Dissipative stabilization of quantum-feedback-based multipartite entanglement with Rydberg atoms

A quantum-feedback-based scheme is proposed for generating multipartite entanglements of Rydberg atoms in a dissipative optical cavity. The Rydberg blockade mechanism efficiently prevents double excitations of the system, which is further exploited to speed up the stabilization of an entangled state with a single Rydberg state excitation. The corresponding feedback operations are greatly simplified, since only one regular atom needs to be controlled during the whole process, irrespective of the number of particles. The form of the entangled state is also adjustable via regulating the Rabi frequencies of driving fields. Moreover, a relatively long lifetime of the high-lying Rydberg level guarantees a high fidelity in a realistic situation.

Figure 1

Schematic view of the feedback setup and atomic-level configuration. The system consists of many cascade three-level atoms simultaneously interacting with nonresonant classical fields and a quantized cavity field. Each atom is constituted by a Rydberg state $|r\rangle$, an optical state $|p\rangle$, and a ground state $|g\rangle$. The intermediate state $|p\rangle$ can be eliminated adiabatically under the large-detuning condition $|{\mathrm{\Delta }}_{a\left(b\right)}|\gg \{g,{\mathrm{\Omega }}_{c\left(\mathrm{B},\mathrm{R}\right)}\}$, which is then reduced to an effective two-level atom resonantly coupled to a damped optical cavity with coupling strength $-g{\mathrm{\Omega }}_c/{\mathrm{\Delta }}_b$ and driven by a laser field with Rabi frequency ${\mathrm{\Omega }}_{\mathrm{R}}{\mathrm{\Omega }}_{\mathrm{B}}/{\mathrm{\Delta }}_a$. The single-atom feedback control $U_{\mathrm{fb}}$ is triggered right after the leakage photon is measured by the detector $D$.

Figure 2

Analysis of bipartite entanglement under quantum feedback control. (a) The target-state fidelity obtained from the effective master equation of Eq. (20) (undersurface) is in excellent agreement with the result derived from the original master equation of Eq. (25) (upper surface), under the given parameters $\kappa =25\mathrm{\Gamma }, g_{\mathrm{eff}}=2.5\mathrm{\Gamma }, U=500\mathrm{\Gamma }$, and $t=100/\mathrm{\Gamma }$. (b) The fidelity calculated from the effective master equation is not consistent with the original one (inset) out of the weak driving regime ($N$ denotes the considered dimension of cavity mode), where we have chosen $\omega =-0.5\pi$, and other parameters are the same as in (a). (c) The Rydberg blockade provides a way to speed up convergence of the target-state population originating from a two-atom ground state $|gg\rangle$ (solid line), compared with the population in the absence of Rydberg blockade (dash-dotted line). This speedup effect will be more pronounced for a stronger driving field $\mathrm{\Omega }=10\mathrm{\Gamma }$ as shown in the inset. (d) Time evolution of the fidelity for the target-state $(cos\theta |gr\rangle +sin\theta |rg\rangle )$ with unity (solid line) and 0.5 (dash-dotted line) detection efficiency, respectively. The effective Rabi frequency has been set as $\mathrm{\Omega }=0.1g_{\mathrm{eff}}$.

Figure 3

The populations of quantum states are simulated with and without quantum feedback control during the preparation of (a) the three-qubit DFS state and (b) the $W$ state, where $\omega =-0.5\pi , \kappa =100\mathrm{\Gamma }, g_{\mathrm{eff}}=5\mathrm{\Gamma }, \mathrm{\Omega }=5\mathrm{\Gamma }$, and $U=2500\mathrm{\Gamma }$.

Figure 4

The fidelities of four-qubit DFS state and $W$ state are simulated numerically with dimension of cavity mode $N=2$ (solid line) and $N=5$ (dotted line), where $\omega =-0.5\pi , \mathrm{\Omega }=\mathrm{\Gamma }, \kappa =25\mathrm{\Gamma }, g_{\mathrm{eff}}=2.5\mathrm{\Gamma }$, and $U=500\mathrm{\Gamma }$.

Figure 5

(a) The effects of Stark shifts and Rydberg blockade on preparation of the antisymmetric Bell state $\left(\right|gr\rangle -|rg\rangle )/\sqrt{2}$, where $\omega =-0.5\pi , \mathrm{\Omega }=g_{\mathrm{eff}}, \kappa =10g_{\mathrm{eff}}$, and $U=200g_{\mathrm{eff}}$. (b) The influences of Stark shifts on preparation of the tripartite DFS state (solid line) and tripartite $W$ state (dash-dotted line), with the same parameters as in (a). (c) Populations of quantum states versus time during generation of the Bell state (tripartite DFS state) with spontaneous emission rates ${\gamma }_r=0.008g_{\mathrm{eff}}$ and ${\gamma }_p=8g_{\mathrm{eff}}$ (${\mathrm{\Delta }}_{a\left(b\right)}=80g$). (d) Fidelities of the bipartite Bell state and tripartite DFS state versus time under experimental parameters.