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The exact quantum solution of N-radiators in cooperative interaction with a cavity field

An exact analytical approach to study the time evolution of the interaction between a system of N two-level atoms with a single-mode cavity field is proposed. Supposing that at the initial moment the system of radiators is prepared in a superposition of states, the wave functions of the atom–field interaction for three and four atoms are presented exactly. In the case of an odd number of atoms in the cavity, the quantum Rabi frequencies have non-zero values; otherwise one zero value of the quantum Rabi frequency is characteristic of the even number of atoms. It was shown that with an increase of the odd number of atoms the non-zero quantum Rabi frequencies in the system increase too. It was established that a single atom has one quantum Rabi frequency, three atoms have two frequencies and five atoms have three. For the even number of atoms, a zero value of eigenstates of radiators in the interaction with a single cavity mode is observed. In this case the number of collective levels in the system is odd and the trapping states of atoms in the cavity are possible. From analytical solutions it follows that it is possible to realize the micro-maser regimes for three atoms choosing the flying time for which the radiators pass totally in the ground state. The dependence of the flying time on the number of atoms is established as well as the cavity state for which the atomic ensemble comes back to its initial state. The behavior of quantum fluctuations of photon numbers and atomic inversion during collective absorption and radiation is exactly analytically and numerically studied. The exact quantum solutions for these qubits, formed from two-level atoms, can be used in quantum processing of information encapsuled in the initial states of the atomic ensemble and the cavity field.