Multipartite stationary entanglement generation in the presence of dipole-dipole interaction in an optical cavity

Engineering a stationary entanglement between atoms or ions placed at small distances is a challenging problem in quantum information science. In this paper, the stationary and dynamics of entanglement in a system of dipole-coupled qubits interacting with a single-mode optical cavity in the strong coupling regime are theoretically investigated. We find that the entanglement in the steady state can be induced and tuned in nonresonance cases by considering the dipole-dipole interaction (DDI) between the qubits. We also point out that the novel measure used in this study can quantify the net multiqubit entanglement in the system. By increasing the DDI intensity, the behavior of the system depends on the number of qubits. Interestingly, with only two qubits, the amounts of steady-state entanglement enhances as the DDI intensity increases. For a system with more than two qubits, we find that DDI in the weak range of intensity plays a constructive role in the entanglement between qubits. However, it could destroy the stationary entanglement as the interaction between qubits intensifies. The entanglement of a system consisting of more than two qubits tends to disappear for an ensemble of qubits with smaller interatomic distances that lead to stronger dipolar interaction between nonadjacent qubits. Finally, we study the atomic population transfer as well as the transmitted spectrum.

Figure 1

Schematic diagram of $N$ qubits coupled to each other by dipole-dipole interaction. The cavity is pumped by a coherent laser field with strength $\eta$. The decay rates of atoms and cavity field are $\gamma$ and $\kappa$, respectively. The interatomic distance is $d$ and $\phi$ denotes the angle between the dipole moment ($\mu$) and atomic position vector ($R_i$). Each atom can be considered as a two-level system with the ground ($|g\rangle$) and excited ($|e\rangle$) states.

Figure 2

The normalized stationary generated entanglement as a function of $J$ and ${\mathrm{\Delta }}_c$ for (a) $N=2$, (b) $N=3$, (c) $N=4$, and (d) $N=5$. The other parameters are $\eta =0.12g, \kappa =0.12g$, and $\gamma =0.076g$.

Figure 3

The distance between two peaks as a function of $g$ for several values of the number of the atoms in the cavity. Other parameters are the same as Fig. 2.

Figure 4

Dynamics of entanglement as a function of the scaled $\tau =gt$. Top plots (bottom plots) correspond to initially entangled (ground) state of the qubits. In these plots $J=0$ (right plots) and $J=2g$ (left plots). We have chosen the values of ${\mathrm{\Delta }}_c$ for which the stationary state could have a peak. Other parameters are the same as in Fig. 2.

Figure 5

Comparing the entanglement distance measure (dashed red line) and the von Neumann entropy (solid line) in the stationary state as functions of ${\mathrm{\Delta }}_c$. The first, second, and third columns correspond to $J=0, J=2g$, and $J=10g$, respectively. Furthermore, the first, second, and third rows correspond to $N=2, N=3$, and $N=4$, respectively. The other parameters are ${\mathrm{\Delta }}_a=0, \eta =0.12g, \kappa =0.12g$, and $\gamma =0.076g$.

Figure 6

$\langle S_z\rangle$ at stationary state as a function of ${\mathrm{\Delta }}_c$ for several values of $J$ and for (a) $N=2$, (b) $N=3$, (c) $N=4$, and (d) $N=5$. The other parameters are similar to those in Fig. 2.

Figure 7

The normal-mode spectrum for different numbers of qubit (a) $N=2$, (b) $N=3$, (c) $N=4$, and (d) $N=5$. The other parameters are the same as those in Fig. 2.