Frequency shifts as a reflection of ground state squeezing and entanglement in two coupled harmonic oscillators

Abstract: It is often argued that two 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 can encode genuinely quantum features. In particular, we demonstrate that the characteristic frequency shifts observed in such systems act as a signature of non-classical correlations and ground-state entanglement at zero temperature, specifically two-mode squeezing between the uncoupled modes. From a complementary perspective, these two effects -- frequency shifts and squeezing -- represent the same underlying phenomenon expressed in different mode bases. What appears as a spectral renormalization in one description manifests as entanglement in another. These shifts therefore can serve as an entanglement witness accessible via standard frequency measurements. Furthermore, we show that this underlying squeezing, although not directly measurable, can be exploited to enhance the signal-to-noise ratio in precision frequency measurements of individual oscillators without requiring squeezed quantum noise. Our results uncover a new route to quantum-enhanced sensing within systems traditionally regarded as classical, offering fresh insight into how signatures of quantumness persist across the quantum-to-classical boundary.