Synchronization Dynamics in a Designed Open System

We theoretically propose a unifying expression for synchronization dynamics between two-level constituents. Although synchronization phenomena require some substantial mediators, the distinct repercussions of their propagation delays remain obscure, especially in open systems. Our scheme directly incorporates the details of the constituents and mediators in an arbitrary environment. As one example, we demonstrate the synchronization dynamics of optical emitters on a dielectric microsphere. We reveal that the whispering gallery modes (WGMs) bridge the well-separated emitters and accelerate the synchronized fluorescence, known as superfluorescence. The emitters are found to overcome the significant and nonuniform retardation, and to build up their pronounced coherence by the WGMs, striking a balance between the roles of resonator and intermediary. Our work directly illustrates the dynamical aspects of many-body synchronizations and contributes to the exploration of research paradigms that consider designed open systems.

Figure 1

(a) Schematic view of four-level optical emitters randomly positioned on a polystyrene sphere, and an energy diagram of them. We assume that all the dipoles are aligned in one direction. Utilizing the pump light resonating with none of the WGMs, we first excite the higher state (or continuum) $|E{⟩}_i$, which relaxes into the state $|1{⟩}_i$ by fast and radiationless decay. This prepares the initial state, whereby each two-level state $\{|0{⟩}_i,|1{⟩}_i\}$ is incoherent and population inverted. (b) Scattering cross section of the polystyrene sphere plotted against its radius $r_s$. Each peak corresponds to the WGM that resonates with the $|0{⟩}_i-|1{⟩}_i$ excitations.

Figure 2

Fluorescent dynamics from the optical emitters. Here, $I_1=1/\tau$ is the fluorescence intensity from an isolated emitter in a vacuum. (a) Spontaneous emission accelerated by the resonator effect of the WGM in the polystyrene sphere ($r_s/\lambda =3.184$). Inset: We plot the synchronized fluorescence (superfluorescence) where all the emitters are close packed in a vacuum. (b) Synchronized fluorescence when the polystyrene sphere radii is $r_s/\lambda =3.112$. Inset: We plot the intensity when the microsphere is removed. (c) The same plot for the case using the sphere with $r_s/\lambda =2.461$.

Figure 3

(a) Delay times of the synchronization ${\tau }_d$ in the case of close-packed emitters and of those on the polystyrene spheres. The red line represents the analytical result ${\tau }_d^{\mathrm{DS}}/\tau =(\ln N)/N$, whereas the other two lines represent constant multiples of the red line to fit the plots. (b) Influence of the resonant-level distribution for the $N=100$ emitters. The radii of the polystyrene spheres are $r_s/\lambda =3.112$. (c) The same plot for $r_s/\lambda =2.461$. Here, $\sigma$ is the standard deviation from the peak resonant energies 3.0 eV, and $\mathrm{\Delta }{E}_{\mathrm{WGM}}$ is the full width of the corresponding WGM.