Three-mode Entanglement And Squeezing Of Coupled Harmonic Oscillators In The Non-Markovian Reservoir

Quantum entangled states play an important role in the quantum teleportation and quantum information science.But due to the unavoidable interaction with the environment, we should take decoherence and dissipation into account when we analyze continuous variable quantum channels.Therefore the research of the dynamics of entanglement in open quantum systems has become a vital topic and has attracted much attention.Within the theory of open quantum systems,the dissipative dynamics are mainly described by master equations of the reduced density matrix.We use the perturbative method to derive master equation of the density matrix of a system composed of three coupled identical harmonic oscillators simultaneously interacting with a common environment. We do not make the rotating-wave and markovian approximations on the interaction Hamiltonian and treat the environment as a non-markovian reservoir to the oscillators. To solve the master equation in a simple way,we make a unitary transformation of the position and momentum operators,and we find that in the new 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 get a set of coupled first order ordinary differential equation of the covariance matrix elements whose coefficients changed with time.We find that working with the differential equations for the covariance matrix elements it is easier to find the density operator of the system than working with the Fokker-Planck equation.Solving for the time evolution of the covariance matrix elements by the numerical methods of Runge-Kutta and setting three-mode squeezed vacuum state as initial state,we can then obtain the entanglement dynamics through the computation of the negativity and get the sum of the variances of the position and momentum operators through local squeezing transformation.The results show that three-mode entanglement not only depends on the initial state but is also closely related to the nature of the environment and the coupling strength between system and the environment.In addition,we show that the three-mode entanglement can be predicted by suitably transformed squeezed fluctuations so that anywhere there is squeezing between the oscillator modes there is an entanglement.