An exact analytical approach for studying the time evolution of equidistant system of radiators prepared in a superposition of states in interaction with single cavity mode it is proposed. This method gives us the possibility to find the analytical representation of wave function for a large system of atoms. Exact solution for interaction of four radiators with quantum cavity field is presented. The representation of cavity field is studied as function of the number of atoms in the system. For odd numbers of radiators was established the following role: one atom have one quantum Rabi frequency, three atoms have two frequencies; fife atoms-thee nonzero frequencies. The zero value Rabi frequencies in the system is absent. For even atomic numbers the zero value of Rabi frequencies appear: two, four, six and so one. In the last case the number of collective level in the system is odd and the trapping states of the atoms in the cavity are possible.