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Two mode theory of BoseEinstein condensates: interferometry and the Josephson model
Please use this identifier to cite or link to this item: http://hdl.handle.net/1959.3/214617
 Title
 Two mode theory of BoseEinstein condensates: interferometry and the Josephson model
 Author(s)
 Dalton, B. J.; Ghanbari, S.
 Abstract
 This topical review provides an overview of the key theoretical features of BoseEinstein condensates (BECs) in cold atomic gases at near zero temperature in the situation where all the bosons occupy at most two single particle states or modes. This situation applies to singlecomponent BECs in double well trap potentials and to twocomponent BEC in single well trap potentials, such as occur when BEC are used in interferometry experiments. The Hamiltonian is introduced in terms of field operators and mode expansions are restricted to a total of two modes. Spin operators and their eigenstates are introduced as the fundamental basis states for describing the twomode N boson quantum system. The spin states have a macroscopic angular momentum quantum number of N/2 and the magnetic quantum number k specifies the relative number of bosons in the two modes. The treatment presented involves an extensive use of angular momentum theory, including unitary rotation operators. Important states of the twomode system such as binomial or coherent states, relative phase eigenstates are discussed. Boson position measurements are specified via quantum correlation functions, and the use of these functions in describing coherence properties, interference patterns and fragmentation effects in BECs is presented. The Bloch vector is defined and related to the quantum correlation functions, with quantum fluctuations of the Bloch vector being treated in terms of the covariance matrix. Applications to important twomode states are made. Spin squeezing is discussed. Based on applying variational principles, the general dynamical behaviour of the twomode BEC is determined via generalised Gross Pitaevskii equations for the modes and matrix mechanics equations for the probability amplitudes of the relative number basis states, the mode and amplitude equations being coupled and selfconsistent. The single mode equations are also presented. The Hamiltonian is written in terms of the spin operators and the Josephson Hamiltonian obtained as a simplification in which the dynamical behaviour of the mode functions is ignored  for the onecomponent case the mode functions are also required to be localised and separate. Coefficients in the Josephson Hamiltonian describe tunneling/intercomponent coupling, asymmetry and collisions and these are defined via integrals involving the mode functions. The Josephson model involves using the Josephson Hamiltonian to give simple predictions of the energy states and dynamical behaviour of the twomode system, dynamical effects on the mode functions being ignored. The three regimes  Rabi, Josephson and Fock are described, and the energy states obtained for the Fock and Rabi regimes. Dynamical behaviour treatments based on the Josephson model are outlined. In the situation where all bosons are in the same single particle state, semiclassical Bloch equations are derived and their solutions given in terms of elliptic functions. The quantum regime is treated using matrix mechanics equations for the probability amplitudes. Two representative applications of the Josephson model dynamics are treated, with graphs showing the results of numerical work being displayed. The first is in describing Heisenberg limited BEC interferometry for a singlecomponent BEC in a double well, the treatment showing collapses and revivals in the probability distribution for the relative phase. The second treats Ramsey interferometry for a twocomponent BEC in a single well, the study revealing that oscillations of the Bloch vector collapse and revive, with te Bloch vector's departure from the Bloch sphere during the collapse period revealing that the BEC has fragmented. In both cases collisions cause the dephasing effects that result in the collapse, revival phenomena. The review ends with a brief outline of phase space and other approaches that extend the treatment beyond the twomode theory, enabling decoherence effects associated with bosons in noncondensate modes to be studied. A summary of the review contents is included. Detailed mathematical derivations are included in several appendices, available as online supplementary material.
 Publication type
 Journal article
 Research centre
 Swinburne University of Technology. Faculty of Engineering and Industrial Sciences. Centre for Atom Optics and Ultrafast Spectroscopy
 Source
 Journal of Modern Optics, Vol. 59, no. 4 (Feb 2012), pp. 287353
 Publication year
 2012
 FOR Code(s)
 0205 Optical Physics; 0206 Quantum Physics; 1007 Nanotechnology
 Keyword(s)
 BoseEinstein condensates; Interferometry; Josephson model; Twomode theory
 Publisher
 Taylor & Francis
 ISSN
 09500340
 Publisher URL

http://dx.doi.org/10.1080/09500340.2011.632100
 Copyright
 Copyright © 2012 Taylor & Francis.
 Peer reviewed