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Physical Review A

Action-angle variables and fluctuation-dissipation relations for a driven quantum oscillator

S. Krinsky

Published Mar 1, 1985• 4 citations
Publisher

Abstract

For a classical oscillator subject to a time-dependent perturbation, the average change in the action variable 〈\ensuremath{\Delta}J〉=〈J-${J}_{0}$〉 is related to its fluctuation 〈(\ensuremath{\Delta}J${)}^{2}$〉 by 〈\ensuremath{\Delta}J〉=(1/2)\ensuremath{\partial}〈(\ensuremath{\Delta}J${)}^{2}$〉/\ensuremath{\partial}${J}_{0}$, where ${J}_{0}$ is the initial value of J and the average is over the initial value ${\ensuremath{\theta}}_{0}$ of the angle variable. In this paper, the quantum-mechanical generalization of this relation is derived, and we discuss the correspondence between our work and the usual treatment of the fluctuation-dissipation theorem for a system which initially is described by a canonical distribution.

DOI
10.1103/physreva.31.1267

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For a classical oscillator subject to a time-dependent perturbation, the average change in the action variable 〈\ensuremath{\Delta}J〉=〈J-J0{J}_{0}J0​〉 is related to its fluctuation 〈(\ensuremath{\Delta}J)2{)}^{2})2〉 by 〈\ensuremath{\Delta}J〉=(1/2)\ensuremath{\partial}〈(\ensuremath{\Delta}J)2{)}^{2})2〉/\ensuremath{\partial}J0{J}_{0}J0​, where J0{J}_{0}J0​ is the initial value of J and the average is over the initial value \ensuremathθ0{\ensuremath{\theta}}_{0}\ensuremathθ0​ of the angle variable. In this paper, the quantum-mechanical generalization of this relation is derived, and we discuss the correspondence between our work and the usual treatment of the fluctuation-dissipation theorem for a system which initially is described by a canonical distribution.
The fluctuation-dissipation theorem (FDT) is a central result in statistical physics, both for classical and quantum systems. It establishes a relationship between the linear response of a system under a time-dependent perturbation and time correlations of certain observables in equilibrium. Here we derive a generalization of the theorem which can be applied to any Markov quantum system and makes use of the symmetric logarithmic derivative (SLD). There are several important benefits from our approach. First, such a formulation clarifies the relation between classical and quantum versions of the equilibrium FDT. Second, and more important, it facilitates the extension of the FDT to arbitrary quantum Markovian evolutions, as given by quantum maps. Third, it brings out the full connection between the FDT and the Quantum Fisher information, the figure of merit in quantum metrology.
We analytically describe the decay to equilibrium of generic observables of a nonintegrable system after a perturbation in the form of a random matrix. We further obtain an analytic form for the time-averaged fluctuations of an observable in terms of the rate of decay to equilibrium. Our result shows the emergence of a fluctuation-dissipation theorem corresponding to a classical Brownian process, specifically, the Ornstein-Uhlenbeck process. Our predictions can be tested in quantum simulation experiments, thus helping to bridge the gap between theoretical and experimental research in quantum thermalization. We test our analytic results by exact numerical experiments in a spin chain. We argue that our fluctuation-dissipation relation can be used to measure the density of states involved in the nonequilibrium dynamics of an isolated quantum system.
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 develop a perturbation scheme to treat the dynamics of a quantum Brownian particle (coordinate xxx) coupled nonlinearly to a bath of oscillators (coordinates qn{q}_{n}qn​) with an interaction action in the form $\ensuremath{\lambda}f(x){q}_{n}^{k}$. We derive the influence functionals for the k=2,3,4k=2, 3, 4k=2,3,4 cases to order ${\ensuremath{\lambda}}^{2}$ and derive the master equations for the special cases of local dissipation and white noise, as well as the general cases of nonlocal dissipation and colored noise for f(x)=xandx2f(x)=x \mathrm{and} {x}^{2}f(x)=xandx2. We show that a generalized fluctuation-dissipation relation exists between the lthl\mathrm{th}lth-order noise kernels of the kthk\mathrm{th}kth-order coupling and their corresponding dissipation kernels.
For a time-dependent classical quadratic oscillator we introduce pairs of real and complex invariants that are linear in position and momentum. Each pair of invariants realize explicitly a canonical transformation from the phase space to the invariant space, in which the action-phase variables are defined. We find the action operator for the time-dependent quantum oscillator via the classical-quantum correspondence. Candidate phase operators conjugate to the action operator are discussed, but no satisfactory ones are found.