In addition to the acoustic (phonon) excitations described above, specific excitations called ‘quantum vortices’ may exist in superfluid Bose systems. Attempts to incorporate the vortex excitations have already been made in the original version of Landau's theory (Landau, 1941). The existence of a periodical lattice of quantum vortices in a superconductor located in a magnetic field was conjectured by Abrikosov (1952, 1957). The idea of quantum vortices occurring in rotating helium below the λ-point was introduced independently by Onsager (1949) and Feynman (1955). In this section we present the description of quantum vortices in the functional integral formalism. We will also discuss the role of quantum vortices in the phase transition for the superfluid to the normal state. Our starting point is the functional integration scheme over fast and slow variables. We will apply this formalism mainly to two-dimensional Bose systems and will try to take into account vortex-like configurations of the Bose field. We thus come to the conclusion that the system of vortices is equivalent to the system of charged particles interacting via a new field which play the role of electromagnetic field. At small temperatures there are only pairs with opposite signs, bounded by the long-range logarithm potential. It turns out that the phase transition in the two-dimensional Bose system is nothing but the dissociation of bounded pairs. A similar approach to the three-dimensional systems leads to the conclusion that phase transition is accompanied here by creation of long vortex filaments.
In addition to the acoustic (phonon) excitations described above, specific excitations called ‘quantum vortices’ may exist in superfluid Bose systems. Attempts to incorporate the vortex excitations have already been made in the original version of Landau's theory (Landau, 1941). The existence of a periodical lattice of quantum vortices in a superconductor located in a magnetic field was conjectured by Abrikosov (1952, 1957). The idea of quantum vortices occurring in rotating helium below the λ-point was introduced independently by Onsager (1949) and Feynman (1955). In this section we present the description of quantum vortices in the functional integral formalism. We will also discuss the role of quantum vortices in the phase transition for the superfluid to the normal state. Our starting point is the functional integration scheme over fast and slow variables. We will apply this formalism mainly to two-dimensional Bose systems and will try to take into account vortex-like configurations of the Bose field. We thus come to the conclusion that the system of vortices is equivalent to the system of charged particles interacting via a new field which play the role of electromagnetic field. At small temperatures there are only pairs with opposite signs, bounded by the long-range logarithm potential. It turns out that the phase transition in the two-dimensional Bose system is nothing but the dissociation of bounded pairs. A similar approach to the three-dimensional systems leads to the conclusion that phase transition is accompanied here by creation of long vortex filaments.
The experimental realization of quantum-degenerate Bose gases made of atoms with sizeable magnetic dipole moments has created a new type of fluid, known as a quantum ferrofluid, which combines the extraordinary properties of superfluidity and ferrofluidity. A hallmark of superfluids is that they are constrained to rotate through vortices with quantized circulation. In quantum ferrofluids the long-range dipolar interactions add new ingredients by inducing magnetostriction and instabilities, and also affect the structural properties of vortices and vortex lattices. Here we give a review of the theory of vortices in dipolar Bose-Einstein condensates, exploring the interplay of magnetism with vorticity and contrasting this with the established behaviour in non-dipolar condensates. We cover single vortex solutions, including structure, energy and stability, vortex pairs, including interactions and dynamics, and also vortex lattices. Our discussion is founded on the mean-field theory provided by the dipolar Gross-Pitaevskii equation, ranging from analytic treatments based on the Thomas-Fermi (hydrodynamic) and variational approaches to full numerical simulations. Routes for generating vortices in dipolar condensates are discussed, with particular attention paid to rotating condensates, where surface instabilities drive the nucleation of vortices, and lead to the emergence of rich and varied vortex lattice structures. We also present an outlook, including potential extensions to degenerate Fermi gases, quantum Hall physics, toroidal systems and the Berezinskii-Kosterlitz-Thouless transition.
In this work we studied two sistems of bosonic superfluids at zero temperature,
ultra-cold Bose gases and superfluid helium, within a simple classical field formalism and we
underlined differences and similarities between them. We also analyzed how the presence
of strong interactions, as in the case of helium, modifies the quasiparticle excitation spectrum and
subsequently the vortex line density profile. After a general review about these bosonic systems properties
we started in general from the Hartree equation of a bosonic system and,
with different hypotesis on the interaction potential between bosons, we obtained the equations
describing the two above mentioned systems, mainly focusing on elementary
quasiparticle excitations and quantized vortices.
Furthermore throughout this work the results were compared with experimental
data and aftermaths of the main theories in literature.
In 1995 two different research groups observed for the first time the Bose-Einstein condensation (BEC) in ultracold gases. When the confining magnetic trap was turned off the gas was left free to expand, and the velocity of the particles showed a clear peak: most of the particles were occupying the same single particle state, the one of lowest energy. The Bose-Einstein condensation had been predicted in 1925 by Einstein, written by inspiration of a work on the statistic of the photons by Bose (1924). In this work Bose described the behavior of an ensemble of photons, treating them as massless particles, with no number conservation associated. Einstein extended this approach to particles with a mass and with fixed number, creating what is now called the Bose-Einstein distribution. The particles that follow such a description are called ``bosons'', as opposed to the ``fermions'' of the Fermi-Dirac statistics. Einstein predicted that in a gas of bosons - under a critical temperature - a finite fraction of the total number of particles would have been in the ground state, and act as a single entity. This amusing theoretical discovery found its utility a few years later. In the late thirties, new techniques allowed to cool Helium-4 at few Kelvins above the absolute zero. The properties of the resulting liquid were a puzzlement to the scientific community: among others, it could flow without experiencing friction. The liquid was called a ``superfluid''. A first explanation was given by London in 1938, which linked the superfluid behavior to the presence of a BEC among the bosonic Helium particles. The fermions cannot condense by themselves. On the other hand, they can form bound pairs and act as bosons, as it happens in a metal at low temperature. Using this approach, in 1957 Bardeen, Cooper and Schrieffer created a successful model of superconductivity by describing a superconductor as a superfluid in a charged system. During the course of these years we explored the superfluid properties of Bosons and Fermions in different settings. The original contributions of the thesis are described starting from the third chapter, where we speak about the generation and stability of solitons in a periodic optical lattices, both fixed or in motion. In the fourth chapter we study the generation of giant vortices in cold fermions, by using a generalized hydrodynamical approach. In chapter 5 we study the effect of a quasiperiodic lattice and the glassy phase it produces on a gas of bosons. Finally, we study the interaction of normal matter and superfluids with gravitational waves. While this interaction is seen to be extremely small, we believe that the resulting formalism is interesting by itself.
The aim of this paper is to derive the equations of motion of quantum vortices in a weakly non-ideal Bose–Einstein condensate (BEC) at finite temperatures and in the presence of macroscopic superfluid and normal velocities. The motivation for this work is that the dynamics of quantum vortices in superfluid helium has been studied at the phenomenological level, and a number of issues require clarification, which can be addressed by studying the corresponding problem in BEC. In BEC, quantum vortices are located at points where the macroscopic wave function (MWF) ψ(r,T), which describes the BEC dynamics, vanishes ψ(r,T)=0. The intersection of the surfaces Reψ(r,T)=0 and Imψ(r,T)=0 forms a one-dimensional line, which is just the quantum vortex line. During the evolution of the MWF, the position of the zero surfaces changes, that is, the vortex lines move. Conversely, the motion of these vortex filaments determines the evolution of the MWF. Thus, there is a mutual correspondence, and the quantity ψ(r,T) can be considered as a functional of the entire vortex configuration, i.e., {s(ξ,t)}. This viewpoint allows us to derive the equations of motion for vortices from the Gross–Pitaevskii equation (GPE), which describes the dynamics of MWF. The key finding is the confirmation that the equation derived from the fundamental quantum description (the GPE) for a BEC has the same mathematical structure as the equation derived from macroscopic, phenomenological reasoning for superfluid helium.
Abstract Vorticity in two-dimensional superfluids is subject to intense research efforts due to its role in quantum turbulence, dissipation and the BKT phase transition. Interaction of sound and vortices is of broad importance in Bose–Einstein condensates and superfluid helium. However, both the modelling of the vortex flow field and of its interaction with sound are complicated hydrodynamic problems, with analytic solutions only available in special cases. In this work, we develop methods to compute both the vortex and sound flow fields in an arbitrary two-dimensional domain. Further, we analyse the dispersive interaction of vortices with sound modes in a two-dimensional superfluid and develop a model that quantifies this interaction for any vortex distribution on any two-dimensional bounded domain, possibly non-simply connected, exploiting analogies with fluid dynamics of an ideal gas and electrostatics. As an example application we use this technique to propose an experiment that should be able to unambiguously detect single circulation quanta in a helium thin film.
Bose-Einstein condensation, first predicted by Bose and Einstein in 1924, was finally realized in dilute, weakly interacting gases of alkali atoms in 1995. A few years later, quantized vortices - intimately related to the phenomenon of superfluidity - were created and observed in these novel quantum many-body systems.\n\nIn this Thesis, vortices in dilute atomic Bose-Einstein condensates are studied theoretically and computationally using microscopic mean-field theory formalisms. Special emphasis is on the structure and the stability of vortices at finite temperatures. The formation of quantized circulation in Bose-Einstein condensates is investigated, as well as multiply quantized vorticity and the adiabaticity of the precessing motion of a vortex.\n\nThis research strongly relies on numerically solving the Gross-Pitaevskii equation for the macroscopic condensate wavefunction and the coupled Bogoliubov equations and their variants for the collective quasiparticle excitation eigenmodes. The computed elementary excitation spectra are used, for instance, in determining the local stability of various condensate states. The vortex formation process is studied at finite temperatures in terms of the surface modes of the condensate providing the Landau critical velocities for the breakdown of superfluidity. Free energy calculations are used for determining the thermodynamic equilibrium states.\n\nSingly quantized, axisymmetric vortex states are found to be locally energetically stable even in the zero-temperature limit within the self-consistent finite-temperature approximations. This would imply for the vortex a precession direction opposite to that observed in the experiment. Adiabatic approximation of quantum mechanics is formulated for Bose-Einstein condensed systems and is applied to a case of a moving vortex line. The experimentally observed precessing motion of the vortex is found to be nonadiabatic. A method to create stable multiply quantized vortices using an additional pinning potential is presented. The computed critical trap rotation frequencies for vortex nucleation at finite temperatures are in agreement with experiments and the predictions of zero-temperature field theories.