Non-Poissonian photon statistics from macroscopic photon cutting materials

In optical materials energy is usually extracted only from the lowest excited state, resulting in fundamental energy-efficiency limits such as the Shockley–Queisser limit for single-junction solar cells. Photon-cutting materials provide a way around such limits by absorbing high-energy photons and ‘cutting' them into multiple low-energy excitations that can subsequently be extracted. The occurrence of photon cutting or quantum cutting has been demonstrated in a variety of materials, including semiconductor quantum dots, lanthanides and organic dyes. Here we show that photon cutting results in bunched photon emission on the timescale of the excited-state lifetime, even when observing a macroscopic number of optical centres. Our theoretical derivation matches well with experimental data on NaLaF4:Pr3+, a material that can cut deep-ultraviolet photons into two visible photons. This signature of photon cutting can be used to identify and characterize new photon-cutting materials unambiguously.

The submitted paper propose to use time correlation spectroscopy to demonstrate without any ambiguity the occurrence of quantum cutting effect in NaLaF4:Pr3+. Indeed, former works in that field always tend to prove the effect using comparison of excitation spectra showing then that quantum yield can be over 1. Nevertheless excitation spectra are always experimental conditions rendering comparison hardly fully reliable. Since quantum cutting corresponds to the successive cascade emission of 2 photons from a single emitter, the author are using start-stop measurement where the first photon from the cascade is the triggering the electronics (start) while the second photon down to the ground state is the stop. This approach enables thus, to measure the decay time of the intermediate level and to demonstrate that the 2 photons arise from the same emitter. It corresponds to the bunching technics. Playing with 2 experimental conditions, with spectral separation and without spectral selection on the 2 detections channel, the authors achieve for the first time to estimate the yield of the quantum cutting effect which is, beyond the demonstration of the correlation between red/blue photon, the main result. I consider that the experimental evidence is clear and that this paper brings new insights in the field of quantum cutting . I nevertheless invite the authors to revise their manunuscript for the following reasons. The Monte Carlo simulation is, from my point of view useless and bring more confusion than help for the understanding. Indeed, bunching technic is rather classical, and eq 2 is well admitted. In addition, the population eq can be solved without MC. Last remark for Fig1-g, what does mean N=1 when keeping Nx=2?? Why the simulation is diverging from the analytical solution. It seems even that that it does not reproduce the curve correctly for N=10. I suggest to entirely remove the section on MC and the fig:1-g which does not bring additional information. For clarity, it would be better to use another sign for excitation rate, since X is also the label of the level. When comparing the case of various Nx, the authors claim that the excitation has no effect on the signal to noise ration which is not clear. It suggests that Nx has no effect on the results. In such a situation, 2 goals might be forseen. Either, the experiment tends to demonstrate the occurrence of the quantum-cutting effect, or the goal is to extract parameter from the g2 fitting. In the latter case, I am not sure that the confidence on the deduced kx is the same for all the Nx. A small contrast with a high S/N ration might not be equivalent to a high contrast with a weak S/N. I suggest to improve the statistical analysis from that respect. The information on N about 10^17-10^18 / 10mg is useless since the spot size is not given. The conclusion is confusing: "macroscopic number of optical center….", having a large N is not supposed to kill the bunching effect, it can only add a background of uncorrelated photons. Nx is supposed to change the contrast, not N.
Reviewer #2 (Remarks to the Author): report on "Non-Poissonian photon statistics from macroscopic photon cutting materials" by de Jong et al.
The authors report on the theoretical and experimental evidence that photon-cutting materials have a photon emission statistics that is non-poissonian. More precisely, they prove that the correlation function between photons emitted from different energy level of the photon-cutting material exhibit a bunching-like curve. The paper is very well written, the experiments and the theory are sound and convincing, the subject is exciting and quite timely, and last but not least, the findings are new and original. I can't but warmly recommend the paper for publication.
1. I could not find the procedure for normalizing the g2. it is important to give it, and possibly justify it if this is not a fully coherent normalization (ie, using a laser as an input of their HBT interferometer) 2. It turns out that a very similar effect has been recently observed (Meuret et al., PRL, (2015))in a quite different situation, namely intensity interferometry of the photons produced by an electron beam in interaction with a solid (cathodoluminescence). Although the physics rationale is quite different (1 electron create 1 plasmon that is able to excite several individual emitters in synchronization; therefore, the bunching is observed for any material), the maths seem very similar. However, in these experiments, the bunching amplitude is much higher than what observed by the authors, which is puzzling. Finally, this effect is used to measure the lifetime of several materials (see Meuret et al., ACS Photonics, 2016). In cathodoluminescence, lifetimes measurements are complicated, therefore this technique is useful. A possible application of the authors finding is a novel method to measure lifetime, *if* this is competitive with time resolved PL.

Our response to the reviewer report of Feb 7, 2017 on manuscript NCOMMS-16-27637-T:
We thank both Reviewers for carefully reading our manuscript and their constructive comments, which have been helpful to improve the manuscript. Below we reproduce the Reviewers' comments in black italic, and give our point-by-point response in blue. Changes to the manuscript based on the Reviewers' comments and suggestions are highlighted in red, both in this document and in the "Marked Manuscript" file attached.

Response to Reviewer #1
The submitted paper propose to use time correlation spectroscopy to demonstrate without any ambiguity the occurrence of quantum cutting effect in NaLaF4:Pr3+. Indeed, former works in that field always tend to prove the effect using comparison of excitation spectra showing then that quantum yield can be over 1 Our response: We are pleased to read that the reviewer recognizes the novelty and importance of our work.
The Monte Carlo simulation is, from my point of view useless and bring more confusion than help for the understanding. Indeed, bunching technic is rather classical, and eq 2 is well admitted. In addition, the population eq can be solved without MC. Last remark for Fig1-g, what does mean N=1 when keeping Nx=2?? Why the simulation is diverging from the analytical solution. It seems even that that it does not reproduce the curve correctly for N=10. I suggest to entirely remove the section on MC and the fig:1

-g which does not bring additional information.
Our response: We apologise that our discussion of the Monte Carlo simulations was too concise. The Reviewer correctly points out that this could lead to confusion about (i) the deviations between the Monte Carlo simulation and the analytical equation (Eq. 2) and about (ii) the meaning of a constant steady-state population of ! = 2.
As mentioned by the Reviewer, Figure 1g shows differences between the analytical and the Monte Carlo model at = 1 and = 10. These arise because the analytical model neglects depletion of the ground state of optical centres. This is a good approximation for macroscopic at low excitation rate, but not for a small number of centres. The Monte Carlo simulation thus validates the analytical model in the limit of large , but also shows its break-down at small . Action 1: On page 2 of the revised manuscript we clarify what we mean by a constant steadystate population ! of 2: "The excitation rate is set at Φ = 2 !" / , which corresponds to a constant steady-state population of ! = 2 in the limit of large ." And a few lines down we discuss the deviations between analytical and Monte Carlo model: "The analytical model (equation (2)) matches well with the Monte Carlo results for an ensemble of emitters with > 10, including macroscopic materials containing a number of optical centres on the order of Avogadro's constant (Fig. 1g, green and blue), clearly revealing the occurrence of photon-pair emission in the photon statistics. The analytical model is less accurate for a small (yellow and red), because the approximation of no ground-state depletion is justified only for large and low excitation rates." For clarity, it would be better to use another sign for excitation rate, since X is also the label of the level.
Our response: This is a good suggestion of the Reviewer.

Action 2:
We use "Φ" as the symbol for excitation rate throughout the revised manuscript and the Supplementary Information.
When comparing the case of various Nx, the authors claim that the excitation has no effect on the signal to noise ration which is not clear. It suggests that Nx has no effect on the results. In such a situation, 2 goals might be forseen. Either, the experiment tends to demonstrate the occurrence of the quantum-cutting effect, or the goal is to extract parameter from the g2 fitting. In the latter case, I am not sure that the confidence on the deduced kx is the same for all the Nx. A small contrast with a high S/N ration might not be equivalent to a high contrast with a weak S/N. I suggest to improve the statistical analysis from that respect.
Action 4: On page 3 of the revised manuscript we write: "We use ~10 mg of the material, containing approximately = 10 !" − 10 !" optical centres that in the experiment are excited more or less homogeneously." And we revised the Methods section to read: "~10 mg of powder was glued as a thin layer of a few mm 2 to a non-luminescent background using SPI silver paint. The spectral output of a Micropack DH-2000 deuterium lamp filtered using an Acton Optics & Coatings 180-N 180±10 nm bandpass filter illuminated the sample homogeneously, exciting Pr 3+ to a 4f 1 5d 1 level, from which rapid non-radiative relaxation to the 1 S 0 level takes place." The conclusion is confusing: "macroscopic number of optical center….", having a large N is not supposed to kill the bunching effect, it can only add a background of uncorrelated photons. Nx is supposed to change the contrast, not N.