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arXiv:1711.05100 · Physical Review A

Effective three-body interactions for bosons in a double-well confinement

Jacek Dobrzyniecki, Xikun Li, Anne E. B. Nielsen, Tomasz Sowiński

Published Jan 12, 2018• 16 citations
arXivHTMLPDF

Abstract

When describing the low-energy physics of bosons in a double-well potential with a high barrier between the wells and sufficiently weak atom-atom interactions, one can, to a good approximation, ignore the high-energy states and thereby obtain an effective two-mode model. Here we show that the regime in which the two-mode model is valid can be extended by adding an on-site three-body interaction term and a three-body interaction-induced tunneling term to the two-mode Hamiltonian. These terms effectively account for virtual transitions to the higher-energy states. We determine appropriate strengths of the three-body terms by an optimization of the minimal value of the wave-function overlap within a certain time window. Considering different initial states with three or four atoms, we find that the resulting model accurately captures the dynamics of the system for parameters where the two-mode model without the three-body terms is poor. We also investigate the dependence of the strengths of the three-body terms on the barrier height and the atom-atom interaction strength. The optimal three-body interaction strengths depend on the initial state of the system.

DOI
10.1103/physreva.97.013609

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When describing the low-energy physics of bosons in a double-well potential with a high barrier between the wells and sufficiently weak atom-atom interactions, one can, to a good approximation, ignore the high-energy states and thereby obtain an effective two-mode model. Here we show that the regime in which the two-mode model is valid can be extended by adding an on-site three-body interaction term and a three-body interaction-induced tunneling term to the two-mode Hamiltonian. These terms effectively account for virtual transitions to the higher-energy states. We determine appropriate strengths of the three-body terms by an optimization of the minimal value of the wave-function overlap within a certain time window. Considering different initial states with three or four atoms, we find that the resulting model accurately captures the dynamics of the system for parameters where the two-mode model without the three-body terms is poor. We also investigate the dependence of the strengths of the three-body terms on the barrier height and the atom-atom interaction strength. The optimal three-body interaction strengths depend on the initial state of the system.
In this work, our primary goal has been to explore the range of validity of two-mode models for Bose-Einstein condensates in double-well potentials. Our derivation, like others, uses symmetric and antisymmetric condensate basis functions for the Gross-Pitaevskii equation. In what we call an ``improved two-mode model'' (I2M), the tunneling coupling energy explicitly includes a nonlinear interaction term, which has been given previously in the literature but not widely appreciated. We show that when the atom number (and hence the extent of the wave function) in each well vary appreciably with time, the nonlinear interaction term produces a temporal change in the tunneling energy or rate, which has not previously been considered to our knowledge. In addition, we obtain a parameter, labeled ``interaction tunneling,'' that produces a decrease of the tunneling energy when the wave functions in the two wells overlap to some extent. Especially for larger values of the nonlinear interaction term, results from this model produce better agreement with numerical solutions of the time-dependent Gross-Pitaevskii equation in one and three dimensions, as compared with models that have no interaction term in the tunneling energy. The usefulness of this model is demonstrated by good agreement with recent experimental results for the tunneling oscillation frequency [Albiez et al., Phys. Rev. Lett. 95, 010402 (2005)]. We also present equations and results for a multimode approach, and use the I2M model to obtain modified equations for the second-quantized version of the Bose-Einstein double-well problem.
We investigate the tunneling dynamics of a few bosons with both two- and three-body interactions in a doublewell potential. Uncorrelated tunneling of Rabi oscillation with the minimum period can happen only when the two- and three-body interactions satisfy a critical condition, i.e., the effective interaction energy is minimized. When the atomic interactions are slightly away from the critical condition in the weak interaction regime, the uncorrelated tunneling exhibits collapse-revival character. When the atomic interactions are strong and far away from the critical condition, the correlated tunneling with Rabi oscillation occurs. The tunneling period (the period of collapse-revival) increases (decreases) when the rate between the two-body and three-body interactions is away from the corresponding critical condition or when the number of bosons increases. Further, the tunneling properties are understood with the help of the energy spectrum of the system. Eventually, the effect of the initial configuration on the tunneling dynamics of a few bosons for both odd and even numbers of bosons is studied, which results in intriguing consequences.
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.
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 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.