Frequency shifts heralding ground state squeezing and entanglement of two coupled harmonic oscillators

TL;DR

This work shows that two linearly coupled harmonic oscillators, despite their Gaussian, near-classical behavior, host genuine quantum features such as ground-state entanglement and two-mode squeezing. By analyzing frequency shifts and transforming between bare and normal-mode representations, the authors demonstrate that frequency renormalization serves as an entanglement witness and that latent squeezing can enhance frequency estimation without direct noise squeezing. The paper provides explicit criteria and measures, including the Duan–Simon inseparability condition and the logarithmic negativity, and extends the analysis to finite temperature, showing how thermal fluctuations constrain entanglement. Collectively, the results reveal a bridge between spectral properties and quantum resources, suggesting new quantum-enhanced sensing strategies in systems traditionally deemed classical, and hint at extensions to spin systems and many-body models.

Abstract

It is often argued that two linearly coupled quantum harmonic oscillators, even when cooled to their ground state, display no inherently quantum features beyond quantized energy levels. Here, we challenge this view by showing that their classical observables encode genuinely quantum features. In particular, we demonstrate that the characteristic frequency shifts observed in coupled oscillators signal non-classical correlations and ground-state entanglement at zero temperature corresponding to two-mode squeezing between the uncoupled modes. From a complementary perspective, these two effects, frequency shifts and squeezing, represent the same underlying phenomenon but expressed in different mode bases. What appears as a spectral renormalization in one description manifests itself as entanglement in the other. Frequency shifts therefore constitute an entanglement witness accessible via standard spectroscopy. While the underlying squeezing is not directly measurable, it can be exploited to enhance the signal-to-noise ratio in precision frequency measurements of individual oscillators without requiring squeezed quantum noise. This uncovers a new route to quantum-enhanced sensing within systems traditionally regarded as classical, offering fresh insight into how signatures of quantumness persists across the quantum-to-classical boundary.

Figures (2)

• Figure 1: Inseparability criterion at $T=0$ ($\beta=\infty$) versus $\omega/\Omega$ and $g/g_c$ (explicit form given in the Appendix). The scale is truncated at $1$ to highlight the entangled regime (enclosed with a dashed line), with maximal entanglement at $\omega=\Omega$.
• Figure 2: Inseparability criterion as a function of $g/g_c$ and the dimensionless temperature $\beta \omega_-\equiv \omega_-/T$ for the resonant case $\omega=\Omega$. Increasing temperature reduces the entangled region, with entanglement vanishing beyond a specific temperature dependent on $g/g_c$. The scale is truncated at $1$ to highlight the entangled regime (enclosed with a dashed line). Note that $\omega_-$ is a function of $g/g_c$.