Quantum diffusion equations featuring explicitly time-dependent transport coefficients are derived from generalized non-Markovian Langevin equations. Generalized fluctuation-dissipation relations and analytic expressions for calculating the friction and diffusion coefficients in nuclear processes are obtained. The asymptotic behavior of the transport coefficients and correlation functions for a damped harmonic oscillator that is linearly coupled in momentum to a heat bath is studied. The coupling to a heat bath in momentum is responsible for the appearance of the diffusion coefficient in coordinate. The problem of regression of correlations in quantum dissipative systems is analyzed.
Beyond the conventional quantum regression theorem, a general formula for non-Markovian correlation functions of arbitrary system operators both in the time- and frequency-domain is given. We approach the problem by transforming the conventional time-non-local master equation into dispersed time-local equations-of-motion. The validity of our approximations is discussed and we find that the non-Markovian terms have to be included for short times. While calculations of the density matrix at short times suffer from the initial value problem, a correlation function has a well defined initial state. The resulting formula for the non-Markovian correlation function has a simple structure and is as convenient in its application as the conventional quantum regression theorem for the Markovian case. For illustrations, we apply our method to investigate the spectrum of the current fluctuations of interacting quantum dots contacted with two electrodes. The corresponding non-Markovian characteristics are demonstrated.
Beyond the conventional quantum regression theorem, a general formula for non-Markovian correlation functions of arbitrary system operators both in the time- and frequency-domain is given. Weapproach the problem by transforming the conventional time-non-local master equation into dispersed time-local equations-of-motion. The validity of our approximations is discussed and we find that the non-Markovian terms have to be included for short times. While calculations of the density matrix at short times suffer from the initial value problem, a correlation function has a well defined initial state. The resulting formula for the non-Markovian correlation function has a simple structure and is as convenient in its application as the conventional quantum regression theorem for the Markovian case. For illustrations, we apply our method to investigate the spectrum of the current fluctuations of interacting quantum dots contacted with two electrodes. The corresponding non- Markovian characteristics are demonstrated.
Equations for correlation functions in nonequilibrium systems are derived using a generalization of the diagrammatic Keldysh method. The four field operators of the correlation function are ordered on a time contour that consists of four axes extending from -\ensuremath{\infty} to +\ensuremath{\infty} and from +\ensuremath{\infty} to -\ensuremath{\infty} (doubled Keldysh contour). For those correlation functions that may be considered as correlation functions of one-particle Green-functions' fluctuations, a closed system of equations is obtained in the case of weak interaction of the particles with the scattering system. Its structure may be related to the Langevin approach: on the left-hand side a linear operator acts on each of two pairs of the correlation-function arguments, each of these two operators being just the operator of the linearized quantum kinetic equation. The expression for the right-hand side of this equation is derived. It has the meaning of the correlation function of extraneous Langevin sources, and the equation may be called the quantum Boltzmann-Langevin equation. In the quasi-classical limit it reduces to the usual Boltzmann-Langevin equation for the particles' distribution-function fluctuations. Possible applications of the method are discussed.
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.
We describe the dynamics of an open system
S in terms of a nonunitary time evolution operator. The latter is defined to solve the generalized master equation for the density operator of
S as obtained using Zwanzig's projector technique. Multitime correlation functions of observables of
S are expressed in terms of this time evolution operator. Our results generalize well-known fluctuation dissipation (or regression) theorems valid for Markovian motions of open systems. Moreover, they can be used to combine density-operator and Green's-function methods in dealing with open systems. As an example we treat the retarded electron-electron interaction in superconductors.