  The Large Charge sector of Conformal Field Theory (CFT) can generically be
described through a semiclassical expansion around a superfluid background. In
this work, focussing on $U(1)$ invariant Wilson-Fisher fixed points, we study
the spectrum of spinning large charge operators. For sufficiently low spin
these correspond to the phonon excitations of the superfluid state. We discuss
the organization of these states into conformal multiplets and the form of the
corresponding composite operators in the free field theory limit. The latter
entails a mapping, built order-by-order in the inverse charge $n^{-1}$, between
the Fock space of vacuum fluctuations and the Fock space of fluctuations around
the superfluid state. We discuss the limitations of the semiclassical method,
and find that the phonon description breaks down for spins of order $n^{1/2}$
while the computation of observables is valid up to spins of order $n$.
Finally, we apply the semiclassical method to compute some conformal 3-point
and 4-point functions, and analyze the conformal block decomposition of the
latter with our knowledge of the operator spectrum.
