  At high magnetic field, the semiclassical approximation which underlies the
Ginzburg-Landau theory of the mixed state of type II superconductors breaks
down. In a quasi-1D superconductor with an {\it open Fermi surface}, a high
magnetic field stabilizes a cascade of superconducting phases which ends in a
strong reentrance of the superconducting phase. From a microscopic mean-field
model, we determine the thermodynamics and the excitation spectrum of these
quantum superconducting phases.
