On some unexplored decoherence aspects in the Caldeira–Leggett formalism: arrival time distributions, identical particles and diffraction in time

Abstract

Some unexplored decoherence aspects within the Caldeira–Leggett master equation are analyzed and discussed. The decoherence process is controlled by the two environment parameters, the relaxation rate or friction and the temperature, leading to a gradual transition from the quantum to classical regime. Arrival time distributions, non-minimum-uncertainty-product or stretching Gaussian wave packets, identical particles and diffraction in time display interesting features during the decoherence process undergone by the time dependent interference patterns. We show that the presence of a constant force field does not affect the decoherence, positive values of the stretching parameter reduce the rate of decoherence, the symmetry of the wave function for identical particles is not robust enough in the gradual decoherence process because indistinguishability is lost, and diffraction in time and space is gradually washed out by increasing the temperature and/or relaxation rate in the zero dissipation limit within the so-called quantum shutter problem.

Appendix A: continuity equation for the single-particle density

Taking time derivative of both sides of Eq. (48) yields

$$\begin{aligned} \frac{ \partial }{ \partial t} P_{ \text {sp} , \pm }(x, t)= & {} {\mathcal {N}}_{\pm }^2 \left\{ \frac{ \partial }{ \partial t}P_{11}(x, t) + \frac{ \partial }{ \partial t}P_{22}(x, t) \pm \text {Re} \{ \int dx' \frac{ \partial }{ \partial t} [ P_{12}(x, t) P_{21}(x', t) ] \} \right\} \end{aligned}$$

(A1)

Now using the continuity Eq. (6) and imposing appropriate boundary condition yielding zero boundary terms we obtain the following continuity equation,

$$\begin{aligned} \frac{ \partial }{ \partial t} P_{ \text {sp} , \pm }(x, t) + \frac{ \partial }{ \partial x} J_{ \text {sp} , \pm }(x, t)= & {} 0 \end{aligned}$$

(A2)

where

$$\begin{aligned} J_{ \text {sp} , \pm }(x, t)= & {} {\mathcal {N}}_{\pm }^2 [ J_{11}(x, t) + J_{22}(x, t) \pm \text {Re} \{J_{12}(x, t) s(t)\} ] \end{aligned}$$

(A3)

and $J_{kl}(x, t)$ given by diagonal elements of (5) satisfies Eq. (6) with $P_{kl}(x, t)$.