# High-precision Quantum Transmitometry of DNA and Methylene-Blue using a
Frequency-Entangled Twin-Photon Beam in Type-I SPDC

Ali Motazedifard  1,2,3,4* and Seyed Ahmad Madani 1,2,3,5  1 Quantum Optics
Group, Iranian Center for Quantum Technologies (ICQTs), Tehran, Iran
2 Quantum Communication Group, Iranian Center for Quantum Technologies
(ICQTs), Tehran, Iran
3 Quantum Sensing and Metrology Group, Iranian Center for Quantum Technologies
(ICQTs), Tehran, Iran
4 Quantum Optics Group, Department of Physics, University of Isfahan, Hezar-
Jerib, 81746-73441, Isfahan, Iran
5 Photonic and Nanocrystal Research Lab (PNRL), Faculty of Electrical and
Computer Engineering, University of Tabriz, Tabriz 5166614761, Iran
<EMAIL_ADDRESS>
###### Abstract

Using the coincidence-count (CC) measurement of the generated frequency-
entangled twin-photons beam (TWB) via the process of type-I spontaneous
parametric-down conversion (SPDC) in BBO nonlinear crystal (NLC), we have
precisely measured the transmittance of very diluted Rabbit- and Human-DNA,
Methylene-Blue (MB), as a disinfectant, and thin-film multilayer at near IR
wavelength 810nm with an accuracy in order of $\%0.01$ due to the quantum
correlation, while accuracy of classical-like measurement, single-count (SC),
is in order of $\%0.1$ in our setup. Moreover, using quantum measurement of
the transmittance, the different types of DNA with the same concentration, and
also very diluted (in order of pg/$\mu$l) different concentrations of DNA and
MB solutions are distinguished and detected with high-reliability.
Interestingly, in case of Human-DNA samples in contrast to our classical-like
measurement we could precisely detect and distinguish two very diluted
concentrations $0.01\rm ng/\mu l$ and $0.1\rm ng/\mu l$ with high reliability
while commercial standard spectrometer device of our DNA-manufacturer never
could detect and distinguish them. Surprisingly, measurement on the thin-film
multilayer illustrates that the introduced method in this work might be
performed to cancer/brain tissues or Stem cells for cancer therapy, and may
hopefully open a pave and platform for non-invasive quantum diagnosis in
future.

††journal: osac††articletype: Research Article

##  1 Introduction

In the context of the nonlinear light-matter interaction, spontaneous
parametric-down conversion (SPDC) is the well-known nonlinear optical process
in which a twin-photons beam, a pair of signal-idler photons, is generated out
of quantum vacuum when a pump laser beam is incident onto an optical nonlinear
material [1, 2, 3, 4]. In SPDC, an input photon of classical laser pump at
higher frequency is annihilated and two entangled photons are created at lower
frequencies under the so-called phase-matching (PM) conditions [1], i.e.,
energy-momentum conservation. SPDC sources in bulk NLCs, as an entangled-
squeezed biphoton source (for more details see Ref. [1]), are very interested
compared to other technologies such as superconductors since they are
inexpensive, user-friendly, work at room-temperature, needs no cooling, and
finally can be on-chip using integrated quantum photonic techniques in order
to be commercialized. That is why SPDC in NLCs has been variety range of
applications [5] from the fundamental to the applied point of view such as
quantum communication [6], entanglement generation and nonlocal realism tests
[7, 8, 9, 10, 11, 12, 13, 14, 15], quantum state teleportation [16, 17],
quantum illumination [18, 19], quantum imaging [20, 21, 22, 23], and quantum
interferometry [24, 25] and finally quantum efficiency and absorption
measurement [26, 27, 28, 29].

Some well-known physical technologies in the context of the quantum light-
matter interaction, such as optical spectroscopy, magnetic resonance imaging,
and tomography have been employed for understanding biological structure and
disease treatment to medical diagnosis. Beyond these, recently, the
SPDC-/entanglement-based methods, have attracted interests to be applied in
the field of biology and medicine from different points of view such as
metrology, phenomenology, philosophy and consciousness [30, 31, 32, 33, 34,
35, 36, 37, 38]. In Refs. [30, 31, 32, 33, 34, 35], the effect of brain
tissues with different thicknesses, type and ages on the entangled state has
been investigated. High-precision protein concentrometry and tomography of the
biological samples such as onion-skin tissue have been done through entangled
photons in Refs. [36] and [38], respectively. More interestingly, in Ref. [37]
the authors have shown that the Rod cells which respond to individual photons
in the human cornea can detect a single-photon incident on with a probability
significantly above chance. But, in all of them they have not provided any
practical optical parameter to characterize the target sample.

Inspired by the above-mentioned investigations, and using the extension of the
Klyshko method [26] we are motivated to measure the transmittance of DNA and
Methylene-Blue (MB) solutions through frequency-entanglement, as a practical
meaningful optical parameter unlike the Refs. [30, 31, 32, 33, 34, 35, 36, 37,
38]. In cancer therapy and genetic sciences, the measurement of DNA
concentrations, identification/detection of the different types of DNAs or
even diagnosis of DNAs with different concentrations are essential to
biologists and medicines. Furthermore, the MB with different doses as a
disinfectant is very important in surgeries. Very recently, it has been shown
[39] a drug developed by Boxin (Beijing) Biotechnology Development LTD based
on MB photochemical technology, the so-called BX-1 drug, can inactive lipid-
enveloped viruses such as HIV-1 in plasma with high efficiency and without
damage to other components in the plasma as a safe and reliable method in
clinical trials of HIV treatment. In addition, it has been challengingly
confirmed the inactivation effect of BX-1 in SARS-CoV-2 [39]. In Ref. [39],
the authors have argued that BX-1 can effectively eliminate SARS-CoV-2 within
2 mins and the virus concentration (titer) decreases and can reach 4.5 log10
$\rm TCID_{50}$/ml (fifty-percent tissue culture infective dose which is the
measure of infectious virus titer). The authors have also argued that BX-1 is
safe for blood transfusion and plasma transfusion therapy in recovery
patients, and the inactivated vaccine preparation has great potential for
treatment in the current Coronavirus outbreak [39]. That is one reason we are
interested in MB and DNA as a biological and medical/chemical solution samples
for its transmittance measurement vs concentration. Despite of the quantum
spectroscopy, here, the frequency of twin-photons are far from the absorption
frequency of the samples which means that it needs no monochromators or very
notch filters. In the following, as an application of quantum measurement in
biology we have shown that how a simple method based on the coincidence-count
rate measurement of the generated frequency-entangled twin-photons beam due to
the type-I SPDC in nonlinear BBO crystal which pass through DNA-and Methylene-
solutions enables us to quantum measurement of the transmittance with $\%0.01$
uncertainty without any advanced or sub-pixel data analysis. Interestingly,
through the quantum transmittometry Human-/Rabbit-DNA solutions with different
concentrations $\sim(0.01-100)\rm ng/\mu l$ have been detected. The DNA
solutions with concentrations in order of $0.01$ and $0.1\rm ng/\mu l$ never
can be detected by the standard (conventional) commercial spectrometer devices
while we could detect them by CC- or transmittance-measurement with more than
at lest 10-standardeviation. Surprisingly, in the same concentration we could
distinguish/identify the Human-DNA and Rabbit-DNA solutions via their CC rates
which never can be distinguished by classical methods such as single-count
measurements. Furthermore, the same procedure has been performed to MB which
led to detection of a very diluted Methylene solutions with concentrations
$\sim 4.5\rm pg/\mu l$ through its quantum transmittometry. Finally, the
transmittance of some known physical thin-films on soda-lime are measured,
that because of similarities to biological tissues, it points out that the
introduced method may be applied to cancer-tissues or Stem Cells towards
quantum diagnosis.

The paper is organized as follows. In Sec. (2), we review shortly on the
physical model of the SPDC and their PM equations. Also, in this section we
show that the output of the SPDC is an frequency-entangled state. The
experimental setup, methods and results are discussed in Sec. (3). Finally, we
summarize our conclusions and discuss our outlooks in Sec. (5) .

##  2 Theoretical model and description

However, there is a large texts on the SPDC physics with more details, but,
here in this section, we again would like to review short on the SPDC physics
to better understanding of the readers especially in the field of biology.

###  2.1 Two-particle entangled state

Entanglement which has no classical counterpart plays the key role in the
quantum technologies as the second revolution of the quantum theory during
these decades. The best definition which can quantify the entangled state is
non-separable property of a quantum state. Let us consider a pure state for a
system composed of two spatially separated subsystems 1 and 2 which are
closed:

$\displaystyle\hat{\rho}=|\Psi\rangle\langle\Psi|,\qquad|\Psi\rangle=\sum_{a,b}\mathcal{C}(a,b)|a\rangle|b\rangle,$
(1)

where $\hat{\rho}$ is the density matrix operator, ${|a\rangle}$ and
${|b\rangle}$ are two sets of orthogonal vectors for subsystems 1 and 2,
respectively. If $\mathcal{C}(a,b)$ cannot be factored into a product of the
form $f(a)\times g(b)$, then, the state of the system is non-separable, and
thus

$\displaystyle\hat{\rho}\neq\hat{\rho}_{1}\otimes\hat{\rho}_{2},$ (2)

which means that it is an entangled state [1], as was defined for the first
time by Schrödinger. The first simplest example known as the Einstein-
Podolsky-Rosen (EPR) state is given by [40]

$\displaystyle|\Psi\rangle=\sum_{a,b}\delta(a+b-\alpha_{0})|a\rangle|b\rangle,$
(3)

where $a$ and $b$ are the momentum or the position of particle 1 and 2,
respectively, and $\alpha_{0}$ is a constant. As can be seen the value of
momentum or position is determined for neither single subsystem. Also, the
Dirac delta function indicates if one of the subsystems is measured to be at a
certain value for that observable the other one is $\%100$ determined
independent of their distance.

The Bell states (or EPR-Bohm-Bell states), which are the most popular
entangled two-particles state, are a set of polarization states for a pair of
entangled photons. They can be represented by

$\displaystyle|\phi_{12}^{(\pm)}\rangle=\frac{1}{\sqrt{2}}[|0_{1}0_{2}\rangle\pm|1_{2}1_{1}\rangle],$
(4a)
$\displaystyle|\psi_{12}^{(\pm)}\rangle=\frac{1}{\sqrt{2}}[|0_{1}1_{2}\rangle\pm|1_{2}0_{1}\rangle],$
(4b)

which form a complete orthonormal basis for polarization of light. $|0\rangle$
and $|1\rangle$ represent the two orthogonal polarization bases, for example,
horizontal and vertical linear polarization, respectively. Note that in Eq.
(4) the two-photon space-time wavepackets have been ignored. Here, it should
be emphasized that to experimentally prepare the Bell states one must overlap
these two wavepackets [14, 10], i.e., to make them quantum mechanically
indistinguishable, to keep valid the quantum state superposition in Eq. (4).

###  2.2 SPDC as a source of two-photon entangled-state

In nonlinear optical process of SPDC [41], when a coherent pump laser beam is
incident onto an optical nonlinear material with second-order nonlinearity,
susceptibility $\bar{\bar{\chi}}^{(2)}$, a pair of signal-idler photons, the
so-called twin-photons, is generated with an efficiency in order of
$10^{-7}-10^{-11}$.

Using the first-order perturbation theory together considering the second-
order nonlinear interaction Hamiltonian of the SPDC, after some algebraic
manipulations the state of the SPDC can be obtained as follows (for more
details see Ref. [1])

$\displaystyle|\Psi\rangle_{\rm SPDC}\simeq-\frac{i}{\hbar}\int
dt\mathcal{\hat{H}}_{int}|0\rangle$
$\displaystyle=\sum_{s,i}\delta(\omega_{p}-\omega_{s}-\omega_{i})\Phi(\Delta_{k}L)\overbrace{\hat{a}_{s}^{\dagger}(\textbf{k}_{s})\hat{a}_{i}^{\dagger}(\textbf{k}_{i})|0\rangle}^{|k_{s}\rangle|k_{i}\rangle}$
(5a)
$\displaystyle=\\!\\!\\!\int\\!\\!\\!g(\omega_{p})d\omega_{p}\\!\\!\int\\!\\!d\omega_{s}f(\omega_{p},\omega_{s})\overbrace{\hat{a}_{s}^{\dagger}(\omega_{s})\hat{a}_{i}^{\dagger}(\omega_{p}-\omega_{s})|0\rangle}^{|\omega_{s}\rangle|\omega_{i}\rangle},$
(5b)

where $\omega_{j}$, $\textbf{k}_{j}$ ($j=p,s,i$) are the frequency and
wavevector of the signal ($s$), idler ($i$), and pump ($p$) photons,
respectively, and also $\hat{a}_{s}^{\dagger}$ ($\hat{a}_{i}^{\dagger}$) is
creation operator for signal (idler) photon. Note that $\omega_{p}$ and
$\textbf{k}_{p}$ can be considered as constants. $g(\omega_{p})$ and
$f(\omega_{s},\omega_{i})$ are the spectral amplitudes of the pump and signal-
idler beams, respectively, which usually determined by the wavevector phase-
matching condition of the SPDC and provide all information about the spectrum
and the correlation properties of the signal-idler pair under the pump
photons. The sinc-like function $\Phi(\Delta_{k}L)$ shows the effect of the
finite length of the crystal ($L$) which is given by [1]

$\displaystyle
L\Phi(\Delta_{k}L)=\int_{0}^{L}dze^{i\Delta_{k}L}\frac{e^{i\Delta_{k}L}-1}{(i\Delta_{k}L)},$
(6)

where $\delta_{k}=k_{p}-k_{s}-k_{i}$. It is easy to show that in the ideal
case, infinite length,
$\Phi(\Delta_{k}L)\to\delta(\textbf{k}_{p}-\textbf{k}_{s}-\textbf{k}_{i})$,
and thus, the state of the SPDC can be simplified as

$\displaystyle|\Psi\rangle_{\rm
SPDC}\simeq\sum_{s,i}\delta(\omega_{p}-\omega_{s}-\omega_{i})\delta(\textbf{k}_{p}-\textbf{k}_{s}-\textbf{k}_{i})|k_{s}\rangle|k_{i}\rangle.$
(7)

The first Dirac delta in the SPDC state implies on the energy conservation and
the latter implies on the momentum conservation. More surprisingly, compared
to Eq. (3) the signal-idler two-photon SPDC state of Eq. (7) is an EPR type
entangled-state only because of the delta functions due to phase-matching. It
is obvious that it cannot be factorized into a product of the signal and the
idler photon state. By looking at the space-time projection of the state which
is equivalent to a Fourier transform, for a pictorial view of the effective
two-photon wavefunction, or wavepacket or biphoton, one can emphasize this
important physics of this EPR-type-entanglement of the two-photon SPDC-state
(for more details see Ref. [1]).

Here, it is worthwhile to remind that the biphoton state which is an EPR-type
entangled state is also usually the so-called ‘frequency-entangled’ state [1,
5] (for better understanding see Eqs. (5a) and (5b)). Furthermore, one should
emphasize that the concept of the biphoton or signal-idler pair photon or
twin-photon is never what usually people think, i.e., the two individual
independent photons.

###  2.3 Phase-matching equations

The two delta functions in biphoton state of Eq. (7) which are the so-called
the phase-matching (PM) equations determine the energy of the signal-idler
pair photons and what they ‘looks’ like and how propagate in space-time. For
example, in type-I PM in a negative uniaxial NLC, a linearly polarized pump
laser beam as an extraordinary ray under the PM condition can generate a
signal-idler pair which both are polarized as the ordinary rays while in the
type-II, a signal-idler pair with one ordinary polarization and another
extraordinary polarization are generated (for more details see Refs. ([1, 42,
43]). The PM equations, the energy-momentum conservation, can be written as

$\displaystyle\omega_{p}=\omega_{s}+\omega_{i},\qquad\textbf{k}_{p}=\textbf{k}_{s}+\textbf{k}_{i}.$
(8)

In our case, the planar type-I SPDC in which the signal-idler pair have the
same ordinary ($o$) polarization and pump has extraordinary ($e$)
polarization, the PM conditions can be simplified as follows [1, 42]

$\displaystyle\\!k_{s}(\lambda_{s})\\!\sqrt{1\\!-\left(\frac{\sin\theta_{s}^{out}}{n_{o}(\lambda_{s})}\right)^{2}}\\!+\\!k_{i}(\lambda_{i})\sqrt{1\\!-\left(\frac{\lambda_{i}}{\lambda_{s}}\frac{\sin\theta_{s}^{out}}{n_{o}(\lambda_{i})}\right)^{2}}\\!\\!=\\!k_{p}(\lambda_{p},{\rm\psi}),$

where $\lambda_{i}=\lambda_{s}\lambda_{p}/(\lambda_{s}-\lambda_{p})$,
$k_{j}(\lambda_{j})=2\pi n_{o}(\lambda_{j})/\lambda_{j}$ with $j=s,i$ and
$k_{p}(\lambda_{p},\rm\psi)=2\pi n_{\rm
eff}(\lambda_{p},\rm\psi)/\lambda_{p}$. The type-I tuning curves, outside
angles versus wavelength, for a NLC can be theoretically plotted using Eq.
(2.3). Note that the output angles $\theta_{s,i}^{out}$ are the angles between
the pump wavevector $\textbf{k}_{p}$ and the signal or idler output beam from
the end-side of the crystal. Also, $\rm\psi$ is the angle between the pump
wavevector with the OA inside the crystal. It should be noted that the
refraction indexes can be calculated from the Sellmeier equations [1, 42, 2,
3] as follows

$\displaystyle
n_{j}(\lambda_{j})=\sqrt{L_{1}+\frac{L_{2}}{\lambda_{j}^{2}-L_{3}}-L_{4}\lambda_{j}^{2}},\qquad(j=e,o)$
(10)

where coefficients $L_{m}$ ($m=1,2,3,4$) are determined experimentally (for
example in the case of BBO crystal see Ref. [44]). Furthermore, we emphasize
that the refractive index for the $e$-ray also depends on the angle
$\varphi_{e}=\rm\psi-\theta_{s}^{int}$ which can be found from Snell’s law as
$\sin\theta_{s}^{int}=\sin\theta_{s}^{out}/n_{o}(\lambda_{s})$. Therefore, we
need to use the effective refractive index for all $e$-rays (here in our case,
e-ray is only the pump beam $p$) which is given by [1, 42, 2, 3]

$\displaystyle n_{\rm
eff}(\lambda_{e},{\rm\psi})=\left[\frac{\cos^{2}\varphi_{e}}{n_{o}^{2}(\lambda_{e})}+\frac{\sin^{2}\varphi_{e}}{n_{e}^{2}(\lambda_{e})}\right]^{-\frac{1}{2}}.$
(11)

###  2.4 Coincidence count rate

The joint detection counting rate of detectors $\rm D_{1}$ and $\rm D_{2}$
(for example the first detector is in signal-arm and the other one is in the
idler-arm), which is also known as the coincidence count (CC) $R_{c}$, on the
acquisition time interval $T$, is the time-integrals over the biphoton
wavepacket $\Psi_{Bp}(r_{1},t_{1};r_{2},t_{2})$ as follows [1]

$\displaystyle R_{c}\\!\propto\\!\frac{\eta_{1}\eta_{2}}{\rm
T}\\!\\!\\!\int_{0}^{T}\\!\\!\\!\\!\\!\int_{0}^{T}\\!\\!\\!\\!dt_{1}dt_{2}|\overbrace{\langle
0|E_{2}^{(+)}(r_{2},t_{2})E_{1}^{(+)}(r_{1},t_{1})|\Psi\rangle_{\rm
SPDC}}^{\equiv\Psi_{Bp}(r_{1},t_{1};r_{2},t_{2})}|^{2},$ (12)

where $E_{j}^{(+)}(r_{j},t_{j})=\int d\omega\hat{a}(\omega){\rm exp}[-i(\omega
t_{j}-k(\omega)r_{j})]$ ($j=1,2$) is the positive-frequency component of the
field operator at detector $\rm D_{j}$ with quantum efficiency $\eta_{j}$. It
should be noted that the CC rate can be seen as a criterion for quantum
correlation or frequency-entanglement in SPDC systems which reads a counting
statistics. Note that in the SPDC-based experiment we usually measure the CC
rate to exploit the entanglement.

Now, it is worthwhile to illustrate the relation between the CC rate as an
experimental quantity and the second-order correlation function
$g^{(2)}(\tau)$, as a photon counting statistics criterion in the theoretical
point of view which is given by

$\displaystyle
g^{(2)}(\tau)=\frac{\langle\hat{a}^{{\dagger}}_{s}(t+\tau)\hat{a}^{{\dagger}}_{i}(t)\hat{a}_{i}(t)\hat{a}_{s}(t+\tau)\rangle}{\langle\hat{a}^{{\dagger}}_{s}(t+\tau)\hat{a}_{s}(t+\tau)\rangle\langle\hat{a}^{{\dagger}}_{i}(t)\hat{a}_{i}(t)\rangle}\quad.$
(13)

The correlation function is proportional to the probability of detecting one
signal-photon at time $t+\tau$, given that another idler-photon was detected
at earlier time $t$. $\tau$ is the delay time of detection between the idler
and signal arm in output of the SPDC. When $g^{(2)}(\tau)<g^{(2)}(0)$, the
photons tend to distribute themselves preferentially in bunches rather than at
random (photon bunching). On the other hand, if $g^{(2)}(\tau)>g^{(2)}(0)$,
fewer photon pairs are detected close together than further apart (photon
antibunching). Using the quantum moment factoring theorem or the Gaussian
properties of the noise forces and the quantum regression theorem (see Ref.
[45]), the correlation function can be simplified as follows

$\displaystyle g^{(2)}(\tau)=1+\frac{R_{si}R_{is}}{R_{s}R_{i}},$ (14)

where $R_{s(i)}=\langle a^{{\dagger}}_{s(i)}(t)a_{s(i)}(t)\rangle$ is the
signal(idler) photon generation rate, and $R_{si}=R_{is}^{\ast}=\langle
a^{{\dagger}}_{s}(t+\tau)a_{i}(t)\rangle$ is the cross-correlation rates or
signal-idler correlation rate. In Eq. (13) we have assumed an infinite time
resolution of the photon detector and ideal efficiency of the detectors.
However, in practice in lab, in the detectors, one can only resolve photons in
a finite time window $\tau_{w}$. In the optical frequency, the time resolution
is generally within a nanosecond while in microwave is of the order of a
microsecond because Raman-absorption and arrival time cannot be distinguished
that it sets the length of the detection time window. The finite time
resolution of the detectors imply that the second-order correlation function
is usually a piecewise function as

$\displaystyle
g^{(2)}(\tau_{j})=1+\frac{\int_{\tau_{j}}^{\tau_{j}+\tau_{w}}|R_{si}(\tau)|^{2}d\tau}{R_{s}R_{i}\tau_{w}},$
(15)

in which $\tau_{j+1}=\tau_{j}+\tau_{w}$ and can be rewritten as

$\displaystyle
g^{(2)}(\tau_{j}):=\frac{R_{cc}(\tau_{j})}{R_{acd}}=\frac{R_{s}R_{i}\tau_{w}+\int_{\tau_{j}}^{\tau_{j}+\tau_{w}}|R_{si}(\tau)|^{2}d\tau}{R_{s}R_{i}\tau_{w}},$
(16)

where the generalized CC rate $R_{cc}(\tau_{j})$ includes two parts: the
accidental rate $R_{acd}=R_{s}R_{i}\tau_{w}$, and the correlated CC rate which
contains the contribution of cross-correlation of the signal-idler which is
also proportional to $R_{c}$ in Eq. (12). It should be noted that what we
measure in the Lab usually is $R_{c}$ in Eq. (12) which can be related to the
second-order correlation function or counting statistics through Eq. (16).

##  3 Experimental results

In this section, we are going to proceed with our experimental results,
quantum measurement of the transmittance of the different samples such as DNA,
MB and thin-films. Also, we compare the obtained transmittance via the CC-
measurement to the SC measurement, as a classical-like measurement, which
clarifies the advantage of entanglement-based measurement.

###  3.1 Experimental Setup and Adjustments

Schematic illustration of our experimental arrangement has been shown in Fig.
1(a). A $180{\rm mW}$ Gaussian beam of a $405{\rm nm}$ diode laser with
$\varnothing 3.8\rm mm$ after passing through a diaphragm SP1, acting as a
spatial filter, passes through a HWP [Newlightphotonic: WPA03-H-810, air-
spaced 0th-order waveplate at 810nm, $\varnothing$15.0mm, and OD 1" mounted]
to satisfy the PM condition. For achieving this, we have set the HWP to
$+3.5^{\circ}$ to have horizontally e-polarize pump beam. The e-polarized pump
beam is reflected from two broadband (380-420nm) high-reflective ($R>\%99.0$)
mirrors M1 and M2 [Newlightphotonic: BHR10-380-420-45; AOI 45 deg,
$\varnothing$25.4mm] and then passes through the SP2. After SP2, the
horizontally e-polarized pump beam has power around $85{\rm mW}$. The pump
beam is incident vertically on the single BBO type-I crystal
[Newlightphotonic: NCBBO5050-405(I)-HA3] with Size $(5\times 5\times 0.5){\rm
mm}$, cut only for type-I PM SPDC and pumped by 405nm diode laser with the
half opening angle of 3 degrees, and also AR coated with OD 1" mounted. Based
on the information of the manufacturer company when the pump beam are incident
vertically on the entrance-side of the crystal the angle between pump
wavevector and the OA is around $\rm\psi=29.3^{\circ}$. In this manner, the
frequency-entangled signal-idler pair are generated in planer case at output
angles around $\theta_{s}=\theta_{i}\simeq 3^{\circ}$. The down-converted
photons after passing through an uncoated long-wave pass red filters LPFj
($j=1,2$) [Newlightphotonic: LWPF1030-RG715; cutoff wavelength 715nm,
$\varnothing$25mm] by the fiber coupler lenses FCj ($j=1,2$) [Thorlabs:
PAF2A-11B; FC/APC, f=11.0 mm, 600-1050 nm, $\varnothing$2.38 mm waist] are
focused into the multimode fibers MMFj ($j=1,2$) [Excelitas: SPCM-QC9FIBER-ND
100 FC], and then transferred via the couplers into the 4-channel single-
photon counting module (SPCM) [Excelitas: SPCM-AQ4C; dark counts 351, 483,
156, 146 cps for channels A,B,C, D respectively, time resolution 600ps, dead-
time 50ns and quantum efficiency about $\%$28 at 810nm]. To obtain the CC rate
of the D1 and D2 we have used the coincidence counter module which we have
made it using Altera-DE2 FPGA board and has coincidence time window
$\tau=7.1$ns with adjustable acquisition time greater than 0.1s. To make sure
its validity and also to calibration, we have compared it to the 8-channel
coincidence-counter/time-tagger made by QuTools Co. [with time resolution
(bin-size) and time-stamp 81ps] such that we found their SCs and CCs were in
good agreement. After online calculations via the C++ software in our CC
module, the CC and SC rates fed to the PC and shown online in the Labview
interface panel on the monitor. All the analysis of the experimental data has
been done in MATLAB software. Finally, to decrease the background noises we
have tried to remove all the reflected and scattered light from the
environment and blocked the pump beam after crystal by blocker B. Moreover, we
control the temperature of the Lab during each experiment. Furthermore, to
keep laser power constant to avoid noise in count rates, we stabilized the
diode laser by a 5V 1A driver with $\lesssim\%0.1$ current-fluctuation which
results $\lesssim\%0.1$ power-fluctuation by assumption constant load on the
adapter. The distance between two detectors is about 120mm and the distance
between detectors and BBO NLC is roughly 98cm. To calibration and have the
optimum PM and maximum CC rate we follow these steps before adding the
samples:

(i) tuning the incident angle and polarization of pump beam to have stable
maximum CC rate;

(ii) adjusting, tilting and moving detectors and BBO crystal which is on
tiltable/rotatable mount. Note that the entrance port of the detectors (filter
+ coupler + entrance edge of fiber are fixed together on a holder as one
element) is in the rotatable and movable mechanical elements.

After calibration and achieving the stable maximum CC rates, samples (DNA or
MB solutions and thin-films on soda-lime) are added into the setup in the
signal-port. The signal beam propagates and transmits through the samples and
then reaches to the D2. In this condition, the CC and also SC rates are
affected by the transmittance of the samples $\mathcal{T}$ which depends on
the concentration of the solutions or refractive index of the thin-film
samples. In the following we show how can measure the transmittance
$\mathcal{T}$ via the CC rate and SC rate as a result of quantum entanglement
and classical-like measurements, respectively, to characterize and detect the
samples.

Figure 1: (Color online) (a) Schematic illustration of the experimental
arrangement for quantum measurement of the transmittance of DNA and Methylene-
Blue solutions which are inside the glass cuvette and also thin-film samples.
(b) Schematic of DNA chain and Methylene- Blue molecule. (c) Top-view of the
experimental setup. (d) Different solution samples of Methylene-blue inside
Erlen and frozen DNA-solutions with different concentrations inside the micro-
tubes at $-20^{\circ}{\rm C}$.

###  3.2 Samples Preparation

DNA is the first sample which we are interested in with different
concentrations (see Fig. 1(b)). DNA (Deoxyribonucleic acid) is a molecule
composed of two chains or strands that are intertwined and wrapped around each
other to form a double helix carrying genetic instructions which are
responsible for the development, functioning, growth and reproduction of all
known organisms and many viruses and plays the key role in genetics. These
strands are also known as polynucleotides and composed of simpler monomeric
units called nucleotides. Each nucleotide includes one of four nitrogen-
containing nucleobases (cytosine [C], guanine [G], adenine [A] or thymine
[T]), a sugar or deoxyribose, and a phosphate group. We choose two different
types of DNA, a Human-muscle and a Rabbit-muscle DNA solutions with different
concentrations. These samples have been prepared by a PCR (Polymerase Chain
Reaction) device. We have prepared Human DNA solutions with concentrations
$0.01,0.1,1,10$ and 100 ${\rm ng/\mu l}$ and Rabbit solution with
concentration 100${\rm ng/\mu l}$. Prepared samples have been identified via
their transmittance using UV-IR Thermo 200C Spectrophotometer device by
manufacturer (Neda Chemistry Co.) It should be noted that using the Thermo
200C Spectrophotometer device DNA solutions with concentrations less than
1${\rm ng/\mu l}$, we have called it the standard limit (SL) of this device,
never can be detected and distinguished. In the other words, our DNA solutions
with concentrations 0.01 and 0.1 ${\rm ng/\mu l}$ never can be detected and
identified via their transmittances by the standard commercial spectrometer
device Thermo 200C. The prepared samples should be maintained at
$-20^{\circ}\rm C$.

MB [$\rm C_{16}H_{8}Cl~{}\\!N_{3}S$] with different concentrations is the
second type of our sample (see Fig. 1(b)). Its different doses act as a
disinfectant. We prepare the master sample by adding 1.5g MB in 100ml Ethanol.
After that, by adding Ethanol into master Methylene-blue (MMB) one can dilute
it and the different concentrations $1.5\rm\mu g/\mu l-4.5pg/\mu l$ are
prepared to use.

Finally, we prepared some different dielectric thin-films: soda-lime glass
without thin-films and also Al and Cr single metal layer and multilayers such
as 3-layers $\rm MgF_{2}/ZnS/MgF_{2}$ and 4-layers $\rm
MgF_{2}/ZnS/MgF_{2}/ZnS$ on the soda-lime glass. We prepared Cr/Al nano layers
by DC-sputtering coating (by MSS110 DC-Sputtering assisted turbo molecular
pump device made by ACECR in IRAN) on soda-lime glass slides at final vacuum
pressure $1\times 10^{-5}$mbar, Ar work-pressure $6.6\times 10^{-3}$mbar,
deposition rate $3.5\rm A^{o}/s$, 7SCCM Ar gas, and with current 120mA and
voltage 500v at temperature $100^{o}C$ around the substrate. For the
dielectric multilayers we used evaporation method (by EDS110 device assisted
diffusion pump made by ACECR) at final vacuum pressure $8\times 10^{-6}$mbar
with deposition rate $1\rm A^{o}/s$. Here, all thickness have been obtained by
the piezo crystal during the coating process with around $\%10$ accuracy.

In our experiments, DNA and MB solutions are inside the glass cuvette (see
inset in Fig. 1(c)). Also, the thin-films are placed in mount. The cuvette and
mount can be fixed on the tiltable and rotatable sample-holder which is placed
in the signal-arm in the nearest distance to D2 such that the signal photons
incident vertically on the cuvette or thin-film surface and thus transmit
without any refraction. To sure it, we couples a 120mW 810nm-diode laser to
the fiber coupler FC2 from the back in the signal-port, D2, and try to tilt
and rotate the samples to retroreflect incident test-beam 810nm on itself.

At this point, the setup is ready to measurement of the transmittance (see
Fig. 1(c) which shows the top-view of experimental arrangement and Fig. 1(d)
shows the MMB and diluted MMB inside Erlen and also freezed DNA solutions
inside microtubes. The inset shows a diluted MB inside cuvette in front of the
D2). In the next subsection, we introduce our simple and short method to
measurement of the transmittance.

###  3.3 Method

As mentioned, the samples with transmittance $\mathcal{T}$ are placed in the
signal-arm or sample-arm. Interestingly, since the absorption frequency of our
samples is roughly far from the signal-idler wavelength 810nm, then in
contrast to the quantum spectroscopy, we need no monochromators and narrow-
band filters to add in our setup. Therefore, one can assume and sure that all
changes in the CC or SC rates is only and mostly due to the transmittance of
the samples because of the changes in their refractive index or
concentrations. In this manner, using the quantum correlation, i.e., CC rate,
we have

$\displaystyle\eta_{2}\mathcal{T}_{cc}=\frac{N_{cc}^{(s)}}{N_{1}^{(s)}},$ (17)

where the quantum efficiency is given by $\eta_{2}=N_{cc}^{(0)}/N_{1}^{(0)}$
[5]. $N_{cc}^{(s)}$ and $N_{cc}^{(0)}$ are the CC rates in the presence and
absence of the samples, respectively, while $N_{1}^{(s)}$ and $N_{1}^{(0)}$
are the SC rates of the idle-arm (which there is no sample in this arm) in the
presence and absence of the sample in sample or signal-arm. Keeping it in mind
that the time average idler SC rate is approximately the same for both cases
of the presence and absence of the sample S, i.e.,
$\bar{N}_{1}^{(s)}\simeq\bar{N}_{1}^{(0)}=\bar{N}_{1}$. Therefore, the
transmittance can be precisely measured using the following relation

$\displaystyle\mathcal{T}_{cc}=\frac{N_{cc}^{(s)}}{N_{1}^{(s)}}\frac{N_{1}^{(0)}}{N_{cc}^{(0)}}~{},$
(18)

which can be approximated as

$\displaystyle\mathcal{T}_{cc}\simeq\frac{N_{cc}^{(s)}}{N_{cc}^{(0)}}~{}.$
(19)

In our experiments we use Eq. (18) to obtain the transmittance by calculating
the time-dependent CC rate matrix from the experimental data versus
acquisition time. To have error in transmittance measurement, we clearly can
use the relation

$\displaystyle\frac{\Delta\mathcal{T}_{cc}}{\mathcal{\bar{T}}_{cc}}=|\frac{\Delta
N_{cc}^{(s)}}{\bar{N}_{cc}^{(s)}}|+|\frac{\Delta
N_{cc}^{(0)}}{\bar{N}_{cc}^{(0)}}|+|\frac{\Delta
N_{1}^{(s)}}{\bar{N}_{1}^{(s)}}-\frac{\Delta
N_{1}^{(0)}}{\bar{N}_{1}^{(0)}}|~{}.$ (20)

The standard method for transmittance measurement, which is used in the
standard commercial spectrometer devices, is measuring the transmitted
intensity divide to incident intensity of light, which is fully classical
method based on the intensity measurement of the light. Now, for comparison to
the classical method for illustration of the advantage of the quantum
correlation, frequency-entanglement, respect to the classical intensity
measurement one can write a relation for transmittance based on SC rate
measurement, which is seen as a best classical-like method, as follows

$\displaystyle\mathcal{T}_{sc}=\frac{N_{2}^{(s)}}{N_{2}^{(0)}}~{},\qquad\frac{\Delta\mathcal{T}_{sc}}{\mathcal{\bar{T}}_{sc}}=|\frac{\Delta
N_{0}^{(s)}}{\bar{N}_{2}^{(s)}}|+|\frac{\Delta
N_{2}^{(0)}}{\bar{N}_{2}^{(0)}}|,$ (21)

where $N_{2}^{(j)}$ (j=s,0) are the SC rates of the signal-arm (sample-arm) in
the presence and the absence of the samples. Note that if one arm of the SPDC
to be blocked, the state of the system to be approximately collapsed to a pure
state and thus the other arm can be seen as a non-ideal single-photon source
or heralded single-photon source [5, 2]. Therefore, if by measuring the
second-order autocorrelation function via HBT-interferometry (Hanbury Brown
and Twiss), one finds $g^{(2)}(0)<0.1$ the source is called the idea single-
photon source [5, 2], otherwise if $g^{(2)}(0)<1$ the source is a non-ideal
single-photon source. Therefore, we would stress that in principle
$\mathcal{T}_{sc}$ which is obtained by measuring the SC rate is more accurate
than the measurement of the classical light beam intensity to obtain
transmittance.

It is worth to note that the classical analog of the CC measurement of
entangled photons is a measurement based on the classically correlated light
beam which can be prepared by Arrechi’s disk [46] or spatially light modulator
(SLM). Surprisingly, it has been shown theoretically and experimentally that
the CC-based measurement of the entangled photons is much more accurate than
its classical counterpart, CC measurement of classically correlated beam [18,
19]. That is why here we are going to show the advantage and compare the
transmittance measurement based on the CC measurement of entangled photons and
SC-based measurement.

Finally, it should be noted that we have to take into account the well-known
error corrections [52] due to the accidental coincidence rate, dark count, and
dead-time. The accidental rate, $N_{Ac}=N_{1}N_{2}\tau_{cc}/T$, and dark
counts $D_{j}$ ($j=1,2$) can be taken into account [52] by substituting
$N_{cc}-N_{Ac}$ and $N_{j}-D_{j}$ instead of experimentally recorded counts
$N_{cc}$ and $N_{j}$, respectively. Another correction due to the dead time of
the detectors can be taken into account by substituting $\gamma N_{j}$ instead
of $N_{j}$ where $\gamma$ is the dead time correction factor,
$\gamma=1-N_{j}\tau_{\rm dead}/T$ [52]. Note that since single count in the
sample arm is variable due to the present of the different samples, then its
$\gamma$-factor is not constant and changes in each case but is around 0.9.
Here, $\tau_{cc}$, $\tau_{\rm dead}$ and $T$ are, respectively, the
coincidence time window, the dead time of the detectors and the interval
(integration) or gate time which in our case their values are, respectively,
7.1ns, 50ns and 0.3s. Other corrections factors which are discussed in Ref.
[52] are ignorable in our setup.

###  3.4 DNA results

After calibration of the setup we checked our methods on DNA solutions with
different concentrations $\mathcal{C}$. The results of the transmittance
measurement based on CC- or SC-count measurement on Human (H) and Rabbit (R)
DNAs are given in Tab. (1). Comparing the four last columns in Tab. (1)
clearly shows that the errors due to transmittance measurement via the CC
rate, as a quantum entanglement-based measurement, are very smaller than the
SC measurement as a classical-like measurement. Moreover, relative error
($\Delta N/\bar{N}$) in CC rate is generally in order of $10^{-4}$ while in SC
rate is in order of $10^{-3}$, i.e., the CC rates are 1-order more accurate
than the SC rates. Furthermore, the average of transmittance obtained by both
CC and SC rates is in very good agreement to each other which shows the self
consistency of the method.

Table 1:  Experimental values for the transmittance measurement on DNA of
Human (H)/Rabbit (R) hand-muscle with different concentrations based on the
SC-measurement and CC-measurement. Here, during the experiment the temperature
of the Lab is fixed at $(18\pm 0.1)^{\circ}\rm C$. The acquisition time and
coincidence time window are respectively $T_{\rm acq}=0.3\rm s$ and
$\tau=7.1\rm ns$. The average single-count rate of the idler-arm during the
experiment is $\bar{N}_{1}=(1.240\pm 0.005)\rm Mcps$. Here, the CC and SC
rates are vs count per second (cps) or Hz. Also, we define a figure of merit
as an entanglement advantage respect to the single-count measurement in count
rates as $\mathcal{G}_{N}={\rm err}^{\rm rel}(N_{2})/{\rm err}^{\rm
rel}(N_{cc})$ and to transmittance measurement as $\mathcal{G}_{T}={\rm
err}^{\rm rel}(\mathcal{T}_{sc})/{\rm err}^{\rm rel}(\mathcal{T}_{cc})$ in
which the relative error is defined as ${\rm err^{rel}}(x)=\Delta
x/\bar{x}$.Note that, here, the standard limit of concentration (SL) to
detection or identification of DNA through its transmittance is defined as
1$\rm ng/\mu l$ which is obtained by the standard commercial Thermo200c UV-
spectrometer device by manufacturer.

Sample | DNA | Concentration | Count Rates | Transmittance | Ent. Adv.
---|---|---|---|---|---
$\\#$ | Type |  $\mathcal{C}$ [$\rm ng/\mu l$] | CC $N_{cc}$[K cps] | SC $N_{2}$[K cps] |  $\mathcal{T}_{cc}$[$\%$] |  $\mathcal{T}_{sc}$[$\%$] | $\mathcal{G}_{T}$ | $\mathcal{G}_{N}$
$1$ | No.samp/cuv. | —– | $107.17\pm 0.05$ | $1005.172\pm 0.721$ | —– | —– | —– | $1.52$
$2$ | R | $100$ | $90.28\pm 0.05$ | $853.118\pm 1.001$ | $84.24\pm 0.09$ | $84.89\pm 0.16$ | $1.89$ | $1.87$
$3$ | H | $100$ | $91.50\pm 0.03$ | $864.208\pm 0.980$ | $85.37\pm 0.07$ | $85.98\pm 0.16$ | $3.85$ | $3.84$
$4$ | H | $10$ | $92.53\pm 0.01$ | $873.128\pm 0.748$ | $86.34\pm 0.05$ | $86.86\pm 0.13$ | $2.60$ | $6.43$
$5$ | H |  $1$ (SL) | $93.28\pm 0.01$ | $877.985\pm 0.920$ | $87.04\pm 0.05$ | $87.35\pm 0.15$ | $3.19$ | $13.42$
$6$ | H | $0.1$ | $94.16\pm 0.03$ | $887.752\pm 0.667$ | $87.86\pm 0.07$ | $88.32\pm 0.13$ | $1.83$ | $2.24$
$7$ | H | $0.01$ | $94.90\pm 0.04$ | $888.613\pm 0.774$ | $88.55\pm 0.08$ | $88.40\pm 0.14$ | $1.84$ | $2.24$

By Comparing the second and the third row in Tab. (1), one can interestingly
figure out the advantage of the use of entanglement or CC rate respect to use
of SC rate. As is seen in Fig. (2) and Tab. (1), at the same concentration
100$\rm ng/\mu l$ the Rabbit-DNA can be detected and distinguished from the
Human-DNA by more than 30$\sigma_{\rm st}$ and 13$\sigma_{\rm st}$ via the CC
rate and its analogous transmittance, respectively, while it can be detected
by more than 9$\sigma_{\rm st}$ and 4.8$\sigma_{\rm st}$ using SC rate and its
analogous transmittance, respectively. This shows the entanglement or quantum
correlation advantage to high-precision distinguishability between Human and
Rabbit DNAs with the same concentration such that the CC-based measurement is
approximately 3-times more accurate than the SC-based measurement. As is seen
in Fig. 2(c) and (d) and H-rows in Tab. (1), below the SL of concentration
($1\rm ng/\mu l$ which is defined for DNA detection using transmittance
measurement via standard spectrometer devices such as Thermo200C) the SC-based
measurements never can detect DNA solutions with high reliability, we called
it unreliable measurement region. While, surprisingly, using the CC-based
measurement (transmittance or CC rate) one can detect and distinguish DNA
solutions with high-reliability more than 20$\sigma_{\rm st}$ and
8$\sigma_{\rm st}$, respectively, via the CC rate and CC-based transmittance
(see Fig. 2(a) and (b)). In other words, frequency-entanglement enables us to
ultra-precise measurement and high-accuracy distinguishability at least
2-order below the SL of DNA-concentration to diagnose and distinguish DNA with
concentrations 0.01 and 0.1 $\rm ng/\mu l$. From the experimental results in
Tab. (1) it is clear that the CC-base measurement of the transmittance has
0.05 accuracy which is at least 3-times more accurate than the SC-based
measurement of the transmittance.

Figure 2: (Color online) (a) and (c) Experimental CC and SC rate measurements
for DNA solutions with different concentrations, respectively. (b) and (d)
Experimental CC-based and SC-based transmittance measurement of DNA solutions
vs their concentrations, respectively. Here, the thick blue circle points and
thick red triangle points are respectively referred to experimental data for
the Human and Rabbit DNAs. Below the SL region the CC-base measurement results
are very reliable in contrast to the SC-based measurement results.

By considering the SNR definition as the signal to noise ratio [21, 22] and
the sensitivity $\mathcal{S}$ as the minimum noise in the system (see Ref.
([47]) and references therein) as

$\displaystyle\rm SNR=10{\rm Log}(\frac{P_{\rm signal}}{P_{\rm
Noise}})~{},\qquad\mathcal{S}=-10{\rm Log}P_{\rm Noise}^{\rm min}$ (22)

where the power of signal and noise is defined as their count rates ($P_{\rm
signal(noise)}:=N_{\rm signal(noise)}$). Using the experimental results in
Tab. (1), one can find the average SNR for the CC and SC rates as $\rm
SNR_{\rm Ent.}\simeq 36.5\rm dB$ and $\rm SNR_{\rm SC}=30.3\rm dB$,
respectively. Moreover, the average sensitivity is respectively
$\mathcal{S_{\rm Ent.}}\simeq-14.2\rm dB$ and $\mathcal{S_{\rm
SC}}\simeq-29.2\rm dB$. This clearly shows that in our experiment the CC-based
measurements which are due to the frequency-entangled state of SPDC has 6dB
SNR, and 15dB sensitivity more than the SC-based measurement. Furthermore, in
case of transmittance measurement: $\rm SNR_{\rm Ent.}^{T}\simeq 31.1\rm dB$
and $\rm SNR_{\rm SC.}^{T}\simeq\rm 27.8dB$ which means 3dB more SNR is
achievable using CC-based transmittance measurement. As is seen in Fig. (2),
we should emphasize that different DNAs with different concentrations are
reliably detectable and distinguishable from each other using the CC-based
measurement with accuracy at least more than 10$\sigma_{\rm st}$.

Figure 3: (Color online) Theoretical fitted curve to the experimental
transmittance points of DNA obtained from the CC-measurement.

In Fig. (3) the theoretical exponential model for transmittance vs
concentration as

$\displaystyle\mathcal{T}_{\rm
fit}=T_{0}e^{-\mathcal{C}/\mathcal{C}_{0}}+T_{\infty}e^{-\mathcal{C}/\mathcal{C}_{\infty}},$
(23)

have been fitted to the experimental transmittance points obtained via the CC-
measurements. Using this model which has good agreement in this concentration
regime, one can estimate concentration of DNA solutions with unknown
concentration. This may lead indirectly to high-accuracy DNA concentration
measurement, concentrometry. If we assume $\%0.01$ uncertainty in
experimentally transmittance measurement using the CC rate, one can expect
$3\rm pg/\mu l$ uncertainty in concentration measurement, while in SC case the
uncertainty in transmittance could be $\%0.1$ which results $30\rm pg/\mu l$
uncertainty in concentration measurement using SC rate. Although, we have
focused on the transmittance measurement, but using the theoretical fitting
curve in Eq. (23), and preparing a database for some DNA solutions in
different concentration regimes, one can indirectly measure concentration of
DNA solutions via their CC-based transmittance measurement. Furthermore, the
concentrometry using CC rate is more accurate than the SC rate notably below
the SL which the SC-based measurement cannot work reliably. But, it is clear
that the method works better and more accurate for the transmittance
measurement instead of concentration measurement.

Finally, this introduced method in this subsection might be used to
manufacture an entanglement-based transmittometer/concentrometer simple-device
for DNA detection via the CC-measurement.

###  3.5 MB results

Table 2:  Experimental values for the transmittance measurement on the
Methylene-blue (MB) with different concentrations based on the SC-measurement
and CC-measurement. Here, during the experiment the temperature of the Lab is
fixed at $(27\pm 0.1)^{\circ}\rm C$. Here, $\bar{N}_{1}=(1.238\pm 0.005)\rm
Mcps$ and the other experimental parameters are the same as Tab. (1).

Sample | MB | Concentration | Count Rates | Transmittance | Ent. Adv.
---|---|---|---|---|---
$\\#$ | Type |  $\mathcal{C}$ [$\rm ng/\mu l$] | CC $N_{cc}$[K cps] | SC $N_{2}$[K cps] |  $\mathcal{T}_{cc}$[$\%$] |  $\mathcal{T}_{sc}$[$\%$] | $\mathcal{G}_{T}$ | $\mathcal{G}_{N}$
$1$ | No.samp/cuv. | —– | $59.87\pm 0.03$ | $931.34\pm 0.33$ | —– | —– | —– | —–
$2$ | MMB | $1.5\times 10^{4}$ | $2.32\pm 0.02$ | $32.13\pm 1.23$ | $3.37\pm 0.03$ | $3.44\pm 0.14$ | $1.02$ | $1.03$
$3$ | $\rm S_{0}$ | $4.5\times 10^{3}$ | $33.23\pm 0.01$ | $506.56\pm 1.39$ | $55.51\pm 0.04$ | $54.39\pm 0.17$ | $4.58$ | $12.54$
$4$ | $\rm S_{1}$ | $15$ | $51.51\pm 0.01$ | $809.26\pm 1.84$ | $86.04\pm 0.06$ | $86.89\pm 0.23$ | $3.81$ | $9.82$
$5$ | $\rm S_{2}$ | $1.5$ | $52.38\pm 0.01$ | $820.02\pm 1.58$ | $87.49\pm 0.06$ | $88.05\pm 0.20$ | $3.37$ | $8.86$
$6$ | $\rm S_{3}$ | $0.15$ | $53.27\pm 0.01$ | $828.51\pm 1.47$ | $88.97\pm 0.06$ | $88.96\pm 0.19$ | $2.96$ | $6.80$
$7$ | $\rm S_{4}$ | $0.075$ | $53.89\pm 0.01$ | $840.56\pm 1.80$ | $90.00\pm 0.06$ | $90.25\pm 0.23$ | $3.56$ | $8.83$
$8$ | $\rm S_{5}$ | $0.045$ | $54.58\pm 0.01$ | $847.24\pm 1.68$ | $91.17\pm 0.06$ | $90.97\pm 0.21$ | $3.42$ | $8.81$
$9$ | $\rm S_{6}$ | $4.5\times 10^{-3}$ | $55.42\pm 0.01$ | $858.15\pm 1.15$ | $92.56\pm 0.06$ | $92.14\pm 0.16$ | $2.52$ | $6.22$

Similar to DNA solutions, the same procedure have been performed on the MB
solutions to obtain their transmittance vs concentration (see Tab. (2) and
Fig. (4)). As is seen the transmittance obtained in both cases CC- and SC-
based measurement are in good agreement. Similar to DNA case the CC-based
measurements have more accuracy and more SNR concerning the SC-based
measurement. For example, very diluted MB solution with concentration $4.5\rm
pg/\mu l$ and $\%92$ transmission has been detected through CC-based
transmittance measurement with $\rm SNR\simeq 37dB$ which is 9dB more than SC-
based measurement. The physical conclusion in case MB is the same as DNA: the
entanglement enables us to more accurate measurement of the transmittance of
the diluted solutions with higher SNR and smaller noise. The uncertainty in
transmittance due to the CC and SC rate measurement is respectively, $\%0.06$
and $\%0.17$. It means that the CC-based measurement is more accurate and more
reliable because of entanglement advantage. Moreover, using the fitting Eq.
(23) a $0.1\rm pg/\mu l$ uncertainty is achievable using CC-based
transmittance measurement due to the frequency-entanglement advantage.

Figure 4: (Color online) (a) and (b) Experimental CC and SC rates for MB
solutions with different concentrations, respectively. (c) and (d)
Experimental CC- and SC-based transmittance measurement of MB solutions vs
their concentrations, respectively.

Here, It should be reminded that as DNA case, one can manufacture an optical
device based on frequency-entanglement to high precision measurement of the
transmittance or indirect measurement of the concentration through the
transmittance.

Furthermore, as we said in the previous section the maximum absorption
wavelength of DNA and MB are about $\lambda_{\rm DNA}^{\rm ab.}\sim 260\rm nm$
[48] and $\lambda_{\rm MB}^{\rm ab.}\sim 668\rm nm$, respectively, which are
far from signal wavelength $\lambda_{s}=810\rm nm$ that passes through the
sample. This means that our method for transmission measurement based on the
CC rate measurement is still valid.

###  3.6 Thin-films results: towards quantum diagnosis of cancer tissues

To test our method for measuring the transmittance of the transparent bulk
object, specially, thin-film (TF) samples, we proceed in this subsection to
measure the transmittance of soda-lime glass, single- and multi-layer
dielectric thin-films and also single metal layers using CC and SC rate
measurement (see Tab. (3)).

To test the validity of our methods, we try to perform measurements on the
Soda lime and Lamel glasses (see samples $\\#2$ and $\\#3$ in Tab. (3)). Our
transmittance results are in agreement with transmittance that we expect for
soda-lime with thickness around 1mm and Lamel with thickness around 150$\mu\rm
m$ based on Ref. [49](around $\%$ 3(2)-4 absorption and $\%$ 3(2)-4 reflection
for soda-lime(Lamel) glass). Moreover, as is seen in Tab.(3) both
transmittances using CC and SC measurement are in agreement but CC-based
measurement has more accuracy than SC-based measurement. Using the Fresnel
coefficient for transmittance at near-normal incident as a rough model
(ignoring the effects of absorption, interference and thickness), $T=4n_{\rm
air}n/(n_{\rm air}+n)^{2}$, one can simply find $n=1.677\pm 0.003$ and
$n=1.530\pm 0.004$ using the CC, respectively, for soda-lime and Lamel
glasses. It seems in this method one can obtain the refractive index with
accuracy in order of $10^{-3}$.

Samples $\\#4$ and $\\#5$ show the transmittance measurement of dielectric
multilayers MgF2/ZnS on soda-lime in which MgF2 and ZnS layers have thickness
around $\lambda_{\tiny{1064}}/4$ with $10\%$ tolerance, respectively. Their
imaginary refractive index are $\tilde{n}_{\rm
MgF2}^{\rm\lambda_{s}}=1.4193+\rm i4.2103\times 10^{-4}$ with absorption
coefficient $\alpha_{\rm MgF2}^{\rm\lambda_{s}}=65.319\rm cm^{-1}$ and
$\tilde{n}_{\rm ZnS}^{\rm\lambda_{s}}=2.3256+\rm i0.027886$ with absorption
coefficient $\alpha_{\rm MgF2}^{\rm\lambda_{s}}=4326.3\rm cm^{-1}$ [49] which
imply that their absorption wavelength is far from signal-wavelength or they
are fully transparent for incident signal-idler wavelength $810\rm nm$. As is
seen in Tab. (3), similar to solution cases both SC- and CC-based
transmittance measurement are in agreement. However, it is clear that the CC-
based measurement is more accurate. Here we remind that in cases of glasses
and multi-layers samples since the absorption wavelength are far from incident
signal-wavelength, thus, our method works well as we expected.

Despite of this to show that our method as we expected cannot work well for
absorber material like metals, we try to preform our method on Al and Cr
nanolayers with thickness around 15nm and 25nm with $10\%$ precision,
respectively. Note that both Al and Cr are good absorbers at wavelength 810nm
which absorb more than $\%50$, especially Al. Refractive index of Al and Cr
are, respectively, $\tilde{n}_{\rm Al}^{\rm\lambda_{s}}=2.7670+\rm i8.2925$
with absorption coefficient $\alpha_{\rm Al}^{\rm\lambda_{s}}=1.2865\times
10^{6}\rm cm^{-1}$, and $\tilde{n}_{\rm Cr}^{\rm\lambda_{s}}=3.1797+\rm
i3.4698$ with absorption coefficient $\alpha_{\rm
Cr}^{\rm\lambda_{s}}=5.3831\times 10^{5}\rm cm^{-1}$. As is seen in Tab. (3),
the results of transmittance based on CC and SC measurement has no agreement
which is due to the absorption. In this manner, to have a precise and valid
transmittance we need to put narrow-band filters or monochromator to receive
exactly the re-emitted photon due to the absorption. That is why the results
have no coincidence together.

Surprisingly, it should be noted that our most motivation to perform the
method on the TFs is that the similarity between TFs and cancer tissues such
as brain tissues which can be prepared as a thin-film on a slide. Therefore,
it may be used for cancer therapy or diagnostic purposes. We are hopeful that
in the soon future this simple method, which is based on the CC-measurement of
generated frequency-entangled biphotons, opens a new pave and simple platform
to cancer therapy and quantum diagnosis of cancer more accurate than the
present classical methods in different types of cancer tissues.

Table 3:  Experimental values for the transmittance measurement of TFs. Here,
the temperature of the Lab is fixed at $T=(20\pm 0.1)^{\circ}\rm C$, and
$\bar{N}_{1}=(1.270\pm 0.008)\rm MHz$.

| Sample | Count Rates | Transmittance | Ent. Adv.
---|---|---|---|---
$\\#$ | Type | CC $N_{cc}$[K cps] | SC $N_{2}$[K cps] |  $\mathcal{T}_{cc}$[$\%$] |  $\mathcal{T}_{sc}$[$\%$] | $\mathcal{G}_{T}$ | $\mathcal{G}_{N}$
$1$ | No sample | $95.50\pm 0.04$ | $933.35\pm 0.74$ | —– | —– | —– | —–
$2$ | Soda Lam | $89.39\pm 0.01$ | $869.99\pm 0.80$ | $93.61\pm 0.05$ | $93.21\pm 0.16$ | 3.50 | 7.01
$3$ | Lamel | $91.30\pm 0.02$ | $893.63\pm 0.76$ | $95.61\pm 0.06$ | $95.74\pm 0.16$ | 2.77 | 3.66
$4$ | MgF2/ZnS/MgF2/Soda Lam | $84.430\pm 0.013$ | $821.96\pm 0.49$ | $88.42\pm 0.05$ | $88.07\pm 0.13$ | 2.72 | 3.96
$5$ | MgF2/ZnS/MgF2/ZnS/Soda Lam | $65.862\pm 0.008$ | $639.97\pm 0.82$ | $68.97\pm 0.03$ | $68.57\pm 0.15$ | 4.39 | 10.44
$6$ | Al/Soda Lam | $16.320\pm 0.002$ | $146.50\pm 0.37$ | $17.09\pm 0.01$ | $15.70\pm 0.05$ | 7.23 | 25.78
$7$ | Cr/Soda Lam | $27.882\pm 0.004$ | $250.51\pm 0.50$ | $29.20\pm 0.01$ | $26.84\pm 0.08$ | 5.48 | 13.60

##  4 Discussion

Let us discuss more in this section to illustrate the ability of the
introduced simple method.

  * (a)

A considerable difference of coincidence count (CC) rate by human- and rabbit-
DNA solutions are actually just due to the difference of their transmittance
which itself originates to difference of their refractive indexes at 810nm. It
is very important to note that the absorption wavelength of DNAs is usually
around 250-300nm [48] while our probe wavelength of the signal-idler biphoton
is at 810nm which is very far from the absorption frequency of Rabbit and
human DNAs.

  * (b)

It should be mentioned that the other important physical parameters which may
influence or affect on the CC or SC rate are temperature, concentration and
inhomogeneity of the solution during the measurement. To avoid the first one,
we have fixed the temperature of Lab during the experiment, then, we are sure
that any small difference of the CC/SC is just due to the difference of
transmittance or small difference of refractive index. Furthermore, to avoid
the second one, we requested from our genetic expert colleagues in Neda Shimi
Ltd to prepare both human and Rabbit DNA with the same concentration as
possible as. Finally, to avoid the effects of inhomogeneity, diffusion, and
turbulence of the solutions during the measurement, after pouring solution in
the cell, we waited for a certain time to be sure that there is no
inhomogeneity, diffusion, and turbulence, and then, we start the data
acquisition to obtain the transmittance.

  * (c)

It should be noted that the introduced method can be applied to other
solutions as far as the absorption wavelength to be far from the signal
wavelength. Note that, by having a tunable source like PPKTP or PPLN SPDC
sources, which exists in many Labs, one can always apply this method for any
solutions or samples. Since it is just enough to tune the SPDC-pump to
generate the output wavelength far from the absorption wavelength of the
target solution. In this case, in addition to tunable source, one needs to a
wide range single photon detectors. In other words, different samples cannot
change the physics, and just change the values of the CC/SC rates and
consequently the transmittances at desired wavelengths.

  * (d)

It should be emphasized that in the present work all transmittance-
measurements are a relative measurement with respect to the prepared
concentrations by our reference external molecular genetic Lab, Neda Shimi
Co., not an absolute measurement. Furthermore, since our goal is transmittance
measurement, then, we have not used or found a theoretical model for behavior
of the transmittance vs concentration, temperature, and refractive index of
the used samples. But, if we can find a convenient theoretical model similar
to Refs. [50, 51] whose authors provided the model versus concentration and
temperature for Ethanol, thus, we are able to perform an absolute measurement
which yields to absolute values of refractive index or concentration at the
fixed temperature. It would be a good future research plan with collaboration
of chemists and geneticists to find a model for the behavior of the DNA and MB
to apply the introduced method to obtain the absolute value of the refractive
index and concentration.

Nevertheless, with no theoretical model, one can still prepare many solution
samples with different concentrations at fixed temperature to prepare a
database for their transmittance behavior. Then, one can find a trend for each
sample versus its concentrations in the fixed temperature. Then, by
calibration respect to the used concentration reference at the determined
temperature, one can provide relatively concentration and refractive index of
an unknown solutions with respect to the reference through their measured
transmittance. This may lead to manufacture of a quantum-based device in the
future.

  * (e)

Let us clarify the probable ambiguity that why we argue that the thin film
multilayer results may lead to quantum diagnosis or application in medicine
and biology in the future.

In Refs. [30, 31, 32, 33, 34, 35, 36, 38, 37], the authors have investigated
how much the brain tissue, biological objects, rat or kidney tissues can
preserve the frequency-entanglement, polarization-entanglement and even
recently the hyper-entanglement such as orbital angular momentum (OAM) which
are very interesting from the philosophy and consciousness point of view.
Their motivations are usually to show that entanglement can be used to
characterize these tissues and can be used to cancer therapy or diagnosis of
the tissues, or quantum interpret of the brain of biological mechanism. But,
as far as we know, they have not provided a practical measure or quantity for
this aim, except Bell parameter S or some well-known nonlocality parameter
such entanglement entropies [30, 31, 32, 33] which quantify the entanglement.
While we have provided an optical parameter, transmittance, which can
practically characterize the sample via its response to the quantum
entanglement, nonclassical light. That is why we have been motivated to
investigate the thin films samples, since we believe that it may open a new
practical way to cancer therapy and quantum diagnosis in medicine and biology.

Let us now illustrate the relation or similarities/differences between thin
films and cancer tissues or rat sections. As presented in mentioned Refs. [30,
31, 32, 33, 34, 35, 36, 38, 37], the rat sections or brain/kidney tissues with
different thickness (10-600$\mu$m) are similar to a thin or thick films. But,
usually the optical thin films or multilayers are homogeneous and uniform
while these biological tissues with different thickness are inhomogeneous and
scatterer which can be modeled theoretically by considering different
scatterer layers. Although there are this differences between optical
multilayers and biological films like brain/kidney/rat tissues, but the thin
film model can still provide a rough approximate model to investigate the
interaction of entanglement light source with these biological tissues. That
is another reason we have been motivated to perform our method on the thin
film layers to measure their transmittance, and we believe that our simple
quantum measurement approach can open a new platform to quantum diagnosis of
biological tissues and its characterization as has been addressed in Refs.
[30, 31, 32, 33]. Besides these, in Refs. [30, 31, 32, 33] the authors have
used the Bell parameter S and entanglement entropy measures which give no
information about the materials, while if our method has been applied, then,
transmittance measurement enables us to precisely characterize. Furthermore,
they have used polarization-entanglement which is very fragile when light
passing through the tissues while our frequency-entanglement biphoton is
robustness with much higher rate. These are reasons we argue that the thin
film results can open a way towards the quantum diagnosis, quantum therapy and
also quantum characterization in the context of medicine and biology.

  * (f)

One of the most important questions which can be addressed here is the genuine
damage of a biological sample that may be caused due to its transmittance
measurement process. Note that we never have shown in this paper that a
genuine damage is avoided in DNA or MB samples through the use of quantum
correlations. But, it is worth to remind that in SPDC process in NLCs a
quantum signal usually has a power in order of pW or smaller which means that
it is very low-power compared to the classical light signal which may be in
order of $\mu$W or nW or mW. Thus, one can urge that the damage or destructive
effects in case of quantum measurement are definitely very smaller than the
case of classical light with very higher power. Therefore, a quantum-based
measurement might be seen as a non-invasive measurement technique which is so
important especially in biological samples. Nevertheless, it definitely needs
to be proven but we leave it as a novel research for the next future works.

##  5 Summary, Conclusion remarks and Outlooks

We have shown that the CC rate measurement of the generated frequency-
entangled biphotons at NIR wavelength $\rm 810nm$ which pass through the
solutions, DNA and MB, and TFs enables us to high precision quantum
transmittometry with 4-digit accuracy for coincidence time-window 7.1ns, while
the SC measurement as the classical measurement leads to 3-digit accuracy.
Furthermore, for very dilute solutions in order of $0.01\rm ng/\mu l$ or
lower, the SC measurement of transmittance never gives reliable answers, while
the CC measurement leads to highly reliable measurement of transmittance and
distinguishability between the very diluted solutions. Also, by obtaining the
transmittance, one can calculate the refractive index by considering the
proper model for the transmittance vs refractive index and parameters sample
yielding to 3-digit accuracy. In contrast to the quantum spectroscopy, the
method needs no monochromators or narrow-band filters for the samples with
absorption frequency far from signal-photon wavelength 810nm. Moreover, by
choosing a convenient model for concentration vs refractive index or
transmittance, one may estimate and obtain the concentration of solutions with
$\rm 0.1pg/\mu l$ precision.

Interestingly, it seems that the method has potential to be improved. If one
can use the coincidence counter with more narrower time-window, for example in
order of ps instead of current ns, using the deep learning methods for noise
cancellation, and using a source with higher rate such as PPKTP Sagnac-based
source (see Ref. [6] and references therein) in which $\Delta N/\bar{N}$ is
usually in order of $10^{-5}$, one may expect to measure the transmittance
based on the CC measurement with at least 5-digit, which may yield to very
more accurate refractive index or concentration.

Finally as outlooks, the method has potential to open a new platform to high-
precision quantum transmittometry of different types of samples. Also, we are
hoping it opens a new pave in the next future to non-invasive quantum
diagnosis of cancer tissues like brain tissues or rat, Retina, Liver tissues
and Stem Cells as an application of quantum measurement based on entangled
photons in the context of quantum biology.

## AUTHOR CONTRIBUTIONS

A.M.F. proposed the primary idea. A.M.F and S.A.M. developed the idea of using
DNA and MB solutions, respectively, and conceived the experiment, built the
experimental setup, experimental measurements and analyzed the data. A.M.F.
and S.A.M. performed the theoretical calculation and numerical analysis (via
MATLAB software), respectively. All authors contributed to prepare the
manuscript. A.M.F. wrote the manuscript, revised and answered to the referees’
comments during the process.

## Acknowledgments

We express our gratitude to ICQTs. We also express our gratitude to Dr. M.
Abbasi who performed the graphical works related to DNA/Methylene via AVOGADRO
software. We also thank Dr. M. Akbari because of his useful comments on
quantum spectroscopy. Furthermore, we express our gratitude to Dr. Neda
Shirin, manager of Neda Chemistry Ltd., and also to Mr. F. Alidoust and Dr. P.
Zardari for preparing DNA and Methylene solution samples, respectively, and
their discussions on biological/chemical properties of these solutions.
Finally, we thank Prof. A. Rostami, Prof. M. H. Naderi and Prof. R. Roknizadeh
because of their useful comments and discussions. We also thank the Dr. N. S.
Vayaghan who is also the supervisor of the quantum optics Lab, and currently
the equipment is at his disposal. Finally, we thank the Dr. J. Sabet and Dr.
M. Aaram.

## Disclosures

The authors declare no conflicts of interest.

## References

  * [1] Y. Shih, “Entangled biphoton source-property and preparation," Rep. Prog. Phys. 66, 1009 (2003).
  * [2] R. Boyd, Nonlinear Optics, (Academic Press, 4th edition, 2020).
  * [3] G. Agrawal, Nonlinear Fiber Optics, (Academic Press, 5th edition, 2012).
  * [4] S. P. Walborna, C. H. Monken. S. Padua, P. H. Souto Ribeiroa “Spatial correlations in parametric down-conversion," Phys. Rep. 495, 87 (2010).
  * [5] D. S. Simon, G. Jaeger, and A.r V. Sergienko, Quantum Metrology, Imaging, and Communication, (Springer, Quantum Science and Technology, 2017).
  * [6] S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. L. Pereira, M. Razavi, J. Shamsul Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, “Advances in quantum cryptography," Adv. Opt. Photon. 12, 1012-1236 (2020).
  * [7] J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein-Podolsky-Rosen Paradox Using Momentumand Position-Entangled Photons from Spontaneous Parametric Down Conversion," Phys. Rev. Lett. 92, 210403 (2004).
  * [8] T.z Paterek, A. Fedrizzi, S.n Gröblacher, Th. Jennewein, M. Z., M. Aspelmeyer, and A. Zeilinger, “Experimental Test of Nonlocal Realistic Theories Without the Rotational Symmetry Assumption," Phys. Rev. Lett. 99, 210406 (2007).
  * [9] S. Groblacher, T. Paterek, R. Kaltenbaek, C. Brukner, M. Zukowski, M. Aspelmeyer and A. Zeilinger, “An exp erimental test of nonlocal realism," Nature 446, 871 (2007).
  * [10] D. Dehlinger and M. W. Mitchell, “Entangled photons, nonlocality, and Bell inequalities in the undergraduate laboratory," Am. J. Phys. 70, 903 (2002).
  * [11] A. Motazedifard, S. A. Madani, and N. S. Vayaghan, “Measurement of entropy and quantum coherence properties of two type-I entangled photonic qubits," arXiv:2012.02658v1 [quant-ph](2020).
  * [12] A. Motazedifard, S. A. Madani, J. J. Dashkasan and N. S. Vayaghan, “Nonlocal realism tests and quantum state tomography in Sagnac-based type-II polarization-entanglement SPDC-source," arXiv:2012.04086v1 [quant-ph] (2020) .
  * [13] M. Genovese, “Research on hidden variable theories: A review of recent progresses," Phys. Rep. 413/6, 319 (2005).
  * [14] P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs," Phys. Rev. Lett. 75, 4337 (1995).
  * [15] K. A. Forbes, J. S. Ford, and D. L. Andrews, “Nonlocalized Generation of Correlated Photon Pairs in Degenerate Down-Conversion," Phys. Rev. Lett. 118, 133602 (2017).
  * [16] Y-H. Kim, S. P. Kulik, and Y. Shih, “Quantum Teleportation of a Polarization State with a Complete Bell State Measurement," Phys. Rev. Lett. 86, 1370 (2001).
  * [17] S. Pirandola, J. Eisert, C. Weedbrook, A. Furusawa, and S. L. Braunstein, “Advances in quantum teleportation," Nature Photon 9, 641 (2015).
  * [18] S. Lloyd, “Enhanced sensitivity of photodetection via quantum illumination," Science 321, 1463 (2008).
  * [19] E. D. Lopaeva, I. Ruo Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, “Experimental Realization of Quantum Illumination," Phys. Rev. Lett. 110, 153603 (2013).
  * [20] M. Genovese, “Real applications of quantum imaging," J. Opt. 18 073002 (2016).
  * [21] L. M. Procopio, O. Rosas-Ortiz, and V. Velázquez, “Spatial correlation of photon pairs produced in spontaneous parametric down-conversion," AIP Conference Proceedings 1287, 80 (2010).
  * [22] A. I. Ruíz, R. Caudillo, V. M. Velázquez, and E. Barrios, “Using Spatial Correlations of SPDC Sources for Increasing the Signal to Noise Ratio in Images," J. Phys.: Conf. Ser. 839, 012025 (2017).
  * [23] Y. Shih,“The physics of ghost imaging,"  arXiv:0805.1166v5 [quant-ph] (2008).
  * [24] X. Y. Zou, L. J. Wang, and L. Mandel, “Induced Coherence and Indistinguishability in Optical Interference," Phys. Rev. Lett. 67, 318 (1991).
  * [25] M. Genovese, “Experimental Quantum Enhanced Optical Interferometry," arXiv:2101.02891v1 [quant-ph] (2021).
  * [26] D. N. Klyshko, “Use of two-photon light for absolute calibration of photoelectric detectors," Sov. J . quantum Electron. 10, 1112 (1980).
  * [27] P.-A. Moreau, J. Sabines-Chesterking, R. Whittaker, S. K. Joshi, P. M. Birchall, A. McMillan, J. G. Rarity and J. C. F. Matthews, “Demonstrating an absolute quantum advantage in direct absorption measurement," Sci. Rep. 7, 6256 (2017).
  * [28] P. R. Tapster, S. F. Seward, and J. G. Rarity, “Sub-shot-noise measurement of modulated absorption using parametric down-conversion," Phys. Rev. A 44, 3266 (1991).
  * [29] Zh. Zhao, K. A. Meyer, W.m B. Whitten, and R. W. Shaw, “optical Absorption Measurements with Parametric Down-Converted Photons," Anal. Chem. 80, 7635 (2008).
  * [30] L. Shi, E. J. Galvez, and R. R. Alfano, “Photon Entanglement Through Brain Tissue," Sci. Rep. 6, 37714 (2016).
  * [31] S. Mamani, L. Shi, T. Ahmed, R. Karnik, A. Rodriguez-Contreras, D. Nolan, and R. Alfano, “Transmission of classically entangled beams through mouse brain tissue," J. Biophoton. 11, e201800096 (2018).
  * [32] S. Mamani, L. Shi, D. Nolan, R. Alfano, “Majorana vortex photons a form of entangled photons propagation through brain tissue," J. Biophoton. 10, e201900036 (2019).
  * [33] E. J. Galvez, L. Shi, B. Khajavi, J. Castrillon, and R. R. Alfano, “Transmission of Polarization-Entangled Photons Through Brain Tissue," Frontiers in Optics 2017, OSA Technical Digest (online) (Optical Society of America, 2017), paper JTu2A.5.
  * [34] S. M. Reyes, D. A. Nolan, L. Shi, and R. R. Alfano, “Special classes of optical vector vortex beams are Majorana-like photons," Opt. Commun. 464, 125425 (2020).
  * [35] S. Golovynskyi, I. Golovynska, L. I. Stepanova, O. I. Datsenko, L. Liu, J. Qu, and T. Y. Ohulchanskyy, “Optical windows for head tissues in near-infrared and short-wave infrared regions: Approaching transcranial light applications," J. biophoton. 11(12), e201800141 (2018).
  * [36] A. Crespi, M. Lobino, J. C. F. Matthews, A. Politi, Ch. R. Neal, R. Ramponi, R. Osellame, and J. L. O’Brien, “Measuring protein concentration with entangled photons," Appl. Phys. Lett. 100, 233704 (2012).
  * [37] J. N. Tinsley, M. I. Molodtsov, R. Prevedel, D. Wartmann, J. Espigulé-Pons, M. Lauwers and A. Vaziri,“Direct detection of a single photon by humans," Nat. Commun. 7, 12172 (2016).
  * [38] M. B. Nasr, D. P. Goode, N. Nguyen, G. Rong L. Yang, B. M. Reinhard, B. E. A. Saleh, and M. C. Teich, “Quantum optical coherence tomography of a biological sample," Opt. Commun. 282,1154 (2009).
  * [39] Ch. Jin, B. Yu, J. Zhang, H. Wu, X. Zhou, H. Yao, F. Liu, X. Lu, L. Cheng, M. Jiang, N. Wu, “Methylene blue photochemical treatment as a reliable SARS-CoV-2 plasma virus inactivation method for blood safety and convalescent plasma therapy for the COVID-19 outbreak," europepmc (2020).
  * [40] A. Einstein , B. Podolsky , and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?," Phys. Rev. 47, 777 (1935).
  * [41] D. N. Klyshko, Photon and Nonlinear Optics (New York: Gordon and Breach Science, 1988); A. Yariv, Quantum Electronics (New York: Wiley, 1989).
  * [42] N. Boeuf, David A. Branning, I. Chaperot, E. Dauler, S. Guerin, G. S. Jaeger, A. Muller, and A. L. Migdall, “Calculating characteristics of noncollinear phase matching in uniaxial and biaxial crystals," Opt. Eng. 39(4), (2000).
  * [43] F. A. Laudenbach, Engineering Spectrally Pure Quantum States with SPDC using Periodically Poled Crystals and Pulsed Laser Sources, (University of Vienna, Master thesis, 2015).
  * [44] D. N. Nikogosyan, “Beta barium borate (BBO)," Appl. Phys. A 52, 359 (1991).
  * [45] Ch. Zhong, X. Han, H. X. Tang, and L. Jiang,Phys. Rev. A 101, 032345 (2020); A. Motazedifard, M. H. Naderi, and R. Roknizadeh, J. Opt. Soc. Am. B 32, 1555 (2015); A. Motazedifard, M. H. Naderi, and R. Roknizadeh, J. Opt. Soc. Am. B 34, 642 (2017); D. F. Walls and G. J. Milburn, Quantum optics (Springer-verlag-1994).
  * [46] F. T. Arecchi, A. Berne, and P. Bulamacchi, “High-Order Fluctuations in a Single-Mode Laser Field," Phys. Rev. Lett. 16, 32 (1966); T. Stangner, H. Zhang, T. Dahlberg, K. Wiklund, and M. Andersson,“Step-by-step guide to reduce spatial coherence of laser light using a rotating ground glass diffuser," App. Opt. 56 (19), 5427 (2017);
  * [47] A. Motazedifard, A. Dalafi, F. Bemani, and M. H. Naderi, “Force sensing in hybrid Bose-Einstein-condensate optomechanics based on parametric amplification," Phys. Rev. A 100, 023815 (2019).
  * [48] J. C. Sutherland, and K. P. Griffin, “Absorption Spectrum of DNA for Wavelengths Greater than 300nm," Radiation Research 86, 399-409 (1981).
  * [49] M. Bass, G. Li, and E. V. Stryland, Handbook of optics (Vol. IV, 3rd edition, sponsored by Optical Society of America, McGraw-Hill 2010). Also, see https://refractiveindex.info/?shelf=glass$\&$book=soda-lime$\&$page=Rubin-clear
  * [50] R. J. J. Rioboo, M. Philipp, M. A. Ramos, and J. K. Krüger, “Concentration and temperature dependence of the refractive index of ethanol-water mixtures: Influence of intermolecular interactions," Eur. Phys. J. E 30, 19 (2009);
  * [51] X. Zhu, T. Mai, Z. Zhao, “Relationship between Refractive Index and Molar Concentration of Multi-Component Solutions," 4th International Conference on Machinery, Materials and Information Technology Applications 2016.
  * [52] G. Brida, M. Genovese, and C. Novero, “An application of two-photon entangled states to quantum metrology," J. Mod. Opt. 47, 2099 (2000).