Non-Markovianity-Assisted Steady State Entanglement

We analyze the steady state entanglement generated in a coherently coupled dimer system subject to dephasing noise as a function of the degree of Markovianity of the evolution. By keeping fixed the effective noise strength while varying the memory time of the environment, we demonstrate that non-Markovianity is an essential, quantifiable resource that may support the formation of steady state entanglement whereas purely Markovian dynamics governed by Lindblad master equations lead to separable steady states. This result illustrates possible mechanisms leading to long-lived entanglement in purely decohering, possibly local, environments. We present a feasible experimental demonstration of this noise assisted phenomenon using a system of trapped ions.

Figure 1

Beating pattern in the dimer population inversion for different values of $f$ (See main text for details). A Markovian environment ($f\gg 1$) will simply wash out the discreteness of the vibrational modes while the presence of beating in the dimer signal is a signature of the persistence of a coherent interaction with a localized vibrational (damped) mode.

Figure 2

Using the non-Markovianity measure introduced in 16 we can rigorously quantify the deviation from strict Markovianity of the dynamical map describing the time evolution of the dimer system. A finite value of the measure ${\mathcal{D}}_{NM}$ indicates that the dynamics cannot be accounted for in terms of a master equation of the Lindblad form. Here the values of $f$ range from $f_{\min }=0.0035$ to $f_{\max }=3.6554$ when the coordinate $x$ varies between 0 and 140. The value of ${\mathcal{D}}_{NM}$ decreases monotonically with $f$ for the selected parameter regime. For $f$ sufficiently large, the dynamical map is divisible, ${\mathcal{D}}_{NM}$ is zero and the dimer’s evolution is fully Markovian.

Figure 3

Time evolution of the entanglement content of the dimer system when the index $f$ varies from the range $f\gg 1$ (Markovian evolution) to $f\ll 1$ (non-Markovian effects). When the evolution is strictly Markovian, the steady state is separable while in the presence of non-Markovianity, the dimer system contains quantum correlations that steadily become close to 1 $e$ bit for $f$ sufficiently small.