Coherent multichannel optical theorem: Quantum control of the total scattering cross section

The optical theorem is a fundamental aspect of quantum scattering theory. Here, we generalize this theorem to the case where the incident scattering state is a superposition of internal states of the collision partners, introducing additional interference contributions and, e.g., providing a route to control the total integral cross section. As in its standard form, forward scattering plays an essential role in the multichannel optical theorem, but with interference terms being related to the inelastic forward scattering amplitudes between states in the initial superposition. Using the resultant control index, we show that extensive control is possible over ultracold collisions of oxygen molecules in their rovibrational ground states, and of ${}^{85}\mathrm{Rb}\text{−}{}^{85}\mathrm{Rb}$ collisions, promising systems for the experimental demonstration of the quantum interference control of the total scattering cross section.

Figure 1

Coherent control of the total integral cross section for the cold ${}^{17}\mathrm{O}_2+{}^{17}\mathrm{O}_2$ collisions at $10\mu \mathrm{K}$ from the initial superposition ${\mathrm{\Psi }}_E$ (upper panels) and ${\mathrm{\Psi }}_{NE}$ (lower panels). [(a),(c)] Control landscape; [(b),(d)] Control by the relative phase $\beta$ with $\eta =\pi /4$ fixed.

Figure 2

Coherent control of the total cross section for the ultracold ${}^{85}\mathrm{Rb}+{}^{85}\mathrm{Rb}$ collisions in their lower hyperfine states $F=2$ at $50\mu \mathrm{K}$ from the initial superpositions (a) ${\mathrm{\Psi }}_E$ and (b) ${\mathrm{\Psi }}_{NE}$. The relative phases $\beta ,\gamma$ are varied, while the relative populations $\eta =11\pi /32$, $\epsilon =3\pi /8$ are fixed for achieving the best control.