Universal Spectra of Coherent Atoms in a Recurrent Random Walk

TL;DR

This paper experimentally measures the universal spectral behavior of coherent atoms undergoing a one-dimensional random walk, confirming theoretical predictions about recurrence probabilities and critical exponents in diffusion processes.

Contribution

It provides the first direct measurement of the Laplace transform of recurrence probability in 1D using EIT in atomic vapors, revealing universal spectral features dependent on effective dimensionality.

Findings

Measured power law spectrum with exponent 0.56 in 1D diffusion

Confirmed universality of the spectrum depending on dimensionality

Demonstrated EIT as a tool for probing diffusion dynamics

Abstract

The probability of a random walker to return to its starting point in dimensions one and two is unity, a theorem first proven by G. Polya. The recurrence probability -- the probability to be found at the origin at a time t, is a power law with a critical exponent d/2 in dimensions d=1,2. We report an experiment that directly measures the Laplace transform of the recurrence probability in one dimension using Electromagnetically Induced Transparency (EIT) of coherent atoms diffusing in a vapor-cell filled with buffer gas. We find a regime where the limiting form of the complex EIT spectrum is universal and only depends on the effective dimensionality in which the random recurrence takes place. In an effective one-dimensional diffusion setting, the measured spectrum exhibits power law dependence over two decades in the frequency domain with a critical exponent of 0.56 close to the expected value 0.5. Possible extensions to more elaborate diffusion schemes are briefly discussed.