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Multi-mode entanglement of N harmonic oscillators coupled to a non-Markovian reservoir

Multi-mode entanglement is investigated in the system composed of $N$ coupled identical harmonic oscillators interacting with a common environment. We treat the problem very general by working with the Hamiltonian without the rotating-wave approximation and by considering the environment as a non-Markovian reservoir to the oscillators. We invoke an $N$-mode unitary transformation of the position and momentum operators and find that in the transformed basis the system is represented by a set of independent harmonic oscillators with only one of them coupled to the environment. Working in the Wigner representation of the density operator, we find that the covariance matrix has a block diagonal form that it can be expressed in terms of multiples of $3\times 3$ and $4\times 4$ matrices. This simple property allows to treat the problem to some extend analytically. We illustrate the advantage of working in the transformed basis on a simple example of three harmonic oscillators and find that the entanglement can persists for long times due to presence of constants of motion for the covariance matrix elements. We find that, in contrast to what one could expect, a strong damping of the oscillators leads to a better stationary entanglement than in the case of a weak damping.