Algorithm Implementation Of The Fourth Order Steklov Resource Problem And Nonlocal Nonreciprocal Transmission In Hybrid Optomechanical System

In this paper,we propose an algorithm implementation of the fourth order Steklov resource problem and nonlocal nonreciprocal transmission in hybrid optomechanical system.For the fourth-order Steklov resource problem,an effective spectral Galerkin approximation and error estimation based on tensor product are studied.Firstly,we introduce an appropriate Sobolev space and derive the weak form of the original problem and the corresponding discrete scheme.Secondly,based on the Lax-Milgram lemma,we prove the existence and uniqueness of the weak solution and the approximation solution.We further prove the error estimate of approximation solution by using the approximation properties of the orthogonal projection operators.In addition,we construct a set of appropriate basis functions in the approximation space and derive the matrix form based on tensor product of the discrete scheme.Finally,some numerical examples are given to show the effectiveness of the algorithm and the correctness of the theoretical results.For the nonlocal circulator protocol is proposed in hybrid optomechanical system.By analogy with quantum communication,using the input-output relationship,we establish the quantum channel between two optical modes with long-range.The three body nonlocal interaction between the cavity and the two oscillators is obtained by eliminating the optomechanical cavity mode.By introducing the phase accumulation between cyclic interactions,the unidirectional transmission of quantum state between optical mode and two mechanical modes are realized.The results show that nonreciprocal transmissions are achieved as long as the accumulated phase reaches a certain value.In addition,the effective interaction parameters in our system are amplified,which reduces the difficulty of the implementation of our protocol.Our research can provide potential applications for nonlocal manipulation and transmission control of quantum platforms.