Time-asymmetric quantum-state-exchange mechanism

We show here that due to nonadiabatic couplings in decaying systems, applying the same time-dependent protocol in the forward and reverse direction to the same mixed initial state leads to different final pure states. In particular, in laser-driven molecular systems, applying a specifically chosen positively chirped laser pulse or an equivalent negatively chirped laser pulse yields entirely different final vibrational states. This phenomenon occurs when the laser frequency and intensity are slowly varied around an exceptional point (EP) in the laser intensity and frequency parameter space where the non-Hermitian spectrum of the problem is degenerate. The protocol implies that a positively chirped laser pulse traces a loop in time in the laser parameters’ space whereas a negatively chirped pulse follows the same loop in the opposite direction. According to this protocol one can choose the final pure state from any initial state. The obtained results imply the intrinsic nonadiabaticity of quantum transport around an EP, and offer a way to observe the EP experimentally in time-dependent quantum systems.

Figure 1

Loops in the parameter space of the laser intensity and wavelength. When the laser intensity is turned on, the ninth and the tenth vibrational states of the molecular ion become resonances. The EP is obtained when the two resonances become degenerate non-Hermitian eigenstates. Note that the red dashed line representing a chirped Gaussian laser pulse closes a loop on the zero intensity axis.

Figure 2

The asymmetric state-exchange mechanism in H${}_2{}^{+}$. $W_{\nu }\left(t\right)$ is the projection of the propagated ${\text{H}}_2{}^{+}$ wave packet (WP) during a chirped Gaussian laser pulse on the field-free vibrational states of the molecular ions, $\nu =9$ and $\nu =10$. The thin black lines describe the solution which starts at the $\nu =9$ vibrational state whereas the thick red lines describe the solution which starts at the $\nu =10$ vibrational state. The chirp changes the laser wavelength and intensity in time according to the dashed red loop encircling the EP shown in Fig. 1. (a) A positive chirp corresponds to a clockwise trajectory on the loop. (b) A negative chirp corresponds to a counterclockwise trajectory on the loop. Similar results are obtained when we follow a trajectory along the blue solid line in Fig. 1.

Figure 3

The population $P_j$ of the ninth (black) and tenth (red) vibrational states of ${\text{H}}_2{}^{+}$ at the end of the laser pulse on a logarithmic scale as a function of the pulse duration $T_{\text{loop}}$. The solid lines correspond to the solid-line loop in Fig. 1, which encircles the EP, while the dashed lines correspond to the dotted-line loop in Fig. 1, which excludes the EP. The initial state for all calculations is an equal weight superposition of the two states. (a) displays the results of the clockwise path on the loops while (b) portrays the results of the counterclockwise path on the loops.