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.