Confinement-induced resonances in two-center problem via a pseudopotential approach

We study confined scattering of a quantum particle by two centers fixed on the longitudinal axis of a harmonic-waveguide-like trap. The conditions of confinement-induced resonances (CIRs) appearing in these systems, when scattering cross section approaches the unitary limit, are derived for a regularized pseudopotential describing particle interaction with scattering centers. In the limit of a single center, the position of CIR for an even state tends to the well-known result obtained by Olshanii [Phys. Rev. Lett. 81, 938 (1998)]. Our result can be applicable to confined atomic scattering by fixed impurities, such as ions or Rydberg atoms, with possible extension to $N$ impurities, or by two-atomic molecules.

Figure 1

A schematic representation of an atom with the wave number $k$ confined in a waveguide and scattered from two impurities.

Figure 2

Total transmission coefficient $T_{tot}(\frac{a_{\perp }}{a_{{}_{3D}}},a,k)$ (in logarithmic scale) for three different values of $a=0.01a_{\perp }$ (blue curves), $a=0.10a_{\perp }$ (red curves), and $a=0.50a_{\perp }$ (black curves). Dashed (solid) curves belong to $k{a}_{\perp }=0.0707$ ($k{a}_{\perp }=7.07\times {10}^{-5}$). We indicate by the arrow the CIR position obtained by Olshanii [8] for the scattering in a one-center problem.

Figure 3

CIR position vs $a/a_{\perp }$ when both even and odd wave scattering are included and $k{a}_{\perp }=0.0707$. (a) The CIR condition when $T_{tot}=0$ [obtained from Eq. (72) with plus sign before the square root]. The CIR position obtained by Olshanii [8] for the scattering in a one-center problem is indicated with an arrow. (b) Dual CIR position when $T_{tot}=1$.

Figure 4

Total transmission coefficient $T_{tot}$ (in logarithmic scale) along with (a) pure even transmission $T_e$ and (b) pure odd transmission $T_o$ for the impurity separation $2a=a_{\perp }$ as a function of $a_{\perp }/a_{{}_{3D}}$ when $k{a}_{\perp }=0.0707$.

Figure 5

Even CIR position vs $a/a_{\perp }$ when $k{a}_{\perp }=0.0707$. The CIR position obtained by Olshanii [8] for the scattering in a one-center problem is here indicated with an arrow.

Figure 6

Odd CIR position vs $a/a_{\perp }$ when $k{a}_{\perp }=0.0707$.

Figure 7

(a) $\left[1+\mathrm{Re}\left\{f_e\right\}\right]$ (blue dashed curve), $\mathrm{Re}\left\{f_o\right\}$ (red dashed curve), and the resultant $\left[1+\mathrm{Re}\left\{f_e\right\}+\mathrm{Re}\left\{f_o\right\}\right]$ (black solid curve), and (b) $\mathrm{Im}\left\{f_e\right\}$ (blue dashed curve), $\mathrm{Im}\left\{f_o\right\}$ (red dashed curve), and the resultant $\left[\mathrm{Im}\left\{f_e\right\}+\mathrm{Im}\left\{f_o\right\}\right]$ (black solid curve) for the impurity separation $2a=a_{\perp }$ as a function of $a_{\perp }/a_{{}_{3D}}$ when $k{a}_{\perp }=0.0707$. Both black solid curves pass zero axis at $a_{\perp }/a_{{}_{3D}}=0.42,3.04$, representing the CIR positions.