Operational aspects of time-propagators in periodically driven systems- An analytic perspective

Abstract

Understanding dynamics of quantum systems (specifically spin systems) governed by time dependent Hamiltonians has remained an active area of research due to its wide-ranging appli cations in problems ranging from chemical physics to quantum computation. The controlled manipulation of spin interactions has been exploited in a wide-range of real world scenarios and hence quantifying the response of spin systems subjected to time-dependent interactions serves as the main motivation behind the development of analytic methods tailored for study ing the time-evolution in such systems. From an analytic perspective, the time-evolution of quantum systems is governed by the time-evolution operator/time-propagator, U (t). The present thesis examines the nuances of analytic methods (based on perturbation theory) employed in the derivation of evolution operators in periodically driven quantum systems in terms of expansion schemes based on: (i) exponential ansatzes and (ii) Floquet theo rem. Given the wide-variety of expansion schemes available for describing the time-evolution of quantum systems, an in-depth study of the applicability and advantages of the various schemes (by employing experimental scenarios as a case study) is a practical and essential task. While the mathematical intricacies of the expansion schemes are well established, the operational aspects of the same in time-evolution studies remain less explored and authen ticated in physical problems of relevance. In this work, the operational inconsistencies and limitations observed in time-evolution studies based on the various expansion schemes are identified and corroborated through rigorous comparisons with simulations emerging from exact numerical methods for suitable examples. Specifically, the operational aspects and exactness of the time-propagators in the Schrödinger and Interaction representation are ex amined in systems modulated by multiple frequencies. Through experimentally relevant ex amples (from magnetic resonance spectroscopy), the interplay between the natural frequency, driving amplitude and driving frequency in time evolution studies is discussed at both stro boscopic and non-stroboscopic time intervals. Additionally, the factors governing the long time behaviour and convergence properties of the time-propagators are also elucidated