Collective atomic recoil laser as a synchronization transition

Abstract

We consider here a model previously introduced to describe the collective behavior of an ensemble of cold atoms interacting with a coherent electromagnetic field. The atomic motion along the self-generated spatially periodic force field can be interpreted as the rotation of a phase oscillator. This suggests a relationship with synchronization transitions occurring in globally coupled rotators. In fact, we show that whenever the field dynamics can be adiabatically eliminated, the model reduces to a self-consistent equation for the probability distribution of the atomic "phases." In this limit, there exists a formal equivalence with the Kuramoto model, though with important differences in the self-consistency conditions. Depending on the field-cavity detuning, we show that the onset of synchronized behavior may occur through either a first- or second-order phase transition. Furthermore, we find a secondary threshold, above which a periodic self-pulsing regime sets in, that is immediately followed by the unlocking of the forward-field frequency. At yet higher, but still experimentally meaningful, input intensities, irregular, chaotic oscillations may eventually appear. Finally, we derive a simpler model, involving only five scalar variables, which is able to reproduce the entire phenomenology exhibited by the original model.

Synchronized states in chaotic systems coupled indirectly through a dynamic environment

“…Global oscillations of concentration of neurotransmitter released by each cell can stimulate collective rhythms in a population of circardian oscillators [29]. Moreover, in an ensemble of cold atoms interacting with a coherent electromagnetic field, by controlling field cavity detuning, synchronized behavior with self-pulsating periodic and chaotic oscillations are found to occur [30]. In all these cases, the coupling function has a dynamics modulated by the system dynamics.…”

Irregular Collective Behavior of Heterogeneous Neural Networks

“…A second difference is that, at variance with the KM, the coupling contributes also to slowing down the spiking activity of the single neurons (a somehow similar mechanism operates in ensemble of cold atoms [13]) and drives a subset of neurons below the firing threshold -a phenomenon reminiscent of oscillator-death [14]. However, the most striking difference concerns the above-threshold regime, as the overall neural activity is not simply periodically modulated, but exhibits irregular, seemingly chaotic, oscillations (still in the presence of a negative "microscopic" second Lyapunov exponent).…”

“…On the other hand, the study of relatively small networks shows that the time needed to approach a periodic orbit is exponentially long with the system size, implying that the "transient" extends over increasingly longer time scales. In other words, this is an instance of stable chaos [9], a phenomenon already detected in networks of pulse coupled oscillators without delay [10], although its onset in systems with delayed coupling is controversial [11,12].A second difference is that, at variance with the KM, the coupling contributes also to slowing down the spiking activity of the single neurons (a somehow similar mechanism operates in ensemble of cold atoms [13]) and drives a subset of neurons below the firing threshold -a phenomenon reminiscent of oscillator-death [14]. However, the most striking difference concerns the above-threshold regime, as the overall neural activity is not simply periodically modulated, but exhibits irregular, seemingly chaotic, oscillations (still in the presence of a negative "microscopic" second Lyapunov exponent).…”

Self-organization in cold atomic gases: a synchronization perspective

“…Broadly speaking, the spatial ordering into a periodic structure and the synchronization of oscillators on a limit cycle can be thought of as the same phenomenon, if the extended nature of the spatial coordinate is ignored and only the spatial phase is considered. Implementing light-mediated atom-atom interactions opens the possibility for tunable and controllable realizations of long-range interacting and mean-field models for synchronization [20], and indeed this was exploited in [7,21] to connect the viscous CARL dynamics to the Kuramoto model for synchronization of coupled oscillators [22]. We show in the following that this connection is not limited to CARL, but applies also to the symmetry-breaking transverse instabilities studied in [15,16].…”

“…We show in the following that this connection is not limited to CARL, but applies also to the symmetry-breaking transverse instabilities studied in [15,16]. Moreover, the connection made in [7,21] referred to the case where strong damping is present in the system (hence the denomination 'viscous' CARL), which in the Kuramoto analogy translates into the case where the oscillators have zero natural frequencies (their distribution is a Dirac delta function). We extend here the Kuramoto analogy to the situation analysed in [15,16], where no damping is present and a finite spread exists in the natural frequency spread of the fictitious oscillators.…”