The Importance of The Difference in Adiabatic Phases in Non-Cyclic Evolution on Various Time Scales

To understand the importance of the interference effect of non-physical non-cyclic phases difference on the expectation value of physical operators that do not commute with the time-dependent adiabatic Hamiltonian, we compare the value of ⟨ ψ ( t ) | s z | ψ ( t ) ⟩ of the exact soluble two-level model with its adiabatic approximation in a non-cyclic evolution. This expectation value is calculated in different orders of magnitude of time periods. For time intervals of the same order as the adiabatic Hamiltonian’s period, the inability to account for the contribution of non-physical, non-cyclic adiabatic phases in the temporal evolution of this quantum system may introduce a percentage disagreement of up to 60 % with the precise result of measurable physical quantities. The results obtained in the article are independent of the Hamiltonian instantaneous eigenstate basis that we use to decompose the vector state in adiabatic evolution. For times greater than any integer number of periods T 0 , in which the classical parameters return to their initial values, we only obtain the correct result of the expected value of the physical operators in state vectors with adiabatic evolution when we include in the vector state its Berry phase plus the non-cyclic phase acquired in the elapsed time interval after this integer number of periods T 0 . This simple model emphasizes the relevance of the interference effects of physical or non-physical adiabatic phases in calculating the time evolution of the expectation value of physical operators in an adiabatically evolving quantum state with non-cyclic development.