Dilute Atomic Gases Research Articles

Number-conserving approach to a minimal self-consistent treatment of condensate and noncondensate dynamics in a degenerate Bose gas

We describe a number-conserving approach to the dynamics of Bose-Einstein condensed dilute atomic gases. This builds upon the works of Gardiner [Phys. Rev. A 56, 1414 (1997)] and Castin and Dum [Phys. Rev. A 57, 3008 (1998)]. We consider what is effectively an expansion in powers of the ratio of noncondensate to condensate particle numbers, rather than inverse powers of the total number of particles. This requires the number of condensate particles to be a majority, but not necessarily almost equal to the total number of particles in the system. We argue that a second-order treatment of the relevant dynamical equations of motion is the minimum order necessary to provide consistent coupled condensate and noncondensate number dynamics for a finite total number of particles, and show that such a second-order treatment is provided by a suitably generalized Gross-Pitaevskii equation, coupled to the Castin-Dum number-conserving formulation of the Bogoliubov--de Gennes equations. The necessary equations of motion can be generated from an approximate third-order Hamiltonian, which effectively reduces to second order in the steady state. Such a treatment as described here is suitable for dynamics occurring at finite temperature, where there is a significant noncondensate fraction from the outset, or dynamics leading to dynamical instabilities, where depletion of the condensate can also lead to a significant noncondensate fraction, even if the noncondensate fraction is initially negligible.

Phase diagram of a dilute fermion gas with density imbalance

We map out the phase diagram of a dilute two-component atomic fermion gas with unequal populations and masses under a Feshbach resonance. As in the case of equal masses, no uniform phase is stable for an intermediate coupling regime. For majority component heavier, the unstable region moves towards the BEC side. When the coupling strength is increased from the normal phase, there is an increased parameter space where the transition is into the FFLO state. The converse is true if the majority is light.

Dissipative structures of diffuse molecular gas

Aims. This paper belongs to a series of four, dedicated to the analysis of the dynamical, thermal and chemical properties of translucent molecular gas, with the perspective of characterizing the processes driving the dissipation of supersonic turbulence, an anticipated prerequisite of dense core formation. Methods. We analyze the small scale morphology and velocity structure of the parsec-scale environment of a low mass dense core (1 M� ). Our work is based on large maps made with the IRAM-30 m telescope in the two lowest rotational transitions of 12 CO and 13 CO with high angular (20 �� or 0.015 pc at 115 GHz) and spectral (0.055 km s −1 ) resolutions. The field is translucent, hence providing strong constraints on the column density and physical conditions in the gas. Results. More than one third of the field mass (6.5 M� ) lies in an elongated tail of dense and cold gas, possibly extending beyond the edge of the map and connected to the core in space and velocity. This core tail is highly turbulent and sub-structured into narrow filaments of aspect ratio up to 20. These are pure velocity structures with velocity shears in the range 2–10 km s −1 pc −1 . Another third of the mass, according to the weak extinction of the field, lies in more dilute molecular and atomic gas. Its molecular fraction, largely traced by optically thick 12 CO lines, is even more turbulent than the dense core tail. The gas emitting in the broad wings of the 12 CO lines is organized into a conspicuous network of narrow criss-crossed filaments, whose pattern at the parsec scale is seen for the first time. The gas there is optically thin in the 12 CO(1–0) line (τ12 < 0.2), warmer than 25 K and more dilute than 1000 cm −3 . These optically thin 12 CO-filaments, though contributing to about 10% of the mass of the environment, have a CO cooling rate a few times larger than that of the whole field on average. Whether dense or dilute, all the filamentary structures in the field (with transverse sizes 0.015–0.03 pc), are preferentially oriented along the direction of the magnetic fields, as measured a few parsecs away. Using the Chandrasekhar-Fermi method, we estimate the intensity of the magnetic fields intensity in the dilute molecular gas to be Bpos = 15 µG. We infer that the turbulent motions in the dilute gas are in the trans-Alfvenic range. –––

Conditions of low dimensionality for strongly interacting atoms under a transverse trap

For a dilute atomic gas in a strong transverse trapping potential, one normally expects that, in the ground state, the gas will populate only the lowest transverse level. We show, however, that for the strongly interacting gas under a Feshbach resonance, the ground state includes a large fraction of atoms in excited levels of the trap, even if the gas is in the dilute limit and the trap is very strong. This is because the effective atom-molecule coupling is typically enhanced to many times the trap mode spacing by an induced confinement along the untrapped dimension(s). Thus one cannot ``freeze out'' the transverse degrees of freedom except under certain conditions.

Superradiant Rayleigh scattering of high-intensity short light pulses from a Bose-Einstein condensate of dilute atomic gases

Equations of a semiclassical model of superradiant Rayleigh scattering of high-intensity short light pulses from a Bose-Einstein condensate of dilute atomic gases are solved numerically taking into account the excitation of atoms by coherent Rayleigh radiation and their recoil in the backward direction. The evolution of the populations of coherent atomic states with a particular momentum is studied, and the pulse shape and the structure of the spectrum of such scattering are found in relation to the laser beam intensity and the recoil kinetic energy of atoms.

Design and construction of an efficient electro-optic modulator for laser spectroscopy

We discuss design considerations and construction of a home-built electro-optic phase modulator (EOM) that can be used for locking a laser to an atomic transition. The EOM is designed to operate at a resonant frequency of ≈20 MHz and imposes a phase modulation on a laser beam. The phase-modulated light is sent through a reference cell containing a dilute gas of rubidium atoms. When the laser is scanned over an atomic resonance, the absorption of light through the cell can be detected and mixed down to DC to produce a dispersion shaped “error” signal. The error signal can be used to lock the laser to the atomic resonance. We also describe tests of the basic properties of the resonant circuit. The Q of the circuit is measured to be 10, resulting in 60% efficiency for the first-order sidebands at RF drive powers of 0.7 W. Applications include the spectroscopy of laser-cooled atoms.PACS Nos.: 01.50.Pa, 39.25.+k 32.80.Pj, 42.62.Fi

Bose–Einstein condensation of exciton polaritons

Phase transitions to quantum condensed phases--such as Bose-Einstein condensation (BEC), superfluidity, and superconductivity--have long fascinated scientists, as they bring pure quantum effects to a macroscopic scale. BEC has, for example, famously been demonstrated in dilute atom gas of rubidium atoms at temperatures below 200 nanokelvin. Much effort has been devoted to finding a solid-state system in which BEC can take place. Promising candidate systems are semiconductor microcavities, in which photons are confined and strongly coupled to electronic excitations, leading to the creation of exciton polaritons. These bosonic quasi-particles are 10(9) times lighter than rubidium atoms, thus theoretically permitting BEC to occur at standard cryogenic temperatures. Here we detail a comprehensive set of experiments giving compelling evidence for BEC of polaritons. Above a critical density, we observe massive occupation of the ground state developing from a polariton gas at thermal equilibrium at 19 K, an increase of temporal coherence, and the build-up of long-range spatial coherence and linear polarization, all of which indicate the spontaneous onset of a macroscopic quantum phase.

Super-shell structure in harmonically trapped fermionic gases and its semi-classical interpretation

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 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 semi-classical trace formula that uniformly describes the symmetry breaking U(3) → SO(3) in a three-dimensional harmonic oscillator potential perturbed by an anharmonic term ∝r4 with arbitrary strength. We show that at low Fermi energies (or particle numbers), the beating gross-shell structure of this system is dominated solely by the twofold degenerate circular and (diametrically) pendulating orbits.

Asymmetric Fermi superfluid in a harmonic trap

We consider a dilute two-component atomic fermion gas with unequal populations in aharmonic trap potential using the mean field theory and the local density approximation.We show that the system is phase separated into concentric shells with the superfluid in thecore surrounded by the normal fermion gas both on the weak-coupling BCS side and nearthe Feshbach resonance. On the strong-coupling BEC side, the composite bosonsand left-over fermions can be mixed. We calculate the cloud radii and compareaxial density profiles systemically for the BCS, near resonance and BEC regimes.

Symmetry aspects in non-relativistic multi-scalar field models and application to a coupled two-species dilute Bose gas

We discuss unusual aspects of symmetry that can happen due to entropic effects in the context of multi-scalar field theories at finite temperature. We present their consequences, in special, for the case of nonrelativistic models of hard core spheres. We show that for nonrelativistic models phenomena like inverse symmetry breaking and symmetry non-restoration cannot take place, but a reentrant phase at high temperatures is shown to be possible for some region of parameters. We then develop a model of interest in studies of Bose-Einstein condensation in dilute atomic gases and discuss about its phase transition patterns. In this application to a Bose-Einstein condensation model, however, no reentrant phases are found.

Superfluid stability in the BEC-BCS crossover

We consider a dilute atomic gas of two species of fermions with unequal concentrations under a Feshbach resonance. We find that the system can have distinct properties due to the unbound fermions. The uniform state is stable only when either (a) beyond a critical coupling strength, where it is a gapless superfluid, or (b) when the coupling strength is sufficiently weak, where it is a normal Fermi gas mixture. Phase transition(s) must therefore occur when the resonance is crossed.

Bose–Einstein condensates: Analytical methods for the Gross–Pitaevskii equation

We present simple analytical methods for solving the Gross–Pitaevskii equation (GPE) for the Bose–Einstein condensation (BEC) in the dilute atomic alkali gases. Using a soliton variational Ansatz we consider the effects of repulsive and attractive effective nonlinear interactions on the BEC ground state. We perform a comparative analysis of the solutions obtained by the variational Ansatz, the perturbation theory, the Thomas–Fermi approximation, and the Green function method with the numerical solution of the GPE finding universal ranges where these solutions can be used to predict properties of the condensates. Also, a generalization of the soliton variational approach for two-species of alkali atoms of the GPE is performed as a function of the effective interaction λi(i=1,2) and the inter-species λ12 and λ21 constants.

Coherent backscattering of light with nonlinear atomic scatterers

We study coherent backscattering of a monochromatic laser by a dilute gas of cold two-level atoms in the weakly nonlinear regime. The nonlinear response of the atoms results in a modification of both the average field propagation (nonlinear refractive index) and the scattering events. Using a perturbative approach, the nonlinear effects arise from inelastic two-photon scattering processes. We present a detailed diagrammatic derivation of the elastic and inelastic components of the backscattering signal for both scalar and vectorial photons. In particular, we show that the coherent backscattering phenomenon originates in some cases from the interference between three different scattering amplitudes. This is in marked contrast with the linear regime where it is due to the interference between two different scattering amplitudes. In particular we show that, if elastically scattered photons are filtered out from the photodetection signal, the nonlinear backscattering enhancement factor exceeds the linear barrier of 2, consistently with a three-amplitude interference effect.

Superradiant scattering of light from a Bose-Einstein condensate of dilute atomic gases

A generalization of the semiclassical model of superradiant scattering of light from a Bose-Einstein condensate of dilute atomic gases is proposed. A number of effects connected with the backward recoil of atoms in this scattering are explained.

Superradiant Rayleigh scattering of light from a Bose-Einstein condensate of dilute atomic gases, taking into account backward recoil momentum

An extension of the semiclassical model is proposed for superradiant Rayleigh scattering of light from a Bose-Einstein condensate of dilute atomic gases. In contrast to the familiar models, this model takes into account the fact that the atom may be excited by the scattered light and may acquire backward recoil momentum. The presented equations are solved analytically for the initial (linear) stage of the process. The intensity of the scattered light and the efficiency of populating atomic states with different momentum values are studied as a function of detuning from resonance.

Bose-Einstein Condensates and the Numerical Solution of the Gross-Pitaevskii Equation

The achievement of Bose-Einstein condensation in ultracold vapors of alkali atoms has accelerated the study of dilute atomic gases in condensed quantum states. This review introduces some key issues in the area.

Gross‐Pitaevskii equation: Variational approach

In this paper we present an analytical method for solving the Gross-Pitaevskii equation for the Bose-Einstein condensation in the dilute atomic alkali gases. Using a variational ansatz, we are able to obtain an analytical solution for the order parameter and for the chemical potential as a function of a unique universal parameter: λ/(ħ ω l) (l is the oscillator length, ω the trap frequency and λ an effective coupling constant). We also perform a comparative analysis between our method of solution and two other methods with defined range of validity: the perturbation theory and the Thomas-Fermi Approximation (TFA). The presented method is valid for negative and for positive self interactions as well. In the positive coupling constant region it is consistent with the TFA, extending its validity beyond the limits of validity of this well known approximation, towards small and negative values of the self-interaction coupling constant. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Spatially periodic structures of an atomic Bose-Einstein condensate

The conditions providing the formation of periodic vortex lattices of an interference nature in an atomic Bose-Einstein condensate (i.e., in the absence of rotation of the condensate) are determined. Spatially periodic exact solutions of the nonlocal nonlinear Schrodinger equation (the generalized Gross-Pitaevskii equation) that describes the Bose-Einstein condensate of a dilute gas of alkali metal atoms with due regard for the nonlocality of interatomic interactions are obtained in the form of a set of two or three plane waves. It is shown that periodic vortex lattices can be produced in interference experiments with a Bose-Einstein condensate of a dilute gas of alkali metal atoms.

Photon Recoil Momentum in Dispersive Media

A systematic shift of the photon recoil momentum due to the index of refraction of a dilute gas of atoms has been observed. The recoil frequency was determined with a two-pulse light grating interferometer using near-resonant laser light. The results show that the recoil momentum of atoms caused by the absorption of a photon is n variant Planck's k, where n is the index of refraction of the gas and k is the vacuum wave vector of the photon. This systematic effect must be accounted for in high-precision atom interferometry with light gratings.

VORTICES IN MULTICOMPONENT BOSE–EINSTEIN CONDENSATES

We review the topic of quantized vortices in multicomponent Bose–Einstein condensates of dilute atomic gases, with an emphasis on the two-component condensates. First, we review the fundamental structure, stability and dynamics of a single vortex state in a slowly rotating two-component condensates. To understand recent experimental results, we use the coupled Gross–Pitaevskii equations and the generalized nonlinear sigma model. An axisymmetric vortex state, which was observed by the JILA group, can be regarded as a topologically trivial skyrmion in the pseudospin representation. The internal, coherent coupling between the two components breaks the axisymmetry of the vortex state, resulting in a stable vortex molecule (a meron pair). We also mention unconventional vortex states and monopole excitations in a spin-1 Bose–Einstein condensate. Next, we discuss a rich variety of vortex states realized in rapidly rotating two-component Bose–Einstein condensates. We introduce a phase diagram with axes of rotation frequency and the intercomponent coupling strength. This phase diagram reveals unconventional vortex states such as a square lattice, a double-core lattice, vortex stripes and vortex sheets, all of which are in an experimentally accessible parameter regime. The coherent coupling leads to an effective attractive interaction between two components, providing not only a promising candidate to tune the intercomponent interaction to study the rich vortex phases but also a new regime to explore vortex states consisting of vortex molecules characterized by anisotropic vorticity. A recent experiment by the JILA group vindicated the formation of a square vortex lattice in this system.