The Research On The Eigenstates Of Periodically Driven Two-Level Atomic Systems

Periodically driven quantum systems have attracted extensive attention of researchers recently,because periodically driving is an effective means to achieve coherent control,and has shown broad application prospects in quantum precision measurement,quantum simulation,quantum system control and other fields.Floquet theory is a commonly used theory to deal with periodic driven quantum systems.Based on this theory,people can implement Floquet engineering in quantum systems.Based on Floquet theory,the eigenstates of a periodically driven two-level atomic system are studied in detail in this paper.Firstly,for a general periodically driven quantum system,we introduce two methods based on Floquet theory.One is to map the time-dependent Schrodinger equation into the time independent quasienergy eigenequation based on the periodic Fourier expansion of the function.The other is to solve the Schrodinger equation in the extended Hilbert space based on the composite basis vector.Using these methods,we can change the timedependent problem into the time-independent problem,and further discuss the periodicity of quasienergy and other system characteristics.Then,for a specific periodically driven two-level atom system,we further define the specific form of the composite basis vector as the direct product of the eigenstates of free atoms and Fourier series.Using Floquet theory to project the wave function and Hamiltonian to the new basis vector,we can obtain the eigenequation of the Floquet Hamiltonian and the quasienergy of the system.We assume that the eigenstates are related to Bessel functions.The results show that the coupling coefficient between the two levels is only the same as the Fourier series of the same order and the coupling between sublevels of different Fourier orders caused by periodic driving is related to the Bessel function.When the coupling coefficient is zero,the decoupling can be carried out using the recursive formula of the Bessel function,it is proved that the eigenstates are Bessel functions and the results agree well with the numerical results.For the case where the coupling coefficient between different energy level is not zero,we have carried out numerical calculation.By calculating the change of quasi-energy with coupling coefficient,it is found that the level structure show crosses at some values of g.For Hamiltonian in the numerical calculation of truncation problem,we use the basis vector modulated by the Bessel function and re-transform the Hamiltonian.According to the properties of Bessel function,the effective order of the truncation of the Hamiltonian under the new basis vector is related to the amplitude of the periodically driving.Therefore,the solution of infinite-dimensional matrix can be equivalent to the eigenstate problem of finite-dimensional matrix.In conclusion,this article simply reviews the periodically driven quantum systems Floquet theory used in two ways,and make use of these methods in the composite Hilbert space.Using these method in the composite Hilbert space,we study the eigenstate of periodically driven two-level atom system problems,proves that the coupling coefficient between level is zero,the eigen state of the system can be represented with Bessel function,in the case of coupling coefficient between non-zero level,the effective Hamiltonian truncation order number related to the driving range.