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Elementary excitations and quantized vortices in bosonic superfluids

Alessandro Lenoci

Published Aug 24, 2018• 0 citations
OpenAlex

Abstract

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.

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The spectrum of elementary excitations in the system of interacting Bose- and Fermi-particles was obtained at a temperature of absolute zero (Dose- condensation being taken into account by means of the Bogolyabov canonic transformations. In terms of a model of permeable spheres'' with finite repulsion it(is shown that the boson spectrum omega (p) in a definite region of parameter has the minimum at p = 0. By selecting parameters one succeeds in obtaining a quite satisfactory quantitative agreement of the theoretical and experimental spectra for superfluid helium. In the case when the Fermi energy is considerably higher than the bosorins maximum energy, a gap characterizing bond energy of the copper pairs can appear in the fermion speetrum due to exchange of virtual supercondensate'' bosons. If in this case the interaction between fermions at infinity is of attraction character, then under certain conditions sustained longitudinal collective excitations of Pose-type with a finite gap'' in the energy spectrum may exist in the superfluid fermion subsystem. Possibility of appearance of similar excitations in superfluid nuclear matter of neutron stars (pulsars) due to gravitational interaction between nucleons is discussed. (auth)

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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.
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.
Here, it was shown that the chain of equations, which is similar to the Bogoliubov chain in the theory of classical gases, may be obtained for the quasi-averages of the products of field operators in the system of Bose particles. The closed system of dynamic equations, which accounts for both the one-particle condensate and pair correlations, was obtained at zero temperature for the case where it is sufficient to confine ourselves to taking into account the quasi-averages of one field operator and the products of two field operators. The spectrum of excitations in a spatially homogeneous state was studied, and it was shown that this spectrum has two branches: the sound branch and the branch with an energy gap at zero momentum. The possibility of the existence of quasiparticle excitations with an energy gap in superfluid helium was discussed in connection with experiments on the absorption of microwave radiation.
The paradoxes and disparities in the contemporary microscopic theory of superfluid helium (He–II) are discussed along with possible ways of resolving them by taking pair correlations of He4 atoms into consideration. It is shown that most paradoxes are associated with the commonly accepted initial assumption concerning the dominating role of single-particle Bose condensate (SPBC) in the quantum microstructure of the superfluid component ρs. The existence of intensive SPBC leads to a strong hybridization of the elementary excitation branches and to a common dispersion law for all boson branches, which is identified with the quasiparticle spectrum E(p) observed experimentally from slow neutron scattering in liquid helium. However, the stability of this spectrum during a transition through the λ-point and the large value of the gap in the vicinity of the “rotonic” minimum contradict both the Landau theoretical criterion of superfluidity and the small value of experimentally measured critical velocity. At the same time, a strong interaction between particles in the Bose liquid He4 strongly suppresses the SPBC which amounts to less than 1% of all He4 atoms and hence cannot be the main constituent of the superfluid component, unlike the case of a weakly nonideal Bose gas. Moreover, for a quite strong attraction between particles in a certain region of the momentum space, bound pairs of bosons can be formed in the superfluid Bose liquid, and a coherent pair condensate (CPC) analogous to the Cooper pair condensate in superconductors may appear. Such a strong CPC may completely suppress the weak SPBC. In this case, the one-particle spectrum ε(p) of elementary excitations does not hybridize with the collective (two-particle) spectrum and does not appear in the structure of the dynamic form factor S(p,ε), i.e., does not coincide with the spectrum measured from neutron scattering. The dispersion of one-particle spectrum is defined by the momentum dependence of the pair order parameter Ψ̃(p) and may have a minimum or a point of inflection at p≠0. This peculiarity in the one-particle spectrum of a Bose liquid with CPC but without SPBC vanishes together with Ψ̃(p) at the temperature Tc=Tλ of the phase transition from the superfluid to the normal state (unlike the rotonic minimum in the collective spectrum), while the corresponding critical velocity vc=min[ε(p)/p] vanishes at the λ-point in accordance with the Landau criterion and the experimental data. The assumption that the strong “Cooper-like” CPC is responsible for the quantum structure of the superfluid component ρs is confirmed indirectly by the successful application of the Justrow approximation (based on strong pair correlations) for describing the properties of liquid He4 and quantum liquid mixtures He3–4He on one hand, and by an anomalously large effective mass of He3 impurity atoms in He4, which is approximately equal to total mass of He3 and He4 atoms, thus pointing to the existence of helium atoms in superfluid liquid He–II. The value of the superfluid velocity circulation quantum in the Onsager–Feynman vortices in a Bose liquid with CPC but without SPBC is discussed as well as the critical velocities of superfluid He4 in ultrathin films and channels in which the creation and motion of quantum vortices are ruled out, and the quasiparticle spectrum undergoes dimensional quantization.
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 this chapter a basic introduction to the theory of vortices in ultra-cold (superfluid) atomic gases is given. The main focus will be on bosonic atomic gases, since these contain the same basic physics, but with simpler formulas. Towards the end of the chapter, the difference between bosonic and fermionic atomic gases is discussed. This discussion will allow the reader to make the conceptual step from bosonic to fermionic gases, while pinpointing the main differences and difficulties when working with fermionic gases rather than bosonic gases. The goal of this chapter is to provide a good and general starting point for researchers, or other interested parties, who wish to start exploring the physics of ultra-cold gases.
The spectrum of elementary excitations in the system of interacting Bose- and Fermi-particles was obtained at a temperature of absolute zero (Dose- condensation being taken into account by means of the Bogolyabov canonic transformations. In terms of a model of permeable spheres'' with finite repulsion it(is shown that the boson spectrum omega (p) in a definite region of parameter has the minimum at p = 0. By selecting parameters one succeeds in obtaining a quite satisfactory quantitative agreement of the theoretical and experimental spectra for superfluid helium. In the case when the Fermi energy is considerably higher than the bosorins maximum energy, a gap characterizing bond energy of the copper pairs can appear in the fermion speetrum due to exchange of virtual supercondensate'' bosons. If in this case the interaction between fermions at infinity is of attraction character, then under certain conditions sustained longitudinal collective excitations of Pose-type with a finite gap'' in the energy spectrum may exist in the superfluid fermion subsystem. Possibility of appearance of similar excitations in superfluid nuclear matter of neutron stars (pulsars) due to gravitational interaction between nucleons is discussed. (auth)