63 research outputs found
The two-site Bose--Hubbard model
The two-site Bose--Hubbard model is a simple model used to study Josephson
tunneling between two Bose--Einstein condensates. In this work we give an
overview of some mathematical aspects of this model. Using a classical
analysis, we study the equations of motion and the level curves of the
Hamiltonian. Then, the quantum dynamics of the model is investigated using
direct diagonalisation of the Hamiltonian. In both of these analyses, the
existence of a threshold coupling between a delocalised and a self-trapped
phase is evident, in qualitative agreement with experiments. We end with a
discussion of the exact solvability of the model via the algebraic Bethe
ansatz.Comment: 10 pages, 5 figures, submitted for publication in Annales Henri
Poincar
Quantum Dynamics of Atom-molecule BECs in a Double-Well Potential
We investigate the dynamics of two-component Bose-Josephson junction composed
of atom-molecule BECs. Within the semiclassical approximation, the multi-degree
of freedom of this system permits chaotic dynamics, which does not occur in
single-component Bose-Josephson junctions. By investigating the level
statistics of the energy spectra using the exact diagonalization method, we
evaluate whether the dynamics of the system is periodic or non-periodic within
the semiclassical approximation. Additionally, we compare the semiclassical and
full-quantum dynamics.Comment: to appear in JLTP - QFS 200
Chemical inhibition of xylem cellular activity impedes the removal of drought-induced embolisms in poplar stems – new insights from micro-CT analysis
The extended Heine-Stieltjes polynomials associated with a special LMG model
New polynomials associated with a special Lipkin-Meshkov-Glick (LMG) model
corresponding to the standard two-site Bose-Hubbard model are derived based on
the Stieltjes correspondence. It is shown that there is a one-to-one
correspondence between zeros of this new polynomial and solutions of the Bethe
ansatz equations for the LMG model.A one-dimensional classical electrostatic
analogue corresponding to the special LMG model is established according to
Stieltjes early work. It shows that any possible configuration of equilibrium
positions of the charges in the electrostatic problem corresponds uniquely to
one set of roots of the Bethe ansatz equations for the LMG model, and the
number of possible configurations of equilibrium positions of the charges
equals exactly to the number of energy levels in the LMG model. Some relations
of sums of powers and inverse powers of zeros of the new polynomials related to
the eigenenergies of the LMG model are derived.Comment: 11 pages, LaTe
Quantum dynamics of a model for two Josephson-coupled Bose--Einstein condensates
In this work we investigate the quantum dynamics of a model for two
single-mode Bose--Einstein condensates which are coupled via Josephson
tunneling. Using direct numerical diagonalisation of the Hamiltonian, we
compute the time evolution of the expectation value for the relative particle
number across a wide range of couplings. Our analysis shows that the system
exhibits rich and complex behaviours varying between harmonic and non-harmonic
oscillations, particularly around the threshold coupling between the
delocalised and self-trapping phases. We show that these behaviours are
dependent on both the initial state of the system as well as regime of the
coupling. In addition, a study of the dynamics for the variance of the relative
particle number expectation and the entanglement for different initial states
is presented in detail.Comment: 15 pages, 8 eps figures, accepted in J. Phys.
On quantum group symmetry and Bethe ansatz for the asymmetric twin spin chain with integrable boundary
Motivated by a study of the crossing symmetry of the `gemini' representation
of the affine Hecke algebra we give a construction for crossing tensor space
representations of ordinary Hecke algebras. These representations build
solutions to the Yang--Baxter equation satisfying the crossing condition (that
is, integrable quantum spin chains). We show that every crossing representation
of the Temperley--Lieb algebra appears in this construction, and in particular
that this construction builds new representations. We extend these to new
representations of the blob algebra, which build new solutions to the Boundary
Yang--Baxter equation (i.e. open spin chains with integrable boundary
conditions).
We prove that the open spin chain Hamiltonian derived from Sklyanin's
commuting transfer matrix using such a solution can always be expressed as the
representation of an element of the blob algebra, and determine this element.
We determine the representation theory (irreducible content) of the new
representations and hence show that all such Hamiltonians have the same
spectrum up to multiplicity, for any given value of the algebraic boundary
parameter. (A corollary is that our models have the same spectrum as the open
XXZ chain with nondiagonal boundary -- despite differing from this model in
having reference states.) Using this multiplicity data, and other ideas, we
investigate the underlying quantum group symmetry of the new Hamiltonians. We
derive the form of the spectrum and the Bethe ansatz equations.Comment: 43 pages, multiple figure
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