Quantum Synchronization of Perturbed Oscillating Coherences

Abstract

Synchronization in quantum systems has been recently studied through persistent oscillations of local observables, which stem from undamped modes of the dissipative dynamics. However, the existence of such modes requires fine-tuning the system to satisfy specific symmetry constraints. We investigate the response of spin systems that possess such oscillating modes to generic, weak perturbations. We show that even when these perturbations break the symmetry and lead to a single steady state, the phase correlations in the resulting state exhibit signatures of synchronization. Our results therefore connect the persistent oscillation notion (dynamical) and the notion based on phase correlations (steady-state) of synchronization, which so far have been regarded as distinct phenomena. Furthermore, we demonstrate that steady-state synchronization in these systems can exhibit features that are absent in the dynamical synchronization. Our work suggests robustness of synchronization and points toward a potential unifying framework of quantum synchronization.

Figures (5)

• Figure 1: The two domains of synchronization. At the perturbation strength $\eta=0$, oscillating coherences may be present either when the subsystems are decoupled ($\mathcal{L}_I = 0$) or when they are coupled through an engineered Liouvillian ($\mathcal{L}_I \neq 0$) that fulfills the criteria of dynamical synchronization (illustrated via the green region). Both cases exhibit steady-state synchronization (illustrated via the yellow region) once a generic perturbation $\eta>0$ is introduced, breaking the fine-tuned constraints of dynamical synchronization.
• Figure 2: (a), (d) Distribution $S_\mathrm{d}(\phi_1', \phi_2')$ [see below Eq. \ref{['equ:S_phi_spin1']}] of the angular differences $\phi_1' = \phi_1-\phi_3$, $\phi_2' = \phi_2-\phi_3$ for the spin-1 model with $N=3$. The distributions are for a single randomly picked sample at weak perturbation strengths $\eta \ll \omega$. Panel (a) corresponds to $c_{0+} > c_{0-}$ while (d) represents $c_{0+} < c_{0-}$. (b), (e) represent the angular configurations of the three spins that maximize $S_\mathrm{d}$. The maxima are marked in the panels (a) and (d) with yellow symbols, each type of which corresponds to an angular configuration drawn in the panels (b) and (e) below. (c), (f) represent the probability of occurrence of $\chi$ for $c_{0+} > c_{0-}$ and $c_{0+} < c_{0-}$, respectively.
• Figure 3: (a) Average distribution of $\chi$ as sampled from $S_\mathrm{d}(\phi_1', \phi_2')$ above $0.95\max S_\mathrm{d}$, at varying perturbation strengths $\eta$. At $\eta\ll \omega$, the distribution shows three distinct peaks at the $\chi$ values of the spin angle configurations illustrated in Fig. \ref{['fig:S_phis']}. (b) The probability $P_a$ that $\chi$ falls within the three same-sized intervals $\mathcal{I}_a$, $a=1, 2, 3$, that surround the three peaks, respectively, in the $\eta\ll\omega$ limit. (c) The ratio $P_2/(3P_3)$, which in the $\eta\ll\omega$ limit equals the ratio of $\pi/3$- to $(0, \pi)$-type of synchronization. (d) The distribution and average of the steady-state synchronization measure $S_\mathrm{max}$ defined in Eq. \ref{['equ:phase_measure_def']}. We randomize over $10^4$ samples and choose $\omega=1$, $J=\Delta=0.5$, and $\gamma=2$.
• Figure 4: The $l=\sqrt{\chi}$ values of the angular differences $\phi_1', \phi_2'$, each ranging from $-2\pi$ to $2\pi$. The region $\Omega_d$ at $d=1$ is enclosed in dashed lines and has an area equal to (as can be seen by shifting $\phi_1'$ or $\phi_2'$ by $2\pi$) that of the square $[0, 2\pi]\times [0, 2\pi]$, enclosed by the yellow solid lines. The dotted lines illustrate, at $d=1$, the separation of the region $\Omega_{d}$ into the triangles $T^a_d$.
• Figure 5: The model of two spin-$1/2$s [described in Eqs. \ref{['equ:spin1/2:H']} and \ref{['equ:spin1/2:L']}, with $J=1$, $\Delta = 1$, $B = 0.3$, $\gamma = 0.5$] under generic random perturbations at strenghs $\eta$. Data represent average over $10^4$ realizations of randomness. (a) The distribution of $\chi_2(\phi_1')$ characterizing the phase difference $\phi_1'=\phi_1-\phi_2$, sampled from angular configurations such that $S_\mathrm{d}$ achieves higher than $0.95 S_\mathrm{max}$. (b) The distribution of $S_\mathrm{max}$, the measure of steady-state synchronization, and its average, $\braket{S_\mathrm{max}}$.