High-Speed Driving of a Two-Level System

TL;DR

This paper derives the optimal control protocols for driving a two-level quantum system between states in minimal time, revealing a transition from bang-off-bang to bang-bang controls depending on constraints.

Contribution

It provides an analytical solution for the minimal-time control of two-level systems, extending previous results and exploring the effects of bounded control strengths.

Findings

Optimal protocols are bang-off-bang for large control bounds.

For smaller bounds, optimal protocols switch to bang-bang.

Multistep protocols can be optimal for general states.

Abstract

A remarkably simple result is found for the optimal protocol of drivings for a general two-level Hamiltonian which transports a given initial state to a given final state in minimal time. If one of the three possible drivings is unconstrained in strength the problem is analytically completely solvable. A surprise arises for a class of states when one driving is bounded by a constant $c$ and the other drivings are constant. Then, for large $c$, the optimal driving is of type bang-off-bang and for increasing $c$ one recovers the unconstrained result. However, for smaller $c$ the optimal driving can suddenly switch to bang-bang type. It is also shown that for general states one may have a multistep protocol. The present paper explicitly proves and considerably extends the author's results contained in Phys. Rev. Lett. {\bf 111}, 260501 (2013).