Classical and Quantum Shortcuts to Adiabaticity for Scale-Invariant Driving

Abstract

A shortcut to adiabaticity is a driving protocol that reproduces in a short time the same final state that would result from an adiabatic, infinitely slow process. A powerful technique to engineer such shortcuts relies on the use of auxiliary counterdiabatic fields. Determining the explicit form of the required fields has generally proven to be complicated. We present explicit counterdiabatic driving protocols for scale-invariant dynamical processes, which describe, for instance, expansion and transport. To this end, we use the formalism of generating functions and unify previous approaches independently developed in classical and quantum studies. The resulting framework is applied to the design of shortcuts to adiabaticity for a large class of classical and quantum, single-particle, nonlinear, and many-body systems.

Nearly optimal quantum control: an analytical approach

“…Many techniques to do this have been developed for spin-1/2 systems [6][7][8]. Here we will focus on the class of such solutions called shortcuts to adiabaticity [8,[11][12][13][14][15]. This class covers not only spin-1/2 but also many practically interesting multistate situations [16].…”

Suppression of work fluctuations by optimal control: An approach based on Jarzynski's equality

“…[29] showed, STA in known examples can be understood as a result of a canonical transformation of FFAD (or vice versa). Though Campo and Jarzynski extended STA to more complicated systems such as many-body systems [32], a general discussion by them also indicates that this method can only apply to systems with the so-called "scale-invariant driving" [29]. For these reasons both FFAD and STA require full knowledge of the system Hamiltonian, and this requirement presents a limitation when we attempt to use accelerated adiabatic processes to suppress work fluctuations in general situations.…”

“…For the sake of discussions later on, we divide accelerated adiabatic processes known to date into two types (though these two types can even be regarded as being equivalent upon a transformation [29] and there is no clear distinction in the literature). In the first type, an additional control Hamiltonian is introduced to drive a system (within a short time scale), such that the evolution of the system, either classical or quantum, still follows the adiabatic evolution of the original bare system.…”

Construction of exact Ermakov-Pinney solutions and time-dependent quantum oscillators

“…The charged scalar field in a time-dependent, homogeneous magnetic field is equivalent to that of an infinite system of coupled oscillators for Landau levels [5]. The quantum theory of time-dependent oscillators provides shortcuts to adiabaticity with an important, analytical model [6,7].…”