Characterizing quantum synchronization in the van der Pol oscillator via tomogram and photon correlation

TL;DR

This paper addresses quantum synchronization (QS) in a driven quantum van der Pol oscillator by introducing two experimentally accessible probes: the nonclassical area δ derived from homodyne tomograms and the equal-time second-order correlation g^(2)(0). It develops a tomographic framework and analyzes QS in both classical (κ2→0) and deep quantum (κ2→∞) regimes, deriving steady-state behavior and, in the deep quantum limit, analytic expressions for the density matrix and the tomogram ω(X_θ, θ). The results reveal Arnold tongue structures in the quadrature plane, with δ signaling phase locking that is strong in the classical regime but saturates in the deep quantum regime, while g^(2)(0) provides a complementary, regime-dependent signal. A key contribution is the analytic solution for the steady state in the deep quantum limit and the reformulation of the master equation in tomographic terms, enabling direct experimental access to QS signatures without full state reconstruction. The work offers a practical bridge between theory and experiment for quantum synchronization in nonlinear open systems and points toward experimental realization in platforms supporting homodyne tomography and photon-correlation measurements.

Abstract

We access the 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 $g^{(2)}(0)$, which are both viable in experimental architectures. The nonclassical area quantifier rooted in homodyne tomography, allows us to assess the nonclassical nature of the vdPo's state directly from the tomogram without requiring full state reconstruction or the 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 behavior, enabling the identification of the synchronization region (Arnold tongue) directly in terms of experimentally measurable quantities. Furthermore, the behaviour of $g^{(2)}(0)$ provides a statistical perspective that reinforces the tomographic signatures of QS. By analyzing the interplay between these metrics, we can gain more profound insights into the underlying mechanisms that govern QS in such systems.

Figures (5)

• Figure 1: (Color online) Different behaviors of the phase locking measure $\delta$ at the steady state. The nonclassical area $\delta$ measures synchronization at the steady state of an externally driven van der Pol oscillator. In panels (a) and (b), red (blue) indicates maximum (minimum) synchronization as measured by $\delta$. The damping ratio is set to (a) $\kappa_2=0$, (b) $\kappa_2=10^3$. Inside the tongue: strong phase-locking, collective dynamics $(\delta>>1)$. Outside the tongue: Weak or no phase-locking, less collective emission $(\delta>1)$. The lower panel (c) represents the synchronization measure across various regimes of the quantum vdP, as mentioned in Reference Mok2020 in Table 1. The different drive strengths $F$, indicated in the legend, are selected from the top panels, implying a scenario of undriven to driven quantum vdPo, with $\Delta=2$. Increasing the driving strength is seen to produce qualitative nonclassicality up to the quantum regime ($\kappa_2=1$). The inset illustrates that the behavior of nonclassicality in the deep quantum regime ($\kappa_2 >> 1$) results in synchronization similar to that observed in the classical limit.
• Figure 2: (color online). At steady-state, the equal-time second-order correlation $g^{(2)}(0)$ exhibits phase locking as a function of driving strength $F$ and detuning $\Delta$. In panels (a, b), the red (blue) region indicates the maximum (minimum) synchronization measurement. Inside the tongue: strong phase-locking indicates bunching with collective dynamics $(g^{(2)}(0)>1)$. Outside the tongue: Weak or no phase-locking indicates antibunching with nonclassicality due to $(g^{(2)}(0)<1)$. In panel (c), the absence of Arnold's tongue shape, indicated by $g^{(2)}(0) \rightarrow 0$, highlights a strikingly diminishing correlation with strong nonclassicality. The damping ratio is set to (a) $\kappa_2=0$, (b) $\kappa_2=1$, and (c) $\kappa_2=10^3$.
• Figure 3: (color online). Top row: Tomographic probability distributions $\omega(X_\theta,\theta)$ at steady state of the driven quantum vdPo in the deep quantum regime, shown for different drive strengths: (a) weak drive $F=0$, (b) intermediate drive $F=1$, and (c) strong drive $F=10$. For weak driving [panel (a)], the tomogram remains broadly distributed, indicating low phase preference and weak synchronization. At intermediate driving [panel (b)], the distribution becomes anisotropic with visible deformation, signaling the onset of partial phase locking. Under strong driving [panel (c)], the tomogram exhibits a pronounced localized shape, characteristic of enhanced phase concentration and stronger synchronization. Bottom row: Wigner functions (d-f) corresponding to the tomograms in panels (a–c). At weak drive (d), the Wigner function retains near-rotational symmetry, consistent with the absence of synchronization. For intermediate drive (e), angular modulation emerges, marking the onset of rotational symmetry breaking. Under strong drive (f), the Wigner function shows pronounced phase-space asymmetry, mirroring the tomogram’s localization and providing clear evidence of QS. Here, we choose $\Delta=2$ for all panels.
• Figure 4: (color online). (a) Steady state coherence $|\rho_{01}|$ as a function of driving strength $F$ and detuning $\Delta$ for the quantum vdPo in the limit $\kappa_2\rightarrow\infty$, showing sensitivity of phase locking. Outside of the red area, the vdPo exhibits more classical (mixed) behavior, loses quantum features, and fails to exhibit synchronization. (b) The gradient $\partial_F\mathcal{S}$ is used to locate the synchronization threshold, or the region where quantum control is most effective. By fine-tuning the parameters, the phase relationship between $|0\rangle$ and $|1\rangle$ is optimized, enabling more reliable qubit operations. In both panels, the black curve represents the critical drive $F_c$ that describes the peak of maximum coherence $|\rho_{01}|$, which inherently demonstrates a two-level qubit.
• Figure 5: (color online). (a) Quantum tomogram against quadrature eigenvalues $X_\theta$ for an undriven quantum vdPo showing rotational symmetry, i.e., no preferred phase direction. From the synchronization perspective, that corresponds to no phase locking, as shown in panel \ref{['Fig:Tomogram_Wigner']}(a). The limit-cycle amplitude Eq. (\ref{['eq13']}) is shown in panel (b) at steady state. The color scale indicates the degree of phase locking: deep-blue regions correspond to weak synchronization, while the transition to red marks enhanced synchronization-near resonance ($\Delta\approx0$) and grows with increasing drive strength.