The continued-fraction equation which determines the quasienergy of a two-level system in an external monochromatic field is derived by a simple semiclassical argument. The perturbation expansion of the quasienergy is extended to tenth order in the coupling constant.
The Hill method is used in an analysis of the Schrodinger equation for a quatum system subjected to a time-periodic nonmonochromatic field. An algebraic equation is obtained and its degree in respect of exp(iET), where E is the quasienergy and T is the period of the external field, is equal to the number of levels. The coefficients of this equation are expanded in powers of the intensity parameter into a series with a factorial convergence even when one of the field frequencies is in exact resonance with one of the transitions. The Fourier components of the density matrix are expressed in terms of quasienergy considered as a function of the parameters of the problem and this gives approximate expressions outside the perturbation theory framework. Various results are obtained for a two-level system and, in particular, a formula is derived for the description of the natural lifetime of a quasienergy state.
We obtain expressions for the radiative level shifts of a two-level system in terms of (i) recurrence relations, (ii) ratios of determinants, (iii) continued fractions, and (iv) a Lidstone expansion. These expressions are shown to be very useful for numerical computations. It is pointed out that perturbation series in powers of the coupling constant are not the most appropriate way of representing the solutions of the problem, but if they are to be used, different series should be employed depending on the relative value of the frequency of the field to the frequency of the two-level system. The significance of these results in the general theory of perturbation is discussed.
Analytical expressions for the semiclassical dressed states and corresponding quasienergies are obtained for a two-level quantum system driven by a non-resonant and/or strong laser field in a coherent state. These expressions are of first order in a proper perturbative expansion, and already contain all the relevant physical information on the dynamical and spectroscopic properties displayed by these systems under such particular conditions. The influence of the laser field parameters on transition frequencies, selection rules and line intensities can be easily understood in terms of the quasienergy-level diagram and the allowed transitions between the different semiclassical dressed states.
Analytical expressions for the quasienergies and corresponding semiclassical dressed states are obtained for a two-level quantum system driven by a nonresonant and/or strong laser field in a coherent state. These expressions are of first order in a proper perturbative expansion, and already contain all the relevant physical information on the dynamical and spectroscopic properties displayed by these systems under such particular conditions. The influence of the laser field parameters on transition frequencies, selection rules, and line intensities is easily understood in terms of the quasienergy-level diagram and the allowed transitions between the different semiclassical dressed states.
The semiclassical interaction of a two-level system with a multimode electromagnetic field is discussed in detail. The equations of motion for the macroscopic density matrix are transformed into a single integral equation for the population density, with a complex kernel which can take into account both running- and standing-wave configurations. The iterative solution to this integral equation, which corresponds to an expansion in terms of powers of the electromagnetic field amplitudes, turns out to be divergent when the field amplitudes are greater than some critical value. In the case in which the kernel is a periodic or quasiperiodic function of time, a general solution to the integral equation is found, and this solution turns out to be unique and convergent for every value of the electromagnetic (e.m.) field amplitude. This solution is expressed through a continued-fraction expansion, whose terms are matrices, which can be readily obtained from the kernel. This solution is a generalization of the continued-fraction expansion usually met in the strong-signal semiclassical theory of lasers. Some particular cases are then treated in detail; among them, the solution to the multimode operation in laser devices is given in which the excited level lifetime is long compared with the period of the beating term between two adjacent modes. Although the theory presented is directed towards the steady-state regime in multimode lasers, it may also treat the transient regime of laser operations, or it may be extended to include phenomena where a strong e.m. field, made up of several modes equally spaced in frequency, interacts with a two-level system.