In this work we investigate the energy gap between the ground state and the first excited state in a model of two single-mode Bose–Einstein condensates coupled via Josephson tunnelling. The energy gap is never zero when the tunnelling interaction is non-zero. The gap exhibits no local minimum below a threshold coupling which separates a delocalized phase from a self-trapping phase that occurs in the absence of the external potential. Above this threshold point one minimum occurs close to the Josephson regime, and a set of minima and maxima appear in the Fock regime. Expressions for the position of these minima and maxima are obtained. The connection between these minima and maxima and the dynamics for the expectation value of the relative number of particles is analysed in detail. We find that the dynamics of the system changes as the coupling crosses these points.
In this work we investigate the energy gap between the ground state and the first excited state in a model of two single-mode Bose–Einstein condensates coupled via Josephson tunnelling. The energy gap is never zero when the tunnelling interaction is non-zero. The gap exhibits no local minimum below a threshold coupling which separates a delocalized phase from a self-trapping phase that occurs in the absence of the external potential. Above this threshold point one minimum occurs close to the Josephson regime, and a set of minima and maxima appear in the Fock regime. Expressions for the position of these minima and maxima are obtained. The connection between these minima and maxima and the dynamics for the expectation value of the relative number of particles is analysed in detail. We find that the dynamics of the system changes as the coupling crosses these points.
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.
We study the exact solution for a two-mode model describing coherent coupling between atomic and molecular Bose–Einstein condensates (BEC), in the context of the Bethe ansatz. By combining an asymptotic and numerical analysis, we identify the scaling behaviour of the model and determine the zero temperature expectation value for the coherence and average atomic occupation. The threshold coupling for production of the molecular BEC is identified as the point at which the energy gap is minimum. Our numerical results indicate a parity effect for the energy gap between ground and first excited state depending on whether the total atomic number is odd or even. The numerical calculations for the quantum dynamics reveals a smooth transition from the atomic to the molecular BEC.
In this thesis, I present the first experimental implementation of a Josephson junction for Bose-Einstein condensates. The weak link between two BECs constitutes the nonlinear generalization of the well known Josephson junction of weakly-coupled superconductors that are separated by a thin insulating barrier. In our experiment, the required overlap of two macroscopic wavefunctions is provided by loading a BEC into an optical double well potential. It is realized by a superposition of a one-dimensional optical lattice with a focused laser beam optical dipole trap. The tunneling dynamics between the two potential wells exhibits two distinct dynamical regimes. For small initial population imbalances of the two wells we observe nearly sinusoidal Josephson tunneling oscillations, which are characterized by an oscillating population and relative phase. The situation changes drastically, if the initial population imbalance is chosen above a critical value. In this case, resonant tunneling between the two wells is prohibited because the difference between the on-site particle interaction energies in the two wells exceeds the tunneling energy splitting. As a consequence, the atomic distribution becomes self-locked and the relative phase evolves unbound in time. This regime of prohibited tunneling, which has no analogon in superconducting Josephson junctions, is called “macroscopic quantum self-trapping”.
In this paper we theoretically investigate the nonlinear tunnelling of two weakly linked Bose–Einstein condensates in a double-well trap in a strong interaction regime with the two-body interaction extended to neighbouring lattice sites. As a consequence the boson Josephson-junction tunnelling depends on the atom–atom interaction and the total atom number as well. A crucial atom-pair tunnelling term, obtained directly from the extension, results in significant energy spectrum corrections and an abrupt change of the ground state viewed as quantum phase transition. An atom-number oscillation state with a tunable relative phase between two Bose–Einstein condensates predicted for the first time is seen to be a degenerate ground state in an interaction region. The quantum phase transitions between degenerate and non-degenerate ground states driven by the atom-pair tunnelling are analysed. We show the scaling behaviours and the critical exponents of fidelity susceptibility, which can classify the universality.
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 study the quantum dynamics of a Bose Josephson junction(BJJ) made up of two coupled Bose-Einstein condensates. Apart from the usual ac Josephson oscillations, two different dynamical states of BJJ can be observed by tuning the inter-particle interaction strength, which are known as 'π-oscillation' with relative phase π between the condensates and 'macroscopic self-trapped' (MST) state with finite number imbalance. By choosing appropiate intial state we study above dynamical branches quantum mechanically and compare with classical dynamics. The stability region of the 'π-oscillation' is separated from that of 'MST' state at a critical coupling strength. Also a significant change in the energy spectrum takes place above the critical coupling strength, and pairs of (quasi)-degenerate excited states appear. The original model of BJJ can be mapped on to a simple Hamiltonian describing quantum particle in an 'effective potential' with an effective Planck constant. Different dynamical states and degenerate excited states in the energy spectrum can be understood in this 'effective potential' approach. Also possible novel quantum phenomena like 'macroscopic quantum tunneling'(MQT) become evident from the simple picture of 'effective potential'. We study decay of metastable 'π-oscillation' by MQT through potential barrier. The doubly degenerate excited states in the energy spectrum are associated with the classically degenerate MST states with equal and opposite number imbalance. We calculate the energy splitting between these quasi-degenerate excited states due to MQT of the condensate between classically degenerate MST states.