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Journal of Physics Communications

Superfluidity of fermionic pairs in a harmonic trap. Comparative studies: Local Density Approximation and Bogoliubov-de Gennes solutions

Agnieszka Cichy, Andrzej Ptok

Published Apr 30, 2020• 2 citations
Publisher

Abstract

Abstract Experiments with ultracold gases on the lattice give the opportunity to realize superfluid fermionic mixtures in a trapping potential. The external trap modifies the chemical potential locally. Moreover, this trap also introduces non-homogeneity in the superconducting order parameter. There are, among other approaches, two methods which can be used to describe the system of two-component mixtures loaded into an optical lattice: the Local Density Approximation (LDA) and the self-consistent Bogoliubov–de Gennes equations. Here, we compare results obtained within these two methods. We conclude that the results can be distinguishable only in the case of a small value of the pairing interaction.

DOI
10.1088/2399-6528/ab8f02

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Abstract Experiments with ultracold gases on the lattice give the opportunity to realize superfluid fermionic mixtures in a trapping potential. The external trap modifies the chemical potential locally. Moreover, this trap also introduces non-homogeneity in the superconducting order parameter. There are, among other approaches, two methods which can be used to describe the system of two-component mixtures loaded into an optical lattice: the Local Density Approximation (LDA) and the self-consistent Bogoliubov–de Gennes equations. Here, we compare results obtained within these two methods. We conclude that the results can be distinguishable only in the case of a small value of the pairing interaction.
Superconductivity and superfluidity of fermions require, within the BCS theory, matching of the Fermi energies of the two interacting Fermion species. Difference in the number densities of the two species leads either to a normal state, to phase separation, or---potentially---to exotic forms of superfluidity such as Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, Sarma state, or breached pair state. We consider ultracold Fermi gases with polarization, i.e., spin-density imbalance. We show that, due to the gases being trapped and isolated from the environment in terms of particle exchange, exotic forms of superfluidity appear as a shell around the Bardeen-Cooper-Schrieffer (BCS) superfluid core of the gas and, for large density imbalance, in the core as well. We obtain these results by describing the effect of the trapping potential by using the Bogoliubov-de Gennes equations. For comparison to experiments, we calculate also the condensate fraction, and show that, in the center of the trap, a polarized superfluid leads to a small dip in the central density difference. We compare the results to those given by local density approximation and find qualitatively different behavior.
It is quite common that several different phases exist simultaneously in a system of trapped quantum gases of ultra-cold atoms. One example is the strongly interacting Fermi gas with two imbalanced spin species, which has received a great deal of attention owing to the possible occurrence of exotic superfluid phases. By using novel numerical techniques and algorithms, we self-consistently solve the Bogoliubov–de Gennes equations, which describe Fermi superfluids in the mean-field framework. From this study, we investigate the novel phases of spin-imbalanced Fermi gases and examine the validity of local density approximation (LDA), which is often invoked in the extraction of bulk properties from experimental measurements within trapped systems. We show how the validity of the LDA is affected by the trapping geometry, the number of atoms and the spin imbalance.
We present an inhomogeneous theory for the low-temperature properties of a resonantly interacting Fermi mixture in a trap that goes beyond the local-density approximation. We compare the Bogoliubov-de Gennes and a Landau-Ginzburg approach and conclude that the latter is more appropriate when dealing with a first-order phase transition. Our approach incorporates the state-of-the-art knowledge on the homogeneous mixture with a population imbalance exactly and gives good agreement with the experimental density profiles of Shin {\it et al}. [Nature {\bf 451}, 689 (2008)]. We calculate the universal surface tension due to the observed interface between the equal-density superfluid and the partially polarized normal state of the mixture. We find that the exotic and gapless superfluid Sarma phase can be stabilized at this interface, even when this phase is unstable in the bulk of the gas.
We use the Bogoliubov-de Gennes formalism to analyze the ground state phases of harmonically trapped two-species fermion mixtures with unequal masses. In the weakly attracting limit and around unitarity, we find that the superfluid order parameter is spatially modulated around the trap center, and that its global maximum occurs at a finite distance away from the trap center where the mixture is locally unpolarized. As the attraction strength increases towards the molecular limit, the spatial modulations gradually disappear while the Bardeen-Cooper-Schrieffer (BCS) type nonmodulated superfluid region expands until the entire mixture becomes locally unpolarized.
The Higgs mode is a key component in the spontaneous breaking of a continuous symmetry along with the Nambu-Goldstone mode, and has been studied extensively for homogeneous systems. We consider it for inhomogeneous systems, using the superfluid of harmonically trapped ultracold Fermi atomic gas. The Fermionic field operators are expanded in a complete set of wave functions corresponding to inhomogeneous situation. Within the Hatree-Fock approximation, we derive integral equations from the Bogoliubov-de Gennes equations, which lead to the frequencies of the collective modes, including the Higgs and Nambu-Goldstone modes. The results show that the frequency of the Higgs mode equals twice the absolute value of the order parameter at the center of trap. This feature is robust against variations in the interaction strength, trap potential, and temperature. These results are consistent with previous theoretical and experimental studies.