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).
We present a systematic study of the phenomena of number squeezing and fragmentation for a repulsive Bose-Einstein condensate (BEC) in a three-dimensional double-well potential over a range of interaction strengths and barrier heights, including geometries that exhibit appreciable overlap in the one-body wave functions localized in the left and right wells. We compute the properties of the condensate with numerically exact, full-dimensional path-integral ground-state (PIGS) quantum Monte Carlo simulations and compare with results obtained from using two- and eight-mode truncated basis models. The truncated basis models are found to agree with the numerically exact PIGS simulations for weak interactions, but fail to correctly predict the amount of number squeezing and fragmentation exhibited by the PIGS simulations for strong interactions. We find that both number squeezing and fragmentation of the BEC show nonmonotonic behavior at large values of interaction strength
a. The number squeezing shows a universal scaling with the product of number of particles and interaction strength (
Na), but no such universal behavior is found for fragmentation. Detailed analysis shows that the introduction of repulsive interactions not only suppresses number fluctuations to enhance number squeezing, but can also enhance delocalization across wells and tunneling between wells, each of which may suppress number squeezing. This results in a dynamical competition whose resolution shows a complex dependence on all three physical parameters defining the system: interaction strength, number of particles, and barrier height.
Compared to the atomic chip, which has been a powerful platform to perform an astonishing range of applications from rapid Bose-Einstein condensate (BEC) production to the atomic clock, the molecular chip is only in its infant stages. Recently a one-dimensional electric lattice was demonstrated to trap polar molecules on a chip. This excellent work opens up the way to building a molecular chip laboratory. Here we propose a two-dimensional (2D) electric lattice on a chip with concise and robust structure, which is formed by arrays of squared gold wires. Arrays of microtraps that originate in the microsize electrodes offer a steep gradient and thus allow for confining both light and heavy polar molecules. Theoretical analysis and numerical calculations are performed using two types of sample molecules,
ND3 and SrF, to justify the possibility of our proposal. The height of the minima of the potential wells is about 10 \ensuremath{\mu}m above the surface of the chip and can be easily adjusted in a wide range by changing the voltages applied on the electrodes. These microtraps offer intriguing perspectives for investigating cold molecules in periodic potentials, such as quantum computing science, low-dimensional physics, and some other possible applications amenable to magnetic or optical lattice. The 2D adjustable electric lattice is expected to act as a building block for a future gas-phase molecular chip laboratory.