Entanglement of a bipartite multiparticle mesoscopic superposition

Entanglement of an open bipartite system is investigated. In particular, we calculate the entanglement of a subsystem $A$ with $k$ particles in an $N$ atoms of two-level system $(k⩽N∕2)$ initially prepared in a mesoscopic superposition $\mid \psi ⟩=[{\mid {\varphi }_1\left({\theta }_0\right)⟩}^{\otimes N}+{\mid 0⟩}^{\otimes N}]$, with $\mid {\varphi }_1\left({\theta }_0\right)⟩=\cos {\theta }_0\mid 0⟩+\sin {\theta }_0\mid 1⟩$, subject to the time evolution described by a dephasing channel. For an arbitrary number of particles $N$, numerical results are given for the full time evolution up to ten particles and $1⩽k⩽5$. Analytical results are obtained for short times and asymptotic time regimes. Entanglement is maximum when ${\theta }_0=\pi ∕2$ [Greenberger-Horne-Zeilinger (GHZ) state], independently of the number of particles in each partition. As $N$ gets larger, entanglement tends to $1∕2$, and becomes independent of ${\theta }_0$. The “thermodynamic” regime ($N\to \infty$ with $fN$ finite, where $f=k∕N$) is reached already for $N=1000$ particles. In the regime of $k⪡N$ and $N\to \infty$, the entanglement has its maximum value at ${\theta }_0=\pi ∕2$ and for ${\theta }_0=0,\pi$ its minimum value. The time evolution reduces the maximum value of entanglement, showing that the subsystem $A$ loses coherence to the environment. Finally, we point out the differences between the entanglement rate obtained here from that one in Phys. Rev. Lett. 89, 210402 (2002), where only one $N$-particle subsystem is used.

Figure 1

Negativity for $\tau =0$ (units of ${\gamma }^{-1}$) as a function of ${\theta }_0$ for $N=10$ and $k=1,5$ and $N=100$ and $k=1$, 10, and 50 for $\tau =0$.

Figure 2

Negativity is plotted as a function of ${\theta }_0$ for $N=10$, $k=1,5$ when $\tau =0$ and $\tau =0.05$ (in units of ${\gamma }^{-1}$).

Figure 3

Negativity is plotted as a function of ${\theta }_0$ for different numbers of particles $N$. This curve illustrates the approach to the “thermodynamic limit” as $N$ increases.

Figure 4

Negativity is plotted as a function of ${\theta }_0$ in two different regimes, namely small $k$ ($k⪡N$ and $N\to \infty$) and the “thermodynamic limit” (TL). We see that for $N=1000$, negativity is already quite close to that which corresponds to the “thermodynamic limit.” The curve shows negativity as a function of ${\theta }_0$ for $\tau =0.0005$ (units of ${\gamma }^{-1}$), $N=1000$, and $k=1$, 2, and 3 “TL.”

Figure 5

The full curve represents the exact numerical solution for the time evolution of the negativity for all times. The dashed curve represents the analytical result of negativity for short-time regimes $(\gamma \tau ⪡1)$. The main figure is a linear plot while the zoom in the interesting part is a log-linear plot. In both figures, we consider $k=1$, ${\theta }_0=\pi ∕3$, and $N=10$.