Two-level system with nonlinear coupling

We present a numerical study of a two-level system coupled quadratically to a harmonic oscillator. The possibility of an instability as a function of the coupling strength is studied. The model might be relevant for the behavior of quadratically coupled polarons in solids. In the adiabatic limit the thermodynamical properties of the system can be calculated exactly over the whole coupling range. In addition, we discuss computations based on a discrete path-integral representation for the partition function. In the adiabatic limit there is a transition as a function of the coupling strength from the weak-coupling regime to a state where the oscillator mode is soft.