Synchronization effects in a periodically driven two-level system

TL;DR

This work analyzes phase synchronization in a periodically driven two-level system coupled to a non-Markovian bosonic bath, solving the full dynamics without RWA using the numerically exact HEOM method. The key finding is that synchronization robustly emerges when the drive amplitude-to-frequency ratio Ω/ω coincides with a zero z_k of the Bessel function J_0, i.e., ω = Ω / z_k, causing the static term in the rotating-frame Hamiltonian to vanish and the system–bath coupling to commute, which preserves Re{c(t)} and enables a stationary phase reference. The authors connect this resonant-ratio condition to a degeneracy of Floquet quasienergies and demonstrate, via phase-space measures based on the Husimi Q-function, that a nonzero synchronization measure S(φ,t) develops and stabilizes at long times under the resonance. The results show that resonance-induced coherence preservation and synchronization are robust to non-Markovian bath parameters, offering a simple mechanism for resilient quantum control in open driven systems across various settings.

Abstract

We study phase-synchronization in a driven two-level system coupled to a non-Markovian bosonic reservoir. The dynamics is described by treating the system-bath coupling and the coherent drive without invoking the rotating-wave approximation, and simulated using the numerically exact hierarchical equations of motion. We observe that a robust phase-locking develops and that the corresponding synchronization measure rapidly acquires a finite value when the system is tuned to what we identify as a resonant-ratio condition, namely when the ratio between the drive amplitude and its frequency coincides with a zero of the Bessel function $J_0$. We provide an explanation for this phenomenon by means of a static approximation derived from a Fourier analysis of the periodically driven Hamiltonian.

Figures (6)

• Figure 1: Time evolution of the reduced system under the Hamiltonian of Eq. \ref{['eq:Htot']} for an environment initially in a thermal state, plotted as trajectories on the Bloch sphere. The model parameters are set to $\Delta=0$, $\Omega=60\,\omega_0$, $\gamma=\omega_0/2$, $\lambda=\omega_0$, and $T=0.5$. The final time of the simulation is $\omega_0\, t_{\rm max}=30$. Lighter colors correspond to later times, and the starting point of each trajectory is denoted with a dot. The trajectories converge to a limit cycle around the $x$ axis retaining the same $x$ component as the initial state.
• Figure 2: Dynamics of $Q$ as a function of the angles $\theta$ and $\varphi$. Top to bottom: $\omega = \omega_{\text{rrc}}-\delta\omega$, $\omega=\omega_{\text{rrc}}$, and $\omega = \omega_{\text{rrc}}+\delta\omega$. Left to right: initial, intermediate, and final time. We fix $\delta\omega = 10\omega_0$. The other parameters are the same as in Fig. \ref{['fig:trajs']}.
• Figure 3: Time evolution of the angles $\theta$, $\varphi$ maximizing $Q$. Time is encoded in the color (gray) gradient of the points, with darker tones corresponding to earlier times. Left to right: $\omega=\omega_{\text{rrc}}-\delta\omega$, $\omega=\omega_{\text{rrc}}$, $\omega=\omega_{\text{rrc}}+\delta\omega$.
• Figure 4: Time evolution of $\max_\varphi \lvert S(\varphi,t) \rvert$ for $\omega=\omega_{\text{rrc}}$ and $\omega=\omega_{\text{rrc}}\pm\delta\omega$. Inset: $S(\varphi,t)$ as a function of $\varphi$ evaluated at the final time $t=t_{\text{max}}$.
• Figure 5: Maximum of $\lvert S(\varphi, t_{\text{max}}) \rvert$ as a function of $\Omega$ and $\omega$. The white lines are the first three RRCs $\omega = \omega_{\text{rrc}}, \omega_{\text{rrc}}^2,\omega_{\text{rrc}}^3$. The white dots and crosses correspond to the parameters used in Figs. \ref{['fig:Q']}-\ref{['fig:S_max']}. In order to improve the stability, the simulation is done by taking an average around $t_{\text{max}}$, in a time window comparable to the duration of a limit cycle. Left panel: $\Delta = 0$, right panel: $\Delta = 0.2\hbar\omega_0$.