Dynamical generation of chiral $W$ and Greenberger-Horne-Zeilinger states in laser-controlled Rydberg-atom trimers

Motivated by the significantly improved scalability of optically trapped neutral-atom systems, extensive efforts have been devoted in recent years to quantum-state engineering in Rydberg-atom ensembles. Here we investigate the problem of engineering generalized (“twisted”) $W$ states, as well as Greenberger-Horne-Zeilinger (GHZ) states, in the strongly interacting regime of a neutral-atom system. We assume that each atom in the envisioned system initially resides in its ground state and is subject to several external laser pulses that are close to being resonant with the same internal atomic transition. In particular, in the special case of a three-atom system (Rydberg-atom trimer) we determine configurations of field alignments and atomic positions that enable the realization of chiral $W$ states—a special type of twisted three-qubit $W$ states of interest for implementing noiseless-subsystem qubit encoding. Using chiral $W$ states as an example we also address the problem of deterministically converting twisted $W$ states into their GHZ counterparts in the same three-atom system, thus significantly generalizing recent works that involve only ordinary $W$ states. We show that starting from twisted—rather than ordinary—$W$ states is equivalent to renormalizing downward the relevant Rabi frequencies. While this leads to somewhat longer state-conversion times, we also demonstrate that those times are at least two orders of magnitude shorter than typical lifetimes of relevant Rydberg states.

Figure 1

(a) Schematic illustration of a Rydberg-atom trimer ($N=3$). The three Rydberg atoms, located at the positions specified by the vectors ${\mathit{x}}_n$ ($n=1,2,3$), form an equilateral triangle. The ground and Rydberg states of each atom are denoted by $|g\rangle$ and $|r\rangle$, respectively, while $V$ stands for the magnitude of their pairwise van der Waals interaction. (b) Energy-level scheme of a Rydberg-atom trimer. The origin of the energy scale is chosen such that $E_g=0$. We are considering long interaction times $T$, i.e., $|E_r-E_g|T/\hslash \gg 1$, $\left|V\right|T/\hslash \gg 1$, and large electronic excitation energies, i.e., $|E_r-E_g|\gg |V|$.

Figure 2

Schematic drawing of the orientation of the laser field and the atomic plane for the $N=3$ case. The atoms form an equilateral triangle with interatomic distance $d$. The orientation of a plane-wave laser field with wave vector $\mathit{k}$ is defined through the angles ${\vartheta }_k$ and ${\phi }_k$.

Figure 3

Fidelity ${\text{F}}_{\left|W\right(\mathrm{\Phi })\rangle }=|\langle W\left(\mathrm{\Phi }\right)|{\zeta }_{1s}\rangle |$ ($s=0,\pm$) of the twisted $W$ state $\left|W\right(\mathrm{\Phi })\rangle$ corresponding to the chiral basis states $|{\zeta }_{10}\rangle =|W_3(\mathit{k}=0)\rangle$ (solid line), $|{\zeta }_{1+}\rangle$ (dashed line), and $|{\zeta }_{1-}\rangle$ (dotted line) as the target states, for different polar angles ${\vartheta }_k=arcsin(\mathrm{\Phi }/2\pi )$ and azimuthal angle ${\phi }_k-{\phi }_2=\pi /2$. The vertical dotted lines indicate the configurations required for the preparation of the two chiral states.

Figure 4

Typical time evolution of fidelities ${\text{F}}_{\left|\psi \right(t)\rangle }=|\langle \psi \left(t\right)|\mathrm{\Psi }\rangle |$ ($|\mathrm{\Psi }\rangle =|ggg\rangle ,\left|\psi \right(T)\rangle$) corresponding to a $\pi$ pulse with ${\mathrm{\Omega }}_1=\pi /\left(2\sqrt{3}T\right)$, where $T$ is the conversion time and $|\psi \left(T\right)\rangle =|W_3\left({\mathit{k}}_1\right)\rangle$ the resulting twisted $W$ state. The time evolution corresponds to the $N=3$ Hamiltonian [cf. Eq. (37) in Sec. 6a] and randomly chosen phases ${\mathit{k}}_1·{\mathit{x}}_n$, with $V/\hslash =3000/T$ and ${\mathrm{\Omega }}_2={\mathrm{\Omega }}_3=0$.

Figure 5

Energies of the atomic ensemble with $N=3$ corresponding to the states $|{\zeta }_{as}\rangle$ [cf. Eqs. (14)]. The solid arrows indicate transitions driven in the case when all laser fields are aligned such that $({\mathit{k}}_j-{\mathit{k}}_1)·{\mathit{x}}_n=0$, while the dotted ones indicate those corresponding to the choice of laser-field alignments described by the upper-sign case of Eq. (36). Here $E_g=0$ is chosen as origin for the energy scale.

Figure 6

Time dependence of target-state fidelities ${\text{F}}_{|\text{GHZ}\rangle }=\sqrt{\langle \text{GHZ}\left|\varrho \left(t\right)\right|\text{GHZ}\rangle }$ [cf. Eq. (39)] for the $W$-to-GHZ state conversions via two different intermediate states $|{\zeta }_{20}\rangle$ and $|{\zeta }_{2,-}\rangle$, both shown for $V{T}_{+}/\hslash =3000$. The solid lines correspond to the unitary dynamics where $\varrho \left(t\right)=\left|\psi \right(t)\rangle \langle \psi (t\left)\right|$. The dotted lines correspond to the open-system dynamics with the dephasing and spontaneous-decay rates $\mathrm{\Gamma }=\gamma =0.1/T_{+}$. The two conversion pathways are adjusted such that their respective total laser-pulse energy consumptions are mutually equal.

Figure 7

Mean values of the GHZ-state fidelity ${\text{F}}_{|\text{GHZ}\rangle }$ and their corresponding standard deviations ${\sigma }_{\text{F}}$ (shaded area) corresponding to the $W$-to-GHZ state conversion via $|{\zeta }_{20}\rangle$, computed from a sample of $S=500$ results, for varying standard deviation ${\sigma }_d$ of the interatomic distance. The error bars show the standard error ${\sigma }_{\text{F}}/\sqrt{S}$ of the mean fidelity. The parameter values used are $V/\hslash =30.86/T_0$ and $d=40{\lambda }_0$, where $T_0$ is the state-conversion time and ${\lambda }_0$ is the resonance wavelength.

Figure 8

Energy-level scheme corresponding to the Hamiltonian $H_A+H_3^{\text{off}}\left({\mathit{k}}_0\right)$ for Rydberg trimers. $E_g$ is chosen as origin of the energy scale. The arrows indicate the transitions driven by laser fields with detunings ${\mathrm{\Delta }}_i+{\delta }_i$ as determined by Eq. (46) [cf. Eq. (47)].

Figure 9

Expectation values $\langle P_a\rangle =\langle \psi \left(t\right)\left|P_a\right|\psi \left(t\right)\rangle$ for the atomic excitation number $a$ over the conversion time for a conversion scheme $|ggg\rangle \stackrel{\pi -\text{pulse}}{⟶}|D_1^3\left({\mathit{k}}_0\right)\rangle \stackrel{\left[45\right]}{⟶}$ GHZ state for different amounts of relative twisting corresponding to $|{\mathrm{\Sigma }}_{{\mathit{k}}_0-{\mathit{k}}_j}|=3,2,1$ [(a)–(c)]. $T_{0,0.5,0.75}$ is the respective conversion time under the assumption of equal laser-pulse energy consumption [cf. Eq. (38)].