Three-Axis Spin Squeezed States Associated with Excited-State Quantum Phase Transitions

TL;DR

The paper introduces a three-axis spin squeezing scheme within an anisotropic Lipkin-Meshkov-Glick model, unifying one-axis and two-axis twisting into an asymmetric quantum rotor with tunable ellipticity. By combining semiclassical Euler-top dynamics, Majorana stellar representations, and Husimi-Q phase-space analysis, it demonstrates squeezing scalings of $\xi^2 \sim N^{-2/3}$ for OAT-like regimes and $\xi^2 \sim N^{-1}$ at TAT-like points, along with enhanced entanglement in low-spin systems. Tuning the anisotropy parameters drives ground-state QPTs and excited-state QPTs (ESQPTs), manifested as level clustering and density-of-states singularities, with accompanying changes in dynamics and metrological performance. The work also provides geometric and phase-space perspectives on the squeezing process and suggests practical implementations in Rydberg arrays and cavity-QED systems for high-precision sensing and quantum simulation of critical phenomena.

Abstract

Spin squeezing in collective atomic ensembles enables quantum-enhanced metrology by reducing noise below the standard quantum limit through nonlinear interactions. Extending the one-axis and two-axis twisting paradigms of Kitagawa and Ueda, we introduce a general class of three-axis spin squeezed states within the anisotropic Lipkin-Meshkov-Glick model. The model features direction-dependent quadratic couplings that interpolate between uniaxial and biaxial regimes and can be interpreted as an asymmetric quantum rotor. Using semiclassical dynamics, Majorana representations, and Husimi-Q distributions, we analyze the structure and metrological properties of the resulting states. The three-axis framework reproduces the known N^(-2/3) scaling of one-axis twisting and the Heisenberg-limited N^(-1) scaling of two-axis twisting, while allowing additional tunability and enhanced entanglement generation in low-spin systems. We further show that tuning the anisotropy parameters induces ground-state and excited-state quantum phase transitions, including a second-order transition associated with level clustering and critical dynamics. These results unify spin squeezing, quantum criticality, and rotor analogies, and suggest implementations in Rydberg arrays and cavity-QED platforms for precision sensing and quantum simulation.

Figures (3)

• Figure 1: Energy spectrum $E_k$ as a function of the normalized one-axis twisting parameter $\mu_0$ for $j=10$, illustrating an excited-state quantum phase transition (ESQPT) at $\mu_0^c \approx 1.5$ (vertical dashed line). The level clustering and avoided crossings highlight the transition from stable rotational modes to critical dynamics.
• Figure 2: Husimi-$Q$ quasi-probability distribution $Q(\theta,\phi)$ for the one-axis spin squeezed state $|j,\mu\rangle$ with $j=20$, where $x\equiv Q(\theta,\phi)\sin\theta\cos\phi$, $y\equiv Q(\theta,\phi)\sin\theta\sin\phi$, and $z\equiv Q(\theta,\phi)\cos\theta$. The spin squeezed states $|j,\mu\rangle$ are shown in viridian, and the initial spin coherent state $|\pi/2,0\rangle$ is depicted in half-transparent moss green and lavender.
• Figure 3: Husimi-$Q$ quasi-probability distribution $Q(\theta,\phi)$ for the two-axis spin squeezed state $|j,\nu\rangle$ with $j=20$, where $x\equiv Q(\theta,\phi)\sin\theta\cos\phi$, $y\equiv Q(\theta,\phi)\sin\theta\sin\phi$, and $z\equiv Q(\theta,\phi)\cos\theta$. The upper two panels display $xy$-plane projections of the $Q$ distribution for the two-axis spin squeezed states in blue, with the $Q$ distribution for the initial state $|j,-j\rangle$ shown in half-transparent grass green. The lower two panels present the three-dimensional $Q$ distributions for the two-axis squeezed states.