Characterizing quantum synchronization in the van der Pol oscillator via tomogram and photon correlation
Scalable methods for detecting and quantifying the nonclassical nature of a quantum state in noisy environments are challenging due to a complex relationship between noise and quantum coherence. In particular, identifying experimentally accessible signatures of synchronization in such regimes remains an open problem. By leveraging promising experimental implementation, we underpin what possible direct measures of nonclassicality are available. This work outlines accessing quantum synchronization (QS) in the steady state of a driven quantum van der Pol oscillator (vdPo) using two distinct figures of merit: (i) the nonclassical area $δ$ and (ii) the second-order correlation function $g^{(2)}(0)$, both of which are viable in experimental architectures. The nonclassical area quantifier based on homodyne tomography allows us to assess the nonclassical nature of the vdPo state directly from the tomogram without requiring full state reconstruction or Wigner function negativity. Within a well-defined parameter regime of drive strength and detuning, both $δ$ and $g^{(2)}(0)$ exhibit pronounced signatures of synchronization that complements the phase coherence between the drive and the vdPo. We derive an analytical expression for the steady state density matrix and the corresponding tomogram of the system, valid for arbitrary strengths of the harmonic drive. Analysis of the quantum tomogram uncovers clear phase locking behaviour, enabling the identification of the synchronization region (Arnold tongue) directly in terms of $g^{(2)}(0)$ and $δ$. Furthermore, the behaviour of $g^{(2)}(0)$ provides a statistical perspective that reinforces the tomographic signatures of QS. By analyzing the interplay between the aforementioned metrics, our findings indicate a scalable and experimentally relevant framework for characterizing QS in the driven vdPo.