Stochastic simulation algorithm for the quantum linear Boltzmann equation

We develop a Monte Carlo wave function algorithm for the quantum linear Boltzmann equation, a Markovian master equation describing the quantum motion of a test particle interacting with the particles of an environmental background gas. The algorithm leads to a numerically efficient stochastic simulation procedure for the most general form of this integrodifferential equation, which involves a five-dimensional integral over microscopically defined scattering amplitudes that account for the gas interactions in a nonperturbative fashion. The simulation technique is used to assess various limiting forms of the quantum linear Boltzmann equation, such as the limits of pure collisional decoherence and quantum Brownian motion, the Born approximation, and the classical limit. Moreover, we extend the method to allow for the simulation of the dissipative and decohering dynamics of superpositions of spatially localized wave packets, which enables the study of many physically relevant quantum phenomena, occurring e.g., in the interferometry of massive particles.

Figure 1

CPU time of the Monte Carlo unraveling as a function of the number $N$ of basis states involved in the initial superposition. The curve is almost a straight line with slope $a\simeq 1.1$ in the logarithmic plot. It follows that the CPU time grows almost linearly with $N$, that is $T\propto N^{1.1}$.

Figure 2

The first four phase shifts ${\delta }_l$ as a function of the relative momentum $p$ for the interaction potential [Eq. (68)] and mass ratio $m/M=1$. Their asymptotic behavior is in agreement with [Eq. (71)]. (There are two bound states for $l=0,1$, one for $l=2$, and no bound state for $l=3$.)

Figure 3

Jump rate $\Gamma$ as a function of the momentum $U$ assuming the Gaussian interaction potential with $V_0=1$ (left) and $V_0=20$ (right). The solid line corresponds to the exact scattering amplitude, while the dashed line gives the Born approximation. As one expects, the results deviate strongly for large interaction potentials.

Figure 4

Semilogarithmic plot of the “coherence” $C\left(t\right)$ defined in Eq. (75) for the state of Eq. (72) with $U_0=\sqrt{6}$. The interaction is described by the Gaussian potential [Eq. (68)] with $V_0=1$ (left) and $V_0=20$ (right). The solid line is obtained using the exact scattering amplitude, while the dashed line corresponds to the Born approximation. The predictions of the decoherence rates differ substantially in case of the large interaction potential with $V_0=20$.

Figure 5

Evolution of the density matrix in the position representation for an initial superposition of two Gaussian wave packets, obtained by solving the three-dimensional QLBE for $s$-wave hard-sphere scattering. The spatial coherences $\rho \left(x/{\Lambda }_{\text{th}},y/{\Lambda }_{\text{th}}\right)$ are expressed in units of the thermal wavelength.

Figure 6

Decay rate of the spatial coherences as a function of the wave packet separation for a Gaussian interaction potential and the mass ratios $M/m=100$ (left) and $m=M$ (right). The solid line shows the prediction of pure collisional decoherence, Eq. (14), and the dots give the result of the stochastic simulation of the QLBE. One observes that the predictions of the two models agree for $M⪢m$, while they deviate for $m=M$. The localization rate saturates in all cases at the average collision rate ${\Gamma }_{\text{eff}}$. We note that the decay rate of the QLBE does not vanish for $m=M$ as $x\to 0$, since there is a loss of the populations due to diffusion.

Figure 7

Evolution of the position diagonal elements of the density matrix $\rho \left(x/{\lambda }_{\text{dB}},x/{\lambda }_{\text{dB}}\right)$ for an initial superposition of two counterpropagating minimum-uncertainty wave packets. The figure, obtained by solving the QLBE for $s$-wave hard-sphere scattering, shows three snapshots of the dynamics at times ${\Gamma }_0\left(t_0,t_1,t_2\right)=\left(0,9,18\right)$. The increasing influence of decoherence manifests itself as a reduced visibility of the interference fringes as time progresses.

Figure 8

(Color online) Energy relaxation as obtained by using the exact scattering amplitude of the Gaussian interaction potential, compared to the corresponding Born approximation and the solution of the CL Eq. (20). We choose the potential strengths $V_0=1$ (left) and $V_0=20$ (right) and the mass ratios $M/m=1$ (top) and $M/m=10$ (bottom). The correct equilibrium values [Eq. (86)] of 3/2 (top) and 3/20 (bottom) are obtained in all cases. One observes that the exact result agrees with the solution of the CL equation for a heavy tracer particle (bottom), while it deviates strongly for $M=m$ (top). The Born approximation gives reliable results only when the kinetic energy is much greater than the potential one (top left), which indicates that it may underestimate the energy relaxation even for weak interactions if the test particle has a large mass (bottom left).

Figure 9

The solid lines give the time dependence of the spatial variance, as obtained by solving the QLBE for $s$-wave hard-sphere scattering and mass ratios of $M/m=100$ (left) and $M/m=1$ (right). Left: after an initial quadratic increase the variance displays the straight line behavior expected of classical diffusion. The fit displayed by the dotted line has a slope close to the diffusion constant predicted by the Kramers equation (relative error: 12%). For comparison, the dashed line gives the dispersive broadening of the initial wave packet in absence of a gas. Right: for a test particle with a mass equal to that of the gas particles the variance growth is dominated by classical diffusion even on the time scale of a single collision.