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arXiv:quant-ph/0503196

Quantum Scattering in Strong Cylindrical Confinement

J. I. Kim, Jörg Schmiedmayer, Peter Schmelcher

Published Mar 24, 2005• 0 citations
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Abstract

A Green's function formalism to analyze the scattering properties in confined geometries is developed. This includes scattering from a central field inside the guide created e.g. by impurities. For atomic collisions our approach applies to the case of parabolic confinement and, with certain restrictions, also to an anharmonic one. The coupling between the angular momentum phase shifts $δ_l$ of a spherically symmetric scattering potential $V(r)$ due to the cylindrical confinement is analysed. Under these general conditions, a broad range of scattering energies covering many transversal excitations is considered and changes to the bound states of $V(r)$ are derived. For collisions between identical atoms, the boson-fermion and fermion-boson mappings are demonstrated.

DOI
10.48550/arxiv.quant-ph/0503196

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A Green's function formalism to analyze the scattering properties in confined geometries is developed. This includes scattering from a central field inside the guide created e.g. by impurities. For atomic collisions our approach applies to the case of parabolic confinement and, with certain restrictions, also to an anharmonic one. The coupling between the angular momentum phase shifts δlδ_lδl​ of a spherically symmetric scattering potential V(r)V(r)V(r) due to the cylindrical confinement is analysed. Under these general conditions, a broad range of scattering energies covering many transversal excitations is considered and changes to the bound states of V(r)V(r)V(r) are derived. For collisions between identical atoms, the boson-fermion and fermion-boson mappings are demonstrated.
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.
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.
Abstract A mechanism to modify the energy band structure is proposed by considering a chain of periodic scatterers forming a linear lattice (akin to an optical lattice for ultracold atoms) around which an external cylindrical trapping potential is applied along the chain axis. It is analytically shown that, when this trapping (confining) potential is tight enough so as to reduce the dimensionality of the problem to an effective one dimension, it may modify the bound and scattering states of the local lattice potential, whose three-dimensional nature around each site is fully taken into account and going beyond the zero-range contact-potential approximation. Since these states contribute to the formation of the energy bands, the latter could thereby be continuously tuned by manipulating the confinement without the need to change the lattice potential. Such dimensionality reduction by quantum confinement is analyzed by using a Green’s functions method that can describe the scattering off the lattice sites in the presence of strong confinement and that can collect the contribuitions of several scattering partial waves.
Subject of this thesis is the investigation of the quantum dynamics of ultracold atoms in confined geometries. We discuss the behavior of ground state atoms inside a quadrupole magnetic field. Such atoms in enough weak magnetic fields can be approximately treated as neutral point-like particles. Complementary to the well-known positive energy resonances, we point out the existence of short-lived negative energy resonances. The latter originate from a fundamental symmetry of the underlying Hamiltonian. We drive a mapping of the two branches of the spectrum. Moreover, we analyze atomic hyperfine resonances in a magnetic quadrupole field. This corresponds to the case for which both the hyperfine and Zeeman interaction, are comparable, and should be taken into account. Finally we develop a general grid method for multichannel scattering of two atoms in a two-dimensional harmonic confienment. With our approach we analyze transverse excitations/deexcitations in the course of the collisional process (distinguishable or identical atoms) including all important partial waves and their couplings due to the broken spherical symmetry. Special attention is paid to suggest a non-trivial extension of the CIRs theory developed so far only for the single-mode regime and zero-energy limit.
Abstract The advances in cold atom experiments have allowed construction of confining traps in the form of curved surfaces. This opens up the possibility of studying quantum gases in curved manifolds. On closed surfaces, many fundamental processes are affected by the local and global properties, i.e. the curvature and the topology of the surface. In this paper, we study the problem of potential scattering on a spherical surface and discuss its difference with that on a 2D plane. For bound states with angular momentum m , their energies ( E m ) on a sphere are related to those on a 2D plane ( − ∣ E m , o ∣ ) as E m = − ∣ E m , o ∣ + E R m 2 − 1 3 + O r o 2 R 2 , where E R = ℏ 2 / ( 2 MR 2 ) , and R is the radius of the sphere. Due to the finite extent of the manifold, the phase shifts on a sphere at energies E ∼ E R differ significantly from those on a 2D plane. As energy E approaches zero, the phase shift in the planar case approaches 0, whereas in the spherical case it reaches a constant that connects the microscopic length scale to the largest length scale R .
We develop a grid method for multichannel scattering of atoms in a waveguide with harmonic confinement. This approach is employed to extensively analyze the transverse excitations and deexcitations as well as resonant scattering processes. Collisions of identical bosonic and fermionic as well as distinguishable atoms in harmonic traps with a single frequency $\ensuremath{\omega}$ permitting the center-of-mass (c.m.) separation are explored in depth. In the zero-energy limit and single mode regime we reproduce the well-known confinement-induced resonances (CIRs) for bosonic, fermionic, and heteronuclear collisions. In the case of the multimode regime up to four open transverse channels are considered. Previously obtained analytical results are extended significantly here. Series of Feshbach resonances in the transmission behavior are identified and analyzed. The behavior of the transmission with varying energy and scattering lengths is discussed in detail. The dual CIR leading to a complete quantum suppression of atomic scattering is revealed in multichannel scattering processes. Possible applications include, e.g., cold and ultracold atom-atom collisions in atomic waveguides and electron-impurity scattering in quantum wires.