Dissipative preparation of steady Greenberger-Horne-Zeilinger states for Rydberg atoms with quantum Zeno dynamics

Inspired by a recent work [F. Reiter, D. Reeb, and A. S. Sørensen, Phys. Rev. Lett. 117, 040501 (2016)], we present a simplified proposal for dissipatively preparing a Greenberger-Horne-Zeilinger (GHZ) state of three Rydberg atoms in a cavity. The $Z$ pumping is implemented under the action of the spontaneous emission of $\mathrm{\Lambda }$-type atoms and the quantum Zeno dynamics induced by strong continuous coupling. In the meantime, a dissipative Rydberg pumping breaks up the stability of the state $|{\mathrm{GHZ}}_{+}\rangle$ in the process of $Z$ pumping, making $|{\mathrm{GHZ}}_{-}\rangle$ the unique steady state of the system. Compared with the former scheme, the number of driving fields acting on atoms is greatly reduced and only a single-mode cavity is required. The numerical simulation of the full master equation reveals that a high fidelity $\sim 98\%$ can be obtained with the currently achievable parameters in the Rydberg-atom-cavity system.

Figure 1

Protocol for preparation of the GHZ state: a simplified $Z$ pumping first transforms one and two atoms in state $|1\rangle$ into $|000\rangle =\left(\right|{\mathrm{GHZ}}_{+}\rangle \pm |{\mathrm{GHZ}}_{-}\rangle )/\sqrt{2}$, and in the eigenstates of the parity operator $\mathcal{P}={\mathrm{\Pi }}_{i=1}^3{\left(\right|1\rangle }_{ii}\langle 0|+{|0\rangle }_{ii}\langle 1\left|\right)$ the state $|{\mathrm{GHZ}}_{+}\rangle$ can again be rewritten in the form $|{\mathrm{GHZ}}_{+}\rangle =\left(\right|+++\rangle +|+--\rangle +|-+-\rangle +|--+\rangle )/2$. Then a resonant Rydberg pumping couples to the transition from $|+++\rangle$ to $|rrr\rangle$; thereby the stability of state $|{\mathrm{GHZ}}_{+}\rangle$ under the $Z$ pumping is destroyed, leaving the unique steady state $|{\mathrm{GHZ}}_{-}\rangle$ unchanged.

Figure 2

Schematic view of the four-level Rydberg atom. The quantum bit is encoded into the ground states $|0\rangle$ and $|1\rangle$. A quantized cavity mode is coupled to the transition between $|0\rangle$ and $|e\rangle$ with strength $g$, while a classical field of Rabi frequency $\mathrm{\Omega }$ drives the atomic transition from $|1\rangle$ to $|e\rangle$. In the meantime, the ground states $|0\rangle$ and $|1\rangle$ can also be pumped upwards to the Rydberg state $|r\rangle$ under the actions of two independent classical fields with the same Rabi frequency ${\mathrm{\Omega }}_r$, and commonly detuned by $-\mathrm{\Delta }$. For the sake of convenience, we have assumed the excited state $|e\rangle$ ($|r\rangle$) spontaneously decays downwards to $|0\rangle$ and $|1\rangle$ with the same rate ${\gamma }_e/2({\gamma }_r/2)$, respectively. Note that the atom-dependent light shift ${\delta }_i$ plays an important role in the process of $Z$ pumping.

Figure 3

The effective transitions of quantum states for the simplified $Z$ pumping. The evolution of quantum states is frozen in a Zeno subspace corresponding to the vacuum state of the cavity field. The quantum states with one and two atoms in state $|1\rangle$ are coupled to the excited states by the near-resonant classical fields, and the population of state $|000\rangle$ is accumulated asymptotically to a steady value due to the spontaneous emission of excited states and the coherent pumping of laser fields. During the process, the state $|111\rangle$ is unaffected by the weak driving fields because of the Zeno requirement $\mathrm{\Omega }\ll g$.

Figure 4

Numerical simulation of the $Z$-pumping process. The initial state is a fully mixed state in the basis of quantum bits: ${\rho }_0={\mathrm{\Sigma }}_{i,j,k=0,1}|ijk\rangle \langle ijk|/8$, and the corresponding parameters are set as $\mathrm{\Omega }=0.02g, {\delta }_1={\delta }_3=-0.01g, {\delta }_2=0.02g, {\gamma }_e=0.1g$, and $\kappa =0$. The final state is stabilized into ${\rho }_s=7/8|000\rangle \langle 000|+1/8|111\rangle \langle 111|$ after a short time $t=2000/g$.

Figure 5

Numerical simulation of the population for the GHZ state starting from the fully mixed state. The effective master equation (solid line) and the full master equation (dashed line and dash-dotted line) are both simulated with the experimentally achievable parameters: $g=2\pi \times 50$ MHz, $\mathrm{\Omega }=0.01g, {\mathrm{\Omega }}_r=g, {\delta }_1={\delta }_3=-0.005g, {\delta }_2=0.01g, U=\mathrm{\Delta }=58g, \kappa =2\pi \times 1$ MHz, ${\gamma }_e=2\pi \times 3$ MHz, and ${\gamma }_r=2\pi \times 0.144$ MHz.

Figure 6

Numerical simulation of the populations for quantum states using the full Hamiltonian $H_k$ in Eq. (2) in (a) and (c), compared with the results obtained from utilizing the effective Hamiltonian of Eq. (3) in (b) and (d). The initial states are chosen as $|001\rangle |0_c\rangle$ and $|011\rangle |0_c\rangle$, respectively, and the corresponding parameters are $\mathrm{\Omega }=0.02g, {\delta }_1={\delta }_3=-0.01g$, and ${\delta }_2=0.02g$.