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arXiv:quant-ph/0506192 · Physical Review A

Quantum scattering in quasi-one-dimensional cylindrical confinement

J. I. Kim, Jörg Schmiedmayer, Peter Schmelcher

Published Oct 17, 2005• 35 citations

Abstract

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 $p$- and $d$-wave pairings.

DOI: 10.1103/physreva.72.042711

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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

p

- and

d

-wave pairings.

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.

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.

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 study the three-body problem for both fermionic and bosonic cold-atom gases in a parabolic transverse trap of length scale ${a}_{\ensuremath{\perp}}$. For this quasi-one-dimensional (quasi-1D) problem, there is a two-body bound state (dimer) for any sign of the 3D scattering length

a

and a confinement-induced scattering resonance. The fermionic three-body problem is universal and characterized by two atom-dimer scattering lengths

{a}_{\mathit{ad}}

and

{b}_{\mathit{ad}}

. In the tightly bound ``dimer limit'' ${a}_{\ensuremath{\perp}}∕a\ensuremath{\rightarrow}\ensuremath{\infty}$, we find

{b}_{\mathit{ad}}=0

and

{a}_{\mathit{ad}}

is linked to the 3D atom-dimer scattering length. In the weakly bound ``BCS limit'' ${a}_{\ensuremath{\perp}}∕a\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$, a connection to the Bethe ansatz is established, which allows for exact results. The full crossover is obtained numerically. The bosonic three-body problem, however, is nonuniversal:

{a}_{\mathit{ad}}

and

{b}_{\mathit{ad}}

depend both on ${a}_{\ensuremath{\perp}}∕a$ and on a parameter

{R}^{*}

related to the sharpness of the resonance. Scattering solutions are qualitatively similar to fermionic ones. We predict the existence of a single confinement-induced three-body bound state (trimer) for bosons.