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Physical review. B, Condensed matter

Superfluid B3 in a weak link

Salman Ullah, Alexander L. Fetter

Published Mar 1, 1989• 49 citations
Publisher

Abstract

We use the Ginzburg-Landau theory to consider the flow of superfluid $^{3}\mathit{B}$ 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.

DOI
10.1103/physrevb.39.4186

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Physical Review A•Jun 1, 1974•414 citations
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 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 study the current-phase relation of a Bose-Einstein condensate flowing through a repulsive square barrier by solving analytically the one-dimensional Gross-Pitaevskii equation. The barrier height and width fix the current-phase relation $j(\ensuremath{\delta}\ensuremath{\phi})$, which tends to $j~\mathrm{cos}(\ensuremath{\delta}\ensuremath{\phi}/2)$ for weak barriers and to the Josephson sinusoidal relation $j~\mathrm{sin}(\ensuremath{\delta}\ensuremath{\phi})$ for strong barriers. Between these two limits, the current-phase relation depends on the barrier width. In particular, for wide-enough barriers, we observe two families of multivalued current-phase relations. Diagrams belonging to the first family, already known in the literature, can have two different positive values of the current at the same phase difference. The second family, new to our knowledge, can instead allow for three different positive currents still corresponding to the same phase difference. Finally, we show that the multivalued behavior arises from the competition between hydrodynamic and nonlinear-dispersive components of the flow, the latter due to the presence of a soliton inside the barrier region.
In this thesis we demonstrate the power of the Gross-Pitaevskii and the time-dependent Ginzburg-Landau equations by numerically solving them for various fundamental problems related to superfluidity and superconductivity. We start by studying the motion of a massive object through a quantum fluid modelled by the Gross-Pitaevskii equation. Below a critical velocity, the object does not exchange momentum or energy with the fluid. This is a manifestation of its superfluid nature. We discuss the effect of applying a constant force to the object and show that for small forces a vortex ring is created to which the object becomes attached. For a larger force the object detaches from the vortex ring and we observe periodic shedding of rings. All energy transfered to the system is contained within the vortex rings and the drag force on the object is due to the recoil of the vortex emission. If we exceed the speed of sound, there is an additional contribution to the drag from sound emission. To make a link to superconductivity, we then discuss vortex states in a rotating system. In the ground state, regular arrays of vortices are observed which, for systems containing many vortices, mimic solid-body rotation. In the second part of the thesis, we initially review solutions to the Ginzburg-Landau equations in an applied magnetic field. For superconducting disks we observe vortex arrays similar to those in rotating superfluids. Finally, we study an electrical current flow along a superconducting wire subject to an external magnetic field. We observe the motion of flux lines, and hence dissipation, due to the Lorentz force. We measure the V - I curve which is analogous to the drag force in a superfluid. With the introduction of impurities, flux lines become pinned which gives rise to an increased critical current. (author)
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 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.