Lewis-Riesenfeld invariants and transitionless quantum driving

Abstract

Different methods have been recently put forward and implemented experimentally to inverse engineer the time dependent Hamiltonian of a quantum system and accelerate slow adiabatic processes via non-adiabatic shortcuts. In the "transitionless tracking algorithm" proposed by Berry, shortcut Hamiltonians are designed so that the system follows exactly, in an arbitrarily short time, the approximate adiabatic path defined by a reference Hamiltonian. A different approach is based on designing first a Lewis-Riesenfeld invariant to carry the eigenstates of a Hamiltonian from specified initial to final configurations, again in an arbitrary time, and then constructing from the invariant the transient Hamiltonian connecting these boundary configurations. We show that the two approaches, apparently quite different in form and so far in results, are in fact strongly related and potentially equivalent, so that the inverse-engineering operations in one of them can be reinterpreted and understood in terms of the concepts and operations of the other one. We study as explicit examples the expansions of time-dependent harmonic traps and state preparation of two level systems.

Shortcuts to adiabaticity: Fast-forward approach

“…Our second aim has been to relate it to other inverse engineering methods. In a previous publication, the inverse-engineering method based on invariants, was related to the transitionless tracking algorithm, and their potential equivalence was demonstrated [33]. Similarly we have established in this paper the connection between the fast-forward method and the invariant-based method for quadratic-in-momentum invariants.…”

“…These relations do not imply the full identity of the methods but their overlap and equivalence in a common domain. They are still useful heuristically as separate approaches since they are formulated in rather different terms [28,33]. Moreover they facilitate extensions beyond their common domain, as exemplified by the wave-splitting processes discussed in the previous section.…”

“…with OC and invariantbased methods, or because of their different domains. Clarifying the features, overlaps, and relations among these approaches is important to apply the ones which are best suited for specific systems and objectives [33], or to develop new ones. In this paper we shall establish in particular the connection between the fast-forward and the invariant-based methods.…”

Efficient shortcuts to adiabatic passage for fast population transfer in multiparticle systems

“…To speed up the transfer by using the dynamics of invariant-based inverse engineering, we need to introduce an invariant Hermitian operator I S 2 (t), which satisfies i∂I S 2 (t)/∂t = [H S 2 (t), I S 2 (t)] [10,13,14,23,24], for H S 2 (t) possesses the SU(2) dynamical symmetry. And I S 2 (t) is given by…”

“…Recently, a lot of work has been done in finding shortcuts to adiabaticity for the two-or three-level atomic system [8][9][10][11][12][13][14][15]. By means of resonant laser pulses, Chen and Muga have successfully performed fast population transfer (FPT) in three-level systems via invariantbased inverse engineering [13].…”