Asymmetric power dissipation in electronic transport through a quantum point contact

We thank the referee for his/her positive overall judgment and useful remarks. We have taken into account the suggestions as explained in detail below.

1. The referee feels that the paper is not self-contained. We indeed use concepts and results from the literature, describing the main lines, and indicating the references upon which we build our analysis. We feel that a more detailed presentation of those aspects would not be useful and might divert the reader from the essential new physics that we present in our work. For example, we have outlined the main lines of the derivation of the transmission probability for an abrupt constriction in Sec. 3.1, but without rewriting the full description of the approach that can be found in Ref. [9]. The case of the damping rate is similar. We have presented a way to estimate the distance of the point with the strongest heat dissipation from the QPC in Sec. 4. from the knowledge of the damping rate in the 2DEG. Such a parameter is extracted from the literature taking into account the typical electron densities and the bias voltages used in the experiment of Ref. [13]. We believe that a detailed description of the evaluation of that parameter is not crucial for the understanding of our approach. Moreover, our approach remains valid in systems where other processes might dominate the damping rate. Of course, for a reader interested in the details of the damping rate in a 2DEG, we provide references [34-39], including a book.
2. The referee suggests a more detailed discussion of the choice of parameters. While we have provided general results in Sec. 2, we have indeed chosen a particular model system in Sec. 3. Following the comment of the referee, we have extended the discussion of the consequences of the particular choice of an abrupt QPC by a few sentences at the end of the first paragraph of Sec. 3.2.1. However, the quantitative results in Sec. 3 and Sec. 4 are quite general since the electrochemical potential, bias voltage, temperature and dissipated power are expressed in units of the main system parameter (the lowest transverse energy E_1 in the narrow part of the constriction). Only when we discuss the results beyond the qualitative features, we mention estimates of parameter values for the situation of the experiment of Ref. [13] in order to confront our results with the observations.

We thank the referee for his positive overall judgment and useful remarks. We have taken into account the suggested changes as explained in detail below.

1. Our model and our results are not limited to left-right symmetric systems since the equality between transmission coefficients for both directions of carrier transport does not require such a symmetry. It was only when presenting the low-voltage expansion of Sec. 2.2 and in a sentence that was placed in the 2nd paragraph of Sec. 2 in the original version that we adopted this simplifying hypothesis, which might have triggered the referee's comment. Indeed, the lowest order correction to the transmission coefficient is of second order in the bias voltage for left-right symmetric systems, but a first-order term cannot be excluded in the absence of such a symmetry. Following the remark of the referee, and in order to clarify the generality of our results, we have included in the revised version a first-order term in the low-voltage expansions in Sec. 2.2 and shown that it does not affect the leading order terms for the power dissipation and its asymmetry, which scales with the third power of the bias voltage. The results presented in Sec. 2 are therefore also valid for systems without left-right symmetry. While the example of the system chosen to obtain quantitative results in Sec. 3 has left-right symmetry, we have demonstrated that there is no strong influence of such a symmetry on the power dissipation asymmetry at low bias voltage. To make this point clear, in the revised version of our paper we have explicitly included a first-order term in bias in the expansion of the transmission (6), and we have introduced a discussion of its consequences for the expansions (7) and (8) of the dissipated power and the asymmetry below Eq. (8). Moreover, we have moved the comment about the absence of a first-order correction in bias voltage to the transmission for the special case of left-right symmetric systems from the paragraph above Sec. 2.1 to the end of Sec. 2.2.
2. As mentioned above, our results are not restricted to the case of geometrical or electrical symmetry. In the revised version, we mention the particular case of the electric asymmetry in the discussion of the low-voltage expansion at the end of Sec. 2.2.
3. For the example model of Sec. 3, we have chosen an abrupt QPC since it is a 2d model with a low number of parameters and for which an analytic solution for the energy-dependent transmission coefficient is available. We agree with the referee that the plateau oscillations would not appear for adiabatic QPC models with smooth width variation or smooth saddle-point shaped potential. We included a discussion of the issue at the end of the first paragraph in Sec. 3.2.1. indicating which are the specific features of Fig. 2 linked with the abrupt character of of the QPC.
4. The discussion of the effect of electron-electron interaction and the possible implementation of a self-consistent treatment were already presented in the second paragraph of Sec. 2, but not in Sec. 3, where some plots are extended to relatively large voltage values. Following the referee's suggestion, we added two sentences at the end of the second paragraph of Sec. 3.2.1. (where Fig. 3 is discussed) to alert the reader about the possible influence of electron-electron interactions at the strongest bias therein considered.