Cost of counterdiabatic driving and work output

Unitary processes allow for the transfer of work to and from Hamiltonian systems. However, to achieve nonzero power for the practical extraction of work, these processes must be performed within a finite time, which inevitably induces excitations in the system. We show that depending on the time scale of the process and the physical realization of the external driving employed, the use of counterdiabatic quantum driving to extract more work is not always effective. We also show that by virtue of the two-time energy measurement definition of quantum work, the cost of counterdiabatic driving can be significantly reduced by selecting a restricted form of the driving Hamiltonian that depends on the outcome of the first energy measurement. Lastly, we introduce a measure, the exigency, that quantifies the need for an external driving to preserve quantum adiabaticity which does not require knowledge of the explicit form of the counterdiabatic drivings, and can thus always be computed. We apply our analysis to systems ranging from a two-level Landau-Zener problem to many-body problems, namely, the quantum Ising and Lipkin-Meshkov-Glick models.

Figure 1

(a), (b) Instantaneous cost of counterdiabatic driving for two spins (a) and eight spins (b). The red circles represent ${\partial }_t{C}_t^1$ while blue continuous lines depict ${\partial }_t{C}_{{}_W}^1$ for decreasing values of the inverse temperature (as indicated by the arrow) $\beta =\infty$ (top) to $\beta =0$ (bottom). The inset of (a) shows the instantaneous exigency ${\partial }_t{C}_0$ also from $\beta =\infty$ (top) to $\beta =0$ (bottom).

Figure 2

(a) Ratio of costs $C_W^1/C_t^1$ versus inverse temperature $\beta$. The arrows indicate the asymptotic value computed for the ground state (at $\beta =\infty$). (b) $C_0/C_t^1$ versus $\beta$. The blue continuous line represents $L=4$, green dashed line $L=6$, and red dotted line $L=8$.

Figure 3

Comparison of exigency to the cost of transitionless driving in the Landau-Zener model [Eq. (5)] as $g\left(t\right)$ is driven from $g\left(t_0\right)=-10\mathrm{\Delta }$ to $g\left(t_1\right)=5\mathrm{\Delta }$ following also Eq. (9). Red continuous line represents ${\partial }_t{C}_t^1$ while the other lines show ${\partial }_t{C}_0$ for an initial ground-state occupation of $p_g=1$ (blue dotted-dashed line) and $p_g=0.75$ (light blue dashed line). Here, $\tau =(t-t_0)/(t_1-t_0)$ and $(t_1-t_0)={\mathrm{\Delta }}^{-1}\hslash$.

Figure 4

First derivative of the exigency, Eq. (14), when driving the ground state of the LMG model through its quantum phase transition. The curves are for increasingly large system sizes from the bottom: $N=100$ (bottom, purple continuous curve), $N=200$ (red dashed curve), $N=300$ (green dotted-dashed curve), and $N=400$ (blue dotted curve). The topmost black thin continuous line is derived from the HP mapping with $N=400$. Inset: the second derivative of the exigency. The pronounced dip that approaches the critical point as system size increases is clearly visible. In both figures, $\tau =(t-t_0)/(t_1-t_0)$ and $(t_1-t_0)={\mathrm{\Delta }}^{-1}\hslash$.