Generalized coherent states for discrete and continuous dynamics and their applications

We investigate new forms of generalized coherent states for several systems with no direct reference to their group structures. A general procedure of constructing a set of coherent states, as proposed by Perelomov, is to use an irreducible representation of a Lie group acting on an extremal state (e.g., vacuum state). However, Klauder put forth a set of conditions to define coherent states in a way free from any underlying group structure. First we critically examine Klauder's coherent states, as modified by Gazeau and Klauder, which are defined over energy eigenstates of a physical system, and construct a new modified class of coherent states satisfying Klauder's criteria. Our set of coherent states is a natural generalization of the harmonic oscillator coherent states and encompasses both the discrete and continuous spectra in a unified manner. We apply our formulation to a particle in a one-dimensional box, a particle in the Poschl-Teller potential, and the compactified Coulomb problem in order to demonstrate explicitly the limiting procedure for going from the coherent states for discrete dynamics to those for continuous dynamics. We also discuss the path integral with our coherent states. Secondly, we construct two different classes of coherent states for a cubic SU(2) or Higgs algebra. The first class is constructed by following a Perelomov-like procedure so as to retain the usual SU(2) limit; and the second one is the cubic SU(2) version of our generalized coherent states similar to those of Gazeau and Klauder. The final topic we discuss in this thesis is related to quantum computation with coherent states. We show that simulation of a dynamical quantum system can be performed on a coherent quantum computer as a Quantum Fourier Transform operator to the extremal state belonging to the irreducible representation of the group. The states obtained in this manner are termed as Perelomov type coherent states. Our construction here is based on applying the deformed operators directly to the extremal state as opposed to the previous work in which the undeformed operators are constructed from the deformed ones and then the states are constructed in the Perelomov sense. The states so constructed reduce smoothly to the usual SU(2) coherent states in the flat space limit, whereas, this feature is not evident in the previously proposed states. Another subject of our discussion in this thesis is the quantum computation with optical coherent states. In the last part of this work, we show, following Zalka's prescription, that simulation of a quantum system can be carried out efficiently on a coherent state quantum computer.