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arXiv:2010.10115 · Physical review. B./Physical review. B

Josephson effect at finite temperature along the BCS-BEC crossover

V. Piselli, Stefano Simonucci, Giancarlo Calvanese Strinati

Published Oct 19, 2020• 14 citations
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Abstract

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.

DOI
10.1103/physrevb.102.144517

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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.
We study the stationary Josephson effect for neutral fermions across the BCS-BEC (Bose-Einstein condensate) crossover, by solving numerically the Bogoliubov-de Gennes equations at zero temperature. The Josephson current is found to be considerably enhanced for all barriers at about unitarity. For vanishing barrier, the Josephson critical current approaches the Landau limiting value which, depending on the coupling, is determined by either pair-breaking or sound-mode excitations. In the coupling range from the BCS limit to unitarity, a procedure is proposed to extract the pairing gap from the Landau limiting current.
A realistic description of the Josephson effect at finite temperature with ultra-cold Fermi gases embedded in nontrivial geometrical constraints (typically, a trap plus a barrier) requires appropriate consideration of pairing fluctuations that arise in inhomogeneous environments. Here, we apply the theoretical approach developed in the companion article [Pisani \emph{et al.}, Phys. Rev. B {\bf 108}, 214503 (2023)], where the inclusion of pairing fluctuations beyond mean field across the BCS-BEC crossover at finite temperature is combined with a detailed description of the gap parameter in a nontrivial geometry. In this way, we are able to account for the experimental results on the Josephson critical current, reported both at low temperature for various couplings across the BCS-BEC crossover and as a function of temperature at unitarity. Besides validating the theoretical approach of the companion article, our numerical results reveal generic features of the Josephson effect which may not readily emerge from an analysis of corresponding experiments with condensed-matter samples owing to the unique intrinsic flexibility of experiments with ultra-cold gases.
A realistic description of the Josephson effect at finite temperature with ultracold Fermi gases embedded in nontrivial geometrical constraints (typically, a trap plus a barrier) requires appropriate consideration of pairing fluctuations that arise in inhomogeneous environments. Here, we apply the theoretical approach developed in a companion paper [Pisani et al., Phys. Rev. B 108, 214503 (2023)], where the inclusion of pairing fluctuations beyond mean field across the BCS--Bose-Einstein-condensate (BEC) crossover at finite temperature is combined with a detailed description of the gap parameter in a nontrivial geometry. In this way, we are able to account for the experimental results on the Josephson critical current, reported both at low temperature for various couplings across the BCS-BEC crossover and as a function of temperature at unitarity. Besides validating the theoretical approach of the companion paper, our numerical results reveal generic features of the Josephson effect which may not readily emerge from an analysis of corresponding experiments with condensed-matter samples owing to the unique intrinsic flexibility of experiments with ultracold gases.
We develop a microscopic model to describe the Josephson tunneling between two superfluid reservoirs of ultracold fermionic atoms which accounts for the dependence of the critical current on both the barrier height and the interaction strength along the crossover from the BCS to the BEC regime. Building on a previous study [F. Meier and W. Zwerger, Phys. Rev. A 64, 033610 (2001)] of weakly interacting bosons, we derive analytic results for the Josephson critical current at zero temperature for homogeneous and trapped systems at arbitrary coupling. The critical current exhibits a maximum near the unitarity limit which arises from the competition between the increasing condensate fraction and a decrease in the chemical potential along the evolution from the BCS to the BEC limit. Our results agree quantitatively with numerical simulations and recent experimental data.
The proximity effect, which arises at the interface between two fermionic superfluids with different critical temperatures, is examined with a nonlocal (integral) equation whose kernel contains information about the size of Cooper pairs that leak across the interface. This integral approach avoids reference to the boundary conditions at the interface that would be required with a differential approach. The temperature dependence of the pair penetration depth on the normal side of the interface is determined over a wide temperature range also varying the interparticle coupling along the BCS side of the BCS-BEC crossover independently on both sides of the interface. In this way, the size of Cooper pairs evolves from being much larger than (BCS limit) the interparticle distance to being comparable with (unitarity limit, halfway between the BCS and BEC limits) the interparticle distance. Conditions are then found for which the proximity effect is optimized in terms of the extension of the pair penetration depth.
We calculate the pair correlation function and the order parameter correlation function, which probe, respectively, the intrapair and interpair correlations of a Fermi gas with attractive interparticle interaction, in terms of a diagrammatic approach as a function of coupling throughout the BCS-Bose-Einstein condensation (BEC) crossover and of temperature, both in the superfluid and normal phase across the critical temperature Tc{T}_{c}Tc​. Several physical quantities are obtained from this calculation, including the pair coherence and healing lengths, the Tan's contact, the crossover temperature T∗{T}^{*}T∗ below which interpair correlations begin to build up in the normal phase, and the signature for the disappearance of the underlying Fermi surface which tends to survive in spite of pairing correlations. A connection is also made with recent experimental data on the temperature dependence of the normal coherence length as extracted from the proximity effect measured in high-temperature (cuprate) superconductors.