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AIP conference proceedings

Density correlations in ultracold atomic Fermi gases

Wolfgang Belzig

Published Jan 1, 2005• 9 citations
Publisher

Abstract

We investigate density fluctuations in a coherent ensemble of interacting fermionic atoms around the BEC‐BCS crossover. Adapting the concept of full counting statistics, well‐known from quantum optics and mesoscopic electron transport, we study second‐order as well as higher‐order correlators of density fluctuations. This method is applied to the crossover from a molecular BEC state to a fermionic BCS state and yields a transition from Poissonian statistics to binomial statistics. The statistics can thus be used as an experimental tool to gain information on the many‐body ground state in the crossover regime.

DOI
10.1063/1.2036873

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We investigate density fluctuations in a coherent ensemble of interacting fermionic atoms around the BEC‐BCS crossover. Adapting the concept of full counting statistics, well‐known from quantum optics and mesoscopic electron transport, we study second‐order as well as higher‐order correlators of density fluctuations. This method is applied to the crossover from a molecular BEC state to a fermionic BCS state and yields a transition from Poissonian statistics to binomial statistics. The statistics can thus be used as an experimental tool to gain information on the many‐body ground state in the crossover regime.
We investigate density fluctuations in a coherent ensemble of interacting fermionic atoms. Adapting the concept of full counting statistics, well known from quantum optics and mesoscopic electron transport, we study second-order as well as higher-order correlators of density fluctuations. Using the mean-field BCS state to describe the whole interval between the BCS limit and the Bose-Einstein condensate (BEC) limit, we obtain an exact expression for the cumulant-generating function of the density fluctuations of an atomic cloud. In the two-dimensional case, we obtain a closed analytical expression. Poissonian fluctuations of a molecular condensate on the BEC side are strongly suppressed on the BCS side. The size of the fluctuations in the BCS limit is a direct measure of the pairing potential. We also discuss the BEC-BCS crossover of the third cumulant and the temperature dependence of the second cumulant.
We propose a new method of detecting the onset of superfluidity in a two-component ultracold fermionic gas of atoms governed by an attractive short-range interaction. By studying the two-body correlation functions we find that a measurement of the momentum distribution of the density and spin-response functions allows one to access separately the normal and anomalous densities. The change in sign at low momentum transfer of the normal-ordered part of the density response function signals the transition between a BEC and a BCS regime, characterized by small and large pairs, respectively. This change in sign of the density response function represents an unambiguous signature of the BEC-to-BCS crossover. Spin rotational symmetry breaking due to the magnetic field, if observed, can be used to validate the one-channel model.
We consider the superradiant phase transition of a two-component Fermi gas in a cavity across a Feshbach resonance. It is known that quantum statistics plays a crucial role for the superradiant phase transition in atomic gases; in contrast to bosons, in a Fermi gas this transition exhibits strong density dependence. We show that across a Feshbach resonance, while the two-component Fermi gas passes through the BEC-BCS crossover, the superradiant phase transition undergoes a corresponding crossover from a fermionic behavior on the weakly interacting BCS side, to a bosonic behavior on the molecular Bose-Einstein condensate (BEC) side. This intricate statistics crossover makes the superradiance maximally enhanced either in the unitary regime for low densities, in the BCS regime for moderate densities close to Fermi surface nesting, or in the BEC regime for high densities.
A debated issue in the physics of the BCS-BEC crossover with trapped Fermi atoms is to identify characteristic properties of the superfluid phase. Recently, a condensate fraction was measured on the BCS side of the crossover by sweeping the system in a fast (nonadiabatic) way from the BCS to the Bose-Einstein condensation (BEC) sides, thus "projecting" the initial many-body state onto a molecular condensate. We analyze here the theoretical implications of these projection experiments, by identifying the appropriate quantum-mechanical operator associated with the measured quantities and relating them to the many-body correlations occurring in the BCS-BEC crossover. Calculations are presented over wide temperature and coupling ranges, by including pairing fluctuations on top of the mean field.
We present a theoretical study of the ground state of the BCS-BEC crossover in dilute two-dimensional Fermi gases. While the mean-field theory provides a simple and analytical equation of state, the pressure is equal to that of a noninteracting Fermi gas in the entire BCS-BEC crossover, which is not consistent with the features of a weakly interacting Bose condensate in the BEC limit and a weakly interacting Fermi liquid in the BCS limit. The inadequacy of the two-dimensional mean-field theory indicates that the quantum fluctuations are much more pronounced than those in three dimensions. In this work, we show that the inclusion of the Gaussian quantum fluctuations naturally recovers the above features in both the BEC and the BCS limits. In the BEC limit, the missing logarithmic dependence on the boson chemical potential is recovered by the quantum fluctuations. Near the quantum phase transition from the vacuum to the BEC phase, we compare our equation of state with the known grand canonical equation of state of two-dimensional Bose gases and determine the ratio of the composite boson scattering length aB{a}_{B}aB​ to the fermion scattering length a2D{a}_{2\mathrm{D}}a2D​. We find ${a}_{B}\ensuremath{\simeq}0.56{a}_{2\mathrm{D}}$, in good agreement with the exact four-body calculation. We compare our equation of state in the BCS-BEC crossover with recent results from the quantum Monte Carlo simulations and the experimental measurements and find good agreements.
This thesis presents a series of experimental studies that leverage atom-resolved imaging to probe fundamental properties of ultracold Fermi gases in continuous space. We demonstrate the viability of this continuum microscopy by reliably observing the expansion of a Gaussian wave packet and the Fermi hole of a two-dimensional ideal Fermi gas, measuring its second- and third-order correlation functions. Building upon this foundation, we introduce a novel local thermometer based on the relationship between fluctuations and correlations. This method enables accurate global and local temperature measurement without requiring trap calibration. Further investigations reveal sub-extensive fluctuations absent from the well-known fluctuation-dissipation theorem. We describe how these fluctuations are linked to entanglement entropy and observe a violation of the area law. Following this we connect the spatial organization of non-interacting fermions to random matrix theory via the Fredholm determinant. Measurements of the counting statistics and nearest-neighbor spacing are compared with numerical predictions, showing excellent agreement. Finally, we explore the physics of two-dimensional interacting fermions in the BEC-BCS crossover. Our measurements of two- and three-point correlation functions show significant discrepancies with mean-field BCS theory, but are corroborated by exact quantum Monte Carlo calculations. We also perform a precise measurement of Tan's contact. As a whole, this work provides a unique and unprecedented microscopic perspective on the fundamentals of Fermi gases, and establishes quantum gas microscopy as a powerful and versatile tool for exploring quantum gases in the continuum.