Contribution of adiabatic phases to noncyclic evolution

Abstract

We show that the difference of adiabatic phases, that are basis-dependent, in noncyclic evolution of nondegenerate quantum systems have to be taken into account to give the correct interference result in the calculation of physical quantities in states that are a superposition of instantaneous eigenstates of energy. To verify the contribution of those adiabatic phases in the interference phenomena, we consider the spin-1/2 model coupled to a precessing external magnetic field. In the model, the adiabatic phase increases in time up to reach the difference of the Berry's phases of the model when the external magnetic field completes a period.Keywords: Berry's phase, adiabatic phase, noncyclic adiabatic evolution, spin-1/2 model In 1928 Born and Fock[1] proofed the Adiabatic Theorem. In a quantum system with non-degenerate energy spectrum, this theorem says that if the system at t = 0 is an eigenstate of energy with quantum numbers {n}, along an adiabatic evolution it continues to be in an eigenstate of energy at time t with the same initial quantum numbers {n}. As a consequence of this theorem, the vector state of the quantum system acquires an extra phase besides the dynamical phase. This extra phase is actually named geometric phase. Before the important work by MV Berry in 1984[2] with cyclic adiabatic hamiltonian, this extra phase was realized to be dependent on the choice of the basis of instantaneous eigenstates of energy. This extra phase was considered non-physical since it could be absorbed in the choice of the states in the instantaneous basis. [3].In Ref.[2], MV Berry showed that the adiabatic phase acquired by the instantaneous eigenstates of energy, after a closed evolution in the classical parameter space, is physical due to its independence to the chosen basis to describe the state vector at each instant. Since the publication of the Ref.[2], the study of Berry's phase has followed very interesting and broad directions. More recently, the geometric phases have been proposed as a prototype for a quantum bit (qubit) [4][5][6][7]. In 1988 Samuel and Bhandari[8] generalized the geometric phase to noncyclic evolution. Many others interesting papers appear to discuss those physical phases in noncyclic evolution in the classical parameter space [9][10][11]. Experimental verification to the presence of those noncyclic geometric phases have been realized [12].The interference effect is a keystone in the linearity of the Quantum Mechanics. In the present letter we address to the question of the effect of the adiabatic evolution on the phases in quantum systems leaves a physical trace in measurable quantities associated to the noncyclic evolution of states described by a superposition of instantaneous eigenstates of energy. The same question was proposed in the nice Ref. [9], but differently from them we do not look for a physical noncyclic geometric phase. * Electronic address: mtt@if.uff.br Let us consider a time-dependent hamiltonian H(t) that evolves adiabatically.Following Ref.[2], we leave open the pos...

Contribution of the basis-dependent adiabatic geometric phase to noncyclic evolution

“…We are correcting our statement in Ref. [8] where we affirmed that this difference of adiabatic phases "is gauge invariant at any time t". However the dependence of γj (t) − γk (t) on the functions α j (t) and α k (t) cancels out the contributions of these functions in eq.…”

“…In Ref. [8] it was shown that no special recipe was needed to calculate a noncyclic adiabatic phase to obtain the correct result of measurable physical quantities in a quantum state evolving under the action of any non-degenerate adiabatic Hamiltonian.…”

“…For a quantum state |ψ(t) given by eq. ( 4), the expectation value of any operator O (time-dependent or not) associated to a physical quantity has interference effects due to the presence of phases [8]…”

“…We revisit the simple two-level model [10] to exemplify the previous result. The Hamiltonian of the spin-1/2 in the presence of an external classical magnetic field B(t) is [8],…”

“…In Ref. [8] the eigenstates of Hamiltonian (13) correspond to the choices: f (t) = π and g(t) = 1. For these functions, eqs.…”