Circuit Simulation Of Non-hermitian System

In the basic assumptions of quantum mechanics,we assume that the Hamiltonian H and the mechanical quantity operator are Hermitian,that is,~+=.The eigenvalues of the corresponding operators obtained from the Schrodinger equation are real numbers,which correspond to the observable measurements in the experiment.Hermitian Hamiltonians are often related to isolated systems,but in practice,open systems are very common.Describing such systems requires non-Hermitian Hamiltonians,such as systems with gain or loss in optical systems,systems with finite-lifetime particles that decay with time relaxation,etc.When the Hamiltonian no longer satisfies the Hermitian symmetry,the eigenvalue may no longer be a real number.Besides that,there will be some other novel phenomena.For example,the occurrence of energy exceptional points(EPs)in non-Hermitian systems,and in non-Hermitian non-reciprocal systems,the conventional bulk–boundary correspondence is broken,and the bulk states are all localized to the boundary(non-Hermitian skin effect).Non-Hermitian topological systems have attracted more and more attention.The first chapter will introduce the concepts of non-Hermitian and non-reciprocity,and mention the importance of PT symmetry.Because the skin effect produced by non-Hermitian is closely related to the generalized Brillouin zone,it will also be briefly introduced in the first chapter.For the simulation of physical phenomena,it is very important to use an appropriate platform.Due to the flexibility and simplicity of design,circuit system has become a powerful platform for studying topology.The second chapter will introduce the Laplace transform of the circuit system and the instruments and principles used in the experiment.With the development of research,the interaction of non-Hermitian systems,non-Hermitian skin effect(NSHE)and other quantum/topological phenomena attract attention.For example,whether the NSHE can still exist when it encounters localization items.In the third chapter,the skin effect is realized by non-reciprocal circuit.Different from the traditional non-reciprocal circuit system,we realize non-reciprocity by adjusting the numerical comparison relationship of electronic components.Previously,the circuit scheme to realize non-reciprocity was mainly based on logic devices such as operational amplifiers or diodes.In addition,when we add an inductance representing Anderson localized phase to the circuit system,the platform simulates the interplay of non-Hermitian skin effect and Anderson localization.The third and fourth chapters of this paper will study the non-Hermitian skin effect,size-dependent boundary effects,adjustable high-order EP and the interplay of skin effect and Anderson localization respectively from two circuit system design methods based on Hatano-Nelson model(HN model).The simulation of non-Hermitian skin effect adopts the way of dynamic measurement to obtain static information,while the high-order EP uses the construction of non-reciprocal admittance matrix to replace Hamiltonian.The two circuit systems realize non-reciprocity in different ways.The former is based on the proportion of parameters of adjacent devices,and the latter uses an operational amplifier.The fifth chapter will introduce the superconducting qubit measurement system,the basic characterization of a single superconducting qubit,and the optimization of measurement equipment I have done in it,including superconducting bias tee exploration,copper powder and iron powder filters,and the final results.In addition,we propose a preliminary scheme based on two qubits non-Hermitian system simulation.The research on non-Hermitian systems is still in its infancy.In the near future,a large number of theories are proposed,and how to verify them with experimental simulation is also crucial.In the subsequent research on the critical skin effect and high-dimensional non-Hermitian system,we believe that the circuit system will be a potential platform.At the same time,the superconducting quantum system will also become a powerful platform for the study of PT symmetry in non-Hermitian systems.