Abstract
We study the entanglement dynamics of a system consisting of a large number of coupled harmonic oscillators in various configurations and for different types of nearest-neighbour interactions. For a one-dimensional chain, we provide compact analytical solutions and approximations to the dynamical evolution of the entanglement between spatially separated oscillators. Key properties such as the speed of entanglement propagation, the maximum amount of transferred entanglement and the efficiency for the entanglement transfer are computed. For harmonic oscillators coupled by springs, corresponding to a phonon model, we observe a non-monotonic transfer efficiency in the initially prepared amount of entanglement, i.e. an intermediate amount of initial entanglement is transferred with the highest efficiency. In contrast, within the framework of the rotating-wave approximation (as appropriate, e.g. in quantum optical settings) one finds a monotonic behaviour. We also study geometrical configurations that are analogous to quantum optical devices (such as beamsplitters and interferometers) and observe characteristic differences when initially thermal or squeezed states are entering these devices. We show that these devices may be switched on and off by changing the properties of an individual oscillator. They may therefore be used as building blocks of large fixed and pre-fabricated but programmable structures in which quantum information is manipulated through propagation. We discuss briefly possible experimental realizations of systems of interacting harmonic oscillators in which these effects may be confirmed experimentally.
Highlights
For harmonic oscillators coupled by springs, we observe a non-monotonic transfer efficiency in the initially prepared amount of entanglement, i.e. an intermediate amount of entanglement is transferred with the highest efficiency
We give an example for the time evolution of the logarithmic negativity between the 0th and the 30th oscillator for both interactions in figure 4. For both interactions we obtain qualitatively the same behaviour but we observe that under the rotating-wave approximation (RWA) interaction the entanglement propagates somewhat faster but as expected this difference decreases with decreasing coupling constant c. Another difference is the fact that the entanglement under the RWA interaction does not exhibit the small-amplitude oscillations that the interaction due to harmonic oscillators coupled by springs exhibits due to the existence of counter-rotating terms of the form akak+1
We have investigated the entanglement dynamics of systems of harmonic oscillators both analytically and numerically
Summary
We present the systems under consideration, namely coupled harmonic oscillators, together with the Hamiltonians that describe the various models for their interaction. We will restrict our attention to Hamiltonians that are quadratic in position and momentum operators. This will be crucial for the following analysis as it permits us to draw on the results and techniques from the theory of Gaussian continuous variable entanglement. The most important results from this theory will be reviewed here briefly
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