Wave-packet dynamical analysis of ultracold scattering in cylindrical waveguides

A wave-packet propagation method is developed and applied to investigate the quantum dynamics of scattering processes of identical and distinguishable atoms in harmonic waveguides. The quantum dynamics of the confinement-induced resonances (CIRs) for ultracold collisions of identical particles, $s$-wave CIRs for bosons and $p$-wave CIRs for fermions, is explored in detail. Our multigrid approach allows us to fully take into account the coupling between the center-of-mass (c.m.) and relative motions in the case of distinguishable atoms. The latter includes, in particular, $s$- and $p$-partial-wave mixing, caused by the confining trap, which acts differently on the different atomic species. Specifically, we explore in detail the recently discovered [J. I. Kim, V. S. Melezhik, and P. Schmelcher, Phys. Rev. Lett. 97, 193203 (2006)] dual CIR, which is based on a destructive interference mechanism leading to complete transmission in the waveguide, although the corresponding scattering in free space exhibits strong $s$- and $p$-wave scattering.

Figure 1

(a) $s$- and $p$-wave scattering wave functions ${\psi }_{1s}\left(kr\right)$ and ${\psi }_{2p}\left(kr\right)$ (normalized to unit $\text{amplitude}\to \text{sin}\left(kr-\pi ∕2l+{\delta }_l\right)∕\left(kr\right)$ for $r\to \infty$), together with the corresponding interatomic potentials $V\left(r\right)+l\left(l+1\right)∕\left(2\mu r^2\right)$. Calculations are performed for $V_0=-1.136$, $\mu =1$, leading to a single bound state in the $s$-wave potential (the scattering function possesses a single node as a function of $r$ in the region where the potential acts) and a completely repulsive interaction for the $p$ wave (no nodes with respect to $r$ in the same region). The collision energy is $E=5\times {10}^{-4}$. Cases shown correspond to the following $s$- and $p$-wave scattering parameters: $a_s\left(E\right)=+4.543$ and $V_p\left(E\right)=-1.677$. (b) The same as in (a) but now for a deeper interatomic potential $V_0=-8.45$, leading to three bound states in the $s$-wave potential (see the three nodes with respect to $r$ in the region where the potential acts) and one bound state in the $p$-wave potential (a single node with respect to $r$ in the same region). Collision energy is $E=5\times {10}^{-4}$. The resulting $s$- and $p$-wave scattering parameters are $a_s\left(E\right)=+4.799$ and $V_p\left(E\right)=-48.20$. All quantities are given in units according to Eqs. (6, 7).

Figure 2

(Color online) Time evolution of the scattering amplitude $f^{+}\left(t\right)=f_g\left(t\right)$, the effective quasi-1D coupling constant $g_{1D}\left(t\right)$, and the transmission coefficients $T\left(t\right)$ and $\tilde{T}\left(t\right)$ for different values of $a_{\perp }∕a_s$: far below the CIR $a_{\perp }∕a_s=0.248$, near the CIR $a_{\perp }∕a_s=1.469$, and far above the CIR $a_{\perp }∕a_s=2.080$. The calculations are performed for scattering of identical bosons in a harmonic trap with ${\omega }_1={\omega }_2=0.02$, for the longitudinal collision energy $\epsilon =0.004$. $V_0$ is varied within the region $-1.3\le V_0\le -0.85$. Time $t$ is given in units of $t_0=2\pi ∕\left(\omega +\epsilon \right)$; the effective quasi-1D coupling constant $g_{1D}$ in units of $1∕\left(\mu a_{\perp }\right)$.

Figure 3

(Color online) Scattering amplitude $f^{+}\left(a_{\perp }∕a_s\right)=f_g\left(a_{\perp }∕a_s\right)$, effective quasi-1D coupling constant $g_{1D}\left(a_{\perp }∕a_s\right)$, and transmission coefficient $T\left(a_{\perp }∕a_s\right)$ calculated for the symmetric case $f_u=0$ of bosonic scattering for $V\left(r\right){\delta }_{l0}$ in a harmonic confinement ${\omega }_1={\omega }_2=0.02$, for $\epsilon =0.004$. The coupling constant $V_0$ is varied in the region $-1.3\le V_0\le -0.85$. The effective quasi-1D coupling constant $g_{1D}$ is provided in units of $1∕\left(\mu a_{\perp }\right)$.

Figure 4

(Color online) Scattering amplitude $f^{+}\left(a_{\perp }∕a_p\right)=f_u\left(a_{\perp }∕a_p\right)$, mapped coupling constant $g_{1D}^{map}\left(a_{\perp }∕a_p\right)$, and transmission coefficient $T\left(a_{\perp }∕a_p\right)$ for the antisymmetric case $f_g=0$ of fermionic scattering defined by the interatomic operator $V\left(r\right){\delta }_{l1}$ in a harmonic confinement ${\omega }_1={\omega }_2=0.02$ for $\epsilon =0.004$. The coupling constant $V_0$ is varied in the region $-9.5\le V_0\le -7.5$. The mapped coupling constant $g_{1D}^{map}$ is provided in units of $1∕\left(\mu a_{\perp }{k}^2\right)$.

Figure 5

(Color online) (a) Scattering amplitudes $f^{+}\left(a_{\perp }∕a_s\right)$ and $f_g\left(a_{\perp }∕a_s\right)$ and corresponding transmission coefficients $T\left(a_{\perp }∕a_s\right)={\left|1+f^{+}\left(a_{\perp }∕a_s\right)\right|}^2$ and $T_g\left(a_{\perp }∕a_s\right)={\left|1+f_g\left(a_{\perp }∕a_s\right)\right|}^2$ for scattering of distinguishable and identical (bosonic) atoms in a harmonic confinement ${\omega }_1={\omega }_2=0.02$ for $\epsilon =0.004$. In both cases the interatomic interaction is described by the same potential function (21); however, the interaction operator is acting on both even and odd waves for scattering of distinguishable atoms and exclusively on the $s$ waves for scattering of identical bosons. The coupling constant $V_0$ is varied in the region $-1.3\le V_0\le -0.85$. (b) Scattering amplitudes $f^{+}\left(a_{\perp }∕a_p\right)$ and $f_u\left(a_{\perp }∕a_p\right)$ and corresponding transmission coefficients $T\left(a_{\perp }∕a_p\right)={\left|1+f^{+}\left(a_{\perp }∕a_p\right)\right|}^2$ and $T_u\left(a_{\perp }∕a_s\right)={\left|1+f_u\left(a_{\perp }∕a_p\right)\right|}^2$ for scattering of distinguishable and fermionic atoms in harmonic confinement ${\omega }_1={\omega }_2=0.02$, at $\epsilon =0.004$. In both cases the interatomic interaction is described by the same potential function (21); however, the interaction operator is acting on both even and odd waves for the scattering of distinguishable particles and exclusively on the $p$ waves for scattering of identical fermions. The coupling constant $V_0$ varies within the region $-9.5\le V_0\le -7.5$.

Figure 6

(Color online) (a) $s$- and $p$-wave scattering lengths $a_s\left(V_0\right)$ and $a_p\left(V_0\right)$; (b) corresponding $s$-wave and total 3D scattering cross sections ${\sigma }_s\left(V_0\right)$ and $\sigma \left(V_0\right)={\sigma }_s\left(V_0\right)+{\sigma }_p\left(V_0\right)$ in free space; and (c) $s$-wave and total reflection coefficients $R_s\left(V_0\right)$ and $R\left(V_0\right)$ describing the scattering for the separable case ${\omega }_1={\omega }_2=0.02$ and the nonseparable case ${\omega }_1∕1.35={\omega }_2=0.02$ of harmonic confinement. All quantities are given in units according to Eqs. (6, 7); $\epsilon =0.004$.

Figure 7

Time evolution of the intersection of the probability density distribution ${\left|\psi \left(z,\rho =0,t\right)∕r\right|}^2$ of the wave packet in the harmonic trap ${\omega }_1={\omega }_2=0.02$ for two distinguishable particles for three cases: (a) nonresonant scattering in free space for $V_0=-0.6$, (b) $s$-wave resonant scattering in free space for $V_0=-1.12$ (creating a CIR in a harmonic trap), (c) simultaneous $s$- and $p$-wave resonances in free space at $V_0=-8.45$ (creating a dual CIR in a harmonic trap). Time $t$ is given in units of $t_0=2\pi ∕\left(\omega +\epsilon \right)$ and other quantities in units according to Eqs. (6, 7); $\epsilon =0.004$.

Figure 8

(Color online) Time evolution of the probability density distribution in the harmonic trap ${\omega }_1∕1.35={\omega }_2=0.02$ averaged over the angular variables $W\left({\rho }_R,r,t\right)=\int {\left|\psi \left({\rho }_R,r,\theta ,\varphi ,t\right)∕\right|}^2\left(r^2{\rho }_R\right)\text{sin}\theta d\theta d\varphi$ for two distinguishable particles $m_1∕m_2=40∕87$ for the same three cases as in Fig. 7: (a) nonresonant free-space scattering $V_0=-0.6$; (b) $s$-wave resonant scattering in free space for $V_0=-1.12$ (creating a CIR in a harmonic trap); (c) simultaneous $s$- and $p$-wave resonant behavior in free space for $V_0=-8.45$ (creating a dual CIR in a harmonic trap). Time $t$ is given in units of $t_0=2\pi ∕\left(\omega +\epsilon \right)$ and other quantities in units according to Eqs. (6, 7); $\epsilon =0.004$.