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Acta Physica Sinica

COHERENT-STATE TREATMENT OF NONEQUILIBRIUM BOSE CONDENSATION

HUANG HONG-BIN, 东南大学物理系,南京210018

Published Jan 1, 1993• 3 citations
Publisher

Abstract

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.

DOI
10.7498/aps.42.1385

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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 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.
To study the coherent states of the Fermi-Bose superfluid mixture system,by using the mean-field approximation and Lie Algebra,it shows that the system is in the SU(2)SU(1,1)coherent-states at T=0.The result of the ground state energy is the same as that of Bogoliubov transformation.However,the coherent state method is convenient than Bogoliubov transformation in treating the BEC-BCS systems because this method can give the both the state function and the corresponding eigenvalues of the systems.
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.
Abstract The review presents the methods of generation of nonlinear coherent excitations in strongly nonequilibrium Bose-condensed systems of trapped atoms and their properties. Non-ground-state Bose–Einstein condensates are represented by nonlinear coherent modes. The principal difference of nonlinear coherent modes from linear collective excitations is emphasized. Methods of generating nonlinear modes and the properties of the latter are described. Matter-wave interferometry with coherent modes is discussed, including such effects as interference patterns, internal Josephson current, Rabi oscillations, Ramsey fringes, harmonic generation, and parametric conversion. Dynamic transition between mode-locked and mode-unlocked regimes is shown to be analogous to a phase transition. Atomic squeezing and entanglement in a lattice of condensed atomic clouds with coherent modes are considered. Nonequilibrium states of trapped Bose-condensed systems, starting from weakly nonequilibrium state, vortex state, vortex turbulence, droplet or grain turbulence, and wave turbulence, are classified by means of effective Fresnel and Mach numbers. The inverse Kibble–Zurek scenario is described. A method for the formation of directed beams from atom lasers is reported.