# Fermi polaron laser in two-dimensional semiconductors

### Submission summary

 As Contributors: Tomasz Wasak Arxiv Link: https://arxiv.org/abs/2103.14040v2 (pdf) Date submitted: 2022-04-21 09:22 Submitted by: Wasak, Tomasz Submitted to: SciPost Physics Academic field: Physics Specialties: Atomic, Molecular and Optical Physics - Theory Approach: Theoretical

### Abstract

We study the relaxation dynamics of driven, two-dimensional semiconductors, where itinerant electrons dress optically pumped excitons to form two Fermi-polaron branches. Repulsive polarons excited around zero momentum quickly decay to the attractive branch at high momentum. Collisions with electrons subsequently lead to a slower relaxation of attractive polarons, which accumulate at the edge of the light-cone around zero momentum where the radiative loss dominates. The bosonic nature of exciton polarons enables stimulated scattering, which results in a lasing transition at higher pump power. The latter is characterized by a superlinear increase of light emission as well as extended spatiotemporal coherence. As the coherent peak is at the edge of the light-cone and not at the center, the many-body dressing of excitons can reduce the linewidth below the limit set by the exciton nonradiative lifetime.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission 2103.14040v2 on 21 April 2022

## Reports on this Submission

### Report

The manuscript by Wasak et al. presents calculations of the dynamics of populations in a system of exciton polarons. These calculations, following from deriving a quantum Boltzmann equation for the polaron populations, describe relaxation from a population of polarons in the (upper) repulsive state to the (lower) attractive state, and then relaxation within the attractive branch. The main result of the manuscript is that this system shows nonlinear behavior of the relaxation process, due to stimulated scattering, analogous to a laser. Key to this is that relaxation from the repulsive to the attractive branch involves transitions to higher momentum states, due to energy-momentum conservation in scattering with electrons. This means that these attractive branch polarons start outside the light cone. When the population is low, most exciton-polarons are lost by non-radiative decay before they relax into the light cone. When population is high, stimulated scattering means that most exciton-polarons relax to the light cone. This leads to a laser-like transition in the radiative efficiency.

The work appears novel and in general the methods seem appropriate (see specific questions below). On balance, there is a reasonable case that this work "opens a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work", consistent with the SciPost Physics criteria. This is because it shows the potential for interesting results in studying the dynamics of this polaron system, and its response to selectively pumping the repulsive branch. While there has been significant recent interest on this exciton-polaron system, this has not generally explored the dynamics or nonlinear response to pumping.

This current work does rely on some approximations that may require further investigation. For example, the role of the trion-hole continuum is mostly ignored, and effects of broadening due to coupling between polaron resonances and the continuum is not considered in the dynamics. Further, relaxation is assumed to only occur via the excitonic part of the polaron scattering from electrons, neglecting the role of electron-electron scattering on the hybrid polaron. Nevertheless, despite these issues (which future work might address), this manuscript does provide reasonable evidence that interesting behavior should arise in this system, prompting future work.

As noted below, there are a number of issues that require clarifying in the current manuscript.

### Requested changes

I list first here both points that require addressing before publication, and then some optional suggestions the authors may wish to consider.

1. I have some concern about describing the behavior as being a laser, rather than laser-like. I agree there is a strong similarity, in that the transition is driven by a change of radiative efficiency. However the behavior is not driven by light amplification nor by stimulated emission of radiation. The term laser-like may be clearer; the authors may wish to consider whether they think the current term is appropriate.

2. I note that the references in the manuscript generally appear to date from before the initial submission of this paper in March 2021. The authors may wish to consider whether there are more recent references that should be included. For example, work on other aspects of interactions between polarons and electrons such as 10.1103/PhysRevB.103.075417 and 10.1103/PhysRevB.105.L041401

3. Regarding Eq. 9, the authors may wish to consider including a figure that shows the comparison between this analytic prediction and the results of the numerics for the form of the tail.

4. In section VI, it is stated that "the linewidth rapidly increases as the radiative loss becomes more prominent" at large pump strength. While this seems plausible from the physics described, it is hard to see how this appears in the expression given for S(omega). Could the authors explain where this appears in the equation more clearly?

5. In defining g^{(1)}(r,t), the definition written appears to suggest this is just the phase of G^{(1)}(r,t). I would assume the definition should be G^{(1)}(r,t)/|G^{(1)}(0,0)|

6a. In the start of Appendix A (after Eq. A2) A few points are unclear about the model of pumping. First, it is not clear why the pump term consists of both a term that increases polaron population and one that decreases it. Could this be explained further?

6b. Second, it is stated here that the assumption of frequency-independent pump is relaxed later. While it is clear in the main text that this is so (in that only the repulsive branch is pumped), the relaxation of this assumption does not seem to be explicitly discussed in the appendix. Should this be discussed in Appendix B, when writing the pumping term in the bare Green's functions?

7. Before Eq. A6, "Habbard"->"Hubbard"

8. The diagrams shown in Figure 5 do not seem to correspond directly to the equations actually used, as given in Eq. A8. Most notably,in the equations, the molecular channel is treated as a specific resonance, leading to a Green's function with a single argument. In contrast, the diagrams suggest it is dealt with by a T matrix, which would be a function of three momenta, that includes both the bound molecular channels and the scattering continuum. The equations written would seem instead to correspond to those in the attached file, where red and blue lines are as in the manuscript and grey lines indicate the molecule Green's function.

This should be clarified before publication.

9. In Appendix A.5, the sentence "In Fig. 1(b) in the main text..." is unclear. I suggest this should read "In Fig.1(b) in the main text we present the results of the self-consistent calculations of the excitation spectrum, using the parameters as given above".

10. In Appendix A.6 "These function are" -> "These functions are"

11. In Appendix A.6, in discussing Z_alpha(k), the method described of dividing the energy range by the maximum of the molecular spectral function presumably has the effect of imposing that Z_att+Z_rep=1. That is, this assumes there is no transfer of spectral weight to the trion-hole continuum. Is this assumption justified?

12. Throughout Appendix A7 and Appendices B, C, the manuscript keeps switching whether to label real-space coordinates as r or x. Given that x is used for the exciton channel label, it might be clearer to use r throughout. In any case, the notation should be consistent.

13. After Eq. A12, the discussion of momentum arguments is confusing, and seems likely to be wrong. The equation involves three momenta, k, q, q', but the discussion refers also to a fourth momentum p. This should be checked and clarified.

I also note that in A12, each line has a separate equation number, even though this is only one equation. The same applies to A15. (Other equations, such as E1, E8 do not do this.

14. In Figure 6, it would be helpful to add a legend or colorscale to label the meaning of the line colors.

15. In Appendix E, the subscript e on \epsilon_e is missing in Eq. E1, Eq. E3.

16. I Appendix E, after E14, the discussion changes from calculating W to calculating Gamma. However the text is not very clear. It would help to change "In the next step" to instead say "Having now calculated $W...$ we now turn to calculating $\Gamma....$" or equivalent.

• validity: good
• significance: good
• originality: good
• clarity: ok
• formatting: reasonable
• grammar: good

### Report

The manuscript by Wasak et al. presents calculations of the dynamics of populations in a system of exciton polarons. These calculations, following from deriving a quantum Boltzmann equation for the polaron populations, describe relaxation from a population of polarons in the (upper) repulsive state to the (lower) attractive state, and then relaxation within the attractive branch. The main result of the manuscript is that this system shows nonlinear behavior of the relaxation process, due to stimulated scattering, analogous to a laser. Key to this is that relaxation from the repulsive to the attractive branch involves transitions to higher momentum states, due to energy-momentum conservation in scattering with electrons. This means that these attractive branch polarons start outside the light cone. When the population is low, most exciton-polarons are lost by non-radiative decay before they relax into the light cone. When population is high, stimulated scattering means that most exciton-polarons relax to the light cone. This leads to a laser-like transition in the radiative efficiency.

The work appears novel and in general the methods seem appropriate (see specific questions below). On balance, there is a reasonable case that this work "opens a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work", consistent with the SciPost Physics criteria. This is because it shows the potential for interesting results in studying the dynamics of this polaron system, and its response to selectively pumping the repulsive branch. While there has been significant recent interest on this exciton-polaron system, this has not generally explored the dynamics or nonlinear response to pumping.

This current work does rely on some approximations that may require further investigation. For example, the role of the trion-hole continuum is mostly ignored, and effects of broadening due to coupling between polaron resonances and the continuum is not considered in the dynamics. Further, relaxation is assumed to only occur via the excitonic part of the polaron scattering from electrons, neglecting the role of electron-electron scattering on the hybrid polaron. Nevertheless, despite these issues (which future work might address), this manuscript does provide reasonable evidence that interesting behavior should arise in this system, prompting future work.

As noted below, there are a number of issues that require clarifying in the current manuscript.

### Requested changes

I list first here both points that require addressing before publication, and then some optional suggestions the authors may wish to consider.

1. I have some concern about describing the behavior as being a laser, rather than laser-like. I agree there is a strong similarity, in that the transition is driven by a change of radiative efficiency. However the behavior is not driven by light amplification nor by stimulated emission of radiation. The term laser-like may be clearer; the authors may wish to consider whether they think the current term is appropriate.

2. I note that the references in the manuscript generally appear to date from before the initial submission of this paper in March 2021. The authors may wish to consider whether there are more recent references that should be included. For example, work on other aspects of interactions between polarons and electrons such as 10.1103/PhysRevB.103.075417 and 10.1103/PhysRevB.105.L041401

3. Regarding Eq. 9, the authors may wish to consider including a figure that shows the comparison between this analytic prediction and the results of the numerics for the form of the tail.

4. In section VI, it is stated that "the linewidth rapidly increases as the radiative loss becomes more prominent" at large pump strength. While this seems plausible from the physics described, it is hard to see how this appears in the expression given for S(omega). Could the authors explain where this appears in the equation more clearly?

5. In defining g^{(1)}(r,t), the definition written appears to suggest this is just the phase of G^{(1)}(r,t). I would assume the definition should be G^{(1)}(r,t)/|G^{(1)}(0,0)|

6a. In the start of Appendix A (after Eq. A2) A few points are unclear about the model of pumping. First, it is not clear why the pump term consists of both a term that increases polaron population and one that decreases it. Could this be explained further?

6b. Second, it is stated here that the assumption of frequency-independent pump is relaxed later. While it is clear in the main text that this is so (in that only the repulsive branch is pumped), the relaxation of this assumption does not seem to be explicitly discussed in the appendix. Should this be discussed in Appendix B, when writing the pumping term in the bare Green's functions?

7. Before Eq. A6, "Habbard"->"Hubbard"

8. The diagrams shown in Figure 5 do not seem to correspond directly to the equations actually used, as given in Eq. A8. Most notably,in the equations, the molecular channel is treated as a specific resonance, leading to a Green's function with a single argument. In contrast, the diagrams suggest it is dealt with by a T matrix, which would be a function of three momenta, that includes both the bound molecular channels and the scattering continuum. The equations written would seem instead to correspond to those in the attached file, where red and blue lines are as in the manuscript and grey lines indicate the molecule Green's function.

This should be clarified before publication.

9. In Appendix A.5, the sentence "In Fig. 1(b) in the main text..." is unclear. I suggest this should read "In Fig.1(b) in the main text we present the results of the self-consistent calculations of the excitation spectrum, using the parameters as given above".

10. In Appendix A.6 "These function are" -> "These functions are"

11. In Appendix A.6, in discussing Z_alpha(k), the method described of dividing the energy range by the maximum of the molecular spectral function presumably has the effect of imposing that Z_att+Z_rep=1. That is, this assumes there is no transfer of spectral weight to the trion-hole continuum. Is this assumption justified?

12. Throughout Appendix A7 and Appendices B, C, the manuscript keeps switching whether to label real-space coordinates as r or x. Given that x is used for the exciton channel label, it might be clearer to use r throughout. In any case, the notation should be consistent.

13. After Eq. A12, the discussion of momentum arguments is confusing, and seems likely to be wrong. The equation involves three momenta, k, q, q', but the discussion refers also to a fourth momentum p. This should be checked and clarified.

I also note that in A12, each line has a separate equation number, even though this is only one equation. The same applies to A15. (Other equations, such as E1, E8 do not do this.

14. In Figure 6, it would be helpful to add a legend or colorscale to label the meaning of the line colors.

15. In Appendix E, the subscript e on \epsilon_e is missing in Eq. E1, Eq. E3.

16. I Appendix E, after E14, the discussion changes from calculating W to calculating Gamma. However the text is not very clear. It would help to change "In the next step" to instead say "Having now calculated $W...$ we now turn to calculating $\Gamma....$" or equivalent.

### Attachment

• validity: good
• significance: good
• originality: good
• clarity: ok
• formatting: reasonable
• grammar: good