Spectral considerations of Entangled two-photon absorption effects in Hong-Ou-Mandel interference experiments

The nonlinear optical phenomenon of two-photon absorption (TPA) is of current great interest, as it finds various scientific and technological applications, such as laser scanning multiphoton microscopy, photodynamic therapy, microengraving, etc [1, 2, 3, 4, 5]. Typically, TPA is achieved with nonlinear materials of large TPA cross-sections $(\delta_{c})$ through the use of pulsed lasers delivering high density of random photons. However, from recent theoretical [6, 7, 8] and experimental works [9, 10], the interest for implementing TPA has extended to the use of correlated (entangled) photons, namely entangled two-photon absorption (ETPA) process.

The advantage of ETPA compared with its counterpart, the classical TPA effect, is that in the former the total rate of absorbed photons $(R_{TPA})$ has a linear dependence on the photon excitation flux $(\phi)$, while the dependence is quadratic in the case of classical light [7, 11, 12]: $R_{TPA}=\sigma_{e}\phi+\delta_{c}\phi^{2}$, where $\sigma_{e}$ is the entangled TPA cross-section. Different authors claim there is a difference up to $\sim 32$ orders of magnitude between $\sigma_{e}$ [10, 13, 14] and $\delta_{c}$ [15, 16, 17], with estimated values of $\sigma_{e}$ for molecules used as nonlinear models falling within the wide range from $10^{-22}$ to $10^{-18}[cm^{2}/molecule]$ [19, 13, 20]. Although it is not completely correct compare directly these two parameters because they have different units $(\sigma_{e}[cm^{2}/molecule],~{}\delta_{c}[cm^{4}s/molecule])$, in principle it would be possible to achieve ETPA in the low flux regime by illuminating the sample with correlated photon pairs produced by spontaneous parametric down conversion (SPDC) [18]. Nevertheless, there is a recent debate about the actual magnitude of $\sigma_{e}$ or even if the ETPA effects has been really detected experimentally [20, 21].

Diverse experimental configurations have been employed to study the ETPA activity, based on measuring the transmittance [10, 14, 22, 19, 21] or fluorescence exhibited by the sample upon excitation with SPDC photon pairs [9, 23, 20, 24]. For the purposes of the present work, we highlight the case where the difference in transmittance between a solution of the molecule under test and the solvent alone is recorded as a function of the excitation intensity, where the temporal delay ($\delta t$) between the entangled photons is set to zero. In some of these works, the zero delay configuration is assured by implementing a HOM interferometer [21] before the photons interact with the sample. It is recognized that in these experimental attempts to detect ETPA the results can be biased by artifacts[20, 25]. The artifacts emulating the ETPA process are: optical losses induced by linear absorption, scattering, molecular aggregation, hot band absorption and others. The existence of these artifacts[20, 26, 27] have fed the controversy about if the ETPA has been or not demonstrated unambiguously in experiments. In this context it is worth to further investigate ETPA, imposing experimental conditions, including the discrimination of artifacts, that contribute to discard or confirm the possibility of observing the effect.

Recently, the HOM interference based in a Mach-Zehnder interferometer was used to analyze changes in entangled photon pairs, where the time delay shifting induced by a sample in the HOM interferogram (HOM dip) is utilized as a parameter to perform linear spectroscopy studies [28]. Likewise, it has been shown that the features of the HOM dip can provide a signature of the linear absorption spectrum of a resonant sample with the two photon spectrum[29, 30]. Although the HOM dip has already been used to explore ETPA experiments[31], the visibility change in a HOM dip has not received attention as a parameter aimed to analyze the ETPA phenomenon. So, with the motivation of detecting nonlinear effects, new approaches in the use of HOM interference deserves detailed examination, taking advantage of its sensibility and robustness to discriminate photon states. In this work, we present a detailed study of the effects produced on the visibility of a HOM dip by a nonlinear sample illuminated with entangled photons. Experimentally, the pairs of entangled photons interact with the sample under test before they are launched into the HOM interferometer. This approach allows to explore directly the changes in the entangled photon states, while by discriminating the linear optical losses the conditions and possibilities to detect ETPA are established. This is the first time, to the best of our knowledge, that the visibility of a HOM dip is fully investigated intending to detect ETPA.

The experimental setup is shown in Figure 4$A$. A CW laser (Crystalaser DL-405-100), centered at $\lambda_{p}=403nm$ and with a $\Delta\lambda_{p}\sim 1nm$ bandwidth FWHM, is focused by lens $L_{1}$ (focal length $f_{1}=500mm$) into a BBO crystal to produce collinear cross-polarized frequency-degenerate Type-II SPDC photons pairs around $806nm$. In order to optimize the phase-matching conditions, which maximize the SPDC emission process, the polarization of the pump beam is aligned with the plane defined by the crystal’s optical axis. We do so by rotating the zero order half wave plate $HWP_{2}$, which also works as a control of the density of down-converted photons produced by the crystal. Prior being focused to enter the crystal, the laser pump power is controlled by a half-plate-wave ($HWP_{1}$) and a Glan-Thompson polarizer ($Polarizer$). The residual pump after the crytsal is eliminated by using a filter element $F_{1}$, composed of a longpass filter (Thorlbas FELH0500) and a bandpass filter (Thorlbas FBH800-40). To make sure that the residual pump was eliminated with $F_{1}$, a picture of the entangled-photons spatial mode was taken in two configurations of photon density obtained by varying $HWP_{2}$: 1) with maximum photon density and 2) minimum photon density. In the first case, a spatial mode like the one shown on the right side of Figure 5 is obtained, while in the second case no spatial mode is detected and the image is essentially dark, which indicates that there is no residual presence of pumping, more details can be found in the Supporting Information, Fig S3. Then, the filtered down-converted photons propagates through a Michelson interferometer (starting at $PBS_{1}$) which introduces a controllable temporal delay ($\delta t$) between them.

In addition to the control of frequency ($F_{1}$) and temporal delay ($\delta t$) indistinguishability, the final requisite to obtain an optimal HOM interference pattern at $PBS_{2}$ is to eliminate the polarization distiguishability that is characteristic in photon pairs produced by Type-II SPDC process. To do so, a half-waveplate ($HWP_{3}$) is introduced to rotate $45^{\circ}$ the horizontal and vertical axis of polarization of the down-converted photons. This element works as a control to turn “on” or “off” the HOM interference. Then, the down-converted photons are focused into the $1cm$ quartz-cuvette containing the sample ($S$) with a $5cm$ focal length lens ($L_{2}$), producing a $W_{0}=58\mu m$ spot diameter which is then collimated with a second lens of the same focal length ($L_{3}$). A Rayleigh length of $Z_{R}=1.3cm$ is generated, so the interaction volume of the photons with the sample can be considered as a cylinder of $l=1cm$ length [20].

The HOM effect results from the interference of indistinguishable photons at a beam splitter $PBS_{2}$ [38, 39, 40, 41, 37]. In our setup we obtain the HOM interferogram by recording the $CC$ as a function of the time delay between the photons, registered by a time-to-digital converter module (ID Quantique id800) and the avalance photodiodes $APD_{1}$ and $APD_{2}$ (Excelitas SPCM-AQRH). As a crucial property of the ETPA transmittance experiments discussed below, in our setup we can change the the degree of photons indistinguishability in all the relevant degrees of freedom (frequency, polarization, arrival time), allowing to fully control the interference process. A detailed experimental analysis of the interference process under different conditions of photons indistinguishability, noise and errors are found in the Supporting Information, Fig S4.

As an initial calibration step, we measured the HOM dip produced by the down-converted photons in free space propagation ($HOM_{ref}$). As shown in Figure 5, the distinctive characteristic of the $HOM_{ref}$ is a steep dip around $\delta t=0$ and a high level of $CC_{max}$ for $\delta t$ values larger that the coherence length of the down-converted photons. In this figure two different configurations of photon indistinguishability are presented. The first configuration was set to optimize the spectral indistinguishability of the photon pairs by using a bandpass filter (Thorlbas FBH810-10) of narrow bandwidth ($10$nm, centered at $810$nm) and longpass filter (Thorlbas FELH0500) in the element $F_{1}$ in combination with a single-mode fiber (SMF), obtaining a $V_{ref}$ of $94\%$ and a temporal FWHM of $181$fs. For the second configuration a bandpass filter of $40$nm bandwidth centered at $800$nm (Thorlabs FBH800-40) and longpass filter (Thorlbas FELH0500) was used in combination with a multi-mode fiber (MMF), obtaining a $V_{ref}$=$60$% and a FWHM temporal width of $71$fs. The reduction in $V_{ref}$ is due to asymmetric spectrum of each photon proper of a SPDC Type-II process [34]. Hereafter, in all the experiments the second filter configuration is employed.

In the experiments, the laser pump power was fixed at $43.9$mW. Figure 6$A$ presents four HOM dips of interest: $HOM_{ref}$ (free space propagation), $HOM_{sol}$ (solvent into the cuvette) and $HOM_{sam}$ (sample at the highest concentration); in addition, the case of a empty cuvette (without solvent) is also shown.

This figure shows that $CC_{max}$ decrease as the cuvette, solvent or samples are introduced in the path of propagating photons. As it can be clearly seen, well defined offsets are detected with respect to $HOM_{ref}$. The offset is large between $HOM_{ref}$ and $HOM_{sol}$ because the focused beam in the latter case travels through a denser medium changing the conditions of light coupling to the detection devices (APDs); meanwhile, the offset between $HOM_{sam}$ and $HOM_{sol}$ is small. This small offset between solvent and sample, clearly detectable at long $\delta t$, are ascribed to Fresnel losses, scattering and residual linear absorption, which are larger in the sample than in solvent. It is worth to remark that in our model the linear losses do not change the visibility of an interferogram. To see this, Figure S5 presents the simulation of a HOM dip obtained for a sample with different levels of lineal optical losses: the visibility remains unaltered when the losses varied in the range $0\leq\eta^{\prime}<1$. This outcome was expected as the lineal losses are independent of frequency, and this is exactly the experimental result observed for the HOM dips of Figure 6$A$, where all HOM dips have nearly the same visibility. Since $\eta^{\prime}$ accounts in the model for linear losses unable to change the visibility while $\eta$ is related to the ETPA effect that inherently would change the visibility, then the experimental results suggest that the sample is not producing non-linear absorption. Notice that the experimental HOM dips are reproduced accurately by the model (Eq. S1) considering the nonlinearity of the sample negligible ($\eta=0$) and different values of $\eta^{\prime}$, as depicted by the continuous lines in Figure 6$A$ and Figure 6$B$. The $\eta^{\prime}$ values (0 for free space, 0.32 for cuvette, 0.66 for solvent and 0.69 for most concentrated sample) generated by the model are in good agreement with the level of linear losses expected for each case.

In Figure 6$B$ a zoom of the $HOM_{sol}$ and $HOM_{sam}$ dips for the entire sample concentrations set is displayed, where the black dots represent data from the solvent and red dots from the sample with the highest concentration (100mM); cyan dots are data for the other utilized concentrations in the range $0.1\mu$M - $58$mM. It is interesting to note that as the concentration increases, the offset in $CC$ is greater outside the dip (large delays) than inside the dip (zero delay), which is an intrinsic effect of the HOM dip as it was recently discussed[21]. This intriguing effect depicted markedly in Figure 6$B$, apparently implies that the HOM visibilities decrease from solvent to samples of higher concentrations, however, by using Eq. 8 it is found that $V_{ref}$, $V_{sol}$, and all $V_{sam}$ are essentially the same within the range of uncertainties of the experimental error, the latter determined by the noise fluctuations of the $CC$ signal. To visualize the level of experimental uncertainties, Figure 6$C$ presents as an example the data corresponding to the sample with the highest concentration; here the $HOM_{sam}$ is plotted within a known range of confidence bands, whose width represents 2 standard deviations (2$\sigma$) of the experimental data. Based in the 2$\sigma$ and the propagation of uncertainties when Eq. 8 is employed, the visibility of each experimental HOM dip and its corresponding uncertainty was calculated. For instance, the visibility of the HOM dip for the sample at the heights concentration resulted in $V_{sam}=57.8\pm 1.4\%$ while for the air and solvent was $V_{ref}=60.4\pm 1.3\%$ and $V_{sol}=58.3\pm 1.2\%$. Table 1 summarizes the visibilities obtained for each sample and the values obtained for the ratio $V_{sam}/V_{sol}$.

Figure 7 shows the $V_{sam}/V_{sol}$ ratio as a function of the sample concentration, showing just marginal changes in the visibility of samples respect to the solvent; the absence of a well defined trend in the variation of $V_{sam}/V_{sol}$ avoids to conclude unambiguously the presence of ETPA activity, at least with the sensitivity of our experimental apparatus.

Using the mathematical model here introduced, we computed the changes of visibility predicted under our experimental conditions: sample bandwidth $\Delta\lambda_{H}=30nm$ (FWHM), central wavelength of the sample transfer function $2\lambda_{H}=816nm$ (taken directly from the TPA spectrum of Rhodamine B), pump bandwidth $\Delta\lambda_{p}\sim 1nm$ (FWHM) centered at $\lambda_{p}=403nm$, and filter bandwidth $\Delta\lambda_{F}=40nm$ (FWHM) centered at $\lambda_{F}=800nm$. After making a run of the model, no significant changes in visibility are found; only considering the sample as a perfect ETPA absorber (Figure 7$B$ for $\eta=1$) produces a $V_{sam}/V_{sol}=0.987$ value, which is just in the border of the experimental uncertainties of our apparatus. When a broadband pump is assumed, the ratio $V_{sam}/V_{sol}=0.935$ is obtained; such a change in the visibility is in principle detectable in our experimental approach, but it implies the unrealistic condition of a perfect ETPA absorber.

From the mathematical and experimental considerations above discussed, it is difficult and probably unworkable to detect ETPA in Rhodamine B in the $800nm$ region, because the effect strongly depends on the detuning between the sample transfer function and the filtered photons function. Physically, this means that as the spectral sample bandwidth differs markedly from the corresponding spectrum of photons, less is the probability of inducing JSI asymmetries susceptible of being detected as changes in the HOM dip. When the spectral considerations are not fulfilled, the ETPA interaction can not be reached, so the sample behaves as a photon attenuator which eliminates equally photons of all frequencies (a neutral density filter characterized by $\eta^{\prime}$), producing an offset between $HOM_{sol}$ and $HOM_{sam}$.

This study suggests that previous works [19] in which Rhodamine B was used as a model to study ETPA might involve signals not coming from two-photon absorption, but from artifacts or spurious signals, i.e., linear losses. Transmittance experiments intended to measure $R_{TPA}$ based exclusively on photon counting have procedural and theoretical problems to discriminate optical losses other than TPA. In this context, a system based on the interference of two photons, in which visibility of a HOM dip is not affected by linear optical losses, represents a novel and alternative scheme to detect the changes in the photon symmetry of the correlated photon-pairs state produced by the non-linear absorption effect.

In this paper we have used the two-photon interference effect as a mechanism to study and establish the spectral considerations for which ETPA transmittance experiments can be carried out, analyzing the visibility of the HOM dip, being the first time, to the best of our knowledge, that this approach is explored.

We presented a mathematical model for the interaction of down-converted photons with a nonlinear sample. The model uses the parameter $\eta$ to account for the strength of the ETPA process. In order predict the spectral conditions that would lead to measurable ETPA by means of our experimental proposal, theoretical simulations were generated using as variables the sample bandwidth and the detuning of the photon spectrum, under either narrow or broad bandwidth pump. By considering the visibility of the HOM dips as figure of merit, we did not find experimental evidence of ETPA, in agreement with the simulations that compare the visibilities a solution of Rhodamine B and the solvent alone. Thus, it can be inferred that the significant detuning between the sample spectrum and the photon pair spectrum, in combination with possibly weak sample nonlinearities, precluded introducing asymmetries in the JSI, a necessary condition to induce changes in the visibility of HOM dips.

Despite the ETPA was not unequivocally determined in our experiments for the case of Rhodamine B, our model allowed to explore the combinations of pumping bandwidth, detuning, sample bandwidth and sample nonlinear strength that would favor changes in the visibility of HOM dips, and in turn the detection of ETPA signals under a transmission scheme. For this scheme, the effect of linear losses independent of frequency was also explored. Considering this, we believe that our experimental and theoretical results represent a step forward in the application of quantum sensing techniques as ultra-sensitive devices for the study of elusive nonlinear optical effects, such as ETPA in molecular systems.

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

We acknowledge support from CONACYT, Mexico. This work was supported by CONACYT, Mexico grant FORDECYT-PRONACES 217559.

The authors declare no conflicts of interest.