Nonequilibrium dissipation-driven steady many-body entanglement

We study an ensemble of two-level quantum systems (qubits) interacting with a common electromagnetic field in the proximity of a dielectric slab whose temperature is held different from that of some far surrounding walls. We show that the dissipative dynamics of the qubits driven by this stationary and out of thermal equilibrium field allows the production of steady many-body entangled states, different from the case at thermal equilibrium where steady states are always nonentangled. By studying up to ten qubits, we point out the role of symmetry in the entanglement production, which is exalted in the case of permutationally invariant configurations. In the case of three qubits, we find a strong dependence of tripartite entanglement on the spatial disposition of the qubits, and in the case of six qubits we find several highly entangled bipartitions where entanglement can, remarkably, survive for large qubit-qubit distances up to $100\mu \mathrm{m}$.

Figure 1

The $N$ qubits are placed in the same plane $x-y$ at a distance $z$ from a dielectric slab of thickness $\delta$ whose temperature $T_M$ is kept constant and different from that of the surrounding blackbody radiation, $T_W$, which is also constant. At the bottom of the figure we show the case when the qubits occupy the vertices of regular polygons inscribed in a circle of radius $r$, for $N=2,3,6$.

Figure 2

The three qubits lie on a plane parallel to the slab of $\delta =0.01\mu \mathrm{m}$ at a distance $z=8\mu \mathrm{m}$. The qubits frequency is fixed at ${\omega }_0=0.05\times {10}^{14}\mathrm{rad}/\mathrm{s}$, and the two temperatures are fixed at $T_{\mathrm{W}}=5$ K and $T_{\mathrm{M}}=300$ K. (a) Qubits 1 and 3 are kept fixed at a distance $d_{13}=2\mu \mathrm{m}$. Qubit 2 is moved along the line passing from $O$ and $A$. When qubit 2 is in $O$, the three qubits are aligned and $l=1\mu \mathrm{m}$ ($l/d_{13}=0.5$). When qubit 2 is in $A$, the three qubits form an equilateral triangle of side $l=2\mu \mathrm{m}$ ($l/d_{13}=1$). The steady global negativity, ${\mathcal{N}}_{123}$, and the steady negativity between the external qubits 1 and 3 after tracing out qubit 2, ${\mathcal{N}}_{1-3}$, are plotted as a function of $l/d_{13}$. (b) For the case $l=d_{13}$, we report populations and scaled eigenvalues of the collective states. The real part of the eigenvalues gives the position with respect to the various frequencies $n{\tilde{\omega }}_0$, while the imaginary part is connected to the lifetime of the different eigenstates, whose inverse is also depicted in a scaled way.

Figure 3

The six qubits form a regular hexagon parallel to the slab and inscribed in a circle of radius $r$. The values of $\delta ,z,{\omega }_0,T_{\mathrm{W}}$, and $T_{\mathrm{M}}$ are the same as Fig. 2. The maximal negativity among two-qubit negativities (${\mathcal{N}}_{1-3}$) and among each type of bipartite negativities (${\mathcal{N}}_{1/23456}$, ${\mathcal{N}}_{14/2356}$, and ${\mathcal{N}}_{124/356}$) are plotted as a function of the circle radius $r$.

Figure 4

The six qubits almost form a regular hexagon parallel to the slab and inscribed in a circle of radius $r\approx 0.33\mu \mathrm{m}$. The values of $\delta ,z,{\omega }_0,T_{\mathrm{W}}$, and $T_{\mathrm{M}}$ are the same as Fig. 3. The same negativities of Fig. 3 (${\mathcal{N}}_{1-3}$, ${\mathcal{N}}_{1/23456}$, ${\mathcal{N}}_{14/2356}$, and ${\mathcal{N}}_{124/356}$) are plotted as a function of the ratio $x/r$. For $x=0$, a regular hexagon is formed and the corresponding maxima of negativities coincide with the values found in Fig. 3 for $r\approx 0.33\mu \mathrm{m}$

Figure 5

For each $N$, the qubits form a regular polygon inscribed in a circle of radius 0.5 $\mu \mathrm{m}$, parallel to the slab. The values of $\delta ,z,{\omega }_0,T_{\mathrm{W}}$, and $T_{\mathrm{M}}$ are the same as Fig. 2. The maximal negativity is plotted as a function of the number of qubits $N$, which varies between 2 and 10. For each $N$ the maximal negativity is found between all the two-qubit negativities and all the bipartite negativities.