Photon pumping, photodissociation and dissipation at interplay for the fluorescence of a molecule in a cavity

In this paper the Authors study theoretically the time-resolved fluorescence spectrum of a model molecule placed inside an optical cavity.
Many effects are simultaneously considered, including electron-electron and electron-nuclear interactions, the nuclear motion, the quantum nature of the light, and the coupling of the system with a Caldeira-Leggett bath. The presented results are obtained within a virtually exact time-dependent numerical methods based on configuration interaction.
The Authors are able to assess, among the several and complex competing effects, what are the physical processes that tend to enhance or quench the fluorescence emission.
In particular they find that fluorescence is enhanced by a fast pumping rate while is suppressed by cavity leakage, electronic correlations and molecular dissociation.
The manuscript is well written and the results are presented in a clear and comprehensive way, with the help of explanatory figures, and by including additional details in the appendices.

Despite the model is very simple, exact results are highly valuable as they serve as a reference point to develop and benchmark approximations for the study of more realistic systems where the exact solution is out of reach.
For this reason the presented work can potentially guide future works on the topic, and I therefore recommend it for publication after the Author address the minor points listed below.

1) The model contains several parameters. While I understand that it is not possible to present a systematic study across the whole phase space, the Authors should explain why the chosen values of the model parameters are relevant to capture the qualitative behavior of a realistic molecule in an optical cavity.

2) The Caldeira-Leggett bath responsible for the cavity leakage is treated within the Ehrenfest approximation. The Authors should discuss the validity of this approximation as it has been shown to violate the principle of detailed balance. This is could be relevant when considering the energy transfer from the cavity modes to the bath.

3) As one of the main aspects of the paper is the study of fluorescence and SHG spectra on the pumping rate, the Authors should show and discuss explicitly the pumping protocol that they have considered. They only mention the value of the pumping speed.

1-The model accounts for different competing light-matter phenomena: leakage, photon confinement, nuclear motion and electronic correlations.
2-Exact time-dependent treatment of all degrees of freedom.

1-The molecular model is quite limited.
2-The interatomic interactions seem unrealistic.
3-Relevance of the model unclear.

In this work Gopalakrishna et al, develop a simple model for a one dimensional molecular dimer coupled to cavity photons and driven by a fluorescent field. Despite its simplicity on the molecular side, the model has a diverse character as it incorporates several different competing light-matter phenomena and mechanisms, like the confinement of photons via the cavity, photon pumping, photon leakage, nuclear motion and electron interactions. All degrees of freedom are treated quantum mechanically via an exact time-dependent configuration interaction approach.

The most important findings of this work are: (i) the second harmonic generation becomes larger with faster photon pumping. (ii) The electronic interactions suppress the fluorescence signal. (iii) The fluorescent and second harmonic generations signals depend significantly on the atomic mass and on the cavity leakage.

The manuscript it is definitely interesting and well-written. It incorporates many different aspects of light-matter interaction and several interesting phenomena take place as I summarized above. However, in my opinion the model molecule is clearly limited, it is one-dimensional which implies that rotational degrees of freedom are not taken into account, but most importantly only a single orbital per atom/site is included which essentially allows only a hopping to take place for the electrons between the two atoms/sites. Thus, the relevance of the model for realistic systems to me it is not obvious but rather unclear.

Further, the significance of the results is not evident to me. In my opinion this work does not meet the criteria for publication in Scipost Physics. I do not see how the results of this work can be considered as groundbreaking or opening a new pathway or a new research direction.

In my opinion, this is an interesting work which fits better to the publication criteria of Scipost Physics Core. I could recommend publication in Scipost Physics Core once the following points are adequately addressed.

1) The Hamiltonian, in the un-numbered equation, where the model is introduced it is written in a mixture between first and second quantized notation. This is rather unusual and it makes difficult to intuitively understand the model. It would be beneficial for the accessibility of the work a pure first or second quantized definition of the Hamiltonian to be given. I am particularly referring to the fact that the matter part is at the same time described with momentum and position operators in configuration space and fermionic annihilation and creation operators.

2) The un-numbered equation should be numbered as it is the defining equation of the model.

3) The interatomic potential used in the model is of the form 1/x^4. However, interatomic potentials typically are Lennard-Jones or Morse potentials. None of these potentials has such a form. The authors should clarify why they use the 1/x^4 potential, it seems rather unphysical. Also they should clarify to what extent the obtained results depend on the form of the chosen potential.

4) For the treatment of the bath the authors employ the well-known Caldeira-Leggett model. As they describe in Appendix A.4, the authors drop the quadratic ~(x)^2 term from the Caldeira-Leggett model, and they treat only the bilinear coupling between the molecule and the bath. However, dropping the quadratic term violates the translational invariance of the model, rendering their results dependent on the reference frame. To put differently performing a translation r -> r+a the model Hamiltonian does not remain invariant. This makes questionable the obtained results. The authors need to substantiate the validity of their approximation or they should include the quadratic term.

In this interesting manuscript, Gopalakrishna et al. study second harmonic generation (SHG) in a model molecule in an optical cavity. The Authors employ an exact time-dependent configuration interaction method to include all quantum degrees of freedom (electrons, photons and relative atomic motion) on equal footing. Even if the description is for a model molecular system, the theoretical formulation at the exact level is nonetheless impressive. The reported findings shed light on several optical response scenarios: Electron-correlation effects on Mollow spectra, a competition between SHG and photo-dissociation, and photon-pumping effects on fluorescence.

While the methodological aspects and the reported findings certainly deserve to be published, the manuscript in its present form contains some issues (mostly straightforward), which would require a revision before I can recommend publication in SciPost Physics. I elaborate my observations below.

(1) The manuscript has been structured with an appendix for technical details to better streamline the core message. This is fine as long as important concepts are discussed so that the development is properly justified. For example, the total Hamiltonian contains external fields V_ext, for which the Authors write: ''will be specified later''. However, starting on line 96 the Authors already state time-scale arguments which would require knowing something about the external fields. The specific form of the driving is given only later as a footnote [45], which in my opinion would deserve a proper description in the main text since it relates to, e.g., the accumulated photon number (central concept).

(2) The photon Hamiltonian on line 88 relates omega to fluorescent photons. In Eq. (1) these are referred to as omega'. This is of course a choice (and a very minor issue) but it would be helpful to remain consistent. The primed symbols were generally linked to the fluorescent part. The spectrum P is also sometimes written P(w,t) and sometimes P(t,w).

(3) The fluorescent coupling is studied in two ways: phenomenologically with an exponential damping ~exp(-Gamma*t) and more systematically with a bath of harmonic oscillators. I feel this comparison is an important contribution (as many studies have been conducted using the phenomenological approach), and the agreement in Figure 3 shows that the value Gamma=0.02 is ''appropriate'' in this situation. Is there a way to estimate the strength of Gamma from the Caldeira-Legget bath? [See also point (7) below.]

(4) The Authors mention they have ''ensured numerical convergence with respect to these parameters'' referring to the number of cavity photons, fluorescence photons, and nuclear motion grid points. It would be interesting to know from a computational point of view what these numbers are roughly to appreciate the computational cost.

(5) Figure 1 discussion in the text refers to panels a-d but the figure contains only panels a-b. The associated results are distinguished with empty/filled curves. In the Figure 1 caption there could be some punctuation between 9 and t_1 so the photon number does not get mixed with the pumping time parameter. In relation to my first point, here it is difficult to appreciate the role of g_d.

(6) Figure 2 and 6 references seem to have been mixed. The text in section 3.1 refers to Figure 6 although it is clear Figure 2 is described. Regarding Figure 2, the Authors write about the excited state populations: ''very small for the pumped cases, but noticeably large for the coherent case''. I do not think the figure really supports this. It is true that in the coherent case n_1 is about twice as large as in the pumped case. To further assess the validity of this statement, in Figure 6 (Appendix A.3), it is not so easy to compare the peak heights.

(7) Is it not possible (or reasonably justifiable) to try to obtain the long-time limit of the fluorescence spectra for the Caldeira-Legget bath? I understand the motivation to compare the peak positions in Figure 4 with the long-time limit of the exponential dissipation. The peak-height comparison however seems more difficult: The Authors write ''for bath dissipation the intensity of P(w',t) is considerably weaker''. In this regard, is there some insight why the Mollow triplet is not as prominent (or not there at all) for the harmonic bath?

(8) The Authors conclude by a discussion on scaling their development up to more realistic systems. It might be useful for the reader to get an idea what this would amount to: E.g., what are the Gross-Pitaevskii limit or nonequilibrium Green's functions (electronic, photonic etc). Since this is probably not the point of this manuscript, I think references to literature would suffice.