Vortex Dynamics in Trapped Bose-Einstein Condensate

We perform numerical simulations of vortex motion in a trapped Bose-Einstein condensate by solving the two-dimensional Gross-Pitaevskii Equation in the presence of a simple phenomenological model of interaction between the condensate and the finite temperature thermal cloud. At zero temperature, the trajectories of a single, off-centred vortex precessing in the condensate, and of a vortex-antivortex pair orbiting within the trap, excite acoustic emission. At finite temperatures the vortices move to the edge of the condensate and vanish. By fitting the finite-temperature trajectories, we relate the phenomenological damping parameter to the friction coefficients α and α′, which are used to describe the interaction between quantised vortices and the normal fluid in superfluid helium.     

Vortex Precession in a Rotating Nonaxisymmetric Trapped Bose-Einstein Condensate

We study the precession of an off-axis straight vortex in a rotating non-axisymmetric harmonic trap in the Thomas-Fermi (TF) regime. A time-dependent variational Lagrangian analysis yields the dynamical equations of the vortex and the precessional angular velocity in two-dimensional (2D) and three-dimensional (3D) condensates.     

Thermodynamics of a Trapped Bose Condensate with Negative Scattering Length

We study the Bose–Einstein condensation (BEC) for a system of ⁷Li atoms, which have negative scattering length (attractive interaction), confined in a harmonic potential. Within the Bogoliubov and Popov approximations, we numerically calculate the density profile for both condensate and non-condensate fractions and the spectrum of elementary excitations. In particular, we analyze the temperature and number-of-boson dependence of these quantities and evaluate the BEC transition temperature T _BEC. We calculate the loss rate for inelastic two- and three-body collisions. We find that the total loss rate is strongly dependent on the density profile of the condensate, but this density profile does not appreciably change by increasing the thermal fraction. Moreover, we study, using the quasi-classical Popov approximation, the temperature dependence of the critical number N _c of condensed bosons, for which there is the collapse of the condensate. There are different regimes as a function of the total number N of atoms. For N<N _c the condensate is always metastable but for N>N _c the condensate is metastable only for temperatures that exceed a critical value T _c.     

Stability of a Vortex in Spinor Bose-Einstein Condensate

We propose a method to create a vortex in BEC utilizing the spin degree of freedom. We consider the optical plug at the center of the vortex, which makes the vortex-creation process stable. We also investigate the instability of the halfway state to complete vortex state without the optical plug.     

The Condition for Existence of Complex Modes in Trapped Bose–Einstein Condensate with a Highly Quantized Vortex

We consider a trapped Bose–Einstein condensate with a highly quantized vortex. Pu et al. found numerically the parameter region in which complex eigenvalues arise. Recently, the splitting of a highly quantized vortex into two singly quantized vortices is observed in the experiment. We derive analytically the condition for the existence of complex eigenvalues by using the small coupling constant expansion and the two-mode approximation. We check that our results agree with those by Pu et al.     

Atomic Phase Conjugation From a Bose Condensate

We discuss the possibility of observing atomic phase conjugation from Bose condensates, and using it as a diagnostic tool to access the spatial coherence properties and to measure the lifetime of the condensate. We argue that since phase conjugation results from the scattering of a partial matter wave off the spatial grating produced by two other waves, it offers a natural way to directly measure such properties, and as such provides an attractive alternative to the optical methods proposed in the past.     

The Two-Dimensional Bose–Einstein Condensate

We study the Hartree–Fock–Bogoliubov mean-field theory as applied to a two-dimensional finite trapped Bose gas at low temperatures and find that, in the Hartree–Fock approximation, the system can be described either with or without the presence of a condensate; this is true in the thermodynamic limit as well. Of the two solutions, the one that includes a condensate has a lower free energy at all temperatures. However, the Hartree–Fock scheme neglects the presence of phonons within the system, and when we allow for the possibility of phonons we are unable to find condensed solutions; the uncondensed solutions, on the other hand, are valid also in the latter, more general scheme. Our results confirm that low-energy phonons destabilize the two-dimensional condensate.     

Simplified System for Creating a Bose–Einstein Condensate

We designed and constructed a simplified experimental system to create a Bose–Einstein condensate in ⁸⁷Rb. Our system has several novel features including a mechanical atom transfer mechanism and a hybrid Ioffe–Pritchard magnetic trap. The apparatus has been designed to consistently produce a stable condensate even when it is not well optimized.     

Quasiparticle Kinetic Equation in a Trapped Bose Gas at Low Temperatures

Recently the authors used the Kadanoff–Baym non-equilibrium Green's function formalism to derive kinetic equation for the non-condensate atoms, in conjunction with a consistent generalization of the Gross–Pitaevskii equation for the Bose condensate wavefunction. This work was limited to high temperatures, where the excited atoms could be described by a Hartree–Fock particle-like spectrum. Following the approach of Kane and Kadanoff in 1965, we present the generalization of our recent work which is valid at low temperatures, where the input single-particle spectrum is now described by the Bogoliubov–Popov approximation. We derive a kinetic equation for the quasiparticle distribution function with collision integrals describing scattering between quasiparticles and the condensate atoms. From the general expression for the collision integral for the scattering between quasiparticle excitations, we find the quasiparticle distribution function corresponding to local equilibrium. This expression includes a quasiparticle chemical potential that controls the non-diffusive equilibrium between condensate atoms and the quasiparticle excitations. We derive a generalized Gross–Pitaevskii equation for the condensate wavefunction that also includes the damping effects due to collisions between atoms in the condensate and the thermally excited quasiparticles. For a uniform Bose gas, our kinetic equation for the thermally excited quasiparticles reduces to that found by Eckern, as well as by Kirkpatrick and Dorfman.     

The Condensate Wave Function of a Trapped Atomic Gas

We discuss various properties of the ground state of a Bose-condensed dilute gas confined by an external potential. We devote particular attention to the role played by the interaction in determining the kinetic energy of the system and the aspect ratio of the velocity distribution. The structure of the wave function near the classical turning point is discussed and the drawback of the Thomas-Fermi approximation is explicitly pointed out. We consider also states with quantized vorticity and calculate the critical angular velocity for the production of vortices. The presence of vortex states is found to increases the stability of the condensate in the case of attractive interactions.     

The Structure of a Quantized Vortex in a Bose-Einstein Condensate

The structure of a quantized vortex in a Bose-Einstein Condensate is investigated using the projection method developed by Peierls, Yoccoz, and Thouless. This method was invented to describe the collective motion of a many-body system beyond the mean-field approximation. The quantum fluctuation has been properly built into the variational wave function, and a vortex is described by a linear combination of Feynman wave functions weighted by a Gaussian distribution in their center positions. In contrast to the solution of the Gross-Pitaevskii equation, the particle density is finite at the vortex axis and the vorticity is distributed in the core region.     

Simulation of a Stationary Dark Soliton in a Trapped Zero-Temperature Bose-Einstein Condensate

We discuss a computational mechanism for the generation of a stationary dark soliton, or black soliton, in a trapped Bose–Einstein condensate (BEC) using the Gross–Pitaevskii (GP) equation for both attractive and repulsive interaction. It is demonstrated that the black soliton with a “notch" in the probability density with a zero at the minimum is a stationary eigenstate of the GP equation and can be efficiently generated numerically as a nonlinear continuation of the first vibrational excitation of the GP equation in both attractive and repulsive cases in one and three dimensions for pure harmonic as well as harmonic plus optical-lattice traps. We also demonstrate the stability of this scheme under different perturbing forces.     

Dynamics of Trapped Bose Gases at Finite Temperatures

Starting from an approximate microscopic model of a trapped Bose-condensed gas at finite temperatures, we derive an equation of motion for the condensate wavefunction and a quantum kinetic equation for the distribution function for the excited atoms. The kinetic equation is a generalization of our earlier work in that collisions between the condensate and non-condensate (C ₁₂ ) are now included, in addition to collisions between the excited atoms as described by the Uehling–Uhlenbeck (C ₂₂ ) collision integral. The continuity equation for the local condensate density contains a source term ₁₂ which is related to the C ₁₂ collision term. If we assume that the C ₂₂ collision rate is sufficiently rapid to ensure that the non-condensate distribution function can be approximated by a local equilibrium Bose distribution, the kinetic equation can be used to derive hydrodynamic equations for the non-condensate. The ₁₂ source terms appearing in these equations play a key role in describing the equilibration of the local chemical potentials associated with the condensate and non-condensate components. We give a detailed study of these hydrodynamic equations and show how the Landau two-fluid equations emerge in the frequency domain is a characteristic relaxation time associated with C ₁₂ collisions. More generally, the lack of complete local equilibrium between the condensate and non-condensate is shown to give rise to a new relaxational mode which is associated with the exchange of atoms between the two components. This new mode provides an additional source of damping in the hydrodynamic regime. Our equations are consistent with the generalized Kohn theorem for the center of mass motion of the trapped gas even in the presence of collisions. Finally, we formulate a variational solution of the equations which provides a very convenient and physical way of estimating normal mode frequencies. In particular, we use relatively simple trial functions within this approach to work out some of the monopole, dipole and quadrupole oscillations for an isotropic trap.     

Two-Fluid Hydrodynamics in Trapped Bose Gases and in Superfluid Helium

A review is given of recent theoretical work on the superfluid dynamics of trapped Bose gases at finite temperatures, where there is a significant fraction of non-condensate atoms. One can now reach large enough densities and collision cross-sections needed to probe the collective modes in the collision-dominated hydrodynamic region where the gas exhibits characteristic superfluid behavior involving the relative motions of the condensate and non-condensate components. The precise analogue of the Landau-Khalatnikov two-fluid hydrodynamic equations was recently derived from trapped Bose gases, starting from a generalized Gross-Pitaevskii equation for the condensate macroscopic wavefunction and a kinetic equation for the non-condensate atoms.     

Vortex Nucleation and Array Formation in a Rotating Bose-Einstein Condensate

We study the dynamics of vortex lattice formation of a rotating trapped Bose-Einstein condensate by numerically solving the two-dimensional Gross-Pitaevskii equation, and find that the condensate undergoes elliptic deformation, followed by unstable surface-mode excitations before forming a quantized vortex lattice. The dependence of the number of vortices on the rotation frequency is obtained.     

Quantum Turbulence in a Trapped Bose-Einstein Condensate under Combined Rotations around Three Axes

In this paper, we study quantum turbulence in trapped Bose-Einstein condensates by performing a numerical simulation of the Gross-Pitaevskii equation. Combining rotations around three axes, we successfully induce quantum turbulence in which quantized vortices are not crystallized but fully tangled. Compared to our previous work with using combined rotations around two axes, turbulence is more isotropic and the obtained energy spectrum is more consistent with the Kolmogorov law.      

Vortex Lattice Structures of a Bose-Einstein Condensate in a Rotating Lattice Potential

We study vortex lattice structures of a trapped Bose-Einstein condensate in a rotating lattice potential by numerically solving the time-dependent Gross-Pitaevskii equation. By rotating the lattice potential, we observe the transition from the Abrikosov vortex lattice to the pinned vortex lattice. We investigate the transition of the vortex lattice structure by changing conditions such as angular velocity, strength, and lattice constant of the rotating lattice potential.     

Coarse-Grained Finite-Temperature Theory for the Bose Condensate in Optical Lattices

In this paper, we derive a coarse-grained finite-temperature theory for a Bose condensate in a one-dimensional optical lattice, in addition to a confining harmonic trap potential. We start with a two-particle irreducible (2PI) effective action on the Schwinger-Keldysh closed-time contour path. In principle, this action involves all information of equilibrium and non-equilibrium properties of the condensate and noncondensate atoms. In constructing a theory for the condensate and noncondensate in a periodic lattice potential, a difficulty arises from the rapid spatial variation due to a lattice potential, compared to the length scale of the harmonic potential. We employ a coarse-graining procedure to eliminate this rapid variation. By introducing a variational ansatz for the condensate order parameter in an effective action, we derive a coarse-grained effective action, which describes the dynamics on the length scale much longer than a lattice constant. Using the variational principle, coarse-grained equations of motion for condensate variables are obtained. These equations include a dissipative term due to collisions between condensate and noncondensate atoms, as well as noncondensate mean-field. As a result of a coarse-graining procedure, the effects of a lattice potential are incorporated into equations of motion for the condensate by an effective mass, a renormalized coupling constant, and an umklapp scattering process. To illustrate the usefulness of our formalism, we discuss a Landau instability of the condensate moving in optical lattices by using the coarse-grained generalized Gross-Pitaevskii hydrodynamics. We find that the collisional damping rate due to collisions between the condensate and noncondensate atoms changes its sign when the condensate velocity exceeds a renormalized sound velocity, leading to a Landau instability consistent with the Landau criterion. Our results in this work give an insight into the microscopic origin of the Landau instability.