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}+i
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.
The heat-bath frequency distribution, necessary to maintain the canonical commutation relations for all time for a quantum mechanical oscillator with non-Markovian features like memory-dependent damping, is shown to satisfy certain constraint relations. An algorithm is given to explicitly find the heat-bath frequency distribution in terms of a series expansion for all processes where the timescale for the non-Markovian memory kernel is much smaller than the inverse of the strength of damping term. In the non-Markovian case, the heat-bath distribution function exhibits dependence on the system characteristics. The KS periodicity conditions on the system Green functions as the system approaches equilibrium are established for the present case.
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.
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.
The reduced dynamics of a quantum system interacting with a linear heat bath finds an exact representation in terms of a stochastic Schrödinger equation. All memory effects of the reservoir are transformed into noise correlations and mean-field friction. The classical limit of the resulting stochastic dynamics is shown to be a generalized Langevin equation, and conventional quantum state diffusion is recovered in the Born-Markov approximation. The non-Markovian exact dynamics, valid at arbitrary temperature and damping strength, is exemplified by an application to the dissipative two-state system.
The system of nonlinear Langevin equations was obtained by using Hamiltonian's operator of two coupling quantum oscillators which are interacting with heat bath. By using the analytical solution of these equations, the analytical expressions for transport coefficients was found. Generalized Langevin equations and fluctuation-dissipation relations are derived for the case of a nonlinear non-Markovian noise. The explicit expressions for the time-dependent friction and diffusion coefficients are presented for the case of linear couplings in the coordinate between the collective two coupled harmonic oscillators and heat bath.