We consider a problem of interference between two independent condensates that lack true long-range order. We show that their interference pattern contains information about correlation functions within each condensate. As an example, we analyze the interference between a pair of one-dimensional interacting Bose liquids. We find universal scaling of the average fringe contrast with system size and temperature that depends only on the Luttinger parameter. Moreover, the full distribution of the fringe contrast, which is also equivalent to the full counting statistics of the interfering atoms, changes with interaction strength and lends information on high-order correlation functions. We also demonstrate that the interference between two-dimensional condensates at finite temperature can be used as a direct probe of the Kosterlitz-Thouless transition. Finally, we discuss the generalization of our results to describe the interference of a periodic array of independent fluctuating condensates.
We study the buildup of quantum coherence between two Bose-Einstein condensates that are initially in mixed states. We consider in detail the two cases where each condensate is initially in a thermal or a Poisson distribution of atom number. Although initially there is no relative phase between the condensates, a sequence of spatial atom detections produces an interference pattern with arbitrary but fixed relative phase. The visibility of this interference pattern is close to one for the Poisson distribution of two condensates with equal counting rates but it becomes a stochastic variable in the thermal case, where the visibility will vary from run to run around an average visibility of $\ensuremath{\pi}/4.$ In both cases, the variance of the phase distribution is inversely proportional to the number of atom detections in the regime where this number is large compared to one but small compared with the total mean number of atoms in the condensates.
Interference of two independently prepared ideal Bose gases is discussed, on the basis of the idea of measurement-induced interference. It is known that, even if the number of atoms in each gas is individually fixed finite and the symmetry of the system is not broken, an interference pattern is observed on each single snapshot. The key role is played by the Hanbury Brown and Twiss effect, which leads to an oscillating pattern of the cloud of identical atoms. Then, how essential is the Bose-Einstein condensation to the interference? In this work, we describe two ideal Bose gases trapped in two separate three-dimensional harmonic traps at a finite temperature
T, using the canonical ensembles (with fixed numbers of atoms). We compute the full statistics of the snapshot profiles of the expanding and overlapping gases released from the traps. We obtain a simple formula valid for finite
T, which shows that the average fringe spectrum (average fringe contrast) is given by the purity of each gas. The purity is known to be a good measure of condensation, and the formula clarifies the relevance of the condensation to the interference. The results for
T=0, previously known in the literature, can be recovered from our analysis. The fluctuation of the interference spectrum is also studied, and it is shown that the fluctuation is vanishingly small only below the critical temperature
Tc, meaning that interference pattern is certainly observed on every snapshot below
Tc. The fact that the number of atoms is fixed in the canonical ensemble is crucial to this vanishing fluctuation.
We study theoretically the interference patterns produced by the overlap of an array of Bose-Einstein condensates that have no phase coherence among them. We show that density-density correlations at different quasimomenta, which play an important role in two-condensate interference, become negligible for large
N, where
N is the number of overlapping condensates. In order to understand the physics of this phenomenon, it is sufficient to consider the periodicity of the lattice and the statistical probability distribution of a random-walk problem. The average visibility of such interference patterns decreases as ${N}^{\ensuremath{-}1∕2}$ for large
N.
We study the full counting statistics (FCS) of quantum gases in samples of thousands of interacting bosons, detected atom-by-atom after a long free-fall expansion. In this far-field configuration, the FCS reveals the many-body coherence from which we characterize iconic states of interacting lattice bosons, by deducing the normalized correlations
g(n)(0) up to the order
n=6. In Mott insulators, we find a thermal FCS characterized by perfectly-contrasted correlations
g(n)(0)=n!. In interacting Bose superfluids, we observe small deviations to the Poisson FCS and to the ideal values
g(n)(0)=1 expected for a pure condensate. To describe these deviations, we introduce a heuristic model that includes an incoherent contribution attributed to the depletion of the condensate. The predictions of the model agree quantitatively with our measurements over a large range of interaction strengths, that includes the regime where the condensate is strongly depleted by interactions. These results suggest that the condensate component exhibits a full coherence
g(n)(0)=1 at any order
n up to
n=6 and at arbitrary interaction strengths. The approach demonstrated here is readily extendable to characterize a large variety of interacting quantum states and phase transitions.
The full statistical distribution of the superfluid fraction characterizing one-dimensional Bose gases in random potentials is discussed. Rare configurations with extreme fluctuations of the disorder potential can fragment the condensate and reduce the superfluid fraction to zero. The resulting bimodal probability distribution for the superfluid fraction is calculated numerically in the quasi-one-dimensional mean-field regime of ultracold atoms in laser speckle potentials. Using extreme value statistics, an analytical scaling of the zero-superfluid probability as a function of disorder strength, disorder correlation length, and system size is presented. It is argued that similar results can be expected for point-like impurities and that these findings are in reach for present-day experiments.