We evaluate the zero temperature properties and Bose-Einstein condensation (BEC) of hard sphere bosons in a harmonic trap using Quantum Monte Carlo methods. The chief purpose of this work is to go beyond the dilute limit, to test the limits of the Gross-Pitaevskii equation and related mean field approximations and to explore the role of interactions in determining the zero temperature properties of the trapped Bose gas. The density is increased by adjusting both the number of trapped bosons, N , and the s-wave scattering length, a, to study systems from the highly dilute region corresponding to early experiments in trapped BEC [2, 3] up to liquid He densities and beyond. Rather than relying on the perturbative Bogoliubov approximation (which assumes a large condensate fraction) to describe the condensate, we use the one-body density matrix (OBDM) formulation of BEC so that the properties of the condensate for systems with arbitrarily large interactions may be studied. In this formulation of BEC, condensate properties are obtained by diagonalizing the OBDM and obtaining the corresponding single particle “natural orbitals” and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single particle orbital(s) with macroscopic occupation (N0 >> 1). Within this framework, we calculate the effects of interactions and increased density on the ground state energy, the density profile, the momentum distribution, the condensate fraction and condensate “wave-function” and several other properties.
The thermodynamics of a free Bose gas with effective temperature scale
T~ and hard-sphere Bose gas with the
T~ scale are studied.
T~ arises as the temperature experienced by a single particle in a quantum gas with 2-body harmonic oscillator interaction
Vosc, which at low temperatures is expected to simulate, almost correctly, the attractive part of the interatomic potential
VHe between
4He atoms. The repulsive part of
VHe is simulated by a hard-sphere (HS) potential. The thermodynamics of this system of HS bosons, with the
T~ temperature scale (HSET), is investigated, first, by the Bogoliubov-Huang method and next by a modified version of this method, which takes approximate account of those terms of the 2-body repulsion which are linear in the zero-momentum Bose operators
a0,a0∗ (originally rejected by Bogoliubov). Theoretical heat capacity
CV(T) exhibits good agreement, below 2.1 K, with the experimental heat capacity graph observed in
4He at saturated vapour pressure. The phase transition to the low-temperature phase, with a Bose-Einstein condensate, occurs in the HSET at $T_λ=$2.17 K, and is accompanied, in the modified HSET version, by a singularity of
CV(T). Other thermal properties of HSET, such as the momentum distribution function, the fraction of atoms in the momentum condensate and normal fluid density, agree qualitatively with those of
4He, but improve those of the free Bose gas.
We investigate the dynamic structure factor of a system of Bose particles at zero temperature using quantum Monte Carlo methods. Interactions are modeled using a hard-sphere potential of size
a and simulations are performed for values of the gas parameter
na3 ranging from the dilute regime up to densities
n where the thermodynamically stable phase is a solid. With increasing density, we observe a crossover of the dispersion of elementary excitations from a Bogoliubov-type spectrum to a phonon-maxon-roton curve and the emergence of a broad multiphonon contribution accompanying the single-quasiparticle peak. In particular, for
na3=0.2138, which corresponds to superfluid
4He at equilibrium density, the extracted spectrum turns out to be in good agreement with the experimental energy-momentum dispersion relation in the roton region and for higher momenta. The behavior of the spectral function at the same density in the stable solid and metastable gas phase above the freezing point is also discussed.
A method developed by the authors for treating Bose-Einstein particles at low temperature is specialized to the case of hard sphere interactions. It is shown that the integral equations determining the energy can in this case be replaced by an integral involving the unknown excitation spectrum. The energy is first determined as a power series in ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$, where $\ensuremath{\rho}$ is the density and
a the hard-sphere radius. This solution is shown to diverge badly at high density. A result is then obtained which is valid over the full range of density, reducing to the exact expressions at low density and to a good approximation at high density. The excitation spectrum of the system is shown to have a peculiar nonmonotonic relation between energy and momentum very similar to that deduced by Landau from the low-temperature properties of
He4.
We obtain ab-initio estimations of the dynamic structure factor, S ( q ,ω), of Bose gases at zero temperature. More precisely, we use the Genetic Inversion via Falsification of Theories (GIFT) algorithm to perform analytic continuations of imaginary time correlation functions computed via an exact Path Integral projector method. Using the hard-sphere potential to model the two-body interactions between the atoms, we compute S ( q ,ω) changing the gas parameter from the dilute regime ( na 3 = 10 −4 ) up to the density corresponding to superfluid 4 He at equilibrium ( na 3 = 0.2138). With increasing density, we observe the emergence of a broad multiphonon contribution accompanying the quasiparticle peak and a crossover of the dispersion of elementary excitations from a Bogoliubov-like spectrum to a phonon-maxon- roton curve. Apart from the low wave vector region, for na 3 = 0.2138 the energy-momentum dispersion relation and the static density response function, χ( q ), turns out to be in good agreement with the superfluid 4 He experimental data at equilibrium density.
One considers the superfluid (SF) state of a Bose liquid with a strong repulsion between bosons, in which at T=0, along with a weak single-particle Bose-Einstein condensate (BEC), there exists an intensive pair coherent condensate (PCC), analogous to the Cooper condensate in a Fermi liquid with an attraction between the fermions. Such a PCC emerges in a system of bosons due to an oscillating sign-changing momentum dependence of the Fourier component of the pair interaction potential, which is characteristic of a certain family of repulsion potentials. In such cases, the Fourier component is negative in some domain of nonzero momentum transfer, which corresponds to an effective attraction. The collective effects of renormalization (``screening'') of the initial interaction lead to a suppression of the repulsion and an enhancement of the effective attraction. It is the ratio of the BEC density to the full density of the liquid
n0/n≪1 that is used as a small parameter---unlike in the Bogolyubov theory for a quasi-ideal Bose gas, in which the small parameter is the ratio of the number of supracondensate excitations to the number of particles in an intensive BEC,
(n−n0)/n0≪1. A closed system of nonlinear integral equations for the normal and anomalous self-energy parts is obtained, in the framework of a renormalized perturbation theory built on combined hydrodynamic and field variables. In the framework of the hard-spheres model, a spectrum of quasiparticles is obtained, which is in good accordance with the experimental spectrum of elementary excitations in superfluid
4He. The question of applicability of the Landau criterion in the absence of quantum vortices is discussed.