Anharmonicity of the vacuum Rabi peaks in a many-atom system

The interaction of a single mode of the electromagnetic field with a collection of atoms has been extensively studied in cavity quantum electrodynamics. Since the seminal work of Sanchez Mondragon et al. @1#, where the term vacuum field Rabi splitting was coined and Agarwal's calculations of the microwave absorption by Rydberg atoms in a cavity @2# were introduced, there have been extensive investigations of the transmitted spectrum of the atoms-cavity system @3-6#. The experiments have been carried out in a regime in which the excitation is extremely weak. This means that the number of energy quanta in the system is very small, much less than 1, so it cannot sustain an appreciable population in the ex- cited state. The weak excitation regime is characterized by the appearance of two peaks in the transmitted spectrum of the composite atoms-cavity system. This doublet is a mani- festation of the degeneracy in the frequency of two oscilla- tors being lifted once they are coupled. The rate for the ex- change of energy between the two oscillators is precisely the frequency splitting. One oscillator is the single mode of the electromagnetic field, while the other is the collective polar- ization of the N atoms. Not surprisingly, quantum- mechanical and classical calculations predict exactly the same spectrum. Even when dissipation is taken into account, the doublet is clearly resolved as long as the two oscillators fulfill the following condition: Their decay rates should be approximately equal and smaller than the rate of energy ex- change. The purpose of this Rapid Communication is to report an experimental study of the behavior of an atom-cavity system as the excitation increases, from low to high intensities, away from the linear regime so far explored. We follow the changes in the transmitted spectrum starting with the vacuum Rabi doublet at low intensities. The presence of more energy in the system, and the possibility of coherent exchange of that energy between the atoms and the cavity, modifies the simple harmonic-oscillator structure. The transmitted spec- trum presents hysteresis followed by the merging of the dis- torted peaks into a single peak. This work is an exploration in frequency space of the underlying structure for the dynam- ics of the atoms-cavity system with arbitrary excitation. For an elementary theory we may start with the Maxwell- Bloch equations for the atom-cavity system, following the literature in optical bistability @7#. To simplify the discus- sion, for the present we do not take into account the trans- verse character of the mode in the cavity, nor the standing waves of the Fabry-Perot interferometer. We derive the transmitted spectra for arbitrary input intensity from the steady-state solution to the Maxwell-Bloch equations when the frequency of the driving field changes by an amount V from the resonance condition. The purely radiative decay rate of the atoms is characterized by g' , while the cavity field decays with a rate k. The dipole coupling between N two-level atoms and the cavity mode is gAN, where g5(m 2 v/2\e 0 V) 1/2 ; m is the transition-dipole moment of the atom, v the resonance frequency of both atoms and cav- ity, and V the cavity mode volume. The input field amplitude y and the output field amplitude x are normalized to the square root of the saturation intensity for the atomic transi- tion. They represent the intracavity field in the absence and presence of atoms, respectively. Guided by the behavior of the system in the low-intensity regime, where there are clearly two normal modes, we write the transmission as consisting of two parts: