  The propagation of monomode photons inside a coiled optical fibre was
regarded as a time-dependent quantum evolution process, which gives rise to a
geometric phase. It is well known that the investigation of non-adiabatic
geometric phases ought to be performed only in the Schr\"{o}dinger picture. So,
the projections of photon spin operators onto the fixed frame of reference is
discussed in this paper. In addition, we also treat the non-normal-order spin
operators and consider the potential effects (e.g., quantum-vacuum geometric
phases) of quantum fluctuation fields arising in a curved optical fibre. The
quantum-vacuum geometric phase, which is of physical interest, can be deducted
by using the operator normal product, and the doubt of validity and
universality for the normal-normal procedure applied to time-dependent quantum
systems is thus proposed. In the Appendix, the discussion of possible
experimental realizations of quantum-vacuum geometric phases is briefly
presented.
