Specifying the unitary evolution of a qudit for a general nonstationary hamiltonian via the generalized gell-mann representation

Abstract

Optimal realizations of quantum technology tasks lead to the necessity of a detailed analytical study of the behavior of a d-level quantum system (qudit) under a time-dependent Hamiltonian. In the present article, we introduce a new general formalism describing the unitary evolution of a qudit (d ≥ 2) in terms of the Bloch-like vector space and specify how, in a general case, this formalism is related to finding time-dependent parameters in the exponential representation of the evolution operator under an arbitrary time-dependent Hamiltonian. Applying this new general formalism to a qubit case (d = 2), we specify the unitary evolution of a qubit via the evolution of a unit vector in R4, and this allows us to derive the precise analytical expression of the qubit unitary evolution operator for a wide class of nonstationary Hamiltonians. This new analytical expression includes the qubit solutions known in the literature only as particular cases.