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arXiv:1404.2250 · Physical Review A

Number-squeezed and fragmented states of strongly interacting bosons in a double well

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

Published Nov 22, 2017• 3 citations
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Abstract

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.

DOI
10.1103/physreva.96.053627

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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 aaa. The number squeezing shows a universal scaling with the product of number of particles and interaction strength (NaNaNa), 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.
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.
We present path integral ground state (PIGS) quantum Monte Carlo calculations for the ground state (T = 0) properties of repulsively interacting bosons in a three-dimensional external double well potential over a range of interaction strengths and potential parameters. We focus our calculations on ground state number statistics and the one-body density matrix in order to understand the level of squeezing and fragmentation that the system exhibits as a function of interaction strength, and we compare our PIGS results to both a standard and an improved two-mode model and a recently-proposed eight-mode model. In general, the various models agree with the PIGS simulations for weak interactions, but the full quantum Monte Carlo treatment is required to correctly predict the amount of squeezing and fragmentation exhibited for strong interactions. One novel and somewhat surprising result from our simulations involves the relationship between squeezing and interaction strength: rather than a monotonic relationship between these quantities, we find that for certain double well barrier heights the squeezing increases as a function of interaction strength until it reaches a maximum, after which it decreases again. We also see a similar relationship between fragmentation and interaction strength. We propose novel physical mechanisms to account for this behavior.
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).
Abstract The out-of-equilibrium quantum dynamics of an interacting Bose gas trapped in a one-dimensional asymmetric double-well potential is studied by solving the many-body Schrödinger equation numerically accurately. We examine how the gradual loss of symmetry of the confining trap affects the macroscopic quantum tunneling dynamics of the system between the two wells. In an asymmetric double well, the two wells are not equivalent anymore, say, the left well is deeper than the right one. Accordingly, we analyze the dynamics by initially preparing the condensate in both the left and the right wells. The dynamics of the system is characterized by the time evolution of a few physical quantities of increasing many-body complexity, namely, the survival probability, depletion and fragmentation, and the many-particle position and momentum variances. In particular, we have examined the frequencies and amplitudes of the oscillations of the survival probabilities, the time scale for the development of fragmentation and its degree, and the growth and oscillatory behavior of the many-particle position and momentum variances. There is an overall suppression of the oscillations of the survival probabilities in an asymmetric double well, except for resonant values of asymmetry for which the one-body ground state energy in the right well matches with one of the one-body excited states in the left well, thereby resulting in resonantly enhanced tunneling from the right well ground state. Overall, depending on whether the condensate is initially prepared in the left or right well, the repulsive inter-atomic interactions affect the survival probabilities differently. For a sufficiently strong repulsive interaction, the system is found to become fragmented. The degree of fragmentation depends both on the asymmetry of the trap and the initial well in which the condensate is prepared in a non-trivial manner. Furthermore, we show that the phenomenon of resonantly enhanced tunneling can be accompanied by a large degree of fragmentation (depletion) for the strong (weak) interaction. The many-particle position and momentum variances follow the density oscillations of the system in the asymmetric double well and bears prominent signatures of the degree of depletion or fragmentation, depending on the strength of the interactions. These quantities further exhibit a fine structure signifying a breathing-mode oscillation. Finally, a universality of fragmentation for systems made of different numbers of particles but the same interaction parameter is also found and its dependence on the asymmetry is investigated. The phenomenon is robust despite the asymmetry of the junction and admits a macroscopically-large fragmented condensate characterized by a diverging many-particle position variance. This is as far as one can get from the dynamics of the density in the junction.
We compare and contrast the mean-field and many-body properties of a Bose-Einstein condensate trapped in a double-well potential with a single impurity atom. The mean-field solutions display a rich structure of bifurcations, as parameters such as the boson-impurity interaction strength and the tilt between the two wells are varied. In particular, we study a pitchfork bifurcation in the lowest mean-field stationary solution, which occurs when the boson-impurity interaction exceeds a critical magnitude. This bifurcation, which is present for both repulsive and attractive boson-impurity interactions, corresponds to the spontaneous formation of an imbalance in the number of particles between the two wells. If the boson-impurity interaction is large, the bifurcation is associated with the onset of a Schr\"odinger-cat state in the many-body ground state. We calculate the coherence and number fluctuations between the two wells, and also the entanglement entropy between the bosons and the impurity. We find that the coherence can be greatly enhanced at the bifurcation.
We study the ground state of a large bosonic system trapped in a symmetric double-well potential, letting the distance between the two wells increase to infinity with the number of particles. In this context, one should expect an interaction-driven transition between a delocalized state (particles are independent and all live in both wells) and a localized state (particles are correlated, half of them live in each well). We start from the full many-body Schr{ö}dinger Hamiltonian in a large-filling situation where the on-site interaction and kinetic energies are comparable. When tunneling is negligible against interaction energy, we prove a localization estimate showing that the particle number fluctuations in each well are strongly suppressed. The modes in which the particles condense are minimizers of nonlinear Schr{ö}dinger-type functionals.