Motor-Pump Efficiencies

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As we stated above, the sign of the external force, together with the rotor's stationary condition and the first law of thermodynamics, determines the direction of the energy flow and, with it, the operation mode of the device. Obviously, as no other power sources are involved in this example, the efficiency of this energy conversion, defined as *η* = (output power)/(input power), is always equal to one. This, however, only establishes those regions in the parameter space where we can expect the device operating as a quantum motor or a quantum energy pump. In this section, we discuss the particular conditions that appear when the device operates through a specific type of current. In this sense, the motor regime corresponds to the situation in which a transport current (e.g., charge, heat, spin, etc.) flowing through the leads in response to a bias (voltage, temperature, spin polarization, etc.) delivers some amount of energy into the local system, which can be used as mechanical work. The pump regime, on the other hand, corresponds to the inverse operation in which the external work is exploited to produce a current flowing against the imposed bias. This topic was also discussed in [87] for charge and heat currents in a DQD device, where limitations to the efficiency of the considered processes were attributed to the different orders appearing in the frequency expansion of the currents. We here provide a similar analysis in terms of our explicit model for the mechanical system. The inclusion of the external force in the description of the model, as we shall see next, appears as the key ingredient in bridging the motor and pump regimes for a given choice of the bias.

For the device acting as a motor, the output power should be given by Wext/*<sup>τ</sup>*, under the condition Wext > 0, but we still need to specify the input power. If we consider that the mechanical rotor is driven by the electric current, i.e., due to some applied bias voltage and no thermal gradient applied, the input power is given by −*Q<sup>I</sup>* · *δV*/*<sup>τ</sup>*, and hence, the efficiency of this type of motor is:

$$\eta = -\frac{\mathcal{W}\_{\text{ext}}}{Q\_I \cdot \delta V} = 1 + \frac{Q\_I}{Q\_I \delta V}. \tag{72}$$

Equation (33) establishes that the maximum efficiency for this device is *η* ≤ 1, and the above equation tells us that the heat current produced by the bias voltage reduces the motor's performance. As in this work, we calculate such currents through an expansion in the angular velocity, the efficiency is also limited by this expansion. If in the calculation of ˙ *θ*, we consider Equation (60) up to first order in C(*k*) *F* , then, as discussed in Section 6, order-by-order energy conservation demands that the currents are to be considered up to second order, and in terms of the current coefficients, this takes the form:

$$\eta = -\frac{\mathcal{C}\_{\text{ext}} / \delta V}{\mathcal{C}\_{I}^{(0)} \dot{\theta}^{-1} + \mathcal{C}\_{I}^{(1)} + \mathcal{C}\_{I}^{(2)} \dot{\theta}}.\tag{73}$$

In the limits of the motor's operation regime, given by Cext = 0 and Cext = C(0) *F* , it is easy to see that the efficiency goes to zero, since for Cext = 0, the numerator in the above expression is zero, and for Cext = C(0) *F* the rotor's velocity goes to zero, so the denominator grows to infinity due to the contribution C(0) *I* / ˙*θ* from the leakage current. The same happens if we consistently include higher order terms in this expression. For example, if we use Equation (63) for the rotor's angular velocity, then we should add C(3) *I*˙*θ*2 in the denominator.

Away from this region, we enter in the "pumping domain" characterized by a charge current, which opposes the "natural" direction dictated by the bias voltage and Wext < 0. In this sense, the input power and the output power are inverted with respect to the motor region, and consequently, the efficiency of this "battery charger" device is given by:

$$\eta = -\frac{\mathcal{C}\_I^{(0)} \dot{\theta}^{-1} + \mathcal{C}\_I^{(1)} + \mathcal{C}\_I^{(2)} \dot{\theta}}{\mathcal{C}\_{\text{ext}} / \delta V}. \tag{74}$$

In this regime there is, however, an additional condition to be fulfilled, which is *QIδV* > 0. Regarding the different orders in *QI*, it usually happens that close to the transition point Cext = C(0) *F* , the charge current still flows in the bias direction, since it is dominated by the leakage current. In this region, we say that the pumping mechanism is "frustrated" as the energy delivered by the rotor is not enough to reverse the direction of the charge current. Going away from this region, the angular velocity acquires some finite value, reducing the zeroth-order contribution C(0) *I* / ˙*θ* to the point where it is equal to the higher order contributions C(1) *I* + C(2) *I* ˙*θ*, thus marking the activation point of the charge pump. In Figure 4a, we show the efficiency of the device as a function of the external force for different orbit sizes and fixed bias *δV* = 2*k*B*T*. In all cases, the device starts from Cext = 0 as a motor, and its efficiency reaches a maximum, which increases with the orbit size. Soon after this point, the motor's efficiency decreases to zero due to the leakage current effect, which becomes dominant at Cext = C(0) *F* . From this point, we can observe the gapped region for the frustrated pump, which is more pronounced for small orbits, since the first-order pumped charge *Q*(1) *I* is smaller than its quantized limit, and thereby, it takes a larger value of Cext to compensate the amount of pumped leakage current *Q*(0) *I* in a cycle.

**Figure 4.** Normalized efficiencies as a function of the normalized external work (Cext/C(0) *F* ) and orbit radius: *δ* = 10 (red), 20 (blue), 50 (green), and 100 (orange), in units of *k*B*T*. Panel (**a**) shows the electric motor/pump lowest order efficiencies (solid lines) for the device driven by a bias voltage *δV* = 2*k*B*T* and *δT* = 0. The next-order efficiency for the smallest orbit is shown in the dotted red and in the inset for negative Cext. Panel (**b**) shows the heat engine/refrigerator lowest order efficiencies for the device driven by a temperature gradient *δT* = 0.5*T* and *δV* = 0. The other parameters coincide with those of Figure 2.

Up to this point, we have discussed only the effect of the lowest order terms of the expansion in ˙ *θ*, given by Equations (62), (73), and (74). For the smallest orbit, in addition, we show in the dotted red line the next-order efficiency, obtained from Equation (63) and adding the *Q*(3) *I* term to Equations (73) and (74). We can see that for the motor regime, there are no significant changes, but for the pump regime, important differences appear. Firstly, in the Cext > 0 region, there is a cut-off for the external force in which the pumping mechanism is again frustrated, i.e., the charge current again points in the bias direction. As can be seen in Figure 5a,c, this decreasing in the efficiency is not attributed to the extra heat dissipated to the reservoirs, as one may expect. Note in the figure that the extra contribution to the pumped heat, *Q*(3) *J* , is negligible as compared to the lowest order terms. What happens here is that the third-order contribution to the pumped charge *Q*(3) *I* rapidly becomes dominant in the charge pump region, causing a sudden drop in the efficiency and, with it, the appearance of a second frustrated-pump region. Secondly, another higher order effect appears in the Cext < 0 region. There, the efficiency is nonzero for Cext/C(0) *F* < −8 (see the inset in Figure 4a), meaning that the pumping mechanism can be activated even when the external force points in the same direction as that of the current-induced force. Given the convention used for Fext in Equation (7) and the chosen parameters, in this region, the sign of ˙ *θ* remains the same as that when Fext = 0. There, the zeroth- and first-order contributions to the charge current flow in the same direction. In the analyzed case, again the third-order term is the one that reverses the direction of the total charge current; see Figure 5b. It is important to mention that the purpose of the present discussion is only to highlight deviations from the linear solution of Equation (62), not to analyze the convergence of the total pumped current. For the larger values of *δ* used in Figure 4a, we do not show the next-order corrections, as they are negligible in the shown range of Cext.

An analysis similar to the above one can be carried out for the device driven by a temperature gradient *δT*, defined through *TL* = *T* + *δT*/2 and *TR* = *T* − *δT*/2 and no bias voltage applied, such that for *δT* > 0, we have *T*hot = *TL* and *T*cold = *TR*. When Wext > 0, we have a motor device driven by a heat current in response to a thermal gradient (heat engine), then the input power should be given by <sup>−</sup>*QJ*hot/*<sup>τ</sup>*, and Equation (35) implies:

*η* = − Wext *QJ*hot = 1 + *QJ*cold *QJ*hot ≤ 1 − *T*cold *T*hot = *η*carnot. (75)-2 0 2 4 0 2.5 5 7.5 10 -2 0 2 4 -30 -20 -10 0 0 0.5 1 -30 -20 -10 0 10

**Figure 5.** Different order contributions to the pumped charge and heat as a function of the normalized external work for the case *δ* = 10 *k*B*T* (dotted red in Figure 4a). Panels (**<sup>a</sup>**,**b**) show the *k*th-order pumped charge *Q*(*k*) *I* = (*Q*(*k*) *IL* − *Q*(*k*) *IR* )/2. The sum of all these contributions is shown in solid black and denoted as *Q*(total) *I* . (**c**) Pumped heat contributions to the left and right reservoirs *Q*(*k*) *J* = *Q*(*k*) *JL* + *Q*(*k*) *JR* , divided by *Q*(total) *J* , i.e., the sum of all contributions from *k* = 0 to *k* = 3. The vertical gray lines mark different transition points: Cext/C(0) *F* = 1 is the motor/pump (energy pump) transition, while the other lines correspond to transitions between frustrated-pump/charge-pump regimes, i.e., when *Q*(total) *I* changes its sign.

As compared with Equation (73), we can see that the efficiency of the quantum heat engine, now given by:

$$\eta = -\frac{\mathcal{C}\_{\text{ext}}}{\mathcal{C}\_{\text{f\_{\text{hot}}}}^{(0)}\theta^{-1} + \mathcal{C}\_{\text{f\_{\text{hot}}}}^{(1)} + \mathcal{C}\_{\text{f\_{\text{hot}}}}^{(2)}\theta}{}'\tag{76}$$

is defined in the same range for Cext as in the electric motor. Regarding Figure 4b, the engine's normalized efficiency *η*/*η*carnot looks similar to that of the electric motor. Perhaps the only difference here is that for the smallest orbit *δ* = 10*k*B*T*, the efficiency maximum is very low, such that it cannot be appreciated on the employed scale of the plot.

Now, we move to the heat pump region where the device acts as a refrigerator, as we demand that the heat current flows against the direction dictated by *δT*. Therefore, in addition to the Wext < 0 condition, the overall amount of pumped heat in the cold reservoir should be negative, i.e., *QJ*cold < 0. The efficiency of the refrigerator, or coefficient of performance (COP), is then given by:

$$\text{COP} = \frac{Q\_{l\_{\text{cold}}}}{\mathcal{W}\_{\text{ext}}} = \frac{Q\_{l\_{\text{hot}}}}{Q\_{l}} - 1 \le \frac{T\_{\text{cold}}}{T\_{\text{hot}} - T\_{\text{cold}}} = \text{COP}\_{\text{carnot}} \tag{77}$$

where we can consistently expand *QJ*cold in terms of ˙ *θ*. In Figure 4b, we show the lowest order contribution from Equation (62), as the next-order calculation does not change significantly the efficiencies in the considered regimes of the parameters. Again, we can observe in Figure 4b a gap region where the device is frustrated since the work delivered by the rotor is not enough to reverse the direction of the heat current. One of the differences with the electric counterpart is that, for the refrigerator, the normalized COP develops a maximum that is always smaller than that of the quantum heat engine, while the obtained efficiency maxima (motor and pump) for a fixed orbit in Figure 4a are very similar. Additionally, for the chosen value *δT* = 0.5 *T* and small orbit radius, the device can only work as a heat engine, and the refrigerator cannot be activated even if the external force is large, as happens for *δ* = 20 *k*B*T* (solid blue line). The reason for this relies on the competition between the different orders in the pumped heat *QJ*cold : the reduction of the leakage pumped current, *Q*(0) *J*cold , by increasing the magnitude of ˙*θ*, is accompanied by an increase of the second-order contribution, *Q*(2) *J*cold , such that the first-order term, *Q*(1) *J*cold , may not be able to compensate these two and the requirement *QJ*cold < 0 cannot be fulfilled; see Figure 6.

**Figure 6.** Different order contributions to the pumped heat as a function of the normalized external work for the case *δ* = 20 *k*B*T* (solid blue line in Figure 4b). Here, *Q*(total) *J*refers to the sum of all contributions.
