Quantizing Lévy flights

The Caldeira-Leggett model of quantum Brownian motion is generalized using a generic velocity-dependent coupling. That leads to the description of a set of models able to capture Markovian and non-Markovian versions of Brownian and Lévy statistics, depending on the functional form of the coupling and on the spectral function of the reservoir. One specific coupling force is found that establishes a connection with Lévy statistics of cold atoms in Sisyphus laser cooling. In the low-velocity limit, this also gives rise to additional inertia of the Brownian particle, reproducing the Abraham-Lorentz equation from first principles for a superohmic bath. Through path-integral quantization in Euclidean time, the environment is integrated out, leaving a set of nonlocal effective actions. These results further serve as starting points for several numerical calculations, particularly decoherence properties of nonohmic baths.

Figure 1

Comparison between (a) Brownian motion and (b) Lévy motion. For Lévy motion, long steps are more frequent and make the dominant contribution to transport.

Figure 2

Sketch of the Gaussian distribution (red) and of the Lévy distribution (blue). Because the support of the Lévy distribution is decaying at a lower rate than that of the Gaussian curve, a property called “heavy tails,” the probability for a step deviating much from the mean is higher. Such a highly deviating step, then, corresponds to an event occurring in the tails of Lévy distribution.

Figure 3

The coupling function of Eq. (23). One can see that in both the high- and the low-velocity limits, $F^{\prime \prime }\left[\dot{Q}\right]$ is consistently negligible.