Fast Generation of GHZ-like States Using Collective-Spin $\mathrm{XYZ}$ Model

The Greenberger-Horne-Zeilinger (GHZ) state is a key resource for quantum information processing and quantum metrology. The atomic GHZ state can be generated by one-axis twisting (OAT) interaction $H_{\mathrm{OAT}}=\chi J_z^2$ with $\chi$ the interaction strength, but it requires a long evolution time $\chi t=\pi /2$ and is thus seriously influenced by decoherence and losses. Here we propose a three-body collective-spin XYZ model which creates a GHZ-like state in a very short timescale $\chi t\sim \ln N/N$ for $N$ particles. We show that this model can be effectively produced by applying Floquet driving to an original OAT Hamiltonian. Compared with the ideal GHZ state, the GHZ-like state generated using our model can maintain similar metrological properties reaching the Heisenberg-limited scaling, and it shows better robustness to decoherence and particle losses. This Letter opens the avenue for generating GHZ-like states with a large particle number, which holds great potential for the study of macroscopic quantum effects and for applications in quantum metrology and quantum information.

Figure 1

Dynamics of the XYZ model and the TAT model, with ${\chi }_{\mathrm{XYZ}}={\chi }_{\mathrm{TAT}}$. (a) Classical trajectories of (a1) the XYZ model and (a2) the TAT model on the Bloch sphere. The red trajectories are separatrixes shared by the XYZ model and the TAT model, while the blue one is one that only exists in the XYZ model. (b) Evolution of the quantum states of (b1) the XYZ model and (b2) the TAT model, represented by the Husimi $\mathcal{Q}$ function on the Bloch spheres. The evolution time is labeled with $t_c=\ln N/(N{\chi }_{\mathrm{XYZ}})$. (c) Husimi $\mathcal{Q}$ function of (c1) the XYZ model and (c2) the TAT model at $t=t_c$, represented with polar coordinates $(\theta ,\varphi )$.

Figure 2

Generation of the catlike state with Floquet driving. (a) An illustration of the proposed pulse sequences. One period is $3\tau$ (shaded), consisting of $\pm \pi /2$ pulses (up for “$+$” and down for “$-$”) along the $x$ axis (red one) and $y$ axis (blue one). (b) Time evolution of the optimal quantum Fisher information $F_Q$ of the original OAT model (green dotted), the effective XYZ model (blue dashed), and the proposed Floquet-driving scheme with driving parameter $\alpha =0.4$ (red solid) for $N=100$ particles. Horizontal lines indicate the standard quantum limit $F_Q=N$ (black dash-dotted line) and the Heisenberg limit $F_Q=N^2$ (black dash-dot-dotted line). The time when $F_Q$ reaches its maximum of the XYZ model and the OAT model are marked as $t_{\max }^{\mathrm{XYZ}}$ and $t_{\max }^{\mathrm{OAT}}$, respectively. (c) Probability distribution $P_m={|⟨m|\psi ⟩|}^2$ on eigenstates of $J_y$ at $t=t_{\max }^{\mathrm{XYZ}}$ for the XYZ model. (d) Optimal time $t_{\max }$ when $F_Q$ reaches its maximum for the OAT model (green dotted) and the proposed Floquet-driving scheme (red circles) versus particle number $N$, compared with the predicted value $t_c\simeq 3\ln N/(\alpha N)$ (blue dashed line).

Figure 3

Parity oscillation of the GHZ-like state for $N=100$ particles. (a) The expectation value of parity $\mathrm{\Pi }$ versus the azimuth angle $\theta$ of the rotation axis, for the GHZ-like state generated through the XYZ dynamics (red solid line) and the perfect GHZ state (black dashed line). (b) The precision $\mathrm{\Delta }\theta$ at ${\theta }_0=\pi /(2N)$ versus evolution time $t$ of the XYZ dynamics (red solid line), in comparison with the quantum Cramér-Rao bound $F_Q^{-1/2}$ (blue dashed line). (c) The optimal precision $\mathrm{\Delta }{\theta }_{\min }$ versus particle number $N$ of the GHZ-like state (red circles), comparing with the Heisenberg limit $1/N$ (blue dashed line).

Figure 4

The effective XYZ dynamics and the obtained GHZ-like state in the presence of decoherence and losses. (a1) The quantum Fisher information $F_Q$ of the GHZ(-like) state created through our Floquet-driving scheme (red solid line) and OAT dynamics (blue dashed line) for $N=100$ particles with respect to $\mathrm{\Gamma }$. (a2) The ratio of the quantum Fisher information for two dynamics $F_Q^{(\mathrm{OAT}+\mathrm{FD})}/F_Q^{(\mathrm{OAT})}$ versus $N$ and $\mathrm{\Gamma }$. Data are calculated at $\chi t=3\ln N/(\alpha N)$ with $\alpha =0.4$ for Floquet-driving scheme and $\chi t=\pi /2$ for OAT scheme. (b1) The optimal quantum Fisher information of the perfect GHZ state (blue dashed line) and the GHZ-like state generated through XYZ dynamics at $t=t_c$ (red solid line), $t=0.8t_c$ (orange dotted), and $t=0.6t_c$ (green dash-dotted line) when $\mathrm{\Delta }N$ particles are lost. (b2) The probability distribution $P_m$ for the GHZ-like state generated through XYZ dynamics at $t=t_c$ (gray), $t=0.8t_c$ (red) and $t=0.6t_c$ (blue).