Dynamics of correlations in atomic Bose-Einstein condensates

Abstract

The Gross-Pitaevskii equation has been extremely successful in the theory of weakly interacting Bose-Einstein condensates. However, present-day experiments reach beyond the regime of its validity due to the significant role of correlations. We review a method for tackling the dynamics of correlations in Bose condensed gases, in terms of non-commutative cumulants. This new approach has a wide applicability in the areas of current interest, e.g. the production of molecules and the manipulation of interactions in condensates. It also offers an interesting perspective on the classical-field methods for partly condensed Bose gases.     

Coherence of bose condensates

Abstract

We study the coherence of interacting Bose condensates in recent magnetic trap experiments. The coherent evolution manifests itself in the macroscopic interference of two independent Bose condensates. The theoretical predictions from the time-dependent Gross–Pitaevskii equation are in excellent agreement with the measured interference patterns. A coherent coupling of two condensates represents the atomic analogon of a Josephson junction. The dependence of the magnetic confinement on the nuclear spin orientation allows one to build a controllable beam splitter by magnetic resonance. The application of this beam splitter to realize an atom laser is studied theoretically. The coherence of the output beam is limited only by phase diffusion of the condensate.     

Probing first-order spatial coherence of a Bose-Einstein condensate

Abstract

We probe the spatial coherence properties of a magnetically trapped Bose gas. Two matter wave beams are extracted from two spatially separated regions of the trap and overlap outside the trapping region. The visibility of the resulting interference pattern measures the phase coherence between the regions of extraction. By varying the spatial separation between the two regions the first-order spatial correlation function of the trapped Bose gas can be measured. The location of the minima of the interference pattern is reproducible, which experimentally confirms that the trapped Bose-Einstein condensate is not fragmented into individual condensates.     

Quantum dynamics of the phase of a Bose–Einstein condensate

Abstract

In this review we discuss the dynamics of the phase of trapped Bose–Einstein condensates. In particular we consider the phenomena of phase decoherence (termed also as phase collapse, or diffusion), and phase revival in systems of interacting atoms. We analyse the dependence of the collapse and revival times on the trap potential, dimensionality of the gas, atom number fluctuations, and on the coherent dynamics of the condensate. We show that in a class of experimentally relevant systems, the collapse time is relatively short, and in some cases vanishes in the limit of a large number of atoms, implying that the trapped Bose gas cannot sustain a well-defined quantum phase, and that the phase memory is lost on a relatively short time scale. Furthermore, we calculate the relative atom number fluctuations or a model of two interacting condensates, and show that the fluctuations are generically sub-Poissonian.     

Vortices Observed and to be Observed

Linear defects are generic in continuous media. In quantum systems they appear as topological line defects which are associated with a circulating persistent current. In relativistic quantum vacuum they are known as cosmic strings, in superconductors as quantized flux lines, and in superfluids and low-density atomic Bose-Einstein condensates as quantized vortex lines. We discuss unconventional vortices in unconventional superfluids and superconductors, which have been observed or have to be observed, such as continuous singly and doubly quantized vortices in ³ He-A and chiral Bose condensates; half-quantum vortices (Alice strings) in ³ He-A and in nonchiral Bose condensates; Abrikosov vortices with fractional magnetic flux in chiral and d-wave superconductors; vortex sheets in ³ He-A and chiral superconductors; the nexus—combined object formed by vortices and monopoles. Some properties of vortices related to the fermionic quasiparticles living in the vortex core are also discussed.     

Particle number fluctuations in an ideal Bose gas

Abstract

We analyse occupation number fluctuations of an ideal Bose gas in a trap which is isolated from the environment with respect to particle exchange (canonical ensemble). We show that in contrast to the predictions of the grandcanonical ensemble, the counting statistics of particles in the trap ground state changes from monotonously decreasing above the condensation temperature to single-peaked below that temperature. For the exactly solvable case of a harmonic oscillator trapping potential in one spatial dimension we extract a Landau–Ginzburg functional which–despite the non-interacting nature of the system–displays the characteristic behaviour of a weakly interacting Bose gas. We also compare our findings with the usual treatment which is based on the grand-canonical ensemble. We show that for an ideal Bose gas neither the grand-canonical and canonical ensemble thermodynamically equivalent, nor the grand-canonical ensemble can be viewed as a small system in diffusive contact with a particle reservoir.     

Magnetic Field Dependence of Internal Josephson Effects in Spinor Dipolar Bose–Einstein Condensates

We study internal Josephson effects in spin-1 Bose–Einstein condensates with magnetic dipole–dipole interactions by calculating the Gross–Pitaevskii equations with single mode approximations. The effects represent various oscillating modes given by combinations of 0, π, and running phase modes in Josephson effects in Bose gasses. Especially, we investigate the dependences of a magnetic field on the effects, presenting new dynamics which is not shown elsewhere.     

Mixtures of Bose liquids at finite temperature

The superfluidity of two weakly interacting Bose gases at finite temperature is studied with the Bogoliubov model. At low temperatures, only the lowest energy normal modes are relevant, exciting particles of each species from their respective condensates. The low-temperature depletion of the zero-momentum condensate and superfluid density are calculated for arbitrary mixtures. For identical short-range interactions between all particles and low concentrations of one species, curves of constant depletion and superfluid density have a form similar to those of a free Bose gas. These considerations suggest that superfluid flow of the unstable isotope ⁶He at low concentrations ( 10^–9) in liquid ⁴He would be detectable only at temperatures below 10^–6 K. Other, more general interaction potentials predict similar behavior.Suported in part by NSF grant DMR 75-08516.     

Excitation of Bound Modes in an Optical Fibre

Abstract

Simple approximate formula for the excitation of an optical fibre by a plane wave focused through a lens are presented. The formula are valid for any mode of a weakly guiding fibre, and depend only on the angle of incidence of the plane wave striking the lens and a single parameter which contains all the waveguide information. The same results apply to the case of a uniform beam exciting a fibre. The fraction of power excited in the first two modes is compared with numerical results.     

Proposals for creating Schrödinger cat states in bose-einstein condensates

Abstract

Three different schemes for producing Schrödinger cat states in Bose-Einstein condensates are outlined and the effects of loss in each of them compared. The first scheme involves coupled interacting condensates and proves to be very fragile to loss. This is improved upon with a second scheme which first evolves a cat state in phase space and then rapidly transforms it to a number cat state. Finally a third scheme is discussed which makes use of number correlated condensates and is remarkably robust to loss. It may prove to be valuable for experimentally creating such states.     

The Schrödinger Cat Family in Attractive Bose Gases

We show that the ground state of an attractive Bose gas in a double well evolves from a coherent state to a Schrödinger Cat like state as the tunneling barrier is decreased. The latter exhibits super-fragmentation similar to the ground state of a spin-1 Bose gas with antiferromagnetic interactions. We also show that the fragmented condensates of attractive and repulsive Bose gases in double wells lead to very different interference patterns.     

Degenerate Quantum Gases in Microgravity

Clouds of ultra-cold atoms and especially Bose–Einstein condensates (BEC) provide a source for coherent matter-waves in numerous earth bound experiments. Analogous to optical interferometry, matter-wave interferometers can be used for precision measurements allowing for a sensitivity orders of magnitude above their optical counterparts. However, in some respects the presence of gravitational forces in the lab limits experimental possibilities. In this article, we report about a compact and robust experiment generating Bose–Einstein condensates in the drop tower facility in Bremen, Germany. We also present the progress of building the succeeding experiment in which a two species atom interferometer will be implemented to test the weak equivalence principle with quantum matter.     

Vortex–Antivortex-Pair Lattices in Spin–Orbit Coupled Bose–Einstein Condensates

We investigate theoretically the ground states of Bose–Einstein condensates with Rashba spin–orbit coupling in optical lattices within mean-field framework. We obtain numerically the Bloch states and energy spectrum for the single particle Hamiltonian, meanwhile the analytical solution of Bloch states is also presented. For a spin–orbit coupling Bose–Einstein condensates with a weak interaction, we show the existence of the vortex–antivortex-pair lattices state by simulating the Gross–Pitaevskii equation.     

Chromatic Modulation for Optical Fibre Sensing Electromagnetic and Speckle Noise Analysis

Abstract

Changes in the chromaticity of the transmitted electromagnetic field in an optical fibre are considered. An electromagnetic analysis that predicts the radial dependence of the chromaticity at the exit is presented and coupled to a speckle noise analysis which provides an estimate of the error. Theoretical predictions are compared with experimental results and good agreement between the two is demonstrated. The results can be useful in the design of optical fibre sensors which use the chromatic modulation method.     

Tachyon Condensation and Brane Annihilation in Bose-Einstein Condensates: Spontaneous Symmetry Breaking in Restricted Lower-Dimensional Subspace

In brane cosmology, the Big Bang is hypothesized to occur by the annihilation of the brane–anti-brane pair in a collision, where the branes are three-dimensional objects in a higher-dimensional Universe. Spontaneous symmetry breaking accompanied by the formation of lower-dimensional topological defects, e.g. cosmic strings, is triggered by the so-called ‘tachyon condensation’, where the existence of tachyons is attributable to the instability of the brane–anti-brane system. Here, we discuss the closest analogue of the tachyon condensation in atomic Bose–Einstein condensates. We consider annihilation of domain walls, namely branes, in strongly segregated two-component condensates, where one component is sandwiched by two domains of the other component. In this system, the process of the brane annihilation can be projected effectively as ferromagnetic ordering dynamics onto a two-dimensional space. Based on this correspondence, three-dimensional formation of vortices from a domain-wall annihilation is considered to be a kink formation due to spontaneous symmetry breaking in the two-dimensional space. We also discuss a mechanism to create a ‘vorton’ when the sandwiched component has a vortex string bridged between the branes. We hope that this study motivates experimental researches to realize this exotic phenomenon of spontaneous symmetry breaking in superfluid systems.     

A Monte Carlo formulation of the Bogolubov theory

Abstract

We propose an efficient stochastic method to implement numerically the Bogolubov approach to study finite-temperature Bose-Einstein condensates. Our method is based on the Wigner representation of the density matrix describing the non-condensed modes and a Brownian motion simulation to sample the Wigner distribution at thermal equilibrium. Allowing it to sample any density operator Gaussian in the field variables, our method is very general and it applies both to the Bogolubov and to the Hartree-Fock Bogolubov approach, in the equilibrium case as well as in the time-dependent case. We think that our method can be useful to study trapped Bose-Einstein condensates in two or three spatial dimensions without rotational symmetry properties, as in the case of condensates with vortices, where the traditional Bogolubov approach is difficult to implement numerically due to the need to diagonalize very big matrices.     

Instability of Overlapped Vortices Rotating in Opposite Directions in Binary Bose–Einstein Condensates

We theoretically study the instability of vortices in pancake-shaped trapped binary Bose–Einstein condensates. We consider that a quantized vortex is at the center of each condensate and two condensates rotate in the opposite directions. The total circulation is zero in BECs having the overlapped vortices because there is relative rotation, which is the rotation of one component in relation to the rotation of the other, but no total rotation, which is the sum of the rotation in both components. We think that the zero-quantum vortices are unstable because this system is locally countersuperflow, two counterpropagating miscible superflows. In a uniform system, the countersuperflow is unstable when the relative velocity between the two condensates exceeds a critical value. We investigate the dynamics of the zero-quantum vortices by numerically solving the Gross–Pitaevskii equations. To understand the results of the numerical calculations, we apply the countersuperflow instability to our present system.     

The dynamics of three-mode coupling in a trapped Bose gas

ABSTRACT

Multiple mode couplings in topological coherent modes of Bose–Einstein condensate are considered, by introducing an external alternating (resonating) field in the system. This analysis is based on the analytical solutions of nonlinear Gross–Pitaevskii equation for a trapped Bose gas at nearly absolute zero temperature. The dynamics of fractional populations of the generated coherent modes are analysed, particularly for a three-level system in the limit of small to large detuning of the intermediate state. These coupled topological modes, though nonlinear, are analogous to a resonant atom and exhibit a variety of significant non-trivial phenomena (effects), like: dynamic phase transitions, interference patterns, critical phenomena, mode-locking and chaotic motion.     

Collective Shape Oscillation and Domain Formation of Two-Component Bose–Einstein Condensates

In this paper, we describe two kinds of characteristic nonlinear dynamics in two-component Bose–Einstein condensates. For two overlapping components, we analyzed the collective modes and a variety of nonlinear mode couplings, which were then confirmed by numerical simulation of the time-dependent Gross–Pitaevskii equations. Next, we consider the nonlinear dynamics of two condensates after abruptly turning on the intercomponent coupling strength. For strong intercomponent interactions the out-of-phase density wave of the condensates became unstable, leading to mugtiple domain formation.