# Entangled two-photon absorption detection through a Hong-Ou-Mandel
interferometer

Freiman Triana-Arango1<EMAIL_ADDRESS>Gabriel Ramos-Ortiz1<EMAIL_ADDRESS>Roberto Ramírez-Alarcón1<EMAIL_ADDRESS>1Centro de Investigaciones en
Óptica AC, Apartado Postal 37150, León, Gto, México

###### Abstract

Recently, different experimental methods to investigate the entangled two-
photon absorption (ETPA) process in a variety of materials have been reported.
The present work reports on a different approach on which the ETPA process is
detected based on the changes induced on the Hong-Ou-Mandel (HOM)
interferogram of two photons produced by a spontaneous parametric down
conversion (SPDC) Type-II process. Using an organic solution of the laser dye
Rhodamine B as a model, it is demonstrated that ETPA at $800nm$ can be
monitored by the changes in the temporal-width and visibility of the HOM
interferogram produced by the photons at a beam splitter after interacting
with the sample. Additionally, we present a detail model in which the sample
is modeled as a filtering function which fullfills the energy conservation
conditions required by ETPA, allowing to explain the experimental observations
with excellent agreement. We believe that this work represents a new
perspective to study the ETPA process, by using an ultra-sensitive quantum
interference technique and a detailed mathematical model of the process.

##  I Introduction

The non-linear 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, microengraving, photodynamic
therapy, etc [1, 2, 3]. Typically, TPA is achieved in third order materials
with large TPA cross-sections $(\delta_{c})$ through the use of pulsed lasers
delivering high density of random photons. However, from recent theoretical
[4, 5] and experimental work [6, 7], the interest for implementing TPA with
correlated (entangled) photons, namely entangled two-photon absorption (ETPA)
process, has increased notoriously.

The advantage of the ETPA process compared to the classical TPA effect is that
the former contributes to the total rate of absorbed photons $(R_{TPA})$ with
a linear dependence on the photon excitation flux $(\phi)$, while the
dependence is quadratic in the case of classical light [5, 8, 9]:
$R_{TPA}=\sigma_{e}\phi+\delta_{c}\phi^{2}$, where $\sigma_{e}$ is entangled
two-photon absorption cross-section. Notoriously, there can be $~{}32$ orders
of magnitude difference between $\sigma_{e}$ [7, 10, 11] and $\delta_{c}$ [12,
13, 14], $(\sigma_{e}\sim 10^{-18}[cm^{2}/molecule],~{}\delta_{c}\sim
10^{-50}[cm^{4}s/molecule])$, so in principle it is possible to achieve TPA in
the low flux regime by illuminating the sample with correlated photon pairs
produced by quantum processes such as spontaneous parametric down conversion
(SPDC) [15].

The calculated values of $\sigma_{e}$ reported in the literature for molecules
used as models lies in a wide range from $10^{-22}$ to
$10^{-18}[cm^{2}/molecule]$ [16, 10, 17], however there is a recent debate
about the real magnitude of this parameter or even the possibility to
experimentally observe the effect [17, 18].

Different experimental configurations have been employed to detect and
quantify the ETPA activity in different materials, based on measuring the
transmittance [7, 11, 19, 16, 18] and fluorescence exhibited by the sample
upon excitation with SPDC down-converted photon pairs [6, 20, 17, 21].
Although it has been pointed out that the ETPA quantification in transmittance
experiments could be influenced by artifacts [17], it is expected that
$\sigma_{e}$ lays within the aforementioned range. In such experiments the
difference in transmittance from a solution of the molecule under test and the
solvent alone is recorded, as a function of the excitation intensity, with the
experimental condition of the temporal delay ($\delta t$) between the down-
converted photons set to zero. In many of these works, the zero delay
condition is assured by implementing a HOM interferometer before the photons
interact with the sample. Recently, the HOM interference have been used to
analyze the changes in one of the photons of the down-converted pair by using
a Mach-Zehnder interferometer to analyze a additional time delay introduced by
a sample [22], however the temporal-width and visibility changes in the HOM
dip, once the down-converted photons interacted with the sample, has not been
consider as a measurement device to study the ETPA interaction.

Figure 1: A) Experimental configuration for the transmittance ETPA experiments
designed to use the HOM interferometer as a sensing device. B) Rhodamine B
molecule. C) Rhodamine B TPA cross-section (blue dots) and Rhodamine B linear
absorption spectrum (black solid curve) as a function of wavelength. The gray
rectangle corresponds to the the excitation region in our ETPA transmittance
experiment. The experimental data represented as blue dots was taken from
[12].

In this context, the present work proposes a transmittance experiment in which
the ETPA interaction in a sample is identified and quantified by the changes
induced in the visibility and temporal-width of the HOM interferogram produced
by two indistinguishable photons, generated in a SPDC Type-II process, after
they both interact with the sample.

##  II Experiment

The experimental setup is shown in Fig.(1A). A CW laser (Crystalaser
DL-405-100), centered at $403nm$ and with a $\sim 1nm$ bandwidth, is focused
by lens L1 (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 fulfill of the phase-matching
conditions, which maximize the SPDC emission process, it is necessary to align
the polarization of the pump beam 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 emitted
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
($Polarizer$). After the crystal we use the filter element $F_{1}$, composed
of a longpass filter (Thorlbas FELH0500) and a bandpass filter (Thorlbas
FBH800-40), in order to eliminate the residual pump. 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.

Additionally to control the frequency (bandapass filter) and temporal (delay)
indistinguishability, the final requisite to obtain an optimal HOM
interference pattern at $PBS_{2}$ is to eliminate the polarization
distiguishability, proper of photon pairs produced in a 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 in the experiments described below. Then, the down-converted
photons are focused into the $1cm$ quartz-cuvette containing the sample ($S$)
with a $5cm$ focal length lens ($L2$), producing a $W_{0}=58\mu m$ spot
diameter which is then collimated with a second lens of the same focal length
($L3$). 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.

The HOM effect results from the interference of two indistinguishable photons
at a beam splitter [23, 24, 25, 26, 27]. In our setup we obtain the HOM
interferogram by recording the coincidence counts rate $CC$ as a function of
the delay time $\delta t$ between the photons, registered by a time-to-digital
converter module (ID Quantique id800), between the $APD_{1}$ and $APD_{2}$
avalance photodiodes (Excelitas SPCM-AQRH), after the frequency and
polarization indistinghishable SPDC down-converted photons interact at
$PBS_{2}$. As a crucial property of the ETPA transmittance experiments
discussed below, in our setup we can change the degree of indistinguishability
of the photons in all the relevant degrees of freedom (frequency,
polarization, arrival time), allowing to fully control the interference
process. As shown in Fig.(2), the distinctive characteristic of the HOM
interferogram is a steep dip around $\delta t=0$ and a high coincidence counts
level for $\delta t$ values larger that the coherence length of the down-
converted photons.

As an inital calibration step, we measured the HOM interferogram for free
space propagation of the down-converted photons (cuvette removed), for two
different configurations of the filter element $F_{1}$. The first
configuration was set to optimize the spectral indistinguishability of the
photon pairs and maximize the HOM dip visibility by using a bandpass filter
with $10$nm bandwidth centered at $810$nm (Thorlbas FBH810-10). With this
filter we obtained a HOM dip with $94\%$ visibility and a temporal FWHM of
$181$fs. For the second configuration we used a bandpass filter with $40$nm
bandwidth, centered at $800$nm (Thorlbas FBH800-40), obtaining a $61$%
visibility and a FWHM temporal width of $70$fs, as shown in Fig.(2). The
reduction in the visibility is due to asymmetric spectrums of each photon
proper of a SPDC Type-II process [28]. The second filter configuration will be
used in the rest of the experiments and the obtained HOM dip is denoted in
this work as the reference HOM interferogram $(HOM_{ref})$.

Figure 2: HOM interferogram obtained for free space propagation of the down-
converted photons (cuvette removed), for two filter configurations. The black
dots correspond to the HOM dip obtained with a $10$nm bandaps filter (Thorlabs
FBH800-10), while the red dots where obtained with a $40$nm bandpass filter
(Thorlabs FBH800-40). The insets show an image of the Type-II SPDC rings
distribution as taken by a CCD camera (Thorlabs DCU224M), for both filter
configurations. The collinear photons used in the experiments were obtained
from the region where the SPDC cones ovelap. The solid lines were obtained
from $Eq.(\ref{HOM_interferogram_2}$) by setting the ETPA sample efficiency
parameter to zero ($\eta=0$), as explained below. In our experiments we use
the $40$nm filters and the obtained HOM dip is denoted as $HOM_{ref}$

### Sample and solvent configuration

The sample used as a model in our ETPA transmittance experiments was Rhodamine
B $(RhB)$ dissolved in methanol at different concentrations. As a reference,
the well-known linear absorption spectrum of this molecule is displayed in
Fig.(1C) along with the nonlinear absorption spectrum obtained from the
classical TPA effect [12]. As it can be seen, around $800$nm there is region
of two-photon resonance (indicated in gray color in the figure), which we will
aim at in our experiments, unlike of previous work where the ETPA in $RhB$ was
associated to the excitation of the state $S_{2}$ corresponding to a one-
photon energy of $355$nm [16].

Two experiments were carried out to study the ETPA effect in $RhB$. In the
first case we monitored the transmittance of the cuvette containing the
solvent with a sample concentration of $100$mM, by detecting the coincident
counts $CC$ as a function of the laser pump power, for two different $\delta
t$ values $\delta t=0$ (center of HOM dip) and $\delta t=167fs$ (out of HOM
dip). In the second case the laser pump power was fixed at $43.9$mW and the
$CC$ were registered as a function of $\delta t$, obtaining HOM interferograms
for: 1) free space propagation ($HOM_{ref}$), 2) the cuvette containing only
the solvent ($HOM_{sol}$) and 3) the cuvette containing the solvent-sample
solution for concentrations: $0.1\mu$M, $1\mu$M, $0.1$mM, $1$mM, $10$mM,
$58$mM, $100$mM ($HOM_{sam}$). In order to avoid additional refractive errors
induced by the manipulation of the cuvette, in all of these measurements the
solvent-sample solutions for different concentrations were deposited without
moving the cuvette, which was fixed at all time. Additionally, a cleaning and
drying process was implemented every time we changed the sample under study,
assuring any perturbation of the cuvette.

Figure 3: Noise HOM interferogram measurements. The first row present
measurements taken for a configuration of indistinguishable photons and high
density of down-converted photons, the second row has distinguishable photons
and high density of down-converted photons and the third row has
indistinguishable photons and low density of down-converted photons. $\uparrow
PM$ and $\downarrow PM$ indicate high and low photon density, respectively.
”$Two-cha$” indicates both arms of the interferometer unblocked, ”$Static-
cha$” indicates only the static arm unblocked and ”$Movable-cha$” only movable
arm unblocked. Red dots correspond to the measurements taken for the sample at
a concentration of $100$mM ($HOM_{sam}$), black dots for the solvent
($HOM_{sol}$) and blue dots for the empty cuvette taken as the reference
($HOM_{ref}$)

### Noise measurement

In order to provide certainty that changes in the visibility of the HOM
interferograms can be ascribed to ETPA, special care was taken to identify
possible sources of noise, systematic errors and artifacts. The versatility of
the experimental apparatus allows to control various parameters independently:
laser pump power ($HWP_{1}$), density of down-converted photons ($HWMP_{2}$),
polarization indistinguishability ($HWP_{3}$) and temporal delay ($\delta t$)
between the SPDC photon pairs. Then, the possible presence of errors and
artifacts in the HOM interferograms were assesed by monitoring the $CC$ as a
function of $\delta t$, when the density of down-converted photons and the
photon pairs polarization indistinguishability were set to be either high or
low under the following three configurations of the HOM inteferometer: 1) both
arms of the interferometer unblocked; 2) only the static arm unblocked; 3)
only the delay arm unblocked. This resulted in nine different situations in
which the $CC$ were plotted versus $\delta t$. In all this measurements the
laser pump power was fixed constant at the maximal value of $43.9mW$.

The first column of Fig.(3) presents the HOM interferograms obtained when both
arms of the interferometer are unblocked ($Two-channel$). Blue dots represent
the free space propagation ($HOM_{ref}$), black dots the solvent ($HOM_{sol}$)
and red dots represent the sample $RhB$ in solution with methanol for a
concentration of $100$mM ($HOM_{sam}$). Panel A) corresponds to the condition
of polarization indistinguishable photons where, as expected, we observe a
maximum HOM dip visibility, contrasting with the zero visibility produced in
the case of polarization distinguishable photons (panel B)). The level of
sensitivity of the apparatus can be seen in panel C) where the characteristic
interference dip at $\delta t=0fs$ is obtained even under the condition of
very low density of indistinguishable photons. It must be observed that for
long $\delta t$, where the interference effect tend to vanish, there exist an
offset between the curves in the following order:
$HOM_{ref}>HOM_{sol}>HOM_{sam}$. The offset is small between sample and
solvent, but significantly larger for the reference. These offsets, clearly
detectable at long $\delta t$, are ascribed to linear absortion, scattering
and Fresnel losses, which are larger in the sample than in solvent. Hereafter
we refer to these losses as linear losses. Intriguingly, however, is the fact
that the difference in $CC$ between samples and solvent decreases as the delay
decreases, as observed in panel A); in particular, when the delay is zero the
offset between them is minimum. It is of paramount importance to determine if
such a change in the visibility of the $HOM_{sam}$ with respect to the
$HOM_{sol}$ is produced by an ETPA process or it is due to artifacts in the
experiment, as discussed below.

Figure 4: ETPA transmittance measurements. Panels A) - C) present data in
which linear losses are not eliminated (raw measurements), while plots in
panels D) - F) show recomputed data when those linear losses are considered.
A) and D) display $R_{sol}$ and $R_{sam}$ as a function of the laser pump
power. B) and E) are the $R_{TPA}$ as a function of the $R_{sol}$ for a time
delay of $\delta t=0$fs. Panels C) and F) are the same experiments than B) and
E) but for a time delay of $\delta t=167$fs. Data in B), C), E), F) have
$95\%$ predictions bounds of the fit plotted by the dashed lines.

Panels D)-I) in Fig.(3) present the detected $CC$ when either one of the arms
in the HOM interferometer is blocked. In this case the second (third) column
in the figure corresponds to the detcted signal when the static (movable) arm
is unblocked. As expected, the level of $CC$ is very low in these conditions
as one photon of the pairs of generated entangled photons is blocked to freely
propagate in the optical system, leading then to the detection of accidental
$CC$. The panels D) and G) show the case of high density of indistinguishable
photons, but the blocking of one of the arms leads to a level of $CC$ than can
be considered basically noise. The noise is further reduced when the
conditions are set to generate distinguishable photons (panels E) and H)) and
becomes negligible when the density of indistinguishable photons is low
(panels F) and I)). As it can be observed, in all these six cases the noise
detected as accidental $CC$ is identical in each arm, demonstrating that each
arm is optically equivalent. Further, it must be remarked that a dip is not
registered in any of the cases in the vicinity of zero delay. Thus, the
reduction of the difference in $CC$ between solvent and sample at $\delta
t=0$fs, shown in panel A), should have a physical origin while it can not be
explained by errors or artifacts. This suggest that the small reduction of
visibility in the HOM interferograms of the sample at $\delta t=0$fs, relative
to the solvent could be assigned to ETPA, being the difference of $CC$ at long
delay due to a combination of absortion, scatering and diffraction losses
mechanisms which do no depend on the temporal delay between the photons.

##  III Results

### Variable pump-power measurements

Transmittance experiments were performed to evaluate the ETPA in $RhB$. To
vary the fluence of down-converted photons, the CW laser was configured to
generate a pump power from $0.25mW$ to $43.9mW$. The rate of $CC$ as a
function of the laser pump power was evaluated at $\delta t=0$fs and $\delta
t=167$fs, for two cases: 1) cuvette with solvent and 2) cuvette with a
solution of $RhB$ in methanol at a concentration of $100$mM. All these
measurmentes were taken without manipulating the cuvette, which was fixed in
the same position all the time, as explained before. The rate of transmitted
down-converted photons detected as $CC$ for solvent ($R_{sol}$) and sample
($R_{sam}$) are presented in Fig.(4A) for delay values of $\delta t=167$fs in
black (solvent), red (sample), and $\delta t=0$fs in blue (solvent) and
magenta (sample). The difference $|R_{sol}-R_{sam}|$ is typically defined as
the TPA rate of the sample $(R_{TPA})$ [16], but special care must be taken in
the use of this definition in order to discriminate the possible presence of
linear losses.

Let us calculate the quotient of $R_{TPA}$ with $R_{sol}$ for the whole pump-
power interval in transmittance experiments, which define the slope $m_{\delta
t}=\frac{R_{TPA}}{R_{sol}}$. This slope is related with the $\sigma_{e}$ value
by means of $m_{\delta t}=(CVN_{A}\sigma_{e})/(A)$ [16], where $C$ is the
concentration of the sample in the solvent, $V$ and $A$ are the interaction
volume and area, respectively, and $N_{A}$ is the Avogadro’s number. Fig.(4B)
presents the plot of $R_{TPA}$ versus $R_{sol}$ and the obtained fitting for
$m_{\delta t}$ at $\delta t=0$fs while Fig.(4C) presents the same relation for
$\delta t=167$fs. As it can be seen, effectively there is a difference between
$R_{sol}$ and $R_{sam}$, where $R_{TPA}$ in Fig.(4C) is larger for $\delta
t=167$fs than for $\delta t=0$fs in Fig.(4B). However, the ETPA process should
lead to a $R_{TPA}$ value larger at $\delta t=0$fs since it is a photon-delay
dependent process, which takes place mainly when both photons arrive
simultaneously at the sample [6, 10]. Thus, there must be detrimental effects
influencing the calculated $R_{TPA}$.

Notice that the transmittances at the maximum power in the Fig.(4A)
corresponds to the condition used to generate the interferograms $HOM_{sol}$
and $HOM_{sam}$ in Fig.(3A) so, the difference between solvent and sample
signal at $\delta t=167$fs is ascribed to linear losses which do not depend on
$\delta t$. Thereby, the value of $R_{TPA}$ at $\delta t=167$fs, for each pump
power value, is due exclusively to $\delta t$ independent linear losses that
can be eliminated as an offset in the data of Fig.(4A). When eliminating the
linear losses to re-compute $R_{sam}$, the plots of panels A), B) and C) in
Fig.(4) produce the plots D), E) and F). In these three new graphs, various
facts are noticed: first, $R_{TPA}$ for $\delta t=167$fs (panel F)) tends to
be null since the down-converted photons present a high delay between them and
the sample seems to be unable to produce the ETPA process. Second, the
computed value of $R_{TPA}$ increase substantially for $\delta t=0$fs (panel
E)) as compared with the case when the linear losses are not eliminated (panel
B)). Interestingly, in figure Fig.(4D) the fact that $R_{sam}$ appears larger
than $R_{sol}$ for $\delta t=0$fs could be considered as a contradiction,
since we expect a reduction in the transmitted photon pairs due to the ETPA
interaction, but this is a direct consequence of registering the CC after the
HOM interference process takes place in the sense that, as will be explained
below, the ETPA interaction occurring in the sample results in a visibility
decrease (CC increment) of the $HOM_{sam}$ respect to the $HOM_{sol}$, as
shown in Fig.(3A). Third, the slope for the sample at $\delta t=0$fs is larger
than at $\delta t=167$fs, meaning that if the temporal delay of the photons
arriving to the sample is long, the ETPA process does not occur, conversely if
the delay tends to zero, the ETPA process is detected, that is to say ETPA
works as an optical switch since it can be turned ”off” if the down-converted
photons involved in the process arrive at the sample with a temporal delay
longer than their coherence time, as measured in the HOM effect.

Various experiments on ETPA in transmission [16, 7, 29] and in
fluorescence[19, 10] configurations have been implemented, these works have
reported different values of $\sigma_{e}$ for different materials, however not
considering the linear losses could have produced underestimated values of
$\sigma_{e}$. In our case, when the linear losses are not taken into account a
value of $\sigma_{e}=(1.6978\times 10^{-21})\pm 1.32\%[cm^{2}/molecule]$ is
obtained while a larger value of $\sigma_{e}=(5.6874\times 10^{-21})\pm
1.37\%[cm^{2}/molecule]$ is calculated by eliminating the linear losses, which
is in good agreement with reported values [16, 18].

### HOM interferogram measurements results

Based on the results discussed above, the HOM interferograms shown in Fig.(3A)
are revisted and presented as Fig.(5A), eliminating the linear losses from the
raw data and normalizing the interferograms. In this figure we can appreciate
two main effects: first we notice that the temporal width $\Delta t_{FWHM}$ of
the $HOM_{sam}$ and $HOM_{sol}$ dip increases as compared to $HOM_{ref}$.
Second, it is observed that the visibility suffered a reduction for
$HOM_{sam}$ and $HOM_{sol}$ as compared with $HOM_{ref}$. Both effects are
present in the HOM dips of the sample and the solvent, but they are
notoriously enhanced for the case of the sample. So, if ETPA did not exist,
the two signals (solvent and sample) would behave similarly and the quotient
between them would be a constant, but this is not the case. In Fig.(5B) the
quotient between $R_{TPA}$ and $R_{sol}$ as a function of $\delta t$ is shown.
As expected, it is observed that the ETPA signal depends on the delay, showing
a Gaussian-like profile with a temporal width (FWHM) of $58.5$fs.

Now, in a fundamental level, an increase in the temporal width of the HOM dip
is related to a reduction of the spectral bandwidth of the interfering
spectrally indistinguishable photon pairs, then the observed change in the
temporal width for the $HOM_{sol}$ and $HOM_{sam}$ with respect to the
$HOM_{ref}$ should be due to an absortion process which reduces the spectral
bandwidth of the photons. Hence, the width increase of the solvent signal in
time domain is due to a non-negligible TPA process, while in the sample the
width is further increased due to a higher TPA. Then, the width change of the
HOM interferogram establishes the bandwidth reduction of the down-converted
photons and the visibility change establish the amount of photons absorbed
(efficiency of the process), corresponding to that bandwidth reduction. In the
next section we will provide a mathematical justification for the width and
visibility changes of the HOM interferogram, through the modeling of the
spectral interaction of the down-converted photons with the sample.

Figure 5: HOM interferogram measurements. A) Normalized HOM interferograms for
sample (red line), solvent (black line) and reference (blue line). B) The ETPA
signal retrieved from the solvent and sample HOM interferograms as a function
of the down-converted photons delay. The solid lines on each measurement are
obtained with the theoretical model presented below. Each data point of these
graphs represent a measurement of 4 seconds.

##  IV Mathematical model

In order to broaden the discussion of the experimental results, we propose a
model based on the fundamental interaction of the down-converted photons with
the two-photon transitions of the sample. With this model we find that during
the ETPA interaction, the sample produces a measurable effect over the joint
spectral intensity ($JSI$) function, which determines all the properties of
the SPDC photon pair quantum state. Now, since the HOM dip is directly related
to the JSI function of the interfering photons, the effect of the sample over
the JSI can be observed as a modification in the visibility and width of the
HOM interferogram, when the interference is produced after the down-converted
photons interact with the sample.

The state generated by a Type-II SPDC process can be written as [15, 30, 31]

$\lvert\psi(t)\rangle=\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{s}d\omega_{i}\zeta(\omega_{s},\omega_{i})\lvert\omega_{s}\rangle_{s}\lvert\omega_{i}\rangle_{i},$
(1)

where $\lvert\ \omega_{j}\rangle$ represents a single photon with frequency
$\omega_{j}$ for the signal (s) or idler (i) mode, with $j=s,i$. The function
$\zeta(\omega_{s},\omega_{i})$ represents the joint spectral amplitud function
(JSA) and
$S(\omega_{s},\omega_{i})=\left\|\zeta(\omega_{s},\omega_{i})\right\|^{2}$ is
the join spectral intensity (JSI). The JSA function contains all the relevant
information of the photon pair quantum state and, for the case of spectrally
filtered photons interacting with a sample in an ETPA process, it can be
written as

$\zeta(\omega_{s},\omega_{i})=\alpha(\omega_{s},\omega_{i})\phi(\omega_{s},\omega_{i})f(\omega_{s})f(\omega_{i})h(\omega_{s},\omega_{i}),$
(2)

where the pump envelope function $\alpha(\omega_{s},\omega_{i})$ is given by

$\alpha(\omega_{s},\omega_{i})=e^{-\frac{\left(\omega_{s}+\omega_{i}-\omega_{p}\right)^{2}}{2\Delta\omega_{p}^{2}}},$
(3)

with $\Delta\omega_{p}$ and $\omega_{p}$ the bandwidth and central frequency
of the pump spectrum respectively, while the phase-matching function
$\phi(\omega_{s},\omega_{i})$ is represented by

$\phi(\nu_{s},\nu_{i})=e^{-\frac{\gamma}{4}\left(\tau_{s}\nu_{s}+\tau_{i}\nu_{i}\right)^{2}}=e^{-\left(P\nu_{s}^{2}+Q\nu_{i}^{2}+2R\nu_{s}\nu_{i}\right)},$
(4)

where $\nu_{s}=\omega_{s}-\omega_{0}$, $\nu_{i}=\omega_{i}-\omega_{0}$, being
$\omega_{0}=\frac{1}{2}\omega_{p}$ the central frequency of the down-converted
photons wavepackets. Also,
$P=\frac{\gamma\tau_{s}^{2}}{4},~{}~{}Q=\frac{\gamma\tau_{i}^{2}}{4},~{}~{}R=\frac{\gamma\tau_{s}\tau_{i}}{4}$
and $\gamma\simeq 0.19$ is the usual constant for the sinc to gaussian
function FWHM approximation. The constants $\tau_{\mu}~{}(\mu=s,i)$ are
related with the group velocity of the photons wavepackets by means of
$\tau_{\mu}=l\left[k^{\prime}_{\mu}(\omega_{0})-k^{\prime}_{p}(2\omega_{0})\right]=l\left(\frac{1}{u_{\mu}}-\frac{1}{u_{p}}\right)$,
being $l$ the crystal length, $k^{\prime}_{\mu}$ the derivatives of the linear
momentums $k_{\mu}$ of the signal and idler photons respectively and $k_{p}$
the linear momentum of the pump beam. Eq.(4) shows that the phase-matching
function is not symmetric in its frequency arguments since $P\neq Q$
$(\tau_{s}\neq\tau{i})$, as a consequence of the different refractive index
experimented by each photon wavepacket inside the nonlinear uniaxial crystal
[28].

The filter function $f(\omega_{\mu})$ accounts for the effect of the bandpass
filter used to control the bandwidth of the down-converted photons ($F_{1}$),
which has an intensity profile that can be obtained from the manufacturer
datasheet and can be modeled as

$F(\omega_{\mu})=\left\|f(\omega_{\mu})\right\|^{2}=e^{-\left[\frac{(\omega_{\mu}-\omega_{F})^{2}}{2\Delta\omega_{F}^{2}}\right]},$
(5)

where $\mu=(s,i)$ and $\omega_{F}$ and $\Delta\omega_{F}$ are the central
frequency and the bandwidth of the filter respectively.

In our analysis we model the effect of the ETPA process as a filtering
function $h(\omega_{s},\omega_{i})$ which fulfills the energy conservation
condition required by ETPA, that is, the frequency sum of the two down-
converted photons must be equal to a frequency transition of the material
$\omega_{H}$, namely, $\omega_{s}+\omega_{i}=\omega_{H}$. In the frequency
domain, such a sample transfer function can be modeled as a kind of ”notch”
filter with a Gaussian profile and bandwidth $\Delta\omega_{H}$, proposed as:

$H(\omega_{s},\omega_{i})=\left\|h(\omega_{s},\omega_{i})\right\|^{2}=1-\eta
e^{-\frac{\left(\omega_{s}+\omega_{i}-\omega_{H}\right)^{2}}{2\Delta\omega_{H}^{2}}},$
(6)

where the parameter $\eta$ ($0\leq\eta\leq 1$) quantifies the quantum
efficiency of the entangled two-photon absorption process, which depends on
the sample used or on the concentration of the solvent-sample solution. In the
experiments presented here we use the two-photon interference effect as an
ultra-sensitive device, capable to detect slight changes in the quantum state
of the photon pairs, induced by an ETPA process occurring in the sample. Then,
in order to model our measurements we need to calculate the HOM interferogram,
as given by [28, 32]

$R_{c}(\delta
t)=K\iint\limits_{-\infty}^{+\infty}\,d\omega_{s}\,d\omega_{i}\left[\left\|\Omega(\omega_{s},\omega_{i})\phi(\omega_{s},\omega_{i})\right\|^{2}-\left\|\Omega(\omega_{s},\omega_{i})\right\|^{2}\phi(\omega_{s},\omega_{i})\phi^{*}(\omega_{i},\omega_{s})e^{-i(\omega_{i}-\omega_{s})\delta
t}\right],$ (7)

where $K$ is a normalization constant and
$\Omega(\omega_{s},\omega_{i})=\alpha(\omega_{s},\omega_{i})f(\omega_{s})f(\omega_{i})h(\omega_{s},\omega_{i})$.
The first integral of Eq.(7) is the total spectral content of the down-
converted photons interacting with the sample, and the second integral is the
Fourier transform of the spectral product of the sample transfer function and
the filtered down-converted photons function, therefore it is time-delay
dependent. When the time-delay between the photons is long, the second
integral of the Eq.(7) tends to zero, and $R_{c}(\delta t)$ will tend to the
maximum level of normalized CC. Now, as the time-delay between the down
converted photons decreases the subtraction of the integrals decreases until
the point of zero time-delay, where $R_{c}(\delta t)$ will reach the minimum
level of normalized CC, and it will be zero only if phase-matching function is
symmetric with respect to its frequency arguments
$\phi(\omega_{i},\omega_{s})=\phi(\omega_{i},\omega_{s})$. Nevertheless, if
the phase-matching function is not symmetric with respect to its frequency
arguments, nor if any of the functions in Eq.(7) introduces asymmetry, the
subtraction of the integrals will be different from zero, and will be
significantly different from zero if the asymmetry is large. The visibility of
the HOM dip is given by [27]:

$V=\frac{R_{c}(\delta t)_{max}-R_{c}(\delta t)_{min}}{R_{c}(\delta
t)_{max}+R_{c}(\delta t)_{min}},$ (8)

where $R_{c}(\delta t)$ represent the $maximal$ and $minimal$ value of the
normalized CC when scanning $\delta t$, then the visibility value will be
dependent on the asymmetry of the SPDC photon pairs quantum state.

Using the phase-matching, pump, filter and sample transfer functions described
above, we can compute a final expression for the normalized HOM interferogram,
which accounts for the effects of the interaction of the down-converted
photons with the sample:

$R_{c}(\delta t)=1-\left[\kappa e^{-\frac{\Delta\omega_{\Lambda}^{2}}{2}\delta
t^{2}}-\eta^{\prime}e^{-\frac{\Delta\omega_{J}^{2}}{2}\delta t^{2}}\right],$
(9)

where the constant $\kappa$ defines the visibility of the free space HOM dip
($HOM_{ref}$), obtained when sample is not present ($\eta=0$). The function
$\Delta\omega_{\Lambda}$ represents the bandwidth of the filtered down-
converted photons and $\Delta\omega_{J}$ is a function which relates the
bandwidths of the filtered down-converted photons function and the sample
transfer function, then the width of the HOM dip will be defined and
controlled by these two bandwidths. The parameter

$\eta^{\prime}=\frac{J_{0}\Delta\omega_{J}}{J_{\Lambda}\Delta\omega_{\Lambda}}\eta,$
(10)

is a modified ETPA quantum efficiency, which quantifies the two-photon
interaction with the sample and considers the parameters:

$\displaystyle J_{0}$
$\displaystyle=e^{-\left[\frac{(\omega_{\Lambda}-\omega_{H})^{2}}{2(\Delta\omega_{\Lambda}^{2}+\Delta\omega_{H}^{2})}\right]},$
(11a) $\displaystyle J_{\Lambda}$
$\displaystyle=e^{-\left[\frac{(\omega_{0}-\omega_{F})^{2}}{2(\Delta\omega_{0}^{2}+\Delta\omega_{F}^{2})}\right]},$
(11b) $\displaystyle\Delta\omega_{J}$
$\displaystyle=\Delta\omega_{\Lambda}\left[1+\left(\frac{\Delta\omega_{\Lambda}}{\Delta\omega_{H}}\right)^{2}\right]^{-1/2},$
(11c)

where $\omega_{0}$ and $\Delta\omega_{0}$ are the central frequency and
bandwidth of the down-converted photons and $\omega_{\Lambda}$ is the central
frequency of the filtered photons. The quotient $(J_{0}/J_{\Lambda})$ presents
dependence on the frequency detuning
$(\delta\omega=|\omega_{\Lambda}-\omega_{H}|)$ between the central frequencies
of the filtered down-converted photons and the sample transfer function. The
parameter $J_{\Lambda}$ can be considered equal to one, since in general the
central frequency of the filter is selected equal to the central frequency of
the down-converted photons ($\omega_{F}=\omega_{0}$), thereby
$0\leq(J_{0}/J_{\Lambda})\leq 1$. Also, in Eq.(11c)
$0\leq(\Delta\omega_{J}/\Delta\omega_{\Lambda})\leq 1$ and since
$0\leq\eta\leq 1$, the value of $\eta^{\prime}$ must be in the range
$0\leq\eta^{\prime}\leq 1$. Then, the effect of the modified ETPA efficiency
$\eta^{\prime}$ is to reduce the visibility obtained for the reference HOM
dip, as a consequence of the ETPA process.

The model shows that there are three ways to modify the HOM interferogram
related to the interaction of the down-converted photons with the sample
(Eq.9): first, by changing the frequency detuning between the central
frequencies of the filtered down-converted photons function and the sample
transfer function $(\delta\omega=|\omega_{\Lambda}-\omega_{H}|)$, which has
the effect to modulate $\eta^{\prime}$ through $J_{0}$ (Eq. (10)), since for
$\delta\omega=0$ $J_{0}$ is maximal and when the detuning is large
$\eta^{\prime}\rightarrow 0$ so we can turn “off” the ETPA process. Second, by
varying the bandwidth of the functions $\Delta\omega_{\Lambda}$ and
$\Delta\omega_{H}$ we can also modulate $\eta^{\prime}$, but more importantly
we can change the $\Delta\omega_{J}$ function which determines the temporal
width of the HOM dip associated to the sample. Third, by changing the $\eta$
parameter of the sample which directly affects $\eta^{\prime}$. In our
experiments it is possible to modify $\eta$ by changing the concentration of
the sample, while the other parameters are implicitly determined when we
defined the experimental setup and when $RhB$ was chosen as a model for these
studies.

Figure 6: A) $HOM_{sam}$ for a concentration of $100$mM (red) and $HOM_{sol}$
(black). B) HOM interferogram for seven different concentrations of $RhB$ and
its corresponding $\eta$ values obtained by fitting the model. The points
represent the measurements and the solid lines corresponds to the theoretical
curve generated by using the fitted $\eta$ value. The inset shows a zoom
around the $\delta t=0$ region to appreciate how the $HOM_{sam}$ approaches to
the $HOM_{sol}$ for lower concentrations. The inset also allows to observe the
excellent agreement between the measurements and the theoretical model. Each
data point represents a measurement of 4 seconds.

To contrast our model with experimental data, we measured the HOM
interferogram of the solvent-sample solution of $RhB$ for the concentrations
defined in the experiment section. In total we measured eight HOM
interferograms: seven for the solutions of $RhB$ ($HOM_{sam}$) and one more
for the methanol solvent ($HOM_{sol}$). During these experiments we used a
single 1cm long quartz-cuvette containing the solvent or the solvent-sample
solutions. The cuvette was always fixed in the same position and between each
successive measurement we deposited the solvent-sample solution without
perturbing the cuvette, by implementing the cleaning and drying procedure
described before. By fixing all the control parameters with the experimental
values used in our setup, Eq.(9) predicts specific values of $\eta$ for each
concentration of $RhB$, namely, a particular quantification of the ETPA
process.

In Fig.(6) we present the experimental results for the HOM dip measurements in
dots and in solid lines the corresponding plotting of Eq.(9) considering the
obtained ETPA process efficiency $\eta$ values. Fig.(6A) shows the HOM
interferogram obtained for the solvent and for the most concentrated solution
of $RhB$ ($100$mM), presenting the visibility and temporal-width values
measured, along with the calculated $\eta$ values. For this sample
concentration we obtain a fitted value of $\eta=0.1247$. We also present a
calculated $\eta_{s}=0.0069$ associated with a weak ETPA process occurring in
the solvent. All of the measurements and simulations, with their corresponding
$\eta$ values, are presented in (6B), showing and excellent agreement as can
be appreciated in the inset. In this figure we notice how for low
concentrations the HOM dips obtained for the solvent and the solvent-sample
solutions are almost the same, with and approaching measured visibility of
$V=57.8\%$ and a corresponding predicted value of $\eta=0.0294$, for the
solution at $0.1\mu$M. We can also appreciate how by increasing the
concentration, or the value of the $\eta$ parameter, the HOM dip visibility
correspondingly reduces, as predicted by the model. Remarkably, the model also
allows to reproduce the measured ETPA signal as a function of the temporal
delay between the down converted photons, presented as the solid line in Fig.
(5B).

##  V Conclusions

In this paper we have used the two-photon interference effect, as a mechanism
to study and quantify the ETPA process for the molecular system Rhodamine B in
the $800nm$ region, being the first time, to the best of our knowledge, that
this approach is explored. In our setup, we measure the HOM interferogram
produced by indistinguishable Type-II collinear SPDC photon pairs for three
cases: 1) free space propagation, 2) after interacting with the methanol
solvent, 3) after interacting with the solvent-sample solution at different
concentrations. The HOM dip obtained after free space propagation is used for
device optimization while the measurements obtained after the down-converted
photons interact with the solvent or the solvent-sample solution are used to
study the ETPA process in Rhodamine B.

As a calibration stage we present a detailed experimental analysis of the
source of errors in the coincidence counts rate, as a function of the time-
delay between the down-converted photons. By direct comparison between the HOM
interferograms obtained for the solvent and the solvent-sample solution, for
zero temporal delay and a long temporal delay (longer than the coherence time
of the down-converted photons), we can deduce the linear losses that must be
compensated in order to avoid underestimations in the calculations of the ETPA
cross-section. In our case, when the linear losses are not taken into account
a value of $\sigma_{e}=(1.6978\times 10^{-21})\pm 1.32\%[cm^{2}/molecule]$ is
obtained, while a larger value of $\sigma_{e}=(5.6874\times 10^{-21})\pm
1.37\%[cm^{2}/molecule]$ is calculated by eliminating the linear losses.

Then, by implementing the standard measurements of transmittance as a function
of the pump power, we obtained clear evidence of the dependence of the TPA
rate of the sample with the temporal delay, when considering linear losses,
showing the expected behaviour of switching ”on” for zero delay times and
switching ”off” for long delay times. Furthermore, when comparing the HOM
interferogram measurements of the solvent and the solvent-sample solution it
was possible to extract the ETPA signal behavior as a function of the temporal
delay between the down-converted photon pairs which, also as expected, shows a
clear signal peak around zero temporal delay between the indistinguishable
photon pairs. This delay dependence of the ETPA signal has only been obtained
in fluorescence experiments, but never from a transmission measurement.

We also present a detailed mathematical model which accounts for the ETPA
interaction of the down-converted photons with a sample in a certain
concentration, by modeling the sample as filtering function which fulfills the
energy conservation conditions required by ETPA. In the model we propose the
$\eta$ parameter with values $0\leq\eta\leq 1$, which accounts for the quantum
efficiency of the ETPA process and, interestingly, it can be experimentally
controlled by changing the concentration of the solvent-sample solution under
study. We also introduce the $\Delta\omega_{J}$ bandwidth function which
relates the bandwidths of the filtered down-converted photons function and the
sample transfer function. From the model we can see that the
$\Delta\omega_{J}$ function, fixed when the experimental setup was defined and
when RhB was chosen, determines the measured width for the HOM dip associated
with the ETPA process, while the $\eta$ parameter has a direct effect on the
HOM dip visibility in the sense that an increasing value for $\eta$ reduces
controllably the visibility, starting from a maximal value of the visibility
for the HOM interferogram of the solvent.

Then, we performed HOM dip measurements for the solvent and seven different
concentrations of the solvent-sample solution and fitted the model in order to
obtain predicted $\eta$ values, showing excellent agreement with the
measurements. This analysis shows that for low concentration of the solvent-
sample solution at $0.1\mu$M, the HOM interferograms are almost overlapped,
with approaching measured visibilities of $V=57.8\%$ for the solvent-sample
solution and $V=59.7\%$ for the solvent and corresponding predicted values of
$\eta=0.0294$ for the sample and $\eta_{s}=0.0069$ for the solvent, which is
considered to present a weak ETPA process. For the highest concentrated
solvent-sample solution of $100m$M the measured visibility of the HOM dip
reduces to $50.4\%$ with a predicted greater value of $\eta=0.1247$. It is
important to mention that in the presented analysis it is missing the relation
between $\eta$ and fundamental parameters of the molecular system under study,
such as the $\sigma_{e}$ or the concentration. To find this fundamental
relation will be part of another work currently under development.

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.

##  VI Funding

This work was supported by CONACYT, Mexico grant Fronteras de la Ciencia
217559.

##  VII Acknowledgments

We acknowledge support from CONAYCT, Mexico.

##  VIII Disclosures

The authors declare no conflicts of interest.

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