We derive 15 classes of time-dependent two-state models solvable in terms of the confluent Heun functions. These classes extend over all the known families of three-and twoparametric models solvable in terms of the hypergeometric and the confluent hypergeometric functions to more general four-parametric classes involving three-parametric detuning modulation functions. In the case of constant detuning the field configurations describe excitations of two-state quantum systems by symmetric or asymmetric pulses of controllable width and edge-steepness. The classes that provide constant detuning pulses of finite area are identified and the factors controlling the corresponding pulse shapes are discussed. The positions and the heights of the peaks are mostly defined by two of the three parameters of the detuning modulation function, while the pulse width is mainly controlled by the third one, the constant term. The classes suggest numerous symmetric and asymmetric chirped pulses and a variety of models with two crossings of the frequency resonance. We discuss the excitation of a two-level atom by a pulse of Lorentzian shape with a detuning providing one or two crossings of the resonance. We derive closed form solutions for particular curves in the 3D space of the involved parameters which compose the complete return spectrum of the considered two-state quantum system.

We derive five classes of quantum time-dependent two-state models solvable in terms of the double-confluent Heun functions, five other classes solvable in terms of the biconfluent Heun functions, and a class solvable in terms of the tri-confluent Heun functions. These classes generalize all the known families of two-or three-parametric models solvable in terms of the confluent hypergeometric functions to more general four-parametric classes involving three-parametric detuning modulation functions. The particular models derived describe different non-linear (parabolic, cubic, sinh, cosh, etc.) level-sweeping or levelglancing processes, double-or triple-level-crossing processes, as well as periodically repeated resonance-glancing or resonance-crossing processes. Finally, we show that more classes can be derived using the equations obeyed by certain functions involving the derivatives of the confluent Heun functions. We present an example of such a class for each of the three discussed confluent Heun equations.