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Progress of Theoretical Physics

Theory of Bose-Condensed Quantum Crystal

Makio Uwaha

Published May 1, 1978• 0 citations
Publisher

Abstract

Superfluidity and long wavelength excitation in Bose-condensed quantum crystal are discussed. The expression for superfluid density is derived in the framework of lattice model by a projection method. In the case of particles with hard core a modified quantum lattice model is used to derive four sound-like excitations. They are coupled modes of three phonons and a mode related to the breakdown of gauge symmetry.

DOI
10.1143/ptp.59.1461

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Superfluidity and long wavelength excitation in Bose-condensed quantum crystal are discussed. The expression for superfluid density is derived in the framework of lattice model by a projection method. In the case of particles with hard core a modified quantum lattice model is used to derive four sound-like excitations. They are coupled modes of three phonons and a mode related to the breakdown of gauge symmetry.
A theory of collective excitations in Bose-Einstein condensation in a trap is developed based on the quantum Hamilton-Jacobi equation of Bohm and the phase coherence along with the idea of off-diagonal long range order of Penrose and Onsager. First, we show that a free surface behaves like a normal fluid - a breakdown of superfluidity. Second, inside the free surface it is shown that the spectrum of phonons is of the form ω=ckω=ckω=ck scaled with the external potential, where the speed of (first) sound, c=[4πaρℏ2]1/2/Mc=[4πaρ\hbar^{2}]^{1/2}/Mc=[4πaρℏ2]1/2/M and kkk is the wave number. Third, in the limit a→0a\to 0a→0, the hard spheres in the Bose-Einstein condensation collapse to a close-packed classical lattice with the zero-point vibrational motion about fixed points.
The spectrum of phonon-like collective excitations in the system of Bose-atoms in optical lattice (more generally, in the system of quantum particles described by the Bose-Hubbard model) is investigated.Such excitations appear due to displacements of particles with respect to their local equilibrium positions.The two-level model taking into account the transitions of bosons between the ground state and the first excited state in potential wells, as well as interaction between them, is used.Calculations are performed within the random phase approximation in the hard-core boson limit.It is shown that excitation spectrum in normal phase consists of the one exciton-like band, while in the phase with BE condensate an additional band appears.The positions, spectral weights and widths of bands strongly depend on chemical potential of bosons and temperature.The conditions of stability of a system with respect to the lowering of symmetry and displacement modulation are discussed.
The dynamics of the collective excitations of a lattice Bose gas at zero temperature is systematically investigated using the time-dependent Gutzwiller mean-field approach. The excitation modes are determined within the framework of the linear-response theory as solutions of the generalized Bogoliubov--de Gennes equations valid in the superfluid and Mott-insulator phases at arbitrary values of parameters. The expression for the sound velocity derived in this approach coincides with the hydrodynamic relation. We calculate the transition amplitudes for the excitations in the Bragg scattering process and show that the higher excitation modes make significant contributions. We simulate the dynamics of the density perturbations and show that their propagation velocity in the limit of week perturbation is satisfactorily described by the predictions of the linear-response analysis.
We begin our study of the superfluid Bose liquid with a brief review of the experimental and theoretical background of the liquid 4 He, the only Bose liquid found naturally in the laboratory. We then consider, in Chapter 2 , the quasi-particle excitations and show how at T = 0 the macroscopic occupation of the zero momentum state leads, in the long wavelength limit, to an identical excitation spectrum for quasi-particles and density fluctuations. The spectrum consists in phonons with a velocity given by the macroscopic velocity of sound. We turn in Chapter 3 to a consideration of the observed elementary excitation spectrum of He II. We discuss both the experimental measurements and the theories that have thus far been proposed to explain them.