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Theoretical Studies of dilute Bose-Einstein condensates in a double-well potential

E. Sakellari

Published Jan 1, 2004• 0 citations

Abstract

In this Thesis we apply the Gross-Pitaevskii equation (GPE) to describe properties of a dilute, near zero temperature Bose gas for various confining geometries. We start by reviewing some basic information about the density, the chemical potential and elementary excitations of a dilute atomic condensate confined in a single harmonic trap for a Bose condensate with repulsive and attractive interactions and we also discuss the stability in the case of attractive interactions. We extend our study to a one and three dimensional double-well trap. We investigate the eigenenergy levels and show that the nonlinearity leads to triangular structures which appear either in the ground or excited states for the case of a Bose condensate with attractive or repulsive interactions respectively. We apply the eigenenergy level picture to analyse Josephson effects induced when the barrier IS moved at a constant velocity across the trapping potential or by the application of a time-dependent potential gradient. The GPE simulations are compared to the predictions of a nonlinear two state model. Above a critical velocity there is a transition to a superposition of ground and excited states which leads to sudden changes in the population difference. The direction of Josephson flow depends critically on the initial state of the system and we discuss the feasibility of experimental control of the atomic flow using phase-imprinting. The stability of a low temperature Bose-Einstein condensate with attract interactions in one and three dimensional double-well potentiate is discussed. The condensate is shown to collapse at a critical potential gradient which corresponds to a critical number of atoms in one of the two wells. Finally we investigate the stability and tunnelling effects in a multi-well system.

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In this Thesis we apply the Gross-Pitaevskii equation (GPE) to describe properties of a dilute, near zero temperature Bose gas for various confining geometries. We start by reviewing some basic information about the density, the chemical potential and elementary excitations of a dilute atomic condensate confined in a single harmonic trap for a Bose condensate with repulsive and attractive interactions and we also discuss the stability in the case of attractive interactions. We extend our study to a one and three dimensional double-well trap. We investigate the eigenenergy levels and show that the nonlinearity leads to triangular structures which appear either in the ground or excited states for the case of a Bose condensate with attractive or repulsive interactions respectively. We apply the eigenenergy level picture to analyse Josephson effects induced when the barrier IS moved at a constant velocity across the trapping potential or by the application of a time-dependent potential gradient. The GPE simulations are compared to the predictions of a nonlinear two state model. Above a critical velocity there is a transition to a superposition of ground and excited states which leads to sudden changes in the population difference. The direction of Josephson flow depends critically on the initial state of the system and we discuss the feasibility of experimental control of the atomic flow using phase-imprinting. The stability of a low temperature Bose-Einstein condensate with attract interactions in one and three dimensional double-well potentiate is discussed. The condensate is shown to collapse at a critical potential gradient which corresponds to a critical number of atoms in one of the two wells. Finally we investigate the stability and tunnelling effects in a multi-well system.

The stability of a low temperature Bose-Einstein condensate with attractive interactions in one and three dimensional double-well potentials is discussed. In particular, the tunnelling dynamics of a condensate under the influence of a time-dependent potential gradient is investigated. The condensate is shown to collapse at a critical potential gradient which corresponds to a critical number of atoms in one of the two wells. The sensitivity of this tunnelling induced collapse could provide a useful tool in the study of condensates with attractive interactions.

The stability of a low-temperature Bose–Einstein condensate with attractive interactions in one- and three-dimensional double-well potentials is discussed. In particular, the tunnelling dynamics of a condensate under the influence of a time-dependent potential gradient is investigated. The condensate is shown to collapse at a critical potential gradient which corresponds to a critical number of atoms in one of the two wells. The sensitivity of this tunnelling-induced collapse could provide a useful tool in the study of condensates with attractive interactions.

Bose–Einstein condensation has been realized as dilute atomic vapors. This achievement has generated immense interest in this field. This article review of recent theoretical research into the properties of trapped dilute-gas Bose–Einstein condensates. Among these properties, stability of Bose–Einstein condensates confined in traps is mainly discussed. Static properties of the ground state are investigated by using the variational method. The analysis is extended to the stability of two-component condensates. Time-development of the condensate is well-described by the Gross–Pitaevskii equation which is known in nonlinear physics as the no nlinear Schrödinger equation. For the case that the inter-atomic potential is effectively attractive, a singularity of the solution emerges in a finite time. This phenomenon which we call collapse explains the upper bound for the number of atoms in such condensates under traps.

The stability of a low temperature Bose-Einstein condensate with attractive interactions in one and three dimensional double-well potentials is discussed. In particular, the tunnelling dynamics of a condensate under the influence of a timedependent potential gradient is investigated. The condensate is shown to collapse at a critical potential gradient which corresponds to a critical number of atoms in one of the two wells. The sensitivity of this tunnelling induced collapse could provide a useful tool in the study of condensates with attractive interactions. PACS numbers: 03.75.Lm, 03.75.kkTunnelling induced collapse in time-dependent double-well potential 2

We discuss Bose-Einstein condensation in harmonic traps where the confinement has undergone a splitting along one direction. We mostly consider three-dimensional potentials consisting of two cylindrical wells separated a distance

2a

along the z axis. For ideal gases, the thermodynamics of the confined bosons has been investigated, performing exact numerical summations to describe the major details of the transition and comparing the results with the semiclassical density-of-states approximation. We find that for large particle number and increasing well separation, the condensation temperature evolves from the thermodynamic limit value

{T}_{c}^{(0)}(N)

to

{T}_{c}^{(0)}(N/2).

The effects of adding a repulsive interaction between atoms has been examined resorting to the Gross-Pitaevskii-Popov procedure, and it is found that the shift of the condensation temperature exhibits different signs according to the separation between wells. In particular, for sufficiently large splitting, the trend opposes the well-known results for harmonic traps, since the critical temperature appears to increase with growing repulsion strength.

In this work, we study phase transitions in dilute Bose-Einstein Condensates theoretically. The Gross-Pitaevskii equation (GPE) is applied to describe the properties of dilute Bose gases near zero temperature for various confining geometries. Then, using the harmonic trap, the Thomas Fermi equation has been investigated. The Bose-Hubbard Model has been also investigated using the mean field approach. It is indicated that Bose-Einstein condensation is a second order phase transition. We also presented an exactly solvable phase transition model in which the phase transition is purely statistically derived. It is found out that the mean field theory can be applied to a number of physical systems so as to study the phenomena of Berezinsky-Kosterlitz- Thouless (BKT) phase transitions. Keywords : Phase Transition, BEC, GPE, Bose-Hubbard Model, Berezinsky-Kosterlitz-Thouless (BKT), Mean Field equation