**1. Introduction**

In recent years, there has been a sustained growth in the interest in different forms of nanomachines. This was boosted by seminal experiments [1–8], the blooming of new theoretical proposals [9–30], and the latest developments towards the understanding of the fundamental physics underlying such systems [31–45]. Quantum mechanics has proven to be crucial in the description of a broad family of nanomachines, which can be put together under the generic name of "quantum motors" and "quantum pumps" [13,46–53]. They typically consist of an electromechanical device connected to electronic reservoirs and controlled by nonequilibrium sources; see Figure 1. These nonequilibrium sources may include temperature gradients and bias voltages among the reservoirs or even an external driving of the internal parameters of the system. The dimensions of the electronic component of these devices are normally within the characteristic coherence length of the electrons flowing through them, hence the essential role of quantum mechanics in their description.

Various aspects of quantum motors and pumps have been extensively studied in the literature. For example, it has been shown that quantum interferences can be exploited to boost the performance of these devices. Remarkably, some systems operate solely due to quantum interference, e.g., quantum pumps and motors based on chaotic quantum dots [13,39,54], Thouless quantum pumps and motors [13,22,55], or Anderson quantum motors [56], among others. On the other hand, the strong Coulomb repulsion between electrons in quantum-dot-based pumps and motors has shown to enhance the performance (or even induce the activation) of these nanodevices [26,42,57–59]. The effect of decoherence has also been addressed [22,39,54,60], as well as the influence of the friction forces and the system-lead coupling in the dynamics of quantum motors and pumps [22]. Indeed, the thermodynamics of those systems has proven to be a key aspect to study. In the last few years, different individual efforts have coalesced to give rise to a new field dubbed "quantum thermodynamics" [48–53,61], which studies the relations among the different energy fluxes that drive the motion of those machines where quantum mechanics plays a fundamental role.

**Figure 1.** Examples of local systems (enclosed by dashed lines) where the movement of a mechanical piece (in blue) is coupled to the flux of quantum particles traveling from/to infinite reservoirs (black hemiellipses). (**a**) A Thouless' adiabatic quantum motor made of charges periodically arranged on the surface of a rotational piece and interacting with a wire coiled around it [22]. (**b**) An Anderson's adiabatic quantum motor made of a multi-wall nanotube where the outer one, with random impurities, is shorter than the inner one. Another example of it can be made with a rotating piece as in (**a**), but with charges randomly distributed. (**c**) A double quantum dot capacitively coupled to a rotor with positive and negative permanent charges. The dots are assumed to be weakly coupled to the electron reservoirs [42]. (**d**) As a result of an external agent, a tip hits a conductive wire capacitively coupled to permanent charges underneath. This starts the oscillation of the wire, which in turn pumps electrons between the reservoirs [23].

Despite the progress in the theoretical description of quantum motors and pumps, most of the research has focused on parameter conditions that lie close to the thermodynamic equilibrium, i.e., small bias voltages, temperature gradients, or frequencies of the external driving [48–53,61]. This is reasonable since under such conditions, the linear response regime of the nonequilibrium sources gives an accurate description of the problem, greatly simplifying its general treatment. For example, in this regime, it is common to define dimensionless figures of merit made by some combination of linear response coefficients, which give a measurement of the efficiency or the maximum power that quantum devices can achieve. It is also known that such figures of merit fail in nonlinear regime conditions [52]. Although efforts have been made in this direction, currently, there is not a nonlinear version for the figures of merit, and the performance must be calculated from the microscopic details of the system's dynamics. One strategy to deal with such situations is to use phenomenological models where the linear response coefficients are

parameterized with respect to the voltage biases, the temperature gradients, or to other relevant parameters of the system; see [52] and the references therein. However, such parameterizations usually hide the physics behind the nonlinearities and require the optimization of a number of variables that grow very fast with the complexity of the model. The weakly nonlinear regime of transport has also been explored within the scattering matrix formalism. Under these conditions, it is enough to expand the response coefficients up to second order of the voltage biases and temperature gradients, which can be done by using standard quantum transport techniques. This approach has been applied to a variety of situations, where it proved to be a valuable strategy; see [48] and the references therein. However, it would be also important to extend this method to more general situations without hindering the description of the physical processes that take part in the nonlinear effects, while keeping the deep connections between the response coefficients.

Regarding the dynamics of quantum motors and pumps, one can notice that most of the works in the literature assume a constant terminal velocity of the driving parameters without a concrete model for them. A typical problem is that when these devices are coupled to nonequilibrium sources, nonconservative current-induced forces (CIFs) appear. These CIFs come, in the first place, by assuming a type of Born–Oppenheimer approximation where the electronic and mechanical degrees of freedom can be treated separately and, secondly, by evaluating the mean value of the force operator [10,11,22,24,26,34,35,37,39,42,44,49–51,62–71]. Because of the delayed response of the electronic degrees of freedom to the mechanical motion, one should include the so-called nonadiabatic corrections with the CIFs. This phenomenon is translated into a possible complex dependency of the CIFs on the velocity of the mechanical degrees of freedom. When the effect of the mechanical velocities on CIFs can be treated in linear response, it is clear whether it is adequate or not to assume a constant terminal velocity [22,39,42]. However, in far-from-equilibrium conditions, this subject has not been fully addressed.

In this article, we discuss two key aspects of far-from-equilibrium quantum motors and pumps: their steady-state dynamics, especially when CIFs present nontrivial dependencies on the terminal velocities; and their nonequilibrium thermodynamical properties, when a linear response description is not enough. We provide a systematic expansion to study the relations between the different energy fluxes that drive the quantum device. These aspects are illustrated in a concrete example where nonlinear effects due to nonequilibrium sources play a major role in the steady-state properties of the system. We show that these nonlinearities may result in, e.g., negative friction coefficients or motor/pump coexistence regimes.

Our work is organized as follows: In Section 2, we present the general model that describes the considered type of systems, and we derive an effective Langevin equation that characterizes the dynamics of the mechanical degrees of freedom, treated classically in the present context. In Section 3, we discuss in general terms the steady-state dynamics of quantum motors and pumps, highlighting some key aspects that differentiate close-to and far-from equilibrium conditions. In Section 4, we derive, on general grounds, the first law of thermodynamics for the kind of systems treated. Then, we expand the different energy fluxes passing through the system in terms of the nonequilibrium sources (temperature gradients, bias voltages, and velocities) for an arbitrary number of reservoirs. In this way, we obtain an order-by-order relation between the different energy fluxes entering and leaving the device. In Section 5, we perform a derivation of the rate of entropy production from first principles. Then, based on the second law of thermodynamics, we discuss the limits of the efficiency for different forms of quantum motors and pumps in general nonequilibrium conditions. In Section 6, we analyze and give physical interpretation to some of the relations obtained in Section 4. Finally, in Section 7, we consider the CIFs for strongly-interacting electrons in a particular example based on a double quantum dot system coupled to a mechanical rotor. We then analyze in detail the effects of higher order terms in the CIFs on the final steady state of the electromechanical system.

### **2. Current-Induced Forces and Langevin Equation**

In this section, we introduce the generic model for the treatment of CIFs and the standard method employed in the description of the dynamics of the mechanical degrees of freedom. As a starting point, we consider as the local system the region where electronic and mechanical degrees of freedom are present and coupled to each other, like the examples shown in Figure 1. Such a local system is generically modeled by the following Hamiltonian:

$$
\hat{H}\_{\rm local} = \hat{H}\_{\rm s}(\hat{X}) + \frac{\mathbf{P}^2}{2m\_{\rm eff}} + \mathcal{U}(\hat{X}, t),
\tag{1}
$$

where *X*ˆ = ( *X*ˆ 1, ..., *X*ˆ *N*) is the vector of mechanical coordinates and *P*ˆ = ( *P*ˆ 1, ..., *P*ˆ *N*) collects their associated momenta, *<sup>m</sup>*eff is the effective mass related to *X*ˆ , and *U*(*X*<sup>ˆ</sup> , *t*) represents some external potential, of a mechanical nature, that may be acting on the local system. The explicit time dependence on this potential thus emphasizes the fact that an external agen<sup>t</sup> can exert some effective work on the local system. The Hamiltonian *H*ˆ s includes both the electronic degrees of freedom and their coupling to the mechanical ones through:

$$
\hat{H}\_{\mathsf{F}}(\hat{\mathbf{X}}) = \sum\_{i} E\_{i}(\hat{\mathbf{X}}) \left| i \right\rangle \left\langle i \right| \,\tag{2}
$$

where the sum runs over all possible electronic many-body eigenstates |*i*. The local system is then coupled to macroscopic reservoirs, and the total Hamiltonian, including the mechanical degrees of freedom, reads:

$$
\hat{H}\_{\text{total}} = \hat{H}\_{\text{local}} + \sum\_{r} \hat{H}\_{r} + \sum\_{r} \hat{H}\_{\text{s},r} \,. \tag{3}
$$

Each lead *r* is described as a reservoir of noninteracting electrons through the Hamiltonian:

$$
\hat{H}\_r = \sum\_{k\sigma} \epsilon\_{rk} \mathfrak{E}\_{rkr}^\dagger \mathfrak{E}\_{rkr\prime} \tag{4}
$$

where *c*ˆ † *rkσ* (*c*<sup>ˆ</sup>*rkσ*) creates (annihilates) an electron in the *r*-reservoir with state-index *k* and spin projection *σ*. As usual, the reservoirs are assumed to be always in equilibrium, characterized by a temperature *Tr* and electrochemical potential *μ<sup>r</sup>*. The coupling between the local system and the *r*-lead is determined by the tunnel Hamiltonian:

$$
\hat{H}\_{\rm s,r} = \sum\_{kr\ell} \left( t\_{r\ell} d\_{\ell\sigma}^{\dagger} \pounds\_{rkr} + \text{h.c.} \right),
\tag{5}
$$

where *tr*- denotes the tunnel amplitude, assumed to be *k* and *σ* independent for simplicity, and the fermion operator ˆ *d*† *σ* (ˆ *dσ*) creates (annihilates) one electron with spin *σ* in the --orbital of the local system. The tunnel-coupling strengths Γ*r*- = <sup>2</sup>*πρr*|*tr*-| 2 then characterize the rate at which the electrons enter/leave the local system from/to the *r*-reservoir, where *ρr* is the density of states in the *r*-lead. Note that *H*ˆ s is defined in the eigenstate basis, while *H*ˆ s,*r* is written in terms of single-particle field operators. The tunnel matrix elements accounting for transitions between different eigenstates can then be obtained from linear superpositions of the above tunnel amplitudes [72].

To obtain an effective description of the dynamics of the mechanical degrees of freedom through a Langevin equation, we start from the Heisenberg equation of motion for the *P*ˆ operator, which yields:

$$m\_{\text{eff}} \frac{d\dot{\tilde{X}}}{dt} + \nabla \dot{\mathcal{U}} \mathcal{U}(\hat{\mathcal{X}}, t) = -\nabla \hat{H}\_s(\hat{\mathcal{X}}).\tag{6}$$

The measured value *A*measured of an observable described by an operator *A*ˆ can always be taken as its mean value *A* = *A*<sup>ˆ</sup> plus some fluctuation *ξA* around it, i.e., *A*measured = *A* + *ξ<sup>A</sup>*. We will work under the nonequilibrium Born–Oppenheimer approximation [13,22,34,35,37,39,49,73, 74] (or Ehrenfest approximation [33,63,75–77]), where the dynamics of the electronic and mechanical degrees of freedom can be separated and the latter is treated classically. This allows us to neglect the fluctuations of the terms appearing in the left-hand side of Equation (6) and describe the mechanical motion only through the mean value *X*, which is reasonable for large or massive objects. With this in mind, we obtain the following Langevin equation of motion:

$$m\_{\rm eff} \frac{d\dot{X}}{dt} + F\_{\rm ext} = F + \mathcal{J}\_{\prime} \tag{7}$$

where *F* = − ∇<sup>ˆ</sup> *H*s = *i* [*H*ˆ s(*X*<sup>ˆ</sup> ), *P*ˆ] and *ξ* account for the mean value and the fluctuation of the CIF, respectively (throughout this manuscript, we take *h*¯ = 1 for simplicity). As we shall see later on, the external force applied to the mechanical part of the local system, *F*ext, plays the role of an eventual "load" force for a quantum motor or a "driving" force for a quantum pump. As this force will be typically opposed to the CIF, we define *F*ext with a minus sign for better clarity in future discussions. The main task, therefore, relies on the calculation of the CIFs from appropriate formalisms capable of describing the dynamics of the electronic part of the system. Once these forces are calculated, we can use Equation (7) to integrate the classical equations of motion and obtain the effective dynamics of the complete electromechanical system.

In most previous works, *F* is expanded up to first order in *X*˙ , i.e., *F* ≈ *F*(0) − *γ* · *X*˙ . The resulting CIF is then the sum of an adiabatic contribution *F*(0) and its first nonadiabatic correction *F*(1) = − *γ X*˙ , respectively. Under this approximation, Equation (7) turns into:

$$m\_{\rm eff} \frac{d\dot{X}}{dt} + F\_{\rm ext} = F^{(0)} - \gamma \cdot \dot{X} + \mathfrak{F}.\tag{8}$$

Explicit formulas for the calculation of *<sup>F</sup>*(0), *γ*, and *ξ* in terms of Green functions and scattering matrices were derived in [10,11,34,35,37] and extended in [22,39] to account for decoherent events. Although these expressions were obtained in the context of noninteracting particles, they can be used in effective Hamiltonians derived from first principles calculations [62,64]. In [49–51], the CIFs were obtained from the Floquet–Green's function formalism. The role of Coulomb interactions was addressed through different formalisms and methods like, e.g., many-body perturbation theory based on nonequilibrium Green's functions [44]; modeling the system as a Luttinger liquid [24]; and using a time-dependent slave-boson approximation [26]. In [42], explicit expressions for the CIFs within the Coulomb blockade regime of transport were obtained using a real-time diagrammatic approach [78], which we present in more detail in Section 7 when considering the example of Figure 1c.
