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Physical review. B, Condensed matter

Generalized SU(1,1) coherent-state theory and quantum characteristics of a weakly interacting Bose system

Hongbin Huang

Published Jun 1, 1993• 7 citations
Publisher

Abstract

We expand the Hamiltonian of a weakly interacting Bose system to second order in the Bose operators of the elementary excitations and use the mean-field approximation to linearize it. It is found that the dynamical algebra of the ``pairing'' approximated Hamiltonian is SU(1,1) algebra. By using the generalized SU(1,1) coherent-state theory we show that both the condensate states and the weakly excited states are generalized SU(1,1) coherent states. Based on these states, the quantum fluctuations and the second-order correlation functions of the Bose system are calculated and discussed.

DOI
10.1103/physrevb.47.14348

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We expand the Hamiltonian of a weakly interacting Bose system to second order in the Bose operators of the elementary excitations and use the mean-field approximation to linearize it. It is found that the dynamical algebra of the ``pairing'' approximated Hamiltonian is SU(1,1) algebra. By using the generalized SU(1,1) coherent-state theory we show that both the condensate states and the weakly excited states are generalized SU(1,1) coherent states. Based on these states, the quantum fluctuations and the second-order correlation functions of the Bose system are calculated and discussed.
Based on the mean field and Bogoliubov approximations, we linearize the Hamiltonian of a nonequilibrium Bose condensed system, which is created by an external coherent pump field, and coupled to a reservoir. It is found that the dynamical algebra of the linearized Hamiltonian is Heisenberg and SU(1,1) Lie algebra if the coupling to reservoir is not taken into account. By using the methods of coherent state, we obtain its eigenvalues and eigen-functions-direct product of displaced squeezed-number-states and generalized SU(1,1) coherent states. Then we introduce the retarded Green's functions of elementary excitations to calculate its spectrum in the coupling situation, and further give the state functions of exciton system in weakly coupling situation.
Based on the mean field and Bogoliubov approximations, we linearize the Hamiltonian of a nonequilibrium Bose condensed system, which is created by an external coherent pump field, and coupled to a reservoir. It is found that the dynamical algebra of the linearized Hamiltonian is Heisenberg and SU(1,1) Lie algebra if the coupling to reservoir is not taken into account. By using the methods of coherent state, we obtain its eigenvalues and eigen-functions-direct product of displaced squeezed-number-states and generalized SU(1,1) coherent states. Then we introduce the retarded Green's functions of elementary excitations to calculate its spectrum in the coupling situation, and further give the state functions of exciton system in weakly coupling situation.
We generalize the perturbation theory of weakly interacting bosons to Bose-Einstein condensates with hyperfine spin F=1.F=1.F=1. Analytical expressions of second-order self-energies beyond the mean-field approximation are derived at zero temperature. We further extend the perturbation theory to the finite-temperature case, and derive the corresponding self-energies. We evaluate these self-energies near the poles of the first-order (mean-field) Green's functions within the on-shell approximation, and expand the results in powers of wave number in the long-wavelength limit. Excitation spectra for each mode are obtained analytically at low temperatures. We also derive the ground-state energy and thermodynamic quantities such as free energy of the system.
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 Bose-Einstein 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 signal-to-quantum 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 q-parametrized harmonic oscillator. Using the eigenstates of a nonlinear density-dependent annihilation operator of the deformed boson algebra, we propose general time covariant coherent states for any time-independent quantum system. Using the ladder operator approach similar to that of binomial states, we construct interpolating number-coherent states, intermediate states which are generalizations of some fundamental states in quantum optics. Salient statistical properties and non-classical features of these interpolating numbercoherent states are investigated and the interaction with an atomic system in the framework of the Jaynes-Cummings model and the scheme to produce these states are also studied in detail. After briefly reviewing the realization of Bose-Einstein Condensates and relevant theoretical research using mean-field theory, we present a dynamical group approach to Bose-Einstein condensation and the atomic tunnelling between two condensates which interact via a minimal coupling term. First we consider the spectrum of one Bose-Einstein condensate and show that the mean-field dynamics is characterised by the semi-direct product of the 8U(1,1) and Heisenberg-Weyl 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 semi-direct product of 80(3,2) and two independent Heisenberg-Weyl groups. From this we obtain the energy spectrum and eigenstates for the two interacting Bose-Einstein condensates, as well as the Josephson current between the two coupled condensates.
The state function of the two-mode SU(1,1) coherent states field interacting with a four-level atom is derived by means of quantum theory.It is studied that the two-mode intensity correlation function,the Cauchy-Schwartz inequality and the second-order coherence properties for each mode of the light field in the system of the two-mode SU(1,1) coherent states field interacting with four-level atom.It is discussed that the influence of the coupling coefficient between the light field and atom and the initial intensity of the light field on file quantum dynamics of the light field.The results of the studies indicate that the statistic properties of the photon depend on the initial state of the system,the correlation of the two-mode of light is both positive and non-classical,the photon anti-bunching of the first mode appear at all time,and the photon bunching and anti-bunching of the second mode appear alternately.