Shannon entropy and avoided crossings in optical microcavities

We investigated the relation between Shannon entropy and avoided crossings under strong coupling in optical microcavities. Before our work, the relation between Shannon entropy and avoided crossing was investigated in atomic... [ view full abstract ]

We investigated the relation between Shannon entropy and avoided crossings under strong coupling in optical microcavities. Before our work, the relation between Shannon entropy and avoided crossing was investigated in atomic physics, and the result was opposite to ours, i.e., Shannon entropy for an electron decreases due to electron ionization as we move close to the center of the avoided crossing. On the contrary, Shannon entropy of the probability density for optical microcavities (quantum billiards) increases due to the coherent superposition of wave functions as the center of the avoided crossing is approached, but both cases show exchanges of Shannon entropy as well as mode exchanges. Shannon entropy of the probability density for a closed elliptic billiard changes little with the eccentricity, while Shannon entropy of the probability density for an open elliptic billiard is maximized at the center of the avoided crossing. This maximization and exchange of Shannon entropy in an open elliptic billiard comes from the collective Lamb shift, which is an energy-level shift due to the interaction of energy levels with each other via the bath, and it can also induce an avoided crossing and coherent superposition of wave functions. In a closed quadrupole billiard, Shannon entropy is also maximized as the center of the avoided crossing is approached with both exchange of Shannon entropies as well as mode patterns. This maximization and exchange of Shannon entropy in a closed quadrupole billiard comes from the nonlinear dynamical effects in a chaotic system. Irrespective of the origin of the avoided crossings, the open elliptic cavity and the closed quadrupole cavity show similar behaviors to Shannon entropy. That is, the collective Lamb shift of open quantum systems and the symmetry breaking in the closed chaotic quantum systems have equivalent effects on the Shannon entropy.