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How to optimize the absorption of two entangled photons
by E. G. Carnio, A. Buchleitner, F. Schlawin
- Published as SciPost Phys. Core 4, 028 (2021)
|As Contributors:||Edoardo Carnio|
|Date submitted:||2021-09-16 18:16|
|Submitted by:||Carnio, Edoardo|
|Submitted to:||SciPost Physics Core|
We investigate how entanglement can enhance two-photon absorption in a three-level system. First, we employ the Schmidt decomposition to determine the entanglement properties of the optimal two-photon state to drive such a transition, and the maximum enhancement which can be achieved in comparison to the optimal classical pulse. We then adapt the optimization problem to realistic experimental constraints, where photon pairs from a down-conversion source are manipulated by local operations such as spatial light modulators. We derive optimal pulse shaping functions to enhance the absorption efficiency, and compare the maximal enhancement achievable by entanglement to the yield of optimally shaped, separable pulses.
Published as SciPost Phys. Core 4, 028 (2021)
Author comments upon resubmission
Answer to Report 1
We thank the Referee for their report. The role of interfering pathways is indeed minimal, but rather because our system consists of three levels, not because the field is weak. The presence of multiple pathways in a more complex spectrum is discussed in J. Chem. Phys. 154, 214114 (2021), which we now cite as the new Ref. 44 in the manuscript.
Answer to Report 2
We thank Prof. Mukamel for his detailed and helpful feedback. We agree that our manuscript does not present sufficiently groundbreaking results to be published in SciPost Physics; for this reason, we did submit it to SciPost Physics Core. As Prof. Mukamel pointed out, our work “is an interesting study since the field of quantum light spectroscopy is rapidly growing.” While it’s certainly true that the foundation of this study was laid in New J. Phys. 19 013009 (2017), the first half of this manuscript does not merely repeat these earlier calculations. In particular, it * presents a more detailed and pedagogical derivation of the matter response function, starting from the case of a light-matter interaction of finite duration, and of the optimization problem; * actually quantifies the entanglement in the optimal two-photon state and relates it to the maximum enhancement it can yield, and derives limits for the possible enhancement. This analysis is entirely new. Finally, to mirror the words of Prof. Mukamel, the study adapts the optimization problem to realistic experimental settings.
All in all, we think the first half of this manuscript significantly improves and extends the ideas introduced in the NJP, while the second half applies the methods to realistic scenarios (of course starting from a theoretical point of view). The result is a self-contained package where the physics behind the optimization of two-photon absorption is clearly conveyed despite the mathematical artillery necessary in this type of problems.
(The simplicity of the three-level model we discuss in 2. below)
As for the additional comments: 1. Indeed, that we consider two-photon states only is our working assumption across the manuscript, not a consequence of the Lagrange multiplier. We have added a statement in 3.1 to make this completely transparent. 2. We agree with the Referee. The three-level model is, however, the simplest to discuss two-photon absorption, in particular in the aspects that we present in the manuscript. A more structured manifold implies an additional layer of complexity that we address in another publication: J. Chem. Phys. 154, 214114 (2021) [Ref. 44 in the manuscript]. We refer to these results in Sec. 1. 3. We have added the subsection 2.5 in the revised manuscript to discuss the relevance of our model for, e.g., fluorescence measurements. 4. We agree with these remarks, which are compatible with the message of the manuscript: our aim is to gauge the role of quantum correlations between the two photons, and to do so we have to compare the ETPA with the optimal unentangled two-photon state. In NJP (2017) [Ref. 40 in the manuscript], we showed, using the same optimization procedure as for entangled photons, that this is precisely the first pair of Schmidt modes. We fully agree that in practice these will be difficult, if not impossible, to implement. Yet they represent the best possible states and therefore, they indicate the boundary between states, whose efficiency could be matched by classical means, and those entangled states, for which this is no longer possible. We expanded Section 3.4 to highlight this point more clearly. Taking experimental difficulties into consideration in this comparison would not allow us to make general statements about the role of entanglement. 5. According to Landes et al., in the paper referred to by the referee, Ref. 33 in our manuscript, “ETPA events are not frequent enough to produce detectable signals in typical molecular systems using currently-applied SPDC sources”. Nevertheless, this conclusion is drawn “in cases in which ETPA occurs only via far-off-resonant (virtual) intermediate states”, which is not the case of our manuscript. We address this difference in Sec. 1.
List of changes
(All changes to the text are highlighted in red in the manuscript)
* Expanded Secs. 1, 3.1, 3.4 with additional comments.
* Added section 2.5.
* Added references 35, 36, 44, 47.
* Updated references 23 and 33.
Submission & Refereeing History
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