Molecular level crossing and the geometric phase effect from the optical Hanle perspective

Level-crossing spectroscopy involves lifting the degeneracy of an excited state and using the interference of two nearly degenerate levels to measure the excited-state lifetime. Here we use the idea of interference between different pathways to study the momentum-dependent wave-packet lifetime due an excited-state level crossing (conical intersection) in a molecule. Changes in population from the wave-packet propagation are reflected in the detected fluorescence. We use a chirped pulse to control the wave-packet momentum. Increasing the chirp rate increases the transition to the lower state through the conical intersection. It also increases the interference of different pathways in the upper electronic state due to the geometric phase acquired. Therefore, increasing the chirp rate decreases the population of the upper electronic state and its fluorescence yield. This suggests that there is a finite momentum-dependent lifetime of the wave packet through the level crossing as a function of chirp. We dub this lifetime the wave-packet-momentum lifetime.

Figure 1

(a) Level crossing due to the lifting of the degeneracy. (b) Level crossing between electronic states in a molecule (conical intersection). The wave-packet propagation acquires a geometric phase when encircling the conical intersection (red-solid line).

Figure 2

(a) The peak intensity of a chirped Gaussian pulse with width 16 fs. (b) The initial population of the excited states $1-P_{S_0}$ as a function of chirp rate calculated using a 16-fs Gaussian pulse with $\mathrm{\Gamma }=3\mathrm{ps}, {\lambda }_{\alpha }=545\mathrm{nm}, {\lambda }_{S2}=581\mathrm{nm}, C=0.044$, and field strength $E_0=0.044$.

Figure 3

Illustration of the excited-state wave-packet pathways. $S_1$ and $S_2$ are connected by a conical intersection. The wave packet is initially excited in $S_2$. Low-momentum wave-packet components propagate directly to the bottom of $S_2$ (black dashed). Higher-momentum wave-packet components either overcome the barrier and go to the $S_1$ state (yellow-dashed) or encircle the conical intersection returning to the $S_2$ minimum (red-dotted) and create destructive interference, quenching the fluorescence from $S_2$.

Figure 4

The nonadiabatic potential energy surfaces used to calculate the nonadiabatic transition between $S_1$ and $S_2$.

Figure 5

(a) The simulated fluorescence from $S_1$ and $S_2$ using Eqs. (5)) and (6). See the text for the parameters. (b) The initial population ($1-P_{S_0}$), the same as shown in Fig. 2 (blue-dashed), is compared to $P_{S_1}+P_{S_2}$ (red-solid) from (a).