We consider an ultracold Bose gas in a half-open potential well, otherwise confined to a quasi-one-dimensional geometry. The Bogoliubov equations for its elementary excitations are solved in the continuous spectrum to explore the particle and energy content near the edge of the gas, beyond the often applied local density approximation. In particular, gradients in the condensate density enhance density-dominated excitations in the border region. We discuss excess (missing) particles and their energy by comparing to suitable reference solutions for quasi-homogeneous systems. The density profile near the edge shows no Friedel oscillations, but a dipolar feature from the spill-out of thermally excited particles. The calculations are performed in the grand-canonical ensemble and in the thermodynamic limit.
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12 pages, 10 figures, submitted to special issue of J. Chem. Phys. in memoriam Rudolf Podgornik
We consider an ultracold Bose gas in a half-open potential well, otherwise confined to a quasi-one-dimensional geometry. The Bogoliubov equations for its elementary excitations are solved in the continuous spectrum to explore the particle and energy content near the edge of the gas, beyond the often applied local density approximation. In particular, gradients in the condensate density enhance density-dominated excitations in the border region. We discuss excess (missing) particles and their energy by comparing to suitable reference solutions for quasi-homogeneous systems. The density profile near the edge shows no Friedel oscillations, but a dipolar feature from the spill-out of thermally excited particles. The calculations are performed in the grand-canonical ensemble and in the thermodynamic limit.
We investigate the novel density distributions acquired by a dipolar Bose-Einstein condensed gas confined in a box potential, with special focus on the effects of supersolidity. Different from the case of harmonic trapping, the ground-state density reveals a strong depletion in the bulk region and an accumulation of atoms near the walls, well separated from the bulk, as a consequence of the competition between the attractive and the repulsive nature of the dipolar force. In a quasi-two-dimensional geometry characterized by cylindrical box trapping, we observe the emergence of a ringlike configuration near the boundary of the box, revealing peculiar supersolid and crystal effects in a useful range of parameters. In the case of square box trapping, the density oscillations along the edges, caused by the enhanced accumulation of atoms near the vertices, exhibit interesting analogies with the case of box-trapped quasi-one-dimensional configurations. For sufficiently large values of the atom number, the bulk region can also exhibit supersolidity, the resulting geometry reflecting the symmetry of the confining potential even for large systems.
Abstract We derive and analyze the coupled equations of quadratic approximation of the Bogoliubov model for a weakly interacting Bose gas. The first equation determines the condensate density as a variational parameter and ensures the minimum of the grand thermodynamic potential. The second one provides a relation between the total number of particles and chemical potential. Their consistent theoretical analysis is performed for a number of model interaction potentials including contact (local) and nonlocal interactions, where the latter provide nontrivial dependencies in momentum space. We demonstrate that the derived equations have no solutions for the local potential, although they formally reproduce the well-known results of the Bogoliubov approach. At the same time, it is shown that these equations have the solutions for physically relevant nonlocal potentials. We show that in the regimes close to experimental realizations with ultracold atoms, the contribution of the terms originating from the quadratic part of the truncated Hamiltonian to the chemical potential can be of the same order of magnitude as from its c -number part. Due to this fact, in particular, the spectrum of single-particle excitations in the quadratic approximation acquires a gap. The issue of the gap is also discussed.
A theory is developed for the density profile of a weakly interacting Bose gas confined within an arbitrary potential well at nonzero temperature. This problem is of interest in connection with magnetically confined spin-aligned atomic hydrogen (H\ensuremath{\downarrow}). The approach taken is conceptually simpler and of comparable accuracy as compared to previous approaches. It is based on the statement of constancy of the chemical potential in equilibrium. This in turn involves the internal chemical potential, a functional of the density, which we treat in a local-density approximation. For a given temperature we develop a form for the local internal chemical potential for a wide range of density using known low-density, intermediate-density (i.e., near the Bose-condensation density), and high-density analytic forms, with appropriate smooth interpolants connecting these three forms. Density profiles are evaluated within the Bogoliubov approximation for the local internal chemical potential for parameters appropriate to magnetically confined H\ensuremath{\downarrow} for both symmetric linear and quadratic one-dimensionally varying wells. Comparison is made to previous work. The nonanalytic region of the density profile near the Bose-condensation density is analyzed. We also calculate density profiles including the next-highest-order interaction correction (i.e., beyond Bogoliubov) in the internal chemical potential. For conditions relevant to hypothetical H\ensuremath{\downarrow} at moderate densities this leads to a small but discernable effect on the density profile.
Abstract The Bogoliubov–de Gennes equations for the elementary excitations of a Bose gas are solved for an inhomogeneous, one-dimensional system in the vicinity of a turning point, addressing the full continuous spectrum. A basis change in the space of the two Bogoliubov ‘particle’ and ‘hole’ amplitudes is introduced that decouples them approximately. We find a spatially extended mode that governs mainly excitations in the condensate phase, while another mode is spatially localised to the boundary zone where the condensate density gradient is significant. An analytical and numerical discussion of the phase shift is provided that incident matter waves suffer upon reflection at the turning point, forming standing waves. We find phase shifts with relatively strong energy dispersion that show significant differences among four model potentials. As applications, we discuss the boundary-induced change in the density of states and compute eigenfrequencies in a gravitational trap, without recourse to the local density approximation. The non-condensate density at finite temperature and its quantum depletion are discussed in a companion paper.
We use the Bogoliubov theory of Bose-Einstein condensation to study the properties of dipolar particles (atoms or molecules) confined in a uniform two-dimensional geometry at zero temperature. We find equilibrium solutions to the dipolar Gross-Pitaevskii equation and the Bogoliubov-de Gennes equations. Using these solutions we study the effects of quantum fluctuations in the system, particularly focussing on the instability point, where the roton feature in the excitation spectrum touches zero. Specifically, we look at the behaviour of the noncondensate density, the phase fluctuations, and the density fluctuations in the system. Near the instability, the density-density correlation function shows a particularly intriguing oscillatory behaviour. Higher order correlation functions display a distinct hexagonal lattice pattern formation, demonstrating how an observation of broken symmetry can emerge from a translationally symmetric quantum state.
We analyze the equilibrium properties of a weakly interacting, trapped quasi-one-dimensional Bose gas at finite temperatures and compare different theoretical approaches. We focus in particular on two stochastic theories: a number-conserving Bogoliubov (NCB) approach and a stochastic Gross-Pitaevskii equation (SGPE) that have been extensively used in numerical simulations. Equilibrium properties like density profiles, correlation functions, and the condensate statistics are compared to predictions based upon a number of alternative theories. We find that due to thermal phase fluctuations, and the corresponding condensate depletion, the NCB approach loses its validity at relatively low temperatures. This can be attributed to the change in the Bogoliubov spectrum, as the condensate gets thermally depleted, and to large fluctuations beyond perturbation theory. Although the two stochastic theories are built on different thermodynamic ensembles (NCB, canonical; SGPE, grand-canonical), they yield the correct condensate statistics in a large Bose-Einstein condensate (BEC) (strong enough particle interactions). For smaller systems, the SGPE results are prone to anomalously large number fluctuations, well known for the grand-canonical, ideal Bose gas. Based on the comparison of the above theories to the modified Popov approach, we propose a simple procedure for approximately extracting the Penrose-Onsager condensate from first- and second-order correlation functions that is both computationally convenient and of potential use to experimentalists. This also clarifies the link between condensate and quasicondensate in the Popov theory of low-dimensional systems.