Inverted harmonic oscillator dynamics of the nonequilibrium phase transition in the Dicke model

We show how the dynamics of the Dicke model after a quench from the ground-state configuration of the normal phase into the superradiant phase can be described for a limited time by a simple inverted harmonic oscillator model and that this limited time approaches infinity in the thermodynamic limit. Although we specifically discuss the Dicke model, the presented mechanism can be also used to describe dynamical quantum phase transitions in other systems and opens a new avenue in simulations of physical phenomena associated with an inverted harmonic oscillator.