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Physical review. B, Condensed matter

Flow of superfluid B3 past a specular wall: A finite-temperature calculation

Salman Ullah, Louis J. Buchholtz

Published Nov 1, 1989• 3 citations
Publisher

Abstract

We report the first numerical solution of the quasiclassical equations for superfluid $^{3}\mathit{B}$ flowing past a single specular wall. The superfluid gap function has (for finite flow) three nondegenerate components: The two components perpendicular to the flow are nearly unaffected by the flow; the component parallel to the flow is suppressed. We find that the critical current density given by our finite-temperature calculation is almost equal to the Ginzburg-Landau value down to temperatures of approximately T/${T}_{c}$=0.5.

DOI
10.1103/physrevb.40.8791

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arXiv:2010.10115•Physical review. B./Physical review. B•Oct 19, 2020•14 citations
We report the first numerical solution of the quasiclassical equations for superfluid 3B^{3}\mathit{B}3B flowing past a single specular wall. The superfluid gap function has (for finite flow) three nondegenerate components: The two components perpendicular to the flow are nearly unaffected by the flow; the component parallel to the flow is suppressed. We find that the critical current density given by our finite-temperature calculation is almost equal to the Ginzburg-Landau value down to temperatures of approximately T/Tc{T}_{c}Tc​=0.5.
We use the Ginzburg-Landau theory to consider the flow of superfluid 3B^{3}\mathit{B}3B through a constriction (a ``weak link'') separating two bulk regions of superfluid. The boundary conditions that describe the applied flow (specified phase gradient) at the bulk ends differ from the analogous situation in a superconducting weak link (specified phase difference). We compute the critical current, which depends upon the width and length of the weak link.
We have observed that the motion of a vibrating-wire resonator in superfluid 3B^{3}\mathit{B}3B at temperatures below Tc{\mathrm{T}}_{\mathrm{c}}Tc​/8 is associated with a sharp critical velocity, vc{\mathrm{v}}_{\mathrm{c}}vc​, above which very high dissipation sets in. We identify this velocity with the Landau critical velocity for pair breaking. That the observed value of vc{\mathrm{v}}_{\mathrm{c}}vc​ is a factor of 2 lower than the critical velocity expected from the BCS value of the gap we take as evidence that the gap is suppressed by about 50% of its bulk value near a moving boundary.
The order parameter of superfluid 3B^{3}\mathit{B}3B is calculated near surfaces with the quasiclassical theory and a thin-dirty-layer surface-scattering model. The surface structures depend on the temperature, the surface roughness, the Fermi-liquid parameters, and the mass superflow tangential to the surface. All these are studied with special emphasis on the roughness and the superflow. A numerical technique of solving the quasiclassical equations is discussed.
A discussion of an anisotropic (ppp state) Fermi superfluid with a spatially varying order parameter is presented. We restrict ourself to the Landau-Ginzburg region and consider spatial variations governed by the linear gap equation. The kernel of the linearized gap equation is shown to be related to a current-current correlation function of quasiparticles in the normal state. This relation permits the calculation of the two coherence lengths ${\ensuremath{\xi}}_{L}$ and ${\ensuremath{\xi}}_{T}$ defined earlier in the framework of a phenomenological Landau-Ginzburg theory, and a discussion of boundary conditions for the order parameter at interfaces. A significant anisotropy in the coherence lengths is found (${\ensuremath{\xi}}_{L}=\sqrt{3}{\ensuremath{\xi}}_{T}$), which reflects itself in the nature of the supercurrents. The boundary conditions at specularly and diffusely reflecting surfaces lead us to expect that the vector $\stackrel{^}{l}$, which describes the orbital angular momentum of a Cooper pair in an "axial" state, is anchored normal to the walls. Some consequences of these results on various experimental problems (fourth sound, monodomain production, Josephson couplings, behavior in thin capillaries) are briefly discussed.
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 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.