The hyperbolic, the arithmetic and the quantum phase

accepted for publication for the special issue of J. Opt. B, in relation to ICSSUR , Puebla (Mexico): Foundations of Quantum Optics, to be published in June 2004We develop a new approach of the quantum phase in an Hilbert space of finite dimension which is based on the relation between the physical concept of phase locking and mathematical concepts such as cyclotomy and the Ramanujan sums. As a result, phase variability looks quite similar to its classical counterpart, having peaks at dimensions equal to a power of a prime number. Squeezing of the phase noise is allowed forspecific quantum states. The concept of phase entanglement for Kloosterman pairs of phase-locked states is introduced