Universal Work Fluctuations During Shortcuts to Adiabaticity by Counterdiabatic Driving

Counterdiabatic driving (CD) exploits auxiliary control fields to tailor the nonequilibrium dynamics of a quantum system, making possible the suppression of dissipated work in finite-time thermodynamics and the engineering of optimal thermal machines with no friction. We show that while the mean work done by the auxiliary controls vanishes, CD leads to a broadening of the work distribution. We derive a fundamental inequality that relates nonequilibrium work fluctuations to the operation time and quantifies the thermodynamic cost of CD in both critical and noncritical systems.

Figure 1

Thermodynamic cost of counterdiabatic driving (CD). Numerical calculation for a time-dependent harmonic oscillator. (a) Instantaneous value of the average work cost. The orange dots are obtained via CD for $\tau =0.8$ and the green curve is obtained from the adiabatic protocol. (b) Time dependence of the variance of work for different values of the duration $\tau$ of the process. (c) Time dependence of each term in Eq. (16) for $\tau =0.8$. (d) Time average of the work fluctuations ${⟨\delta \mathrm{\Delta }W⟩}_{\tau }$. The green curve is a ${\tau }^{-1}$ fit with the best fit coefficient found to be 0.653, which is larger than the Bures length $\mathcal{L}=0.476$ consistent with inequality (21).

Figure 2

Work fluctuations across a quantum phase transition. (a) Instantaneous work fluctuations during the CD evolution of a quantum Ising chain as a function of time for different choices of the duration of the protocol $\tau =0.5$, 1 and 2 with $\delta =1$. (b) Divergence of the time-integrated work fluctuations with the system size $N$.