Starting from the spectrum of the radially symmetric quantum harmonic oscillator in two dimensions, we create a large set of nonlinear solutions. The relevant three principal branches, with $n_r=0,1$ and 2 radial nodes respectively, are systematically continued as a function of the chemical potential and their linear stability is analyzed in detail, in the absence as well as in the presence of topological charge $m$, i.e., vorticity. It is found that for repulsive interatomic interactions {\it only} the ground state is {\it linearly stable} throughout the parameter range examined. Furthermore, this is true for topological charges $m=0$ or $m=1$; solutions with higher topological charge can be unstable even in that case. All higher excited states are found to be unstable in a wide parametric regime. However, for the focusing/attractive case the ground state with $n_r=0$ and $m=0$ can only be stable for a sufficiently low number of atoms. Once again, excited states are found to be generically unstable. For unstable profiles, the dynamical evolution of the corresponding branches is also followed to monitor the temporal development of the instability.
It was recently shown in self-consistent Hartree-Fock calculations that a harmonically trapped dilute gas of fermionic atoms with a repulsive two-body interaction exhibits a pronounced {\it super-shell} structure: the shell fillings due to the spherical harmonic trapping potential are modulated by a beat mode. This changes the ``magic numbers'' occurring between the beat nodes by half a period. The length and amplitude of the beating mode depends on the strength of the interaction. We give a qualitative interpretation of the beat structure in terms of a semiclassical trace formula that uniform…
We show that a variety of cosmologically motivated effective quasiparticle space-times can be produced in harmonically trapped superfluid Bose and Fermi gases. We study the analogue of cosmological particle production in these effective space-times, induced by trapping potentials and coupling constants possessing an arbitrary time dependence. The WKB probabilities for phonon creation from the superfluid vacuum are calculated, and an experimental procedure to detect quasiparticle production by measuring density-density correlation functions is proposed.
The attractive Fermi-Hubbard Hamiltonian is solved via the Bogoliubov-de Gennes formalism to analyze the ground state phases of population imbalanced fermion mixtures in harmonically trapped two-dimensional optical lattices. In the low density limit the superfluid order parameter modulates in the radial direction towards the trap edges to accommodate the unpaired fermions that are pushed away from the trap center with a single peak in their density. However in the high density limit while the order parameter modulates in the radial direction towards the trap center for low imbalance, it also m…
Motivated by the recent article of P. Shea {\it et al.} [Am. J. Phys. {\bf 77} (6), 2009] we examine the exactly solvable problem of two harmonically trapped ultra-cold bosonic atoms interacting {\it via} a short range potential in one and two dimensions. A straightforward application in one dimension shows that the energy spectrum is universal, provided that the range of the potential is much smaller than the oscillator length, in addition to clearly illustrating why regularization is not required in the limit of zero range. The two dimensional problem is less trivial, requiring a more carefu…
We discuss the model of a $D$-dimensional confined electron gas in which the particles are trapped by a harmonic potential. In particular, we study the non-interacting kinetic and exchange energies of finite-size inhomogeneous systems, and compare the resulting Thomas-Fermi and Dirac coefficients with various uniform electron gas paradigms. We show that, in the thermodynamic limit, the properties of this model are identical to those of the $D$-dimensional Fermi gas.
We theoretically study the collective excitations of an ideal gas confined in an isotropic harmonic trap. We give an exact solution to the Boltzmann-Vlasov equation; as expected for a single-component system, the associated mode frequencies are integer multiples of the trapping frequency. We show that the expressions found by the scaling ansatz method are a special case of our solution. Our findings, however, are most useful in case the trap contains more than one phase: we demonstrate how to obtain the oscillation frequencies in case an interface is present between the ideal gas and a differe…
We study the splitting of a harmonically trapped atomic Bose-Einstein condensate when we continuously turn up an optical lattice (or a double-well) potential. As the lattice height is increased, quantum fluctuations of atoms are enhanced. The resulting nonequilibrium dynamics of the fragmentation process of the condensate, the loss of the phase coherence of atoms along the lattice, and the reduced atom number fluctuations in individual lattice sites are stochastically studied within the truncated Wigner approximation. We perform a detailed study of the effects of temperature and lattice height…
I discuss a Bogoliubov inequality for obtaining a rigorous bound on the maximal axial extension of inhomogeneous one-dimensional Bose-Einstein condensates. An explicit upper limit for the aspect ratio of a strongly elongated, harmonically trapped Thomas-Fermi condensate is derived.
We investigate the harmonically trapped 2D fermionic systems with a effective spin-orbit coupling and intrinsic s-wave superfluidity under the local density approximation, and find that there is a critical value for Zeeman field. When the Zeeman field larger than the critical value, the topological superfluid phases emerge and coexist with the normal superfluid phase, topological phase separation, in the trapped region. Otherwise, the superfluid phase is topologically trivial.
We extend the Projected Gross Pitaevskii equation formalism of Davis et al. [Phys. Rev. Lett. \bf{87}, 160402 (2001)] to the experimentally relevant case of harmonic potentials. We outline a robust and accurate numerical scheme that can efficiently simulate this system. We apply this method to investigate the equilibrium properties of a harmonically trapped three-dimensional Bose gas at finite temperature, and consider the dependence of condensate fraction, position and momentum distributions, and density fluctuations on temperature. We apply the scheme to simulate an evaporative cooling proce…
We investigate the harmonically trapped interacting Bose gas in a quasi-2D geometry using the classical field method. The system exhibits quasi-long-range order and non-classical rotational inertia at temperatures below the Berezinskii-Kosterlitz-Thouless cross-over to the superfluid state. In particular, we compute the scissors-mode oscillation frequencies and find that the irrotational mode changes its frequency as the temperature is sweeped across the cross-over thus providing microscopic evidence for the emergence of superfluidity.
We present a theoretical study of vortices within a harmonically trapped Bose-Einstein condensate in a rotating optical lattice. We find that proximity to the Mott insulating state dramatically effects the vortex structures. To illustrate we give examples in which the vortices: (i) all sit at a fixed distance from the center of the trap, forming a ring, or (ii) coalesce at the center of the trap, forming a giant vortex. We model time-of-flight expansion to demonstrate the experimental observability of our predictions.
We examine the spin asymmetry of ground states for two-dimensional, harmonically trapped two-component gases of fermionic atoms at zero temperature with weakly repulsive short range interactions. Our main result is that, in contrast to the three-dimensional case, in two dimensions a non-trivial spin-asymmetric phase can only be caused by shell structure. A simple, qualitative description is given in terms of an approximate single particle model, comparing well to the standard results of Hartree-Fock or direct diagonalization methods.
We present results from a detailed Quantum Monte Carlo study of BEC applied to JILA experiment [Jin et al Phys. Rev. Lett. $\bf 78$, 764, 1997][1]. This is the first Monte Carlo approach(based on Feynman-Kac path integral method) to the above problem where good qualitative agreement is found for both the lowest lying $m=2$ and $m=0$ modes. We found an upward shift of the experimental data for $m=0$ mode at around $ T=0.7 T_0 $ ($T_0$ is defined as the predicted BEC transition temperature for harmonically confined ideal gas) when the effect of noncondensate was considered.
We provide an in depth analysis of the theory proposed by Holzmann, Chevallier and Krauth (HCK) [Europhys. Lett., {\bf 82}, 30001 (2008)] for predicting the temperature at which the Berezinskii-Kosterlitz-Thouless (BKT) transition to a superfluid state occurs in the harmonically trapped quasi-two-dimensional (2D) Bose gas. Their theory is based on a meanfield model of the system density and we show that the HCK predictions change appreciably when an improved meanfield theory and identification of the transition point is used. In this analysis we develop a consistent theory that provides a lowe…
We use quantum Monte Carlo simulations to obtain zero-temperature state diagrams for strongly correlated lattice bosons in one and two dimensions under the influence of a harmonic confining potential. Since harmonic traps generate a coexistence of superfluid and Mott insulating domains, we use local quantities such as the quantum fluctuations of the density and a local compressibility to identify the phases present in the inhomogeneous density profiles. We emphasize the use of the "characteristic density" to produce a state diagram that is relevant to experimental optical lattice systems, rega…
Quantum fluctuations in time-dependent, harmonically-trapped Bose-Einstein condensates are studied within Bogoliubov theory. An eigenmode expansion of the linear field operators permits the diagonalization of the Bogoliubov-de Gennes equation for a stationary condensate. When trap frequency or interaction strength are varied, the inhomogeneity of the background gives rise to off-diagonal coupling terms between different modes. This coupling is negligible for low energies, i.e., in the hydrodynamic regime, and an effective space-time metric can be introduced. The influence of the inter-mode cou…
We show that a dilute harmonically trapped two-component gas of fermionic atoms with a weak repulsive interaction has a pronounced super-shell structure: the shell fillings due to the spherical harmonic trapping potential are modulated by a beat mode. This changes the ``magic numbers'' occurring between the beat nodes by half a period. The length and amplitude of this beating mode depend on the strength of the interaction. We give a simple interpretation of the beat structure in terms of a semiclassical trace formula for the symmetry breaking U(3) --> SO(3).
We calculate the critical rotation frequency at which a vortex state becomes energetically favorable over the vortex-free ground state in a harmonically trapped Bose-Einstein condensate whose atoms have dipole-dipole interactions as well as the usual s-wave contact interactions. In the Thomas-Fermi (hydrodynamic) regime, dipolar condensates in oblate cylindrical traps (with the dipoles aligned along the axis of symmetry of the trap) tend to have lower critical rotation frequencies than their purely s-wave contact interaction counterparts. The converse is true for dipolar condensates in prolate…
We study the conservative dynamics and stationary configurations of a vortex-antivortex pair in a harmonically trapped two-dimensional Bose-Einstein condensate. We establish the conceptual framework for understanding the stationary states and the topological defect trajectories, through considerations of different mechanisms of vortex motion and the bifurcation of soliton-like stationary solutions. Our insights are based on Lagrangian-based variational calculations, numerical solutions of both the time-dependent and time-independent Gross-Pitaevskii equations, and exact solutions for the non-i…
The local density approximation is used to study the ground state superfluid properties of harmonically trapped p-wave Fermi gases as a function of fermion-fermion attraction strength. While the density distribution is bimodal on the weakly attracting BCS side, it becomes unimodal with increasing attraction and saturates towards the BEC side. This non-monotonic evolution is related to the topological gapless to gapped phase transition, and may be observed via radio-frequency spectroscopy since quasi-particle transfer current requires a finite threshold only on the BEC side.
We discuss the transition from a fully decoherent to a (quasi-)condensate regime in a harmonically trapped weakly interacting 1D Bose gas. By using analytic approaches and verifying them against exact numerical solutions, we find a characteristic crossover temperature and crossover atom number that depend on the interaction strength and the trap frequency. We then identify the conditions for observing either an interaction-induced crossover scenario or else a finite-size Bose-Einstein condensation phenomenon characteristic of an \textit{ideal} trapped 1D gas.
