Reduced dynamics of two oscillators collectively coupled to a thermal bath

We study the reduced dynamics of a pair of nondegenerate oscillators coupled collectively to a thermal bath. The model is related to the trilinear boson model where the idler mode is promoted to a field. Due to nonlinear coupling, the Markovian master equation for the pair of oscillators admits non-Gaussian equilibrium states, where the modes distribute according to the Bose-Einstein statistics. These states are metastable before the nonlinear coupling is taken over by linear coupling between the individual oscillators and the field. The Gibbs state for the individual modes lies in the subspace with infinite occupation quantum number. We present the time evolution of a few states to illustrate the behaviors of the system.

Figure 1

Energy level diagram and the rate of transitions between different levels in the $f^{\left(N\right)}$ subspace (between diagonal elements) [see Eq. (21) for the definition of $f_r^{\left(N\right)}$]. The transition rates between two levels are proportional to the numerical constants on the left of the levels. Transitions between different irreducible $f^{\left(N\right)}$ subspaces are forbidden by the SU(2) symmetry of the reduced dynamics.

Figure 2

Initial and equilibrium configurations of the individual modes of example (1) for several temperatures, with parameters $a=2$ and $N_1/N_2=2$ in Eq. (46).

Figure 3

$P^{({\nu }_1,{\nu }_2)}{\psi }_2{P}^{({\nu }_1-2,{\nu }_2+2)}$ transition. It is one of the diagrams giving the contribution to the collision operator ${\psi }_2$ when virtual mode ${\omega }_v$ is considered. The operators on the right vertex cause the virtual transition.