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arXiv:quant-ph/0410163 · Physical Review A

Two atoms in an anisotropic harmonic trap

Zbigniew Idziaszek, Tommaso Calarco

Published May 23, 2005• 79 citations
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Abstract

We consider the system of two interacting atoms confined in axially symmetric harmonic trap. Within the pseudopotential approximation, we solve the Schr\"odinger equation exactly, discussing the limits of quasi-one-and quasi-two-dimensional geometries. Finally, we discuss the application of an energy-dependent pseudopotential, which allows us to extend the validity of our results to the case of tight traps and large scattering lengths.

DOI
10.1103/physreva.71.050701

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We consider the system of two interacting atoms confined in axially symmetric harmonic trap. Within the pseudopotential approximation, we solve the Schr"odinger equation exactly, discussing the limits of quasi-one-and quasi-two-dimensional geometries. Finally, we discuss the application of an energy-dependent pseudopotential, which allows us to extend the validity of our results to the case of tight traps and large scattering lengths.
We discuss exact solutions of the Schr\"odinger equation for the system of two ultracold atoms confined in an axially symmetric harmonic potential. We investigate different geometries of the trapping potential, in particular we study the properties of eigenenergies and eigenfunctions for quasi-one-dimensional and quasi-two-dimensional traps. We show that the quasi-one-dimensional and the quasi-two-dimensional regimes for two atoms can be already realized in the traps with moderately large (or small) ratios of the trapping frequencies in the axial and the transverse directions. Finally, we apply our theory to Feshbach resonances for trapped atoms. Introducing in our description an energy-dependent scattering length we calculate analytically the eigenenergies for two trapped atoms in the presence of a Feshbach resonance.
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.
We derive analytical solutions for the system of two ultracold spin-polarized fermions interacting in ppp wave and confined in an axially symmetric harmonic trap. To this end we utilize ppp-wave pseudopotential with an energy-dependent scattering volume. This allows to describe the scattering in tight trapping potentials in the presence of scattering resonances. We verify predictions of the pseudopotential treatment for some model interaction potential obtaining an excellent agreement with exact energy levels. Then we turn to the experimentally relevant case of neutral atom interactions in the vicinity of a ppp-wave Feshbach resonance. In the framework of the multichannel quantum-defect theory we derive relatively simple formula for an energy-dependent scattering volume, and later we apply it to investigate the energy spectrum of trapped atoms close to the ppp-wave Feshbach resonance.
We compute the ground-state energy of two atoms in a one-dimensional geometry of a harmonic optical trap. We obtain a dependence of the energy on a one-dimensional scattering length, which corre-sponds to various strengths of the interaction potential V int (x) = V 0 exp {−2cx 2}. The calculation is performed by numerical and analytical methods. For the analytical method we choose the oscillator representation method (OR), which has been successfully applied to computations of bound states of various few-body systems. The main results of this paper are (1) a numerical investigation of the validity range of the previously used pseudopotential method and (2) an investigation of the validity range of the OR for the potential V(x) = V conf (x) + V int (x) = x 2/2 + V 0 exp {−2cx 2}.
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.