Multichannel effects near confinement-induced resonances in harmonic waveguides

We analyze the impact of multichannel scattering in harmonic waveguides on the positions and widths of confinement-induced resonances for both isotropic and anisotropic transversal confinement. Multichannel scattering amplitudes and transmission coefficients are calculated and used to characterize the resonant behavior of atomic collisions with varying anisotropy. A mechanism is established which leads to a splitting of the confinement-induced resonance in the presence of anisotropy.

Figure 1

Dependence of the partial transmission coefficients $T_{n_1,n_2}(a_{\perp }/a_s,E_{\parallel })$ on $a_{\perp }/a_s$ and $E_{\parallel }$ in an isotropic waveguide ${\omega }_1/{\omega }_2=1$.

Figure 2

Transition probability $P_{02}(E,a_{\perp }/a_s)$ from the initial ground $n=n_1+n_2=0$ into the transversally excited states $n=n_1+n_2=2$ for a few values of $a_{\perp }/a_s$.

Figure 3

Partial transmission coefficients $T_{n_1,n_2}(a_{\perp }/a_s,E_{\parallel })$ in an anisotropic waveguide ${\omega }_1/{\omega }_2=1.05$ as functions of $a_{\perp }/a_s$ calculated for a near-threshold collision energy $E_{\parallel }/E_{\perp }^{\left(0\right)}=5\times {10}^{-3}$.

Figure 4

Total transmission coefficients $T(a_{\perp }/a_s,E_{\parallel })$ in the region $a_{\perp }/a_s\sim 1.4603\cdots$ of the CIR for the isotropic case ${\omega }_1/{\omega }_2=1$ as well as for anisotropic waveguides with ${\omega }_1/{\omega }_2=1.05$ and $1.1$ calculated for $W_2/W_0=0.05$. For ${\omega }_1/{\omega }_2\ne 1$ the splitting of the minimum of the transmission coefficient can be observed.

Figure 5

Total transmission coefficients $T(a_{\perp }/a_s,W_2/W_0)$ in the region $a_{\perp }/a_s\sim 1.4603\cdots$ of the CIR for anisotropic case ${\omega }_1/{\omega }_2=1.1$ as a function of relative population $W_2/W_0$ of the first excited manifold $n=2$. The splitting of the minimum of the transmission coefficient is still present even for only 1$\%$ population of the excited states.

Figure 6

Illustration of the spectrum of the atomic dimer in harmonic isotropic ${\omega }_1/{\omega }_2=1$ and anisotropic ${\omega }_1/{\omega }_2=1.2$ waveguides as a function of $a_{\perp }/a_s$. Below the continuum threshold $E_{\perp }^{\left(0\right)}$ the calculated binding energy of the weakly bound state is shown. Between the thresholds $E_{\perp }^{\left(0\right)}$ and $E_{\perp }^{\left(4\right)}$ the calculated resonant energies $E_r$ were determined by the minimum of the transmission coefficient $T_{00}(a_{\perp }/a_s,E)$. Between the $E_{\perp }^{\left(0\right)}$ and $E_{\perp }^{\left(2\right)}$ thresholds the resonant energy $E_r$ coincides with the location of the zero of $Im\left\{f_{00}(a_{\perp }/a_s)\right\}$. Bold solid curves with the full dots correspond to the isotropic case ${\omega }_1/{\omega }_2=1$ and the thin curves with open circles to the anisotropic case ${\omega }_1/{\omega }_2=1.2$. Indices $(n_1,n_2)$ label the splitted sub-levels of the threshold energies $E_{\perp }^{(n_1,n_2)}({\omega }_1/{\omega }_2=1.2)$ of the excited states with $n=n_1+n_2$. For the second excited threshold with $n=4$ only the lowest sublevel $(0,4)$ is shown.