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arXiv:0911.3121 · Journal of Physics A Mathematical and Theoretical

Universality of the energy spectrum for two interacting harmonically trapped ultra-cold atoms in one and two dimensions

Aaron Farrell, Brandon P van Zyl

Published Dec 7, 2009• 25 citations
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Abstract

Motivated by the recent article of Shea et al (2009 Am. J. Phys. 77 511) we examine the exactly solvable problem of two harmonically trapped ultra-cold bosonic atoms interacting via a short-range potential in one and two dimensions. A straightforward application in one dimension shows that the energy spectrum is universal, in addition to clearly illustrating why regularization is not required in the limit of zero range. The two-dimensional problem is less trivial, requiring a more careful treatment as compared to the one-dimensional case. Our two-dimensional analysis reveals that the low-energy physics is also universal, in addition to providing a simple method for obtaining the appropriately regularized two-dimensional pseudopotential.

DOI
10.1088/1751-8113/43/1/015302

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Motivated by the recent article of Shea et al (2009 Am. J. Phys. 77 511) we examine the exactly solvable problem of two harmonically trapped ultra-cold bosonic atoms interacting via a short-range potential in one and two dimensions. A straightforward application in one dimension shows that the energy spectrum is universal, in addition to clearly illustrating why regularization is not required in the limit of zero range. The two-dimensional problem is less trivial, requiring a more careful treatment as compared to the one-dimensional case. Our two-dimensional analysis reveals that the low-energy physics is also universal, in addition to providing a simple method for obtaining the appropriately regularized two-dimensional pseudopotential.
In this undergraduate thesis, we discuss the two-body quantum mechanical problem of two ultra- cold bosonic atoms at low energy limit, interacting through a square potential with and without an external confinement. The two-body s-wave scattering length is introduced in terms of the radial wavefunction. In free space with no confinement, the interaction-scattering length relation is calculated and compared in 3- and 1-dimensions; we show the recurring resonance regions where the scattering length diverges and the existence of dimer bound state with positively large scattering length. Lastly, in the case of a 3-dimensional isotropic harmonic confinement, again under the ultracold limit, we obtain the universal (s-state) energy spectrum of the two atoms in terms of the scattering length.
The exactly solvable model of two indistinguishable quantum particles (bosons or fermions) confined in a one-dimensional harmonic trap and interacting via finite-range soft-core interaction is presented and many properties of the system are examined. Particularly, it is shown that independently on the potential range, in the strong interaction limit bosonic and fermionic solutions become degenerate. For sufficiently large ranges a specific crystallization appears in the system. The results are compared to predictions of the celebrated Busch et al. model and those obtained in the Tonks-Girardeau limit. The assumed inter-particle potential is very similar to the potential between ultra-cold dressed Rydberg atoms. Therefore, the model can be examined experimentally.
We consider the spectrum of two ultracold harmonically trapped atoms interacting via short-range interactions. The Green function approach is used to unify the two- and three-dimensional cases. We derive criteria for the universality of the spectrum, i.e. its independence of the details of the short-range interaction. The results in three dimensions are exemplified for narrow s-wave Feshbach resonances and we show how effective range corrections can modify the rearrangement of the level structure. However, this requires extremely narrow resonances or very tight traps that are not currently experimentally available. In the two-dimensional case, we discuss the p-wave channel in detail and demonstrate how the non-universality of the spectrum arises within the Green function approach. We then show that the spectrum is not particularly sensitive to the short-distance details in the case when the two-body interaction has a bound state.
We examine the problem of two particles confined in an isotropic harmonic trap, which interact via a finite-range Gaussian-shaped potential in two spatial dimensions. We derive an approximative transcendental equation for the energy and study the resulting spectrum as a function of the interparticle interaction strength. Both the attractive and repulsive systems are analyzed. We study the impact of the potential's range on the ground-state energy. We also explicitly verify by a variational treatment that in the zero-range limit the positive $\ensuremath{\delta}$ potential in two dimensions only reproduces the noninteracting results, if the Hilbert space in not truncated, and demonstrate that an extremely large Hilbert space is required to approach the ground state when one is to tackle the limit of zero-range interaction numerically. Finally, we establish and discuss the connection between our finite-range treatment and regularized zero-range results from the literature. The present results indicate that a finite-range interparticle potential is numerically amenable for treating the statics and the nonequilibrium dynamics of interacting many-particle systems (bosons) in two dimensions.
Abstract This work analyzes the influence of an external confinement on the energy spectrum of two interacting ultracold atoms, considering either a regularized contact or a more realistic Born-Oppenheimer potential. For two harmonically confined particles with contact interaction, the known analytical solution exhibits universality in the sense that the energy spectra solely depend on the ratio of the confinement length to the scattering length. In this work it is shown that this type of universality breaks down for realistic interatomic interactions, leading to an explicit trap-size dependence of the energy spectrum. This effect is especially pronounced for very tight confinement. On the other hand, using first-order perturbation theory it is shown analytically that the here discussed universality persists for the contact interaction even in the case of anharmonic confinement, this result being also verified numerically beyond perturbation theory for a sextic trap potential. These findings demonstrate limitations of the popular contact-interaction model and highlight the necessity of considering more realistic finite-range interactions under tight trapping conditions.