Classical diffusion of a quantum particle in a noisy environment

We study the spreading of a quantum-mechanical wave packet in a tight-binding model with a noisy potential and analyze the emergence of classical diffusion from the quantum dynamics due to decoherence. We consider a finite correlation time of the noisy environment and treat the system by utilizing the separation of fast (dephasing) and slow (diffusion) processes. We show that classical diffusive behavior emerges at long times and we calculate analytically the dependence of the classical diffusion coefficient on the noise magnitude and correlation time. This method provides a general solution to this problem for arbitrary conditions of the noisy environment. The calculation can be done in any dimension, but we demonstrate it in one dimension for clarity of representation. The results are relevant to a large variety of physical systems, from electronic transport in solid-state physics to light transmission in optical devices, diffusion of excitons, and quantum computation.

Figure 1

(Color online) The spread of a quantum-mechanical wave packet in a perfectly periodic lattice $\left(T=1,W=0\right)$ showing ballistic spreading (a,b), versus a noisy lattice with $T=1$, $\tau =0.01$, and $W=20$ showing diffusive spread (c,d). The wave function is confined to a single lattice site at $t=0$. The probability distribution of the quantum particle after some propagation is plotted on a semilog scale in (a) and (c) showing a ballistic profile and a diffusive (Gaussian) profile, correspondingly.

Figure 2

(Color online) A plot of the width versus time of a quantum-mechanical wave function spreading in a noisy potential showing diffusive propagation. The simulation parameters are $T=1$, $\tau =0.01$, and $W=20$. The data are averaged over 100 realizations of the disordered potential. The numerical results (open circles) fit the theory (line), with no fitting parameters.

Figure 3

(Color online) A fit of the theoretical results to numerical calculations. The diffusion constant was found numerically, for correlation times $\tau =0.01$ (triangles), $\tau =0.1$ (squares), and $\tau =1$ (circles) and noise magnitudes ranging from $W=2$ to $W=20$. Each point in the graph is an average over 50 realizations. As expected theoretically, there is a crossover from a regime with slope $-2$ (i.e., $1/x^2$ dependence) to a regime with slope $-1$ ($1/x$ dependence). The different curves overlap at points with the same value of $W\tau$. The form of the crossover is given by Eq. (16) integrated numerically for the intermediate regimes. No fitting parameters are used in the comparison.