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arXiv:2311.00540 · Physical Review A

Critical current throughout the BCS-BEC crossover with the inclusion of pairing fluctuations

L. Pisani, V. Piselli, Giancarlo Calvanese Strinati

Published Mar 5, 2024• 6 citations
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Abstract

The present work aims at providing a systematic analysis of the current density versus momentum characteristics for a fermionic superfluid throughout the BCS-BEC crossover, even in the fully homogeneous case. At low temperatures, where pairing fluctuations are not strong enough to invalidate a quasiparticle approach, a sharp threshold for the inception of a back-flow current is found, which sets the onset of dissipation and identifies the critical momentum according to Landau. This momentum is seen to smoothly evolve from the BCS to the BEC regimes, whereby a single expression for the single-particle current density that includes pairing fluctuations enables us to incorporate on equal footing two quite distinct dissipative mechanisms, namely, pair breaking and phonon excitations in the two sides of the BCS-BEC crossover, respectively. At finite temperature, where thermal fluctuations broaden the excitation spectrum and make the dissipative (kinetic and thermal) mechanisms intertwined with each other, an alternative criterion due to Bardeen is instead employed to signal the loss of superfluid behavior. In this way, detailed comparison with available experimental data in linear and annular geometries is significantly improved with respect to previous approaches, thereby demonstrating the crucial role played by quantum fluctuations in renormalizing the single-particle excitation spectrum.

DOI
10.1103/physreva.109.033306

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The present work aims at providing a systematic analysis of the current density versus momentum characteristics for a fermionic superfluid throughout the BCS-BEC crossover, even in the fully homogeneous case. At low temperatures, where pairing fluctuations are not strong enough to invalidate a quasiparticle approach, a sharp threshold for the inception of a back-flow current is found, which sets the onset of dissipation and identifies the critical momentum according to Landau. This momentum is seen to smoothly evolve from the BCS to the BEC regimes, whereby a single expression for the single-particle current density that includes pairing fluctuations enables us to incorporate on equal footing two quite distinct dissipative mechanisms, namely, pair breaking and phonon excitations in the two sides of the BCS-BEC crossover, respectively. At finite temperature, where thermal fluctuations broaden the excitation spectrum and make the dissipative (kinetic and thermal) mechanisms intertwined with each other, an alternative criterion due to Bardeen is instead employed to signal the loss of superfluid behavior. In this way, detailed comparison with available experimental data in linear and annular geometries is significantly improved with respect to previous approaches, thereby demonstrating the crucial role played by quantum fluctuations in renormalizing the single-particle excitation spectrum.
We derive an expression for the superfluid density of a uniform two-component Fermi gas through the BCS-BEC crossover in terms of the thermodynamic potential in the presence of an imposed superfluid flow. Treating the pairing fluctuations in a Gaussian approximation following the approach of Nozi\`eres and Schmitt-Rink, we use this definition of ${\ensuremath{\rho}}_{s}$ to obtain an explicit result which is valid at finite temperatures and over the full BCS-BEC crossover. It is crucial that the BCS gap $\ensuremath{\Delta}$, the chemical potential $\ensuremath{\mu}$, and ${\ensuremath{\rho}}_{s}$ all include the effect of fluctuations at the same level in a self-consistent manner. We show that the normal fluid density ${\ensuremath{\rho}}_{n}\ensuremath{\equiv}n\ensuremath{-}{\ensuremath{\rho}}_{s}$ naturally separates into a sum of contributions from Fermi BCS quasiparticles $({\ensuremath{\rho}}_{n}^{F})$ and Bose collective modes $({\ensuremath{\rho}}_{n}^{B})$. The expression for ${\ensuremath{\rho}}_{n}^{F}$ is just Landau's formula for a BCS Fermi superfluid but now calculated over the BCS-BEC crossover. The expression for the Bose contribution ${\ensuremath{\rho}}_{n}^{B}$ is more complicated and only reduces to Landau's formula for a Bose superfluid in the extreme BEC limit, where all the fermions have formed stable Bose pairs and the Bogoliubov excitations of the associated molecular Bose condensate are undamped. In a companion paper, we present numerical calculations of ${\ensuremath{\rho}}_{s}$ using an expression equivalent to the one derived in this paper, over the BCS-BEC crossover, including unitarity, and at finite temperatures.
The superfluid density is a fundamental quantity describing the response to a rotation as well as in two-fluid collisional hydrodynamics. We present extensive calculations of the superfluid density ${\ensuremath{\rho}}_{s}$ in the BCS-BEC crossover regime of a uniform superfluid Fermi gas at finite temperatures. We include strong-coupling or fluctuation effects on these quantities within a Gaussian approximation. We also incorporate the same fluctuation effects into the BCS single-particle excitations described by the superfluid order parameter $\ensuremath{\Delta}$ and Fermi chemical potential $\ensuremath{\mu}$, using the Nozi\`eres--Schmitt-Rink approximation. This treatment is shown to be necessary for consistent treatment of ${\ensuremath{\rho}}_{s}$ over the entire BCS-BEC crossover. We also calculate the condensate fraction Nc{N}_{c}Nc​ as a function of the temperature, a quantity which is quite different from the superfluid density ${\ensuremath{\rho}}_{s}$. We show that the mean-field expression for the condensate fraction Nc{N}_{c}Nc​ is a good approximation even in the strong-coupling BEC regime. Our numerical results show how ${\ensuremath{\rho}}_{s}$ and Nc{N}_{c}Nc​ depend on temperature, from the weak-coupling BCS region to the BEC region of tightly bound Cooper pair molecules. In a companion paper [Phys. Rev. A 74, 063626 (2006)], we derive an equivalent expression for ${\ensuremath{\rho}}_{s}$ from the thermodynamic potential, which exhibits the role of the pairing fluctuations in a more explicit manner.
The Josephson current-phase characteristics, that arise when a supercurrent flows across two fermionic superfluids separated by a potential barrier, can be controlled by varying either the interparticle coupling or the temperature. While the coupling dependence has been addressed in detail both theoretically and experimentally for an attractive Fermi gas undergoing the BCS-BEC crossover, a corresponding study of the temperature dependence of the Josephson characteristics is still lacking in this context. Here, we investigate the combined coupling and temperature dependence of the Josephson characteristics in a systematic way for a wide set of barriers, within ranges of height and width that can be experimentally explored. Our study smoothly connects the two limiting cases, of nonoverlapping composite bosons at low temperature described by the Gross-Piatevskii equation, and of strongly overlapping Cooper pairs near the critical temperature described by the Ginzburg-Landau equation. In this way, we are able to explore several interesting effects related to how the current-phase characteristics evolve along the BCS-BEC crossover as a function of temperature and of barrier shape. These effects include the coherence length outside the barrier and the pair penetration length inside the barrier (which is related to the proximity effect), as well as the temperature evolution of the Landau criterion in the limit of a vanishingly small barrier. A comparison is also presented between the available experimental data for the critical current and our theoretical results over a wide range of couplings along the BCS-BEC crossover.
A mechanism is proposed to explain the puzzling damping of collective excitations, which was recently observed in the experiments of strongly interacting Fermi gases below the superfluid critical temperature on the fermionic (BCS) side of Feshbach resonance. Sound velocity, superfluid density, and damping rate are calculated with effective field theory. We find that a dominant damping process is due to the interaction between superfluid phonons and thermally excited fermionic quasiparticles, in contrast to the previously proposed pair-breaking mechanism. Results from our effective model are compared quantitatively with recent experimental findings, showing a good agreement.
We address the thermodynamics, density profiles, and superfluid density of trapped fermions undergoing BCS-BEC crossover. We consider the case of zero and finite population imbalance and apply the local density approximation (LDA) to include the trap potential. Our approach represents a fully self-consistent treatment of ``pseudogap effects.'' These effects reflect the distinction between the pair formation temperature T∗{T}^{*}T∗ and the pair condensation temperature Tc{T}_{c}Tc​. As a natural corollary, this temperature difference must be accommodated in the fermionic excitation spectrum Ek{E}_{\mathbf{k}}Ek​ to reflect the fact that fermions are paired at and above Tc{T}_{c}Tc​. It is precisely this natural corollary which has been omitted from all other many body approaches in the literature. At a formal level, we show how enforcing this corollary implies that pairing fluctuation or self-energy contributions enter into both the gap and the number equations; this is necessary in order to be consistent with a generalized Ward identity. At a less formal level, we demonstrate that we obtain physical results for the superfluid density ns(T){n}_{s}(T)ns​(T) at all temperatures. In contrast, previous work in the literature has led to nonmonotonic, or multivalued or discontinuous behavior for ns(T){n}_{s}(T)ns​(T). Because it reflects the essence of the superfluid state, we view the superfluid density as a critical measure of the physicality of a given crossover theory. In a similarly unique fashion, we emphasize that in order to properly address thermodynamic properties of a trapped Fermi gas, a necessary first step is to demonstrate that the particle density profiles are consistent with experiment. Without a careful incorporation of the distinction between the pairing gap and the order parameter, the LDA-derived density profiles (of the unpolarized gas) tend to exhibit sharp features at the condensate edge, which are never seen experimentally in the crossover regime. The lack of demonstrable consistency between theoretical and experimental density profiles, along with problematic behavior found for the superfluid density, casts doubt on previous claims in the literature concerning quantitative agreement between thermodynamical calculations and experiment.
We investigate single-particle excitations and strong-coupling effects in the BCS-BEC crossover regime of a superfluid Fermi gas. Including phase and amplitude fluctuations of the superfluid order parameter within a TTT-matrix theory, we calculate the superfluid density of states (DOS), as well as single-particle spectral weight, over the entire BCS-BEC crossover region below the superfluid transition temperature Tc{T}_{\mathrm{c}}Tc​. We clarify how the pseudogap in the normal state evolves into the superfluid gap, as one passes through Tc{T}_{\mathrm{c}}Tc​. While the pseudogap in DOS continuously evolves into the superfluid gap in the weak-coupling BCS regime, the superfluid gap in the crossover region is shown to appear in DOS after the pseudogap disappears below Tc{T}_{\mathrm{c}}Tc​. In the phase diagram with respect to the temperature and interaction strength, we determine the region where strong pairing fluctuations dominate over single-particle properties of the system. Our results would be useful for the study of strong-coupling phenomena in the BCS-BEC crossover regime of a superfluid Fermi gas.