, we derive a noise average representation of the transition probability W ( X", s"|X', s' ) in terms of the solutions to a Langevin equation with a Gaussian white noise. Upon analytic continuation back to real time W ( X", s"|X', s' ) becomes the propagator of the original Schrödinger equation, namely we have ⟨ X", t"|X', t' ⟩ = W ( X", it"|X', it' ). The main advantage of this approach is to permit a systematic application to quantum-dynamical problems of the mathematical techniques developed in the theory of classical stochastic processes, even when vector potentials and/or time-dependent potentials are present.
We derive a ccc-number generalized Langevin equation (GLE) describing the evolution of the expectation values ${\ensuremath{\langle}{x}_{i}\ensuremath{\rangle}}_{t}$ of the atomic position operators xi{x}_{i}xi​ 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 ttt 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 ccc-number quantum GLE can be used in direct molecular dynamic simulations of open systems under general equilibrium or nonequilibrium conditions.
Extremization of entropy, usually subject to an average energy constraint,(Maxwell-Boltzmann gas) is often associated with maximization of the number of microstates N gas particles may represent. The math procedure of extremization (variational calculus) is then associated with the ln of a factorial expression which reduces to n(ei)ln(n(ei)) using Stirling’s approximation. The central limit theorem, on the other hand, states that the distribution of sample means of a random variable approaches a Gaussian, i.e exp(-pp/(2sig sig)). Here sig sig is the standard deviation. For a random variable, the important or characteristic information is the standard deviation associated with the average of the square pp. We suggest here that an alternative way of looking at maximization of entropy (which is a stability statement we argue) other than ln of permutations is to consider the stability of the variation of the variance of pp subject to a change in the probability, with pp written in terms of P(p). Average kinetic energy in a statistical problem represents such a variance and should be invariant under changes of the probability subject to an average kinetic energy constraint, we suggest. How does this idea apply to quantum mechanics? In quantum mechanics, one has a statistical function exp(ipx) and for bound states creates W(x)=wavefunction = Sum over p a(p)exp(ipx). a(p)a(p) is formally the probability for p, but this does not mean that a(p)a(p) is a random variable because one has the constraint: Sum over p a(p)a(p) pp2/m + Integral dx V(x) W(x)W(x) = En. In the case of a harmonic oscillator, however, one knows a priori that the two summands are each equal to En/2. Furthermore there are a set of En values, starting at a ground state and increasing. All En values, except the ground state, are metastable. Thus the ground state is a special state. Given that Sum over p a(p)a(p)pp2/m = Eo/2, a priori, one may consider that all of the information of a(p)a(p) is associated with Eo. This is exactly the case for a Gaussian from the central limit theorem. Alternatively, from information theory, information = ln(probability) which yields the same result. If this is the case, then pp/2m a(p)a(p) written strictly in terms of P(p)=a(p)a(p) should be stationary upon changing P(p) subject to the E/2 constraint. This special situation, however, only holds for the ground state oscillator. What about average potential energy for the ground state oscillator? If the probability for p (momentum) is a Gaussian due to the Central Limit theorem, this means that the standard deviation is the information (and is linked with randomness). Taking a Fourier transform of a(p) cannot add more information to the picture, we argue. W(x)W(x) should also be a Gaussian, as it is. As a result, xxW(x)W(x), written in terms of P(x)=W(x)W(x) subject to the same Eo/2 constraint should represent a maximization or stationary situation, we argue.