Parameter Regulation Of Simple Time-dependent Hamiltonian Systems

In recent years,due to the rapid development of science and technology,there is no technical difficulty in fabricating artificial metamaterials composed of various special micro-nano structural units.However,the precise driving and control on these micro-nano units are the basic premise for the realization of the final function of quantum devices.The control or drive on the quantum systems is time-dependent dynamics essentially,and it should be dealt with the time-dependent quantum theory.The traditional quantum timedependent theory is based on perturbation theory,it requires a very weak actuation.And then developed the adiabatic theory,although it overcomes the limitation of weak driving,it still requires the change for Hamiltonian to be extremely slow.However,for a general time-dependent Hamiltonian system,the time-dependent dynamic evolution is often hard to be solved strictly.Researchers have proposed many solutions to solve this problem,one scheme is to solve the problem of time-dependent Hamiltonian systems by using the corresponding time-dependent transformation method of Lie algebra according to some symmetries of the time-dependent Hamiltonian systems.Another effective scheme is to solve the analytic solution of the evolution operator by diagonalizing the Hamiltonian of the system.Thus,the dynamic evolution of the system can be solved strictly under the framework of the non-perturbation theory.In this thesis,the advantages of the above two schemes are combined to construct the time-dependent Hamiltonian that can be reversely solved by the Lie transformation method.The reversely designed method is then applied to the shortcuts to adiabaticity(STA)and quantum logic gate control.The contents are as follows:1.A unified method under the SU(2)algebraic framework is developed to design shortcuts to adiabaticity in both transitionless quantum driving and invariant-based inverse engineering problems.This method neatly removes the requirements of finding instantaneous states in the transitionless driving method and the invariant quantities in the invariant-based inverse engineering approach.Taking harmonic trap expansion and a free particle inside an expanding one-dimensional box as examples,the design processes are shown in detail.The general STA schemes for different potential expansions are concisely achieved with the aid of this method.The Lie transformation method presents a powerful tool to design STA for a general time-varying parameter-driven system.2.A simple yet versatile protocol to inversely engineer time-dependent Hamiltonians is proposed.By utilizing SU(2)transformation,a speedup goal of gate operation is achieved and which satisfies both the parallel transport condition and the cyclic evolution condition for the geometric gate operations.As an application,this protocol is adopted to realize the conventional and unconventional nonadiabatic geometric quantum gates with desired evolution paths by controlling the microwave pulses in the diamond nitrogenvacancy center system.Meanwhile,by using the basic definition of linear algebra,the SU(2)transformation is generalized to a general unitary transformation,and the parameter control of a high-fidelity nonadiabatic geometric quantum gate is realized by a more general reverse engineering design method...