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arXiv:1910.10616 · Physical Review A

Spatial structure of the pair wave function and the density correlation functions throughout the BEC-BCS crossover

Juan Carlos Obeso-Jureidini, Vı́ctor Romero-Rochı́n

Published Mar 27, 2020• 7 citations
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Abstract

Within the BCS-Leggett mean-field model of a balanced mixture of two different spin fermionic atoms, we calculate the BCS pair probability distribution and the density correlation functions both between atoms with the same and different spin states, being able to accurately extract their spatial large-distance behavior along the BEC-BCS crossover. It is found that, after different initial transients, these distributions show an exponential decay and a well-defined periodic oscillatory behavior, which we analyze throughout the whole crossover. In addition, we derive analytic expressions for the mean pair size and the correlation lengths of the same and different density correlation functions. While in general, in the BCS side, there are long-range correlations and, in the BEC region, the behavior is dominated by tight-pair formation, each distribution shows its own characteristic behavior, yielding all together a quite complete description of the spatial structure of the superfluid along the crossover.

DOI
10.1103/physreva.101.033619

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Within the BCS-Leggett mean-field model of a balanced mixture of two different spin fermionic atoms, we calculate the BCS pair probability distribution and the density correlation functions both between atoms with the same and different spin states, being able to accurately extract their spatial large-distance behavior along the BEC-BCS crossover. It is found that, after different initial transients, these distributions show an exponential decay and a well-defined periodic oscillatory behavior, which we analyze throughout the whole crossover. In addition, we derive analytic expressions for the mean pair size and the correlation lengths of the same and different density correlation functions. While in general, in the BCS side, there are long-range correlations and, in the BEC region, the behavior is dominated by tight-pair formation, each distribution shows its own characteristic behavior, yielding all together a quite complete description of the spatial structure of the superfluid along the crossover.
The spatial structure of a two-dimensional homogeneous mixture of fermionic atoms in two hyperfine states is analyzed throughout the BEC-BCS crossover. Within the BCS-Leggett mean-field model we consider three functions: the pair wave function and the density-density correlation functions between atoms of the same and of different hyperfine states. For the correlation functions we derive analytical expressions which allow one to unveil the rich spatial structure of the superfluid. Mainly, we are able to study the large-distance behavior of the three functions, which exhibits an exponential decay and a well-defined oscillatory behavior. We report closed-form expressions for the correlation lengths and mean pair radius. Differences and similarities emerge when comparing with the three-dimensional case. Particularly, we find an expression for the large-distance correlation length, in terms of the chemical potential and the gap, valid in two and three dimensions, but whose dependence on the corresponding scattering lengths differ significantly.
Abstract Universality of the long-distance behavior across the Bardeen–Cooper–Schrieffer (BEC)-Bose–Einstein condensate (BCS) smooth transition for two-body density correlation functions and the Cooper-pair probability density is demonstrated in a balanced mixture of a two-component Fermi gas at T = 0. It is numerically shown at the mean-field level that these two-body quantities exhibit an exponential decay in terms of the chemical potential and the low-energy behavior of the gap. A general expression is found for the two-body distributions holding for different features of finite-range potentials, such as divergences at the origin, discontinuities at a finite radius, power-law decay, and exponential decay. The correlation length characterizing the long-distance behavior unravels the dependence on the energy needed to break pairs along the BEC-BCS crossover, a quantity meaningful to the stability of the many-body state.
In a superfluid system, off-diagonal long-range order is expected to be exhibited in the appropriate reduced density matrices when the relevant particles (either bosons or fermion pairs) are considered to recede sufficiently far apart from each other. This concept is usually exploited to identify the value of the condensate density, without explicit concern, however, as to the spatial range over which this asymptotic condition can effectively be achieved. Here, based on a diagrammatic approach that includes beyond-mean-field pairing fluctuations in the broken-symmetry phase at the level of the ttt-matrix also with the inclusion of the Gorkov-Melik-Barkhudarov (GMB) correction, we present a systematic study of the two-particle reduced density matrix for a superfluid fermionic system undergoing the BCS-BEC crossover, when the entities to recede far apart from each other evolve with continuity from largely overlapping Cooper pairs in the BCS limit to dilute composite bosons in the BEC limit. By this approach, we not only provide the coupling and temperature dependence of the condensate density at the level of our diagrammatic approach, which includes the GMB correction, but we also obtain the evolution of the spatial dependence of the two-particle reduced density matrix, from a power law at low temperature to an exponential dependence at high temperature in the superfluid phase, when the interparticle coupling spans the BCS-BEC crossover. Our results put limitations on the minimum spatial extent of a finite-size system for which superfluid correlations can effectively be established.
By using the diffusion Monte Carlo method we calculate the one- and two-body density matrix of an interacting Fermi gas at T = 0 in the BCS to Bose-Einstein condensate (BEC) crossover. Results for the momentum distribution of the atoms, as obtained from the Fourier transform of the one-body density matrix, are reported as a function of the interaction strength. Off-diagonal long-range order in the system is investigated through the asymptotic behavior of the two-body density matrix. The condensate fraction of pairs is calculated in the unitary limit and on both sides of the BCS-BEC crossover.
We analyze the BCS-BEC crossover transition of a balanced two component mixture of fermions interacting via a finite range potential, within a mean field approach. For the analysis we consider three finite range potentials cases describing the interaction between different Fermi species: a square well, an exponential and a Yukawa potential. The T = 0 thermodynamics analysis along the BCS-BEC crossover allow us to recognize the proper variables, for finite range interactions, that capture the transition from a thermodynamic non-universal behavior at unitarity, to its universal restoration. On the other side, the determination of the pair functions along the crossover suggests that the smooth transition occurs always between the scattering resonance and the change of sign of the chemical potential. This identification follows directly from the pair wave functions behavior, which in the BCS and BEC sides become exponentially localized and oscillatory slowly decaying respectively.
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.