We complete the program started in [4] by proving that, in the weak coupling limit, the matrix elements, in the collective coherent vectors, of the Heisenberg evolved of an observable of a system coupled to a quasi-free reservoir through a laser type interaction, converge to the matrix elements of a quantum stochastic process satisfying a quantum Langevin equation driven by a quantum Brownian motion. Our results apply to an arbitrary quasi-free reservoir so, in particular, the finite temperature case is included.We complete the program started in [4] by proving that, in the weak coupling limit, the matrix elements, in the collective coherent vectors, of the Heisenberg evolved of an observable of a system coupled to a quasi-free reservoir through a laser type interaction, converge to the matrix elements of a quantum stochastic process satisfying a quantum Langevin equation driven by a quantum Brownian motion. Our results apply to an arbitrary quasi-free reservoir so, in particular, the finite temperature case is included.

Publications of the Research Institute for Mathematical SciencesAug 31, 19954 citations

On the quantum langevin equation

G. W. Ford, Mark Kac

Journal of Statistical PhysicsMar 1, 1987243 citations

Quantum Langevin Equation

Rafael D. Benguria, Mark Kac

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.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.

Physical Review LettersJan 5, 1981305 citations

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.