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Quantum Electronics and Laser Science Conference

Three coupled atomic Bose-Einstein condensates

Kae Nemoto, Gerard J Milburn, William J. Munro

Published Jan 20, 2003• 0 citations
Publisher

Abstract

Summary form only given. We present a quantum and semiclassical analysis of the dynamics of three coupled atomic Bose-Einstein condensates (BEC). The quantum description is most conveniently analyzed in terms of the operator algebra of SU(3). The space of pure states corresponds to a real manifold of five dimensions corresponding to the parameters of the coset SU(3)/U(2). This is the higher dimensional analogue of the Bloch sphere which has been used to describe the dynamics of two coupled BECs. The equations of motion in the Heisenberg picture can be mapped onto equations in this real manifold. The semiclassical description also involves five real parameters, and corresponds to the mean values of the Heisenberg equations with a factorisation assumption for higher order moments. The symmetries of the problem may be used to construct the ground states of the quantum system for both positive and negative scattering lengths. The structure of the ground states change as the elastic scattering strength changes with respect to the linear tunnelling coupling between the condensate components.

DOI
10.1109/qels.1999.807321

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•Jan 1, 2001•0 citations
Summary form only given. We present a quantum and semiclassical analysis of the dynamics of three coupled atomic Bose-Einstein condensates (BEC). The quantum description is most conveniently analyzed in terms of the operator algebra of SU(3). The space of pure states corresponds to a real manifold of five dimensions corresponding to the parameters of the coset SU(3)/U(2). This is the higher dimensional analogue of the Bloch sphere which has been used to describe the dynamics of two coupled BECs. The equations of motion in the Heisenberg picture can be mapped onto equations in this real manifold. The semiclassical description also involves five real parameters, and corresponds to the mean values of the Heisenberg equations with a factorisation assumption for higher order moments. The symmetries of the problem may be used to construct the ground states of the quantum system for both positive and negative scattering lengths. The structure of the ground states change as the elastic scattering strength changes with respect to the linear tunnelling coupling between the condensate components.
The simplest model of three coupled Bose-Einstein condensates is investigated using a group theoretical method. The stationary solutions are determined using the SU(3) group under the mean-field approximation. This semiclassical analysis, using system symmetries, shows a transition in the dynamics of the system from self trapping to delocalization at a critical value for the coupling between the condensates. The global dynamics are investigated by examination of the stable points, and our analysis shows that the structure of the stable points depends on the ratio of the condensate coupling to the particle-particle interaction, and undergoes bifurcations as this ratio is varied. This semiclassical model is compared to a full quantum treatment, which also displays a dynamical transition. The quantum case has collapse and revival sequences superimposed on the semiclassical dynamics, reflecting the underlying discreteness of the spectrum. Nonzero circular current states are also demonstrated as one of the higher-dimensional effects displayed in this system.
We present an approach for the analysis of Bose-Einstein condensates in a few mode approximation. This method has already been used to successfully analyze the vibrational modes in various molecular systems and offers a perspective on the dynamics in many particle bosonic systems. We discuss a system consisting of a Bose-Einstein condensate in a triple well potential. Such systems correspond to classical Hamiltonian systems with three degrees of freedom. The semiclassical approach allows a simple visualization of the eigenstates of the quantum system referring to the underlying classical dynamics. From this classification we can read off the dynamical properties of the eigenstates such as particle exchange between the wells and entanglement without further calculations. In addition, this approach offers insights into the validity of the mean-field description of the many particle system by the Gross-Pitaevskii equation, since we make use of exactly this correspondence in our semiclassical analysis. We choose a three mode system in order to visualize it easily and, moreover, to have a sufficiently interesting structure, although the method can also be extended to higher dimensional systems.
We study the dynamics of two Bose–Einstein condensates (BECs) tunnel-coupled by a double-well potential. A real three-body interaction term is considered and a two-mode approximation is used to derive two coupled equations, which describe the relative population and relative phase. By solving the equations and analyzing the stability of the system, we find the stable stationary solutions for a constant atomic scattering length. When a periodically time-varying scattering length is applied, Melnikov analysis and numerical calculation demonstrate the existence of chaotic behavior and the dependence of chaos on the three-body interaction parameters.
We present a theoretical analysis of three-dimensional (3D) matter-wave solitons and their stability properties in coupled atomic and molecular Bose-Einstein condensates (BECs). The soliton solutions to the mean-field equations are obtained in an approximate analytical form by means of a variational approach. We investigate soliton stability within the parameter space described by the atom-molecule conversion coupling, the atom-atom sss-wave scattering, and the bare formation energy of the molecular species. In terms of ordinary optics, this is analogous to the process of sub- or second-harmonic generation in a quadratic nonlinear medium modified by a cubic nonlinearity, together with a phase mismatch term between the fields. While the possibility of formation of multidimensional spatiotemporal solitons in pure quadratic media has been theoretically demonstrated previously, here we extend this prediction to matter-wave interactions in BEC systems where higher-order nonlinear processes due to interparticle collisions are unavoidable and may not be neglected. The stability of the solitons predicted for repulsive atom-atom interactions is investigated by direct numerical simulations of the equations of motion in a full 3D lattice. Our analysis also leads to a possible technique for demonstrating the ground state of the Schr\"odinger-Newton and related equations that describe Bose-Einstein condensates with nonlocal interparticle forces.
We model an atom-molecule Bose-Einstein condensate (AMBEC) using simplified set of coupled Gross-Pitaevskii equations (GPE), where we neglect the background (elastic) scattering length of the atoms. We analyze the ground state numerically and analytically, and construct its twin state through transformation δ→−δ,K→−Kδ\rightarrow-δ, K\rightarrow-Kδ→−δ,K→−K. We find that the ground state is a collection of three local eigenstates: all-atom state, mixed (atom-molecule) state, and all-molecule state, while the twin state comprises of mixed state with tunable fraction of atoms including unity but excluding all-molecules. We find the analytic boundaries of the local eigenstates from the stability analysis of the underlying all-molecule and all-atom eigenstate. In the ground state we find either regular oscillations in size and fraction of both condensates, or shrinking of the atomic condensate that resembles the collapse. In the twin state we find rapid irregular disintegration of both condensates. We contrast the properties of the mean-field parametric excitations found in AMBEC with the experimentally observed excitations of the BEC. We discuss some enticing possibilities for creation of atomic BEC of controllable half-width by performing the parameter sweep around the boundary.
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.