For over two decades, quantum interference phenomena have been associated with several intriguing processes [1], observed in atomic systems when coherently prepared, enabling the execution of protocols for manipulation and information processing [2]. The electromagnetically induced transparency (EIT) [3], [4], [5], the coherent population trapping (CPT) [6], [7], [8], [9] and lasing without inversion [10], [11], are well-known examples in which quantum interference plays a fundamental role in the modification of optical properties in atomic systems. These phenomena are observed in typically three-level atomic systems driven simultaneously by a strong coupling field and weak probe field.

Coherent optical spectroscopy of the degenerate two-level system (DTLS) has shown surprising phenomena such as the observation of Raman absorption and gain in laser cooled atoms as well as the resonant increase in the absorption of the medium, an effect known as electromagnetically induced absorption (EIA) [12], [13]. On the other hand, a DTLS associated with Zeeman transitions of alkali atoms can provide a four-level system configuration, triply degenerate in the ground state, depending on the relative polarizations of the coupling and probe fields [14], [15], [16], [17]. In such systems, the effect of a transverse magnetic field (TMF), defined as a magnetic field perpendicular to the propagation direction of the excitation light, on the absorption and fluorescence spectra of DTLS in alkali metals has been investigated in the past, both experimentally [7], [8], [9] and theoretically [6], [18], [19], using the Hanle configuration.

Frometa et al. [14], Aquino Carvalho et al. [15] and Moreira et al. [16] reported observations of magnetically assisted narrow band probe beam amplification in Zeeman degenerate closed two-level system associated with cold cesium atoms. In the experiment, the closed cesium D2 hyperfine transition Fg=3→Fe=2 is coherently excited by a probe and a coupling beam, with opposite circular polarizations, in the presence of dc TMF. Single-pass gain of 100% is observed for moderate values of magnetic field and coupling beam power. However, the simplified theory presented in [14] considering that the TMF introduces an incoherent recycling of atoms in the ground state, does not explain, for example, the limitations of the optical gain, the frequency dependence of the gain peak on the magnetic field, the observation of laser oscillation in [15] and the line shape recovered in light storage in [16].

In the present work we develop an approximate theory to explain the mechanism of magnetically assisted optical gain, considering that the TMF interacts with the atoms in the ground states, leading to a coherent recycling of the atoms in the system. The magnetically assisted gain mechanism is based on manipulation of the dark state created by a coupling and a probe beam in the Zeeman sublevels of a degenerate two-level system. This gain without population inversion is the result of the modification of the original dark state by the applied magnetic field, which leads to a transfer of power from the coupling beam to the probe beam. The theoretical model showed good agreement with the results obtained via numerical integration of the density matrix.