Koopman analysis of nonlinear systems with a neural network representation

The observation and study of nonlinear dynamical systems has been gaining popularity over years in different fields.The intrinsic complexity of their dynamics defies many existing tools based on individual orbits,while the Koopman operator governs evolution of functions defined in phase space and is thus focused on ensembles of orbits,which provides an alternative approach to investigate global features of system dynamics prescribed by spectral properties of the operator.However,it is difficult to identify and represent the most relevant eigenfunctions in practice.Here,combined with the Koopman analysis,a neural network is designed to achieve the reconstruction and evolution of complex dynamical systems.By invoking the error minimization,a fundamental set of Koopman eigenfunctions are derived,which may reproduce the input dynamics through a nonlinear transformation provided by the neural network.The corresponding eigenvalues are also directly extracted by the specific evolutionary structure built in.

Quantum coherence transfer between an optical cavity and mechanical resonators

This study highlights the theoretical investigation of quantum coherence in mechanical oscillators and its transfer between the cavity and mechanical modes of an optomechanical system comprising an optical cavity and two mechanical oscillators that,in this study,were simultaneously coupled to the optical cavity at different optomechanical coupling strengths.The quantum coherence transfer between the optical and mechanical modes is found to depend strongly on the relative magnitude of the two optomechanical couplings.The laser power,decay rates of the cavity and mechanical oscillators,environmental temperature,and frequency of the mechanical oscillator are observed to significantly influence the investigated quantum coherences.Moreover,quantum coherence generation in the optomechanical system is restricted by the system's stability condition,which helps sustain high and stable quantum coherence in the optomechanical system.

Optomechanical ratchet resonators

We describe an optomechanical ratchet scheme to realize nonreciprocal transmission of a light field, which is based on the bias of the optical cavity’s frequency spectrum caused by mechanical ratchet interactions. This approach to break the time-reversal symmetry of light propagation is universally valid in various optomechanical systems with ratchet-oscillator structures. We discuss specifically the implementation of an on-chip Casimir-ratchet optomechanical protocol and demonstrate the optical nonreciprocity with an extremely high isolation ratio and flexible controllability, which does not require external additional optical engineering. Our study opens a door for manipulating flexibly light propagation by using mechanical ratchet resonators, and has potential applications in the on-chip integration of nonreciprocal devices and harness of lateral Casimir forces.