Generalized speed and cost rate in transitionless quantum driving

Transitionless quantum driving, also known as counterdiabatic driving, is a unique shortcut technique to adiabaticity, enabling a fast-forward evolution to the same target quantum states as those in the adiabatic case. However, as nothing is free, the fast evolution is obtained at the cost of stronger driving fields. Here, given the system initially get prepared in equilibrium states, we construct relations between the dynamical evolution speed and the cost rate of transitionless quantum driving in two scenarios: one that preserves the transitionless evolution for a single energy eigenstate (individual driving), and the other that maintains all energy eigenstates evolving transitionlessly (collective driving). Remarkably, we find that individual driving may cost as much as collective driving, in contrast to the common belief that individual driving is more economical than collective driving in multilevel systems. We then present a potentially practical proposal to demonstrate the above phenomena in a three-level Landau-Zener model using the electronic spin system of a single nitrogen-vacancy center in diamond.