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Shortcut to adiabaticity (STA) is a speed way to produce the same final statethat would result in an adiabatic, infinitely slow process. Two typicaltechniques to engineer STA are developed by either introducing auxiliarycounterdiabatic fields or finding new Hamiltonians that own dynamicalinvariants to constraint the system into the adiabatic paths. In this paper, aconsistent method is introduced to naturally connect the above two techniqueswith a unified Lie algebraic framework, which neatly removes the requirementsof finding instantaneous states in the transitionless driving method and theinvariant quantities in the invariant-based inverse engineering approach. Thegeneral STA schemes for different potential expansions are concisely achievedwith the aid of this method.

Lie transformation on shortcut to adiabaticity in parametric driving quantum system