Time delays in ultracold atomic and molecular collisions

We study the behavior of the Eisenbud-Wigner collisional time delay around Feshbach resonances in cold and ultracold atomic and molecular collisions. We carry out coupled-channel scattering calculations on ultracold Rb and Cs collisions. In the low-energy limit, the time delay is proportional to the scattering length and so exhibits a pole as a function of applied field. At high energy, it exhibits a Lorentzian peak as a function of either energy or field. For narrow resonances, the crossover between these two regimes occurs at an energy proportional to the square of the resonance strength parameter $s_{\text{res}}$. For wider resonances, the behavior is more complicated and we present an analysis in terms of multichannel quantum defect theory.

Figure 1

Time delay $Q$ as a function of magnetic field $B$ at various energies for the four example resonances: (a) resonance 1, (b) resonance 2, (c) and (d) resonance 3, and (e) and (f) resonance 4. Dashed and dotted lines show energies 2 and 5 times those of the corresponding solid lines.

Figure 2

The QDT functions $C^{-2}\left(E\right)$ and $tan\lambda \left(E\right)$ for various background scattering lengths $a_{\text{bg}}$.

Figure 3

Resonance parameters for resonances 1 (blue), 3 (orange), and 4 (green). (a) Width ${\mathrm{\Gamma }}_{\text{B}}\left(E\right)$. (b) $B_{\text{res}}^{\mathrm{BW}}\left(E\right)$ (solid lines), $B_{\text{res}}^{\mathrm{pole}}\left(E\right)$ (dotted lines), and peak and trough in $Q(E,B)$ (dashed and dot-dashed lines, respectively). The axes are scaled such that the bare bound state (black line) coincides for all three resonances. (c) Energy derivatives $d{\mathrm{\Gamma }}_{\text{B}}/dE$ (solid lines) and $d{B}_{\text{res}}/dE$ (dashed lines); thicker lines show the absolute values when the quantities are negative.