Excess work in counterdiabatic driving

TL;DR

The paper investigates the energetic cost of counterdiabatic driving, a prominent shortcut to adiabaticity, by reframing the cost in terms of instantaneous and time-averaged excess work. It connects the quantum speed limit, particularly the Mandelstam-Tamm bound, to energy dispersion and state transitions, and shows that under standard CD the excess work vanishes, challenging its use as a cost metric. By treating the CD strength as fixed over the protocol and accounting for the dependence of total-H eigenenergies on the driving, the authors obtain a nonzero time-averaged excess work, demonstrated explicitly in the Landau-Zener model. They argue that the two roles of protocol time—setting the CD amplitude and governing the evolution rate—allow excess work to serve as a meaningful energetic-cost quantifier, with desirable properties such as decay with longer protocols and protocol-dependent cost. The work thus provides a framework to quantify the energetic demands of CD driving and highlights the conditions under which excess work can meaningfully quantify energy consumption in shortcuts to adiabaticity.

Abstract

Many years have passed since the conception of the quintessential method of shortcut to adiabaticity known as counterdiabatic driving (or transitionless quantum driving). Yet, this method appears to be energetically cost-free and thus continually challenges the task of quantifying the amount of energy it demands to be accomplished. This paper proposes that the energy cost of controlling a closed quantum system using the counterdiabatic method can also be assessed using the instantaneous excess work during the process and related quantities, as the time-averaged excess work. Starting from the Mandelstam-Tamm bound for driven dynamics, we have shown that the speed-up of counterdiabatic driving is linked with the spreading of energy between the eigenstates of the total Hamiltonian, which is necessarily accompanied by transitions between these eigenstates. Nonetheless, although excess work can be used to quantify energetically these transitions, it is well known that the excess work is zero throughout the entire process under counterdiabatic driving. To recover the excess work as an energetic cost quantifier for counterdiabatic driving, we will propose a different interpretation of the parameters of the counterdiabatic Hamiltonian, leading to an excess work different from zero. We have illustrated our findings with the Landau-Zener model.

Figures (7)

• Figure 1: Probability of measuring the instantaneous ground state $|- \rangle$ of the total Hamiltonian $H(t) = H_{0}(t)+H_{1}(t)$ of the Landau-Zener model for different protocol times $\tau$. The Hamiltonian $H_{0}(t)$ is given by Eq. (\ref{['eq: LZ_Hamiltonian']}) and $H_{1}(t)$ is given by Eq. (\ref{['eq: CD_LZ']}), when counterdiabatic driving is applied during the protocol $B(t)$ given by Eq. (\ref{['eq: protocol']}). Simulations were performed with $B_{i}/J = -B_{f}/J = -10$ and $J=5$.
• Figure 2: Quantum speed limit in the driven Landau-Zener model as a function of the protocol time $\tau$. In cyan/dashed is $\tau_{QSL}$ given by the Mandelstam-Tamm bound (Eq.(\ref{['eq: QSL']})). In magenta/dashed is the Mandelstam-Tamm bound given in terms of the excess work (Eq.(\ref{['eq: QSL_excess_2']})) using Eq.(\ref{['eq: excess_work_new']}). In yellow/dashed is the Margolus-Levitin bound of reference deffner2013quantum given in terms of the trace norm of $\rho(t)H(t)$. All quantities must be equal or smaller than $\tau$ (solid/black), as can be verified in the figure. The parameters used were $B_{i}/J = -B_{f}/J = 10$ and $J = 5$ with protocol $B(s) = B_{i} + (B_{f} - B_{i})s$. It is noteworthy that the Mandelstam-Tamm bound is tight in this case.
• Figure 3: Time average of expression (\ref{['eq: excess_work_new']}) for the excess work as a function of the driving duration for the protocols given by Eq. (\ref{['eq: protocol']}) (solid/black) and for the linear protocol, $B(s) = B_{i} + (B_{f} - B_{i})s$ (dashed/magenta). Note the non-zero value of the time-averaged excess work and the decaying with the protocol time $\tau$. The parameters used were $B_{i}/J = -B_{f}/J = 10$ and $J = 5$.
• Figure 4: Protocol $B(t)$ of Eq. (\ref{['eq: protocol']}) for different durations. As we increase $\tau$, the relative change of $B$ (which is independent of $\tau$) takes place at smaller rates. The values used for initial and final values were $B_{i}=-50$ and $B_{f}=50$, respectively.
• Figure 5: Variation of the amplitude $C(s)$, given by Eq.(\ref{['eq: CD_LZ']}), of the counterdiabatic Hamiltonian of the Landau-Zener model. The different curves represent processes with different durations. As $\tau$ increases, the intensity of $C(t)$ becomes smaller.