Investigating Lipkin-Meshkov-Glick Model and Criticality-Enhanced Metrology in a Coherent Ising Machine

Abstract

Quantum criticality has received extensive attention due to its ability to significantly enhance quantum sensing. But its realization and control in many-body quantum systems remain challenging. We present an effective scheme to simulate the Lipkin-Meshkov-Glick (LMG) model using a coherent Ising machine (CIM) composed of a network of degenerate optical parametric oscillators (DOPO). In our work, the spin variables of the LMG model are mapped onto the phases of DOPO pulses, and the spin-spin interactions are realized by all-to-all couplings among them. Through our investigation of the critical behavior in the antiferromagnetically coupled LMG model in the thermodynamic limit, i.e., $N\rightarrow\infty$, and its application in quantum sensing near the critical point, we verify that the CIM does not only effectively capture the second-order quantum phase transition (QPT) at the critical point but also reconstructs its complete phase diagram under ferromagnetic coupling. Furthermore, we demonstrate how the critical dynamics of this simulation platform can be utilized for quantum-enhanced metrology, achieving a measurement precision that diverges near the critical point of the LMG model. These results highlight the capability of the CIM as a flexible experimental platform for investigating the QPT in the fundamental quantum magnetic models, providing valuable insights into quantum simulation and critical phenomena.

Figures (7)

• Figure 1: Schematic of a fiber-based large-scale DOPO network. (a) A CIM based on the time-division multiplexed DOPO with mutual coupling implemented by optical delay lines. (b) Simplified illustration of vacuum-squeezing phenomenon in a DOPO below threshold and the performance of binary phases above threshold in in-phase and quadrature-phase coordinates.
• Figure 2: Comparison of exact solutions for the LMG model and CIM. (a) Exact solutions of the LMG model as a function of the magnetic-field strength $h/h_c$ for the rescaled ground-state energy $e_g$ (blue solid line), $de_g/dh$ (red dashed line) and $d^2e_g/dh^2$ (green dashed-dotted line). (b) Exact solutions of the CIM as a function of the renormalized oscillation frequency $\Omega/J$ for the rescaled ground-state energy $e_g$ (blue solid line), $de_g/d\Omega$ (red dahed line) and $d^2e_g/d\Omega^2$ (green dashed-dotted line).
• Figure 3: The numerical evaluation results of the QPT and criticality-enhanced quantum sensing of the LMG model. (a) Quadrature $\langle\hat{P}\rangle_\tau$ of the LMG model after an evolution time $\tau_n=\pi/[\omega(1-g^2_o)^{1/2}]$ as a function of $g$ with $\alpha=1$. The working point $g=g_o$ is marked by filled circles. The inset shows the corresponding susceptibility at the working point $g_o$. (b) The variation relationship of the QFI in the LMG model under different parameters $g$ and coherent amplitude $\alpha$. (c) The inverted variance $\mathcal{F}_g(t)$ of the LMG model vs the evolution time $t$. $\mathcal{F}_g(t)$ shows equidistant peaks at $\sqrt{\Lambda_g}\omega t/(2\pi)$. (d) The peak of the inverted variance $\mathcal{F}_g(t)$ of the LMG model vs the parameter $g$ for an evolution time $\tau_n$ with $\alpha=1$. The inset shows that the local maxima of $\mathcal{F}_g(\tau_n)$ reaches the seam order of $I_g(\tau_n)$.
• Figure 4: The QFI $I_{\tilde{g}}(t)$ of the CIM as a function of the evolution time $t$ for (a) $\tilde{g}=-0.96, -0.94, -0.92$ with $\alpha=1$, (b) $\tilde{g}=0.96, 0.94, 0.92$ with $\alpha=1$, (c) $\alpha=1, 2, 3$ with $\tilde{g}=-0.96$, and (d) $\alpha=1, 2, 3$ with $\tilde{g}=0.96$.
• Figure 5: The inverted variance $\mathcal{F}_{\tilde{g}}(t)$ of the CIM as a function of the evolution time $t$ for (a) $\tilde{g}=-0.92, -0.94, -0.96$ with $\alpha=1$ and (b) $\tilde{g}=0.92, 0.94, 0.96$ with $\alpha=1$