Dynamics of interacting Dicke model in a coupled-cavity array

We consider the dynamics of an array of mutually interacting cavities, each containing an ensemble of $N$ two-level atoms. By exploring the possibilities offered by ensembles of various dimensions and a range of atom-light and photon-hopping values, we investigate the generation of multisite entanglement, as well as the performance of excitation transfer across the array, resulting from the competition between on-site nonlinearities of the matter-light interaction and intersite photon hopping. In particular, for a three-cavity interacting system it is observed that the initial excitation in the first cavity completely transfers to the ensemble in the third cavity through the hopping of photons between the adjacent cavities. Probabilities of the transfer of excitation of the cavity modes and ensembles exhibit characteristics of fast and slow oscillations governed by coupling and hopping parameters, respectively. In the large-hopping case, by seeding an initial excitation in the cavity at the center of the array, a tripartite W state, as well as a bipartite maximally entangled state, is obtained, depending on the interaction time. Population of the ensemble in a cavity has a positive impact on the rate of excitation transfer between the ensembles and their local cavity modes. In particular, for ensembles of five to seven atoms, tripartite W states can be produced even when the hopping rate is comparable to the cavity-atom coupling rate. A similar behavior of the transfer of excitation is observed for a four-coupled-cavity system with two initial excitations.

Figure 1

Sketch of the physical configuration studied in the paper. We consider an array of cavities coupled via hopping fields and containing an ensemble of two-level atoms each. The atoms are collectively coupled with the cavity field. We show a two-site subsystem of a longer array.

Figure 2

Probabilities of excitation transfer vs time for cavity modes (a)–(c) and ensembles (d), where the solid, thick dashed, and thin dotted lines represent the cases of excitations in the first, second, and third cavity, respectively. Here a single excitation is taken only in the ensemble of the first cavity, i.e., $b_1\left(0\right)=1,$ while all the three-cavity modes are initially in their vacuum states. The dynamics of the system corresponds to the resonance case with $J/\Omega =1$ and $N=10$ qubits per ensemble.

Figure 3

Probabilities of excitation transfer vs time for ensembles in three (a) and four (b) coupled cavities. The dynamics of the system correspond to the resonance case with $J=\Omega =3$ and the other parameters are (a) $N=5$, $b_1\left(0\right)=b_3\left(0\right)=0,b_2\left(0\right)=1$ and (b) $N=10$, $b_1\left(0\right)=b_4\left(0\right)=0,b_2\left(0\right)=b_3\left(0\right)=1$.

Figure 4

Probabilities of excitation transfer vs time for cavity modes (a) and ensembles (b) and (c), where the solid, thick dashed, and thin dotted lines represent excitations in the first, second, and third cavity, respectively. The dynamics of the system correspond to the large-hopping case with parameters $J=40,\Omega =1,N=20$, and $b_1\left(0\right)=1$.

Figure 5

Probabilities of excitation transfer vs time for cavity modes (a) and ensembles (b) and (c). The dynamics of the system correspond to the large-hopping case with $J=40,\Omega =1,N=20$, and other parameters are (a) and (b): three cavities with $b_1\left(0\right)=b_3\left(0\right)=0,b_2\left(0\right)=1$ and (c) four cavities, $b_1\left(0\right)=b_4\left(0\right)=0,b_2\left(0\right)=b_3\left(0\right)=1$.

Figure 6

Probabilities of excitation transfer vs time for cavity modes (a)–(c) and ensembles (d), where the solid, thick dashed, and thin dotted lines represent excitations in the first, second, and third cavity, respectively. The dynamics of the system corresponds to the strong-coupling case with parameters $\Omega =10$, $J=2$, $N=5$, and $b_1\left(0\right)=1$.