Preparing spin-squeezed states in Rydberg atom arrays via quantum optimal control

TL;DR

This work develops a gradient-based quantum optimal-control protocol to generate highly spin-squeezed states in Rydberg atom arrays governed by Ising-type van der Waals interactions. By designing time-dependent, piecewise-constant pulses via GRAPE, starting from the product state $|\uparrow_z^{\otimes N}\rangle$, the method targets final states with predefined magnetization and squeezing axis, quantified by the Wineland parameter $ξ_W^2$. For systems up to $N=8$, the protocol approaches the theoretical lower bound $ξ_W^2 \approx 1/(1+N/2)$ and yields significant multipartite entanglement; importantly, pulses optimized at small $N$ transfer to larger systems to achieve $ξ_W^2$ as low as about $0.227$ for $N=50$, outperforming conventional quench dynamics. The results imply a scalable route to metrologically useful entangled states on Rydberg platforms, with robustness to dephasing and potential for experimental realization.

Abstract

We present a quantum optimal control protocol to generate highly spin-squeezed states in Rydberg atom arrays coupled via Ising-type van der Waals interactions. Using gradient-based optimization techniques, we construct time-dependent pulse sequences that steer an initial product state toward highly entangled, spin-squeezed states with predefined magnetization and squeezing axes. We focus on the Wineland parameter $ξ_W^2$ to measure spin squeezing, and our approach achieves near-optimal spin squeezing in one-dimensional ring arrays of up to $N=8$ spins, significantly outperforming conventional quench dynamics for all system sizes studied. Remarkably, optimized pulse sequences can be directly scaled to larger arrays without additional optimization, achieving a squeezing parameter as low as $ξ_W^2 = 0.227$ in systems containing $N=50$ spins. This work demonstrates the potential of quantum optimal control methods for preparing highly spin-squeezed states, opening pathways to enhanced quantum metrology.

Figures (12)

• Figure 1: Sketch of the positions of the spins. $a$ denotes the distance between nearest neighbors spins.
• Figure 2: Summary of the main results. Optimal Wineland parameter $\xi_W^2$ as a function of the system size $N$. Results are shown for the quench protocol, starting from the initial state $\ket{\uparrow_y}^{\otimes N}$ under the Ising van der Waals (VdW) and XY dipolar (Dipolar) Hamiltonians, and for the quantum control protocol developed in this work (VdW+QC), starting from $\ket{\uparrow_z}^{\otimes N}$ with optimized time-dependent fields $\Omega(t)$ and $\Delta(t)$. The dashed line indicates the lower bound $\xi_W^2 = (1+N/2)^{-1}$, corresponding to the minimum achievable value of the Wineland parameter Pezz__2018.
• Figure 3: Spin-squeezed states obtained from different random initial control pulses for $N = 8$. Each point represents a spin-squeezed state obtained through the application of an optimized control pulse sequence. The vertical axis is the value of Wineland parameter ($\xi_W^2$) and the horizontal axis is the normalized mean magnetization ($2|\langle J_z \rangle|/N$). The colored regions indicate the depth of entanglement $k$ according to the criterion developed in Sorensen_2001. Yellow points correspond to states with a fidelity $F$ with the GHZ state below 0.5, and the green points correspond to states with a fidelity larger than 0.5 (see Appendix \ref{['sec:ghz']}). The red square marker highlights the most squeezed state, the green diamond corresponds to the state with the largest magnetization among those with $F \geq 0.5$, and the blue cross is for the state with the highest magnetization.
• Figure 4: Time evolution under two optimized control pulse sequences for $N = 8$. The panels (a) and (b) show the evolution of the minimum variance of the collective angular momentum and the maximum mean magnetization, respectively. For comparison the dashed line represents the variance of a product state (note that the blue curve in (a) has been shifted by a factor 10 for clarity). The blue (resp. red) curves correspond to the pulse sequence leading to the state marked with a blue (resp. red) square in Fig. \ref{['fig:histo8']}. The associated pulse sequences are presented in (c) and (d). The Q-Husumi function in the final state is displayed in the panels (e) (for the state associated with the red marker) and (f) (state associated with the blue marker). Next to these panels, we show the values of the Wineland parameter, spin length, and the duration ($t_{\rm final}$) of the protocols.
• Figure 5: Effect of dephasing noise. Wineland parameter in the final state as a function of the strength $\gamma$ of the dephasing noise (Eq. \ref{['eq:lindblad']}) in a system with $N=8$. The blue (resp. green and red) curve corresponds to the protocol leading to the state marked in blue (resp. green and red) in Fig. \ref{['fig:histo8']}. The pink curve corresponds to a quench protocol (time-independent Hamiltonian and no optimization).