Lasing without inversion

Summary form only given. We present the first self-consistent theory of lasing without inversion in the system with split laser levels taking into account phase considerations in the framework of standard for laser theory, density matrix formalism. Although we applied this approach to the case of lasing in the system with a split upper laser level, it can also be applied to the case of split lower laser level or to more complicated multi-level systems, which may be promising for lasing without inverted population. We formulate general condition of lasing without inversion in the systems with split laser levels: the rate of coupling of split laser levels should to be much greater then the line-widths of split transitions as well as the frequency detuning of laser field from the resonances. Lasing without inversion may be realized in free atoms, molecules, or ions if they contain two levels that are quasi-degenerate with respect to magnetic quantum number or angular moment, and if they exchange very fast with third level that is random degenerate with these two ones. Another widely spread possibility is when these two levels are embedded in continuum and exchange energy with it. Such systems on hydrogen-like ions may serve as an example of a promising laser medium for vacuum ultraviolet and soft X-ray regions. There is also an opportunity for constructing lasers without inversion on solid-state materials, which are in general more rich for the realization of different energetic level configurations. For example, when two narrow quasi -degenerate levels are embedded in quasi-free states (low-density state conduction band), the rate of coherent exchange between these levels with continuum may be very high, that provides the sufficient rate of mixing of quasi-degenerate levels.