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Oxford University Press eBooks

Elementary Bosons

Monique Combescot, Shiue-Yuan Shiau

Published Dec 1, 2015• 0 citations
Publisher

Abstract

Chapter 17 first considers noninteracting elementary bosons for zero and finite temperatures and then discusses the effects of temperature and space dimension on Bose-Einstein condensation. Condensation of massive bosons in a potential trap is also discussed. Next, the chapter discusses condensate fragmentation, showing that the condensate stability against momentum fragmentation requires the introduction of a boson-boson repulsive interaction: exchange processes between the two parts of the fragmented condensate provide a positive extensive contribution to the N-boson energy, which stabilizes the unfragmented condensate. By contrast, spin fragmentation can occur: the condensate is “spin fragmented,” that is, unpolarized, for same-spin bosons undergoing repulsive interactions. The effect of interaction on the elementary boson ground-state energy is also presented, with emphasis on the physical origin of the singularity of the boson correlation energy, and its similarity with the singularity of the electron correlation energy.

DOI
10.1093/acprof:oso/9780198753735.003.0017

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Chapter 17 first considers noninteracting elementary bosons for zero and finite temperatures and then discusses the effects of temperature and space dimension on Bose-Einstein condensation. Condensation of massive bosons in a potential trap is also discussed. Next, the chapter discusses condensate fragmentation, showing that the condensate stability against momentum fragmentation requires the introduction of a boson-boson repulsive interaction: exchange processes between the two parts of the fragmented condensate provide a positive extensive contribution to the N-boson energy, which stabilizes the unfragmented condensate. By contrast, spin fragmentation can occur: the condensate is “spin fragmented,” that is, unpolarized, for same-spin bosons undergoing repulsive interactions. The effect of interaction on the elementary boson ground-state energy is also presented, with emphasis on the physical origin of the singularity of the boson correlation energy, and its similarity with the singularity of the electron correlation energy.
Chapter 16 introduces the last part that addresses the bosonic condensation of composite bosons. To bridge the gap between elementary bosons and composite bosons, elementary fermion systems like semiconductor electrons and holes are also studied. Whereas the previous chapters provided a fairly precise knowledge of bosonic particles relevant for condensed matter physics, namely, Wannier excitons, Frenkel excitons, Cooper pairs, biexcitons, and polaritons, this chapter addresses an important question of current interest: what is the ground state of N composite bosons from the perspective of Bose-Einstein condensation? Or rather, how does the genuine Bose-Einstein condensation for elementary bosons persist in composite boson systems? This chapter lists important results with respect to condensate momentum fragmentation or spin fragmentation, correlation energy, and the dark nature of exciton condensate, to emphasize the connection between elementary bosons, elementary fermions, elementary fermion pairs, and composite bosons, from the composite boson perspective of their condensation.
Chapter 19 addresses bosonic condensation in composite bosons. It first considers the coboson ground-state energy, which is analytically known by now in the case of N Cooper pairs interacting via the reduced BCS potential but not yet known for N Wannier or Frenkel excitons, owing to the complexity of the Coulomb potential. Still, the chapter provides hints on how to derive the density linear term of the N-exciton ground-state energy in the case of Wannier excitons. It also discusses the role of the exciton composite nature in condensate momentum fragmentation, spin fragmentation, and condensate brightness. Indeed, the excitons that have the lowest energy and thus are the ones expected to condense are not coupled to light; so, the exciton condensate must be dark. However, a bright exciton condensate can appear, coherent with the dark exciton condensate, above a density threshold, owing to coupling induced by carrier exchange between dark excitons and bright excitons.
Repulsive Bose-Einstein condensates immersed into a double-well trap potential are studied within the framework of the recently introduced mean-field approach which allows for bosons to reside in several different orthonormal orbitals. In the case of a one-orbital mean-field theory (Gross-Pitaevskii) the ground state of the system reveals a bifurcation scenario at some critical values of the interparticle interaction and/or the number of particles. At about the same values of the parameters the two-orbital mean field predicts that the system becomes twofold fragmented. By applying the three-orbital mean field we verify numerically that for the double-well external potential studied here the overall best mean field is achieved with two orbitals. The variational principle minimizes the energy at a vanishing population of the third orbital. To discuss the energies needed to remove a boson from and the energies gained by adding a boson to the condensate, we introduce boson ionization potentials and boson affinities and relate them to the chemical potentials. The impact of the finite number of bosons in the condensate on these quantities is analyzed. We recall that within the framework of the multiorbital mean-field theory each fragment is characterized by its own chemical potential. Finally, the stability of fragmented states is discussed in terms of the boson transfer energy which is the energy needed to transfer a boson from one fragment to another.
The two-site version of the Bose-Hubbard model can be used as a simplified model of N bosons confined to two quantum wells. We study the behavior of low-temperature Bose-Einstein condensates as a function of the strength of the repulsive interactions between the bosons and of the size of the quantum-mechanical tunneling between the quantum wells. The ground-state of weakly interacting bosons tends to be one in which almost all particles are in the same quantum-mechanical state (‘Bose-Einstein Condensate’, or ‘BEC’). It is found that even very small amounts of interwell tunneling lead to a BEC, but that zero tunneling leads to two separate BECs, one in each well (‘fragmentation’). We study systematically the transition between a single BEC comprised of bosons in both wells, and a fragmented condensate consisting of two BECs, one for each well. Our analysis will use matrix computational methods to solve the Schrödinger equation of the two-well boson system, and obtain the energies and wavefunctions as eigenvalues and eigenvectors of the Hamiltonian matrix. We will compare the results obtained with previous analyses, and against analytical approximations developed for the limiting case of a large number of bosons (large N).
Intensive theoretical studies have recently predicted that a Bose-Einstein condensate will exhibit a variety of novel properties if spin-orbit coupling is present. However, an unambiguous fact has also been pointed out: Rashba coupling destroys a condensate of noninteracting bosons even in high dimensions. Therefore, a conceptually important question arises as to whether or not a condensate exists in the presence of interaction and a general type of spin-orbit coupling. Here we show that interaction qualitatively changes the ground state of bosons under Rashba spin-orbit coupling. Any infinitesimal repulsion forces bosons either to condense at one or two momentum states or to form a superfragmented state that is a superposition of infinite numbers of fragmented condensates. The superfragmented state is unstable against the anisotropy of spin-orbit coupling in systems with large numbers of particles, leading to the revival of a condensate in current experiments.
Bose-Einstein condensation is a quantum statistical phase transition which was theoretically predicted almost a hundred years ago. After years of seminal research, physicists realized the first almost ideal Bose-Einstein condensates in ultracold dilute atomic gases in 1995. Since then, the theoretical and experimental methods concerning such systems have been developing rapidly, and many fascinating phenomena have been found in these novel quantum systems.\n\nBose-Einstein condensation occurs in a system consisting of massive bosons when a single quantum state becomes macroscopically occupied as the temperature is lowered below the transition temperature. In general, condensates consisting of repulsively interacting bosons exhibit superfluidity: Particle currents can flow in the system without dissipation and viscosity. Moreover, the velocity fields of condensates have to be irrotational, which severely restricts the rotational characteristics of these systems. Apart from the center of mass motion, the system may carry angular momentum in the form of elementary excitations or so-called quantized vortices.\n\nThis Thesis is a theoretical study of subjects related to stability and dynamics of quantized vortices in dilute atomic Bose-Einstein condensates. The precession and instability of off-centered vortices in trapped condensates is investigated both in the zero-temperature limit and at finite temperatures. Dynamical stability of multiply quantized vortices and vortex clusters is studied in axisymmetric trap geometries. Splitting of energetically and dynamically unstable multiply quantized vortices into singly quantized vortices is also studied. Finally, as a separate subject, tunneling of a condensate through a potential barrier is investigated. Majority of this work relies on numerical methods for solving the Gross-Pitaevskii and Bogoliubov equations, which are of central importance in the study of dilute atomic Bose-Einstein condensates.