The operation threshold of a double barrier phonon laser

We make an adaptation of laser modelling equations to describe the behavior of a phonon laser (saser). Our saser consists of an AlGaAs/GaAs double barrier het-erostructure designed to generate an intense beam of transversal acoustic (T A) phonons. To study our system, we begin with a Hamiltonian that describes the decay of primary longitudinal optical phonons (LO 1) into secondary (LO 2) and T A (LO 1 → LO 2 + T A) and its inverse process (recombination). Using this Hamilto-nian, a set of coupled equations of motion for the phonons is obtained. We also consider the interaction between the phonons and its reservoirs. These interactions are introduced in the equations of motion leading to a set of coupled Langevin equations. In order to obtain an expression to describe our saser we apply, in the Langevin equations, an adiabatic elimination of some variables of the subsystem. Following the method above we obtain the value of the injection threshold for the operation of our phonon laser. At this threshold occurs a phase transition from a disordered to a coherent state. It is shown that it is not necessary a big " optical " pumping to get a sasing region.