Nonadiabatic Nonlinear Optics and Quantum Geometry --- Application to the Twisted Schwinger Effect

[Reply to the Anonymous Report 1] We would like to deeply thank the referee for reading the manuscript carefully and giving us a positive report. Let us comment on the dissipation and decoherence effect indicated by the referee. Indeed, most of our analysis has been done within the framework of pure quantum evolution without the effect of the environment. The exception is the calculation of the photo-current given in (32), where we have used the steady-state solution of the master equation (31) which phenomenologically incorporates the dissipation effect. A more detailed analysis based on the Liouville von Neumann equation as performed in the reference mentioned by the referee (https://journals.aps.org/prb/abstract/10.1103/PhysRevB.99.214302) would be an interesting future project. We have added this comment in the main text above eq. (31).

[Reply to the Anonymous Report 2] We would like to deeply thank the referee for reading the manuscript carefully and giving us useful comments. We have modified the manuscript accordingly.

1. "The Tunneling probability in Eq. 10 should be defined. " We have added a paragraph describing how we define the tunneling probability below (1).

2. "Comment about v \kappa_{\parallel} `corresponding to the shift vector’ should be either fully explained or removed. Page 5." We thank the referee for pointing this out. The shift vector is an important quantity in the quantum geometric approach to non-linear optics. In order to fully explain this aspect in a self-contained way, we have performed several modifications to the manuscript. [1] We added two paragraphs on page 4 to give the definition and explanation of the shift vector (which is equivalent to the geometric amplitude factor in tunneling theory). The expression Eq. (7) gives the tunneling formula with the geometric amplitude factor. [2] Since quantum geometry is an important aspect of this work, we have changed the title to “Nonadiabatic Nonlinear Optics and Quantum Geometry — Application to Twisted Schwinger Effect” to stress its prominent role. We also changed the section titles (2. and 3.) to fit well with the change of the title. [3] We also added table I and its explanation on page 9 to fully explain the role of quantum geometry and the shift vector in non-linear optics. This gives a contrast between the perturbative theories of optical processes and the nonperturbative tunneling theory (which includes the shift vector contribution). We would like to acknowledge the referee for giving us the opportunity to make these changes, which we think will improve the value of this manuscript.

3. "Why should the expansion of Eq.11 in powers of \Omega to second order be equivalent to the model Eq. 4 on which the results of this paper as based? E.g., why cubic terms in expansion of Eq. 11 are not important near the band minimum?" The concern of the referee is indeed correct. This is the limitation of our work where all the results are obtained within a quadratic approximation. We did not try to go to further order since the derivation of the tunneling formula (2) and (7) was already based on a quadratic approximation (see text above (15)). Thus, it does not make sense to keep higher-order terms of the Hamiltonian while using the tunneling formula. The limitation of the quadratic approximation is already seen in Fig. 1 (c) where the numerical results are compared with the tunneling formula. Although they coincide well in the week field (=slow speed) regime, the deviation increase as the field (speed) becomes stronger (faster). We agree that a more detailed comparison with numerical calculation is necessary. However, we would like to leave this for future study.

4. "Give an order of magnitude estimate of the value of the electric field and \Omega need to observe these effects for a realistic condensed matter realization." For the 2D case, we have added a paragraph on page 11 at the end of section 3, where we estimate the Schwinger limit in a TMD material. It is reachable with current laser fields. For the 3D case, we already have an estimate of the crossover field below (33). This is also achievable with current laser fields.

1. We have added a comment on the role of dissipation and decoherence effect in the main text above eq. (31).
2. We have added a paragraph describing how we define the tunneling probability below (1).
3. In order to make the explanation on the shift vector more self-contained, and highlight the importance of the concept of quantum geometry, we have made the following modifications. [1] We added two paragraphs in page 4 to give the definition and explanation of the shift vector (which is equivalent to the geometric amplitude factor in tunneling theory). The expression Eq. (7) gives the tunneling formula with the geometric amplitude factor. [2] Since quantum geometry is an important aspect of this work, we have changed the title to “Nonadiabatic Nonlinear Optics and Quantum Geometry — Application to Twisted Schwinger Effect” to stress its prominent role. We also changed the section titles (2. and 3.) to fit well with the change of the title. [3] We also added table I and its explanation in page 9 to fully explain the role of quantum geometry and the shift vector in non-linear optics. This gives a contrast between the perturbative theories of optical processes and the nonperturbative tunneling theory (which includes the shift vector contribution).
We added remarks on experimental feasibility in page 11 at the end of section 3.