  We consider the Jaynes-Cummings model of a single quantum spin $s$ coupled to
a harmonic oscillator in a parameter regime where the underlying classical
dynamics exhibits an unstable equilibrium point. This state of the model is
relevant to the physics of cold atom systems, in non-equilibrium situations
obtained by fast sweeping through a Feshbach resonance. We show that in this
integrable system with two degrees of freedom, for any initial condition close
to the unstable point, the classical dynamics is controlled by a singularity of
the focus-focus type. In particular, it displays the expected monodromy, which
forbids the existence of global action-angle coordinates. Explicit calculations
of the joint spectrum of conserved quantities reveal the monodromy at the
quantum level, as a dislocation in the lattice of eigenvalues. We perform a
detailed semi-classical analysis of the associated eigenstates. Whereas most of
the levels are well described by the usual Bohr-Sommerfeld quantization rules,
properly adapted to polar coordinates, we show how these rules are modified in
the vicinity of the critical level. The spectral decomposition of the
classically unstable state is computed, and is found to be dominated by the
critical WKB states. This provides a useful tool to analyze the quantum
dynamics starting from this particular state, which exhibits an aperiodic
sequence of solitonic pulses with a rather well defined characteristic
frequency.
