Multichannel atomic scattering and confinement-induced resonances in waveguides

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 $\omega$ 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.

Figure 1

(Color online) $\mathit{s}$- and $\mathit{p}$-wave scattering parameters ${\mathit{a}}_{\mathit{s}}$ and ${\mathit{V}}_{\mathit{p}}$ as a function of the potential depth ${\mathit{V}}_0$ for the free-space effective potential $V\left(r\right)+l\left(l+1\right)/\left(2r^2\right)$. Divergences correspond to the appearance of new bound state in the case of $\mathit{s}$-wave scattering or shape resonant in the case of $\mathit{p}$-wave scattering, in the effective potential. All quantities are given in the units (7).

Figure 2

(Color online) (a) The effective quasi-1D coupling constant $g_{1\text{D}}$, (b) the transmission coefficient $\mathit{T}$, and (c) the scattering amplitude $f_{00}^e$, as a function of the scattering length ${\mathit{a}}_{\mathit{s}}$ for one-channel scattering of two bosons under a harmonic confining potential with $\omega =0.002$ for a longitudinal relative energy ${ϵ}_{\parallel }=0.0002$. Solid curves shows the analytical results obtained for the $\mathit{s}$-wave pseudopotential zero-energy limit [13, 14, 29]. The constant ${\mathit{V}}_0$ is varied in the region $-2.30<V_0<-0.32$.

Figure 3

(Color online) The scattering amplitude $f_{00}^e$ as a function of the dimensionless energy $\epsilon$ along with the analytical results—solid curves—for $\omega =0.0002$ and $V_0=-3.0$ (i.e., $a_s=-8.95$).

Figure 4

(Color online) The scattering amplitude $f_{00}^e$ as a function of the dimensionless energy $\epsilon$ for several values of $\omega$. The lines show the analytical results. The amplitudes have been calculated for $V_0=-3.0$ (i.e., $a_s=-8.95$).

Figure 5

(Color online) The total transmission coefficient $\mathit{T}$ for a bosonic collision in a harmonic confinement with $\omega =0.002$ as a function of $a_s/a_{\perp }$ for (a) two-open channels with $\epsilon =1.05$ and (b) three-open channels with $\epsilon =2.05$. The system is initially in transverse level $n$.

Figure 6

(Color online) The total transmission coefficients $\mathit{T}$ for bosonic collisions as a function of the dimensionless energy $\epsilon$ for the two cases of the system being initially in (a) the ground transverse state $n=0$ and (b) the first excited transverse state $n=1$, for several ratios of $a_s/a_{\perp }$ and $\omega =0.002$. The black (solid) curve corresponds to $a_s/a_{\perp }=1/C$ for which the zero-energy CIR in the single-mode regime is encountered.

Figure 7

The probability density ${\left|\psi \left(x,z\right)\right|}^2$ for bosonic collisions as a function of $\mathit{x}$ and $\mathit{z}$ at $a_s/a_{\perp }=0.68$ for $\epsilon =0.05$—zero-energy CIR. The corresponding transmission values are also indicated. The result has been obtained for $\omega =0.002$.

Figure 8

The probability density ${\left|\psi \left(x,z\right)\right|}^2$ for bosonic collisions as a function of $\mathit{x}$ and $\mathit{z}$ at $a_s/a_{\perp }=+4.39$ for two cases of the single-mode regime (a) and two-mode regime (b) with different values of $\epsilon$. The corresponding transmission values are also indicated. All subfigures are for $\omega =0.002$ and $n=0$.

Figure 9

(Color online) (a) The transition probabilities $W_{n{n}^{\prime }}$ as a function of $a_s/a_{\perp }$ in the four mode regime for $\epsilon =3.05$. (b) The calculated transition probabilities $W_{n{n}^{\prime }}$ as a function of $\epsilon$ up to four open channels for $a_s/a_{\perp }=-30.08$. $\omega =0.002$ for both subfigures.

Figure 10

(Color online) (a),(b) The mapped coupling constant $g_{1\text{D}}^{\text{map}}$ and the transmission coefficient $\mathit{T}$ as a function of the scattering length ${\mathit{a}}_{\mathit{p}}$ for single-channel scattering of two fermions under a harmonic confinement with $\omega =0.002$ and for the longitudinal energy ${ϵ}_{\parallel }=0.0002$—red (square), $\omega =0.02$, ${ϵ}_{\parallel }=0.002$—black (plus), and $\omega =0.06$, ${ϵ}_{\parallel }=0.002$—blue (dot). (c) The scattering amplitude $f_{00}^o$ as a function of the scattering length ${\mathit{a}}_{\mathit{p}}$ for $\omega =0.002$ and ${ϵ}_{\parallel }=0.0002$. The corresponding constant ${\mathit{V}}_0$ is varied in the region $-4.54<V_0<-4.47$.

Figure 11

(Color online) Total transmission coefficient $\mathit{T}$ in a fermionic collision as a function of ${\mathit{a}}_{\mathit{p}}$ for $\omega =0.002$ being initially in transverse level $\mathit{n}$ for (a) two-open channels with $\epsilon =1.05$ and (b) three-open channel with $\epsilon =2.05$.

Figure 12

(Color online) Total transmission coefficient $\mathit{T}$ for a fermionic collision as a function of the dimensionless energy $\epsilon$ for the two cases of the system being initially in the (a) ground transverse state $n=0$ and (b) first excited transverse state $n=1$, for several ratios of $a_p/a_{\perp }$. We have used $\omega =0.002$.

Figure 13

(Color online) Transition probabilities $W_{n{n}^{\prime }}$ in fermionic collisions as a function of $a_p/a_{\perp }$ for four open channels. $\omega =0.002$ and $ϵ=0.0142$ are employed.

Figure 14

Transmission $\mathit{T}$ as a function of the depth $-V_0$ of the potential (6) for two distinguishable atomic species for $\omega =0.002$ for ${ϵ}_{\parallel }=0.0002$. Units according to Eq. (7).

Figure 15

(Color online) Scattering amplitude as a function of the depth $-V_0$ of the potential (6) for two distinguishable atomic species for $\omega =0.002$ and ${ϵ}_{\parallel }=0.0002$. Units according to Eq. (7).

Figure 16

(Color online) Transmission coefficient $\mathit{T}$ as a function of the depth $-V_0$ of the potential (6) for two distinguishable atomic species for $\omega =0.002$ and $\epsilon =2.05$ in the three-mode regime, being initially in transverse level $\mathit{n}$. Units according to Eq. (7).

Figure 17

(Color online) Transmission coefficient $\mathit{T}$ as a function of the dimensionless energy $\epsilon$ for scattering of two distinguishable particles in a harmonic confinement with $\omega =0.002$ (a) for several ratios $a_s/a_{\perp }$ in the case where ${\mathit{a}}_{\mathit{p}}$ is negligible compared to ${\mathit{a}}_{\mathit{s}}$, (b) for $-V_0=-4.505$ when both scattering lengths ${\mathit{a}}_{\mathit{s}}$ and ${\mathit{a}}_{\mathit{p}}$ are comparable [here the transmission coefficients for bosonic-collision blue (square) and fermionic-collision black (dashed) are also provided] and (c) for $-V_0=-4.505$ the system being initially in the transversal level $\mathit{n}$.

Figure 18

(Color online) Transition probabilities $W_{n{n}^{\prime }}$ for quasi-1D scattering of distinguishable particles in a harmonic trap $\omega =0.002$, as a function of $-V_0$ for four open channels and $\epsilon =3.05$. ${\mathit{V}}_0$ is given in units of Eq. (7).