Driving at the quantum speed limit: Optimal control of a two-level system

TL;DR

This paper derives the minimal time and optimal control protocols for driving a two-level quantum system between states, revealing how constraints on interaction strength affect the control strategy and the quantum speed limit.

Contribution

It provides a simple formula for the quantum speed limit and characterizes the optimal control protocols under different interaction constraints.

Findings

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

For smaller bounds, the protocols switch to bang-bang type.

The quantum speed limit depends on the interaction strength constraint.

Abstract

A remarkably simple result is derived for the minimal time $T_{\rm min}$ required to drive a general initial state to a final target state by a Landau-Zener type Hamiltonian or, equivalently, by time-dependent laser driving. The associated protocol is also derived. A surprise arises for some states when the interaction strength is assumed to be bounded by a constant $c$. 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. We discuss the notion of quantum speed limit time.