Entanglement of quantum oscillators coupled to different heat baths

Abstract

We study the non-equilibrium dynamics of two coupled oscillators interacting with its own heat baths at different temperature T 1 and T 2 with bilinear couplings between them. We particularly focus on the entanglement or inseparability property of their quantum states. Tracing out the bath's degrees of freedom with the method of influence functional, we obtain the stochastic effective action, from which the Heisenberg-Langevin equations for each of the oscillators are derived involving both the damping terms with different Ohmic damping parameters γ 1 and γ 2 and the noise terms manifested from thermal fluctuations of the baths. The general solutions show that the system in the late time reaches the nonequilibrium steady state for damped oscillators with weak oscillatorbath couplings. The entanglement criterions constructed from the covariance matrix elements are computed for the late-time nonequilibrium steady states. The critical temperature T 1c and T 2c , higher than which the entanglement disappears, can also be determined. It is found that when two damping parameters are largely different, say γ 1 ≪ γ 2 , the critical temperature T 1c with respect to the oscillator frequency Ω can be very large, T 1c ≫ Ω, while T 2c ∝ Ω. This is to compare with the case γ 1 = γ 2 where both T 1c , T 2c ∝ Ω. We provide the detailed analysis both numerically and analytically of how the different damping parameters can push the critical temperature of one of the baths to a higher one, and discuss the implications.