Quantum scattering in quasi-one-dimensional cylindrical confinement

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.

Figure 1

(a) Longitudinal profile of the cylindrical guide. The scattering potential $V\left(r\right)$ is limited to within the spherical shaded area at the center. (b) Total potential energy along the transverse $xy$ plane passing through the origin $\mathit{r}=\mathbf{0}$ (thick curve). For an attractive $V\left(r\right)$, one deep $E_0$ and another shallow $E_B$ bound state without confinement are indicated, with their respective probability density profiles (dotted curves).

Figure 2

A pictorial view of a scattering “path” of a particle coming from the left. Outside the effective 1D range $\mid z\mid ⪢R_{1D}^{\prime }$, the particle is stabilized by the confinement back into, e.g., the transversal ground state (for the single-mode regime $n_E=0$). Here the net effect of the scattering is contained in the amplitudes $f_0^{-}$ and $f_0^{+}$ of backward and forward scattering, respectively. Within the dashed circle, however, the spherical symmetry prevails. The phase information of the wave function within this region is then captured by $f_0^{\pm }$.

Figure 3

(Color online) $s$-wave binding energy $E_B$ for parabolic (dashed curve) and square-well (upper solid curve) confinement, scaled by the respective ground-state energies ${ϵ}_0$. The free-space energy (lower solid curve) $E_{Bf}$ is scaled with the square-well value of ${ϵ}_0$ and is defined only for positive $a$. The dotted line marks the continuum threshold under confinement.