Bridging quantum, classical and stochastic shortcuts to adiabaticity

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Adiabatic invariants -- quantities that are preserved under the slow driving of a system's external parameters -- are important in classical mechanics, quantum mechanics and thermodynamics. Adiabatic processes allow a system to be guided to evolve to a desired final state. However, the slow driving of a quantum system makes it vulnerable to environmental decoherence, and for both quantum and classical systems, it is often desirable and time-efficient to speed up a process. {\it Shortcuts to adiabaticity} are strategies for preserving adiabatic invariants under rapid driving, typically by means of an auxiliary field that suppresses excitations, otherwise generated during rapid driving. Several theoretical approaches have been developed to construct such shortcuts. In this dissertation we focus on two different approaches, namely {\it counterdiabatic} driving and {\it fast-forward} driving, which were originally developed for quantum systems. The counterdiabatic approach introduced independently by Dermirplak and Rice [{\it J. Phys. Chem. A}, 107:9937, 2003], and Berry [{\it J. Phys. A: Math. Theor.}, 42:365303, 2009] formally provides an exact expression for the auxiliary Hamiltonian, which however is abstract and difficult to translate into an experimentally implementable form. By contrast, the fast-forward approach developed by Masuda and Nakamura [{\it Proc. R. Soc. A}, 466(2116):1135, 2010] provides an auxiliary potential that may be experimentally implementable but generally applies only to ground states.

The central theme of this dissertation is that classical shortcuts to adiabaticity can provide useful physical insights and lead to experimentally implementable shortcuts for analogous quantum systems. We start by studying a model system of a tilted piston to provide a proof of principle that quantum shortcuts can successfully be constructed from their classical counterparts. In the remainder of the dissertation, we develop a general approach based on {\it flow-fields} which produces simple expressions for auxiliary terms required for both counterdiabatic and fast-forward driving. We demonstrate the applicability of this approach for classical, quantum as well as stochastic systems. We establish strong connections between counterdiabatic and fast-forward approaches, and also between shortcut protocols required for classical, quantum and stochastic systems. In particular, we show how the fast-forward approach can be extended to highly excited states of quantum systems.