Physics – Quantum Physics
Scientific paper
2012-01-13
Physics
Quantum Physics
31 pages
Scientific paper
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, and working to third-order in the free-space scattering length a, defined at zero collision energy. The results take into account the influence of quantum fluctuations to excited orbitals and finite-range effects for realistic two-body interactions. The leading-order, effective four-body interaction energy is U4 = +(2.43317...)(a/l)^3 + O[(a/l)^4]; l and w are the harmonic oscillator length and frequency, and energies are in units of (hbar)(w). After renormalization, the effective dimensionless three-body interaction energy is U3 = -(0.85576...)(a/l)^2 + 2.7921(1)(a/l)^3 + O[(a/l)^4]. The one-standard deviation error 0.0001 for the third-order coefficient in U3 is due to numerical uncertainty in estimating a slowly converging sum; the other two coefficients are either analytically or numerically exact. We then perform numerical simulations for a finite-range boson-boson potential and comparison to the zero-range predictions reveals that finite-range effects must be taken into account. In particular, we show that the energy-dependent pseudopotential accurately captures the finite-range physics, through third order, and in combination with multi-body effective interactions gives excellent agreement with the numerical simulations, validating our theoretical analysis and predictions.
Blume D.
Flynn W. F.
Johnson Randall P.
Tiesinga Eite
Yin Xue-Yu
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