Three-dimensional Monte Carlo simulations of the quantum linear Boltzmann equation

Recently the general form of a translation-covariant quantum Boltzmann equation has been derived, which describes the dynamics of a tracer particle in a quantum gas. We develop a stochastic wave function algorithm that enables full three-dimensional Monte Carlo simulations of this equation. The simulation method is used to study the approach to equilibrium for various scattering cross sections and to determine dynamical deviations from Gaussian statistics through an investigation of higher-order cumulants. Moreover, we examine the loss of coherence of superpositions of momentum eigenstates and determine the corresponding decoherence time scales to quantify the transition from quantum to classical behavior of the state of the test particle.

Figure 1

(Color online) The total transition rate $\Gamma \left(U\right)$ for a constant cross section [Eq. (31)] (continuous line) and for a Gaussian cross section with $a=1$ [Eq. (34)] (broken line).

Figure 2

(Color online) The probability density $R(K,\xi )$ given by Eq. (40) for $U=1$.

Figure 3

(Color online) A single realization of the process $\mathbf{U}\left(t\right)$ for $m∕M=1$.

Figure 4

(Color online) Averages over ${10}^4$ realizations for a constant cross section and $m∕M=0.1$. The broken lines represent the approximate relaxation dynamics according to Eqs. (41, 42), respectively.

Figure 5

(Color online) The same as Fig. 4 for $m∕M=1$.

Figure 6

(Color online) The same as Fig. 4 for $m∕M=10$.

Figure 7

(Color online) Averages over ${10}^4$ realizations for a Gaussian cross section with $m∕M=1$ and $a=1$. The broken lines represent the approximate relaxation dynamics with the rate given by Eq. (45).

Figure 8

(Color online) Cumulants of second, third, and fourth order obtained from averages over ${10}^5$ realizations for a constant cross section and $m∕M=1$.

Figure 9

(Color online) Semilogarithmic plot of the coherence $C\left(t\right)$ obtained from an average over ${10}^5$ realizations for $m∕M=1$ (constant cross section).

Figure 10

(Color online) The decoherence rate ${\gamma }_D$ in units of ${\Gamma }_0$ as a function of the initial momentum $U_0$. Points: Least squares fits of the simulation data for $m∕M=1$. Continuous line: Analytical estimate given by Eq. (67).

Figure 11

(Color online) The same as Fig. 10 for a Gaussian cross section with $a=1$. The continuous line represents the analytical estimate given by Eq. (68).