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arXiv:quant-ph/9809079

Gardiner's phonon for Bose-Einstein Condensation: A physical realization of the $q$-deformed Boson

C. P. Sun, Shi Yu, Yune Gao

Published Sep 26, 1998• 0 citations
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Abstract

The Gardiner's phonon presented for a particle-number conserving approximation method to describe the dynamics of Bose-Einstein Condesation (BEC) (C.W. Gardiner, Phys. Rev. A 56, 1414 (1997)) is shown to be a physical realization of the $q$-deformed boson, which was abstractly developed in quantum group theory. On this observation, the coherent output of BEC atoms driven by a radio frequency (r.f) field is analyzed in the viewpoint of a $q$-deformed Fock space. It is illustrated that the $q$-deformation of bosonic commutation relation corresponds to the non-ideal BEC with the finite particle number $N$ of condensated atoms. Up to order 1/N, the coherent output state of the untrapped atoms minimizes the uncertainty relation like a coherent state does in the ideal case of BEC that $N$ approaches infinity or $q=1-2/N$ approaches one.

DOI
10.48550/arxiv.quant-ph/9809079

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•Jan 1, 2012•1 citation
Gardiner's phonon for Bose-Einstein Condensation: A physical realization of the qqq-deformed Boson
The Gardiner's phonon presented for a particle-number conserving approximation method to describe the dynamics of Bose-Einstein Condesation (BEC) (C.W. Gardiner, Phys. Rev. A 56, 1414 (1997)) is shown to be a physical realization of the qqq-deformed boson, which was abstractly developed in quantum group theory. On this observation, the coherent output of BEC atoms driven by a radio frequency (r.f) field is analyzed in the viewpoint of a qqq-deformed Fock space. It is illustrated that the qqq-deformation of bosonic commutation relation corresponds to the non-ideal BEC with the finite particle number NNN of condensated atoms. Up to order 1/N, the coherent output state of the untrapped atoms minimizes the uncertainty relation like a coherent state does in the ideal case of BEC that NNN approaches infinity or q=1−2/Nq=1-2/Nq=1−2/N approaches one.
The spectrum of light scattered from a Bose–Einstein condensate is studied in the limit of particle-number conservation. To this end, a description in terms of deformed bosons is invoked and this leads to a deviation from the usual predict spectrum’s shape as soon as the number of particles decreases. PACS number(s): 03.65.Fd (Algebraic methods), 03.75.Fi (Bose condensation), 42.50.Ct (Quantum statistical description of interaction of light and matter) 1 The recent achievements of Bose–Einstein condensate (BEC) with a gas of atoms confined by a magnetic trap [1] has stimulated renewed interest in the question as to what signatures Bose–Einstein condensation imprints in the spectrum of light scattered from atoms in such a condensate [2, 3]. As well known, to deal with the dynamics of BEC gas the Bogolubov approximation in quantum many-body theory [4] is an efficient approach, in which the creation
The q -deformed Bose–Einstein distribution derived from the q -boson algebra is used to study the low-temperature behavior of an ideal q -deformed Bose gas with relativistic energy spectrum. The effects of q -deformation on the properties relative to Bose–Einstein condensation (BEC) are discussed. It is shown that q -deformation leads to some novel characteristics different from those of an original Bose gas, which include the criteria on the occurrence of BEC, critical temperature and jump of heat capacity at the critical point. The results obtained here provide a unified description for the properties of q -deformed Bose systems from the nonrelativistic case to the ultrarelativistic limit, so that some important conclusions in the literature are included in this paper.
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.
In this paper, we investigate the spectrum of light scattered from a Bose-Einstein condensate in the framework of -deformed boson.We use an -deformed quantum model in which the Gardiner's phonon operators for BEC are deformed by an operator-valued function,, of the particle-number operator n .We consider the collisions between the atoms as a special kind of -deformation.The collision rateκ is regarded as the deformation parameter and the spectrum of light scattered from the deformed BEC is analyzed.In particular, we find that with increasing the values of deformation parameters κ and N 1 = η ( , total number of condensate atoms) the scattering spectrum shows deviation from the spectrum associated with nondeformed Bose-Einstein condensate.
A recent paper of Gardiner [Phys. Rev. A 56, 1414 (1997)] introduces a particle-number-conserving Bogoliubov method for the excitation spectrum of a Bose-condensed gas, for use in theories of recently experimentally produced trapped atomic Bose condensates. Gardiner's approach is compared and contrasted with the 1959 Girardeau-Arnowitt theory [Phys. Rev. 113, 755 (1959)], to which it is closely related and which is also fully number conserving. The number-conserving Bogoliubov quasiparticle operators of the Girardeau-Arnowitt theory satisfy Bose commutation relations exactly so long as states with the condensate totally depleted are neglected, whereas those of Gardiner satisfy Bose commutation relations only in an approximation that deteriorates progressively as the condensate is depleted.
When a gas of bosonic atoms is cooled sufficiently near to absolute zero, it undergoes a startling transition from a collection of randomly-behaved individual atoms to a phase in which thousands, even millions, of atoms act in collective unison. In this new phase, known as a Bose-Einstein condensate (BEC), the atoms all occupy the exact same quantum mechanical ground state and can be collectively described by a single phase-coherent wavefunction of macroscopic size. But how exactly does this harmonious behaviour arise within a collection of incoherent atoms as their temperature decreases? Does the condensate wavefunction grow from a single central nucleus, or does it emerge simultaneously throughout the system? And how does the situation differ when the gas is cooled suddenly, away from thermodynamic equilibrium? Can non-equilibrium behaviour be inherited by the condensate that forms? In this thesis we aim to answer these and other questions by studying the formation dynamics of Bose-Einstein condensates and the development of phase coherence. The research described herein makes use of several theoretical tools, which we lay out on the workbench in Chapter 2: Semi-classical Hartree-Fock theory provides a mean-field description of a condensate and thermal cloud at equilibrium; Quantum kinetic theory is used to calculate the dynamics of condensate growth into an equilibrium state, from a non-equilibrium thermal cloud; The stochastic projected Gross-Pitaevskii equation (SPGPE) employs a classical-field description to encapsulate the non-equilibrium dynamics of the condensate and low-energy region of the atom cloud, coupled to an equilibrium thermal cloud. This thesis is composed of four principal studies, each constituting a separate chapter.In Chapter 3 we study the formation of a Bose-Einstein condensate in a cigar-shaped 3D harmonic trap, induced by the controlled addition of an attractive ``dimple'' potential along the weak axis in a recent experiment [Garrett et al., Phys. Rev. A 83, 013630 (2011)]. In this manner condensation is induced without cooling due to a localized increase in the phase space density. We perform a quantitative analysis of the thermodynamic transformation in both the sudden and adiabatic regimes for a range of dimple widths and depths. We find good agreement with equilibrium calculations based on self-consistent semi-classical Hartree-Fock theory describing the condensate and thermal cloud. We observe there is an optimal dimple depth that results in a maximum in the condensate fraction. We also study the non-equilibrium dynamics of condensate formation in the sudden turn-on regime, finding good agreement for the observed time dependence of the condensate fraction with calculations based on quantum kinetic theory. In Chapter 4 we study the onset of phase coherence during the formation of a BEC following a sudden quench to below the transition temperature. Using the SPGPE, we simulate a recent experiment performed at ANU in which a sudden RF quench is applied to a dilute gas of metastable Helium atoms confined in an elongated 3D harmonic magnetic trap. At regularly-spaced intervals following the quench, broadband RF pulses are applied to homogeneously outcouple small (<5%) pulses of atoms, which expand and fall to a microchannel plate detector. The second-order correlation function at the detector, postulated to provide a measure of phase coherence of the in-situ cloud from which they were outcoupled, is measured versus time and compared with the condensate growth curve. A delay was observed in the relaxation of the second-order correlation function, relative to the peak in the number of condensate atoms. In our simulations we incorporate the effects of periodic outcoupling both the condensate and thermal cloud, and monitor the time-dependence of first- and second-order correlation functions in both position and momentum space. We observe long-range phase coherence in the final equilibrium state, though mild persistent phase fluctuations are present. We also observe the curious phenomenon of ``superbunching'' - an enhancement of density fluctuations - at specific values of the momentum, which we relate to anomalous correlations of fluctuations. Despite obtaining a good quantitative fit to the experimentally-measured condensate growth curve in our SPGPE simulations, we do not observe any evidence of the delay in coherence observed in the experiment, which remains unexplained. In Chapter 5 we use the SPGPE to study the crossover from a true BEC with long-range phase coherence to a quasicondensate dominated by phase fluctuations, for the case of a Bose gas in an highly-elongated 3D harmonic trap. We use parameters corresponding to an experiment in which a sudden quench resulted in the formation of a non-equilibrium condensate, the dynamics of which we study in the chapter that follows. Sweeping through a range of temperatures and chemical potentials within the crossover region, we reveal the behaviour of phase fluctuations through analysis of correlation functions. We compare the ideal-gas approximation for the transition temperature with the more accurate value obtained via the Binder cumulant. By comparing the phase coherence length with the spatial extent across which density fluctuations are suppressed, we identify the BEC/quasicondensate crossover point. We compare directly with predictions of mean-field theory [Petrov et al., Phys. Rev. Lett. 87, 050404 (2001)] and find that phase fluctuations persist to lower temperatures in our SPGPE simulations. We calculate anomalous correlations of the fluctuation field and show that the resulting correction to the mean-field condensate self-interaction energy is significant within the quasicondensate regime. In Chapter 6 we study the formation of a BEC into a non-equilibrium state by simulating an experiment in which a dilute gas of Rubidium-87 atoms in a highly-elongated 3D harmonic magnetic trap is rapidly quenched to below the transition temperature. Following the quench, a condensate is observed to form with an excessively elongated shape, and subsequently undergo damped quadrupole shape oscillations [Shvarchuck et al., Phys. Rev. Lett. 89, 270404 (2002)]. We simulate this experiment by modifying the SPGPE to incorporate a hydrodynamic thermal cloud, parametrized by axially and temporally-dependent temperature and chemical potential. Comparison against simulations without this axial dependence suggests the hydrodynamic behaviour of the thermal cloud plays a crucial role in the formation of a non-equilibrium condensate. Analyses of the first-order correlation function and the condensate wavefunction reveal the spontaneous formation of quantum vortices, as well as strong phase fluctuations at equilibrium indicating quasicondensate behaviour.