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.
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.
Abstract Chapter 2 describes the basic knowledge of Bose-condensed systems in order to study the quantum hydrodynamics for both the superfluid helium and the atomic-gas BECs in the following sections. Starting from the field theoretical formulation, we will first give a self-contained explanation of the derivation of the most fundamental GP equation for weakly interacting Bose gases at zero temperature. We will also highlight the theoretical treatment for incorporating the concept of elementary excitations to describe low-energy excitations of Bose systems, the definition of normal and superfluid components, and eventually the celebrated model of two-fluid hydrodynamics. Finally, the microscopic structure and dynamics of topological excitations obtained from the GP equation are discussed by exemplifying solitons, vortex lines, and vortex rings.
Abstract This chapter is devoted to discussing relevant dynamical and superfluid phenomena exhibited by interacting Fermi gases at low temperatures, with special focus on the strongly interacting Fermi gas at unitarity. The chapter discusses the predictions of hydrodynamic theory concerning the propagation of sound, the collective oscillations and the expansion of the gas after release of the trap. The Landau criterion for superfluidity and the behaviour of the dynamic structure factor are discussed. It also provides some considerations on the radiofrequency transitions as a method to observe pair-breaking transitions and to measure the pairing gap. The chapter discusses the behaviour of two-fluid hydrodynamics and the propagation of second sound as a tool to measure the superfluid density of interacting Fermi gases. It finally provides a brief summary of the rotational properties of the superfluid Fermi gas, like the quenching of the moment of inertia and the occurrence of quantized vortices.
This thesis examines the manifestation of the critical velocity for superfluidity in dilute gas Bose-Einstein condensates (BECs). Superfluidity is a macroscopic quantum phenomenon characterised by unique properties such as frictionless flow, high thermal conductivity and the presence of quantised vortices. Our study focuses on what is arguably the most well-known attribute: the ability for an object to move through a superfluid without experiencing a drag force when travelling below a threshold critical velocity.The Landau criterion for superfluidity predicts the critical velocity in a BEC is equal to the speed of sound, and this conclusion is supported by a large body of theoretical work. However, a number of experiments have observed lower critical velocities than theory predicts. In the first part of this thesis, we numerically model experiments on the superfluid critical velocity of an elongated, harmonically trapped Bose-Einstein condensate as reported by [P. Engels and C. Atherton, Phys. Rev. Lett. 99, 160405 (2007)]. These experiments swept a repulsive obstacle, formed by a optical dipole potential, through the long axis of a condensate at constant velocity. Their results found an increase in the resulting density fluctuations of the condensate above an obstacle velocity of v _ 0:3 mm s
Abstract Chapter 4 addresses the progress of studies on superfluid dynamics, mainly related to the dynamics of quantized vortices, in cold-atom BECs. First, we give a minimal description of the physics of cold atoms, e.g. cooling, trapping, control of interatomic interactions by external fields. We then turn to the argument of superfluid hydrodynamics in a scalar BEC characterized by a single-component order parameter. The properties of the vortices can be well described on the basis of the GP model, which allows a quantitative comparison between theory and experiment. We will discuss topics such as single vortex dynamics, interacting dynamics of a few vortices, vortex lattice formation in a rotating BEC, and collective properties of a vortex lattice in a rapidly rotating BEC, which have been unraveled experimentally by direct visualization of vortices in condensed gases. Next, we will discuss the properties of vortices in multicomponent BECs characterized by the multicomponent order parameters. We will first explain the vortex physics in two-component BECs as a simple case, finding that their structures and dynamics are highly nontrivial due to the presence of the intercomponent interaction; we will cover the topics of vortex structure composed of two-component order parameters, dynamics of composite vortices, exotic vortex lattices, and hydrodynamic instabilities associated with the interface dynamics between immiscible condensates. Finally, we explain the vortices in a spinor BEC. The vortex structures are extremely rich due to the complicated order parameter manifold, which is clarified by the discussion on vortices in spin-1 and spin-2 spinor BECs.
Abstract Chapter 13 addresses Bose condensation in superfluids (and superconductors), which involves the field operator ψ having a c-number component (<ψ(x,t)>≠0), challenging number conservation. The nonlinear Gross-Pitaevskii equation is derived for this condensate wave function <ψ>=ψ−ψ˜, facilitating identification of the coherence length and the core region of vortex motion. The noncondensate Green’s function G˜1(1,1′)=−i<(ψ˜(1)ψ˜+(1′))+> and the nonvanishing anomalous correlation function F˜∗(2,1′)=−i<(ψ˜+(2)ψ˜+(1′))+> describe the dynamics and elementary excitations of the non-condensate states and are discussed in conjunction with Landau’s criterion for viscosity. Associated concepts of off-diagonal long-range order and the interpretation of <ψ> as a superfluid order parameter are also introduced. Anderson’s Bose-condensed state, as a phase-coherent wave packet superposition of number states, resolves issues of number conservation. Superconductivity involves bound Cooper pairs of electrons capable of Bose condensation and superfluid behavior. Correspondingly, the two-particle Green’s function has a term involving a product of anomalous bound-Cooper-pair condensate wave functions of the type F(1,2)=−i<(ψ(1)ψ(2))+>≠0, such that G2(1,2;1′,2′)=F(1,2)F+(1′,2′)+G˜2(1,2;1′,2′). Here, G˜2 describes the dynamics/excitations of the non-superfluid-condensate states, while nonvanishing F,F+ represent a phase-coherent wave packet superposition of Cooper-pair number states and off-diagonal long range order. Employing this form of G2 in the G1-equation couples the condensed state with the non-condensate excitations. Taken jointly with the dynamical equation for F(1,2), this leads to the Gorkov equations, encompassing the Bardeen–Cooper–Schrieffer (BCS) energy gap, critical temperature, and Bogoliubov-de Gennes eigenfunction Bogoliubons. Superconductor thermodynamics and critical magnetic field are discussed. For a weak magnetic field, the Gorkov-equations lead to Ginzburg–Landau theory and a nonlinear Schrödinger-like equation for the pair wave function and the associated supercurrent, along with identification of the Cooper pair density. Furthermore, Chapter 13 addresses the apparent lack of gauge invariance of London theory with an elegant variational analysis involving re-gauging the potentials, yielding a manifestly gauge invariant generalization of the London equation. Consistency with the equation of continuity implies the existence of Anderson’s acoustic normal mode, which is supplanted by the plasmon for Coulomb interaction. Type II superconductors and the penetration (and interaction) of quantized magnetic flux lines are also discussed. Finally, Chapter 13 addresses Josephson tunneling between superconductors.
Abstract This chapter is devoted to discussing some key superfluid properties exhibited by dilute Bose gases confined in external traps. Key issues concern the behaviour of the critical velocity and the applicability of Landau’s criterion for superfluidity, the moment of inertia and its quenching caused by the irrotationality of the superfluid motion, the scissors mode, and the expansion of a rotating superfluid cloud. Rotational effects at high angular velocities, with the appearance of quantized vortices and vortex lattices, are also considered. The chapter also discusses the consequences for vortices on the collective oscillations of the trapped gas, the stability and the precession of vortex lines, and the nature of the superfluid flow in toroidal traps.