Heralded $W$-state preparation using laser-designed superatoms

We propose a scheme for preparing an ensemble of atoms in a maximally entangled $W$ state by exploiting the Rydberg blockade effect. The success of our protocol is indicated by the detection of an ion, which thus serves as a herald for successful entangled-state preparation. The fidelity of the preparation scheme is independent of the number of atoms in the ensemble. Therefore, a small cloud of atoms in a dipole trap randomly loaded from a background gas can be reliably prepared in a maximally entangled state despite atom number fluctuations.

Figure 1

(a) Illustration of the proposed entangled-state preparation scheme. The detection of an ion indicates the successful preparation of a $W$ state. (b) Level schemes of single-atom states with laser couplings and symmetrized ensemble states (Dicke states). By strongly coupling pairs of collective states the coupling laser causes ac Stark shifts, which differ between different rungs of the Dicke ladder. This makes it possible to tune the probe laser to resonance between the collective ground state $\left|G\right.〉$ and only one dressed excited state.

Figure 2

(a) Level scheme as in Fig. 1, now in the rotating frame. The energy $E$ of the levels are the diagonal elements of the Hamiltonian (1). The coupling ${\mathrm{\Omega }}_c$ leads to a repulsion between pairs of Dicke states. We choose the parameters such that the transition between states $\left|G\right.〉$ and $\left|2+\right.〉$ is resonant. (b) and (c) show the Rabi oscillations between these states. (b) Relatively small $\sqrt{N}{\mathrm{\Omega }}_p/{\mathrm{\Omega }}_c$. Imperfections due to coupling to singly and triply excited states are visible. We compare solutions of the time-dependent Schrödinger equation on the full state space (black dots) to the restriction to the lowest dressed Dicke states (solid green line) and the two-state model with all but states $\left|G\right.〉$ and $\left|2+\right.〉$ eliminated (dashed blue line). The red dotted line shows the ratio of Rydberg excitation probability in states other than $\left|ER\right.〉$ over the total Rydberg probability. Parameters are $N=4,{\mathrm{\Omega }}_c/2\pi =10$ MHz, ${\mathrm{\Omega }}_p/2\pi =0.7$ MHz, ${\mathrm{\Delta }}_c=-{\mathrm{\Omega }}_c/2,{\mathrm{\Delta }}_p=-{\mathrm{\Delta }}_p+{\mathrm{\Delta }}_{\text{eff}}/2$. (c) Different Rabi frequencies and atom numbers. Solid green and dotted red line: ${\mathrm{\Omega }}_c/2\pi =20$ MHz, ${\mathrm{\Omega }}_p/2\pi =1$ MHz. Dot-dashed black and dashed blue line: ${\mathrm{\Omega }}_c/2\pi =50$ MHz, ${\mathrm{\Omega }}_p/2\pi =1.6$ MHz. ${\mathrm{\Omega }}_p$ was adjusted to yield the same period of the effective Rabi oscillations for each ${\mathrm{\Omega }}_c$. For small $\sqrt{N}{\mathrm{\Omega }}_p/{\mathrm{\Omega }}_c$ near-perfect transfer to the state $\left|2+\right.〉$ is achieved.

Figure 3

Parameter study of success probability and infidelity of the $W$-state preparation protocol. (a) Rydberg probability $N_r$ (success probability) and infidelity as a function of ${\mathrm{\Delta }}_c$. The infidelity shows a minimum at approximately ${\mathrm{\Delta }}_c=-{\mathrm{\Omega }}_c/2$. The success probability is $2/3$ at the optimal point and approaches 90% at ${\mathrm{\Delta }}_c/{\mathrm{\Omega }}_c=-2$. Parameters: $N=3,{\mathrm{\Omega }}_c/2\pi =20$ MHz. Here, as in (c) and (d), ${\mathrm{\Omega }}_p$ was adjusted such that $\pi /{\mathrm{\Omega }}_{\text{eff}}=5\mu \mathrm{s}$ and ${\mathrm{\Delta }}_p$ is determined by the resonance condition. In panels (b)–(d) the condition ${\mathrm{\Delta }}_c=-{\mathrm{\Omega }}_c/2$ was imposed. (b) Dependence on the atom number $N$ for fixed laser parameters. The laser parameters are chosen such that the success probability is maximal for $N=100$ atoms. The dot-dashed green line shows the probability $p\left(N\right)$ of preparing an ensemble of $N$ atoms for a Poissonian distribution with a mean of $N=100$ atoms. The protocol shows reasonable success probability and fidelity across the whole atom number distribution. (c) Infidelity as a function of ${\mathrm{\Omega }}_c$ for different atom numbers. At ${\mathrm{\Omega }}_c/2\pi >100$ MHz, infidelities below 1% can be achieved. The black line shows an estimated upper bound on the infidelity. (d) Dependence on different sources of decoherence. The infidelity increases linearly with spontaneous decay rate from intermediate $\left({\mathrm{\Gamma }}_e\right)$ and Rydberg state $\left({\mathrm{\Gamma }}_r\right)$, as well as with the single-atom dephasing rate ${\mathrm{\Gamma }}_d$.