Bose-Einstein Condensation in a Double-Well System

Jesus Caloca

Published Jan 1, 2014• 0 citations

Abstract

The two-site version of the Bose-Hubbard model can be used as a simplified model of N bosons confined to two quantum wells. We study the behavior of low-temperature Bose-Einstein condensates as a function of the strength of the repulsive interactions between the bosons and of the size of the quantum-mechanical tunneling between the quantum wells. The ground-state of weakly interacting bosons tends to be one in which almost all particles are in the same quantum-mechanical state (‘Bose-Einstein Condensate’, or ‘BEC’). It is found that even very small amounts of interwell tunneling lead to a BEC, but that zero tunneling leads to two separate BECs, one in each well (‘fragmentation’). We study systematically the transition between a single BEC comprised of bosons in both wells, and a fragmented condensate consisting of two BECs, one for each well. Our analysis will use matrix computational methods to solve the Schrödinger equation of the two-well boson system, and obtain the energies and wavefunctions as eigenvalues and eigenvectors of the Hamiltonian matrix. We will compare the results obtained with previous analyses, and against analytical approximations developed for the limiting case of a large number of bosons (large N).

Related papers

Fragmented condensation in Bose–Hubbard trimers with tunable tunnelling

Albert Gallemí, M. Guilleumas, Joan Martorell +3 more

We consider a Bose–Hubbard (BH) trimer, i.e. an ultracold Bose gas populating three quantum states. The latter can be either different sites of a triple-well potential or three internal states of the atoms. The bosons can tunnel between different states with variable tunnelling strength between two of them. This will allow us to study; (i) different geometrical configurations, i.e. from a closed triangle to three aligned wells and (ii) a triangular configuration with a π -phase, i.e. by setting one of the tunnellings negative. By solving the corresponding three-site BH Hamiltonian we obtain the ground state of the system as a function of the trap topology. We characterize the different ground states by means of the coherence and entanglement properties. For small repulsive interactions, fragmented condensates are found for the π -phase case. These are found to be robust against small variations of the tunnelling in the small interaction regime. A low-energy effective many-body Hamiltonian restricted to the degenerate manifold provides a compelling description of the π -phase degeneration and explains the low-energy spectrum as excitations of discrete semifluxon states.

arXiv:1501.03306New Journal of PhysicsJul 14, 201537 citations

Ground state and excitations of a Bose gas: From a harmonic trap to a double well

Yonathan Japha, Y. B. Band

We determine the low-energy properties of a trapped Bose gas split in two by a potential barrier over the whole range of barrier heights and asymmetry between the wells. For either weak or strong coupling between the wells, our two-mode theory yields a two-site Bose-Hubbard Hamiltonian with the tunneling, interaction, and bias parameters calculated simply using an explicit form of two mode functions. When the potential barrier is relatively low, most of the particles occupy the condensate mode and our theory reduces to a two-mode version of the Bogoliubov theory, which gives a satisfactory estimate of the spatial shape and energy of the lowest collective excitation. When the barrier is high, our theory generalizes the standard two-site Bose-Hubbard model into the case of asymmetric modes, and correctly predicts a full separation of the modes in the limit of strong separation of the wells. We provide explicit analytic forms for the number squeezing and coherence as a function of particle number and temperature. We compare our theory to other two-mode theories for bosons in a double well and discuss their validity in different parameter regimes.

Physical Review ASep 23, 201113 citations

Bose-Einstein condensates on slightly asymmetric double-well potentials

Bruno Juliá-Díaz, Joan Martorell, A. Polls

An analytical insight into the symmetry-breaking mechanisms underlying the transition from Josephson to self-trapping regimes in Bose-Einstein condensates is presented. We obtain expressions for the ground-state properties of the system of a gas of attractive bosons modeled by a two-site Bose-Hubbard Hamiltonian with an external bias. Simple formulas are found relating the appearance of fragmentation in the condensate with the large quantum fluctuations of the population imbalance occurring in the transition from the Josephson to the self-trapped regime.

arXiv:1005.2203Physical Review AJun 17, 201039 citations

Quantum dynamics of a binary mixture of BECs in a double-well potential: a Holstein–Primakoff approach

R. Citro, Adele Naddeo, E. Orignac

We study the quantum dynamics of a binary mixture of Bose-Einstein condensates (BEC) in a double-well potential starting from a two-mode Bose-Hubbard Hamiltonian. Focussing on the regime where the number of atoms is very large, a mapping onto a SU(2) spin problem together with a Holstein-Primakoff transformation is performed. The quantum evolution of the number difference of bosons between the two wells is investigated for different initial conditions, which range from the case of a small imbalance between the two wells to a coherent spin state. The results show an instability towards a phase-separation above a critical positive value of the interspecies interaction while the system evolves towards a coherent tunneling regime for negative interspecies interactions. A comparison with a semiclassical approach is discussed together with some implications on the experimental realization of phase separation with cold atoms.

arXiv:1104.4965Journal of Physics B Atomic Molecular and Optical PhysicsMay 19, 20115 citations

Squeezed and fragmented states of strongly interacting bosons in a double well. Part I: Theoretical models

Joel Christopher Corbo, Jonathan L. DuBois, K. Birgitta Whaley

We present a systematic study of the phenomena of squeezing and fragmentation for a BoseEinstein condensate (BEC) in a three dimensional double well potential over a range of interaction strengths and barrier heights. We analyze the properties of the condensate in the context of both a standard and an improved two-mode model and a recently-proposed eight-mode model. One fundamental dierence between this and previous two-mode work is that we explicitly examine well geometries that exhibit appreciable overlap in the one-body wavefunctions localized in the left and right wells rather than assuming that they are nearly isolated from each other. One particularly surprising observation that we have made in our study of these models is the existence of an interaction strength that maximizes squeezing, which is in stark contrast to the predictions of the two-mode model with no overlap between the wells for which squeezing monotonically increases with interaction strength. We explain this and other novel results through a careful study of the ground state properties produced by various terms in the model Hamiltonians and the relationship between the strength of these terms and the system parameters, such as well geometry and atomic interaction strength.

Apr 8, 20140 citations

Two mode theory of Bose–Einstein condensates: interferometry and the Josephson model

Bryan J. Dalton, Saeed Ghanbari

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.

Journal of Modern OpticsNov 18, 201148 citations