Counting statistics of interfering Bose-Einstein condensates

TL;DR

This paper introduces a method to predict measurement outcomes in quantum many-body systems, specifically applied to interference patterns of Bose-Einstein condensates, revealing how single-shot images exhibit random interference fringes.

Contribution

It develops a gauge-like transformation approach to construct a probability generating functional for many-body wave functions, applied to Bose-Einstein condensate interference.

Findings

Interference fringes appear in all single-shot measurements.

The probability distribution of images is a sum over phase-difference distributions.

Fringe patterns vary randomly from shot to shot, consistent with spontaneous symmetry breaking.

Abstract

A method is presented that is able to predict the probability of outcomes of snapshot measurements, such as the images of the instantaneous particle density distribution in a quantum many-body system. It is shown that a gauge-like transformation of the phase of the many-body wave function allows one to construct a probability generating functional, the Fourier transform of which with respect to the "gauge" field returns the joint probability distribution to detect any given number of particles at various locations. The method is applied to the problem of interference of two independent clouds of Bose-Einstein condensates, where the initially separated clouds with fixed boson numbers expand and the density profile image of the overlapping clouds is registered. In the limit of large particle numbers, the probability to observe a particular image of the density profile is shown to be given by a sum of partial probability distributions, each of which corresponds to a noisy image of interference of two matter waves with definite phase difference. In agreement with earlier theoretical arguments, interference fringes are, therefore, expected in any single shot measurement, the fringe pattern randomly varying from run to run. These results conform to the physical picture where the Bose-Einstein clouds are in spontaneously symmetry broken states, the hidden phases of which are revealed by the density profile measurement via the position of the interference fringes.