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Coherence behavior of strongly coupled bosonic modes

We study the effect of the intermode coupling in the generation of coherence when two bosonic modes are bilinearly coupled. We consider the case for which the two modes are weakly coupled and the rotating-wave approximation (RWA) applies and the case for which they are strongly coupled and the (RWA) does not apply. Then, in the regime of validity of (RWA), there is no coherence generation solely due to squeezing effects, which means there is an exchange of excitation between the modes and negligible squeezing. On the other hand, if the two modes are strongly coupled coherence is generated by the squeezing interaction. For the system of two bosonic modes weakly coupled with a Markovian bath at temperature $T$ the coherence decreases with $T$. In general, when both kinds of couplings (exchange of excitation and squeezing) are turned on the exchange of excitation contributes to generating more coherence in comparison with a purely squeezing coupling. Thus, the coherence decreases more slowly with the temperature when both couplings are present. Finally, we explore the case when only one of the two coupled modes interacts with a Markovian bath while the other one remains free from the environment. We observe that the intermode coupling induces oscillations in the coherence and fidelity dynamics similar to the behavior for the coupling with a non-Markovian environment.