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
mn, natural frequency
ωn, position
qn and momentum
pn variables) in the form
∑n(vn1(x)qnk+vn2(x)pnl) 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
vn1=λCn1f(x),vn2(x)=−λCn2mn−2ωn−2f(x) where
Cn1,2 are the coupling constants with the
nth oscillator,
f(x) is any arbitrary function of
x, and
λ is a dimensionless constant. Employing the closed-time-path formalism the influence action
SIF is calculated using a perturbative expansion in
λ. It is possible to identify the terms in
SIF quadratic or higher in
Δ(s)≡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.
With this work we elaborate on the physics of quantum noise in thermal equilibrium and in stationary nonequilibrium. Starting out from the celebrated quantum fluctuation-dissipation theorem we discuss some important consequences that must hold for open, dissipative quantum systems in thermal equilibrium. The issue of quantum dissipation is exemplified with the fundamental problem of a damped harmonic quantum oscillator. The role of quantum fluctuations is discussed in the context of both, the nonlinear generalized quantum Langevin equation and the path integral approach. We discuss the consequences of the time-reversal symmetry for an open dissipative quantum dynamics and, furthermore, point to a series of subtleties and possible pitfalls. The path integral methodology is applied to the decay of metastable states assisted by quantum Brownian noise.
Extended quantum systems can be theoretically described in terms of the Schwinger-Keldysh functional integral formalism, whose action conveniently describes both dynamical and static properties. We show here that in thermal equilibrium, defined by the validity of fluctuation-dissipation relations, the action of a quantum system is invariant under a certain symmetry transformation, and thus it is distinguished from generic systems. In turn, the fluctuation-dissipation relations can be derived as the Ward-Takahashi identities associated with this symmetry. Accordingly, the latter provides an efficient test for the onset of thermodynamic equilibrium and it makes checking the validity of fluctuation-dissipation relations unnecessary. In the classical limit, this symmetry reduces to the well-known one that characterizes equilibrium in the stochastic dynamics of classical systems coupled to thermal baths, described by Langevin equations.
We consider the non-Markovian Langevin evolution of a dissipative dynamical system in quantum mechanics in the path integral formalism. After discussing the role of the frequency cutoff for the interaction of the system with the heat bath and the kernel and noise correlator that follow from the most common choices, we derive an analytic expansion for the exact non-Markovian dissipation kernel and the corresponding colored noise in the general case that is consistent with the fluctuation-dissipation theorem and incorporates systematically non-local corrections. We illustrate the modifications to results obtained using the traditional (Markovian) Langevin approach in the case of the exponential kernel and analyze the case of the non-Markovian Brownian motion.
Continuing our inquiry into the conditions when fluctuation-dissipation relations (FDR) may appear in the context of nonequilibrium dynamics of open quantum systems (over and beyond the conventional FDR from linear response theory) we turn to non-Gaussian systems and consider this issue for an anharmonic quantum oscillator interacting with a scalar quantum field bath. We present the general nonperturbative expressions for the rate of energy (power) exchange between the anharmonic oscillator and its thermal bath. For the cases that a stable final equilibrium state exists, and the nonstationary components of the two-point functions of the anharmonic oscillator have negligible contributions to the power balance, we can show nonperturbatively that equilibration implies an FDR for the anharmonic oscillator. We then use a weakly anharmonic oscillator as an example to illustrate the validity of those two assumptions and show that in the weak anhamonicity limit, they are satisfied according to our first-order perturbative results..