Squeezing phenomena in interacting electron-phonon systems

The molecular crystal model of electrons coupled to Einstein phonons is studied as a function of the two parameters: $\lambda$ and the ratio of the electron-phonon coupling energy denoted by $\alpha$. Both the one electron and the many-electron models are studied starting (for the former) from the adiabatic limit and (for the latter) from the ant-adiabatic one. In the "multiphonon" regime $\alpha>1$, the sharp crossover between quasi-free electrons ($\lambda<<1$) and small polarons ($\lambda>>1$) is investigated emphasizing the anomalous lattice fluctuations which occur in the intermediate regime ($\lambda \simeq 1$). These fluctuations are due to the band motion of electrons strongly coupled to the lattice and are shown in turn to weaken the electron mass renormalization inherent to self-trapping. In a relevant part of the intermediate region the electron mass slowly increases with $\lambda$, due to the competition between the phonon dressing effect and the reduction of lattice momentum fluctuations. This reduction is reminiscent of squeezing phenomena occurring in quantum optics. In a gaussian approximation squeezed states imply a dynamical phonon softening.