We consider the precise quantum state of two trapped, coupled Bose-Einstein condensates in the two-mode approximation. We seek a representation of the state in terms of a Wigner-like distribution on the two-mode Bloch sphere. The problem is solved using a self-consistent rotation of the unknown state to the south pole of the sphere. The two-mode Hamiltonian is projected onto the harmonic-oscillator phase plane, where it can be solved by standard techniques. Our results show how the number of atoms in each trap and the squeezing in the number difference depend on the physical parameters. Considering negative scattering lengths, we show that there is a regime of squeezing in the relative phase of the condensates which occurs for weaker interactions than the superposition states found by Cirac et al. (quant-ph/9706034). The phase squeezing is also apparent in mildly asymmetric trap configurations.
We consider the precise quantum state of two trapped, coupled Bose-Einstein condensates in the two-mode approximation. We seek a representation of the state in terms of a Wigner-like distribution on the two-mode Bloch sphere. The problem is solved using a self-consistent rotation of the unknown state to the south pole of the sphere. The two-mode Hamiltonian is projected onto the harmonic-oscillator phase plane, where it can be solved by standard techniques. Our results show how the number of atoms in each trap and the squeezing in the number difference depend on the physical parameters. Considering negative scattering lengths, we show that there is a regime of squeezing in the relative phase of the condensates which occurs for weaker interactions than the superposition states found by Cirac et al. (quant-ph/9706034). The phase squeezing is also apparent in mildly asymmetric trap configurations.
This topical review provides an overview of the key theoretical features of Bose–Einstein 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 single-component BECs in double well trap potentials and to two-component 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 two-mode 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 two-mode 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 two-mode states are made. Spin squeezing is discussed. Based on applying variational principles, the general dynamical behaviour of the two-mode 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 self-consistent. 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 one-component 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 two-mode 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, semi-classical 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 single-component 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 two-component BEC in a single well, the study revealing that oscillations of the Bloch vector collapse and revive, with the 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 two-mode theory, enabling decoherence effects associated with bosons in non-condensate 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.
In this work, we investigate the quantum dynamics of a model for two single-mode Bose–Einstein condensates which are coupled via Josephson tunnelling. Using direct numerical diagonalization of the Hamiltonian, we compute the time evolution of the expectation value for the relative particle number across a wide range of couplings. Our analysis shows that the system exhibits rich and complex behaviours varying between harmonic and non-harmonic oscillations, particularly around the threshold coupling between the delocalized and self-trapping phases. We show that these behaviours are dependent on both the initial state of the system and regime of the coupling. In addition, a study of the dynamics for the variance of the relative particle number expectation and the entanglement for different initial states is presented in detail.
Dynamical evolution of quantum mechanical squeezing and entanglement in a two-mode Bose-Einstein condensate (TBEC) with an adiabatic, time-varying Raman coupling is studied by finding analytical expressions for these quantities. In particular, we study the entanglement between the atoms in the condensate as well as that between the two modes. The nature of the enhanced quantum correlations in TBEC is clarified by considering squeezing and entanglement both with and without the nonlinear interaction turned on; it is found that entanglement approaching maximal value can be achieved with the nonlinear interactions present. Somewhat counterintuitively, greater squeezing is found in the absence of nonlinear interactions. This is due to the collapses and revivals of the TBEC quantum state induced by the nonlinear interactions. In addition, results involving the self-trapping phase state of TBEC indicate potential for creating a dynamically stable, macroscopic entangled quantum state which is relatively robust with respect to atom number fluctuations.
The properties of an atomic Bose-Einstein condensate in a double-well potential are investigated through a two-mode analysis. An analytic solution for the semiclassical tunnelling and self-trapping dynamics is compared with numerical simulations of the quantum dynamics, which exhibit collapses and revivals for a closed system. A continuous non-destructive measurement technique to monitor the Josephson tunnelling oscillations is presented, in which the condensate in one well dispersively shifts the phase of a coherent probe beam in proportion to atom-number. The evolution of the resulting homodyne photocurrent and Bloch Q distributions shows that oscillations develop even when the initial state possesses phase symmetry. The conditional dynamics of the condensate which result from measurement back-action also appear in certain semiclassical formulations. The homodyne measurement technique is incorporated into a proposed weak-force detector. A maximally entangled initial state, which is the ground state for a double condensate with strong attractive atomic interactions, enables a high-precision measurement. The dynamics of quantum many-body multimode systems of interacting bosons are simulated using phase-space methods. The use of the Wigner technique predicts novel noise effects in fibre solitons. The positive-P representation is used to simulate the formation of mesoscopic Bose-Einstein condensates via evaporative cooling in three dimensional atom traps. The results indicate highly non-classical behaviour near the critical point, and provide evidence for the spontaneous formation of vortices. Comparisons with corresponding mean-field calculations reveal large differences between the semiclassical and fully quantum results. Finally, the possibility of future progress with alternative phase-space methods is considered.
We discuss the coherent atomic oscillations between two weakly coupled Bose-Einstein condensates. The weak link is provided by a laser barrier in a (possibly asymmetric) double-well trap or by Raman coupling between two condensates in different hyperfine levels. The boson Josephson junction (BJJ) dynamics is described by the two-mode nonlinear Gross-Pitaevskii equation that is solved analytically in terms of elliptic functions. The BJJ, being a neutral, isolated system, allows the investigations of dynamical regimes for the phase difference across the junction and for the population imbalance that are not accessible with superconductor Josephson junctions (SJJ's). These include oscillations with either or both of the following properties: (i) the time-averaged value of the phase is equal to $\ensuremath{\pi} (\ensuremath{\pi}$-phase oscillations); (ii) the average population imbalance is nonzero, in states with macroscopic quantum self-trapping. The (nonsinusoidal) generalization of the SJJ ac and plasma oscillations and the Shapiro resonance can also be observed. We predict the collapse of experimental data (corresponding to different trap geometries and the total number of condensate atoms) onto a single universal curve for the inverse period of oscillations. Analogies with Josephson oscillations between two weakly coupled reservoirs of ${}^{3}\mathrm{He}\ensuremath{-}B$ and the internal Josephson effect in ${}^{3}\mathrm{He}\ensuremath{-}A$ are also discussed.
We study the coherence fluctuations in a Josephson junction created by two weakly coupled Bose-Einstein condensates. We apply the two-mode approximation in the full quantum theory analysis, and so that the system is described by a two-site Bose-Hubbard Hamiltonian. The coherence fluctuations are determined by the total number of particles N, charging energy EC, junction energy EJ, asymmetry δ and temperature T. For zero-temperature symmetric systems, the coherence fluctuations increase with EC/EJ, however, the number squeezing increases with N2EC/EJ. For zero-temperature asymmetric systems, the coherence fluctuations are suppressed by the balance effects between asymmetry and interparticle interaction and the single-atom resonance effects. The coherence fluctuations grow with temperature, and the formation of a self-trapping state will cause abnormal behaviors at high temperatures.