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Bulletin of the American Physical Society

The role of dimensionality on effective two- and three-body interactions of trapped ultracold bosons

Philip R. Johnson, D. Blume, Eite Tiesinga

Published Jun 5, 2014• 0 citations
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We calculate the renormalized effective two-, three- and four-body interactions for N neutral ultracold bosons in the ground state of an isotropic harmonic trap, assuming two-body interactions modeled with the combination of a zero-range and energy-dependent pseudopotential. We work to third-order in the scattering length a t (0) defined at zero collision energy, which is necessary to obtain both the leading-order effective four-body interaction and consistently include corrections for realistic two-body interactions. The leading-order, effective three- and four-body interaction energies are and , where ω and σ ( ω ) are the harmonic oscillator frequency and length, respectively, and energies are in units of ℏ ω . The one-standard deviation error ±0.0001 for the third-order coefficient in U 3 ( ω ) is due to numerical uncertainty in estimating a slowly converging sum; the other two coefficients are either analytically or numerically exact. The effective three- and four-body interactions can play an important role in the dynamics of tightly confined and strongly correlated systems. We also performed numerical simulations for a finite-range (FR) boson–boson potential, and it was comparison to the zero-range predictions which revealed that finite-range effects must be taken into account for a realistic third-order treatment. In particular, we show that the energy-dependent pseudopotential accurately captures, through third order, the finite-range physics, and in combination with the multi-body effective interactions gives excellent agreement with the numerical simulations, validating our theoretical analysis and predictions.

arXiv:1201.2962•New Journal of Physics•May 25, 2012•29 citations
When quantum particles are confined into lower dimensions, an effective three-body interaction inevitably arises and may cause significant consequences. Here we study bosons in one dimension with weak two-body and three-body interactions and predict the existence of two three-body bound states when both interactions are attractive. Their binding energies are universal functions of the two-body and three-body scattering lengths and have direct relevance to a broad range of quasi-one-dimensional systems realized with ultracold atoms.
When quantum particles are confined into lower dimensions, an effective three-body interaction inevitably arises and may cause significant consequences. Here we study bosons in one dimension with weak two-body and three-body interactions, predict the existence of two three-body bound states when both interactions are attractive, and determine their binding energies as universal functions of the two-body and three-body scattering lengths. We also show that an infinitesimal three-body attraction induces an excited bound state only for 3, 39, or more bosons. Our findings herein have direct relevance to a broad range of quasi-one-dimensional systems realized with ultracold atoms.
We calculate the renormalized effective two-, three- and four-body interactions for N neutral ultracold bosons in the ground state of an isotropic harmonic trap, assuming two-body interactions modeled with the combination of a zero-range and energy-dependent pseudopotential. We work to third-order in the scattering length a t (0) defined at zero collision energy, which is necessary to obtain both the leading-order effective four-body interaction and consistently include corrections for realistic two-body interactions. The leading-order, effective three- and four-body interaction energies are and , where ω and σ ( ω ) are the harmonic oscillator frequency and length, respectively, and energies are in units of ℏ ω . The one-standard deviation error ±0.0001 for the third-order coefficient in U 3 ( ω ) is due to numerical uncertainty in estimating a slowly converging sum; the other two coefficients are either analytically or numerically exact. The effective three- and four-body interactions can play an important role in the dynamics of tightly confined and strongly correlated systems. We also performed numerical simulations for a finite-range (FR) boson–boson potential, and it was comparison to the zero-range predictions which revealed that finite-range effects must be taken into account for a realistic third-order treatment. In particular, we show that the energy-dependent pseudopotential accurately captures, through third order, the finite-range physics, and in combination with the multi-body effective interactions gives excellent agreement with the numerical simulations, validating our theoretical analysis and predictions.