Initial condition sensitivity in photoinduced structural phase transitions- An iterative equation for exciton proliferation during lattice relaxation

We theoretically study lattice relaxation dynamics of photogenerated excito ns, which proliferate during this relaxation, and finally result in a macro scopic excited domain in insulating solids. This macroscopic domain formati on phenomenon is usually called "photoinduced structural phase transition ( PSPT)". According to recent experiments, occurrence or nonoccurrence of thi s PSPT is found to depend very sensitively on the way of the photoexcitatio n, even if the total absorbed photonic energy is same. It is nothing else b ut "initial condition sensitivity" peculiar to nonlinear systems. We concen trate our attention on this sensitivity, and investigate criticality of thi s proliferation, using an iterative equation for its time evolution. As one of the models describing such a sensitivity, we consider a 1-dimensional s trongly coupled many exciton-Einstein phonon system interacting with a rese rvoir. In our system, excitons can proliferate through their third order an harmonicity. This anharmonicity originates from the long range Coulomb inte raction among electrons and holes constituting excitons. Within the Markov approximation for the reservoir, the time evolution of the density matrix o f this exciton-phonon system is investigated full-quantummechanically. In o rder to overcome numerical complexities, we solve our iterative equation fo r the proliferation by the following principle. We al-ways focus only on th e most front of the expanding photoinduced domain, and the contributions fr om the other excitons, not in this front, are approximated by a mean field. As the proliferation proceeds, this front moves. Accordingly, we, step by step, iterate this procedure till the excess phonon energy, initially given by the photoexcitation, is exhausted. Thus, we can investigate the whole p rocess of the time evolution of the domain formation, until this domain con tains a large number of excitons. As for the aforementioned initial conditi on sensitivity, we cast the difference of the initial conditions into the d ifference of the initial distance between adjacent photogenerated excitons. Net proliferation is concluded to occur only when this initial distance is moderate, being not too small nor too large. This sensitivity to the initi al distance results from the aforementioned third order anharmonicity.