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In the year 1999 our group started work in a completely new and very exciting field of the Bose-Einstein condensation (BEC). In November 1999, we organized an international workshop on BEC and Atom Lasers, and hosted several visitors from Spain, New Zealand, and Australia. Since then, we have initiated a number of projects in this rapidly developing area of physics. Nonlinear modes of the BEC. The collective excitations of the BEC confined in a trap can be described in terms of nonlinear modes of the condensate. These are solutions of the nonlinear eigenvalue problem for the condensate wavefunction. Our first project in this field was to characterize nonlinear modes of the BEC in a parabolic trap [Phys. Lett. A 278, 225 (2001)]. The concept of nonlinear modes is vital for our understanding of both steady-state and dynamical properties of condensates. For example, we have demonstrated that a condensate wavefunction in a double-well potential can, with a good accuracy, be represented as a superposition of lowest nonlinear modes of the entire potential [Phys. Rev. A 61, 031601 (2000)]. This work has allowed us to describe the Josephson-like oscillations in a double-well trap for any well separation. It also represents the first step in the theoretical treatment of BEC in a lattice potential beyond the tight-binding approximation. Atomic-molecular BEC. Coherent photoassociation of BEC atoms and formation of atomic-molecular BEC, is the matter-wave analog to the process of second harmonic generation in nonlinear optics. Recent experimental progress in this area and observation of coherent superposition of atoms and molecules, assisted by Feshbah resonance, has revitalised the interest in physics of hybrid condensates. We have applied some ideas and concepts of parametric optical interactions to the theory of atomic-molecular condensates (AMBEC) and successfully completed a project on the dynamics and stability of the AMBEC [Phys. Rev. A 65, 013609 (2002)]. We are now pursuing a research on stable topological states of the AMBEC, such as ring vortices and dark solitons, and investigating the effects of losses on the condensate [Phys. Rev. E 65, 026611 (2002), J. Opt. B Quantum Semicl. Opt. 4, S33 (2002)]. Atom Lasers. The dynamics of the coherent beams of atoms, the so-called atom lasers, formed by continuously coupling the condensate out of the trap is the project which is developing in a close collaboration with the theoretical and experimental Atom Optics groups at the Department of Physics and Theoretical Physics, The Faculties. In May 2001, the Atom Optics Lab at the Faculties have produced the first Australian BEC which takes them a step closer to establishing the Nation-wide Atom Laser Facility in the forthcoming years. A number of important theoretical results that are going to be put to test by the experiment have already been produced in collaboration with our group [cond-mat/0004127, Phys. Rev. A 64, 043605 (2001)]. Spinor BEC. Spinor BECs, or optically trapped ultracold atomic clouds, are subject to parametric coupling between the spin degrees of freedom. They exhibit a number of fascinating physical effects, some of which have already been observed experimentally. Drawing yet another analogy with nonlinear optics, we have performed a detailed study of modulational instability (MI) of the spinor condensates [Phys. Rev. A 64, 021601 (2001)]. The work is now underway to apply the MI analysis to spinor BECs in optical lattices. BEC in optical lattices. Optical lattices are periodic light shift potentials for atoms created by the interference of multiple laser beams. A BEC, loaded into a one-dimensional (1D), 2D, or 3D optical lattice, becomes a testground for a range of fascinating physical effects because the lattice potential can be easily manipulated by changing the geometry, polarization, phase, or intensity of the laser beams. BEC in an optical lattice can be regarded as the reconfigurable analog of a nonlinear Photonics Band Gap (PBG) structure for matter waves - an atomic band gap (ABG) structure. The optical potential of a lattice plays the role of the periodically modulated refractive index of a dielectric and the Kerr nonlinearity is emulated by the atom-atom interactions. The close analogy between light and matter waves suggests the intriguing possibility that the concepts employed in the study of nonlinear PBG structures can be applied to the nonlinear optics of coherent matter waves in optical lattices. One of the phenomena exhibited by single- and multi-component BECs in multi-dimensional optical lattices is the existence of nonlinear localized modes - matter-wave "gap" solitons. Our current work aims to identify the conditions for the formation and stability of such localized matter waves in 1D and 2D optical lattices.
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