We describe the Markovian dynamics of the Gaussian quantum discord of a system composed of two uncoupled bosonic modes in two cases: when they are in contact with a common squeezed thermal bath, and when each of them is interacting with its own squeezed thermal bath. This study is done in the framework of the theory of open quantum systems, based on completely positive quantum dynamical semigroups. We take an initial squeezed thermal state and we show that the behaviour of the Gaussian quantum discord depends on the bath parameters (temperature, dissipation coefficient, squeezing parameter and squeezing phase), and on the initial state of the system (squeezing parameter and average photon numbers). We show that Gaussian quantum discord is decaying in time, tending asymptotically to zero, due to the effect of the environment. We also compare the Gaussian quantum discord with the Gaussian geometric discord.
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