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arXiv:1512.00154 · Journal of Physics B Atomic Molecular and Optical Physics

Exact two-body solutions and quantum defect theory of two-dimensional dipolar quantum gas

Jianwen Jie, Ran Qi

Published Sep 22, 2016• 4 citations
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Abstract

Abstract In this paper, we provide the two-body exact solutions of the two-dimensional (2D) Schrödinger equation with isotropic ± 1 / r 3 interactions. An analytic quantum defect theory is constructed based on these solutions and it is applied to investigate the scattering properties as well as two-body bound states of an ultracold polar molecules confined in a quasi-2D geometry. Interestingly, we find that for the attractive case, the scattering resonance happens simultaneously in all partial waves, which has not been observed in other systems. The effect of this feature on the scattering phase shift across such resonances is also illustrated.

DOI
10.1088/0953-4075/49/19/194003

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Abstract In this paper, we provide the two-body exact solutions of the two-dimensional (2D) Schrödinger equation with isotropic ± 1 / r 3 interactions. An analytic quantum defect theory is constructed based on these solutions and it is applied to investigate the scattering properties as well as two-body bound states of an ultracold polar molecules confined in a quasi-2D geometry. Interestingly, we find that for the attractive case, the scattering resonance happens simultaneously in all partial waves, which has not been observed in other systems. The effect of this feature on the scattering phase shift across such resonances is also illustrated.
Abstract In this paper, we have provided exact two-body solutions to the 2D and 3D Schrödinger equations with isotropic van der Waals potentials of the form ± 1 / r 6 . Based on these solutions, we developed an analytical quantum defect theory (QDT) applicable to both quasi-2D and 3D geometries, and applied it to study the scattering properties and bound-state spectra of ultracold polar molecules confined in these geometries. Interestingly, we find that in the attractive (repulsive) van der Waals potential case, the short-range interaction can be effectively modeled by an infinite square barrier (finite square well), which leads to narrow and dense (broad and sparse) resonance structures in the quantum defect parameter. In the quasi-2D attractive case, shape resonances can appear in an ordered fashion across different partial waves, characterized by sharp phase jumps as the scattering energy is varied. Furthermore, the low-energy analytical expansions derived from QDT show excellent agreement with the exact numerical results, validating the accuracy and usefulness of our analytical approach in describing two-body physics governed by long-range van der Waals interactions.
We analytically derive the general pseudo-potential operator of an arbitrary isotropic interaction for particles confined in two-dimensional (2D) systems, using the same frame work developed by Huang and Yang for 3D scattering [Phys. Rev. {\bf 105}, 767 (1957)]. We also analytically derive the low energy dependence of the phase-shift for an arbitrary potential with a power-law decaying tail, V2D(ρ)∝ρ−αV_{\rm 2D}(\rho)\propto \rho^{-\alpha}V2D​(ρ)∝ρ−α (for α>2\alpha>2α>2). We apply our results to the 2D dipolar gases (α=3\alpha=3α=3) as an example, calculating the momentum and dipole moment dependence of the pseudo-potential for both sss- and p-wave scattering channels if the two scattering particles are in the same 2D layer. We also calculate the result for the s-wave scattering between particles in two different (parallel) layers. Our analytic and numerical results can be directly applied to the study of many-body properties of magnetic dipolar atoms and/or polar molecules in a general 2D system.
We solve the two-particle s-wave scattering problem for ultracold atom gases confined in arbitrary quasi-one-dimensional (1D) trapping potentials, allowing for two different atom species. As a consequence, the centre-of-mass and relative degrees of freedom do not factorize. We derive bound-state solutions and obtain the general scattering solution, which exhibits several resonances in the 1D scattering length induced by the confinement. We apply our formalism to two experimentally relevant cases: (i) interspecies scattering in a two-species mixture, and (ii) the two-body problem for a single species in a non-parabolic trap.
Strongly confining an ultracold atomic gas in one direction to create a quasi-two-dimensional system alters the scattering properties of this gas. We investigate the effects of confinement on Feshbach scattering resonances and show that strong confinement results in a shift in the position of the Feshbach resonance as a function of the magnetic field. This shift, as well as the change of the width of the resonance, is computed. We find that the resonance is strongly damped in the thermal gas, but in the condensate the resonance remains sharp due to many-body effects. We introduce a two-dimensional model system, suited for the study of resonant superfluidity and having the same scattering properties as the tightly confined real system near a Feshbach resonance. Exact relations are derived between measurable quantities and the model parameters.
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 consider a static dipolar impurity immersed in a Fermi gas and show how dipolar interactions shape both the few-body scattering properties and the resulting polaron physics. Using Fumi's theorem, we connect the anisotropic phase shifts directly to the ground-state energy, and compute the exact impurity spectrum via the functional determinant approach. The resulting spectrum is a convolution of independent even- and odd-wave contributions, giving rise to a ground state with simultaneously occupied even- and odd-wave bound states whose separation is tunable via the dipole length. A variational ansatz limited to one particle-hole excitation fails to capture this doubly occupied ground state. We further find that long-range dipolar interactions weaken Anderson's orthogonality catastrophe, reducing the power-law decay of the quasiparticle weight. These effects can be tested with ultracold polar molecules, where microwave shielding and independent tunability of interactions enable experimental access to the predicted regimes.