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Progress of Theoretical Physics Supplement

Quantum Confined Scattering beyond thes-Wave Approximation

J. I. Kim, Vladimir S. Melezhik, Peter Schmelcher

Published Jan 1, 2007• 14 citations
Publisher

Abstract

A dual confinement induced resonance effect first observed numerically in a paper [J. I. Kim, V. S. Melezhik and P. Schmelcher, Phys. Rev. Letters 97 (2006), 193203] predicts the possibility of suppressing the effective two-body interaction in a quasi-one dimensional (1D) cylindrical geometry. Here this low dimensionality effect is analytically modeled by applying a previous formalism to treat low energy quantum scattering processes under tight confinement [J. I. Kim, J. Schmiedmayer and P. Schmelcher, Phys. Rev. A 72 (2005), 042711]. This analytic formalism is improved in order to account non-perturbatively for the three dimensional (3D) scattering phase shift l = 1 for a given spherically symmetric interaction potential between distinguishable colliding particles. The role of the confinement on the scattering properties can then be systematically calculated up to the p-wave.

DOI
10.1143/ptps.166.159

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A dual confinement induced resonance effect first observed numerically in a paper [J. I. Kim, V. S. Melezhik and P. Schmelcher, Phys. Rev. Letters 97 (2006), 193203] predicts the possibility of suppressing the effective two-body interaction in a quasi-one dimensional (1D) cylindrical geometry. Here this low dimensionality effect is analytically modeled by applying a previous formalism to treat low energy quantum scattering processes under tight confinement [J. I. Kim, J. Schmiedmayer and P. Schmelcher, Phys. Rev. A 72 (2005), 042711]. This analytic formalism is improved in order to account non-perturbatively for the three dimensional (3D) scattering phase shift l = 1 for a given spherically symmetric interaction potential between distinguishable colliding particles. The role of the confinement on the scattering properties can then be systematically calculated up to the p-wave.
Finite-size effects not only alter the energy levels of small systems, but can also lead to additional effective interactions within these systems. Here the problem of low-energy quantum scattering by a spherically symmetric short-range potential in the presence of a general cylindrical confinement is investigated. A Green's function formalism is developed which accounts for the full three-dimensional (3D) nature of the scattering potential by incorporating all phase shifts and their couplings. This quasi-1D geometry gives rise to scattering resonances and weakly localized states, whose binding energies and wave functions can be systematically calculated. Possible applications include, e.g., impurity scattering in ballistic quasi-1D quantum wires in mesoscopic systems and in atomic matter-wave guides. In the particular case of parabolic confinement, the present formalism can also be applied to pair collision processes such as two-body interactions. Weakly bound pairs and quasimolecules induced by the confinement and having zero or higher orbital angular momentum can be predicted, such as ppp- and ddd-wave pairings.
It was recently predicted [Phys. Rev. Lett. 81, 938 (1998)]] that atom-atom scattering under transverse harmonic confinement is subject to a "confinement-induced resonance" where the effective one-dimensional coupling strength diverges at a particular ratio of the confinement and scattering lengths. As the initial prediction made use of the zero-range pseudopotential approximation, we now report numerical results for finite-range interaction potentials that corroborate this resonance. In addition, we now present a physical interpretation of this effect as a novel type of Feshbach resonance in which the transverse modes of the confining potential assume the roles of "open" and "closed" scattering channels.
We demonstrate that scattering of particles strongly interacting in three dimensions (3D) can be suppressed at low energies in a quasi-one-dimensional (1D) confinement. The underlying mechanism is the interference of the s- and p-wave scattering contributions with large s- and p-wave 3D scattering lengths being a necessary prerequisite. This low-dimensional quantum scattering effect might be useful in "interacting" quasi-1D ultracold atomic gases, guided atom interferometry, and impurity scattering in strongly confined quantum wire-based electronic devices.
We show that a purely sss-wave interaction in three dimensions (3D) can induce higher partial-wave resonances in mixed dimensions. We develop two-body scattering theories in all three cases of 0D-3D, 1D-3D, and 2D-3D mixtures and determine the positions of higher partial-wave resonances in terms of the 3D sss-wave scattering length assuming a harmonic confinement potential. We also compute the low-energy scattering parameters in the ppp-wave channel (scattering volume and effective momentum) that are necessary for the low-energy effective theory of the ppp-wave resonance. We point out that some of the resonances observed in the Florence group experiment [Phys. Rev. Lett. 104, 153202 (2010)] can be interpreted as the ppp-wave resonances in the 2D-3D mixed dimensions. Our study paves the way for a variety of physics, such as Anderson localization of matter waves under ppp-wave resonant scatterers.
We theoretically investigate the confinement-induced resonance for quasi-one-dimensional quantum systems under transversely anisotropic confinement, using a two-body sss-wave-scattering model in the zero-energy collision limit. We predict a single resonance for any transverse anisotropy, whose position shows a slight downshift with increasing anisotropy. We compare our prediction with the recent experimental result by Haller et al. [Phys. Rev. Lett. 104, 153203 (2010)], in which two resonances are observed in the presence of transverse anisotropy. The discrepancy between theory and experiment remains to be resolved.
We point out that theories describing s -wave collisions of bosonic atoms confined in one-dimensional (1D) or two-dimensional (2D) geometries can be extended to much tighter confinements than previously thought. This is achieved by replacing the scattering length by an energy-dependent scattering length which was already introduced for the calculation of energy levels under 3D confinement. This replacement accurately predicts the position of confinement-induced resonances in strongly confined geometries.