Exact Solution for Chameleon Field, Self-Coupled Through the
  Ratra-Peebles Potential with n = 1 and Confined Between Two Parallel Plates
  We calculate the chameleon field profile, confined between two parallel
plates, filled with air at pressure $P = 10^{-4}\,{\rm mbar}$ and room
temperature and separated by the distance $L$, in the chameleon field theory
with Ratra--Peebles self--interaction potential with index $n = 1$. We give the
exact analytical solution in terms of Jacobian elliptic functions, depending on
the mass density of the ambient matter. The obtained analytical solution can be
used in qBounce experiments, measuring transition frequencies between quantum
gravitational states of ultracold neutrons and also for the calculation of the
chameleon field induced Casimir force for the CANNEX experiment. We show that
the chameleon--matter interactions with coupling constants $\beta \le 10^4$ can
be probed by qBounce experiments with sensitivities $\Delta E \le
10^{-18}\,{\rm eV}$. At $L = 30.1\,{\rm \mu m}$ we reproduce the result $\beta
< 5.8\times 10^8$, obtained by Jenke {\it et al.} Phys. Rev. Lett. {\bf 112},
151105 (2014)) at sensitivity $\Delta E \sim 10^{-14}\,{\rm eV}$. In the
vicinity of one of the plates our solution coincides with the solution,
obtained by Brax and Pignol (Phys. Rev. Lett. {\bf 107}, 111301 (2011)) (see
also Ivanov {\it et al.} Phys. Rev. D {\bf 87}, 105013 (2013)) above a plate at
zero density of the ambient matter.
