Classical and quantum chaotic synchronization in coupled dissipative time crystals

TL;DR

The paper addresses chaotic synchronization in two coherently coupled dissipative time crystals by contrasting classical mean-field dynamics with finite-size quantum dynamics. The authors employ a Lindblad framework and its mean-field limit to derive coupled nonlinear ODEs, then use largest Lyapunov exponents and Pearson correlations to identify a chaotic synchronization regime associated with a uniform magnetization across subsystems. In the quantum regime, quantum trajectories reveal a staggered-to-uniform crossover in trajectory-resolved $z$-magnetizations, accompanied by entanglement fluctuations and a nonequilibrium steady-state whose level spacing agrees with Gaussian unitary ensemble statistics, signaling dissipative quantum chaos. Importantly, the classical and quantum crossover points do not coincide due to noncommuting limits ($S oty$ vs $t oty$) and entanglement effects, yet both sides exhibit a qualitatively similar synchronization phenomenon. The results establish a coherent picture of chaotic synchronization across classical and quantum dissipative time crystals and highlight the role of entanglement in the quantum case, with potential implications for exploring quantum synchronization in open many-body systems.

Abstract

We investigate the dynamics of two coherently coupled dissipative time crystals. In the classical mean-field limit of infinite spin length, we identify a regime of chaotic synchronization, marked by a positive largest Lyapunov exponent and a Pearson correlation coefficient close to one. At the boundary of this regime, the Pearson coefficient varies abruptly, marking a crossover between staggered and uniform $z$-magnetization. To address finite-size quantum dynamics, we employ a quantum-trajectory approach and study the trajectory-resolved expectations of subsystem $z$-magnetizations. Their histograms over time and trajectory realizations exhibit maxima that undergo a staggered-to-uniform crossover analogous to the classical one. In analogy with the classical case, we interpret this behavior as quantum chaotic synchronization, with dissipative quantum chaos highlighted by the steady-state density matrix exhibiting Gaussian Unitary Ensemble statistics. The classical and quantum crossover points are different due to the noncommutativity of the infinite-time and infinite-spin-magnitude limits and the role played by entanglement in the quantum case, quantified via the two-subsystem entanglement entropy.

Figures (8)

• Figure 1: Largest Lyapunov exponent $\Lambda_L$ versus $\Gamma$ and $\Omega$ for initial conditions $m_j^z=1$ and $m_j^{x,y}=0$ for all $j$ (panel a), averaged over random initial conditions as given in Sec. \ref{['ini:sec']} with $a=0.1$ (panel b), and with $a=1$ (panel c). The averages are over $N_{\rm r}=50$ initial conditions. See the main text and Fig. \ref{['fig:correlations_2']} for a discussion about the symbols shown in panel (b).
• Figure 2: Largest Lyapunov exponent $\braket{\Lambda_L}$ (top row) and Pearson correlation coefficient $\overline{C_P}$ (bottom row) versus $\Gamma$ along the line $\Gamma + \Omega = 1.5$ (left column), $2\Omega + \Gamma = 2$ (center column), and $2\Omega + \Gamma = 1.5$ (right column). These values are calculated as averages over $N_{\rm r} = 100$ initial conditions generated on a spherical cap with parameter $a = 0.1$.
• Figure 3: Heat maps of various quantities versus $\Gamma$ and $\Omega$. (a,b) Pearson correlation coefficient. (c,d) Difference between the magnetizations averaged over time and initial conditions, $\langle \overline{\delta m} \rangle = \braket{\overline{m_B^z}}-\braket{\overline{m_A^z}}$. (e,f) Relative sign $s = \mathop{\mathrm{sign}}\nolimits(\braket{\overline{m_z^A}}\braket{\overline{m_z^B}})$ of the two magnetizations. The calculations are performed by averaging over $N_{\rm r}=100$ random initial conditions generated on a spherical cap with parameter $a = 0.1$ (first row), and $a = 1$ (second row). The lines in panel (f) are discussed in Sec. \ref{['qua:sec']}.
• Figure 4: Time and initial-condition average values of $m_j^z(t)$, $\braket{\overline{m_j^z}}$ versus $\Gamma$, along the parameter-space line $\Omega + \Gamma = 1.5$ (a, left column), $2\Omega + \Gamma = 2$ (b, center column), and $2\Omega + \Gamma = 1.5$ (c, right column). We average over a set of $N_{\rm r} = 100$ initial conditions generated either for $a = 0.1$ (top row) or $a=1$ (bottom row). The legend is shared. The background shadowing marks the Lyapunov exponent with the same color code as Fig. \ref{['fig:all_imageslyap_chain']}.
• Figure 5: (a,c) $\overline{S_A^z}$ and $\overline{S_B^z}$ versus $\Omega$ for different $N$. (b,d) Time-averaged connected correlator $\overline{C_{AB}^{zz}}$. Average over $N_{\rm r } \geq 48$ trajectories. On the left column we consider the curves for the line $2\Omega + \Gamma = 2$ and on the right one $2\Omega + \Gamma = 1.5$. The vertical line marks the classical staggered-uniform crossover as found in Fig. \ref{['fig:pearson_heat_map']}(f).