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arXiv:0908.0785 · Brazilian Journal of Physics

Contribution of adiabatic phases to noncyclic evolution

M. T. Thomaz, A.C. Aguiar Pinto, M. Moutinho

Published Jan 1, 2010• 1 citation
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Abstract

We show that the difference of adiabatic phases, that are basis-dependent, in noncyclic evolution of non-degenerate 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.

DOI
10.1590/s0103-97332010000200009

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We show that the difference of adiabatic phases, that are basis-dependent, in noncyclic evolution of non-degenerate 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.
The experimental observation of effects due to Berry's phase in quantum systems is certainly one of the most impressive demonstrations of the correctness of the superposition principle in Quantum Mechanics. Since Berry's original paper in 1984, the spin 1/2 coupled with rotating external magnetic field has been one of the most studied model where those phases appear. We also consider this soluble model. A detailed analysis of the coupled differential equations and comparison with exact results teach us why the usual procedure (of neglecting non-diagonal terms) is mathematically sound.
We evaluate the exact (Pancharatnam) phase differences between the final state \ensuremath{\Vert}\ensuremath{\psi}(t)〉 and various initial states for a spin-1/2 particle in a rotating magnetic field B(t). For initial states \ensuremath{\Vert}n;Bef{\mathbf{B}}_{\mathrm{ef}}Bef​(0)〉, which are eigenstates of the spin component along the direction of the initial effective field Bef{\mathbf{B}}_{\mathrm{ef}}Bef​(0), the exact phase has an energy-dependent part and an energy-independent part. It is shown that these states \ensuremath{\Vert}n;Bef{\mathbf{B}}_{\mathrm{ef}}Bef​(0)〉 are cyclic and their corresponding Aharonov-Anandan phases are evaluated. In the adiabatic limit we discuss different choices of time-dependent bases and the relationship between the exact phase, the Born-Fock-Schiff phase, and Berry's phase. We propose neutron interference experiments to test separately the exact and the adiabatic evolution laws, as well as to measure exact and adiabatic time-dependent phases.
It is found that in a quantum system with cyclic Hamiltonian the time-dependent Schrodinger equation admits the Bloch type solutions which have the form of Bloch function with respect to the time. These solutions constitute an orthonormal basis of the solution space and lead to a new type of quantum phases. The new phases, called Bloch phases by the authors, are the special Aharonov-Anandan phases, and reduce to the Berry phases under the adiabatic approximation if the degeneracy of instantaneous eigenenergy is time-independent. The above conclusions are proved and calculation for the evolution of spin magnetic moments in a magnetic field precessing with constant angular velocity is exemplified.
We study the time evolution of a quantum system under the adiabatic condition using the path integral formulation. We reobtain Berry's phase for systems with nondegenerate and degenerate energy spectra, recovering the well-known results on the literature, and we verify the behavior of the phases under a change of basis of the energy instantaneous eigenstates. We also present a simple derivation of the Adiabatic Theorem[1] using the current notation. We have a comparative summary between the Schrodinger's and the Heisenberg's pictures when the quantum system is driven by a time-dependent hamiltonian and we show how to aply the Adiabatic Theorem to any operator in the Heisenberg's picture that evolves under adiabactic conditions.
The probability amplitude for a spin-1/2 particle in a precessing magnetic field to be in an energy eigenstate is calculated exactly. The probability for the particle to be in a given energy eigenstate and the corresponding phase of the amplitude are obtained. In the adiabatic limit there are no transitions, and the phase is the sum of the dynamical phase and the Berry phase. When a unitary transformation is made, the Hamiltonian and the energy operator transform differently. The energy eigenvalues, Berry phases, and probability amplitudes calculated in a frame rotating with the magnetic field are the same as in the laboratory frame. If the energy operator is incorrectly taken to be the Hamiltonian in the rotating frame, incorrect energy eigenvalues, Berry phases, and probability amplitudes are obtained. Nevertheless, in the adiabatic limit the correct total phase results.