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Physical Review A

s-wave pseudopotential for anisotropic traps

Jochen Pade, Martin Block, Martin Holthaus

Published Dec 2, 2003• 4 citations
Publisher

Abstract

Starting from two identical, interacting particles in an isotropic harmonic trap, we employ first-order perturbation theory to derive the effective s-wave pseudopotential operator for a weakly anisotropic trap. It is shown that such a confinement gives rise to a correction of the s-wave scattering length which is proportional to the square of the ratio of the bare scattering length and the oscillator length. The trap pseudopotential depends on the quantum number of the trapped state, and the effect of a weak anisotropy is dominated by its nearest neighbors. Some implications for experiments with cold bosons in tightly confining potentials are discussed briefly.

DOI
10.1103/physreva.68.063402

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Starting from two identical, interacting particles in an isotropic harmonic trap, we employ first-order perturbation theory to derive the effective s-wave pseudopotential operator for a weakly anisotropic trap. It is shown that such a confinement gives rise to a correction of the s-wave scattering length which is proportional to the square of the ratio of the bare scattering length and the oscillator length. The trap pseudopotential depends on the quantum number of the trapped state, and the effect of a weak anisotropy is dominated by its nearest neighbors. Some implications for experiments with cold bosons in tightly confining potentials are discussed briefly.
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We present a zero-range pseudopotential applicable for all partial wave interactions between neutral atoms. For p and d waves, we derive effective pseudopotentials, which are useful for problems involving anisotropic external potentials. Finally, we consider two nontrivial applications of the p-wave pseudopotential: we solve analytically the problem of two interacting spin-polarized fermions confined in a harmonic trap, and we analyze the scattering of p-wave interacting particles in a quasi-two-dimensional system.
Abstract We study the effects of the effective range of interaction on the eigenvalues and eigenstates of two particles confined in a three-dimensional (3D) isotropic as well as one- or quasi-one dimensional harmonic (1D) traps. For this we employ model potentials which mimic finite-range s -wave interactions over a wide range of s -wave scattering length a s including the unitarity limits a s → ± ∞ . Our results show that when the range is larger than the 3D or 1D harmonic oscillator length scale, the eigenvalues and eigenstates are nearly similar to those of noninteracting two particles in the 3D or 1D trap, respectively. In case of 3D, we find that when the range goes to zero, the results of contact potential as derived by Busch et al (1998 Foundations of Physics 28 549 ) are reproduced. However, in the case of 1D, such reproducibility does not occur as the range goes to zero. We have calculated the eigenvalues and eigenstates in a 1D harmonic trap taking one dimensional finite-range model potential. We have also calculated the bound state properties of two particles confined in a highly anisotropic quasi-1D trap taking three-dimensional finite-range model potential, and examined whether these quasi-1D results approach towards 1D ones as the aspect ratio η of the radial to axial frequency of the trap increases. We find that if the range is very small compared to the axial size of the trap, then one can reach 1D regime for η ≥ 10000 . However, for a large range, one can almost get 1D results for smaller values of η . This study will be important for the exploration of two-body or many body physics of trapped ultracold atoms interacting with narrow Feshbach resonance for which the effective range can be large.
We describe a model for s-wave collisions between ground-state atoms in optical lattices, considering especially the limits of quasi-one- and two-dimensional axisymmetric harmonic confinement. When the atomic interactions are modeled by an s-wave Fermi pseudopotential, the relative motion energy eigenvalues can easily be obtained. The results show that except for a bound state, the trap eigenvalues are consistent with one- and two-dimensional scattering with renormalized scattering amplitudes. For absolute scattering lengths large compared with the tightest trap width, our model predicts a bound state of low energy and nearly-isotropic wave function extending on the order of the tightest trap width.