Transitionless quantum driving achieves adiabatic evolution in a hurry, using a counterdiabatic Hamiltonian to stifle nonadiabatic transitions. Here this shortcut to adiabaticity is cast in terms of a generator of adiabatic transport. This yields a classical analog of transitionless driving, and provides a strategy for constructing quantal counterdiabatic Hamiltonians. As an application of this framework, exact classical and quantal counterdiabatic terms are obtained for a particle in a box and for even-power-law potentials in one degree of freedom.
Transitionless quantum driving achieves adiabatic evolution in a hurry, using a counterdiabatic Hamiltonian to stifle nonadiabatic transitions. Here this shortcut to adiabaticity is cast in terms of a generator of adiabatic transport. This yields a classical analog of transitionless driving, and provides a strategy for constructing quantal counterdiabatic Hamiltonians. As an application of this framework, exact classical and quantal counterdiabatic terms are obtained for a particle in a box and for even-power-law potentials in one degree of freedom.
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.
A shortcut to adiabaticity is a recipe for generating adiabatic evolution at an arbitrary pace. Shortcuts have been developed for quantum, classical and (most recently) stochastic dynamics. A shortcut might involve a counterdiabatic (CD) Hamiltonian that causes a system to follow the adiabatic evolution at all times, or it might utilize a fast-forward (FF) potential, which returns the system to the adiabatic path at the end of the process. We develop a general framework for constructing shortcuts to adiabaticity from flow fields that describe the desired adiabatic evolution. Our approach encompasses quantum, classical and stochastic dynamics, and provides surprisingly compact expressions for both CD Hamiltonians and FF potentials. We illustrate our method with numerical simulations of a model system, and we compare our shortcuts with previously obtained results. We also consider the semiclassical connections between our quantum and classical shortcuts. Our method, like the FF approach developed by previous authors, is susceptible to singularities when applied to excited states of quantum systems; we propose a simple, intuitive criterion for determining whether these singularities will arise, for a given excited state.
A general approach for transitionless quantum driving in open quantum systems is introduced. Under the assumption of adiabatic evolution for time-local master equations, we derive the generalized transitionless Lindbladian required to implement a shortcut to adiabaticity in an open-system scenario. The general counterdiabatic Lindbladian obtained accounts for a phase freedom, which translates into a set of free parameters throughout the dynamics. We discuss how our generalized approach allows us to recover the transitionless Lindbladian introduced by Vacanti et al. [G. Vacanti et al. New J. Phys. 16, 053017 (2014)]. We show how to engineer time-independent master equations that provide the same dynamics as the time-dependent master equation provided by the standard transitionless quantum driving in open systems. We illustrate our results by applying them both to the adiabatic Deutsch algorithm under dephasing and to the Landau-Zener Hamiltonian under a bit phase flip.
Adiabatic quantum state evolution can be accelerated through a variety of shortcuts to adiabaticity . In one approach, a counterdiabatic quantum Hamiltonian, Ĥ CD, is constructed to suppress nonadiabatic excitations. In the analogous classical problem, a counterdiabatic classical Hamiltonian, H CD, ensures that the classical action remains constant even under rapid driving. Both the quantum and classical versions of this problem have been solved for the special case of scale-invariant driving, characterized by linear expansions, contractions, or translations of the system. Here we investigate an example of a non-scale-invariant system, a tilted piston . We solve exactly for the classical counterdiabatic Hamiltonian, H CD ( q, p, t ), which we then quantize to obtain a Hermitian operator, Ĥ CD ( t ). Using numerical simulations, we find that Ĥ CD effectively suppresses nonadiabatic excitations under rapid driving. These results offer a proof of principle, beyond the special case of scale-invariant driving, that quantum shortcuts to adiabaticity can successfully be constructed from their classical counterparts.
We propose a method to produce fast transitionless dynamics for finite-dimensional quantum systems without requiring additional Hamiltonian components not included in the initial control setup, remaining close to the true adiabatic path at all times. The strategy is based on the introduction of an effective counterdiabatic scheme: a correcting Hamiltonian is constructed which approximatively cancels nonadiabatic effects, inducing an evolution tracking the adiabatic states closely. This can be absorbed into the initial Hamiltonian by adding a fast oscillation in the control parameters. We show that a consistent speedup can be achieved without requiring strong control Hamiltonians, using it both as a stand-alone shortcut to adiabaticity and as a weak correcting field. A number of examples are treated, dealing with quantum state transfer in avoided-crossing problems and entanglement creation.
In this thesis, it is presented a set of results in adiabatic dynamics (closed and open system) and transitionless quantum driving that promote some advances in our understanding on quantum control and Hamiltonian inverse engineering. In the context of adiabatic dynamics in closed systems, it is introduced a validation mechanism for the adiabaticity conditions by studing the system dynamics from a non-inertial reference frame. By considering a decohering scenario, validity conditions of the adiabatic approximation are also studied. As a fresh general result with potential applications, it is shown that under decoherence the adiabaticity may still occur in the infinite time limit, as it happens for closed systems, for a class of initial quantum states. To end, the original contributions of this thesis to the theory of shortcuts to adiabaticity refers to a generalized approach of transitionless quantum driving, where one explores the gauge freedom of the quantal phase factors accompanying adiabatic trajectories. A number of theoretical applications are studied, where some theoretical prediction presented in this thesis are experimentally verified through two different experimental setups, namely a qubit encoded in the energy hyperfine structure of a Ytterbium trapped ion, and in nuclear magnetic resonance with a nuclear spin qubit.