Relativistic and Non-Relativistic Quantum Brownian Motion in an Anisotropic Dissipative Medium

Abstract

Using a minimal-coupling-scheme we investigate the quantum Brownian motion of a particle in an anisotropic-dissipative-medium under the influence of an arbitrary potential in both relativistic and non-relativistic regimes. A general quantum Langevin equation is derived and explicit expressions for quantum-noise and dynamical variables of the system are obtained. The equations of motion for the canonical variables are solved explicitly and an expression for the radiation-reaction of a charged particle in the presence of a dissipative-medium is obtained. Some examples are given to elucidate the applicability of this approach.

Appendix: A

In this appendix we evaluate the time-dependent coefficient c(t) in (66) for the spontaneous decay of an initially excited atom embedded in anisotropic dissipative medium. By substituting |ψ(t)〉 from (66) into (67) and using the expansions (7), (22) and (25), we find the following coupled differential equations

$$\dot{c}(t)=-\imath\omega_{0} \, c(t) - \int_{0}^{\infty} {d\omega } \sqrt {\frac{\hbar\omega }{{2 }}}{ q}_{12i}^{*} \,f_{ij}(\omega )M_{j} (\omega,t),$$

(135)

$$\dot M_{i} (\omega ,t) = -\imath\omega \,M_{i} (\omega ,t) + \sqrt {\frac{{ \omega }}{2\hbar^{3}}}{ q}_{12j}\, f_{ji}(\omega )\, \,c(t). $$

(136)

We can solve these coupled differential equations by using Laplace transformation technique. Let $\tilde {c}(s)$ denotes the Laplace transform of c(t). Taking the Laplace transform of (135), combining them and using the relations (33), we find

$$ s\tilde c(s) = c(0)-\imath\omega_{0} \,\tilde c(s) +\frac{1}{\hbar} \left[\mathrm{q}_{\,12,i}^{*} \widetilde{G}_{ij} (\imath s){q}_{12,j}\right]\tilde c(s), $$

(137)

where

$$ \widetilde{G}_{ij}(\imath s) = \int_{0}^{\infty} {d\omega } \frac{{\omega^{2} }}{{\pi (\imath s - \omega )}}\text{Im}\,\chi_{ij} (\omega). $$

(138)

The tensor $\widetilde {G}_{ij}(\imath s)$ gives the spontaneous-emission and frequency-shift of the atom due to the presence of dissipative medium. From definition (138) it is obvious that $\widetilde {G}_{ij}(\imath s)$ is an analytic tensor in the upper half-plane Re(s) > 0, therefore

$$ \dot c(t) = - \imath\omega_{0} c(t) + {\int_{0}^{t}} {K (t - t')\,c(t')\,dt'}, $$

(139)

where

$$ K (t - {t}^{\prime}) = \frac{1}{2\pi\imath\hbar}\int_{-\infty }^{+\infty} {du \,e^{-\imath u(t - {t}^{\prime})}\left[{q}_{\,12,i}^{*} \widetilde{G}_{ij}(u +\imath \tau){q}_{\,12,j}\right]} . $$

(140)

Here we use the Markov’s approximation [80] and replace c(t′) in (139) by

$$ c({t}') = c(t)e^{\imath\omega_{0} (t - {t}')}. $$

(141)

By lengthy but straightforward calculations and using Kramers-Kroning relations we deduce

$$ \dot c(t) = - \imath\omega_{0} c(t) - (\,\,\Gamma + \imath\Delta )\,c(t) $$

(142)

where

$$ {\Gamma} = -\frac{1}{\hbar}\left[{\mathrm{q}_{12,i}^{*}\, \text{Im}[\widetilde{G}_{ij}(\omega_{0}+\imath 0^{+} )]\,\mathrm{q}_{12,j}} \right]=\frac{{\omega_{0}^{2}}}{\hbar}\left[{\mathrm{q}_{12,i}^{*}\, \text{Im}[{ \chi}_{ij}(\omega_{0} )]\,\mathrm{q}_{12,j}} \right], $$

(143)

and

$$ \Delta = -\frac{1}{\hbar}\left[{\mathrm{q}_{12,i}^{*}\, \text{Re}[\widetilde{G}_{ij}(\omega_{0}+\imath 0^{+} )]\,\mathrm{q}_{12,j}} \right]=P\int_{0}^{\infty} {d\omega } \frac{{q}_{12,i}^{*}\, \text{Im}[{ \chi}_{ij}(\omega )]\,\mathrm{q}_{12,j}}{{\pi\hbar(\omega_{0} - \omega )}}. $$

(144)