# Limitations in Fluorescence-Detected Entangled Two-Photon-Absorption
Experiments: Exploring the Low- to High-Gain Squeezing Regimes

Tiemo Landes  Brian J. Smith  Michael G. Raymer  Department of Physics and
Oregon Center for Optical, Molecular, and Quantum Science, University of
Oregon, Eugene, Oregon 97403, USA<EMAIL_ADDRESS>
###### Abstract

We closely replicated and extended a recent experiment (“Spatial properties of
entangled two-photon absorption,” Phys. Rev. Lett. 129, 183601, 2022) that
reportedly observed enhancement of two-photon absorption rates in molecular
samples by using time-frequency-entangled photon pairs, and we found that in
the low-flux regime, where such enhancement is theoretically predicted in-
principle, the two-photon fluorescence signal is below detection threshold
using current state-of-the-art methods. The results are important in the
context of efforts to enable quantum-enhanced molecular spectroscopy and
imaging at ultra-low optical flux. Using an optical parametric down-conversion
photon-pair source that can be varied from the low-gain spontaneous regime to
the high-gain squeezing regime, we observed two-photon-induced fluorescence in
the high-gain regime but in the low-gain regime any fluorescence was below
detection threshold. We supplemented the molecular fluorescence experiments
with a study of nonlinear-optical sum-frequency generation, for which we are
able to observe the low-to-high-gain crossover, thereby verifying our
theoretical models and experimental techniques. The observed rates (or lack
thereof) in both experiments are consistent with theoretical predictions and
with our previous experiments, and indicate that time-frequency photon
entanglement does not provide a practical means to enhance in-solution
molecular two-photon fluorescence spectroscopy or imaging with current
techniques.

††preprint: APS/123-QED

##  I Introduction

The use of the spectral-temporal correlation present in time-frequency
entangled photon pairs to drive nonlinear optical light-matter interactions
more efficiently than can be achieved with classical laser light has been
studied since first being proposed in 1989 [1, 2]. Promising initial work
utilizing squeezed light driving narrow-line atomic systems was demonstrated
in the mid 1990s and 2000s [3, 4]. Recently, the possibility of so-called
entangled two-photon absorption (ETPA) in molecular samples of chemical and
biological interest has been the subject of lively debate. Studies dating back
to 2006 [5, 6, 7, 8] claim to observe ETPA in molecular systems at
efficiencies estimated to be around 10 orders of magnitude larger than
expected from heuristic theoretical arguments [9]. More recent work has
demonstrated functional microscopes purportedly benefiting from time-frequency
entanglement of photons [10, 11]. However, the magnitude of the reported
increase in two-photon absorption (TPA) efficiency by use of photonic
entanglement cannot be easily explained by standard quantum mechanical models,
which indicate that any enhancement should be far more moderate than reported
[12, 13, 14, 15]. The estimates presented by these theories indicate that
measurements of ETPA would be infeasible with currently available entangled-
photon sources based on spontaneous parametric down conversion, two-photon
absorbing samples and current detection methods. Recent studies have placed
experimental upper bounds on the efficiency of ETPA[16, 17] under various
experimental conditions that are consistent with the theoretical estimates.
Separate experiments have described other linear-optical effects that could be
mistaken for ETPA using common experimental methods to search for ETPA
signals[18, 19, 20].

In this work, we present a series of experiments examining TPA in Rhodamine 6G
(R6G) molecular dye in solution and, for comparison, the closely-related
nonlinear optical process of sum-frequency generation (SFG) using various
excitation sources. The TPA experiments aimed to reproduce results from a
recent study that reports ETPA in R6G solution [21] while placing tight
constraints on the source and detection systems. Our experiment had several
key improvements over the original, as described in the methods sections. In
brief, they are: a lower-noise detection system, a flexible laser system
allowing observation of TPA-induced fluorescence using light generated by
high-gain squeezing, and the observation of the low-to-high-gain ’crossover’
in SFG to check consistency of our experimental setup with theory predictions.
The main result is that using this well-calibrated system, no detectable ETPA
was observed, casting doubt or at least questions about the origin of the
signals reported elsewhere.

Here quantum light is generated by multimode spontaneous parametric down
conversion (SPDC) using both continuous-wave and pulsed-pump configurations.
The SPDC process generates squeezed vacuum states of light going from the low-
to high-gain regimes to observe the scaling of TPA and SFG signals arising
from these different input states of light. In the limit of low-gain
squeezing, where the quantum-advantage of using EPP is predicted to be most
pronounced [12, 13, 14, 15], we can access the so-called ’isolated-pair
regime’ in which there is at most one pair of photons distributed across the
correlated source modes. The enhancement in TPA efficiency arising from
spectral-temporal correlations of the source is expected to occur when the
spectral bandwidth of the entangled photon pair is broader than the linewidth
of the two-photon absorption transition. The pulsed version of the experiment
enables us to clearly observe TPA using a high flux of squeezed light. We
verified the presence of the TPA process in this case by studying the
fluorescence emission lineshape and quadratic dependence of fluorescence
intensity with excitation flux. We use squeezed-vacuum light in the high-gain
regime, also called bright squeezed vacuum (BSV) to probe the efficiency of
TPA and determine the minimum flux required to observe fluorescence from TPA
in the pulsed configuration. Separately, we use the pulsed source to confirm
the ability of the standard quantum optical model to accurately predict the
properties of sum-frequency generation (SFG) of squeezed vacuum, which shares
many similarities to TPA and confirm important predictions about the process
driven by squeezed light [12].

With careful calibration of the source and detection system, we were unable to
observe any TPA fluorescence signal when using weak squeezed vacuum (isolated
photon pairs) [20], consistent with predictions of recent calculations of ETPA
efficiency[13, 14]. These calculations predict only a moderate ETPA efficiency
as a result of the exceedingly small TPA cross sections in solvated dye
molecules. The predicted efficiency is insufficient to outweigh the low flux
required to operate in the isolated entangled-photon-pair regime where
enhancement by entanglement is predicted, rather than in the BSV regime where
enhancement by entanglement is predicted to be minimal or nonexistent. The
present experiments provide evidence that the signals attributed to ETPA
reported in previous studies in molecules, in which the TPA linewidth is large
compared to the EPP bandwidth, likely originated from unknown mechanisms, not
clearly related to ETPA. Thus, the present study provides strong evidence that
the sought-after advantages of using ETPA as a practical tool for enhancing
molecular spectroscopy and biological imaging remains elusive.

This paper first brings together and summarizes relevant theoretical
considerations, then presents the results of our experimental study.

##  II Theory summary

The scenario considered is two-photon excitation of a molecular system in
solution by a broadband squeezed-vacuum field having a single linear
polarization and with entanglement in frequency (anti-correlated) and in time
(positively correlated), as commonly generated using spontaneous parametric
down-conversion (SPDC).

The theory of two-photon absorption of time-frequency entangled photons for
atomic or molecular systems with no near-resonant intermediate states has been
described in detail [12, 22]. While the dyes used in our experiments and our
theoretical models correspond to TPA without near-resonant intermediate
states, theoretical work on systems with intermediate states has also been
reported, reaching similar conclusions[15, 23]. Our experiments range from the
low-gain SPDC regime to the regimes of intermediate and high-gain squeezed
vacuum. To distinguish between the regimes we refer to low-gain squeezed
vacuum as the isolated-pair regime in which entangled photon pairs (EPP) are
generated and the probability for multi-pair emission is small. We refer to
the high-gain squeezed vacuum regime as bright squeezed vacuum (BSV)[24]. When
referring to both the low- and high-gain regimes we simply use the term
squeezed vacuum. At intermediate gains, in which the multi-pair emission
becomes comparable to the single-pair emission, a ‘crossover’ regime in the
scaling of the TPA rate is explored. These are different regimes of the same
squeezing process, and are generated in the same nonlinear optical crystal in
our experiments.

Theoretical derivations of the TPA rate point to a positive enhancement
(compared with laser light with the same spectral bandwidth and flux) of the
efficiency of TPA of squeezed vacuum in the isolated-pair regime as a result
of time-frequency correlations and photon-number correlations [22, 12, 13, 14,
25]. This relative increase in TPA efficiency can be large, but only in the
case of ultra-low entangled-pair flux and for narrow molecular or atomic TPA
linewidths compared with the bandwidth of the excitation source. However, due
to the extremely low efficiency of the TPA process itself and challenges in
realizing a high generation rate of isolated entangled photon pairs, the net
ETPA rates are predicted theoretically to be well below current detection
thresholds for common molecular dyes and typical experimental geometries [14].

###  II.1 Linear and Quadratic Flux Scaling

A key experimental feature of ETPA is the predicted linear scaling of the EPTA
rate with excitation flux, not expected for most second-order nonlinear
optical phenomena, which typically display quadratic scaling for classically
driven processes. This linear flux scaling feature alone is cited as evidence
of ETPA in many experiments that have observed signals attributed to ETPA[5,
6, 8].

This linear scaling can be understood in terms of a commonly adopted heuristic
equation for the rate of ETPA for a single absorber (atom or molecule)[9]

$R^{ETPA}=\sigma_{e}\phi+\xi\sigma^{(2)}\phi^{2},$ (1)

where $\phi$ is the photon-flux density [$\text{photons
m}^{-2}\text{s}^{-1}$], $\sigma_{e}$ is the ETPA cross-section
[$\text{m}^{2}$],$\sigma^{(2)}$ is the conventional TPA cross section [GM]. We
have also inserted $\xi$ which is a unitless parameter of order unity
discussed more below. The first term arises from a coherent contribution in
which all photon pairs contributing to TPA are temporally-spectrally
correlated, the second term arises from both correlated photons pairs and
uncorrelated photon pairs. We refer to the contributions from the uncorrelated
pairs as the incoherent contribution.

For molecules in solution $\sigma^{(2)}$ is typically exceedingly small—on the
order of 1 to 1,000 GM, where 1 GM = $10^{-58}\text{m}^{4}\text{s}$ [26]. When
the bandwidth of the exciting squeezed-state light is much greater than the
molecular TPA linewidth, $\sigma_{e}$ is given by

$\sigma_{e}=f\sigma^{(2)}/A_{e}T_{e},$ (2)

where $A_{e}$ and $T_{e}$ are the entanglement area and entanglement time of
the exciting light (defined as the area and time within which photon pairs are
tightly correlated), and $f$ is a phenomenological parameter. This heuristic
description agrees well with in-depth theoretical derivations in cases where
the squeezed light is confined to a single transverse mode (such that $A_{e}$
equals the transverse mode area), which show clearly that $f$ is expected to
be of the order of unity [25, 12].

In contrast to ETPA, TPA driven by coherent or thermal-like light lacks the
linear-in-flux term because in those cases the probability for two photons to
arrive within the molecular response time depends quadratically on the
incident flux. For ETPA the $\sigma_{e}$ term arises due to the fact that
entangled photons are produced in pairs, leading in the low-flux limit to a
linear relationship with the photon flux.

The in-depth derivations also show that when the bandwidth of the exciting
squeezed-state light is much smaller than the molecular TPA linewidth, meaning
the two-photon response is instantaneous, the rate is given by [25],

$R^{ETPA}=g^{(2)}(0)\sigma^{(2)}\phi^{2},$ (3)

where $g^{(2)}(0)$ is the degree of second-order coherence of the light. In
the case of type-0 or type-I SPDC phase matching–where the photons are
indistinguishable–it is given by $g^{(2)}(0)=1/n+3$, in which $n$ is the mean
number of photons per spectral-temporal-spatial mode [12, 23]. In this case
the factor $\xi$ in the heuristic equation is equal to 3 in the high-flux
regime. For type-II phase matching, wherein photons are distinguishable,
$g^{(2)}(0)=1/n+2$ [25]. Note that this equation still maintains linear flux
dependence at low gain due to the implicit dependence of $g^{(2)}(0)$ on
$1/\phi$ , since $\phi\propto n$.

In cases where the squeezed light is confined to a single transverse mode,
such that $A_{e}$ equals the transverse mode area $A_{0}$, we have $n=\phi
A_{0}T_{e}$, where the entanglement time $T_{e}$ equals the inverse of the
full-width-at-half-maximum (FWHM) bandwidth B (in Hz) of the squeezed-light
source $T_{e}=1/B$, assuming the SPDC pump laser bandwidth is much smaller
than the SPDC bandwidth, $\Delta\nu_{p}<<B$. The case when the bandwidth of
the squeezed light source becomes comparable to the molecular TPA linewidth is
described by more complicated formulas, as explored in [22, 25].

If linear loss, e.g., an optical attenuator, is inserted between the SPDC
source and the TPA sample, the flux scaling of the TPA rate in the low-flux
regime is predicted to be quadratic with the amount of loss, because both
photons in a pair are independently subject to the loss. This feature has been
verified in sum-frequency generation (SFG) [27, 17], which is a separate
nonlinear optical process with dynamics similar to that of TPA in the absence
of intermediate resonances [12, 25]. This feature is also readily observable
in photon coincidence counting experiments, which are insensitive to ultra-
fast temporal or spectral effects. Observation of both the linear behavior
(when varying the SPDC pump power) and quadratic behavior (when inserting
loss) in the same experiment is a more robust validation than verifying linear
scaling alone, since linear scaling in itself is a common phenomenon and can
lead to mis-attribution of unrelated linear-optical effects to ETPA [18, 19,
20]. To date, only [21] has reported both effects simultaneously in molecular
TPA.

It is worth noting that, as pointed out in [21], $\sigma_{e}$ is a quantity
that depends on the molecular properties of the TPA sample as well as the
detailed properties of the entangled light driving the interaction. This
includes experimental conditions such as beam focusing, pump bandwidth, as
well as the phase-matching bandwidth of the process generating the entangled
photons. Because of this, it is ambiguous to report $\sigma_{e}$ without
explicit consideration of these parameters. A modified quantity
$\sigma_{e}\times T_{e}\times A_{e}$ is more physically meaningful for
comparison of experiments with varying conditions.

The primary question we wish to address is what values can the parameter $f$
take on in realistic experiments? While theory indicates that $f$ should be on
the order of 1, [14] the value needed to produce the ETPA efficiencies
reported in other ETPA experiments[5, 21] is many orders of magnitude greater.
The present paper reports further concrete experimental evidence that $f$ is
not much greater than $1$ in solvated dye-like molecules.

###  II.2 Crossover Flux

The crossover between the linear and quadratic flux-scaling regimes of two-
photon absorption and SFG rates for a narrowband two-photon absorbing sample
occurs when the two terms in Eq. (1) are comparable, as described in [12, 25,
23]. This corresponds to $\phi=\sigma_{e}/\sigma^{(2)}$. Using $n=\phi
A_{0}/B$, under the condition that $A_{e}=A_{0}$, this criterion is equivalent
to having the number of photons per spectral-temporal mode equal $f$. This
conclusion agrees with expectation when recognizing that $f$ is approximately
equal to 1. The understanding is that when the number of photons per spectral-
temporal mode exceeds 1, we leave the isolated-pair regime. Above the
isolated-pair regime, photons from uncorrelated pairs overlap in time and
generate excitation from TPA at rates comparable to ETPA, eliminating the
benefit of temporal-spectral entanglement and reverting to quadratic flux
scaling. At this flux, a transform-limited pulse of the same duration and
intensity would have nearly the same ETPA efficiency [25].

Unfortunately, it is unfeasible to observe the incident flux at which
crossover occurs directly, due to the extremely low efficiency of TPA.
However, the crossover point in the absence of spatial-coherence effects can
be understood as a feature of the SPDC process and the acceptance bandwidth of
the nonlinear process rather than being a feature unique to the TPA process.
In this regard, the response of SFG is analogous to that of TPA. Because of
this similarity, we use SFG to constrain experimentally the flux at which the
crossover flux occurs for our SPDC (squeezing) source. To do this we measure
directly the crossover that occurs in SFG using broadband entangled photons of
variable flux. The agreement with predictions ensures that our theoretical
description of the interaction captures the relevant dynamics for TPA, giving
us confidence in our reported results for TPA in molecules.

###  II.3 TPA rate with BSV

Here we make a few more clarifying remarks comparing the TPA rate with bright
squeezed vacuum to that with classical light. As noted above, Eq. (3), the TPA
rate of an instantaneous two-photon process is proportional to the degree of
second-order coherence, $g^{(2)}(0)$, of the driving field [24, 28, 29, 30,
31, 25]. For classical coherent-state light arising from laser excitation we
have $g^{(2)}(0)=1$ and for squeezed vacuum $g^{(2)}(0)=3$. For thermal-like
light $g^{(2)}(0)=2$. Values of $g^{(2)}(0)$ greater than 1 can be thought of
as an increase in TPA rate arising from the quadratic nature of the
interaction and the increased fluctuations in the driving field leading to
higher probability for two or more photons present in a single temporal-
spectral mode. However, in the case of interest the two-photon transition has
a finite bandwidth, and thus the bandwidth and spectral correlations of the
incident field are important considerations. In the limiting case of
exceedingly narrow TPA linewidth, $\Delta\nu_{f}$, in comparison to the
bandwidth of the field $B$, the incoherent (spectrally uncorrelated) portion
of the field has an exceedingly small overlap with the transition, and
contributes negligibly to the overall TPA rate [12, 25]. On the other hand,
the TPA rate arising from the coherent (correlated) portion of the field at
high-gain is comparable to the TPA rate arising from classical light. In the
case of a moderate TPA linewidth, the incoherent contribution is no longer
negligible and the final TPA rate will lie somewhere between these two bounds.
A more complete discussion of these effects can be found in [25].

##  III Experimental Overview

Below, we describe our experimental results, which 1) validate the theoretical
prediction that the TPA rate of BSV and coherent excitation are the same, to
within a factor of 3, in Rhodamine 6G; 2) observe the crossover between linear
and quadratic flux-scaling regimes in SFG as a means to characterize the state
of our squeezed light source and as verification of our theoretical models and
experimental techniques; 3) confirm that TPA of pulsed BSV with average fluxes
comparable to maximal achievable entangled photon rates remain in the
quadratic-scaling regime down to state-of-the-art detection thresholds; and 4)
replicate recent experiments that report signatures of ETPA [21] at lower
detection threshold reduced background rates. The replication experiment
demonstrates no detectable ETPA signal despite the reduced background and
detection threshold in comparison with [21].

Based on these results, which are in agreement with our theoretical
predictions, we conclude that the established theory for ETPA is not missing
any crucial effects that could explain the excessively large ETPA efficiencies
reported elsewhere. Importantly, the benefit due to time-frequency
correlations is not generally sufficient to overcome the low fluxes required
to stay in the isolated EPP regime, even in well-suited samples such as
Rhodamine 6G.

##  IV Experiments

ETPA in Rhodamine dyes has been studied by several groups [16, 8, 21, 6].
These dyes are appealing candidates for probing TPA with no resonant
intermediate states (that is, only virtual-state pathways are available), due
to their convenient absorption spectrum relative to available light sources,
high fluorescence quantum yield, and ease of use arising from high solubility
and stability. This allows for sensitive detection at highly concentrated
solutions that can be optimized for fluorescence detection.

###  IV.1 Replication Experiment

In a previous publication we reported bounds on the ETPA efficiency of
Rhodamine 6G (R6G) for entangled photons generated from continuous-wave (CW)
excitation via SPDC centered around a wavelength of 1064 nm [17]. Subsequent
reports observed signals attributed to ETPA in R6G along with the linear and
quadratic dual-scaling behavior characteristic of nonlinear interactions with
entangled photons, as discussed above [21]. This dual-scaling is a key feature
of ETPA not previously observed in any other ETPA study. The reported non-zero
ETPA fluorescence detection rates (on the order of 10 counts/s after
background subtraction of approximately 215 counts/s) [21] did not contradict
the bounds that had been set by our previous work[17]. That report motivated
us to improve the threshold for detecting ETPA relative to our previous report
and to attempt to detect ETPA signals under nearly identical conditions as in
[21].

We made every attempt to replicate the exact experimental conditions of [21],
for example the optical beam parameters, lens focal lengths, interlens
distances, squeezed-light source, detection geometry, etc…, while making minor
modifications to improve detection sensitivity, as described below and in the
supplemental information [32]. As mentioned in the Introduction, we used both
a CW laser as in [21] as well as a picosecond pulsed laser to generate
squeezed vacuum in both low- and high-gain regimes, the latter allowing us to
explore the regime of BSV-induced ETPA for comparison with the CW results and
testing of theory predictions [25]. The pulsed version of the experiment
enabled clear detection of TPA-induced fluorescence. In the case of BSV
excitation, we verified the presence of the TPA process by confirming the
fluorescence emission lineshape [32] and quadratic dependence of fluorescence
intensity with excitation flux. In addition, we used the pulsed source to
confirm the theory’s ability to predict properties of the SFG signal arising
from squeezed vacuum, including observation of the crossover region at
intermediate flux.

Figure 1: Experimental apparatus for the replication experiment. A prism
separates any residual harmonics from the desired 532 nm CW laser light used
to generate the entangled photon pairs at 1064 nm in a 20 mm periodically-
poled lithium niobate (PPLN) crystal. After generation of the entangled pairs,
the pump is separated using a harmonic separator mirror (HS). After this, the
pairs travel through a chopper wheel (CW) with 50% duty cycle. Finally, long-
pass filters (LP) ensure that no pump light passes to the sample. A dichroic
mirror (DM) reflects the pairs to the Rhodamine 6G sample (R6G). Fluorescence
from the sample is collected in the backward geometry before passing through
the dichroic mirror and a set of short-pass filters (SP), and being focused
onto the detector (APD).

Fig. 1 shows a schematic of the replication experiment. The pump generating
the SPDC was a CW laser with a wavelength of 532 nm and bandwidth of
approximately $\Delta\nu_{CW}=5$ MHz (Coherent Verdi V5). A prism was used to
separate any residual 1064 nm light generated by the laser. The beam was then
focused into the down-conversion crystal with a 500 mm focal-length lens,
resulting in an approximately 70 $\mu m$ beam waist. The down-conversion
crystal was a 2 cm long magnesium-oxide doped periodically poled lithium-
niobate (PPLN) crystal (Covesion MSHG1064-0.5-20). The crystal was phase-
matched for collinear, non-degenerate type-0 down-conversion via temperature
tuning in a home-built oven. The SPDC light generated in this process was
collimated using a 100 mm focal-length achromatic lens with anti-reflection
(AR) coating optimized for 532 nm and 1064 nm (Edmund Optics #33-211). The
pump was separated from the SPDC beam via a harmonic separator mirror (Newport
10QM20HB.21). The SPDC beam then passed through a chopper wheel with 50% duty-
cycle operating at 400 Hz. Any remaining 532 nm light was then blocked by two
long-pass filters (Thorlabs FELH0900, FELH0750).

After preparation of the entangled photons, the beam was routed to the
‘sample-and-detection’ apparatus, which collected back-scattered light from
the sample. The SPDC light was reflected off a dichroic mirror (Thorlabs
DMSP0900), and focused onto the sample using a 3 mm focal-length aspheric lens
AR-coated for near-infrared (NIR) light (Thorlabs C330TMD-B). The focal spot
in the sample was near the diffraction limit so that the entanglement area was
approximately equal to the spot size itself, that is $A_{e}=A_{0}$, as assumed
in our theoretical discussions above. The fluorescent sample was mounted on a
linear translation stage enabling optimized placement in the longitudinal
direction of the beam. The results reported in this paper utilized a standard
1-cm quartz cuvette with approximately 1-mm thick quartz window thickness. A
thin sample cell using a microscope coverslip as a window, similar to that
used in [21], was also tested with no significant differences in observed
signal rates in both pulsed and CW experiments. Fluorescence emitted in the
backward direction was collected and collimated by the focusing lens and
transmitted through the dichroic mirror onto a filter stack (Thorlabs
FESH0850, FESH0650, Semrock NF-03-532). An 11 mm focal-length aspheric lens AR
coated for visible wavelengths focused the fluorescence onto a free-space
coupled avalanche photodiode (APD) operating in photon-counting mode (Laser
Components Count-10B). The dark rate of the detector was measured to be
approximately 3 Hz, which is approximately 2 orders of magnitude smaller than
that reported in [21]. All elements of the experiment were light-shielded such
that additional counts from stray light were not distinguishable from the
intrinsic rate of dark counts of the detector alone.

Alignment was optimized by maximizing the measured TPA fluorescence count rate
from the sample when driven by BSV generated from the pulsed laser pumping the
same crystal used to generate the CW low-flux SPDC, as detailed in the next
section. The pulse-pumped BSV source was also used to validate the
differential measurements used to suppress any slow drifts in background
noise. The differential measurements is defined as the difference between
detected counts in time intervals in which the beam was either allowed to pass
through or completely blocked by a rotating shutter, or ’chopper wheel’. There
were approximately 400 open/closed cycles per second. Validation was carried
out both at high signal rates ($10^{3}$ to $10^{5}$ cps) as well as at
fluorescence signal rates below 1 cps, yielding consistent results at all
tested measurement durations, validated up to a maximum duration of 60,000 s.
Background measurements under four different experimental conditions:–with the
laser off; SPDC beam blocked; phase-matching temperature detuned; and on blank
(without dye) ethanol solutions–all yielded no detectable signal above the
dark counts. For a thorough discussion of the differential measurements see
supporting information [32].

The spectral characteristics of the CW SPDC at low gain were characterized via
time-of-flight spectrometry. For this measurement the SPDC beam was coupled
into a single-mode fiber via an 8-mm focal-length aspheric lens AR-coated for
NIR (Newport KGA240-B-MT). The fiber-coupled SPDC light was then split by a
50:50 fiber beam splitter (Thorlabs TW1064R5F2B). One outgoing fiber was sent
directly to a superconducting nanowire single-photon detector (SNSPD)
(IDQuantique), and the second outgoing fiber was sent through 1 km of optical
fiber (Nufern 780HP), which imparted the necessary optical dispersion for a
time-of-flight measurement (converting frequency to time) before arrival at a
second SNSPD. Due to the stochastic nature of the CW SPDC process, the photon
pairs from the source arrive at the detector without reference to any external
clock. However, the relative arrival time of each delayed photon contains
spectral information due to the tight temporal correlation of the pairs [33,
34]. The measurement was calibrated using spectral filters with known sharp
wavelength cutoffs. This measurement confirmed degenerate phase-matching
around a central frequency of 1064 nm and a FWHM spectral bandwidth of 30 nm,
as shown in Fig. 2.

Figure 2: Marginal spectrum of the CW-pumped PDC source at various crystal
temperatures. Spectra were measured on a 1000-m-long dispersive fiber CW time-
of-flight spectrometer. Experiments presented in this paper were operated near
a temperature of 390.1 K.

To minimize the effects of drift and noise, we used ‘chopped’ measurements of
ETPA rates. The ‘chopped’ measurement rates are defined as the difference
between the detector counts measured when the SPDC beam is unblocked (open
channel) by the chopper wheel and when it is blocked (closed channel). For a
signal rate $R_{S}$ arising from fluorescence, the number of counts in a
measurement time $T$ is expected to be $N_{S}=R_{S}T$ with uncertainty $\Delta
N_{S}=\sqrt{R_{S}T}$. To determine whether a measured signal can be attributed
to a process other than noise, we set the detection threshold by requiring the
measured signal should be greater than 5 times the expected noise due to dark
counts, that is we must have $N_{S}\geq 5\Delta
N_{noise}=5\sqrt{N_{noise}}=5\sqrt{DT}$. Here $D$ is the dark count rate
(assumed constant). This allows us to define the detection threshold rate
$R_{det}=5\cdot 2\sqrt{D/T}$. The factor of 2 accounts for the 50% duty cycle
of the chopped measurement scheme. For our experiments, $D=3$ Hz and
$T=60,000$ s, giving $R_{det}=0.07$ Hz. Any signal with fluorescence rate
above this detection threshold should be observable within our experiment. For
comparison, note that the equivalent detection threshold rate in the previous
experiment [21] would be approximately $R_{det}\approx 10\sqrt{200/2\times
10^{4}}=1$ Hz. All cited rates and powers are adjusted to account for the
chopper duty cycle, and correspond to the average values while the beam is
unblocked.

###  IV.2 Replication Result

The PPLN crystal was pumped by a maximum CW power of 1 W, which generated
approximately 200 nW of SPDC power measured after additional bandwidth
filtering (Semrock 1055/70). In order to facillitate comparisons to BSV
experiments, we cite the total SPDC power within this spectral window, without
attempting to adjust for pair-production efficiency. Shorter-duration
measurements at higher powers were also conducted without observing evidence
of TPA, however a power of 1 W was chosen to minimize temperature-dependent
and photorefractive effects in the SPDC crystal. Measurements of fluorescence
generated via TPA of entangled photons in a cuvette containing a 5-mM solution
of Rhodamine 6G (Sigma Aldrich 252174) in ethanol were made with a duration of
60,000 seconds, with no statistically significant difference between open and
closed (chopper) channels. At this measurement duration, our detection
threshold was 0.07 Hz (4.2 counts/minute). The measurement was repeated using
a thin sample cell similar to the one used in [21]. It was also repeated using
a sample of the R6G dye that was used in [21], graciously provided by the
authors to rule out sample-based discrepancies. No significant differences
between samples or sample cells were observed in the repeated measurements
with the CW or the pulsed configuration.

###  IV.3 Moderate-Gain BSV-Induced TPA

Using the same PPLN crystal pumped by a pulsed laser, we investigated the
behavior of TPA driven by the source at moderate gain. We then reduced the
flux in an attempt to approach the low-gain regime with a source known to be
capable of driving observable fluorescence from TPA of squeezed vacuum.

The pulsed SPDC source utilized pump pulses with 532-nm center wavelength,
which were about 8 ps in duration (Lumera Laser Hyperrapid 50). The 8 ps pulse
duration is far longer than the entanglement (correlation) time of the
exciting light ($T_{e}=1/B=\text{110 fs}$), making the theoretical models
discussed earlier valid for our experiments. The repetition rate was varied
between 100 kHz and 5 MHz. Otherwise the experimental apparatus was unchanged
from the CW case, with the beam alignment to the CW beam confirmed by coupling
into single-mode fiber as well as overlap on a CCD camera (Hamamatsu ORCA-
ER-1394) at the location of the crystal, and near the sample itself.

The BSV-induced signal can be unambiguously attributed to fluorescence from
TPA, as verified by spectral measurements and quadratic scaling[32]. This is a
major benefit of our current experiment, which no previous experiments in the
ETPA literature have been able to achieve, due to the limited rates of
detectable fluorescence.

To approach the low-gain regime, the average power of the BSV generated by the
pulsed source was kept constant at 300 nW, which is approximately the maximum
attainable power in the CW regime. At 100 kHz laser repetition-rate, this
provided a clearly measurable TPA signal of 88 Hz. From here, the repetition
rate of the laser was increased incrementally, with the pump intensity
adjusted to ensure the average power of the generated BSV remained constant.
Increasing the laser repetition rate at constant average BSV flux reduces the
number of photons per pulse, and the measurements are reported in terms of
photons per pulse.

Figure 3: Double-logarithmic plot displaying measurements of TPA-induced
fluorescence driven by pulsed BSV at an average power of 300 nW using a
chopped measurement system. Error bars of 3$\sigma$ are shown. Quadratic
scaling dependence (dashed line) is observed over the entire measurement
(power law fit coefficient = 2.1). Repetition rates were varied from 100 kHz
to 5 MHz at constant average power, and values are normalized to per-pulse
rates—higher repetition rate corresponds to lower per-pulse flux. Due to
limitations of the experimental system, the lowest two data points were taken
at 5 MHz at 100 nW and 37 nW BSV flux, respectively. The chopper wheel was
omitted for these two data points, with open/closed channels being defined
relative to the repetition rate of the laser. Measurement durations were
adjusted to ensure $5\sigma$ threshold was met for all measurements.

Due to the quadratic scaling of TPA intensity with peak BSV power, the rate of
TPA is reduced as the repetition rate is increased, despite the average power
remaining constant. This remains true until the linear scaling regime is
reached, after which the repetition rate does not impact the detected
fluorescence rate.

The results of this experiment are shown in Fig. 3 in terms of per-pulse BSV
intensities reported as average photon number per pulse. The quadratic
dependence can be readily observed down to our minimum exposure flux. We were
able to increase the repetition rate to 5 MHz while keeping the average BSV
power constant at 300 nW. At higher repetition rates the pump power and
thereby the BSV gain was insufficient to reach 300 nW of BSV. At 5 MHz
repetition rate and 300 nW BSV power, TPA was still observable at a detection
rate 0.9 Hz. Two further measurements were made at 5 MHz, with reduced average
powers of 100 nW and 37 nW. These measurements took 4 hours, and 4 days
respectively to reach the same statistical significance levels as previous
measurements–indicative of the extremely unfavorable scaling behavior of
measuring weak, shot-noise-limited, quadratic signals. The last two
measurements also omitted the use of the chopper wheel, and instead used the
laser pulse-train to distinguish ‘open’ and ‘closed’ channels in order to
maximize the signal-to-noise ratio of our measurement.

The lowest-flux measurement was made with approximately 40,000 photon pairs
per pulse, or approximately 400 pairs per spectral-temporal mode for our
source. Notably, despite the benefit of being far above the isolated-pair
regime—which is expected to be near 1 pair per spectral mode—at the same
average powers as those attainable in the CW experiment, the fluorescence rate
is well below that reported in [21]. This result is in agreement with the null
result from our CW experiment.

A notable difference between our CW and pulsed experiments is the spectral
bandwidth of the pump pulse, which determines the tightness of the spectral
correlation between the entangled photon pairs [12, 23]. This constitutes a
significant experimental difference between pulsed and CW versions of the
experiment, since the bandwidth of the CW pump is orders of magnitude narrower
than that of the pulsed pump. This difference is not expected to affect the
TPA efficiency in R6G, since the TPA linewidth is much broader than the
pulsed-laser bandwidth, and thus the correlations present in the pulsed case
are sufficient to maximize resonant overlap with the two-photon transition
[22].

###  IV.4 Comparison of BSV and ‘Classical’ Efficiencies

To confirm another relevant point of our understanding of TPA arising from
light generated via SPDC, we designed an experiment to compare the rate of TPA
from a ‘classical’ (coherent-state) laser reference beam to the efficiency of
TPA from BSV generated by our source. As discussed in the theory section, the
relative efficiency for pulses of equal duration is expected to fall between 1
and 3, the former for narrow TPA linewidth and the latter for ultra-broad
phase-matching[25, 24].

To ensure identical spatial characteristics of the beams, a modified
configuration was used that included a spatial-mode filter, as shown in Fig.
4. This configuration used a 10-mm PPLN crystal (Covesion MSHG1064-1.0-10)
rather than the 20 mm PPLN crystal which was used in the above experiments. In
this second configuration, the chopper wheel was omitted and the BSV beam was
coupled into a 5-cm single-mode optical fiber stub (Nurfern 780HP) using an
8-mm focal-length aspheric lens with NIR AR coating (Newport KGA240-B-MT ).
The fiber stub was terminated in an FCPC connector. Its output could be
coupled to a time-of-flight spectrometer, which verified the spectral and
joint-spectral properties of the fiber-coupled BSV, as discussed in the
supplemental information [32].

Figure 4: After generation of BSV in a 10 mm PPLN crystal, the BSV is
separated from the pump via harmonic separator mirror (HS). Residual pump is
blocked by a set of long-pass filters (LP). The BSV is then coupled into a
connectorized 5-cm single-mode fiber stub (SMF). In the characterization step,
the light coupled into the fiber stub was routed to SNSPDs for coincidence
counting and time-of-flight spectrometry measurements. After characterization,
and without adjusting the input coupling, the following experimental step was
performed. The light was collimated out of the fiber stub and sent to a TPA
detection apparatus in the backwards collection geometry. The pairs were
reflected off a dichroic mirror (DM) to a 3-mm-focal-length aspheric lens
which focused the pairs onto the R6G sample. Fluorescence was collected by the
same lens, and passed through the same dichroic mirror (DM) before passing a
set of short-pass optical spectral filters (SP). The fluorescence was then
coupled into a multi-mode fiber (MMF), and sent to a single-photon avalanche
diode (APD) for detection. This configuration was also used to confirm the
spectral properties of the R6G fluorescence in high-gain BSV experiments.
Separately, the classical reference beam was focused into the same fiber stub
for the reference measurement. This ensures perfect spatial-mode matching
between the coherent laser and BSV sources. Figure 5: Comparison of scaling
properties of fluorescence rate with BSV and coherent-state laser input power.
Both show purely quadratic scaling, down to the measurement threshold limit.
The relative efficiency of the BSV is 1.8 times as efficient as coherent
excitation after accounting for pulse-duration effects. This is in good
agreement with expectations. The duration of the 1064 nm laser pulse was
measured to be 9.2 ps via intensity autocorrelation. The duration of the BSV
pulse was measured to be 2.5 ps by the same measurement.

Coupling into the fiber stub was optimized on coincidence counts in the low-
gain SPDC regime. The spectral characteristics of the coupled light were also
verified via measurement of the joint spectral intensity (JSI) of the photons
coupled into the fiber. These were compared with previous JSI
measurements[32]. After verifying the SPDC coupled into the short fiber
displayed the desired JSI, the fiber going to the spectrometer was decoupled
and light was collimated via an 8-mm-focal-length aspheric lens and routed to
the detection apparatus in a backward collection geometry as described above.

Fluorescence emitted in the backwards direction was collected and collimated
by the focusing lens, and passed through the dichroic mirror through a filter
stack (Thorlabs FESH0850, FESH0650, Semrock NF-03-532), before being coupled
into a multi-mode fiber, via an 8-mm aspheric lens (Newport KGA240-B-MT). This
multi-mode fiber was coupled to a spectrometer (Ocean Optics Flame) to verify
that the spectral properties corresponded to known properties of fluorescence
from Rhodamine 6G.

The classical laser reference beam was coupled into the same fiber stub as the
BSV, with all of the subsequent optics remaining unchanged, guaranteeing
identical spatial properties (up to chromatic variations due to the broad BSV
spectrum).

To measure the temporal properties of the BSV, the beam after spatial mode
filtering was routed to an autocorrelation experiment consisting of a
Michelson interferometer prior to a 0.7-mm beta barium borate (BBO) crystal
with broad phase-matching bandwidth. The pulse duration was used to calibrate
the relative efficiency of the BSV compared to that of the classical reference
laser. See further discussion in the supplemental information [32].

The results of the efficiency comparison are summarized in Fig 5. Both sources
showed quadratic scaling as confirmed by power-law fits of the data. After
normalizing by the duration of the pulses, the relative TPA efficiency of the
BSV was measured to be a factor 1.8 times as efficient as the classical
reference laser. This result is in good agreement with expectations for this
factor to be in the range 1 to 3 for a TPA sample with moderately broad TPA
lineshape (resulting from optical dispersion in the system and the finite TPA
molecular linewidth.) Crucially, no evidence of enhanced efficiency is
observable when using BSV compared to classical laser light to drive TPA.

###  IV.5 Sum Frequency Generation

The final experiment we describe uses sum frequency generation (SFG) as a
model for TPA. The theories described in [12, 22] predict TPA scaling behavior
with the squeezed light flux that changes from linear to quadratic at the
crossover flux, as described earlier. The crossover is predicted to occur at
at photon rate of approximately 1 photon per temporal mode. This crossover
rate is an important parameter for estimating the low-gain efficiency if the
low-gain efficiency cannot be measured directly. The moderate-gain BSV results
shown in Fig. 3 confirmed that this crossover for ETPA does not occur until
below 400 photon pairs per mode from our source.

To constrain this expected behavior, which cannot be observed in our ETPA data
for lack of signal, we use SFG as a model as it has similar dynamics to
TPA[12], with higher interaction strengths and thus higher and detectable
efficiency.

This measurement was conducted using the 10-mm PPLN crystal used for the BSV
experiment, as well as the pulsed laser source operating at 10 MHz repetition
rate. The chopper wheel was omitted, and no mode filtering was employed. The
squeezed vacuum was collimated using a 200-mm focal-length achromatic lens.
After spectral filtering (Thorlabs FELH0900, FELH0750), the beam passed
through a dispersion-compensating system (prism pair) to account for the
second-order dispersion accrued in the optical system. The beam was then
focused via a second achromatic, NIR AR-coated 200-mm focal-length lens into a
nominally identical PPLN crystal under the same phase-matching conditions. The
SFG acceptance bandwidth of the PPLN crystal was narrow compared to the
bandwidth of the SPDC light driving the SFG, which ensures contributions
primarily from the coherent contribution to the SFG signal.

Figure 6: The crossover between linear and quadratic scaling regimes for SFG
was measured to occur at approximately 250 photons per pulse, in good
agreement with predictions of 1 photon per temporal mode. The measured result
is higher by a factor of 2-3 than estimates of the number modes for the
source, a difference attributed to spatial correlations present in the bulk
SPDC process, which were not considered in the theoretical description.

After generation of the SFG beam, the remaining squeezed light was blocked by
an interference filter stack (Thorlabs FESH0850, FESH0650, Semrock NF-03-532),
and the resulting light was coupled into a multi-mode optical fiber before
detection on an APD (Laser-components COUNT-Blue 10-B). The intensity of the
BSV was measured via a multi-mode fiber-coupled power meter (Thorlabs S150C).
At low BSV intensities, the sensitivity of the power meter was limiting. The
intensity of the pump was also monitored, to ensure that no errors were
introduced due to measurement limitations of the power meter.

Fig. 6 summarizes the result of this experiment. The crossover is measured to
be near 250 photons per pulse, which is in good agreement with the predicted
crossover for our source with 100 temporal modes. No evidence of a large
discrepancy in the predicted efficiency of the low-gain process for SFG was
observed.

##  V Discussion

Our experiment, which had a substantially lower noise floor and thus lower
detection threshold than the study reported in [21], did not reproduce the
results reported there. At measurement durations up to 60,000 s we did not
detect any signal distinguishable from dark-count noise using a CW source of
entangled photons with 200 nW power. For measurements of this duration,
utilizing the chopped experimental setup, we have a theoretical 5$\sigma$
detection threshold of 0.07 Hz. This methodology was validated by measuring a
0.9 Hz signal over the same duration generated from the pulsed experiment. In
contrast, the results published in [21] indicated an apparent ETPA count rate
of approximately 9 Hz with incident power of 160 nW and otherwise near-
identical conditions.

Our experiment was run in several configurations: utilizing samples contained
in a quartz cuvette as well as in a thin glass cell similar to the one
described in [21]. Both were run utilizing a R6G sample purchased from Sigma-
Aldrich (Sigma Aldrich 252174) as well as with a sample of the same bulk stock
used in [21], which was generously provided by the authors. This rules out
differences due to sample inconsistencies or sample cell setup.

One source of discrepancy between the experiments is the choice of detector.
The experiment in [21] utilized a silicon APD with 500 $\mu$m active area, and
approximately 200 Hz dark rate. Our experiment utilized an APD with 100 $\mu$m
active area and 3 Hz dark rate, allowing for a lower detection threshold. The
25x difference in detector area between the two experiments is worth
considering. Fluorescence collection is a difficult imaging problem, and non-
optimal imaging could lead to overfilling of the detector. To verify that our
experiment was not suffering from poor fluorescence collection, we imaged the
fluorescence generated via TPA of BSV onto a CCD array (Hamamatsu ORCA-
ER-1394). The spot was seen to be tightly focused to 70 ($\mu$m) FWHM spot on
the array. Integrating the accumulated counts for a 100 $\mu$m x 100 $\mu$m
spot and 500$\mu$m x 500$\mu$m spot, respectively, confirmed that the majority
of the fluorescence was contained in the central spot on the CCD.
Additionally, transverse translation of the APD itself yielded results
consistent with the fluorescence underfilling the APD, as desired.

Another crucial difference between the experiments was the choice of anti-
reflection coating for the final focusing and fluorescence collection lenses.
The experiment in [21] utilized an anti-reflection coating optimized for
visible wavelengths (Thorlabs A-coating) coating, which the manufacturer
specifies at approximately 15% reflection at 1064 nm at both the front and
back face of the lens, which results in a significant reduction in pair-rate
arriving at the sample, most of which is reflected directly back into the
collection path. Our experiment utilized the same lens model, but with an
anti-reflection coating optimized for NIR wavelengths (Thorlabs B-coating),
’B’-coating which is specified at ¡ 1 % at 1064 nm. The collection losses at
the central frequency of Rhodamine 6G fluorescence are specified as ¡ 1% for
A-coating and approximately 7% for the B-coating. Finally, we do not observe
any background counts from a cuvette of ethanol alone. In [21], a 10 Hz
background signal was observed from a blank cuvette of ethanol alone and was
subtracted from their experimental runs. In contrast, in our experiment
neither CW nor pulsed BSV—at significantly higher flux than possible in CW
experiments—showed measurable signals from blank ethanol cuvettes.

While we were not able to observe fluorescence from ETPA in the low-gain
regime, we were able to observe TPA of BSV at similar average powers using
pulsed, moderate-gain squeezed vacuum. In agreement with our low-gain null
result, our measurements of TPA driven by moderate-gain squeezed vacuum showed
quadratic scaling at all intensities down to approximately 400 photons per
spectral-temporal mode. At these intensities the signal rates were far below
those reported in [21], despite having the benefit of quadratic intensity
scaling. This measurement would report on large increases in the TPA
efficiency in the low-gain regime if they were present, as these would shift
the expected crossover between linear and quadratic regimes, however no such
shift was observed.

One major contribution of this particular experiment, absent in previous
studies of ETPA, is that the BSV-induced measurements in this experiment can
be unambiguously attributed to TPA-induced fluorescence by observing the
spectral characteristics of the detected light, in addition to standard
scaling properties, all of which could readily be measured in the same
apparatus at high BSV fluxes.

To constrain further the measured value of the TPA crossover flux, we used SFG
to study the characteristics of the squeezed vacuum field. In this measurement
we were able to directly observe the crossover between linear and quadratic
scaling behavior, which occurred at roughly 2.5 photons per mode. This is in
satisfactory agreement with predictions of the theory. The minor discrepancy
between predicted and observed crossover flux is possibly explained by weak
spatial correlations present in the entangled photon beam, which are not
considered in our model. Spatial correlations cannot explain the orders-of-
magnitude differences observed in other ETPA experiments, and are expected to
contribute minimally in tightly focused TPA experiments such as ours.

##  VI Conclusions

In spite of having a lower detection threshold for TPA, our results did not
reproduce the results of the only ETPA experiment to date that has reported
both the linear and the quadratic scaling with flux in ETPA. Our efforts at
direct measurements of TPA at moderate and high gain, generated in the same
experimental apparatus, yielded TPA rates well below those reported in [21]
despite relatively favorable conditions. Our combined results provide strong
evidence that the orders-of-magnitude increases in TPA efficiency by using
entangled light reported in [5, 7, 10, 6, 8] cannot be explained by the
community’s current understanding of ETPA, and may be resulting from some
unexpected linear-optical affect mistakenly being attributed to ETPA. ETPA
experiments that do not confirm the quadratic scaling of the process are
particularly vulnerable to this kind of error [20, 19, 18].

As we have shown theoretically in [25] and experimentally in the present
study, simply increasing the flux of entangled photons does not lead to
greatly enhanced EPTA signals. When the flux is increased, the TPA process
passes the crossover point, above which photons from uncorrelated pairs
overlap. At this point the benefit of individual spectrally correlated pairs
is lost, as the TPA efficiency is dominated by the quadratic scaling, which is
comparable to the efficiency of ‘classical’ (coherent-state) laser excitation.

This is not to say that the phenomenon of ETPA does not exist. Clear, but
minimal evidence of the effect was observed by Georgiades et. al. in an atomic
system where the narrow TPA linewidth gives a large cross section [3]. There
the TPA rate was seen to be increased by around a factor of two at extremely
low fluxes as a result of the number correlations present in low-gain
squeezing, although the narrow bandwidth of the squeezed light in that
experiment precluded any study of enhancement by spectral correlation within
photon pairs. The main barrier to detecting ETPA in solvated molecular samples
is simply the fact that the TPA cross sections are so small as a result of the
large homogeneous TPA linewidths.

##  VII Acknowledgements

We thank Andrew H. Marcus and Markus Allgaier for collaboration in earlier
stages of this research. And we thank Rob Thew and Dmitri Tabakaev for
providing complete information about their experiment and a sample of the dye
they used. This work was supported by the National Science Foundation RAISE-
TAQS Program (Grant No. PHY-1839216).

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