Superfluid Hydrodynamic Equations Research Articles

Hydrodynamics of a Multi-Component Bosonic Superfluid

We obtain the superfluid hydrodynamic equations of a multi-component Bose gas with short-ranged interactions at zero temperature under the local equilibrium assumption and show that the quantum pressure is generally present in the nonuniform case. Our approach can be extended to systems with long-range interactions such as dipole-dipole interactions by treating the Hartree energy properly. For a highly symmetric superfluid, we obtain the excitation spectrum and show that except for the density phonon, all other excitations are all degenerate. The implication of our results is discussed.

Collective oscillation modes of a superfluid Bose–Fermi mixture

In this work, we present a theoretical study for the collective oscillation modes, i.e. quadrupole, radial and axial mode, of a mixture of Bose and Fermi superfluids in the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a molecular Bose–Einstein condensate (BEC) in harmonic trapping potentials with cylindrical symmetry of experimental interest. To this end, we start from the coupled superfluid hydrodynamic equations for the dynamics of Bose–Fermi superfluid mixtures and use the scaling theory that has been developed for a coupled system. The collective oscillation modes of Bose–Fermi superfluid mixtures are found to crucially depend on the overlap integrals of the spatial derivations of density profiles of the Bose and Fermi superfluids at equilibrium. We not only present the explicit expressions for the overlap density integrals, as well as the frequencies of the collective modes provided that the effective Bose–Fermi coupling is weak, but also test the valid regimes of the analytical approximations by numerical calculations in realistic experimental conditions. In the presence of a repulsive Bose–Fermi interaction, we find that the frequencies of the three collective modes of the Bose and Fermi superfluids are all upshifted, and the change speeds of the frequency shifts in the BCS–BEC crossover can characterize the different groundstate phases of the Bose–Fermi superfluid mixtures for different trap geometries.

Super Efimovian expansions of superfluid Fermi gases in the BCS–BEC crossover

We study the super Efimovian expansions of a superfluid Fermi gas in the crossover from Bardeen–Cooper–Schrieffer (BCS) superfluid to a Bose–Einstein condensate (BEC), by using superfluid hydrodynamic equations with the equation of state fitting from the experimental data and a scaling approach. We find that in an isotropic trap, the super Efimovian expansion of the Fermi superfluid in the unitary limit is characterized by the double-log periodicity, while for an axially symmetric trap, the double-log periodicity only emerges in the longer direction for a large anisotropy of the trapping frequencies. Away from unitarity where the scale invariance is broken, the super Efimovian expansions of the superfluid Fermi gases deviate from the double-log oscillations. The oscillations in the BCS regime deviate upward from the double-log oscillations, whereas the oscillations in the BEC regime deviate downward and the deviation in the BEC limit is the most significant. This difference can be used to discriminate different superfluid regimes.

Dynamics of vortex line density and heat transfer processes in superfluid helium

Three dynamics equations for vortex line density are analyzed. It is shown that the Vinen equation gives the values of vortex tangle development time in the case of a constant counterflow more accurately than other alternative equations. Within the system of equations of superfluid turbulence hydrodynamics, obtained using a phenomenological approach, helium boiling times as a function of heat flux density are found, using alternative dynamics equations of vortex tangle density. Unlike the experiments in which different dependences of boiling time tboil on the heat flux density Q (tboil ∝ Qn, −4 ≤ n ≤ −2) are observed, in this case we get only a power-law dependence with an exponent of n = −4. We obtain a velocity distribution of the normal component along the channel, and the temperature dependence of the time near the heater. We conduct a comparison against the numerical and experimental results that were previously obtained in literature.

Path-integral ground state and superfluid hydrodynamics of a bosonic gas of hard spheres

We study a bosonic gas of hard spheres by using the exact zero-temperature path-integral ground-state (PIGS) Monte Carlo method and the equations of superfluid hydrodynamics. The PIGS method is implemented to calculate for the bulk system the energy per particle and the condensate fraction through a large range of the gas parameter $n{a}^{3}$ (with $n$ the number density and $a$ the $s$-wave scattering length), going from the dilute gas into the solid phase. The Maxwell construction is then adopted to determine the freezing at $n{a}^{3}=0.264\ifmmode\pm\else\textpm\fi{}0.003$ and the melting at $n{a}^{3}=0.290\ifmmode\pm\else\textpm\fi{}0.003$. In the liquid phase, where the condensate fraction is finite, the equations of superfluid hydrodynamics, based on the PIGS equation of state, are used to find other relevant quantities as a function of the gas parameter: the chemical potential, the pressure, and the sound velocity. In addition, within Feynman's approximation, from the PIGS static structure factor we determine the full excitation spectrum, which displays a maxon-roton behavior when the gas parameter is close to the freezing value. Finally, the equations of superfluid hydrodynamics with the PIGS equation of state are solved for the bosonic system under axially symmetric harmonic confinement obtaining its collective breathing modes.

Self-similar Expansion of a Turbulent Bose-Einstein Condensate: A Generalized Hydrodynamic Model

We generalize the superfluid hydrodynamic equations to describe the expansion of a turbulent condensate cloud. Our Lagrangian formalism includes the kinetic energy associated with an entangled vortex configuration. The expressions developed here clarify the physics behind the self-similar free expansion as seen in experiments. This feature is a crucial and characteristic signature of turbulence. Our present study improves on our previous model (Caracanhas et al. in J. Low Temp. Phys. 166: 49, 2012).

Dynamical Properties of the Unitary Fermi Gas: Collective Modes and Shock Waves

We discuss the unitary Fermi gas made of dilute and ultracold atoms with an infinite s-wave inter-atomic scattering length. First we introduce an efficient Thomas-Fermi-von Weizsacker density functional which describes accurately various static properties of the unitary Fermi gas trapped by an external potential. Then, the sound velocity and the collective frequencies of oscillations in a harmonic trap are derived from extended superfluid hydrodynamic equations which are the Euler-Lagrange equations of a Thomas-Fermi-von Weizsacker action functional. Finally, we show that this amazing Fermi gas supports supersonic and subsonic shock waves.

Splitting superfluid hydrodynamic equations and instability of solutions in the case of degenerate bosons

Properties of solutions to superfluid hydrodynamic equations as applied to the degenerate Bose gas are considered. The equations are split into two independent pairs of equations. One pair is written for the normal component implies the instability of solutions, which manifests itself in the majorant catastrophe with respect to the total density. The case when the thermodynamic functions depend on the difference of the normal and superfluid velocities is also considered. In that case, the system is not split; however, the instability and the majorant catastrophe occur when the initial temperature tends to absolute zero.

Nonclassical rotational inertia in a two-dimensional bosonic solid containing grain boundaries

We study the occurrence of non-classical rotational inertia (NCRI) arising from superfluidity along grain boundaries in a two-dimensional bosonic system. We make use of a standard mapping between the zero-temperature properties of this system and the statistical mechanics of interacting vortex lines in the mixed phase of a type-II superconductor. In the mapping, the liquid phase of the vortex system corresponds to the superfluid bosonic phase. We consider numerically obtained polycrystalline configurations of the vortex lines in which the microcrystals are separated by liquid-like grain boundary regions which widen as the vortex system temperature increases. The NCRI of the corresponding zero-temperature bosonic systems can then be numerically evaluated by solving the equations of superfluid hydrodynamics in the channels near the grain boundaries. We find that the NCRI increases very abruptly as the liquid regions in the vortex system (equivalently, superfluid regions in the bosonic system) form a connected, system-spannig structure with one or more closed loops. The implications of these results for experimentally observed supersolid phenomena are discussed.

Frequency shift and mode coupling of the collective modes of superfluid Fermi gases in the BCS-BEC crossover

We investigate the dynamical behavior of large-amplitude collective modes in a superfluid Fermi gas in the crossover from Bardeen--Cooper--Schrieffer (BCS) superfluid to Bose--Einstein condensate (BEC) based on a hydrodynamic approach. We first solve the superfluid hydrodynamic equations that describe the time evolution of fermionic condensates in the BCS-BEC crossover and calculate explicitly the frequency shifts of the collective modes induced by nonlinear effects using the Lindstedt--Poincar\'e method. The result shows that the frequency shifts display different features in different superfluid regimes. We then study the second-harmonic generation of the collective modes under a phase-matching condition, which can be fulfilled by choosing appropriate parameters of the system. The analytical results obtained are checked by numerical simulations and good agreement is found.

Strong dipolar effects in a quantum ferrofluid

Symmetry-breaking interactions have a crucial role in many areas of physics, ranging from classical ferrofluids to superfluid (3)He and d-wave superconductivity. For superfluid quantum gases, a variety of new physical phenomena arising from the symmetry-breaking interaction between electric or magnetic dipoles are expected. Novel quantum phases in optical lattices, such as chequerboard or supersolid phases, are predicted for dipolar bosons. Dipolar interactions can also enrich considerably the physics of quantum gases with internal degrees of freedom. Arrays of dipolar particles could be used for efficient quantum information processing. Here we report the realization of a chromium Bose-Einstein condensate with strong dipolar interactions. By using a Feshbach resonance, we reduce the usual isotropic contact interaction, such that the anisotropic magnetic dipole-dipole interaction between 52Cr atoms becomes comparable in strength. This induces a change of the aspect ratio of the atom cloud; for strong dipolar interactions, the inversion of ellipticity during expansion (the usual 'smoking gun' evidence for a Bose-Einstein condensate) can be suppressed. These effects are accounted for by taking into account the dipolar interaction in the superfluid hydrodynamic equations governing the dynamics of the gas, in the same way as classical ferrofluids can be described by including dipolar terms in the classical hydrodynamic equations. Our results are a first step in the exploration of the unique properties of quantum ferrofluids.

Collective oscillations in trapped Bose-Einstein-condensed gases in the presence of weak disorder

The influence of a weak random potential on the collective modes of a trapped interacting Bose-Einstein condensate at zero temperature is calculated in the limit when the correlation length of the disorder is smaller than the healing length of the superfluid. The problem is solved in the Thomas-Fermi limit by generalizing the superfluid hydrodynamic equations to allow for the presence of weak disorder. We find that the disorder-induced frequency-shifts of the low-energy excitations can be of the same order of magnitude as the beyond mean-field corrections in the normal interaction recently observed experimentally.

Formation of a vortex lattice in a rotating BCS Fermi gas

We investigate theoretically the formation of a vortex lattice in a superfluid two-spin component Fermi gas in a rotating harmonic trap, in a BCS-type regime of condensed non-bosonic pairs. Our analytical solution of the superfluid hydrodynamic equations, both for the 2D BCS equation of state and for the 3D unitary quantum gas, predicts that the vortex free gas is subject to a dynamic instability for fast enough rotation. With a numerical solution of the full time dependent BCS equations in a 2D model, we confirm the existence of this dynamic instability and we show that it leads to the formation of a regular pattern of quantum vortices in the gas.

On the description of electrical effects in the two-fluid model of superfluidity

A model description of the electrothermal effect in a superfluid liquid is proposed. It is hypothesized that the superfluid state is an ordered state with an isotropic quadrupole moment which does not manifest electrical properties in an equilibrium liquid. The polarization properties of the liquid are excited by an inhomogeneous distribution of the superfluid flow. The Landau equations of superfluid hydrodynamics are supplemented by an equation relating the electrical polarization of the medium with the density of the superfluid component and the inhomogeneity of the superfluid flow. The value of the expected oscillations of the electrical potential in a second-sound wave agrees with the measurements of A. S. Rybalko, Fiz. Nizk. Temp. 30, 1321 (2004) [Low Temp. Phys. 30, 994 (2004)].

Quadrupole and scissors modes and nonlinear mode coupling in trapped two-component Bose-Einstein condensates

We theoretically investigate quadrupolar collective excitations in two-component Bose-Einstein condensates and their nonlinear dynamics associated with harmonic generation and mode coupling. Under the Thomas-Fermi approximation and the quadratic polynomial ansatz for density fluctuations, the linear analysis of the superfluid hydrodynamic equations predicts excitation frequencies of three normal modes constituted from monopole and quadrupole oscillations, and those of three scissors modes. We obtain analytically the resonance conditions for the second harmonic generation in terms of the trap aspect ratio and the strength of intercomponent interaction. The numerical simulation of the coupled Gross-Pitaevskii equations vindicates the validity of the analytical results and reveals the dynamics of the second harmonic generation and nonlinear mode coupling that lead to nonlinear oscillations of the condensate with damping and recurrence reminiscent of the Fermi-Pasta-Ulam problem.

Transverse Resonances in Aerogel with Liquid 4He

We present preliminary studies of transverse resonances in a thin disk of aerogel filled with normal and superfluid 4 He in the temperature range 1.4 to 3 K. We observed a broad temperature independent mode in the normal phase and three narrow critical modes in the superfluid phase. The system was modeled by combining the equations of superfluid hydrodynamics of helium with those of elasticity of aerogel. Analytical solutions were obtained for a resonator of square profile and two types of boundary conditions at the transducers/aerogel interface. Comparison of the model solutions with the experimental data showed that the dynamics of the oscillation was dominated by compression rather than shear as in pure transverse sound. Recommendations for future improvements are made.

A nonlinear wave equation for second sound in 3He– 4He mixtures

In the vicinity of the tricritical point second sound propagation becomes very nonlinear, primarily as a result of the small superfluid fraction. Applying a reductive perturbation scheme to the full superfluid hydrodynamic equations we obtain a nonlinear wave equation of the Burgers type, and an explicit expression for the nonlinear coefficient.

Interaction of Intense Heat Pulses and Vortices in He II

Dynamics of intense heat pulses propagating in He II and interacting with quantum vortices induced by these pulses has been investigated using numerical methods based on equations of superfluid turbulence hydrodynamics. Investigation has been carried out for the plane geometry up to the second order in amplitude of pulses. The initial generation of vortex filaments has been simulated by the generating term in the Vinen equation. The results on pulse evolution for various heat fluxes are presented for different temperatures. It has been shown that the Feynman–Vinen theory is applicable in the phase-transition region. The results obtained have been compared to experimental data.

Dynamic Measurements of Thermal Transport Coefficients and Boundary Resistance. II.Model for 3He-Superfluid 4He Mixtures

This is the second of a three-part study of the ac response of liquid helium. We derive the temperature response function, ΔT(ω), of a3He-superfluid4He mixture from the equations of superfluid hydrodynamics in the presence of two interfacial boundary resistances,Rb.Specifically, we consider the response ΔT(ω), across a fluid layer of thickness,d, to an ac heat flux,Q(t) = Qoexp(iωt).ΔT(ω) depends on the effective thermal conductivity, κeff, Griffin's diffusion coefficient, Γo(i.e. the thermal diffusivity of3He impurities, Disoin the low3He concentration limit) and the thermal boundary resistance, 2Rb. This analysis provides the basis for experiments to determine these parameters. Although past experiments to measure these properties have been carried out using dc and transient techniques, an ac technique offers significant noise reduction over these techniques. By sweeping the frequency, it is possible for an experimenter to clearly identify different components of the system response to the heat flux. For instance, if τt is the slowest fluid thermal response time, conventional Kapitza boundary effects dominate at frequencies, ωτ≫1. These calculations reveal an interesting analogy to the “Piston Effect” for near-critical classical fluids. In Part I of this work, we used normal liquid4He as a testing ground for developing models of ac heat transport. In Part III of this work, we will present results in which we apply this technique to measurements on dilute mixtures of3He in superfluid4He.

Magnus force in superfluids and superconductors

The forces on the vortex, transverse to its velocity, are considered. In addition to the superfluid Magnus force from the condensate (superfluid component), there are transverse forces from thermal quasiparticles and external fields violating the translational invariance. The forces between quasiparticles and the vortex originate from interference of quasiparticles with trajectories on the left and on the right from the vortex like similar forces for electrons interacting with the thin magnetic-flux tube (the Aharonov-Bohm effect). These forces are derived in the Born approximation for phonons from the equations of superfluid hydrodynamics, and for BCS quasiparticles from the Bogolyubov-de Gennes equations. The effect of external fields violating translational invariance is analyzed for vortices in the two-dimensional Josephson junction array. The symmetry analysis of the classical equations for the array shows that the total transverse force on the vortex vanishes. Therefore the Hall effect which is linear in the transverse force is absent also. This means that the Magnus force from the superfluid component exactly cancels with the transverse force from the external fields. The results of other approaches are also brought together for discussion.