Beyond linear response: Line shapes for coupled spins or oscillators via direct calculation of dissipated power

Absorption spectra are calculated for four model systems using two different methods for the calculation: The first is the standard linear response technique, while the second involves direct calculation of the energy dissipated in the sample interacting with the applied field, thus corresponding exactly to what is in fact measured experimentally. This latter technique, which we call the direct method, agrees with a linear response (as it must) for weakly applied fields for which the first-order perturbation analysis underlying linear response is in fact valid. For cases in which the system/field interaction is stronger, the linear approximation becomes poor, and the direct method, which includes the field-induced dynamics and in itself contains no approximations, is far better. The four systems considered consist of either single or coupled pairs of harmonic oscillators or two-level systems (spins) each in a thermal bath and coupled to the field by a displacement coupling. For both linear-response and direct calculations the system/field dynamics is solved exactly and the system/bath dynamics treated using a semigroup technique. For the harmonic oscillators, for which the population in any single excited state can never become large, the nonlinear effects (those present in the direct but not in the linear response) are quite minor. For the two-level/spin/systems, however, substantial corrections to the linear-response result are found in the direct calculation. These include power (or saturation) broadening and the appearance, when the two spins are coupled strongly to each other and one is coupled strongly to the field, of a new line at the averaged frequency of the two spins. For even stronger coupling, the line shape becomes very complex (corresponding to the full 15D algebra of the two spins plus the field), exhibiting several peaks and Fano antiresonances. The direct method, which is based on work of Lebowitz, seems to offer substantial advantages for the analysis of systems in which the applied field interactions are strong: This situation, common in microwave and magnetic resonance spectroscopy, is also relevant to high-energy laser studies.