Noncondensate Atoms Research Articles

Textural and structural properties and surface acidity characterization of mesoporous silica-zirconia molecular sieves

Homogeneous mesoporous zirconium-containing MCM-41 type silica were prepared by supramolecular templating and their textural and structural properties were studied using powder X-ray diffraction, N 2 porosimetry, atomic force microscopy, EXAFS, XPS, and UV–VIS–NIR diffuse reflectance spectroscopy. Their acid properties were also studied by using IR spectroscopy and by the use of catalytic tests such as the decomposition of isopropanol and the isomerization of 1-butene. The materials prepared show a good degree of crystallinity with a regular ordering of the pores into a hexagonal arrangement and high thermal stability. The specific surface area of the prepared materials decreases as the zirconium content rises. Zirconium atoms are in coordination 7 to 8 and located at the surface of the pores such that a high proportion of the oxygen atoms bonded to zirconium corresponds to surface non-condensed oxygen atoms. Both facts are responsible for the acid properties of the solids that show weak Brønsted and medium strong Lewis acidity.

Dynamic quantum depletion in phase-imprinted generation of dark solitons

We investigate the quantum fluctuations associated with dark-soliton generation in Bose-Einstein condensates. With the phase imprinting procedure treated as a part of the dynamical process, we present numerical solutions of time-dependent quantum depletion. Our results provide evidence of a rapid growth of noncondensate atoms at dark-soliton notches without making assumptions of final quasiparticle states.

Kinetic Theory of a Spin-1/2 Bose-Condensed Gas

We derive a kinetic theory for a spin-1/2 Bose-condensed gas of two-level atoms at finite temperatures. The condensate dynamics is described by a generalized Gross–Pitaevskii equation for the two-component spinor order parameter, which includes the interaction with the uncondensed fraction. The noncondensate atoms are described by a quantum kinetic equation, which is a generalization of the spin kinetic equation for spin-polarized quantum gases to include couplings to the condensate degree of freedom. The kinetic equation is used to derive hydrodynamic equations for the noncondensate spin density. The condensate and noncondensate spins are coupled directly through the exchange mean field. Collisions between the condensate and noncondensate atoms give rise to an additional contribution to the spin diffusion relaxation rate. In addition, they give rise to mutual relaxation of the condensate and noncondensate due to lack of local equilibrium between the two components.

Effect of the thermal gas component on the stability of vortices in trappedBose–Einstein condensates

We study the stability of vortices in trapped single-component Bose–Einsteincondensates within self-consistent mean-field theories; in particular, weconsider the Hartree–Fock–Bogoliubov–Popov theory and its recentlyproposed gapless extensions. It is shown that for sufficiently repulsivelyinteracting systems the anomalous negative-energy modes related to vortexinstabilities are lifted to positive energies due to partial filling of the vortex corewith noncondensed gas. Such behaviour implies that within these theoriesthe vortex states are eventually stable against the transfer of condensatematter to the anomalous core modes. This self-stabilization of vortices,shown to occur under very general circumstances, is contrasted to thepredictions of the non-self-consistent Bogoliubov approximation, which isknown to exhibit anomalous modes for all vortex configurations and thusimplying instability of these states. In addition, the shortcomings of theseapproximations in describing the properties of vortices are analysed, and theneed for a self-consistent theory taking properly into account the coupleddynamics of the condensate and the noncondensate atoms is emphasized.

Coherent tunneling of atoms from Bose-condensed gases at finite temperatures

Tunneling of atoms between two trapped Bose-condensed gases at finite temperatures is explored using a many-body linear-response tunneling formalism similar to that used in superconductors. To lowest order, the tunneling currents can be expressed quite generally in terms of the single-particle Green's functions of isolated Bose gases. A coherent first-order tunneling Josephson current between two atomic Bose-Einstein condensates is found, in addition to coherent and dissipative contributions from second-order condensate-noncondensate and noncondensate-noncondensate tunneling. Our work is a generalization of Meier and Zwerger, who recently treated tunneling between uniform atomic Bose gases. We apply our formalism to the analysis of an out-coupling experiment induced by light wave fields, using a simple Bogoliubov-Popov quasiparticle approximation for the trapped Bose gas. For tunneling into the vacuum, we recover the results of Japha, Choi, Burnett, and Band, who recently pointed out the usefulness of studying the spectrum of out-coupled atoms. In particular, we show that the small tunneling current of noncondensate atoms from a trapped Bose gas has a broad spectrum of energies, with a characteristic structure associated with the Bogoliubov quasiparticle ${u}^{2}$ and ${v}^{2}$ amplitudes.

Finite-temperature theory of the scissors mode in a Bose gas using the moment method

We use a generalized Gross-Pitaevskii equation for the condensate and a semi-classical kinetic equation for the noncondensate atoms to discuss the scissors mode in a trapped Bose-condensed gas at finite temperatures. Both equations include the effect of $C_{12}$ collisions between the condensate and noncondensate atoms. We solve the coupled moment equations describing oscillations of the quadrupole moments of the condensate and noncondensate components to find the collective mode frequencies and collisional damping rates as a function of temperature. Our calculations extend those of Gu\'ery-Odelin and Stringari at T=0 and in the normal phase. They complement the numerical results of Jackson and Zaremba, although Landau damping is left out of our approach. Our results are also used to calculate the quadrupole response function, which is related to the moment of inertia. It is shown explicitly that the moment of inertia of a trapped Bose gas at finite temperatures involves a sum of an irrotational component from the condensate and a rotational component from the thermal cloud atoms.

Quantum effects on dynamics of instabilities in Bose-Einstein condensates

Dynamics of fluctuations in unstable Bose-Einstein condensates is analyzed by the solution of approximate operator equations. In the case of a condensate with a negative scattering length the present treatment describes a delay of collapse, in agreement with recent experiments. In the case of a collision of two condensate wavepackets it is shown that quantum effects lead to a Bose enhancement of elastic-scattering losses. In both cases the noncondensate atoms are formed as entangled pairs in squeezed states.

Observation of Anomalous Spin-State Segregation in a Trapped Ultracold Vapor

We observe counterintuitive spin segregation in an inhomogeneous sample of ultracold, noncondensed rubidium atoms in a magnetic trap. We use spatially selective microwave spectroscopy to verify a model that accounts for the differential forces on two internal spin states. In any simple understanding of the cloud dynamics, the forces are far too small to account for the dramatic transient spin polarizations observed. The underlying mechanism remains to be elucidated.

Temperature-dependent relaxation times in a trapped Bose-condensed gas

Explicit expressions for all the transport coefficients have recently been found for a trapped Bose condensed gas at finite temperatures. These transport coefficients are used to define the characteristic relaxation times, which determine the crossover between the mean-field collisionless and the two-fluid hydrodynamic regime. These relaxation times are evaluated as a function of the position in the trap potential. We show that all the relaxation times are dominated by the collisions between the condensate and the non-condensate atoms, and are much smaller than the standard classical collision time used in most of the current literature. The 1998 MIT study of the collective modes at finite temperature is shown to have been well within the two-fluid hydrodynamic regime.

Bose–Einstein condensation in atomic trap: gravity shift of critical temperature

The 2D Bose condensation of trapped atoms in a gravitational field is considered. The deformation of the finite parabolic potential in this field leads to the cut-off of a trap spectrum and the redefinition of its states. A shift of Bose condensation temperature T c via these factors is calculated in the simplified model. The tunneling of atoms is taken into account, and its contribution to the T c shift by the noncondensate atoms may be found important. The condensate atoms do not contribute to tunneling because they are almost motionless.

Landau-Khalatnikov two-fluid hydrodynamics of a trapped Bose gas

Starting from the quantum kinetic equation for the noncondensate atoms and the generalized Gross--Pitaevskii equation for the condensate, we derive the two-fluid hydrodynamic equations of a trapped Bose gas at finite temperatures. We follow the standard Chapman--Enskog procedure, starting from a solution of the kinetic equation corresponding to the complete local equilibrium between the condensate and the noncondensate components. Our hydrodynamic equations are shown to reduce to a form identical to the well-known Landau-Khalatnikov two-fluid equations, with hydrodynamic damping due to the deviation from local equilibrium. The deviation from local equilibrium within the thermal cloud gives rise to dissipation associated with shear viscosity and thermal conduction. In addition, we show that effects due to the deviation from the diffusive local equilibrium between the condensate and the noncondensate (recently considered by Zaremba, Nikuni, and Griffin) can be described by four frequency-dependent second viscosity transport coefficients. We also derive explicit formulas for all the transport coefficients. These results are used to introduce two new characteristic relaxation times associated with hydrodynamic damping. These relaxation times give the rate at which local equilibrium is reached and hence determine whether one is in the two-fluid hydrodynamic region.

Damping of condensate collective modes due to equilibration with the noncondensate

We consider the damping of condensate collective modes in the collisionless regime at finite temperatures arising from lack of equilibrium between the condensate and the noncondensate atoms, an effect that is ignored in the usual discussion of the collisionless region. As a first approximation, we ignore the dynamics of the thermal cloud. Our calculations should be applicable to collective modes of the condensate that are oscillating out-of-phase with the thermal cloud. We obtain a generalized Stringari equation of motion for the condensate at finite temperatures, which includes a damping term associated with the fact that the condensate is not in diffusive equilibrium with the static thermal cloud. This intercomponent collisional damping of the condensate modes is comparable in magnitude to the Landau damping considered in the recent literature.

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.

Kinetic theory of collective excitations and damping in Bose-Einstein condensed gases

We calculate the frequencies and damping rates of the low-lying collective modes of a Bose-Einstein condensed gas at nonzero temperature. We use a complex nonlinear Schr\"odinger equation to determine the dynamics of the condensate atoms, and couple it to a Boltzmann equation for the noncondensate atoms. In this manner we take into account both collisions between noncondensate-noncondensate and condensate-noncondensate atoms. We solve the linear response of these equations, using a time-dependent gaussian trial function for the condensate wave function and a truncated power expansion for the deviation function of the thermal cloud. As a result, our calculation turns out to be characterized by two dimensionless parameters proportional to the noncondensate-noncondensate and condensate-noncondensate mean collision times. We find in general quite good agreement with experiment, both for the frequencies and damping of the collective modes.

Tunneling dynamics of Bose-Einstein condensates with Feshbach resonances

We study tunneling dynamics of atomic pairs in Bose-Einstein condensates with Feshbach resonances. It is shown that the tunneling of the atomic pairs depends on not only the tunneling coupling between the atomic condensate and the molecular condensate, but also the inter-atomic nonlinear interactions and the initial number of atoms in these condensates. It is found that in addition to oscillating tunneling current between the atomic condensate and the molecular condensate, the nonlinear atomic-pair tunneling dynamics sustains a self-locked population imbalance: macroscopic quantum self-trapping effect. Influence of decoherence induced by non-condensate atoms on tunneling dynamics is investigated. It is shown that decoherence suppresses atomic-pair tunneling.

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.

Two-Fluid Dynamics for a Bose-Einstein Condensate out of Local Equilibrium with the Noncondensate

We extend our recent work on the two-fluid hydrodynamics of a Bose-condensed gas by including collisions involving both condensate and noncondensate atoms. These collisions are essential for establishing a state of local thermodynamic equilibrium between the condensate and noncondensate. Our theory is more general than the usual Landau two-fluid theory, to which it reduces in the appropriate limit, in that it allows one to describe situations in which a state of complete local equilibrium between the two components has not been reached. The exchange of atoms between the condensate and noncondensate is associated with a new relaxational mode of the gas.

Observation of Metastable States in Spinor Bose-Einstein Condensates

Bose-Einstein condensates have been prepared in long-lived metastable excited states. Two complementary types of metastable states were observed. The first is due to the immiscibility of multiple components in the condensate, and the second to local suppression of spin-relaxation collisions. Relaxation via recondensation of noncondensed atoms, spin relaxation, and quantum tunneling was observed. These experiments were done with $F\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ spinor Bose-Einstein condensates of sodium confined in an optical dipole trap.

Bose-Einstein condensate in a double-well potential as an open quantum system

We study the dynamics of a Bose-Einstein condensate in a double-well potential in the two-mode approximation. The dissipation of energy from the condensate is described by the coupling to a thermal reservoir of non-condensate modes. As a consequence of the coupling the self-locked population imbalance in the macroscopic quantum self-trapping decays away. We show that a coherent state predicted by spontaneous symmetry breaking is not robust and decoheres rapidly into a statistical mixture due to the interactions between condensate and non-condensate atoms. However, via stochastic simulations we find that with a sufficiently fast measurement rate of the relative phase between the two wells the matter wave coherence is established even in the presence of the decoherence.

Two-fluid hydrodynamics for a trapped weakly interacting Bose gas

We derive the coupled equations of motion for the condensate (superfluid) and non-condensate (normal fluid) degrees of freedom in a trapped Bose gas at finite temperatures. Our results are based on the Hartree-Fock-Popov approximation for the time-dependent condensate wavefunction and an assumption of local equilibrium for the non-condensate atoms. In the case of a uniform weakly-interacting gas, our formalism gives a microscopic derivation of the well-known two-fluid equations of Landau. The collective modes in a parabolically trapped Bose gas include the analogue of the out-of-phase second sound mode in uniform systems.