Energetic cost for speedy synchronization in non-Hermitian quantum dynamics

TL;DR

The paper develops a framework to quantify finite-time quantum synchronization in continuous-variable systems under non-Hermitian, anti-PT-like interactions, connecting synchronization to thermodynamic resources. It introduces a scale-invariant distance measure for synchronization, derives quantum speed-limit-like bounds, and formulates a dissipative quantum master equation whose rate of synchronization is governed by competition between damping-induced entropy production and non-Hermitian coupling strength. By expressing a rigorous quantum–classical correspondence, it provides a classical master equation and rate bounds, and demonstrates a quantum advantage in a photonic dimer where quantum synchronization extends beyond the classical regime. The results offer experimentally testable predictions in photonic systems and reveal a direct information-theoretic interpretation of synchronization as a resource-costed communication process.

Abstract

Quantum synchronization is crucial for understanding complex dynamics and holds potential applications in quantum computing and communication. Therefore, assessing the thermodynamic resources required for finite-time synchronization in continuous-variable systems is a critical challenge. In the present work, we find these resources to be extensive for large systems. We also bound the speed of quantum and classical synchronization in coupled damped oscillators with non-Hermitian anti-PT-symmetric interactions, and show that the speed of synchronization is limited by the interaction strength relative to the damping. Compared to the classical limit, we find that quantum synchronization is slowed by the non-commutativity of the Hermitian and anti-Hermitian terms. Our general results could be tested experimentally and we suggest an implementation in photonic systems.

Figures (4)

• Figure 1: Quantum ($\chi_{\rm min}$) and classical ($\chi_{\rm min}^{\rm (cl)}$) lower bounds on $\chi$, and convex hull ($\tilde{\chi}_{\rm min}$) of $10^6$ random Gaussian states (1000 states plotted as colored circles). Convex hull is the same for quantum and classical sample states.
• Figure 2: Trajectories in the $\chi$-$D$ plane with $k$ values (from left to right) $k=5$, $k=3$, $k=1$, $k=0.5$, $k=-0.5$, $k=-1$, $k=-3$ are shown in (a) Classical (dotted line) and quantum (dashed line) lower bounds on $\chi$, and convex hull of $10^6$ random Gaussian states (solid line). Time-dependence of $D$ (solid lines) and $\chi$ (dashed lines) with $k$ values (solid lines bottom to top, dashed lines top to bottom) $k=5$, $k=3$, $k=1$, $k=0.5$ in (b) The frequencies of the two oscillators are $\omega_1 = 2\pi$, $\omega_2 = 3\pi$. Both bath temperatures are set to $T=20$. The coupling strengths $\gamma_+= 0.001$ and $\gamma_-$ is determined via $\beta$ and local detailed balance condition as described in the supplemental material noauthor_see_nodate.
• Figure 3: Distance measure $D$ at time $t=10$ as a function of $k$ and $\omega_2-\omega_1$ for quantum (a) and classical (b) evolution. Dashed lines are $k = \pm (\omega_2 - \omega_1)$. Classical synchronous regime is for $k>\left|\omega_2-\omega_1\right|$ (within the dashed lines), whereas for a quantum system the synchronizing regime extends well beyond the classical bounds. All other parameters are same as Fig. \ref{['fig2']}.
• Figure 4: (Color online) Distance measure $D$ at time $t = 10$ for quantum (a) and classical (b) evolution, as a function of the real and imaginary part of $k$. Frequencies are $\omega_1 = \pi$, $\omega_2 = 2\pi$. Classical synchronous regime is $\Re[k]|>\pi$ for $\Im[k]=0$. Quantum system synchronizes for smaller $\Re[k]$.