RESULTS AND DISCUSSION Quantum Physics authors titles recent submissions

5. RESULTS AND DISCUSSION

For an isolated macroscopic quantum system, the tunneling process, according to 4, is manifested by symmetrical oscillations between localized states of the system FIG. 1-a. Since the system is isolated, such symmetrical oscillations are considered as the quantum signature of the system. A quantum system, especially a macroscopic one, is not actually isolated. An environment at thermal equilibrium destroys the quantum coherence between the preferred states of the system. This decoherence process is manifested in the reduction of the amplitude of oscillations, resulting to an equilibrium steady state at long times [37]. In our approach, if we eliminate one of the environments, the expected exponential decay of oscillations is observed FIG. 1-b. 100 200 300 400 500 0.0 0.2 0.4 0.6 0.8 1.0 Time Probability a 2 × 10 6 4 × 10 6 6 × 10 6 8 × 10 6 1 × 10 7 0.0 0.2 0.4 0.6 0.8 1.0 Time Probability b FIG. 1: The dynamics of right-handed probability for the macroscopic quantum system with h = 10 −1 : a for the isolated system b for the system in interaction with an equilibrium environment at T = 25K. Now we turn to the case where the system interacts with two environments at the same temperatures. In order to compare the one-environment dynamics with the two-environment one, we divide the single equilibrium environment into two identical equilibrium environments. In doing so, we should note that, in two-environment dynamics, the effect of one environment is multiplied by the effect of the other one see 26. Hence, to retain the whole environment intact, the multiplication of coupling strengths and the addition of cut-off strengths of two environments should be conserved. At the short-time limit of the dynamics, the interference between two environments is manifested as successive interference patterns FIG. 2-a. Here, the symmetry of the interference patterns can be considered as the signature of the similarity of two environments. At the long-time limit, as expected, the steady state is finally reached. This is in accordance with the work of Li and co-workers, in which a three-level microscopic system in interaction with two heat baths with the same temperature returns to the equilibrium case [25]. Nonetheless, if we compare the one-environment decoherence time FIG. 1b with the two-environment one FIG. 2a, we realize that the interference between two environments through the system intensifis the decoherence process. 200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 Time Probability a 200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 Time Probability b FIG. 2: The dynamics of right-handed probability for the macroscopic quantum system with h = 10 −1 in interaction with two identical environments at T = 100K a with system’s back-action b without system’s back-action. The system being macroscopic exerts a force on environmental particles, which is realized by the second term of 6. It would be interesting to examine the effect of this back-action on the system’s dynamics. The comparison between the dynamics in the presence of this force FIG. 2a and in the absence of it FIG. 2b clearly shows that such a back-action weaken the interference between two environments. The non-equilibrium environment is manifested here by considering two environments at different temperatures. Because of the non-trivial dynamics of the non-equilibrium environment, two consisting environments are not equivalent to a single equilibrium environment. At the short-time limit, the interference between two environments is observed in the dynamics FIG. 3-a, but, unlike the same-temperature dynamics, the interference patterns are asymmetric. This asymmetry can be considered as the dynamical signature of the non-equilibrium environment. At the long-time limit, however, the non-equilibrium feature is suppressed and the system returns to its steady state. Also, the system’s back-action here wash off the interference between two environments FIG. 3-b. 100 200 300 400 0.0 0.2 0.4 0.6 0.8 1.0 Time Probability a 100 200 300 400 0.0 0.2 0.4 0.6 0.8 1.0 Time Probability b FIG. 3: The dynamics of right-handed probability for the macroscopic quantum system with h = 10 −1 in interaction with a non-equilibrium environment, consisting two environments at T A = 1K and T B = 100K a with system’s back-action b without system’s back-action. At a fixed temperature for the first environment, the right-handed probability in terms of the temperature difference between two environments ∆T = T 1 − T 2 is plotted in FIG. 4. It shows that essentially the coherence of the system has an oscillatory dependence with ∆T . At first, for a wide range of temperatures, the coherence increases with ∆T . This is in agreement with the the work of Li and co-workers, in which the coherence increases with the temperature difference of two heat baths coupled to a three-state system [10]. According to our analysis, however, when this difference reaches to a critical value, the coherence is non-monotonically decreased. This is because after the critical point the decoherence feature of the hot environment dominates the process and the coherence is destroyed. 20 40 60 80 100 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ΔT Probability FIG. 4: The right-handed probability in terms of the temperature difference between two environments ∆T = T A − T B at T A = 100K.

6. CONCLUSION