Mesoscopic superposition and sub-Planck-scale structure in molecular wave packets

We demonstrate the possibility of realizing sub-Planck-scale structures in the mesoscopic superposition of molecular wave packets involving vibrational levels. The time evolution of the wave packet, taken here as the SU(2) coherent state of the Morse potential describing hydrogen iodide molecules, produces macroscopic-quantum-superposition-like states, responsible for the above phenomenon. We investigate the phase-space dynamics of the coherent state through the Wigner function approach and identify the interference phenomena behind the sub-Planck-scale structures. The optimal parameter ranges are specified for observing these features.

Figure 1

(Color online) $\mid d_m{\mid }^2$ plotted as a function of $m$ for the Morse potential of the HI molecule for different values of $\alpha$.

Figure 2

(Color online) Wave packet distribution in coordinate space for HI molecules, where $\alpha =1.4$, $\beta =2.07932$. Plotted here is $\mid \chi (\xi ,t){\mid }^2$ as a function of $x$ (where $\xi =2\lambda \exp [-\beta x]$) for (a) $t=0$, (b) $t=T_{\mathrm{rev}}∕8$, (c) $t=T_{\mathrm{rev}}∕4$, and (d) $t=T_{\mathrm{rev}}∕2$.

Figure 3

(Color online) The Wigner function $W(x,p,t)$ and its constituent parts at $t=T_{\mathrm{rev}}∕8$ as a function of $x$ and $p$ for $\alpha =1.4$ (top row) and $\alpha =2.5$ (bottom row). Shown here are the contour plots of (a)$W^{\left(\mathrm{even}\right)}$, (b)$W^{\left(\mathrm{odd}\right)}$, (c)$W^{\left(\mathrm{int}\right)}$, and (d)$W(x,p,t)$.