**8. Summary**

Throughout this manuscript, we revisited some fundamental aspects related to the physics of quantum motors and pumps. Previous results based on the steady-state properties and energy conservation law were extended to deal with arbitrary nonequilibrium conditions in a systematic way. By considering the dynamics of the mechanical degrees of freedom through a Langevin equation, we were able to treat the motor/pump protocols on the same footing. This allowed us to describe the related energy transfer processes through a single parameter: the external force done on the local system.

In the steady-state regime, we treated in general terms the validity of the constant velocity assumption, in Section 3. For arbitrary orders of the nonadiabatic expansion in the CIFs, this was linked to the separation between the electronic and mechanical dynamic scales through a large moment of inertia. We then performed a general expansion (in terms of nonequilibrium sources) of the energy fluxes that took part in the quantum transport problem. This enabled us to derive an order-by-order scheme for the energy conservation law, Equation (25). This equation may be of help in recognizing the physical processes that enter at each order in the expansion, thereby providing a useful tool for the analysis of nonlinear effects. To illustrate this, we discussed the leading orders of the global expansion and showed how different types of expansions of the energy fluxes change the expressions for the efficiency of quantum motors and pumps.

In Section 7, we introduced a specific example of a quantum motor/pump based on a double quantum dot. There, we discussed in more depth how higher order terms of the CIFs affect the stationary state conditions. We found that multiple solutions for the device's terminal velocity could in principle be available for a fixed choice of parameters (voltage and temperature biases and external force). In such a case, the stability of such solutions imposes an additional constraint on the force coefficients, and the final steady state strongly depends on initial conditions. Interestingly, it is possible to obtain more than one stable solution, each of them belonging to a different operation mode of the device. The treated example is also appealing as it is possible to study the transition between different operational modes by continuously moving the external force. This corresponds to the point at which the steady-state velocity changes its sign and, with it, the direction of the energy flow. When considering a specific type of pumped current (charge or heat), there is an intermediate region where the pumping mechanism is "frustrated". In this situation, the energy delivered by the external force is not enough to reverse the natural direction of the charge or heat currents. We found other interesting features of the studied example such as negative friction coefficients at finite voltages or a definite parity of the expansion coefficients with respect to the bias voltage and the temperature gradient, which is a manifestation of the inversion symmetry in the total energy flux. We also used this example to confirm numerically the order-by-order energy conservation law up to third order in the final velocity. Finally, for heat currents, we found parameter conditions under which the device can never work as a "refrigerator", even for large values of the external force. We explained this behavior in terms of the competition between the different orders that participated in the pumped heat of the cold reservoir, highlighting the importance of the order-by-order conservation laws.

**Author Contributions:** Both authors contributed equally to this paper.

**Funding:** This research was funded by Secretaria de Ciencia y Tecnología-Universidad Nacional de Córdoba (Secyt-UNC) and Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET). Both authors are members of CONICET.

**Conflicts of Interest:** The authors declare no conflict of interest.
