Realizing spin Hamiltonians in nanoscale active photonic lattices

Abstract

†These authors contributed equally to this work.Spin models arise in the microscopic description of magnetic materials, where the macroscopic characteristics are governed by exchange interactions among the constituent magnetic moments. Recently, there has been a growing interest in complex systems with spin Hamiltonians1-3 -largely due to the rich behaviors exhibited by such interactions at the macroscale. Along these lines, it has been shown that certain classes of optimization problems involving large degrees of freedom can be effectively mapped into classical spin models. In this vein, the respective extremum can be found by identifying the ground state of the associated spin Hamiltonian. Here, we show both theoretically and experimentally, that the cooperative interplay among vectorial electromagnetic modes in coupled metallic nanolasers4-7 can be utilized as a means to emulate certain types of spin-like systems. The ensuing spin exchange interactions are in general anisotropic, in a way similar to that encountered in magnetic materials involving spin-orbit coupling. For some topologies, we find that these active nanophotonic structures are governed by a classical XY Hamiltonian that exhibits two phases akin to those associated with ferromagnetic (FM) and antiferromagnetic (AF) materials. In addition, we show that in certain configurations, the electromagnetic field distribution can undergo geometrical frustration, depending on the lattice shape and the transverse resonant modes supported by the individual cavity elements. Our results could pave the way towards a new scalable nanophotonic platform to study spin exchange interactions, that can in turn be potentially exploited to investigate more large-scale networks, emulate some magnetic materials, or to address a variety of optimization problems.Spin Hmailtonians arise in a ubiquitous manner in nature. Over the years, such models have been a subject of intense investigation in many and diverse areas of science and technology, ranging from condensed matter physics8 and spintronics9 to quantum information theory3. Of particular interest is geometric frustration that occurs when a certain type of local order, associated with a minimum energy state, cannot extend throughout a system due to geometrical constraints10. This effect appears in a variety of physical problems and settings, ranging from residual entropy in water11 and spin ice12,13, to orbital exchange in Mott insulators14 and the emergence of the blue phases in cholesteric liquid crystals15. In magnetic materials, frustration is typically associated with a set of highly degenerate ground states of a spin Hamiltonian, which in turn leads to complex macroscopic behaviors such as those observed in spin-liquid or spin-ice phases16.

Optical Potts machine through networks of three-photon down-conversion oscillators

“…An interesting approach is to use physical systems as analog computing platforms that emulate spin lattice models, which allow for solving many computationally hard combinatorial problems [2]. Gain-dissipative optical systems have recently attracted much interest as a physical platform for simulating spin lattice models [3][4][5][6][7][8][9][10][11][12][13][14]. In particular, networks of coupled optical parametric oscillators have been used for optically implementing the Ising Hamiltonian [5].…”

Non-Hermitian and topological photonics: optics at an exceptional point

“…Non-Hermiticity and loss management in active lattices is shown to be an effective tool for implementing various spin Hamiltonians in an optical platform [63][64][65][66][67]. In these schemes, the loss that vectorial electromagnetic modes experience on the interface of metallic nanocavities provides an effective way to establish the ferromagnetic and antiferromagnetic type of exchanges.…”

“…In these schemes, the loss that vectorial electromagnetic modes experience on the interface of metallic nanocavities provides an effective way to establish the ferromagnetic and antiferromagnetic type of exchanges. Based on such arrangements, large arrays of nanolasers have been realized, which emit in a single mode and with a desired topological singularity [66,67]. In laser arrays, the interplay between supersymmetry (SUSY) and non-Hermiticity has also been shown to result in single spatial mode operation.…”

“…This precludes a direct generation of a vortex beam with nonzero topological charge since the two opposite azimuthal mode numbers tend to cancel each other. NH schemes based on EPs have recently provided an elegant method to selectively extract only one chiral mode in such lasers [52,[60][61][62][63][64][65][66][67][68][69][70][71][72]. This class of devices can emit in a tunable OAM order while operating in a broadband fashion.…”

Complexity continuum within Ising formulation of NP problems

“…The recent advances in development of physical platforms for optimising combinatorial optimisation problems reveal the future of high-performance computing for the quantum and classical devices. Unconventional computing architectures were proposed for numerous systems including superconducting qubits [1][2][3], CMOS hardware [4], optical parametric oscillators [5,6], memristors [7], lasers [8][9][10], photonic simulators [11,12], trapped ions [13], polariton [14,15] and photon [16] condensates. An attractive opportunity to show the advantageous performance of one system over others becomes a demonstration of the platform's ability to optimise nondeterministic polynomial time (NP) problems that are computationally intractable for the traditional von Neumann architecture machines.…”