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.
