Dynamical invariant formalism of shortcuts to adiabaticity

Abstract

We give a pedagogical introduction to dynamical invariant formalism of shortcuts to adiabaticity. For a given operator form of the Hamiltonian with undetermined coefficients, the dynamical invariant is introduced to design the coefficients. We discuss how the method allows us to mimic adiabatic dynamics and describe a relation to the counterdiabatic formalism. The equation for the dynamical invariant takes a familiar form and is often used in various fields of physics. We introduce examples of Lax pair, quantum brachistochrone and flow equation. This article is part of the theme issue ‘Shortcuts to adiabaticity: theoretical, experimental and interdisciplinary perspectives’.

Shortcuts to adiabaticity: theoretical framework, relations between different methods, and versatile approximations

“…It is also notable that the left-hand side of equation (6) in the classical limit can be regarded as the total derivative of a classical quantity (see, e.g. [58]). It was shown that the solution of the Schrödinger equation (1) can be expressed as…”

“…The integer K is known as the Krylov dimension and it satisfies K ⩽ D 2 − D + 1. By using the Krylov basis, fictitious time evolution of the operator Ô0 = ∂ t Ĥ(t)/b 0 in the Heisenberg picture, Ô(s) = Û † fic (s) Ô0 Ûfic (s), which appears in equation (58), can be expanded as…”

“…In this topical review, we did not consider concrete examples because we would like to clearly emphasize the general and universal properties of shortcuts to adiabaticity. For readers who are interested in simple examples, a topical review [56] and pedagogical introductions [57,58] are useful. Regarding approximate shortcuts to adiabaticity for complicated examples, comprehensive systematic reviews [29,30] are available.…”

“…We can also find a review focusing on fast-forward scaling [56]. Pedagogical introductions to counterdiabatic driving [57] and to invariant-based inverse engineering [58] are also available. However, the theory of shortcuts to adiabaticity is a rapidly growing research area.…”