  The approach of shortcuts to adiabaticity enables the effective execution of
adiabatic dynamics in quantum information processing with enhanced speed. Owing
to the inherent trade-off between dynamical speed and the cost associated with
the transitionless driving field, executing arbitrarily fast operations becomes
impractical. To understand the accurate interplay between speed and energetic
cost in this process, we propose theoretically and verify experimentally a new
trade-off, which is characterized by a tightly optimized bound within
$s$-parameterized phase spaces. Our experiment is carried out in a single
ultracold $^{40}$Ca$^{+}$ ion trapped in a harmonic potential. By exactly
operating the quantum states of the ion, we execute the Landau-Zener model as
an example, where the quantum speed limit as well as the cost are governed by
the spectral gap. We witness that our proposed trade-off is indeed tight in
scenarios involving both initially eigenstates and initially thermal
equilibrium states. Our work helps understanding the fundamental constraints in
shortcuts to adiabaticity and illuminates the potential of under-utilized phase
spaces that have been traditionally overlooked.
