Topological interface states -- a possible path towards a Landau-level laser in the THz regime

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Dear Editor,

thank you very much for sending me the two detailed and globally very positive reports on my manuscript "Topological interface states -- a possible path towards a Landau-level laser in the THz regime". I have amended the manuscript according to the Referees' requests, and please find enclosed a reply to both Referees.

I hope that the manuscript is now deemed suitable for publication in SciPostPhys.

Sincerely yours,
Mark Oliver Goerbig



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Reply to "Anonymous Report 1"


I would like to thank the Referee for her/his detailed and positive report on my manuscript, especially for the indeed relevant question raised in the third paragraph.

The Referee is absolutely right that the opposite surface of the topological insulator, albeit sharp, has the capacity to reabsorb photons emitted in the $(m=0,n=1)->(0,0)$ or $(0,0)->(0,-1)$ transitions since it hosts a chiral surface state that is submitted to Landau-level quantization. In order to suppress such unwanted reabsorption processes, the most promising path is to Pauli-block this transition at the sharp interface. This might be achieved by positioning the Fermi level between the (0,0) and the (0,1) state, e.g. by an electric gate or, more promisingly, by chemical doping with electron donors. I have added a paragraph at the end of Sec. V in response to this point raised by the Referee.

I am also grateful for the minor points (typos) which the Referee pointed out and that have been corrected. Notice just that one may use either $\ell$ or $l_S$ in the argument before Eq. (7), since $l_S=\sqrt{\ell \hbar v/\Delta}$ depends monotonically on $\ell$.

I hope that the Referee now deems the manuscript suitable for publication in SciPostPhys.

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Reply to "Anonymous Report 2"

I would like to thank the Referee for her/his detailed and globally positive report on my manuscript. I am also grateful for the critical questions she/he raised. They are addressed in detail below. The manuscript has been changed accordingly, and her/his comments have helped me to improve it, namely in view of its accessibility to the readers.

1.) The first question is concerned with the Faraday and Voigt geometry, which the Referee prefers to be defined with respect to the polarization plane of the photon. It is true that the wave vector is small within optical processes in matter that are essentially vertical with no wave-vector transfer due to the large speed of light as compared, e.g., to the Fermi velocity. Even if it is small in modulus, the photon nevertheless has a wave vector (and thus a direction of propagation) that is perpendicular to the polarization plane, as a consequence of Maxwell's equations, $\nabla\cdot B=\nabla\cdot E=0$, that read $q\cdot B= q\cdot E=0$ in reciprocal space. Here, E and B are the electric- and magnetic-field components of the photon, respectively, that span the polarization plane, and q is the wave vector. The two different geometries can thus and are often defined with respect to the orientation of the wave vector. A footnote has been added to clarify this point.

2. and 2.bis) Section II provided indeed a succinct introduction to the physical properties of interface states in a topological heterojunction, based on previous studies. In view of the Referee's useful comment, I agree that it might have been too succinct and that further information is useful to render the manuscript "self-contained". The section has therefore been enhanced with the introduction of a generic model Hamiltonian and the main steps in its diagonalization in the presence of a smooth interface. This explicit presentation of the model allows hopefully for a better appreciation of the analogy between the gap variation across the interface and the "fake" magnetic field as well as for understanding how the effective width parameter may be controlled by an inplane magnetic field.

3.) The Referee mentions that the present proposal requires pumping by a source in the THz regime and that she/he does thus not see the interest in the setup as a source of THz light. The interest of the present setup, in the case of resonant pumping, is that the source does not need to be coherent while the outcoming (laser) light is expected to be coherent. I agree that, in order to truly realize the effect, several intermediate steps (such as the measurement of cyclotron emission) are required to test the proposal. However, I would like to point out that one might also possibly use the bulk conduction band as a possible reservoir of electrons for stimulated light emission. In this case, one might pump at higher frequencies. This aspect is discussed in Sec. V below Eq. (15) [former Eq. (11)].

3.bis) The Referee asks for further justification of formula (31). In my response, I understand that she/he refers to the Eq. (11), which is now Eq. (15). The expression is indeed obtained directly from Ref. [33], see e.g. Eq. (20), with the help of Fermi's golden rule. The matrix element is given in terms of the effective Coulomb interaction, which is essentially given by $e^2/\epsilon l_B$ in the lower Landau levels (apart from a numerical prefactor, and in higher Landau levels the magnetic length needs to be replaced by the cyclotron radius $R_C~l_B\sqrt{2n}$) and the Dirac function is broadened by $1/\tau$ in terms of the dephasing time $\tau$. At resonance, when the frequency $\hbar\omega$ coincides with the level spacing $\Delta E$) a Lorentzian-type distribution $\hbar/\tau/[(\hbar\omega-\Delta E)^2+\hbar^2/\tau^2]$ then becomes $\tau/\hbar$, as in Eq. (11). Some intermediate steps have been added to justify the equation, but I have chosen not to provide (known) details about the form factors arising from the wave functions. They give lengthy expressions in terms of Laguerre polynomials that are eventually reduced to a numerical prefactor and that do not affect the argument based on simple orders of magnitude. However, I agree that they may be useful (and will be given) in a more complete study of the transition rates (see below in my response to point 5.bis).

4. and 5.) The figures have been changed in response to the Referee's suggestions. To show that there are no other resonant levels, Fig. 1 has been changed. Both, higher VP surface bands and the bulk conduction and valence bands have been added. I have chosen not to add further levels to Figs. 2 and 3 for reasons of visibility. However, the Fermi level has been added to both of them, in response to the Referee's request, and a different colour code has been adopted for the pump and emission processes (magenta and orange, respectively, not to use the same colours as for the Landau levels associated with the different VP bands).

5.bis) I agree with the Referee that a more detailed and quantitative study of the transition rates and a balance equation is required to test the present proposal. However, I most respectfully disagree with the Referee that such a study, which is beyond the scope of the present manuscript presenting the proposal, be required at this stage. The present manuscript is not meant to be a complete theoretical solution of the THz-laser problem in terms of topological heterojunctions, but it is a condensed-matter proposal that I deem promising, in line with the other Referee who states that she/he is "confident that this convincing idea will stimulate further promising studies". I therefore leave a more detailed study of the transition rates and a balance equation -- along with the necessary experimental studies -- for future work. The scope of the present manuscript and the need for further studies is now more clearly stated at the end of Sec. V.

6.) As a theoretician who is mainly concerned with fundamental research, I am not an expert in technological applications. However, there seems to be an apparent lack of coherent light sources in the THz regime (called the "THz gap") as compared to lasers in the visible and infrared range on the one-hand side and the microwave range on the other hand. Generically, a working Landau-level laser would be precisely situated in this frequency range. A sentence pointing out this technological interest has been added to the conclusions along with a new reference to a so-called "road-map".

7.) I am particularly grateful to the Referee for pointing out the length of the paragraphs, a possible fault related to my German origins, often more prominent in outrageously long sentences [Twain1880]. The revised version of the manuscript takes into account this linguistic criticism, and the paragraphs have been shortened or cut into sub-paragraphs.

I hope that the Referee now deems the revised manuscript suitable for publication in SciPostPhys.

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References (not added to the bibliography of the main text)

[Twain1880] Mark Twain, "The Awful German Language", in "A Tramp Abroad" (1880); open access: https://faculty.georgetown.edu/jod/texts/twain.german.html

Major changes in response to the Referees (minor changes not listed)

* enhanced discussion of the basic model in Sec. II, containing a derivation of the spectra, a clearer discussion of the analogy between the gap variation at the interface and a fake magnetic field, a quantitative reminder of how the effective interface width can be tuned with the help of an in-plane magnetic field
* change of the figures; Fig. 1 contains more levels as well as the bulk bands to allow the reader to appreciate the absence of degenerate transitions; Fermi level has been added to Figs. 2 and 3, and colors for the absorption/emission processes have been changed
* a short argument (and a footnote) added to justify the choice in the definition of the Faraday/Voigt geometry with respect to the direction of the photon's wave vector, which is equivalent (via Maxwell's equations) to a definition in terms of the photon's polarization
* further information provided about Fermi's golden rule in the calculation of rapid interaction-induced relaxation processes around Eq. (15) [former Eq. (11)]; it remains an order-of-magnitude argument
* short discussion about the relevance of relaxation processes added to Sec. V to underline the scope of the present manuscript
* discussion added to Sec. V about possible reabsorption processes by chiral surface at the opposite side, as well as possibility to block such processes
* sentence added to the conclusions about the technological relevance of (coherent) THz radiation source ("THz gap")