Mimicking multichannel scattering with single-channel approaches

The collision of two atoms is an intrinsic multichannel (MC) problem, as becomes especially obvious in the presence of Feshbach resonances. Due to its complexity, however, single-channel (SC) approximations, which reproduce the long-range behavior of the open channel, are often applied in calculations. In this work the complete MC problem is solved numerically for the magnetic Feshbach resonances (MFRs) in collisions between generic ultracold ${}^6\mathrm{Li}$ and ${}^{87}\mathrm{Rb}$ atoms in the ground state and in the presence of a static magnetic field $B$. The obtained MC solutions are used to test various existing as well as presently developed SC approaches. It was found that many aspects even at short internuclear distances are qualitatively well reflected. This can be used to investigate molecular processes in the presence of an external trap or in many-body systems that can be feasibly treated only within the framework of the SC approximation. The applicability of various SC approximations is tested for a transition to the absolute vibrational ground state around an MFR. The conformance of the SC approaches is explained by the two-channel approximation for the MFR.

Figure 1

Scattering length $a_{\mathrm{sc}}$ as a function of the external magnetic field value $B$ for ${}^6\mathrm{Li}$-${}^{87}\mathrm{Rb}$ scattering at $E=50$ Hz. A broad and a narrow MFR are visible at $B_0=1066.917$ G and $B_0=1282.576$ G. The horizontal line marks the background scattering length $a_{\mathrm{bg}}=-17.8a_0$ of the left resonance.

Figure 2

The channel functions ${\psi }_{\chi }(R)$ for the ${}^6\mathrm{Li}$-${}^{87}\mathrm{Rb}$ collision in an off-resonant field $B=1000$ G (a) and a field $B=1066.9$ G close to the resonance (b). The atomic labels (see Table ) are indicated in (a). The insets focus on a region of small internuclear distance.

Figure 3

The channel functions ${\psi }_{\xi }(R)$. The same as Fig. 2 but in MB. The molecular labels are indicated in (a).

Figure 4

Original $X\hspace{0.5em}{}^1{\Sigma }^{+}$ potential $V(R)$ [blue (thick) solid] with applied $s$ variation (black solid), $C_6$ variation (dashes) and $G$ variation (dots). The variation parameters are $s=0.03a_0$, $\delta C_6=C_6/2$, and $\delta G=V(R_{\mathrm{sh}})$ (see Table for the adopted parameters). The insets show some relevant ranges of $R$ on an enlarged scale.

Figure 5

The channel functions of the singlet state $|S_1\rangle$ close to the resonance ($a_{\mathrm{sc}}=-65\hspace{0.5em}450a_0$) and away from the resonance ($a_{\mathrm{sc}}=-14.9a_0$) are depicted together with the channel function of the close to resonant $|S_2\rangle$ state. All three functions differ for $R<30a_0$ only by a constant prefactor.

Figure 6

Comparison of the SC wave functions ${\varphi }^{\upsilon }(R)$ with the MC functions of the singlet state $|S_1\rangle$ (scaled) and the open channel $|a_1\rangle$. The SC potentials are varied to match the asymptotic behavior of the MC channel functions of the open channel. (a) Off-resonant case with $a_{\mathrm{sc}}=-14.9a_0$ ($B=1000$ G). (b) Resonant case with $a_{\mathrm{sc}}=-65\hspace{0.5em}450a_0$ ($B=1066.9$ G). The according values of the $s,\delta C_6,\delta G$, and $\delta \mu$ parameter are given in Table . The smaller plots focus, respectively, on a region of small internuclear distance (left) and a region $R\approx R_{\mathrm{sh}}$ (right).

Figure 7

Sketch of the resonant SC wave functions obtained via $G$ variation and respective BO potentials relevant for the DPA transition to either the singlet ground state (thin, solid) or the triplet ground state (thin, dashed) of the respective $X\hspace{0.5em}{}^1{\Sigma }^{+}$ (thick, leftmost at start) and $a^3{\Sigma }^{+}$ (thick, rightmost at start) potentials. For better visibility, the potentials and wave functions are shifted along the $y$ axis. In reality, singlet and triplet potentials have the same threshold energy.

Figure 8

Comparison of MC and SC results for the transition rate to the absolute vibrational ground state relative to the respective maximal transition rate as a function of the scattering length (a) and the inverse scattering length (b). The MC results are fitted according to the TC approximation [Eq. (20)].

Figure 9

(a) Scattering length $a_{\mathrm{sc}}$ as a function of the $\delta \mu$ parameter of the mass variation. By constraining the variation to one branch (thick line), any scattering length is reached while keeping the number of bound states $N_{\mathrm{b}}$ constant. By a variation between two branches (dashed lines), any scattering length is reached while $N_{\mathrm{b}}$ changes. (b) Resonant SC functions ($a_{\mathrm{sc}}=\infty$) at two different $\delta \mu$ parameters: $\delta \mu \approx -0.01\mu$ (dashed), $\delta \mu \approx 0.03\mu$ (thick). In order to make the relevant phase and amplitude difference at small internuclear distances visible, one of the wave functions is multiplied by $-1$ in the inset.

Figure 10

Comparison of MC and SC results for the transition rate $I^0$ to the absolute vibrational ground state relative to the respective maximal transition rate as a function of the phase shift $\delta$. The different SC variation parameters are either varied around one SC resonance (a) or between two resonances staying on the same $a_{\mathrm{sc}}(\delta \mu )$ branch (b). The MC results are fitted, according to the TC approximation [Eq. (20)]. Note the $\sin {}^2$-like form of the functions.