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

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