34 research outputs found
Linear -Matrix Algebra for Systems Separable\\ in Parabolic Coordinates
We consider a hierarchy of many particle systems on the line with polynomial
potentials separable in parabolic coordinates. Using the Lax representation,
written in terms of matrices for the whole hierarchy, we construct
the associated linear -matrix algebra with the -matrix dependent on the
dynamical variables. A dynamical Yang-Baxter equation is discussed.Comment: 10 pages, LaTeX. Submitted to Phys.Lett.
Quantum Lattice Solitons
The number state method is used to study soliton bands for three anharmonic
quantum lattices: i) The discrete nonlinear Schr\"{o}dinger equation, ii) The
Ablowitz-Ladik system, and iii) A fermionic polaron model. Each of these
systems is assumed to have -fold translational symmetry in one spatial
dimension, where is the number of freedoms (lattice points). At the second
quantum level we calculate exact eigenfunctions and energies of pure
quantum states, from which we determine binding energy , effective
mass and maximum group velocity of the soliton bands as
functions of the anharmonicity in the limit . For arbitrary
values of we have asymptotic expressions for , , and
as functions of the anharmonicity in the limits of large and small
anharmonicity. Using these expressions we discuss and describe wave packets of
pure eigenstates that correspond to classical solitons.Comment: 21 pages, 1 figur
Exact solutions for a family of spin-boson systems
We obtain the exact solutions for a family of spin-boson systems. This is
achieved through application of the representation theory for polynomial
deformations of the Lie algebra. We demonstrate that the family of
Hamiltonians includes, as special cases, known physical models which are the
two-site Bose-Hubbard model, the Lipkin-Meshkov-Glick model, the molecular
asymmetric rigid rotor, the Tavis-Cummings model, and a two-mode generalisation
of the Tavis-Cummings model.Comment: LaTex 15 pages. To appear in Nonlinearit
Small-amplitude excitations in a deformable discrete nonlinear Schroedinger equation
A detailed analysis of the small-amplitude solutions of a deformed discrete
nonlinear Schr\"{o}dinger equation is performed. For generic deformations the
system possesses "singular" points which split the infinite chain in a number
of independent segments. We show that small-amplitude dark solitons in the
vicinity of the singular points are described by the Toda-lattice equation
while away from the singular points are described by the Korteweg-de Vries
equation. Depending on the value of the deformation parameter and of the
background level several kinds of solutions are possible. In particular we
delimit the regions in the parameter space in which dark solitons are stable in
contrast with regions in which bright pulses on nonzero background are
possible. On the boundaries of these regions we find that shock waves and
rapidly spreading solutions may exist.Comment: 18 pages (RevTex), 13 figures available upon reques
Algebraic Bethe ansatz method for the exact calculation of energy spectra and form factors: applications to models of Bose-Einstein condensates and metallic nanograins
In this review we demonstrate how the algebraic Bethe ansatz is used for the
calculation of the energy spectra and form factors (operator matrix elements in
the basis of Hamiltonian eigenstates) in exactly solvable quantum systems. As
examples we apply the theory to several models of current interest in the study
of Bose-Einstein condensates, which have been successfully created using
ultracold dilute atomic gases. The first model we introduce describes Josephson
tunneling between two coupled Bose-Einstein condensates. It can be used not
only for the study of tunneling between condensates of atomic gases, but for
solid state Josephson junctions and coupled Cooper pair boxes. The theory is
also applicable to models of atomic-molecular Bose-Einstein condensates, with
two examples given and analysed. Additionally, these same two models are
relevant to studies in quantum optics. Finally, we discuss the model of
Bardeen, Cooper and Schrieffer in this framework, which is appropriate for
systems of ultracold fermionic atomic gases, as well as being applicable for
the description of superconducting correlations in metallic grains with
nanoscale dimensions. In applying all of the above models to physical
situations, the need for an exact analysis of small scale systems is
established due to large quantum fluctuations which render mean-field
approaches inaccurate.Comment: 49 pages, 1 figure, invited review for J. Phys. A., published version
available at http://stacks.iop.org/JPhysA/36/R6