Effective Hamiltonians in periodically driven systems have received widespread attention for realization of novel quantum phases,non-equilibrium phase transition,and Majorana mode.Recently,the study of effective Hamil-tonian using various methods has gained great interest.We consider a vector differential equation of motion to derive the effective Hamiltonian for any periodically driven two-level system,and the dynamics of the spin vector are an evolution under the Bloch sphere.Here,we investigate the properties of this equation and show that a sudden change of the effective Hamiltonian is expected.Furthermore,we present several exact relations,whose expressions are independent of the different starting points.Moreover,we deduce the effective Hamiltonian from the high-frequency limit,which approximately equals the results in previous studies.Our results show that the vector differential equation of motion is not affected by a convergence problem,and thus,can be used to numeri-cally investigate the effective models in any periodic modulating system.Finally,we anticipate that the proposed method can be applied to experimental platforms that require time-periodic modulation,such as ultracold atoms and optical lattices.

Long Xiong;Wei-Feng Zhuang;Ming Gong

CAS Key Lab of Quantum Information,School of Physics,University of Science and Technology of China,Hefei 230026,China;Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China,Hefei 230026,China

中国物理快报（英文版）

[1]Long Xiong;Wei-Feng Zhuang;Ming Gong-.Dynamics of Quantum State and Effective Hamiltonian with Vector Differential Form of Motion Method)[J].中国物理快报（英文版）,2022(07):36-40

Hamiltonians,tonian

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