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arXiv:0705.2658 · Physical Review Letters

Josephson Effect throughout the BCS-BEC Crossover

Andrea Spuntarelli, P. Pieri, Giancarlo Calvanese Strinati

Published Jul 23, 2007• 64 citations
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Abstract

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.

DOI
10.1103/physrevlett.99.040401

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The thesis analyze Fermionic neutral systems with tunable attractive interaction between the particles whose strenght can be tuned from BCS theory to Bose-Einstein condensate. Below the critical temperature both systems are characterized by an order parameter. A mean-field approach can't describe correctly the phenomenon. This happens if Gaussian Thermal fluctations are included. The aim is to study BCS-BEC crossover around critical temperature through Ginzburg-Landau theory in 3D systems and Josephson effect in 2D systems. In both cases effect of thermal fluctuations have been included.

•Mar 23, 2020•0 citations
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.
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 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.
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.
The Bardeen–Cooper–Schrieffer (BCS) to Bose–Einstein condensate (BEC) crossover problem is solved for stationary gray solitons via the Boguliubov–de Gennes equations at zero temperature. These crossover solitons exhibit a localized notch in the gap and a characteristic phase difference across the notch for all interaction strengths, from BEC to BCS regimes. However, they do not follow the well-known Josephson-like sinusoidal relationship between velocity and phase difference except in the far BEC limit: at unitarity, the velocity has a nearly linear dependence on phase difference over an extended range. For a fixed phase difference, the soliton is of nearly constant depth from the BEC limit to unitarity and then grows progressively shallower into the BCS limit, and on the BCS side, Friedel oscillations are apparent in both gap amplitude and phase. The crossover soliton appears fundamentally in the gap; we show, however, that the density closely follows the gap, and the soliton is therefore observable. We develop an approximate power-law relationship to express this fact: the density of gray crossover solitons varies as the square of the gap amplitude in the BEC limit and as a power of about 1.5 at unitarity.
The thesis analyze Fermionic neutral systems with tunable attractive interaction between the particles whose strenght can be tuned from BCS theory to Bose-Einstein condensate. Below the critical temperature both systems are characterized by an order parameter. A mean-field approach can't describe correctly the phenomenon. This happens if Gaussian Thermal fluctations are included. The aim is to study BCS-BEC crossover around critical temperature through Ginzburg-Landau theory in 3D systems and Josephson effect in 2D systems. In both cases effect of thermal fluctuations have been included.