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

We derive the general formula giving the Berry phase for an arbitrary spin, having both magnetic-dipole and electric-quadrupole couplings with external time-dependent fields. We assume that the effective E and B fields remain orthogonal during the quantum cycles. This mild restriction has many advantages. It provides simple symmetries leading to selection rules and the Hamiltonian-parameter and density-matrix spaces coincide for S=1. This implies the 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. We provide explicit numerical results of Berry phases for S=2,3,4. We give a precise analysis of the non-adiabatic corrections. The accuracy for satisfying adiabaticity is greatly improved if one chooses for the time derivatives of the parameters a time-dependence having a Blackman pulse shape. This has the effect of taming the non-adiabatic oscillation corrections which could be generated by a linear ramping. For realistic experimental conditions, the non-adibatic corrections can be kept < 0.1%. For quantum cycles,involving as sole periodic parameter the precession angle of E around B, the corrections odd upon the reversal of the angular velocity can be cancelled exactly if the quadrupole to dipole coupling ratio takes a "magic" value. The even ones are cancelled by subtraction of the phases relative to opposite velocities. As a possible application of the results of this paper we suggest a route to holonomic entanglement of N non-correlated 1/2-spins by performing adiabatic cycles governed by a Hamiltonian which is a non-linear function of the total spin operator S defined as the sum of the N spin operators. The case N=4 and Sz=1 is treated explicitly and maximum entanglement is achieved.