Dynamics of the Entanglement between Two Oscillators in the Same Environment

We provide a complete characterization of the evolution of entanglement between two resonant oscillators coupled to a common environment. We identify three phases with different qualitative long time behavior: There is a phase where entanglement undergoes a sudden death. Another phase (sudden death and revival) is characterized by an infinite sequence of events of sudden death and revival of entanglement. In the third phase (no sudden death) there is no sudden death of entanglement, which persists for a long time. The phase diagram is described and analytic expressions for the boundary between phases are obtained. These results are applicable to a large variety of non-Markovian environments. The case of nonresonant oscillators is also numerically investigated.

Figure 1

Phase diagram for Ohmic environment ($\Omega =1$, ${\gamma }_0=0.15$, $\Lambda =20$, $m=1$, $C_{12}=0$). The SD, NSD, and SDR phases describe the three different qualitative long time behaviors for the entanglement between two oscillators. For temperatures above the one for which $S_r=r_{\mathrm{crit}}$ the SDR phase is centered about the dashed line $S_r$ and has a width given by the dotted line $r_{\mathrm{crit}}$. Below this temperature the role of $S_r$ and $r_{\mathrm{crit}}$ are interchanged. SDR separates the SD and NSD phases. The low temperature NSD island is non-Markovian and nonperturbative. ${\tilde{E}}_{\mathcal{N}}$ in the NSD phase is the distance to the dashed line for $r>r_{\mathrm{crit}}$, and the distance between the dashed and dotted lines for $r\le r_{\mathrm{crit}}$.

Figure 2

Logarithmic negativity for resonant oscillators in the same environment. (a) For $T=0$ the NSD phase appears both for large and small squeezing. Asymptotic behavior of initially entangled or separable states only depends on $r$. The amplitude of oscillations vanishes when $r\to 0$. (b) The SDR phase appears for intermediate values of squeezing at zero temperature. (c) $T/\Omega =10$, the SD phase appears for small $r$ and NSD phase for large squeezings, oscillations in the steady state are attenuated as the temperature increases.

Figure 3

Entanglement dynamics for nonresonant oscillators initially in a squeezed separable state ($r=3$) and $T/\Omega =10$. (a) Entanglement is created between nonresonant oscillators but it banishes in finite time. (b) $E_{\mathcal{N}}$ for different times as a function of the frequency of the second oscillator, the resonant condition is essential for asymptotic entanglement.