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Soviet Journal of Low Temperature Physics

Hydrodynamics of superfluid crystals

G. A. Vardanyan, K. V. Papoyan, Д. М. Седракян

Published Feb 1, 1984• 0 citations
Publisher

Abstract

It is shown that three-velocity hydrodynamics equations describing the properties of a two-condensate crystal determine the low-frequency spectrum with allowance for superfluid drag. The drag on one superfluid component of density ρ12(s) from another component of density ρ22(s), gives rise to two branches of vibrations of frequencies ω1 and ω2, unlike the case of a one-condensate crystal. The absorption coefficient for transverse sound in a one-condensate crystal is expressed in terms of the quantum-mechanical characteristic quantity that describes the tunneling of atoms.

DOI
10.1063/10.0031056

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It is shown that three-velocity hydrodynamics equations describing the properties of a two-condensate crystal determine the low-frequency spectrum with allowance for superfluid drag. The drag on one superfluid component of density ρ12(s) from another component of density ρ22(s), gives rise to two branches of vibrations of frequencies ω1 and ω2, unlike the case of a one-condensate crystal. The absorption coefficient for transverse sound in a one-condensate crystal is expressed in terms of the quantum-mechanical characteristic quantity that describes the tunneling of atoms.
The equations of motion for the reduced density matrices of an interacting Bose system are treated using the concept of off-diagonal long-range order. The equations are decoupled on the one-particle level. With this approximation, the hydrodynamic equations for the densities and the velocities of the condensate and the depletion are derived. The interaction between the two fluid components depends on density variations and on the relative velocity. The latter interaction terms appear as kinetic pressure terms and a Magnus force term. Via the equation for the total momentum, a connection between the velocities of the condensate and the depletion, and the velocities in Landau's two-fluid model is derived. The excitation spectrum of the system is investigated in the phonon region. A comparison of the resulting velocities for the first and second sound with the experimentally determined ones shows that the functional dependence of the condensate density on temperature is similar to that of the superfluid component.
A system of hydrodynamic equations has been obtained for superfluid Fermi systems with singlet pairing and nonzero momentum which are formed, e.g., in the case of small polarization of the spin system. As a result, the energy gap is a periodic function of the coordinates and a spin-density wave is formed. For such systems the complete set of dynamic variables includes both purely hydrodynamic (superfluid and normal) velocities and the displacement vector corresponding to the violations of the translational invariance. All of the tensors of normal and superfluid densities that appear in the system as well as the elastic moduli have been calculated. A dispersion relation has been obtained for thermal vibrations (second sound) and elastic vibrations, and this relation has turned out to be substantially anisotropic. Observation of the predicted effects is most probable in 3He–He II solutions in magnetic fields of the order of several kilooersteds at low temperatures.
The quantum mechanics of a system of identical interacting particles must lead to the classical hydrodynamic equations of motion at high temperatures, because of the correspondence principle. On the other hand, the behaviour of helium II shows that this is not always the case at low temperatures. In this paper it is shown that in certain cases the quantum description requires an extra parameter, which is the potential of a new velocity field superimposed on the classical motion. Expressed in semi-classical terms, the condition for the existence of this new parameter is that the probability, in the equilibrium state, of a particle having a very large de Broglie wavelength (that is, a negligible momentum) is finite. This condition is satisfied in one model of a superfluid system, a condensed Bose-Einstein gas, but not in a crystal. A tentative theoretical interpretation of two basic equations of the empirical two-fluid theory of helium II is given, in which this new parameter determines the velocity of the superfluid.
We investigate a model of a two-component Bose-Einstein condensate residing in an optical lattice. Within a Bogoliubov approach at the mean-field level, we derive exact analytical expressions for the excitation spectrum of the two-component condensate when taking into account hopping and interactions between arbitrary sites. Our results thus constitute a basis for works that seek to clarify the effects of higher-order interactions in the system. We investigate the excitation spectrum and the two branches of superfluid velocity in more detail for two limiting cases of particular relevance. Moreover, we relate the hopping and interaction parameters in the effective Bose-Hubbard model to microscopic parameters in the system, such as the laser light wavelength and atomic masses of the components in the condensate. These results are then used to calculate analytically and numerically the drag coefficient between the components of the condensate. We find that the drag is most effective close to the symmetric case of equal masses between the components, regardless of the strength of the intercomponent interaction and the lattice well depth.
The phenomena of superfluidity and superconductivity are intimately connected with the existence of a condensate, a macroscopically occupied quantum state. Such condensates occur in a variety of different physical systems, as described in Chapter 1. The foundation for the description of superfluidity is a picture of the system as being comprised of a condensate and elementary excitations. In Chapter 8 we have seen how physical properties such as the energy and the density of a Bose–Einstein condensed system may be expressed in terms of a contribution from the condensate, plus one from the elementary excitations, and in this chapter we shall consider further developments of this basic idea to other situations. As a first application, we determine the critical velocity for creation of an excitation in a homogeneous system (Sec. 10.1). Following that, we show how to express the momentum density in terms of the velocity of the condensate and the distribution function for excitations. This provides the basis for a two-component description, the two components being the condensate and the thermal excitations (Sec. 10.2). In the past, this framework has proved to be very effective in describing the properties of superfluids and superconductors, and in Sec. 10.3 we apply it to dynamical processes. To describe the state of a superfluid, one must specify the condensate velocity, in addition to the variables needed to characterize the state of an ordinary fluid. As a consequence, the collective behaviour of a superfluid is richer than that of an ordinary one. Collective modes are most simply examined when excitations collide frequently enough that they are in local thermodynamic equilibrium.
Abstract This chapter presents a general discussion of the phenomenon of superfluidity, starting from the famous Landau criterion for the critical velocity. The Landau expression for the normal (non-superfluid) component of the gas, expressed in terms of the elementary excitations of the system, is derived. The relationship between Bose–Einstein condensation and the phenomenon of superfluidity is outlined, with special focus on the role played by the phase of the order parameter, whose gradient fixes the velocity of the superfluid. The chapter derives the hydrodynamic equations of superfluids at both zero and finite temperatures and discuss the solutions for first and second sound. The formalism of quantum hydrodynamics is presented, with an application to the Beliaev decay of phonons and to the calculation of the fluctuations of the phase. It also discusses the consequences for superfluidity on the rotational properties and the spatial structure of vortex lines.