The average potential energy can only be extracted partially, as $(k/2){⟨x(0)⟩}_{v_0}^2\le (k/2){⟨x(0{)}^2⟩}_{v_0}={⟨V⟩}_{v_0}$, by the Cauchy-Schwarz inequality.

The diffusive delocalization of the trap center can be prevented by incorporating a recovery step of duration $t_{\mathrm{rec}}$ that resets the system to the origin after any $N$ cycles, yielding a process that is cyclic on the level of average quantities with period length $N{t}_{\mathrm{f}}+t_{\mathrm{rec}}$. If $t_{\mathrm{rec}}$ is chosen to scale sublinearly with $N$, the corrections to both efficiency and power also scale sublinearly. This implies that all our results derived for the operation without recovery steps remain asymptotically valid for large $N$. For further details see Ref. [45].

We can assume that the particle has effectively reached a steady state by the end of the iteration if its duration $t_{\mathrm{f}}$ is greater than the intrinsic relaxation time scales of $v$ and $x$, $\tau$ and $k^{-1}$, so that we can apply Eq. (9).

Taking into account the underlying thermodynamic cost of the self-propulsion as a background dissipation term results in a maximum efficiency at some finite frequency $1/t_{\mathrm{f}}$. For example, assuming a constant rate $\mathcal{A}$ of ATP depletion of the self-propulsion mechanism, the efficiency scales like $\eta \propto -{\mathcal{W}}^{\mathrm{eng}}/(D_v\mathcal{I}+\mathcal{A}{t}_{\mathrm{f}})$.