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arXiv:cond-mat/0702647 · AIP conference proceedings

How to Study Correlation Functions in Fluctuating Bose Liquids Using Interference Experiments

Vladimir Gritsev, Ehud Altman, Anatoli Polkovnikov, Eugene Demler

Published Jan 1, 2006• 2 citations
arXivHTMLPDF

Abstract

Interference experiments with independent condensates provide a powerful tool for analyzing correlation functions. Scaling of the average fringe contrast with the system size is determined by the two‐point correlation function and can be used to study the Luttinger liquid liquid behavior in one‐dimensional systems and to observe the Kosterlitz‐Thouless transition in two‐dimensional quasicondensates. Additionally, higher moments of the fringe contrast can be used to determine the higher order correlation functions. In this article we focus on interference experiments with one‐dimensional Bose liquids and show that methods of conformal field theory can be applied to calculate the full quantum distribution function of the fringe contrast.

DOI
10.1063/1.2400647

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Interference experiments with independent condensates provide a powerful tool for analyzing correlation functions. Scaling of the average fringe contrast with the system size is determined by the two‐point correlation function and can be used to study the Luttinger liquid liquid behavior in one‐dimensional systems and to observe the Kosterlitz‐Thouless transition in two‐dimensional quasicondensates. Additionally, higher moments of the fringe contrast can be used to determine the higher order correlation functions. In this article we focus on interference experiments with one‐dimensional Bose liquids and show that methods of conformal field theory can be applied to calculate the full quantum distribution function of the fringe contrast.
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.
These lecture notes discuss two effects which contribute to the reduction of the interference fringe contrast in matter interferometers. The first effect is the shot noise arising from a finite number of atoms used in experiments. Focusing on a single shot measurement we provide explicit calculations of the full distribution functions of the fringe contrast for the interference of either the coherent or the number states of atoms. Another mechanism of the suppression of the amplitude of interference fringes discussed in these lecture notes is the quantum and thermal fluctuations of the order parameter in low dimensional condensates. We summarize recent theoretical and experimental studies demonstrating that suppression of the interference fringe contrast and its shot to shot variations can be used to study correlation functions within individual condensates. We also discuss full distribution functions of the fringe amplitudes for one and two dimensional condensates and review their connection to high order correlation functions. We point out intriguing mathematical connections between the distribution functions of interference fringe amplitudes and several other problems in field theory, systems of correlated electrons, and statistical physics.
We analyze density-density correlations of expanding clouds of weakly interacting two-dimensional Bose gases below and above the Berezinskii-Kosterlitz-Thouless transition, with particular focus on short-time expansions. During time-of-flight expansion, phase fluctuations of the trapped system translate into density fluctuations, in addition to the density fluctuations that exist in situ. We calculate the correlations of these fluctuations both in real space and in momentum space and derive analytic expressions in momentum space. Below the transition, the correlation functions show an oscillatory behavior, controlled by the scaling exponent of the quasicondensed phase, due to constructive interference. We argue that this can be used to extract the scaling exponent of the quasicondensate experimentally. Above the transition, the interference is rapidly suppressed when the atoms travel an average distance beyond the correlation length. This can be used to distinguish the two phases qualitatively.
We show that the visibility in interference experiments with Bose-Einstein condensates is directly related to the condensate fraction. The probability distribution of the contrast over many runs of an interference experiment thus gives the full counting statistics of the condensed atom number. For two-dimensional Bose gases, we discuss the universal behavior of the probability distribution in the superfluid regime and provide analytical expressions for the distributions for both homogeneous and harmonically trapped samples. They are non-Gaussian and unimodal with a variance that is directly related to the superfluid density. In general, the visibility is a self-averaging observable only in the presence of long-range phase coherence. Close to the transition temperature, the visibility distribution reflects the universal order-parameter distribution in the vicinity of the critical point.
Using a Luttinger-liquid approach we study the quantum fluctuations of a Bose-Josephson junction, consisting of a Bose gas confined to a quasi-one-dimensional ring trap which contains a localized repulsive potential barrier. For an infinite barrier we study the one-particle and two-particle static correlation functions. For the one-body density-matrix we obtain different power-law decays depending on the location of the probe points with respect to the position of the barrier. This quasi-long-range order can be experimentally probed in principle using an interference measurement. The corresponding momentum distribution at small momenta is also shown to be affected by the presence of the barrier and to display the universal power-law behavior expected for an interacting one-dimensional fluid. We also evaluate the particle-density profile, and by comparing with the exact results in the Tonks-Girardeau limit we fix the nonuniversal parameters of the Luttinger-liquid theory. Once the parameters are determined from one-body properties, we evaluate the density-density correlation function, finding a remarkable agreement between the Luttinger-liquid predictions and the exact result in the Tonks-Girardeau limit, even at the length scale of the Friedel-like oscillations which characterize the behavior of the density-density correlation function at intermediate distance. Finally, for a large but finite barrier we use the one-body correlation function to estimate the effect of quantum fluctuations on the renormalization of the barrier height, finding a reduction in the effective Josephson coupling energy, which depends on the length of the ring and on the interaction strength.