**3. Mechanical Steady State**

In this paper, we will restrict ourselves to systems that perform overall cyclic motions. Immediate examples are shown in Figure 1a,c, where the rotation angle of the rotor can be assigned as the natural mechanical coordinate. On the other hand, the examples shown in Figure 1b,d may also, under certain circumstances, sustain cyclic motion, though the general coordinate could be not so obvious. As a possibility for the quantum shuttle of Figure 1b, the cyclic motion would involve a cyclic reversal of the bias voltage (AC-driven). This AC-driven case, though intriguing, goes beyond the scope of the present manuscript, as we are not considering here time-dependent biases. Another scenario would be that of Figure 1d, where the cyclic motion is in principle attainable by periodically hitting the device. Note that we are not dealing with the steady-state of sets of interacting nanomotors, such as those described in, e.g., [79–81]. Instead, here we are interested in the steady-state of the mechanical part of isolated quantum motors and pumps that interact solely with the electrons of a set of reservoirs and where, at most, Coulomb interactions are only taken into account within the local system.

To discuss the dynamics of cyclic motions in simple terms, we start by projecting Equation (7) on a closed trajectory defined in the space of *X*. By assuming a circular trajectory, the dynamics can be described by an angle *θ*, its associated angular velocity ˙ *θ*, the moment of inertia I, and the torques F, Fext, and *ξθ*. Using this, we obtain an effective angular Langevin equation equivalent to Equation (7),

$$\frac{d\theta}{dt} = \frac{1}{\mathcal{T}} \left[ \mathcal{F} - \mathcal{F}\_{\text{ext}} + \mathfrak{J}\_{\theta} \right]. \tag{9}$$

We assume that after a long waiting time, the system arrives at the steady-state regime where the mechanical motion becomes periodic, and it is then characterized by a time period *τ* such that *θ*(*t* + *τ*) = *θ*(*t*) and ˙ *θ*(*t* + *τ*) = ˙ *<sup>θ</sup>*(*t*). Moreover, we will assume that the stochastic force plays a minor role in the above equation, such that it does not affect the mean values of the dynamical variables *θ* and ˙ *θ*, i.e., the mean trajectories with or without the stochastic force approximately coinciding. This occurs, for example, at low temperatures or in mechanical systems with a large moment of inertia [22,39,42]. In the following, we will just ignore *ξθ* for practical purposes (this is the opposite regime of another type of nanomotors, the Brownian motors [1,82]). Under the above assumptions, we integrate both sides of Equation (9) from an initial position *θi* to a final one *θ f* and obtain:

$$\frac{\mathcal{T}}{2} \left[ \dot{\theta}\_f^2 - \dot{\theta}\_i^2 \right] \quad = \int\_{\theta\_i}^{\theta\_f} \left[ \mathcal{F} - \mathcal{F}\_{\text{ext}} \right] \, \text{d}\theta. \tag{10}$$

The torques in this equation are, in general, intricate functions of both *θ* and ˙ *θ*.

Therefore, the calculation of the *θ*-dependent angular velocity usually requires the resolution of a transcendental equation (see, e.g., [42]). Alternatively, one can obtain *θ*(*t*) from the numerical integration of the equation of motion by standard techniques like, e.g., the Runge–Kutta method. All this greatly complicates the study of quantum motors and pumps, to the point where it becomes almost impossible to draw any general conclusion. For this reason, one common simplification consists of taking the terminal velocity as constant during the whole cycle [13,22,24,26,39,49–51,56,74]. Indeed, this description is exact if the external agen<sup>t</sup> compels the constant velocity condition to be fulfilled in a controllable manner, as is often conceived in quantum pumping protocols. However, this is not the case in general, and typically, one expects internal variations for ˙ *θ* in one period. We now address this interesting issue in more detail. First, we take the integral in Equation (10) over the whole period. This gives:

$$\mathcal{W}\_{\rm ext} = \mathcal{W}\_{\rm F} \quad \text{where} \quad \mathcal{W}\_{\rm F} = \int\_0^\tau \mathcal{F}\dot{\theta} \,\mathrm{d}t, \quad \text{and} \quad \mathcal{W}\_{\rm ext} = \int\_0^\tau \mathcal{F}\_{\rm ext}\dot{\theta} \,\mathrm{d}t. \tag{11}$$

The above stationary state condition thus establishes that the work originated from the CIF is always compensated by the external mechanical work in the case that this regime can be reached. Now, let us assume for a moment that the terminal velocity of a nanodevice is constant and positive (we leave the discussion of the effect of the sign of ˙ *θ* for later when treating a concrete example in Section 7.1). If we now expand F in terms of ˙ *θ*, Equation (11) yields:

$$\mathcal{W}\_{\text{ext}} \quad = \sum\_{k} \left( \int\_{0}^{2\pi} \left. \frac{\partial^{k} \mathcal{F}}{\partial \dot{\theta}^{k}} \right|\_{\theta=0} \frac{\text{d}\theta}{k!} \right) \dot{\theta}^{k}. \tag{12}$$

Two important conclusions can be extracted from the above formal solution. First, there may be conditions where some roots of Equation (12) are complex numbers, meaning that the assumption ˙ *θ* = const. is nonsense, as the periodicity condition required for the steady-state regime would not be fulfilled. Second, for real solutions, it was shown in [22,42] that the moment of inertia I not only affects the time that it takes the mechanical system to reach the stationary regime, but also the internal range in the angular velocity, i.e., the difference Δ ˙ *θ* = ˙ *θ*max − ˙ *θ*min in one period. According to Equation (12),

˙ *θ* is independent of I, while the variation of ˙ *θ* scales with I−1, cf. Equation (10). Then, the ratio Δ ˙*θ*/ ˙*θ*, which is the relevant quantity in our analysis, should vanish for large I values, justifying the constant velocity assumption for large or massive mechanical systems.

The value of Fext is supposed to be controllable externally, as well as the voltage and temperature biases, which, in turn, affect the current-induced torque F. Therefore, under the above discussed conditions, ˙ *θ* can be thought as a parameter that surely depends on the internal details of the system, but it is also tunable by external "knobs". Let us analyze a concrete example: Consider a local system connected to two leads at the same temperature and with a small bias voltage *eV* = *μL* − *μ<sup>R</sup>*. By considering the current-induced torque up to its first nonadiabatic correction, i.e., F≈F(0) − *<sup>γ</sup>*˙*θ*, and assuming that Fext is independent of ˙ *θ*, the following relation must hold, according to the above discussion,

$$
\dot{\theta} \approx \frac{Q\_{I\_R}}{2\pi r \overline{\gamma}} \left( V - \frac{\mathcal{W}\_{\text{ext}}}{Q\_{I\_R}} \right) = \frac{Q\_{I\_R} V\_{\text{eff}}}{2\pi r \overline{\gamma}} \, , \tag{13}
$$

where *γ*¯ is the average electronic friction coefficient along the cycle, *QIR* is the pumped charge to the right lead, and we used Onsager's reciprocal relation between F and the charge current *IR* in the absence of magnetic fields [22,39,42,49]. Alternatively, if we assume that the external torque is of the form Fext = *γ*ext ˙ *θ*, one finds:

$$\dot{\theta} \approx \frac{Q\_{I\_R} V}{2\pi} \frac{1}{\left(\bar{\gamma} + \bar{\gamma}\_{\rm ext}\right)} = \frac{Q\_{I\_R} V}{2\pi \gamma\_{\rm eff}}.\tag{14}$$

Note in the above equations that, at least in the present order, the effect of the external forces can be described as a renormalization of the bias voltage *V* or the electronic friction coefficient *γ*. Numerical simulations in [22,42] showed that the above equations agree well in general with the steady-state velocities found by integrating the equation of motion. However, at very small voltages, essential differences may appear. There is a critical voltage below which the dissipated energy per cycle cannot be compensated by the work done by the CIF, and thus, ˙ *θ* = 0. We dubbed this the "nonoperational" regime of the motor. Moreover, when increasing the bias voltage, there is an intermediate region where a hysteresis cycle appears, and two values of the velocity are possible ( ˙ *θ* = 0 and those given by the above equations). Although in Section 7.1, we will take ˙ *θ* as constant when discussing a specific example, the reader should keep in mind that this approximation does not always hold, especially at very small voltages or I.
