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arXiv:cond-mat/0703766

Fundamental noise in matter interferometers

Adilet Imambekov, Vladimir Gritsev, Eugene Demler

Published Mar 28, 2007• 26 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.cond-mat/0703766

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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 study shot (counting) noise of the amplitude of interference between independent atomic systems. In particular, for the two interfering systems the variance of the fringe amplitude decreases as the inverse power of the number of particles per system with the coefficient being a non-universal number. This number depends on the details of the initial state of each system so that the shot noise measurements can be used to distinguish between such states. We explicitly evaluate this coefficient for the two cases of the interference between bosons in number states and in broken symmetry states. We generalize our analysis to the interference of multiple independent atomic systems. We show that the variance of the interference contrast vanishes as the inverse power of the number of the interfering systems. This result, implying high signal to noise ratio in the interference experiments, holds both for bosons and for fermions.
Interferometers with single particles are susceptible for dephasing perturbations from the environment, such as electromagnetic oscillations or mechanical vibrations.On the one hand, this limits sensitive quantum phase measurements as it reduces the interference contrast in the signal.On the other hand, it enables single-particle interferometers to be used as sensitive sensors for electromagnetic and mechanical perturbations.Recently, it was demonstrated experimentally, that a secondorder correlation analysis of the spatial and temporal detection signal can decrease the electromagnetic shielding and vibrational damping requirements significantly.Thereby, the relevant matter-wave characteristics and the perturbation parameters could be extracted from the correlation analysis of a spatially 'washed-out' interference pattern and the original undisturbed interferogram could be reconstructed.This method can be applied to all interferometers that produce a spatial fringe pattern on a detector with high spatial and temporal single-particle resolution.In this article, we present and discuss in detail the used two-dimensional second-order correlation theory for multifrequency perturbations.The derivations of an explicit and approximate solution of the correlation function and corresponding amplitude spectra are provided.It is explained, how the numerical correlation function is extracted from the measurement data.Thereby, the influence of the temporal and spatial discretization step size on the extracted parameters, as contrast and perturbation amplitude, is analyzed.The influence of noise on the correlation function and corresponding amplitude spectrum is calculated and numerically cross-checked by a comparison of our theory with numerical singleparticle simulations of a perturbed interference pattern.Thereby, an optimum spatial discretization step size is determined to achieve a maximum signal-to-noise ratio, which was used in former experiments to identify the perturbation caused by the electrical network.Our method can also be applied for the analysis of broad-band frequency noise, dephasing the interference pattern.Using Gaussian distributed noise in the simulations, we demonstrate that the relevant matter-wave parameters and the applied perturbation spectrum can be revealed by our correlation analysis.
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.
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.
This chapter describes the Bose--Einstein condensate (BEC) interferometry. We first introduce the optical interferometer, briefly discussing the role of fringe contrast in observing interference patterns. Next, we examine a BEC interferometer in a Michelson geometry that consists of a BEC in a trapping potential well. The BEC cloud in the potential well is put into a linear superposition of two clouds that travel along different paths in the trap and are recombined using the same light beams. By studying the population of atoms in the different motional states of the BEC clouds that emerge after recombination, we obtain information about the relative phase shift accumulated by the two traveling superposed condensates during the interferometry. We then characterize the atom distribution that is found in the output of the interferometer by finding the probability density distribution and calculating the features of the probability density such as the mean, the variance, and the interference fringe contrast. Finally, we parameterize the effect of two-body collisional interactions on the observed interference fringes in a way that can be directly controlled in an experiment.