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Abstract
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 at(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 ℏω. The one-standard deviation error ±0.0001 for the third-order coefficient in U3(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.
Highlights
In this paper, we use renormalized quantum field theory [33] to calculate the perturbative ground-state energy for N ultracold neutral bosons in a three-dimensional isotropic harmonic potential with angular frequency ω, and extract from it the effective m-body interaction energies U2(ω), U3(ω), and U4(ω) as a function of ω
We use renormalized perturbation theory, which develops an expansion around physical as opposed to bare coupling parameters, to systematically cancel the multiple divergences that arise at higher-orders in quantum field perturbation theory. (In this paper, the physical coupling parameter is defined in terms of the measured scattering length, or alternatively the measured energy shift, for two interacting ultracold boson in a harmonic trap at a specified trap frequency.) Renormalized perturbation theory, which is more commonly used in high-energy physics, in this context naturally describes how the effective interactions depend on trap frequency
The analysis in this paper provides an explicit example of renormalization physics and running coupling constants that can be directly probed using trapped ultracold bosonic atoms, and used to test central concepts in effective field theory
Summary
Where ri is the position vector of the ith boson. We assume there are no intrinsic three- or higher-body interactions. The two-body coupling constant g2 is related to at(0), at first order in perturbation theory, by g2 = 4π h 2/mA at(0) + O([at(0)]2), where mA is the boson mass, at(0) is the physical s-wave scattering length measured in the limit that the trap frequency and collision energy go to zero, and O([at(0)]2) are terms of order [at(0)]2 and higher. Leading-order contribution of an energy-dependent pseudopotential accurately captures the FR effects, and allows us to validate the analytic and numerical coefficients found from the zero-range perturbation theory. We determine the ground-state energy of N = 3 and N = 4 bosons interacting through the Gaussian model potential under external spherically symmetric harmonic confinement using a basis set expansion that expresses the relative N -body wave function in terms of explicitly correlated Gaussians [64].
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