Trade-Off Between Speed and Cost in Shortcuts to Adiabaticity

Achieving effectively adiabatic dynamics is a ubiquitous goal in almost all areas of quantum physics. Here, we study the speed with which a quantum system can be driven when employing transitionless quantum driving. As a main result, we establish a rigorous link between this speed, the quantum speed limit, and the (energetic) cost of implementing such a shortcut to adiabaticity. Interestingly, this link elucidates a trade-off between speed and cost, namely, that instantaneous manipulation is impossible as it requires an infinite cost. These findings are illustrated for two experimentally relevant systems—the parametric oscillator and the Landau-Zener model—which reveal that the spectral gap governs the quantum speed limit as well as the cost for realizing the shortcut.

Figure 1

We consider the harmonic oscillator with time-dependent frequency ${\omega }_t={\omega }_0+{\omega }_d(t/\tau )$. Main panels: QSL time. The points correspond to the QSL times considered in the insets. Insets: maximal speed $v_{\mathrm{QSL}}$, and instantaneous cost ${\partial }_{t}C$, for $\tau =0.5$ (solid), 1 (dashed), and 2 (dotted). (a) A compression with ${\omega }_0=1$ and ${\omega }_d=4$. (b) An expansion with ${\omega }_0=1$ and ${\omega }_d=-0.75$.

Figure 2

Maximal quantum speed $\log (v_{\mathrm{QSL}})$ for the LZ model evolved through the AC using the linear ramp $g(t)=0.2–0.4(t/\tau )$. (a) $\mathrm{\Delta }=0.001$ and (b) $\mathrm{\Delta }=0.01$. Both insets show the corresponding instantaneous cost ${\partial }_{t}C$.