We develop the Fock-space many-body quantum approach for large number of bosons, when the bosons occupy significantly only few modes. The approach is based on an analogy with the dynamics of a single particle of either positive or negative mass in a quantum potential, where the inverse number of bosons plays the role of Planck constant. As application of the method we consider the Bose-Einstein condensate in a double-well trap. The ground state of positive mass particle corresponds to the mean-field fixed point of lower energy, while that of negative mass to the excited fixed point. In the case of attractive BEC above the threshold for symmetry breaking, the ground state is a cat state and we relate this fact to the double-well shape of the quantum potential for the positive mass quantum particle. The tunneling energy splitting between the local Fock states of the cat state is shown to be extremely small for the nonlinearity parameter just above the critical value and exponentially decreasing with the number of atoms. In the repulsive case, the phase locked ($π$-phase) macroscopic quantum self-trapping of BEC is related to the double-well structure of the potential for the negative mass quantum particle. We also analyze the running phase macroscopic quantum self-trapping state and show that it is subject to quantum collapses and revivals. Moreover, the quantum dynamics just before the first collapse of the running phase shows exponential growth of dispersion of the quantum phase distribution which may explain the growth of the phase fluctuations seen in the experiment on the macroscopic quantum self-trapping.
We develop the Fock-space many-body quantum approach for large number of bosons, when the bosons occupy significantly only few modes. The approach is based on an analogy with the dynamics of a single particle of either positive or negative mass in a quantum potential, where the inverse number of bosons plays the role of Planck constant. As application of the method we consider the Bose-Einstein condensate in a double-well trap. The ground state of positive mass particle corresponds to the mean-field fixed point of lower energy, while that of negative mass to the excited fixed point. In the case of attractive BEC above the threshold for symmetry breaking, the ground state is a cat state and we relate this fact to the double-well shape of the quantum potential for the positive mass quantum particle. The tunneling energy splitting between the local Fock states of the cat state is shown to be extremely small for the nonlinearity parameter just above the critical value and exponentially decreasing with the number of atoms. In the repulsive case, the phase locked (π-phase) macroscopic quantum self-trapping of BEC is related to the double-well structure of the potential for the negative mass quantum particle. We also analyze the running phase macroscopic quantum self-trapping state and show that it is subject to quantum collapses and revivals. Moreover, the quantum dynamics just before the first collapse of the running phase shows exponential growth of dispersion of the quantum phase distribution which may explain the growth of the phase fluctuations seen in the experiment on the macroscopic quantum self-trapping.
We develop a Fock-space WKB method for a Bose-Einstein condensate (BEC) in a double-well trap, using an analogy with a single particle of either positive or negative mass in a quantum potential. The usual mean-field approach is the classical limit, and the inverse number of atoms plays the role of Planck constant. The ground state of positive-mass particles corresponds to the mean-field fixed point of lower energy, while that of negative mass to the excited fixed point. In the case of attractive BECs above the threshold for symmetry breaking, the ground state is the Schr\"odinger cat state and we relate this to the double-well shape of the potential in Fock space. In the repulsive case, we identify the quantum states corresponding to the mean-field macroscopic self-trapping phenomenon. In particular, the phase-locked ($\ensuremath{\pi}$ phase) macroscopic quantum self-trapping of BECs is related to the double-well shape of the potential in Fock space. The running-phase macroscopic quantum self-trapping state is found to be subject to quantum collapses and revivals. The phase dispersion grows exponentially, reaching the absolute maximum just before the first collapse of the running phase, which may explain the growth of the phase fluctuations seen in the experiment on macroscopic quantum self-trapping.
Abstract Bose-Einstein Condensate (BEC) is a paradigmatic representative of macroscopic quantum states. 39 K atoms serve as an ideal system for realizing a nearly ideal BEC owing to their ultra-small background scattering length. This study further investigates the nonlinear dynamics of 39 K BEC confined in a double-well potential under attractive interactions. In this work, a Gaussian barrier is integrated with a double-well potential, and we innovatively adopt a transient dynamics approach to construct a highly non-equilibrium initial state where all atoms are confined to the left well. We track the full evolutionary process of the system from non-equilibrium excitation to stable self-trapping, elucidating wave function contraction and energy conversion. Under high barrier height conditions, tunneling is substantially suppressed: atoms localize in a single well, the longitudinal position expectation value remains constant, and the phase difference is locked, leading to the formation of a macroscopic quantum self-trapping state. Radial oscillations exhibit reduced amplitude and a slower evolution rate owing to density homogenization. A smooth, fluctuation-free momentum density profile corresponds to the static localized regime of self-trapping. Coherence transitions from dynamic oscillation to static locking, and weak attractive interactions in 39 K atoms further prolong the coherence time. This study elucidates key microscopic mechanisms, including the nonlinear coupling mechanism of attractive interactions, the dynamic-static transition rules of coherence, and momentum density acting as a microscopic probe for tunneling and soliton dynamics, thereby providing a quantitative basis for the preparation of tunable quantum states.
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.
We consider a Bose-Einstein condensate in a double-well potential undergoing a dynamical transition from the regime of Josephson oscillations to the regime of self-trapping. We analyze the statistical properties of the ground state (or the highest excited state) of the Hamiltonian in these two regimes for attractive (repulsive) interactions. We demonstrate that it is impossible to describe the transition within the mean-field theory. In contrast, the transition proceeds through a strongly correlated delocalized state, with large quantum fluctuations, and spontaneous breaking of the symmetry.
The realization of Bose-Einstein condensation in dilute atomic gases opens an exciting way to quantum mechanics and begins a new area of quantum simulation. As a macroscopic quantum object and a many-body bosonic system, the Bose-Einstein condensates can show numerous exotic quantum effects and have naturally attracted great attention. One of the simplest quantum many-body systems to be realized experimentally and studied theoretically is ultra-cold atoms in a double-well potential. This system can exhibit a great variety of quantum interference phenomena such as tunneling oscillation, self-trapping and the entanglement of macroscopic superpositions. Specifically, the double-well potentials built by optical or magnetic fields are easy to change and the many-body interaction between ultra-cold atoms can be changed by the method of Feshbach resonance, enabling the precise quantum control of the double-well dynamics of the condensates. In the present work, we study the dynamics of a condensate in a trapping potential consisting of an unalterable double-well trap and an additional moving optical lattice. If the lattice space is much smaller than the size of the double-well trap, the system can be simplified into a double-well trapped condensate with a tunable effective mass. Using the mean-field factorization assumption, together with a two-mode approximation, we obtain the analytic expressions for the dependence of the tunneling rate and the self-collision strength on the effective mass. The tunneling rate decays and the collision strength grows up with the increase of the effective mass. As a consequence of their different changes, we conclude that the adjustment of the effective mass of the ultra-cold atoms, rather than the changing of the trap barrier or adjusting of the atomic scattering length, is an alternative approach to controlling the double-well dynamics of the condensate. Via numerical simulations of the mean-field dynamical equations with some realistic parameters, we show that a transition between the quantum coherent tunneling and the self-trapping behaviors is experimentally realizable with the mass-control approach. Specifically, we show that the approach is still valid for the case of negative mass. Moreover, we find that the negative-mass case can be used even to stimulate the double-well dynamics of the condensate with a negative atomic scattering length.
We study the macroscopic quantum self-trapping effect in a mixture of Bose-Einstein condensate and a large number of quantum degenerate fermions, trapped in a double-well potential with a variable separation between the wells. The large number of fermions localized in each well form quasi-static impurities that affect the dynamics of the bosonic cloud. Our semi-analytical analysis based on a mean-field model shows that main features of macroscopic quantum self trapping in a pure bosonic system are radically modified by the influence of fermions, with both the onset of self-trapping and properties of the self-trapped state depending on the fermion concentration as well as on the type of inter-species interaction. Remarkably, repulsive inter-species interaction leads to population inversion of the bosonic energy levels in the trapping potential and hence to the inversion of symmetry of the macroscopic wavefunction. Realistic physical estimates are given based on experimental parameters for a 40K-87Rb system.