# Entangled two-photon absorption in transmission-based experiments:
deleterious effects from linear optical losses

Freiman Triana-Arango<EMAIL_ADDRESS>[  Roberto Ramírez-Alarcón
<EMAIL_ADDRESS>[  Gabriel Ramos-Ortiz<EMAIL_ADDRESS>[

###### Abstract

Recently different experimental schemes have been proposed to study the
elusive phenomenon of entangled two-photon absorption (ETPA) in nonlinear
materials. The attempts to detect ETPA using transmission-based schemes have
led to results whose validity is currently under debate since the ETPA signal
can be corrupted or emulated by artifacts associated with linear optical
losses. The present work addresses the issue of linear losses and the
corresponding artifacts in transmission-based ETPA experiments through a new
approach that exploits the properties of a Hong-Ou-Mandel (HOM) interferogram.
Here we analyze solutions of Rhodamine B (RhB), commonly used as a model of
nonlinear medium in ETPA studies. Then, by using the HOM interferometer as a
sensing device, we firstly demonstrate the equivalence of the standard
transmission vs pump power ETPA experiments, presented in many reports, with
our novel approach of transmission vs two-photon temporal delay. Secondly, a
detailed study of the effects of optical losses, unrelated to ETPA, over the
HOM interferogram is carried out by: 1) Characterizing RhB in solutions
prepared with different solvents. 2) Considering scattering losses introduced
by silica nanoparticles used as a controlled linear loss mechanism. Our
results clearly expose the deleterious effects of linear optical losses over
the ETPA signal when standard transmission experiments are employed and show
how, by using the HOM interferogram as a sensing device, it is possible to
detect the presence of such losses. Finally, once we showed that the HOM
interferogram discriminates properly linear losses, our study also reveals
that under the specific experimental conditions considered here, which are the
same than those employed in many reported works, the ETPA was not
unequivocally detected.

###### keywords:

American Chemical Society, LaTeX

Centro de Investigaciones en Óptica A.C., A. P. 1-948, 37000 León, Gto,
Mexico] Centro de Investigaciones en Óptica A.C., A. P. 1-948, 37000 León,
Gto, Mexico Centro de Investigaciones en Óptica A.C., A. P. 1-948, 37000 León,
Gto, Mexico] Centro de Investigaciones en Óptica A.C., A. P. 1-948, 37000
León, Gto, Mexico Centro de Investigaciones en Óptica A.C., A. P. 1-948, 37000
León, Gto, Mexico] Centro de Investigaciones en Óptica A.C., A. P. 1-948,
37000 León, Gto, Mexico IR,NMR,UV

##  1 Introduction

Since the discovery and development of the laser1, and subsequently the
nonlinear optical effects2, 3, 4, 5, an advent of photonic technologies have
been aroused 6, 7. Among these, the two-photon absorption (TPA) process,
predicted theoretically by M. Goeppert-Mayer in 1931, has become an useful
tool in different areas such as bioimaging8 and micro-fabrication
applications9. TPA is a third-order nonlinear optical process10, 11 which
requires high-intensity excitation sources and nonlinear optical materials
with large TPA cross-sections ($\delta_{c}$), in order to be observed.
Typically, pulsed lasers are employed as excitation sources to promote TPA
since they provide a high density of random photons. However, the prediction
of TPA through the use of extremely dim (low intensity) non-classical light
12, 13, 14, namely the entangled two-photon absorption (ETPA) effect,
triggered a great interest in understanding the interaction dynamics of this
non-classical excitation source with non-linear active materials. The interest
in detecting ETPA has led to theoretical and experimental proposals whose
results nowadays are still debatable and generate intense scientific research.

Different physical effects and potential applications, without classical
equivalent, have been predicted in this new type of interaction, such as: non-
monotonic behavior of ETPA signals with photon temporal delay15, virtual state
spectroscopy16, 17, 18, induction of forbidden transitions19, entangled two-
photon spectroscopy20, 21, 22, 23, among others. Nevertheless, the effect of
most notoriety is, possibly, the linear behavior of the nonlinear absorption
rate ($R_{TPA}$) with the photon flux ($\phi$)24, contrasting with the
classical case in which the trend is quadratic11:
$R_{TPA}=\sigma_{e}\phi+\delta_{c}\phi^{2}$, where $\sigma_{e}$ is the ETPA
cross-section, that represents, as in the classical case ($\delta_{c}$), the
probability of an electronic transition promoted by the TPA process. The
linear behavior of $R_{TPA}$ with $\phi$ in molecular samples has led to
estimate that $\sigma_{e}\sim 10^{-17}[cm^{2}/molecule]$25, whereas in the
classical case is well known that $\delta_{c}\sim
10^{-50}[cm^{4}s/molecule]$26, 27, 28. Thus, one of the most attractive
features of ETPA is that the same TPA effects, produced with a high flux of
uncorrelated photons (classical excitation), could be achieved by illuminating
the nonlinear material with a low flux of correlated photons (non-classical
excitation). In practice, correlated photons are produced by quantum processes
such as spontaneous parametric down-conversion (SPDC)29, 30.

Some initial experimental configurations were presented to study and quantify
the ETPA activity, and other nonlinear interactions involving entangled
photons, in a variety of materials by means of transmission31, 32, 33 and
fluorescence34 schemes. At the same time, other relevant theoretical works
established some conditions to optimize 35, 36, 37 and perform protocols for
ETPA experiments38. Since then, new schemes and techniques based on both
transmission39, 40 and fluorescence41, 42, 43, 44 have been proposed to
estimate the value of $\sigma_{e}$ in diverse materials. However, some authors
have suggested that the $\sigma_{e}$ values obtained from transmission-based
experimental schemes could be overestimated since the effects of linear losses
emulating45, 46 or contaminating47, 48, 49, 50 the EPTA signal were not
properly discriminated. Therefore, it is worth studying in detail how the
measurements of the ETPA signal and the $\sigma_{e}$ values estimated in
transmission experiments are affected by optical losses.

In a previous work from our group 51, we proposed a novel configuration to
study the ETPA process by using a HOM interferometer as an ultra-sensitive
detection device. In the cited work we presented a theoretical model and
implemented the corresponding measurements of transmission vs two-photon
temporal delay which allowed us to understand the effects of a nonlinear
sample (RhB), modeled as a notch type two-photon spectral filter, over the
asymmetric52 quantum state of photon pairs produced by type-II SPDC, in the CW
and pulsed excitation regimes. In that work, we showed that the ETPA activity
can be evidenced as a noticeable change in the HOM dip visibility, produced by
the interference of the down-converted photons after the interaction with the
sample. Also, we found that the best strategy to observe the ETPA activity in
an organic molecule such as RhB, with the HOM interferogram as a sensing
device, is to exploit a type-II SPDC source in the pulsed pumping regimen and
match the central emission wavelength and bandwidth of the photon pairs to
those of the sample under study. Nevertheless, that study did not show the
equivalence between the proposed HOM-based transmission vs temporal delay
scheme and the standard transmission vs pump power scheme used commonly to
measure ETPA, nor did it present a detailed study of the effects produced by
lineal optical losses over the detected signal in the HOM dip.

In the present work, we implement ETPA transmission experiments using RhB as a
nonlinear sample under CW excitation and exploiting the capabilities offered
by the HOM interferometer as a sensing device, some of them already explored
in linear spectroscopy studies53, 54, 55, 56. We choose the CW pumping regime
for the SPDC type-II photon pairs source since it is a widely used
experimental configuration in reported studies of ETPA activity, for which the
detrimental effects of linear losses has never been considered. By analyzing
the RhB transmission of photon pairs produced around 800nm, we demonstrate
that the scheme of transmission vs two-photon temporal delay, based on the HOM
interferogram, is completely equivalent to the standard transmission vs pump
power configuration reported by various authors. In addition, a detailed study
of the effects of linear optical losses, unrelated to ETPA, is carried out in
both schemes. It is shown that while the linear losses lead to overestimating
$\sigma_{e}$ when the transmission is detected as a function of pump power, in
the case of transmission vs temporal delay the losses only produce a reduction
of the raw base level of coincidence counts of a HOM interferogram, but
without changing the visibility of its dip. Notice that the visibility of a
HOM dip changes upon modifications to the quantum state of the photon pairs,
and changes in the quantum state are expected in the case of ETPA activity 51.
Thus, the transmission vs time delay configuration not only is equivalent to
the standard scheme used to study ETPA but also offers the remarkable
advantage of discriminating the presence of deleterious linear optical losses
over the HOM dip, becoming a good option for studying the elusive ETPA
process.

##  2 Experiment

We aim to perform ETPA transmission vs pump power39 and transmission vs two-
photon temporal delay experiments using a HOM interferometer in the CW
excitation regime, obtaining information about the effects over the quantum
state of type-II SPDC photon pairs, after interacting with a sample. The
experimental setup is shown in Fig. 1$A$, where a CW laser (Crystalaser
DL-405-100), with a center wavelength at $403nm$ and a $\sim 1nm$ bandwidth
(FWHM), is focused by lens $L_{1}$ (focal length $f_{1}=500mm$) into a BBO
crystal to generate collinear, cross-polarized and frequency non-degenerate
Type-II SPDC photons pairs around $806nm$. To control the flux of photon pairs
generated in the BBO crystal we use a half-wave plate ($HWP_{1}$) and a Glan-
Thompson polarizer ($Polarizer$) after the CW laser, regulating the pumping
power of the CW laser in a range from $0.25$mW to $43.9$mW. Then, the photon
pairs are guided to a Michelson interferometer where a controllable temporal
delay ($\delta t$) between them is introduced by a motorized stage. The
delayed photon pairs are focused on the sample ($S$) through lens $L_{2}$
(focal length $f_{1}=50mm$) and collected by $L_{3}$ with the same focal
length. Additionally, to control the frequency and temporal
indistinguishability of the photon pairs, an extra requirement to obtain an
optimal HOM interference pattern at $PBS_{2}$ is to erase the polarization
distinguishability, proper of a Type-II SPDC process. To do so, a half-
waveplate ($HWP_{3}$) is introduced before $L_{2}$ in order to rotate
$45^{\circ}$ the horizontal and vertical axis of polarization of the down-
converted photons. An spectral filtering element, composed of a longpass
filter (Thorlabs FELH0500) and a bandpass filter (Thorlbas FBH800-40), is used
as $F_{1}$ after the BBO crystal to remove the residual $403$nm pump and
control the bandwidth of the down-converted photons, and then $F_{2}$ (with
the same configuration as $F_{1}$) is placed after the lens $L_{3}$ to
eliminate a possible fluorescence residual signal from the sample. After the
interaction of the photons with the sample, they arrive at the beam splitter
($PBS_{2}$) where the HOM interference occurs and then, the photons are
collected with coupling systems ($CS_{1,2}$) into multi-mode fibers ($MMF$) to
be detected by a pair of avalanche photodiodes (Excelitas SPCM-AQRH)
($APD_{1,2}$). Finally, the individual and coincidence counts ($CC$) between
$APD_{1}$ and $APD_{2}$ are recorder by a time-to-digital converter module (ID
Quantique id800) ($TDC$).

Figure 1: $A$) Experimental configuration for the transmission ETPA
experiments using the HOM interferometer as a sensing device, $B$) RhB
molecule, and $C$) calibration HOM dip ($HOM_{ref}$).

A key characteristic of the HOM dip is that it is defined by the quantum state
of the photon pairs that interfere at the beam splitter52, in the sense that
any alteration in the state, due to their interaction with a sample, should be
manifested as a change in the visibility ($V$) or the width of the dip
($\Delta t$)51. In this work, we focus on the changes introduced in the
visibility since, as will be shown below, linear losses do not affect $V$ of
the HOM dip, while it is expected that the nonlinear interaction is able to
induce $V$ changes51. The HOM dip visibility is defined by

$V=\frac{R_{max}-R_{min}}{R_{max}+R_{min}},$ (1)

where $R_{max}$, and $R_{min}$ are the maximum and minimum $CC$ rate,
registered in our HOM apparatus at a temporal delay far from the center
($\delta t=167fs$) and at the center of the dip ($\delta t=0fs$),
respectively.

The experimental setup is calibrated by measuring the HOM two-photon
interference pattern ($HOM_{ref}$) shown in Fig. 1$C$, obtained in free space
propagation (no sample present). The resulting HOM dip has a temporal width of
$\Delta t\sim 79fs$ (FWHM ), corresponding to a bandwidth of the SPDC spectrum
of $\Delta\lambda\sim 24nm$ at $\lambda=806nm$. A good value of $V\sim 58\%$
was obtained, assuring the optimal alignment of the optical system 52. The
spatial mode of Type II SPDC photons is shown in the inset of Fig. 1$C$. The
region where the rings overlap comprises the entangled photons used to excite
the sample.

###  2.1 Sample and transmission measurement procedure

In this study we used as sample ($S$ as shown in Fig. 1$A$) an organic
solution of the laser dye RhB ($\geq 95\%$ purity, Sigma-Aldrich) dissolved in
methanol at ten different concentrations: $0.01\mu$M, $0.1\mu$M, $1\mu$M,
$0.01$mM, $0.1$mM, $1$mM, $4.5$mM, $10$mM, $58$mM, $100$mM. Some of these
concentrations were chosen because they were employed in previous ETPA
reports. 39, 40, 48, 50. The solutions of RhB were contained in a 1 cm-thick
quartz-cuvette.

As a first step in the experimental procedure, the HOM dip of Fig. 1$C$
($HOM_{ref}$) is used to set two temporal delay values of the photon pair: 1)
the center of the HOM dip for zero delayed photons ($\delta t=0fs$), where a
photon-pair flux density ($\phi$), calculated from the reference coincidence
count rate ($R_{ref}$) and the focusing area ($A$) (beam waist $w_{0}=58\mu
m$) of the down-converted photons, is on the order of $\phi=R_{ref}/A=7\times
10^{7}~{}[photons~{}cm^{-2}~{}s^{-1}]$, and 2) far from the center of the dip,
where a delay value of $\delta t=167fs$ is picked and where
$\phi=R_{ref}/A=3\times 10^{8}~{}[photons~{}cm^{-2}~{}s^{-1}]$.

Then, we performed two different transmission experiments. In the first
experiment, the transmission of the down-converted photons after interacting
with the sample is registered as a function of the laser pump power. These
measurements were performed by monitoring the $CC$ at $\delta t=0fs$ and
$\delta t=167fs$ and with two conditions in which the cuvette was containing
either the solvent alone ($HOM_{sol}$) or the sample solution at different
concentrations ($HOM_{sam}$). In the second experiment the $CC$ of the down-
converted photons after interacting with the sample are monitored by
continuously varying the temporal delay. In this case, the pump power was
fixed to the maximum value used in the first experiment. We made this
experiment for the cuvette containing only the solvent or the sample solution
at different concentrations as well.

##  3 Results and Discussion

### Transmission vs power and vs two-photon temporal delay

The measurements of transmission as a function of the pump power ($P_{pump}$)
and transmission as a function of the two-photon temporal delay ($\delta t$)
are presented in Fig. 2, where the rate of transmitted down-converted photons
detected as $CC$ for the solvent and the samples are named $R_{sol}$ and
$R_{sam}$, respectively, to be matched with tags used in previous works39, 50.
The panel $A)$ of the figure shows $R_{(sol,~{}sam)}$ vs $P_{pump}$ for two
specific delay configurations namely $\delta t=0fs$ and $\delta t=167fs$,
while panel $B)$ shows $R_{(sol,~{}sam)}$ vs $\delta t$ for a fixed pump power
of $P_{pump}=43.9mW$. The legend-color confections are selected to be: black-
dots for solvent, red-dots for the most concentrated sample solution
($100$mM), and in a blue-cyan scale color the other sample concentrations in
the range $0.01\mu$M - $58$mM.

Figure 2: Transmission of solutions of the molecule RhB as a function of $A$)
$P_{pump}$ and $B$) $\delta t$. In the case of transmission vs $\delta t$ the
data was acquired by setting the pump power at $P_{pump}=43.9mW$

In Fig. 2 we clearly show the correspondence between the two transmission
schemes when realizing $R_{(sol,~{}sam)}$ measurements, namely the
transmission vs $P_{pump}$ in the standard configuration, or the transmission
vs $\delta t$ of the HOM dip, as proposed in this study. Consequently, the
difference in transmission between the solvent and sample ($R_{sol}-R_{sam}$)
can be indistinctly quantified in the power measurements or in the temporal
delay measurements.

The difference $R_{sol}-R_{sam}$ is typically defined as $R_{TPA}$ 39, but
when this definition is employed special care must be taken to avoid
misleading in the calculation of $\sigma_{e}$. For instance, when $R_{TPA}$ is
computed directly from the data presented in Fig. 2, the largest values are
obtained when the photons arrive at the sample with a temporal delay of
$\delta t=167fs$, which is evidently an incorrect result since the
simultaneous absorption of two photons must be maximum when the temporal delay
between them is zero47, 48, 49. Here it should be observed that the maximum
delay of 167 $fs$ employed in these experiments is larger than the photon pair
coherence time of $\sim$ 79 $fs$, as it can be seen in Fig. 1$C$. In the
transmission vs $P_{pump}$ experiments, the quotient of $R_{TPA}$ with
$R_{sol}$ must be computed for the whole pump-power interval, leading to
straight lines whose slopes are39, 50:

$m=\frac{R_{sol}-R_{sam}}{R_{sol}}=\frac{R_{TPA}}{R_{sol}},$ (2)

The quotient that defines $m$ considers the interferometric effect introduced
by the HOM since it weighs the changes in the value of $R_{TPA}$ with respect
to changes in the signal generated by the solvent alone, for different delay
values. The lines obtained from the experimental data are shown in Fig. 3$A$
for the case of $\delta t=0fs$ and Fig. 3$B$ for $\delta t=167fs$. The black
solid lines in these figures correspond to the hypothetical case of $m=1$, so
they represent the maximum achievable level of ETPA, which occurs when the
transmission of the sample is null ($R_{sam}=0$). The slope of the lines
obtained experimentally are related to the $\sigma_{e}$ value by means of39,
50

$\sigma_{e}=\frac{m}{C}\frac{A}{V_{o}N_{A}},$ (3)

where $C$ is the concentration of the sample, $V_{o}$ and $A$ are the
interaction volume and area, respectively, and $N_{A}$ is the Avogadro’s
number.

Figure 3: ETPA absorption rates of RhB for $A$) $\delta t=0fs$ and $B$)
$\delta t=167fs$. $C$)ETPA cross-section $\sigma_{e}$ (logarithmic scale) and
slope $m$ (inset) as a function of concentration.

The resulting values of $\sigma_{e}$ are presented in Fig. 3$C$ (in
logarithmic scale for clear visualization of data) as a function of sample
concentration, while the inset of the figure shows the behavior of $m$. For
illustration and discussion purposes, the results are presented for both
delays, $\delta t=0fs$ and $\delta t=167fs$. Table 1 summarizes these results
and includes the corresponding uncertainties extracted directly from the
fitting of experimental data in Fig. 3$A$ and Fig. 3$B$.

$C[M/L]$ | $\delta t=0fs$ | $\delta t=167fs$
---|---|---
$m$ | $\sigma_{e}$ | $m$ | $\sigma_{e}$
$1\times 10^{-8}$ | $0.026\pm 0.002$ | $(4.2\pm 0.3)\times 10^{-15}$ | $0.0158\pm 0.0007$ | $(2.5\pm 0.1)\times 10^{-15}$
$1\times 10^{-7}$ | $0.030\pm 0.002$ | $(4.7\pm 0.3)\times 10^{-16}$ | $0.0263\pm 0.0006$ | $(4.2\pm 0.1)\times 10^{-16}$
$1\times 10^{-6}$ | $0.039\pm 0.002$ | $(6.2\pm 0.3)\times 10^{-17}$ | $0.0315\pm 0.0007$ | $(5.0\pm 0.1)\times 10^{-17}$
$1\times 10^{-5}$ | $0.0525\pm 0.0007$ | $(4.2\pm 0.1)\times 10^{-18}$ | $0.0408\pm 0.0008$ | $(6.5\pm 0.1)\times 10^{-18}$
$1\times 10^{-4}$ | $0.0740\pm 0.0006$ | $(1.173\pm 0.009)\times 10^{-18}$ | $0.0556\pm 0.0007$ | $(8.8\pm 0.1)\times 10^{-19}$
$1\times 10^{-3}$ | $0.103\pm 0.001$ | $(1.64\pm 0.02)\times 10^{-19}$ | $0.0786\pm 0.0009$ | $(1.25\pm 0.01)\times 10^{-19}$
$4.5\times 10^{-3}$ | $0.1289\pm 0.0003$ | $(4.54\pm 0.02)\times 10^{-20}$ | $0.1015\pm 0.0007$ | $(3.58\pm 0.03)\times 10^{-20}$
$1\times 10^{-2}$ | $0.1452\pm 0.0003$ | $(2.303\pm 0.005)\times 10^{-20}$ | $0.1163\pm 0.0007$ | $(1.84\pm 0.01)\times 10^{-20}$
$5.8\times 10^{-2}$ | $0.1879\pm 0.0005$ | $(5.14\pm 0.01)\times 10^{-21}$ | $0.1605\pm 0.0009$ | $(4.39\pm 0.03)\times 10^{-21}$
$1\times 10^{-1}$ | $0.203\pm 0.003$ | $(3.22\pm 0.04)\times 10^{-21}$ | $0.1758\pm 0.0006$ | $(2.79\pm 0.01)\times 10^{-21}$

Table 1: Slope ($m$) values and ETPA cross-sections ($\sigma_{e}$) obtained
from RhB solutions of different concentrations at two-photon delay ($\delta
t$) configurations. The units of $\sigma_{e}$ are given in
$[cm^{2}/molecule]$.

Table 1 shows that the difference between $\sigma_{e}$ measured with
simultaneous photons ($\delta t=0fs$) and delayed photons ($\delta t=167fs$)
is rather small. However, it must be noticed that in the case of $\delta
t=167fs$ the two-photon absorption must be null since such temporal delay is
longer than the two-photon coherence time. This fact clearly demonstrates that
the calculated value of $\sigma_{e}$ at long delays can not be attributed to
ETPA but must be the result from an artifact. Therefore, the measurements at
$\delta t=0fs$ might also suffer from the same artifact, i.e., an optical loss
emulating the ETPA process. In spite the transmission vs $P_{pump}$
experiments tend to produce untrue $\sigma_{e}$ values, it results
illustrative to look at possible tendencies in the differences of $\sigma_{e}$
measured at $\delta t=0fs$ and $\delta t=167fs$. We perform an ANOVA test to
evaluate if there is a significant statistical difference in the values of
$\sigma_{e}$ calculated at the two delay configurations across different
concentrations. According to this test, the difference is statistically
significant since $p\leq 0.05$ (for data in Table 1 $p=0.05$ ). We believe
that this apparent difference was biased by an artifact and not from ETPA, as
discussed below considering the inconsistency of big variations of
$\sigma_{e}$ across concentrations.

The values of $\sigma_{e}$ presented in Table 1 varied 6 orders of magnitude
from the lowest to the highest concentration. This might represent an
inconsistency since $\sigma_{e}$ is an intensive (molecular) parameter that
should not depend on concentration, as we can see by differentiating Eq. (3)
and noting that changes in concentration ($\delta C$) must be compensated by
changes in the slope ($\delta m$), namely

$\delta\sigma_{e}=\frac{A}{CVN_{A}}\left(\delta m-\frac{m}{C}\delta
C\right)=0,$ (4)

where,

$\delta m=\frac{m}{C}\delta C,$ (5)

such that the quotient $\frac{m}{C}$ must be a constant. The inset of Fig.
3$C$ presents the relation between $m$ and $C$, showing a clear non-constant
relationship in our transmission vs $P_{pump}$ experiments. Further, it should
be noticed that in Table 1 there is a difference of 7 orders of magnitude in
the concentrations while there is a difference of only 1 order of magnitude in
the slopes, in contradiction with the expected behavior for ETPA where the
slope should grow more rapidly in order to keep the ratio between $m$ and $C$
as a constant. This implies that the increasing difference between $R_{sam}$
and $R_{sol}$ as a function of concentration should be due to optical losses
non-related to ETPA, for instance, it could be the effect of scattering,
agglomeration and an increasing of the residual one-photon absorption
occurring at high molecular concentration, as it has been suggested previously
for the case of classical non-linear absorption57.

We compare some of the $\sigma_{e}$ values previously reported in RhB with the
values here obtained. For example, Villabona-Monsalve et al.39 performed
measurements for concentrations in the range 0.038 to 110 mM, obtaining
$\sigma_{e}$ values from $4.20\times 10^{-18}$ to $1.7\times 10^{-20}$
$[cm^{2}/molecule]$; in particular, for the concentration of 4.5mM they
reported $6.3\times 10^{-20}$ $[cm^{2}/molecule]$, which is similar to the
value of $4.54\times 10^{-20}$ $[cm^{2}/molecule]$ we obtained at the same
concentration when $\delta t=0fs$. In another interesting work50, Corona-
Aquino et al., measured $\sigma_{e}$ values for various concentrations, and
similarly to the present work, they also reported $\sigma_{e}$ at zero and
temporal delays longer than the coherence time; for instance, for a sample of
4.5mM of concentration, they obtained $1.039\times 10^{-20}$ and $1.041\times
10^{-20}$ $[cm^{2}/molecule]$ for $\delta t=0fs$ and $\delta t=667fs$,
respectively. They did not find any trend in the difference of $\sigma_{e}$ in
both delay configurations. In similarity with our work, they also concluded
that the no suppression of the apparent nonlinear absorption at long delays is
due to artifacts.

Lastly, D. Tabakaev et al.61 also presented estimations of $\sigma_{e}$ but
for the case of Rhodamine 6G (a molecule of the Rhodamine family) using an
experimental scheme based on transmission and fluorescence. At the 4.5mM
concentration, the obtained value was $9.9\times 10^{-22}$
$[cm^{2}/molecule]$, which is two orders of magnitude smaller than our result
in RhB. In this cited work, and also in a work by T. Landes et al.48,
agglomeration effects were demonstrated in the fluorescence signal, which is
an indication of optical losses not associated with ETPA that lead to
overestimating the $\sigma_{e}$ values; consequently, authors proposed
corrections resulting in significant smaller values of $\sigma_{e}$.

Overall, all the cited works suggest that ETPA experiments based on
transmission or fluorescence vs power schemes could have reported
overstimulated values of $\sigma_{e}$ or effects unrelated to nonlinear
absorption, due to unaccounted linear optical losses.

To further investigate how the artifacts introduced by lineal optical losses
in transmission experiments can be discriminated, we implemented another set
of experiments in the scheme of transmission vs temporal delay, where a
controllable source of linear optical losses is used, as explained below.

###  3.1 Controllable optical losses introduced in ETPA transmission
experiments

To have a better understanding of how optical losses affect the ETPA
transmission measurements, we designed two complementary experiments. The
first experiment is the characterization of a two-photon interferogram
comprising linear optical losses induced by different solvents. The second
experiment allows us to introduce in a controllable way linear losses by
adding nanospheres of silicon dioxide ($SiO_{2}$) in the solvent. In these
tests, we performed transmission vs $\delta t$ measurements, which as we
already showed, are fully equivalent to the transmission vs $P_{pump}$
measurements. In the first mentioned experiment, HOM dips are obtained for
different solvents (methanol, ethanol, chloroform, ethylene glycol), and the
corresponding linear losses are analyzed; in the second experiment, methanol
is used as a solvent and the linear losses are studied as a function of the
increasing concentration of dispersed nanoparticles. Since the nanoparticles
do not have an absorption band in the wavelength range used in our tests59,
60, they effectively act as a light scattering medium producing controllable
linear losses depending on the nanoparticle’s concentration.

In Fig. 4 we present the obtained HOM dips for the four solvents. As a
reference, the figure also includes the HOM dips from RhB dissolved at the
concentration of 10mM in each of these solvents (solvents data are represented
by dots while $RhB$ samples are represented by starts). This particular
concentration was selected because is similar to those used by other authors
to calculate $\sigma_{e}$ as it was discussed above.

Figure 4: HOM interference experiment for solutions of RhB in different
solvents. Panels A) and B) show raw data and panels C) and D) the
corresponding normalization of panels A) and B. The HOM dips for solvents
alone are shown in panels A) and C), while in panels B) and D) the HOM dips
for solutions of RhB in four different solvents.

Various aspects can be extracted from these results; first, for each solvent
different curves of $R_{sol}$ are registered. This is because each solvent
produces a differentiated level of linear losses associated with Fresnel
reflections, light scattering, etc. According to Fig. 4$A$, the optical losses
produce offsets between the HOM interferograms, where the smallest losses
correspond to methanol, followed by ethanol, then chloroform, and finally
ethylene glycol. Second, Fig. 4$B$ shows the HOM dip of RhB in each solvent,
where it can be noted that there is a small difference between the maximum
value of $R_{sam}$ and its corresponding $R_{sol}$. Such differences are
expected since RhB dissolved in each solvent leads to different levels of
losses. Third, the visibility of the HOM dip for a RhB solution does not show
a significant change compared with the HOM dip of its corresponding solvent.
In fact, and spite the different linear losses involved in each case, the
temporal width ($\Delta t$) and depth of the dips remain unchanged as seen
clearly by plotting normalized interferograms (see Fig. 4$C$ and Fig. 4$D$).
Again, we perform an ANOVA test for the changes in visibility, obtaining a
value of $p=0.56$, therefore the small differences between the visibility of
$HOM_{sam}$ and $HOM_{sol}$ are not statistically significant, while they are
rather associated to random experimental data fluctuations. Table 2 summarizes
these results. Here we recall our previous work on the interference of SPDC
type-II photons51, where it was established that the ETPA activity introduces
asymmetries in the joint spectral intensity (JSI) function, and in turn
changes in the visibility of a HOM dip. Since the visibility of $HOM_{sam}$
shown in Fig. 4 is invariant with respect to $HOM_{sol}$, we conclude that
there is no detection of ETPA signal. This outcome contrasts with that
obtained in the standard transmission vs $P_{pump}$ experiment in which a
value of $2.30\times 10^{-20}$ $[cm^{2}/molecule]$ (see Table 1 ) was
calculated for the sample under test (concentration of 10mM in methanol).

$Solvents$ | $V_{sol}[\%]$ | $R_{sol(max)}[counts/s]$ | $R_{sol(min)}[counts/s]$
---|---|---|---
$Methanol$ | $57.4\pm 1.5$ | $6035\pm 101$ | $1635\pm 101$
$Ethanol$ | $56.8\pm 1.6$ | $5506\pm 96$ | $1516\pm 96$
$Chloroform$ | $57.2\pm 1.6$ | $4552\pm 81$ | $1240\pm 82$
$EthyleneGlycol$ | $57.3\pm 1.7$ | $3636\pm 68$ | $987\pm 68$
$Samples$ | $V_{sam}[\%]$ | $R_{sol(max)}[counts/s]$ | $R_{sam(min)}[counts/s]$
$RhB_{Methanol}$ | $57.7\pm 1.5$ | $5863\pm 95$ | $1573\pm 96$
$RhB_{Ethanol}$ | $56.2\pm 1.7$ | $5152\pm 98$ | $1445\pm 99$
$RhB_{Chloroform}$ | $57.2\pm 1.8$ | $4459\pm 89$ | $1215\pm 89$
$RhB_{EthyleneGlycol}$ | $56.8\pm 1.7$ | $3671\pm 68$ | $1011\pm 68$
Table 2: HOM dip properties for solutions of RhB in different solvents at 10mM
concentration

In the second experiment, we implemented a method of controlled linear losses
by adding silicon dioxide $(SiO_{2})$ nanoparticles (from Sigma Aldrich) of
$10nm$ in diameter into the solvent (methanol) at 5 different concentrations:
$C1=0.1$, $C2=5.2$, $C3=9.6$, $C4=15.3$, $C5=22$ $mg/10mL$. The nomenclature
used for the HOM dips obtained with these samples is $HOM_{SiO_{2}(C)}$, where
$C$ is the concentration of nanoparticles in the solvent. The HOM dips for
nanoparticles solutions (cyan-dots), the HOM dip for solvent (black-dots) and
the HOM dip for the most concentrated sample ($100mM$) of RhB ($HOM_{sam}$,
red-dots) are shown in Fig. 5$A$ for data without any treatment and in Fig.
5$B$ for normalized data.

Figure 5: HOM interference experiment for controlled linear losses mechanism
by adding $SiO2$ nanoparticles. $A)$ HOM dips for methanol solvent (black-
dots), silicon nanoparticles suspended in methanol at different concentrations
(cyan-dots), and RhB sample at the concentration of $100mM$(red-dots). $B)$
normalized HOM dips of the panel $A)$.

Figure 5 displays the dips of $HOM_{SiO_{2}(C)}$, where higher concentration
of nanoparticles implies stronger light scattering, and the subsequent
reduction of $R_{sam}$. Notoriously, the strong transmission losses introduced
by the scattering do not modify the visibility of the interferograms,
introducing only small random fluctuations in acquired data. These experiments
clearly show that linear losses in the transmission vs $\delta t$ measurements
can introduce noticeable offsets in the HOM dips without changing its inherent
visibility, i.e., such losses do not alter the down-converted photon pair
quantum state leaving invariant the dip visibility.

Also, Fig. 5 displays another interesting outcome: the plot of $HOM_{sam}$ at
$100mM$ matches $HOM_{SiO_{2}(C3)}$. In line with our discussion, it follows
that the effect of the light scattering introduced by nanoparticles of silica
dissolved in methanol at the $C$3 concentration is equivalent to the
transmission losses observed for RhB at $100mM$. The type of losses detected
in RhB must be linear because any loss of photon pairs due to ETPA should be
manifested as a change in the visibility of the normalized HOM dip. Indeed,
the visibility is intrinsically related to the quantum state of the down-
converted photons52, 58, while the absorption of entangled photons should
modify the quantum state.51 These observations, and the fact that in the
transmission vs $P_{pump}$ experiments, there is no depletion of $R_{TPA}$
signal even for temporal delay longer than the entanglement time, supports the
conclusion that the calculated $\sigma_{e}$ values presented in Table 1
resulted from artifacts produced by the sample which emulate the ETPA
activity.

##  4 Conclusions

The transmission of entangled photons from a nonlinear sample (RhB) was
examined in two configurations, namely transmission vs $P_{pump}$ and
transmission vs $\delta t$. In our experimental proposal, a HOM interferogram
is registered after a pair of type II down-converted photons produced with a
CW pump laser interact with the sample, with the premise that any effect that
the sample produces over the two-photon quantum state should be observed as a
change in the visibility of the HOM dip.

Remarkably, the proposed scheme of transmission vs $\delta t$ based on the HOM
interferogram results equivalent to the standard transmission vs $P_{pump}$
scheme and with this configuration we observed a persistence of $R_{TPA}$,
even after delaying the photon pairs further than their coherence time,
suggesting that the apparent ETPA signal detected in the transmission vs
$P_{pump}$ experiments is generated by linear optical losses. Likewise, the
variation of six orders of magnitude in the intensive parameter $\sigma_{e}$,
for different RhB concentrations, also supports the conclusion that the
detected $R_{TPA}$ signal can not be ascribed to an ETPA process.

Then, we performed a full analysis of the effects of controllable optical
losses in the transmission vs $\delta t$, through the HOM interferogram. The
obtained results show that linear losses from Fresnel reflections, scattering,
aglomeration or residual one-photon absorption only reduce the counts in
coincidence but without modifying the interferogram visibility. This is due to
the fact that linear losses do not affect the quantum state of photon pairs.

The invariant visibility of the HOM interferograms obtained for the RhB sample
tested at different concentrations stongly suggests that ETPA activity was not
detected in our study. Nevertheless, we show that the HOM dip is capable to
account for the presence of linear losses affecting the ETPA signal, becoming
a good strategy to study this elusive process.

##  5 Associated Content

###  5.1 Data Availability Statement

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.

##  6 Author Information

###  6.1 Corresponding Authors

  * •

Freiman Triana-Arango $-$ Centro de Investigaciones en Óptica A.C., A. P.
1-948, 37000 León, Gto, Mexico; Email<EMAIL_ADDRESS>
  * •

Roberto Ramírez-Alarcón $-$ Centro de Investigaciones en Óptica A.C., A. P.
1-948, 37000 León, Gto, Mexico; Email<EMAIL_ADDRESS>
  * •

Gabriel Ramos-Ortiz $-$ Centro de Investigaciones en Óptica A.C., A. P. 1-948,
37000 León, Gto, Mexico; Email<EMAIL_ADDRESS>
##  7 Disclosures

The authors declare no conflicts of interest.

##  8 Acknowledgments

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

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##  9 TOC graphic