We treat the problem of self-consistently interacting bosons in the presence of a finite (but macroscopic) potential well within a quasiclassical approximation for the normal component and the order parameter. We solve the equilibrium problem and show that condensation actually occurs in two steps: one already at low densities with Bose condensation only in the well, and another one corresponding to the usual condensation in bulk. The peak and width of the distribution of trapped particles in the well display a distinct signature of the local condensation. A possible connection to recent experiments with excitons is discussed.
We treat the problem of self-consistently interacting bosons in the presence of a finite (but macroscopic) potential well within a quasiclassical approximation for the normal component and the order parameter. We solve the equilibrium problem and show that condensation actually occurs in two steps: one already at low densities with Bose condensation only in the well, and another one corresponding to the usual condensation in bulk. The peak and width of the distribution of trapped particles in the well display a distinct signature of the local condensation. A possible connection to recent experiments with excitons is discussed.
The fact that two-dimensional interacting trapped systems do not undergo Bose-Einstein Condensation (BEC) in the thermodynamic limit, though rigorously proved, is somewhat mysterious because all relevant limiting cases (zero temperature, small atom numbers, noninteracting particles) suggest otherwise. We study the possibility of condensation in finite trapped two-dimensional systems. We first consider the ideal gas, which incorporates the inhomogeneity and finite size of experimental systems and can be solved exactly. A semiclassical self-consistent approximation gives us a feel for the temperature scales; diagonalization of the one-body density matrix confirms that the condensation is into a single state. We squeeze a three-dimensional system and see how it crosses over into two dimensions. Mean-field theory, our main tool for the study of interacting systems, prescribes coupled equations for the condensate and the thermal cloud: the condensate receives a full quantum-mechanical treatment, while the noncondensate is described by different schemes of varying sophistication. We analyze the T = 0 case and its approach to the thermodynamic limit, finding a criterion for the dimensionality crossover and obtaining the coupling constant of the two-dimensional system that results from squeezing a three-dimensional trap. We next apply a semiclassical Hartree-Fock approximation to purely two-dimensional finite gases and find that they can be described either with or without a condensate; this continues to be true in the thermodynamic limit. The condensed solutions have a lower free energy at all temperatures but neglect the presence of phonons within the system and cease to exist when we allow for this possibility. The uncondensed solutions, in turn, are valid under a more rigorous scheme but have consistency problems of their own. Path-integral Monte Carlo simulations provide an essentially exact description of finite interacting gases and enable us to study highly anisotropic systems at finite temperature. We find that our two-dimensional Hartree-Fock solutions accurately mimic the surface density profile and predict the condensate fraction of these systems; the equivalent interaction parameter is smaller than that dictated by the T = 0 analysis. We conclude that, in two-dimensional isotropic finite trapped systems and in highly compressed three-dimensional gases, there is a phenomenon resembling a condensation into a single state.
The ideal Bose gas in a highly anisotropic harmonic potential is studied. It is found that Bose-Einstein condensation occurs in two distinct steps as the temperature is lowered. In the first step the specific heat shows a sharp feature, but the system still occupies many one-dimensional quantum states. In the second step, at a significantly lower temperature, the ground state becomes macroscopically occupied. It should be possible to verify these predictions using present-day atom traps. The two-step behavior can occur in a rather general class of anisotropic traps, including the box potential.
Three examples of spatial Bose-Einstein condensations in which the particles macroscopically occupy the lowest localized state of an inhomogeneous external potential are analyzed. The three cases are (a) a small potential well in a large box which causes a spectrum with a gap, (b) a harmonic oscillator potential, and (c) randomly sized trapping potentials caused by ’’impurities.’’ All three cases are two dimensional so that no Bose condensation occurs without the inhomogeneous potentials. An attempt is made to keep the treatments as mathematically simple as possible. A review of much of the literature of spatial Bose condensations is provided. The special problem of the form of the thermodynamic limit in an inhomogeneous potential is discussed for case (b). Numerical examples applying to monolayer 4He adsorbed on a surface are treated.
We present a model for the equilibrium quantum statistics of a condensate of repulsively interacting bosons in a two-dimensional trap. Particle correlations in the ground state are treated exactly, whereas interactions with excited particles are treated in a generalized Bogoliubov mean-field theory. This leads to a fundamental physical picture for the condensation of interacting bosons through an anharmonic oscillator ground state coupled to excited Bogoliubov quasiparticles in which the quantum number statistics of condensate particles emerges self-consistently. Below the Bose-Einstein condensation temperature, our model exhibits a crossover from particle bunching to Poissonian statistics and finally antibunching as the temperature is lowered or as the trapping area is decreased. When applied to Bose condensation of long-lived dressed excitons in a photonic band gap material, our model suggests that this system may serve as a novel tunable source for nonclassical states of light.
We study a non-interacting (perfect) Bose-gas in random external potentials (traps).It is shown that a generalized Bose-Einstein condensation in the random eigenstates manifests if and only if the same occurs in the one-particle kinetic-energy eigenstates, which corresponds to the generalized condensation of the free Bose-gas.Moreover, we prove that the amounts of both condensate densities are equal.This statement is relevant for justification of the Bogoliubov approximation in the theory of disordered boson systems.
Weakly interacting Bose gases in a disorder environment have long been a challenging topic in the field of solid-state physics due to the intriguing interplay between superfluidity and localization. In this thesis we investigate theoretically the equilibrium properties of one- and three-dimensional harmonically trapped dirty Bose-Einstein condensates (BECs). In particular we focus on the decomposition of the total particle density into three components, namely, the condensate density, the thermal density, and the density of fragmented local Bose-Einstein condensates within the respective minima of the random potential landscape. In order to determine the different density components we develop a Hartree-Fock mean-field theory for a weakly interacting trapped BEC in a random environment, then specialize it to contact interaction and delta-correlated disorder. This non-perturbative theory is worked out on the basis of the replica method, which represents a well- established tool to deal with disorder problems. With this we calculate the corresponding free energy at finite temperature and derive from it the underlying self-consistency equations for the respective density components. Then, as a first step, we investigate a quasi one-dimensional Bose-Einstein condensed gas in a harmonic trapping potential with an additional delta- correlated disorder potential at zero temperature. To this end we solve the self-consistency equations within the Thomas-Fermi approximation and find the emergence of a Bose-glass region, where the condensate vanishes. We corroborate this analysis by an elaborate numerical treatment, where the corresponding one-dimensional time-independent Gross-Pitaevskii equation is numerically solved and disorder-ensemble averages of the condensate wave function are performed. The variance of the condensate wave function quantifies the number of bosons which condense in the local minima of the random potential. For weak disorder these mini-condensates turn out to occur preferentially at the border of the condensate, while for intermediate disorder strength this happens in the trap center. Additionally we use a variational ansatz in order to describe analytically the numerically observed redistribution of the fragmented mini-condensates with increasing disorder strength. In close analogy to the one-dimensional case we then treat the three-dimensional dirty Bose gas in an isotropic harmonic trap and a delta- correlated disorder potential at zero temperature using the corresponding self-consistency equations obtained via the Hartree-Fock mean-field theory within the Thomas-Fermi approximation. Additionally we use a variational ansatz, whose results turn out to coincide qualitatively with those obtained from the Thomas-Fermi approximation. Inparticular, a first order quantum phase transition from the superfluid phase, where the condensate density contributes to the total density, to the Bose-glass phase, where all particles are in the mini-condensates, is detected at a critical disorder strength, which agrees with findings in the literature. Furthermore, in a generaltriaxial harmonic trap, we investigate the geometric effect of different trap aspect ratios on the respective properties of the dirty BEC system. Finally, we consider the three- dimensional dirty BEC at finite temperature. This allows us to study the impact of both temperature and disorder fluctuations on the respective components of the density as well as their Thomas-Fermi radii. In particular, we find that the superfluid region, the Bose-glass region, and the thermal region coexist. Furthermore, depending on the system parameters, three phase transitions are detected, namely, one from the superfluid to the Bose-glass phase, one from the Boseglass to the thermal phase, and finally one directly from the superfluid to the thermal phase. All these results could be particularly useful for a quantitative analysis of world-wide on-going experiments with dirty bosons in quasi one-dimensional and three-dimensional harmonic traps.