# Noise-Tolerant Object Detection and Ranging Using Quantum Correlations

Hashir Kuniyil  Helin Ozel  Dept. of Electrical and Electronics Engineering,
Özyeğin University, Istanbul, Turkey, 34794  Kadir Durak<EMAIL_ADDRESS>Dept. of Electrical and Electronics Engineering, Özyeğin University, Istanbul,
Turkey, 34794  Hasan Yılmaz  Institute of Materials Science and
Nanotechnology, National Nanotechnology Research Center (UNAM), Bilkent
University, 06800 Ankara, Turkey

###### Abstract

Imaging, detection and ranging of objects in the presence of significant
background noise is a fundamental challenge in optical sensing. Overcoming the
limitations imposed in conventional methods, quantum light sources show higher
resistance against noise in a time-correlation-based quantum illumination.
Here, we introduce the advantage of using not only time correlations but also
polarization correlations in photon pairs in the detection of an object that
is embedded in a noisy background. In this direction, a time- and
polarization-correlated photon pair source using the spontaneous parametric
down-conversion process is exploited. We found that the joint measurement of
correlated pairs allows distinguishing the signal from the noise photons and
that leads to an improved signal-to-noise ratio. Our comparative study
revealed that using polarization correlations in addition to time correlations
provides improved noise rejection. Furthermore, we show that polarization
correlation allows undoing the detector limitation where high background often
leads to detector saturation.

††preprint: APS/123-QED

Detection of an object which is embedded in a noisy background is a
fundamental problem in sensing and imaging. Conventional techniques explore
the properties of the electromagnetic field to reconstruct the structure of an
object under investigation. This type of detection scheme is limited by the
classical bounds [1]. These limitations are fundamental in the classical
realm, and therefore overcoming its limitations requires non-classical
detection methods [2]. Non-classical correlation measurement has been
recognized as a candidate to surpass classical limits offering advancements in
many technological frontiers such as quantum cryptography [3, 4], quantum
computing [5, 6], quantum imaging [7, 8], quantum ranging [9, 10], and quantum
microscopy [11]. We are specifically looking into the scope of quantum
correlations in detection and ranging of an object under a noisy background,
that is of interest in LIDAR applications.

A quantum-entangled source-based detection scheme called quantum illumination
(QI) enables an exponential improvement in the signal-to-noise ratio (SNR) of
detecting a sample compared to a classical source-based detection within a
noisy environment [12]. QI requires a phase-sensitive joint measurement scheme
that utilizes the spontaneous parametric down-conversion process (SPDC) [13]
to generate photon pairs that have non-classical spatial and temporal
correlations [14]. However, the practical implementation of QI is restricted
with limitations imposed by the losses in the spatial correlations after
scattering by diffusive surfaces in realistic scenarios. Thus, this system is
only appropriate to use for an object that has an optically smooth surface. To
overcome these problems, only non-classical temporal correlation of the pairs
are often employed in QI detection scheme [15, 16, 17, 18]. Although phase-
insensitive scheme does not completely offer the QI’s advantages such as
simple experimental settings, high response time compared to classical systems
[16], and possibility to tune the wavelength of the source [19, 7] (especially
for quantum imaging applications), the quantum phase-insensitive scheme is
still preferable over the classical ones. In phase-insensitive systems, a test
for the value of the second order correlation function ($g^{(2)}$) can be used
to determine if the source is operating in the quantum or classical regime
[15]. If the $g^{(2)}$ value is above the classical regime [20] of 2, it is a
direct indication that the source is operating in the quantum regime and the
correlation of the source is higher than any classical temporal correlation.

In this letter, we use SPDC-generated pairs, correlated both in temporal and
polarization degrees of freedom, as a quantum source. A configurable noise is
externally injected to test the system’s behaviour in the presence of noise. A
test for obtaining $g^{(2)}$ values is carried out by considering only
temporal correlation (TC) of the source and compared to the case of both
temporal and polarization correlations (TPC) in identical settings. Our
results indicate that TPC has higher second order coherence compared to TC in
a noisy environment; therefore, improved resistance can be achieved by using
both temporal and polarization correlations.

Figure 1: The experimental arrangement used for testing the detection schemes
that uses (a) time and polarization correlations and (b) only time
correlations.

A schematic of the experimental setup used for this study is shown in Fig. 1.
The correlated photon pairs generated in a SPDC process are used for the
quantum illumination. Conventionally, these pairs are called signal and idler
photons. For the ease of explanation, we call ”the signal photons” for photons
that are scattered by the target and ”reference photons” for the idler photons
recorded locally. A continuous-wave (CW) narrow bandwidth laser of 405 nm
wavelength and 160 MHz linewidth is used to pump a second-order nonlinear
medium ($\chi^{(2)}$), generating a co-polarized signal and reference photons
in the SPDC process. The twin photons are generated by the SPDC process by
utilizing a type-1 BBO crystal where the signal and the reference have central
wavelengths of 842 nm and 780 nm, respectively. The blue pump is blocked after
the BBO crystal using a long-pass filter with a cut-off wavelength of 750 nm.
Signal and reference photons are also separated by a dichroic mirror with a
cut-off wavelength of 805 nm. The reference photons are directly measured with
the first avalanche photo-detector (APD1) while the signal photons pass
through a polarizing beam splitter (PBS) and a quarter-wave plate (QWP) before
it scatters back over the object (see in Fig. 1a). The horizontally polarized
signal photons become right-hand circularly (RHC) polarized at QWP in the
forward path and returns by left-hand circularly (LHC) polarized after being
scattered back by the object. The back scattering from the object happens at
the $2\pi$ solid angle but only the light that is scattered back withing the
angular range of the illumination optics is collected as the same optics is
used both for back and forth paths. In the return path, the LHC signal photons
become vertically polarized after the QWP consequently reflects at the PBS and
is detected by the APD2. In the time correlated case, which is shown in Fig.
1b, the signal photons are transmitted from the dichroic mirror and then half
of the photons pass through a beam splitter. Half of the back-scattered signal
photons is reflected towards the APD at the beam splitter (BS). However, for a
fair comparison the source intensity in the TC case is increased to obtain the
same number of pair sources for both the TC and the TPC cases. The outputs of
two APDs are fed into a timestamp unit (not shown in Fig. 1) where the timing
information of signals is recorded. An unpolarized and broadband external
noise ($N_{b}$) is injected to study the performance of the system under the
noisy background. In our settings, the quantum state of polarization of the
signal and reference photons are $\lvert H_{s}\rangle$ and $\lvert
H_{r}\rangle$, where $H$ indicates the horizontal polarization. The ’$s$’ and
’$r$’ indices refer to the signal and reference photons, respectively. The
polarization of the signal photons is flipped $90^{\circ}$ in Fig. 1b before
detection at APD2, and turn the combined state of polarization into $\lvert
V_{s}\rangle$ and $\lvert H_{r}\rangle$, where $V$ refers to the vertical
polarization. First, we used an anodized aluminium (reflectance is $0.13\pm
0.03$ [18]) as a target object to test if the detection is possible and the
polarization correlation of the source is maintained after scattering from a
rough surface. Therefore, some amount of polarization state will be sacrificed
even in specular reflection [21, 22]. As a result, the probability of
detection of returned signal photons is extremely low. However, we have
detected a coincidence detection of $200\pm 10$ when the target is positioned
closer (10 cm) to the transmitter (distance from the PBS and object in Fig.
1.(a) with the pump power to be 128 $\mu$m. The anodized aluminium is later
replaced with a semitransparent mirror as a target object to allow the
injection of noise photons externally. An incandescent lamp was used as a
power-tunable broadband unpolarized thermal source to inject noise. Using a
broadband source devises to replicate a more realistic case where the spectral
filtering cannot extinguish the noise. Moreover, such a thermal light source
acts as a proper noise model in a correlation-based detection scheme since it
shows high classical correlations [14]. The noise is unpolarized, which makes
it a difficult scenario to filter out the noise from signal not only in the
presence of only temporal correlations between the signal and reference
photons, but even in the presence of polarization correlations. Here, the
temporal correlation refers to the fact that both the signal and reference
photons are generated almost simultaneously.

Figure 2: The measured signal to noise ratio (SNR) versus the injected noise
$N_{b}$ is plotted for time correlation (TC) (blue dots) and time and
polarization correlation (TPC) (red dots) schemes. The theoretical fit (solid
lines) is made by using equation 1.

In the experimental setup we use only phase-insensitive quantum correlation,
where the signal arm has extra delay with respect to the reference arm. The
timestamp unit is configured to search for joint arrival of photons from both
APDs. If the signal and reference photons statistically coincide at a certain
delay between them it appears on the cross-correlogram. The timestamp unit we
use has a timing resolution (time bin width) of $81$ ps. Therefore, if two
photons arrive within this time interval, that is considered as a coincidence.
In our settings, the signal photons travel $1.2$ m extra path, so a
coincidence is not expected even though both photon pairs are created together
within the 81 ps. To obtain a coincidence in this case, we scan the bin width
until an avalanche of coincidence is observed. In our experimental
configuration, the maximum coincidence value observed, when the pump power is
set to 126, $\mu$m, is $5300\pm 100$ pairs per second with noise-free
settings. We lose some pairs due to the losses in the system and non-unitary
reflection coefficient of the object. These values are set approximately the
same for both TC and TPC (Fig. 1) cases to make a reasonable comparison. To
quantify the quantum correlation exploited by the experimental setup, we
estimate $g^{(2)}$ value from the observed coincidence and singles values of
the signal and reference. The suitable definition of $g^{(2)}$ for our type of
system is identified to be [23, 24, 25]

$\displaystyle g^{(2)}(\tau)=\frac{C(\tau)}{N_{s}N_{r}\Delta tT}$ (1)

where $\tau$ is the timing delay between the signal and reference photons, $C$
the observed number of coincidence at a delay of $\tau$, $N_{s}$ and $N_{r}$
are the detection events per unit time for the signal and reference photons,
respectively. $\Delta t$ is the resolution of the timestamp unit and $T$ is
the duration of the experiment. The maximum $g^{(2)}$ value for the case of TC
and TPC in a noise-free condition is calculated to be $260\pm 16.1$ and
$261\pm 16.2$, respectively. By increasing the noise level, a dramatic
reduction in the $g^{(2)}$ value for both cases are observed. However, in the
TPC case, the reduction is considerably smaller. The classical bound for
$g^{(2)}$ is between 1 and 2 with 2 is an indication of the highest classical
correlation.

Figure 3: Coincidence value showing a sinusoidal trend while rotating the
quarter wave plate, which effectively controls the amount of detected signal
that is reflected from the object. This behaviour is observed only from the
polarized signal photons (since the noise is unpolarized). The increased level
of noise rises the floor height of the sinusoidal plots (minimum coincidence
value is not zero). Therefore, floor height gives rise to the estimation of
the accidental coincidence. The error bars are obtained by assuming the photon
fluctuates by $\sqrt{N}$. Solid lines represent theoretical fit (see
appendix.I for details).

The behavior of signal to noise ratio (SNR) with various input noise levels is
shown in Fig. 2. By consistently increasing the amount of injected noise, we
estimated the $g^{(2)}$ value. With injected noise, a proportional drop in the
SNR values are observed for both TC and TPC settings, with TPC having less
decrease compared to TC. This reduction is influenced by the rate of the
accidental coincidence imposed by the external noise. The ratio of the SNR in
TPC case to TC case is found to be fixed and approximately 2.85. The factor
$N_{s}N_{i}\Delta tT$ in equation 1 is a direct measure of the accidental
coincidence. Accidental coincidences are measured when the signal arm is
blocked before the PBS in Fig. 1a and before the BS in Fig. 1b by allowing the
noise photons to be detected by APD2. Here, note that, $g^{(2)}$ is the total
coincidence value including the accidental ones. Therefore, the signal level
in the SNR is indicated by $g^{(2)}-1$ to subtract the effect of noise. We
observed linear relationship between accidental coincidence value and the
injected noise (see Fig. 4). Therefore, a $g^{(2)}$ value of 261 clearly shows
the quantum nature of the pairs from the SPDC source.

The polarization correlation between the signal and the reference arm, in
addition to the temporal correlations, provides distinguishability in
coincidence measurements between the SPDC and the noise photons. Fig. 3 shows
a sinusoidal behavior of the reflected light from the object due to the
polarization correlations between signal and reference photons. The plot
shifts upward in accordance with the increasing noise. This is realized by
rotating the quarter-wave plate. Combination of PBS and the QWP (see figure
(1)) ensures that only $90^{\circ}$-rotated vertical linearly polarized light
is detected with the APD. In this regard, rotating the QWP in the optical axis
and evaluating the change in coincidence detection allow differentiating the
signal from the noise (hence accidental coincidence). The magnitude of the
sinusoidal plots is $C_{max}-C_{min}$, where $C_{max}$ and $C_{min}$ are the
measure of coincidence when both signal and reference photons are present and
only noise is present cases, respectively. The magnitude reduces as the level
of photons detected is reaching the pinch-off region of the detectors. This
behaviour is attributed to the APDs’ functional limitations when they operate
near the saturation region. To be specific, the detectors will not sense
correlated photons efficiently due to detector’s inability to reset themselves
fast enough when more photons are present [26].

The control of the amount of signal photons in the TPC system opens up a new
way of estimating the presence of noise, leading to the possibility of further
noise rejection. In this context, we introduce a figure of merit parameter
called visibility that can be defined as

Figure 4: Demonstrating the variation of visibility with respect to the
average noise (red). The accidental coincidence increases linearly with the
external noise (blue). Points represent the experimental data and the solid
lines represent the theoretical fit using the equation (3). Error bars in the
accidental coincidence are the order of $\sqrt{N}$. The error in the
visibility is calculated by following the standard error propagation method.

$V=\frac{C_{max}-C_{min}}{C_{max}+C_{min}}$ (2)

In the absence of background noise, $C_{min}$ becomes zero leading to the
visibility value being unity. An estimation of visibility over accidental
coincidence is plotted in Fig.4. The relation of the visibility and the
accidental coincidence can be estimated by following the relation (see the
appendix. I for the derivation):

$V=\frac{C_{corr}(N_{s})}{C_{corr}(N_{s})+2C_{ac}(N_{b})d}$ (3)

where $C_{corr}$ is the coincidences obtained between the signal and the
idler, $N_{s}$ and $N_{b}$ are the number of the signal and noise photons per
unit time, respectively, $C_{ac}$ is the accidental coincidence and $d$ is the
detector’s correction factor (details of the effect of detector parameters on
the visibility is given in the appendix. I). The $d$ parameter is substituted
to the equation to address the change in visibility value due to detector’s
operational change in detection rates close to saturation. The deduced
equation fits with experimental data as shown in Fig. 4. The derived
expression is suitable and useful for finding the visibility value
irrespective of detectors’ characteristic limitations. Previous study shows
that visibility of the single photon detector is highly dependent on the
detector saturation [26] i.e., if the detectors are operating in near or deep
saturation region the actual visibility will be miscalculated. The derived
expression for visibility is applicable in any situation. Because the
saturation of the detector is guided by the dead time of the detector. In the
given expression dead time of the detector is normalized. This calculation
will prevent misleading conclusions based on constraints from the detector’s
internal electronics circuitry in QI detection schemes.

In conclusion, we investigated the prospect of using time and polarization
correlations simultaneously in quantum illumination for sensing an object that
is present in the noisy background. The polarization correlation of the source
is often ignored in the consensus that scattering from an object will lead to
a complete (or substantial) loss of them. Our study shows that the
polarization correlation is maintained even after scattering by a diffusive
surface and it improves the SNR compared to the case where only temporal
correlations are used in the presence of background noise. Although, the
visibility decays with injected noise to the setup, the SNR is improved 2.85
times thanks to the polarization correlations of the pairs. Additionally,
polarization correlation of the signal and reference pairs allows to separate
the signal from the noise. We quantify that and give a useful expression that
also ignores detector parameters such as detector dead time and saturation.
This study suggests the interrogation of polarization entanglement
preservation via violation of Bell’s inequality upon the detection and ranging
of an object in quantum illumination scheme.

## Funding

This work was supported by Scientific and Technological Research Council of
Turkey (TUBITAK) with project number 118E991.

##  I appendix

###  I.1 Quarter wave plate rotation versus sinusoidal behaviour on the
coincidence

In a noise-free setting, the orientation of the principal axis (fast axis) of
the quarter-wave plate defines the maximum and minimum coincidences. Following
Mallus’s law, this condition can be expressed as

$\displaystyle C=C_{max}cos(2\theta)$ (4)

where C is the number of coincidence defined by the rotation of the quarter-
wave plate’s axis, $C_{max}$ is the maximum coincidence value between the
signal and the reference photons; this occurs when the fast axis of the QWP is
at 0, $\pi/2$, $\pi$, $3\pi/2$..etc to linearly polarized signal photons.
Therefore, the coincidence maximum is observed when $\theta=0^{\circ}$ and the
coincidence minimum at $\theta=45^{\circ}$. Intensity of the external noise is
unaffected by the quarter-wave plate’s reference axis ($\theta$) as the noise
is a broadband unpolarized light. This means that the rotation of the quarter-
wave plate will change only the coincidence between signal and idler (not
accidental coincidence).

###  I.2 Detector saturation effect on visibility

The visibility (a contrast parameter) of the polarization correlated
coincidence detection scheme is defined by

$\displaystyle V=\frac{C_{max}-C_{min}}{C_{max}+C_{min}}$ (5)

where $C_{max}$ and $C_{min}$ are the measure of coincidence when both signal
and reference photons are present and only noise is present cases,
respectively. In a practical settings, the $C_{max}$ is the summation of the
coincidence between the signal and reference and the accidental coincidence.
Therefore, the maximum observed coincidence value can be rewritten as
$C_{max}=C_{corr}(N_{s})+C_{ac}(N_{b})$, where $C_{corr}$ is the coincidences
obtained between the signal and idler, $N_{s}$ and $N_{b}$ are the number of
the signal and noise photons, respectively and $C_{ac}$ is the accidental
coincidence. The $C_{min}$ can be equate to the accidental coincidence. After
incorporating these information, Eq. 5 can be rewritten as

$\displaystyle
V=\frac{C_{corr}(N_{s})+C_{ac}(N_{b})-C_{ac}(N_{b})}{C_{corr}(N_{s})+C_{ac}(N_{b})+C_{ac}(N_{b})}$
(6)

turns into,

$\displaystyle V=\frac{C_{corr}(N_{s})}{C_{corr}(N_{s})+2C_{ac}(N_{b})}$ (7)

APD’s parameters also play an important role in the visibility of the source.
It is because that the APDs do not give actual photon count it encounters
above threshold. Figure (1) shows the APD’s output (as photon counts in the
figure) for the various input photons. The experimental results (doted points)
shows a linear trend when less number of photons is sent and it flattens for
high photon number values. The solid line (blue) is an estimated plot
considering APD behaves same regardless of the input value (called actual
count). In the experiment, the intensity of the transmitted signal is constant
and the noise is increased to test for system’s behaviour in different noise
($N_{b}$) conditions. Since the accidental coincidence is a function of the
noise, defined by $C_{ac}=(N+N_{b})N_{r}\tau$, it also follows the trend of
$N_{b}$.

Figure 5: Correction factor versus incident photons

To incorporate this concern we introduce the detector function, called
correction factor, into the Eq. 7 and it turn to

$\displaystyle V=\frac{C_{corr}(N_{s})}{C_{corr}(N_{s})+2C_{ac}(N_{b})d}$ (8)

where d is the detector correction factor which is defined as $d=1/(1-AV\times
t_{d})$, where AV is the actual value observed and $t_{d}$ is the dead time of
the detector. This is consistent with [26], and we observe that correction
factor is the inverse of the effective duty cycle. We have used M/S. Excelitas
make (series no: SPCM-AQRH-TR) actively quenched APD with dead time of 18 ns
as a receiving detector that collect the returned signal from the object. We
have calibrated the APD for input values and found the detector behaviour for
different input photons as shown Fig. 6 The figure shows that if the noise
level (input photons in the figure)is high the detector gives a imprecise
value. That means if we do not account the detectors characteristics the
actual visibility will be miscalculated.

Figure 6: The incident photon versus input photon plot. Red dots are
experimentally obtained photon count rate at the detector. A power meter value
converted into photon number gives rise to input photons count.

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