Coherent Ising machines (CIMs) constitute a promising approach to solve computationally hard optimization problems by mapping them to ground state searches of the Ising model and implementing them with optical artificial spin-networks. However, while CIMs promise speed-ups over conventional digital computers, they are still challenging to build and operate. Here, we propose and test a concept for a fully programmable CIM, which is based on opto-electronic oscillators subjected to self-feedback. Contrary to current CIM designs, the artificial spins are generated in a feedback induced bifurcation and encoded in the intensity of coherent states. This removes the necessity for nonlinear optical processes or large external cavities and offers significant advantages regarding stability, size and cost. We demonstrate a compact setup for solving MAXCUT optimization problems on regular and frustrated graphs with 100 spins and can report similar or better performance compared to CIMs based on degenerate optical parametric oscillators.

For a globally coupled network of semiconductor lasers with delayed optical feedback, we demonstrate the existence of chimera states. The domains of coherence and incoherence that are typical for chimera states are found to exist for the amplitude, phase, and inversion of the coupled lasers. These chimera states defy several of the previously established existence criteria. While chimera states in phase oscillators generally demand nonlocal coupling, large system sizes, and specially prepared initial conditions, we find chimera states that are stable for global coupling in a network of only four coupled lasers for random initial conditions. The existence is linked to a regime of multistability between the synchronous steady state and asynchronous periodic solutions. We show that amplitude-phase coupling, a concept common in different fields, is necessary for the formation of the chimera states. Synchronization is a common phenomenon in interacting nonlinear dynamical systems in various fields of research such as physics, chemistry, biology, engineering, or socioeconomic sciences [1][2][3]. While a lot of knowledge has been gained on the origin of complete synchronization, more complex partial synchronization patterns have only recently become the focus of intense research. We still lack a full understanding of these phenomena, and a very prominent example are chimera states where an ensemble of identical elements self-organizes into spatially separated coexisting domains of coherent (synchronized) and incoherent (desynchronized) dynamics [4,5]. Since their first discovery a decade ago many theoretical investigations of coupled phase oscillators and other simplified models have been carried out [6,7] [7,16,17], e.g., spiral wave chimeras [18,19], FitzHugh-Nagumo neural systems [20], Stuart-Landau oscillators [21][22][23], where pure amplitude chimeras [24] were found, or quantum interference devices [25]. In real-world systems chimera states might play a role, e.g., in the unihemispheric sleep of birds and dolphins [26], in neuronal bump states [27,28], in power grids [29], or in social systems [30].Although no universal mechanism for the formation of chimera states has yet been established, three general essential requirements have been found in many studies: (i) a large number of coupled elements, (ii) non-local coupling, and (iii) specific initial conditions. These were primarily derived from the phase oscillator model [7] but also apply to other systems. If these conditions are not met, the chimera states tend to have very short lifetimes. Recent studies, however, suggest that these paradigms can be broken and chimera states are observed also for small system sizes [31], global coupling [15,23,32,33] and random initial conditions [34].Surprisingly, chimera states appear at the interface of independent fields of research putting together different scientific communities. Recent examples are quantum chimera state [35] or coexistence of coherent and incoherent patterns with respect to the modes of an optical com...

This corrects the article DOI: 10.1103/PhysRevE.91.040901.

Many problems in mathematics, statistical mechanics, and computer science are computationally hard but can often be mapped onto a ground-state-search problem of the Ising model and approximately solved by artificial spin-networks of coupled degenerate optical parametric oscillators (DOPOs) in coherent Ising machines. To better understand their working principle and optimize their performance, we analyze the dynamics during the ground state search of 2D Ising models with up to 1936 mutually coupled DOPOs. For regular as well as frustrated and disordered 2D lattices, the machine finds the correct solution within just a few milliseconds. We determine that calculation performance is limited by freeze-out effects and can be improved by controlling the DOPO dynamics, which allows to optimize performance of coherent Ising machines in various tasks. Comparisons with Monte Carlo simulations reveal that coherent Ising machines behave like low temperature spin systems, thus making them suitable for optimization tasks.