Effective renormalized multi-body interactions of harmonically confined ultracold neutral bosons

We calculate the renormalized effective two-, three- and fourbody 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 fourbody interaction and consistently include corrections for realistic two-body interactions. The leading-order, effective three- and four-body interaction energies are U3 (ω)= -(0.85576...)[at(0)/σ (ω)]2 + 2.7921(1)[at(0)/σ (ω)]3 + O(a4 t) and U4(ω)= +(2.43317...)[at(0)/σ (ω)]3 +O(a4 t), where ω and σ(ω) are the harmonic oscillator frequency and length, respectively, and energies are in units of hω. 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 finiterange 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.