The ability to tailor the hopping interactions between the constituent elements of a physical system could enable the observation of unusual phenomena that are otherwise inaccessible in standard settings 1,2 . In this regard, a number of recent theoretical studies have indicated that an asymmetry in either the short-or long-range complex exchange constants can lead to counterintuitive effects, for example, the possibility of a Kramer's degeneracy even in the absence of spin 1/2 or the breakdown of the bulk-boundary correspondence [3][4][5][6][7][8] . Here, we show how such tailored asymmetric interactions can be realized in photonic integrated platforms by exploiting non-Hermitian concepts, enabling a class of topological behaviors induced by optical gain. As a demonstration, we implement the Haldane model, a canonical lattice that relies on asymmetric long-range hopping in order to exhibit quantum Hall behavior without a net external magnetic flux. The topological response observed in this lattice is a result of gain and vanishes in a passive but otherwise identical structure. Our findings not only enable the realization of a wide class of non-trivial phenomena associated with tailored interactions, but also opens up avenues to study the role of gain and nonlinearity in topological systems in the presence of quantum noise.
We show that, in general, any complex weakly nonlinear highly multimode system can reach thermodynamic equilibrium that is characterized by a unique temperature and chemical potential. The conditions leading to either positive or negative temperatures are explicitly obtained in terms of the linear spectrum of the system, the input power, and the corresponding Hamiltonian invariant. Pertinent examples illustrating these results are provided in various scenarios.
By utilizing notions from statistical mechanics, we develop a general and self-consistent theoretical framework capable of describing any weakly nonlinear optical multimode system involving conserved quantities. We derive the fundamental relations that govern the grand canonical ensemble through maximization of the Gibbs entropy at equilibrium. In this classical picture of statistical photo-mechanics, we obtain analytical expressions for the probability distribution, the grand partition function, and the relevant thermodynamic potentials. Our results universally apply to any other weakly nonlinear multimode bosonic system.
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