Title:

Quantum optical states and BoseEinstein condensation : a dynamical group approach.

The concept of coherent states for a quantum system has been generalized in many
different ways. One elegant way is the dynamical group approach. The subject of this
thesis is the physical application of some dynamical group methods in quantum optics
and BoseEinstein Condensation(BEC) and their use in generalizing some quantum
optical states and BEC states.
We start by generalizing squeezed coherent states to the displaced squeezed phase
number states and studying the signaltoquantum noise ratio for these states. Following
a review of the properties of Kerr states and the basic theory of the deformation
of the boson algebra, we present an algebraic approach to Kerr states and
generalize them to the squeezed states of the qparametrized harmonic oscillator.
Using the eigenstates of a nonlinear densitydependent annihilation operator of the
deformed boson algebra, we propose general time covariant coherent states for any
timeindependent quantum system. Using the ladder operator approach similar to
that of binomial states, we construct interpolating numbercoherent states, intermediate
states which are generalizations of some fundamental states in quantum optics.
Salient statistical properties and nonclassical features of these interpolating numbercoherent
states are investigated and the interaction with an atomic system in the framework of the JaynesCummings model and the scheme to produce these states
are also studied in detail.
After briefly reviewing the realization of BoseEinstein Condensates and relevant
theoretical research using meanfield theory, we present a dynamical group approach
to BoseEinstein condensation and the atomic tunnelling between two condensates
which interact via a minimal coupling term. First we consider the spectrum of one
BoseEinstein condensate and show that the meanfield dynamics is characterised
by the semidirect product of the 8U(1,1) and HeisenbergWeyl groups.We then
construct a generalized version of the BEC ground states and weakly excited states. It
is shown that our states for BEC provide better fits to the experimental results. Then
we investigate the tunnelling between the excitations in two condensates which interact
via a minimal coupling term. The dynamics of the two interacting Bose systems is
characterised by the 80(3,2) group, which leads to an exactly solvable model. Further
we describe the dynamics of the tunnelling of the two coupled condensates in terms
of the semidirect product of 80(3,2) and two independent HeisenbergWeyl groups.
From this we obtain the energy spectrum and eigenstates for the two interacting
BoseEinstein condensates, as well as the Josephson current between the two coupled
condensates.
