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We derive the general formula giving the Berry phase for an arbitrary spin,having both magnetic-dipole and electric-quadrupole couplings with externaltime-dependent fields. We assume that the effective E and B fields remainorthogonal during the quantum cycles. This mild restriction has manyadvantages. It provides simple symmetries leading to selection rules and theHamiltonian-parameter and density-matrix spaces coincide for S=1. This impliesthe identity of the Berry and Aharonov-Anandan phases, which is lost for S>1.We have found that new features of Berry phases emerge for integer spins>2. Weprovide explicit numerical results of Berry phases for S=2,3,4. We give aprecise analysis of the non-adiabatic corrections. The accuracy for satisfyingadiabaticity is greatly improved if one chooses for the time derivatives of theparameters a time-dependence having a Blackman pulse shape. This has the effectof taming the non-adiabatic oscillation corrections which could be generated bya linear ramping. For realistic experimental conditions, the non-adibaticcorrections can be kept < 0.1%. For quantum cycles,involving as sole periodicparameter the precession angle of E around B, the corrections odd upon thereversal of the angular velocity can be cancelled exactly if the quadrupole todipole coupling ratio takes a "magic" value. The even ones are cancelled bysubtraction of the phases relative to opposite velocities. As a possibleapplication of the results of this paper we suggest a route to holonomicentanglement of N non-correlated 1/2-spins by performing adiabatic cyclesgoverned by a Hamiltonian which is a non-linear function of the total spinoperator S defined as the sum of the N spin operators. The case N=4 and Sz=1 istreated explicitly and maximum entanglement is achieved.

Berry's Phases for Arbitrary Spins Non-Linearly Coupled to External Fields. Application to the Entanglement of N > 2 Non-Correlated One-Half Spins