Optimal Control of Spin Squeezing in 2D Finite-Range Interacting Systems

Abstract

Spin squeezing serves as both a fundamental witness of quantum entanglement and a critical resource for quantum-enhanced metrology. While generating substantial spin squeezing in finite-range interacting systems remains challenging, such capability is important for advancing quantum technologies. In this work, we develop an optimal control strategy for achieving enhanced spin squeezing in a two-dimensional XX model with dipolar interactions. Leveraging rotor-spin-wave theory for periodic boundary conditions, we circumvent computational bottlenecks to explore control strategies at unprecedented scales. Remarkably, optimizing a single collective transverse field is sufficient to achieve substantial squeezing enhancement, exceeding the two-axis-twisting benchmark. The optimized control field achieves this breakthrough by dynamically suppressing inter-subspace mixing induced by the finite-range interactions, thereby confining the system evolution predominantly within the maximal spin subspace. We further extend rotor-spin-wave theory to open boundary conditions and incorporate dephasing noise, providing a scalable framework for realistic systems. Under these conditions, the optimized protocol remains effective, highlighting its robustness and suitability for experimental implementation.

Figures (6)

• Figure 1: The optimal squeezing parameter for the control protocol as a function of total evolution time $T$ for different system sizes under periodic boundary conditions (PBC). Filled circles denote results obtained using the RSW-based optimization, while cross markers (‘x’) and diamonds correspond to exact diagonalization (ED) and time-dependent variational Monte Carlo (t-VMC) results, respectively. For comparison, the time evolution of the squeezing parameter for the uncontrolled case (i.e., $h(t) = 0$, obtained by RSW theory) is shown as dotted lines, while the TAT limit is indicated by dashed lines. The solid lines connecting the filled circles serve as visual guides. The inset displays the crossover time $t_{\mathrm{TAT}}$---the point at which the optimized squeezing parameters exceed those of the TAT models---as a function of system size.
• Figure 2: Time evolution of spin system dynamics under optimal control. (a) Normalized spin squared expectation value $\langle\bm{S}^2\rangle/\langle\bm{S}^2\rangle_{\mathrm{max}}$ showing the optimized (blue) versus uncontrolled evolution (orange). (b) Spin squeezing parameter $-10\log_{10}(\xi^2)$ (in dB), with solid lines representing exact evolution and dashed lines showing effective collective spin models. (c) Normalized mean spin $|\langle\bm{S}\rangle|/S_{\mathrm{max}}$. (d) Optimized step-wise constant control field $h(t)$. All quantities are plotted versus dimensionless time $Jt$, where $J$ represents the interaction strength. (e)-(f) Visualization of the spin squeezing dynamics in (b) on the Bloch spheres for the uncontrolled case and optimized case, respectively. We look down at the $y-z$ plane from the positive $x$-axis.
• Figure 3: The optimal squeezing parameter (in dB) as a function of the collective rate. (a) and (b) are for $4\times 4$ and $10\times 10$, respectively. Solid lines are the results from RSW theory. Gray curves denote the results for the uncontrolled case. Orange and blue curves denote the results for the optimized scheme with different total evolution time $T$. The dashed lines in (a) denote the results from ED.
• Figure 4: The optimal squeezing parameter for systems with open boundary conditions as a function of total evolution time (filled circles). The RSW theory generalized to OBCs is employed for the optimization procedure. In comparison, the time evolution of the squeezing parameter for the uncontrolled case are also shown, which are obtained using ED (crosses) or MPS simulations (diamonds) based on the time-dependent variational principle (TDVP, ITensorFishman_2022), with a maximum bond dimension of $300$.
• Figure S1: (a) The time evolution of the squeezing parameter for the control protocol for different system sizes with PBCs. The dashed curves are the RSW results. The solid curves are results from t-VMC. The ED results are denoted by crosses 'x'. (b) The time evolution of the squeezing parameter for a $8\times 8$-size system with OBC. The dashed curves are RSW results. The solid curves are obtained by MPS (TDVP) method.