Adiabatic Product Expansion

The time-evolution operator for an explicitly time-dependent Hamiltonian is expressed as the product of a sequence of unitary operators. These are obtained by successive time-dependent unitary transformations of the Hilbert space followed by the adiabatic approximation at each step. The resulting adiabatic product expansion yields a generalization of the quantum adiabatic approximation. Furthermore, it leads to an infinite class of exactly solvable models. PACS number: 03.65.Bz Consider the dynamics of a quantum mechanical system whose Hamiltonian H = H(τ) is explicitly time-dependent. The evolution of a state vector |ψ(τ)〉 is governed by the Schrödinger equation: ih̄|ψ̇(τ)〉 = H(τ) |ψ(τ)〉 , |ψ(0)〉 = |ψ0〉 , (1) where a dot means a time-derivative. Alternatively, one has |ψ(τ)〉 = U(τ)|ψ0〉, where U(τ) = T e i h̄ ∫ τ 0 H(t) dt (2) is the time-evolution operator. Here T denotes the time-ordering operator. The purpose of this article is to express U(τ) as the product of a sequence of unitary operators each of ∗E-mail: [email protected]