# Demonstration of quantum-enhanced rangefinding robust against classical
jamming

M. P. Mrozowski  R. J. Murchie  J. Jeffers  J. D. Pritchard
<EMAIL_ADDRESS>Department of Physics and SUPA,University of
Strathclyde, Glasgow G4 0NG, United Kingdom

###### Abstract

In this paper we demonstrate operation of a quantum-enhanced lidar based on a
continuously pumped photon pair source combined with simple detection in
regimes with over 5 orders of magnitude separation between signal and
background levels and target reflectivity down to -52 dB. We characterise the
performance of our detector using a log-likelihood analysis framework, and
crucially demonstrate the robustness of our system to fast and slow classical
jamming, introducing a new protocol to implement dynamic background tracking
to eliminate the impact of slow background changes whilst maintaining immunity
to high frequency fluctuations. Finally, we extend this system to the regime
of rangefinding in the presence of classical jamming to locate a target with
an 11 cm spatial resolution limited only by the detector jitter. These results
demonstrate the advantage of exploiting quantum correlations for lidar
applications, providing a clear route to implementation of this system in
real-world scenarios.

Optical lidar is a pivotal technology for achieving precise target detection
and rangefinding with high spatial precision [1, 2], utilised in a range of
applications from performing ground surveys [3], monitoring sea levels [4], to
aiding navigation in autonomous vehicles [5]. Under conditions necessitating
low-light levels and a substantial background, arising from low target
reflectivity, environmental noise, or deliberate jamming, classical lidar
techniques fail to discern between signal and noise photons leading to
diminished signal-to-noise ratio and an inability to detect targets
confidently.

Significant progress has been made towards exploiting lidar at the single
photon level [6] enabled by advances in detector technologies and
computational analysis to enable 3D imaging using single-pixel detection [7]
or single photon cameras [8] suitable for operating in adverse backgrounds [9,
10], however typically these devices operate using strong modulated classical
light sources for target illumination to compensate for low return
probability. In contrast, quantum-enhanced illumination [11] offers a
compelling alternative approach, whereby the utilization of non-classical
heralded photon sources affords the exploitation of coincidence detection
techniques, enabling effective background photon suppression without temporal
modulation of the signal source [12, 13, 14].

The original framework for quantum illumination proposed by Lloyd [15]
demonstrated that by exploiting entanglement it was possible to out-perform
classical systems, with an extension to Gaussian state analysis bounding the
maximum quantum advantage to 6 dB assuming an unknown optimal measurement [16,
17]. Measurement protocols have been proposed offering up to 3 dB advantage
[18, 19], whilst a detection scheme able to exploit the full quantum advantage
[20, 21] remains a significant technical challenge. Experimental
demonstrations of quantum illumination with phase-sensitive detection have
been performed in-fiber with noise added at the detectors [19], with recent
extensions to operation in the microwave domain compatible with radar
applications [22, 23, 24].

Figure 1:  An overview of quantum enhanced lidar based on simple detection. a
A weak classical light source illuminates a weakly reflecting target, which is
embedded within a noisy background environment. A detector is unable to
distinguish between reflected signal photons or background photons, leading to
suppressed signal to noise and increased time required for confident target
detection. b In quantum-enhanced lidar the source is replaced with a non-
classical photon pair source. One photon from each pair is detected locally
(idler) and the signal is sent to probe the target. Strong temporal
correlation between signal and idler can be exploited to perform coincident
detection (shown in c) to both eliminate counts due to background (thereby
enhancing the SNR) and providing distance information from the delay at which
a temporal correlation is observed.

A much simpler approach is to exploit the temporal correlations arising from
photon pairs generated using spontaneous parametric downconversion using
either pulsed or continuous (CW) sources [25]. Experimental demonstrations
have shown enhancement in the signal-to-noise ratio by exploiting these
correlations [26, 27, 12, 13], offering robust operation with respect to
classical jamming [12, 14] and first demonstrations of rangefinding [28, 13,
29]. Recently, using dispersion compensating fibers a further enhancement in
quantum detection was demonstrated by spreading the noise across multiple
time-bins whilst preserving the temporal correlation of the two-photon
coincidence to achieve 43 dB times enhancement in SNR at noise levels up to
three times the signal level [30]. Additionally, detector multiplexing [31]
permits multi-mode range-finding that can enable covert operation using light
that is spectrally and statistically indistinguishable from the background
[29].

In this paper we present experimental demonstration of quantum enhanced lidar
using a CW source of heralded photon pairs with simple detection. Using the
log-likelihood analysis framework [32, 33] we characterise the performance of
our detector, demonstrating operation of the quantum lidar in regimes with
over 5 orders of magnitude separation between signal and background levels and
target reflectivity down to -52 dB. This corresponds to up to 30 dB
enhancement over classical illumination, or 17 times faster target
discrimination to achieve a comparable error rate. Using this system, we
demonstrate the robustness of our quantum enhanced lidar approach to classical
jamming, implementing a dynamic analysis procedure to track slow variations in
background noise and immunity to high frequency fluctuations. Finally, we
extend this system to perform rangefinding, implementing moving target
discrimination with a 11 cm spatial resolution, limited by the timing jitter
of our detectors. These results demonstrate the advantage of exploiting
quantum correlations for lidar applications, and provide a clear route to
realistic use of this system in a scenario comparable with real-world
operations.

##  I Results

###  I.1 Quantum Enhanced lidar

A schematic illustration of classical lidar is shown in Fig. 1a with a
classical light source sending light towards a target, and a detector used to
collect the return signal reflected from the target. The target is embedded
within a noisy background, which here we treat as Poissonian but could be
described by any underlying mode distribution such as thermal light. In the
limit where the noise level is large compared to the returning signal (either
due to using a weak source for covert operation or due to a target at large
distance or with low reflectivity) this leads to a significant reduction in
signal to noise as it is not possible to discriminate between background and
signal photons at the detector, increasing the time required to reach a
confident detection. In this setup, a continuous source can be used to
determine if there is a target, but to obtain information regarding range it
is also necessary to introduce temporal modulation of the source, for example
pulsed operation or amplitude modulation. In a low noise regime this
modulation reduces the covertness of the illumination source, whilst an
uncooporative target can easily jam or spoof the detector by sending an
artificially delayed or modulated signal.

In the quantum enhanced lidar scheme shown in Fig. 1b the light source is
replaced by a non-classical of photon pairs which are generated via
spontaneous parametric down conversion (SPDC) leading to pairs being created
at random times. One photon from each pair is detected locally (the idler),
whilst the signal photon is directed at the target, enabling use of
coincidence detection to discriminate between signal detector counts due to
background photons and those arising from signal reflection (as illustrated in
Fig. 1c). Crucially in this regime we are not exploiting entanglement, but
instead the strong temporal correlations intrinsic to the pair generation
process to suppress background counts, offering a significant improvement in
SNR and a robustness to jamming that we will demonstrate below. Additionally,
the coincident detection permits range-finding without temporal modulation of
the laser, making it much harder to spoof or intercept than classical
techniques.

Figure 2:  a Experimental setup used to conduct the experiment, here the ppKTP
crystal is pumped with 405 nm CW light to generate an 810 nm photon pair in
the process of SPDC Type-II, polarising beam splitter (PBS) is used to
separate signal & idler pair. The idler photons are detected locally, while
the signal photons experience loss passing though a neutral density (ND)
filter before detection. Together with signal, noise is injected into the
signal detector. b-d Measurement in which signal is experiencing 33.5 dB loss
in presence of 1 MHz background noise corresponding to SNR${}_{\text{CI}}$ of
-37.9 dB. In b detection events registered by the signal detector are shown,
while c and d show LLV analysis applied to coincidence detection and classical
detection respectively. In e-g we repeat the measurement after increasing the
loss to 52 dB and reducing the SNR${}_{\text{CI}}$ to -51.5 dB, here to
compensate for the low signal return of 7.1 $\pm$ 0.9 Hz we increase the
integration time per measurement to 1 s. Following the established convention
we show signal detector countrate in e, LLV analysis with 150 measurement
moving average applied to quantum enhanced f and classical g data.

###  I.2 Log-Likelihood Framework

In both target detection and range finding the aim is to evaluate, based on a
finite set of measurements, whether or not a target is present. This reduces
to a state discrimination problem of deciding whether the measured statistics
correspond to the target being present (hypothesis 1 - H1) or absent
(hypothesis 0 - H0). Prior analysis of this problem has focused on the
fundamental bounds given an unknown optimal measurement scheme [15, 16] or
formulated this as a probe transmission estimation problem whereby the
uncertainty of the transmission estimation is described by the Cramér–Rao
bound [13, 14]. In either approach, a fundamental issue lies in defining the
threshold at which to consider a target present or absent.

In this work we use an alternative analysis protocol based on the log-
likelihood value (LLV) $\Lambda$ defined as [32, 33]

$\Lambda(x,k)=\ln\left(\frac{P_{\mathrm{H1}}(x,k)}{P_{\mathrm{H0}}(x,k)}\right),$
(1)

where $x$ represents the measured detector count data, $k$ is the number of
trials and $P_{\mathrm{H{1,0}}}(x,k)$ represents the the probability that the
target is there (or not) given $x$ clicks after $k$ trials. When calculating
$\Lambda$ in the case of quantum illumination (QI) $x$ is the number of
detected coincidence events in a measurement time $T$ for a given coincidence
window $\tau_{c}$, and $k$ is the number of idler clicks, whilst for classical
illumination (CI) $x$ is the number of signal detection events and
$k=T/\tau_{c}$.

An advantage of using the log-likelihood ratio is that it gives a natural
threshold value of $\Lambda=0$ that is independent of the system parameters,
providing a degree of self calibration in the system. If $\Lambda<0$ it is
more likely that the target is absent, whilst for $\Lambda>0$ it is more
likely that a target is present. To evaluate the error associated with making
a decision on wheter a target is present using this threshold, we introduce
the distinguishability
$\phi=1-\left[(1-P_{\mathrm{D}})+P_{\mathrm{FA}}\right]$ where
$P_{\mathrm{D}}$ is the probability of correctly detecting a target that is
there and $P_{\mathrm{FA}}$ is the false alarm probability caused by
incorrectly detecting a target when there is not one present. Analytically
these probabilities are evaluated by integrating over the underlying LLV
probability distributions associated with H1 and H0 respectively, whilst
experimentally we evaluate distinguishability by calculating the fraction of
data points with $\Lambda>0$ when the object is present or absent.

###  I.3 Quantum-Enhanced Detection

The accurate identification of target objects probed with lidar systems is
challenged by loss and noise. Loss encompasses attenuation due to absorption,
scattering, reflection, and transmission, leading to weakened signal return.
Noise involves unwanted signals from detectors (dark counts), environmental
factors (thermal background, jamming), and system imperfections, introducing
inaccuracies and distortions. Addressing these challenges often requires
advanced signal processing, sensor design, and calibration techniques [34].
Here we demonstrate the performance of our simple lidar system using single
photon counting modules based on single photon avalanche diodes (SPADs) for
detection.

A schematic of the setup used to conduct the experiment is shown in Fig. 2a.
Here a photon pair source generates photons at 810 nm using type-II SPDC from
405 nm CW pump in a ppKTP crystal. Details of the source characterisation can
be found in Methods section. During the experimental trials the resulting
photon pair is separated on a polarising beam splitter and the idler photon is
detected directly.

Noise is injected to the system using an 810 nm LED driven by a low-noise
current driver to provide a constant noise independent of target reflectivity
or position. This noise is combined with the attenuated signal and coupled
into a single photon detector. To emulate controllably the loss associated
with finite target reflectivity in the object present (H1) case, the signal
photons pass through a calibrated neutral density filter. For the target
absent case (H0) a beam block is placed in the signal path, allowing only the
noise to be coupled into the signal detector.

Fig. 2 shows the performance of the lidar system for detecting a stationary
target operating under two distinct loss regimes of 33.5 and 52 dB,
approaching values typically encountered in real lidar systems ($\geq$ \- 50
dB) [29]. In both cases, the system operates in a regime with an average
background count rate of 1 MHz, and using a coincidence window of $\tau_{c}=2$
ns.

For the data shown in Fig. 2b-d with a loss of 33.5 dB, the crystal is pumped
at 50 $\mu$W giving a pair production rate of $377\pm 5$ kHz. For the object
present case, this gives an effective signal return rate of $167\pm 1$ Hz,
corresponding to a classical signal to noise of
$\mathrm{SNR}_{\mathrm{CI}}=-37.9\pm 0.1$ dB. To evaluate the quantum SNR
ratio, we take the ratio of the measured coincidence rate with target present
and noise source turned off, against the number of accidental coincidences
recorded with the noise source enabled and target absent. For this data we
find values of $39.1\pm 0.4$ Hz and $200.2\pm 0.5$ Hz respectively giving
$\mathrm{SNR}_{\mathrm{QI}}=-7.1\pm 0.1$ dB. We acquire 3050 consecutive
measurements using an integration time of $T=0.1$ s for both target present
and target absent scenarios. Figure 2b shows the raw signal counts measured in
each case, showing that the additional signal counts with target present are
indistinguishable compared to the $\sim 1000$ count standard deviation of the
background counts.

Figure 3: Characterising system performance for data presented in Fig. 2b-d. a
shows the experimental and theoretical distinguishabilities $\phi$ as a
function of number of averages $N_{\mathrm{av}}$ for both CI and QI data and
theory, with a peak value of $\phi_{\mathrm{QI}}=0.995\pm 0.003$ for an
integration window of 50 (corresponding to 5 s). b shows the experimental and
theoretical receiver-operator curve (ROC) for the optimum average window
length of $N_{\mathrm{av}}=50$. For comparison, CI requires a $17\times$
longer averaging time to achieve an equivalent performance to QI as indicated
by the dashed line. The discrepancy between CI theory and data is due to the
sensitivity to small drifts in background noise level across the measurement
window.

To apply the LLV analysis to the data, the single shot probability
distributions $p_{\text{H}_{1,0}}$ are estimated using the first 50
measurements of each case (see Methods). From this, the LLV $\Lambda(x,k)$ for
each data point can be calculated, resulting in single shot
distinguishabilities of $\phi_{\mathrm{QI}}=0.31\pm 0.01$ and
$\phi_{\mathrm{CI}}=0.086\pm 0.003$ respectively, demonstrating the
enhancement in detection performance using QI. To enhance further the
distinguishability we perform a rolling window average with
$N_{\mathrm{av}}=50$ to smooth the data. Figure 2c-d shows the corresponding
averaged LLV data for QI and CI, which clearly reveals that despite the
relatively small change in signal level on the detector the QI enhanced lidar
is able to discriminate between target present and target absent robustly,
with the corresponding averaged $\Lambda$ values never crossing zero. The
classical LLV however is significantly noisier, with both cases frequently
crossing the detection threshold leading to significant error if using this
for discrimination.

Figure 4:  Two jamming experiments, in a-c we apply slow modulation after
estimating the H1 & H0 probabilities with static background for 200
measurements (grey vertical dashed line) and in d-f we apply fast modulation
of the background and estimate the probabilities with dynamic background (grey
vertical dashed line). Both experiments were performed under identical
conditions of 33.5 dB loss, 2.3 MHz average background, 0.3 MHz modulation
amplitude and integration time of 0.1 s. Here in a and d signal detector count
rates are plotted. In b and e LLV analysis applied to our quantum enhanced
experimental data is shown, here on plot b the inset shows the LLV analysis
without dynamic background tracking, and the main plot shows the results with
us dynamically tracking the background and correcting for the accidental
coincidence counts resulting from increasing/decreasing the background. In c
and f results of classical LLV analysis were plotted. This results show that
quantum enhanced lidar is robust to classical jamming.

In Fig. 2e-g the loss is increased to 52 dB, with the crystal pump power
increased to 150 $\mu$W to increase the pair rate to $1.13\pm 0.02$ MHz whilst
maintaining the same level of background noise. In this regime the effective
signal return rate in the object present case reduces to $7.1\pm 0.9$ Hz,
corresponding to a classical signal to noise of
$\mathrm{SNR}_{\mathrm{CI}}=-51.5\pm 0.6$ dB. The coincidence counts rates
with target present and noise disabled equal to $1.8\pm 0.1$ Hz and for target
absent with noise engaged equal to $577\pm 1$ Hz respectively, leading
$\mathrm{SNR_{QI}}=-25.1\pm 0.2$. As the system is now operating with a signal
over 5 orders of magnitude smaller than the background noise, the integration
time is increased to $T=1$ s for these measurements. As before, Fig. 2e shows
that the raw signal counts are indistinguishable, however using a rolling
window of $N_{\mathrm{av}}=150$ from Fig. 2f,g it is clear that the QI LLV is
able to discriminate between target present and absent cases despite these
challenging operating parameters whilst the CI LLV is entirely unreliable,
with distinguisbabilities of $\phi_{\mathrm{QI}}=0.67\pm 0.22$ and
$\phi_{\mathrm{CI}}=0.33\pm 0.02$ for the window-averaged data.

To characterise system performance, we perform further analysis of the data
presented in Fig. 2b-d for 31.5 dB loss. In Fig. 3a the experimental and
analytical distinguishability are plotted as a function of average window size
$N_{\mathrm{av}}$. These data reveal the for $N_{\mathrm{av}}=50$ (indicated
by the dotted line) the quantum-enhanced lidar reaches a maximum
distinguishability $\phi_{\mathrm{QI}}=0.995\pm 0.003$ in excellent agreement
with theory, whilst the classical system only achieves
$\phi_{\mathrm{CI}}=0.63\pm 0.02$. Extending to longer integration times we
observe the data for both CI and QI begins to deviate from the theoretical
predictions due to long term drift in both the noise source and pump laser
intensities.

Using this optimal average window size, in Fig. 3b we present the experimental
and theoretical receiver operator curve (ROC) obtained by scanning the
threshold value of $\Lambda$ at which a target is deemed to be present. For
the QI data we obtain excellent agreement with theory, and at $\Lambda=0$ this
corresponds to a false-alarm probability of
$P^{\mathrm{QI}}_{\mathrm{FA}}=5\times 10^{-4}$, whilst for CI
$P^{\mathrm{QI}}_{\mathrm{FA}}=0.25$. To quantify the performance advantage
offered by the quantum enhanced lidar in this regime we analytically find the
number of averages required for the CI ROC curve to match the observed QI
data, shown as a black dashed line on the Figure. This analysis shows that for
equivalent performance the classical detector would need to integrate
17$\times$ more data, demonstrating a significant performance advantage when
operating in regimes of significant loss and high background.

###  I.4 Resilience to jamming

Classical jamming of lidar systems refers to intentional interference aimed at
disrupting the operation of lidar technology, as well as operation in
environment where level of background noise fluctuates. Intentional jamming
techniques involve emitting strong modulated light or laser signals, or
deploying countermeasures to confuse or overwhelm the lidar sensor. The
objective of intentional jamming of lidar systems is to hinder accurate data
gathering, compromise situational awareness, or impede target detection and
classification. Such jamming activities can lead to impaired perception and
navigation capabilities in autonomous vehicles and other lidar-dependent
applications, potentially resulting in hazardous scenarios. In the following
we demonstrate the resilience of quantum enhanced lidar to dynamic jamming
using both slow and fast modulation of the background noise. For both of these
experiments the target loss was set to 33.5 dB, pump power to 50 $\mu$W,
$T=0.1$ s, with an average background of 2.3 MHz modulated with an amplitude
of 0.3 MHz.

Figure 4a-c shows the effect of applying a slow sinusoidal background
modulation after an initial period of constant background which is used to
estimate the single shot probabilities $p_{\text{H}_{1,0}}$. In the inset of
Fig. 4b we show the QI LLV evaluated assuming the initial constant background
data, which shows that whilst we maintain a clear separation between the QI
LLV for object present and absent case, the modulation is visible in the data.

To mitigate this effect we implement dynamic background tracking by using the
raw-signal data to estimate the average background level associated with each
measurement (valid in this regime where
$\bar{n}_{\mathrm{bg}}\gg\bar{n}_{\mathrm{sig}}$). Using our model (see
Methods), we create a look-up table (LUT) of probabilities
$P_{\text{H}_{1,0}}$ for different background levels $\bar{n}_{\mathrm{bg}}$
whilst keeping all other parameters constant. For each measurement we then use
the raw signal counts to assign the appropriate probability distribution when
calculating the single shot LLV. This pre-calculated LUT approach can be used
to enable real-time implementation in future experiments using the signal
count-rate to track the background. The resulting QI LLV is shown in the main
plot of Fig. 4b, which has now eliminated the modulation and shows the QI
lidar can be made immune to slow jamming with a single shot distinguishability
of $\phi_{\mathrm{QI}}=0.15\pm 0.03$ despite the 26 % background modulation.
For comparison, Fig. 4c shows the classical LLV (for which no background
correction is possible) is completely unable to distinguish between the two
regimes and has been spoofed by the jamming signal.

In Fig. 4d-f we perform a second experiment where a fast white noise source is
now added to the slow classical modulation, resulting in a pseudo-random noise
level seen from the signal on the number of signal counts in d. As with the
slow modulation, the QI LLV is immune to the fast noise whilst the classical
LLV data is entirely washed out with the fast noise causing the CI LLV to
average to zero making it unable to distinguish if a target is present. These
data highlight the advantage of QI not only in performing faster
discrimination for a given average signal level, but also in providing a
system robust to jamming.

Figure 5: Quantum-Enhanced Rangefinding. We apply a modulated background noise
and translate the target from an initial location of $x_{A}=0$ to $x_{B}=11$
cm after 1000 measurements, moving to $x_{C}=22$ cm after 1500 measurements. a
Raw signal counts showing noise modulation is constant independent of target
location. b-d QI LLV with coincidence channel delays set to probe targets at
$x_{A,B,C}$ respectively. These results show the QI can perform confident
range finding even in the presence of a significant background.

###  I.5 Rangefinding

The main application of lidar based systems involves use of time-of-flight
detection to estimate the distance to targets. In this section we extend our
quantum enhanced lidar to demonstrate active rangefinding even in the presence
of classical noise.

The modified experimental setup used for rangefinding is shown in Fig. 5a. In
order to simulate variations in target position, a mirror serving as the
target was positioned on a motorized translation stage enabling the target to
be moved a total range of 22 cm. We define three locations separated by 11 cm
intervals, denoted as $x_{A}$, $x_{B}$, and $x_{C}$, and assign three parallel
coincidence detection channels with delays of $\tau_{x_{A}}$ = 1.77 ns,
$\tau_{x_{B}}$ = 2.52 ns, and $\tau_{x_{C}}$ = 3.27 ns corresponding to the
round-trip time to each location. As above, the experiment was performed with
loss of 33.5 dB and 50 $\mu$W pump power, but with an average background of
0.1 MHz background and jamming amplitude of 10 kHz. To achieve better
resolution and mitigate the cross-talk between target positions due to the 250
ps jitter of our detectors, the coincidence window $\tau_{c}$ was set to 0.2
ns.

During the experiment we acquire data for each coincidence channel in
parallel, with target moved from an initial position of $x_{A}$ to $x_{B}$
after 1000 measurements and finally to $x_{C}$ after 1500 measurements. The
results are shown in Fig. 5b-e, where as before the raw signal counts are
indistinguishable for both target presence and position whilst the
corresponding QI LLV channels calculated for $N_{\mathrm{av}}=50$ clearly
reveal the quantum-enhanced lidar is able to resolve robustly the location of
and hence track the target as it moves between the three distinct locations,
despite there being no active modulation of the source, and in the presence of
intentional classical noise.

Note that since the time-reference for range finding comes exclusively from
the heralding of the idler photon, the CI detection method is unable to
provide any rangefinding information. Further, the SPDC generation of photon
pairs from a CW pump provides robustness against spoofing due to the
covertness of illumination, which appears to the target as a weak thermal
source within the environmental background.

##  II Discussion

In conclusion, we have presented experimental demonstration of a quantum-
enhanced lidar system utilising a log-likelihood analysis framework for target
detection and rangefinding, robust to classical jamming and dynamic background
changes in challenging regimes of high background and low signal rates. This
work significantly expands beyond prior research on quantum lidar based on
correlated photon sources[12, 29, 13, 14, 28, 30], specifically by operating
in the regime of both large environmental backgrounds (SNR $<$ -50 dB) and low
signal returns ($>$ 50 dB attenuation), compatible with realistic lidar[29].
Our observations reveal a significant improvement in the signal-to-noise ratio
of up to 30 dB when comparing classical and quantum lidar, with the ultimate
limitation in quantum gain related to the second order correlation of the pair
source $g_{s,i}^{2}(0)$ [12]. This improvement represents an approximate 13 dB
increase in SNR compared to the findings in [12], and is comparable to the
results reported in [13]. In this regime we show QI can provide high fidelity
target discrimination on timescales 17$\times$ faster than classical
techniques. In future, further enhancements can be achieved by employing more
sophisticated detection techniques such as dispersion compensation tailored to
pulse sources[30].

Our quantum-enhanced lidar approach is resilient against both sinusoidal and
white-noise modulated classical jamming, and we present a new protocol for
active background tracking to reduce sensitivity to slow drifts or intentional
spoofing attempts, while remaining immune to high-frequency fluctuations.
Specifically, this technique works optimally in the regime of high background
where the instantaneous count-rate measured on the signal detector provides a
real-time probe of the background rate which can then be used to improve the
resilience against noise when analysing small changes in the measured
coincidence rate.

Applying these techniques in a range-finding modality, we demonstrate the
ability not only to perform target detection but also determine the location
of the target in the presence of active jamming. Currently we demonstrate a
resolution of 11 cm, limited only by the timing jitter of the room temperature
SPADs motivated by the ability to incorporate these into a low power, portable
device. Enhanced performance is possible using superconducting nanowire
detectors offering significantly reduced timing uncertainty [35, 36, 37] at
the cost of requiring a cryogenic cooling system.

While the ability to achieve centimeter-level resolutions with room
temperature SPADs is promising for practical applications, it is important to
acknowledge the fundamental limitation that long-range or low-reflectivity
targets require a sufficient photon flux for at least one return photon within
the experiment window. This makes detection of uncooperative targets at
distance challenging, with the demonstrated -52 dB loss comparable to that
expected from a Lambertian scatterer at 15 m using a 10 cm diameter telescope.
Nonetheless, our results demonstrate that quantum-enhanced lidar offers a
practical speedup in time to detection, allowing for operation at lower light
levels, while presenting a low-intensity random thermal signal to observers.
We also note that the detection of cooperative targets could, even with the
results presented here, be accomplished at significant distances.

Compared to classical lidar methods that rely on amplitude-modulated pulses
for rangefinding, our use of continuous-wave sources mitigates the risk of
being spotted and spoofed by observers as peak pulse intensity increases.
These results emphasize the advantages of exploiting quantum correlations for
lidar applications and provide a clear pathway towards the realistic
deployment of this system in scenarios comparable to real-world operations.
Overall, our research contributes to the advancement of quantum-enhanced lidar
technology, paving the way for its integration into practical applications
where enhanced performance and security are crucial factors.

##  III Methods

###  III.1 Experiment Setup

####  III.1.1 SPDC Photon Source

The QI source used in these experiments is a type-II collinear SPDC photon
pair source based on a ppKTP crystal [38], with a polling period of
$10~{}\mu$m. The crystal was pumped using a continuous wave 405 nm pump laser
focused to a $1/e^{2}$ radius of 11 $\mu$m to generate photon pairs at 810 nm.
The source was characterised using a Hanbury-Brown-Twiss configuration [39].
At a pump power $P=0.3~{}$mW, the typical count rates were
$N_{\mathrm{idler}}=0.562$ MHz and $N_{\mathrm{signal}}=0.507$ MHz signal
photons detected using fiber-coupled single-photon counting modules (Excelitas
SPCM-AQRH-14-FC) with a quantum efficiency of $60~{}\%$ at 810 nm operating at
room temperature. Using a $\tau_{c}=2~{}$ns coincidence window we measured the
typical rate of coincidences to be $N_{c}=0.138$ MHz. The second-order
coherence of the signal path conditioned on idler detection was measured to be
$g^{(2)}(0)=0.006$, and the source brightness was $7.55\pm 0.01\times 10^{6}$
pairs/s/mW. For experiments presented above the pump power was varied between
50 and 150$~{}\mu$W.

####  III.1.2 Target Reflectivity and Noise Source

Finite target reflectivity is realised using neutral density filters. To
calibrate the filter loss and thus target reflectivity $\xi$, we take the
ratio of the total signal count-rate with (without) the filter after
subtracting the detector background measured with the SPDC source blocked,
using integration times of up to 1 second to provide sensitivity at the
highest attenuation levels.

To simulate noise in the system, an LED is driven by a low-noise current
driver (Koheron DRV300-A-10) to apply a controllable background count to the
signal detector to provide a constant noise independent of target reflectivity
or position. Use of a low-noise current driver is necessary to make a
comparison between CI and QI detection protocols, as CI is extremely
susceptible to any drift in noise level. For the active jamming presented in
Figs. 4 and 5 a function generator is used to add modulation on top of a DC
offset.

###  III.2 Log-Likelihood Analysis Framework

####  III.2.1 Single-shot detection probabilities

In order to evaluate the log-likelihood ratio, it is necessary to calculate
the underlying probabilities of detection and coincidence events that occur
within a single coincidence window $\tau_{c}$. These probabilities depend on
the average photon number generated in the process of SPDC $\bar{n}$, target
reflectivity $\xi$, the average background on the signal
($\bar{n}_{\mathrm{bg}}$) and idler ($\bar{n}_{\mathrm{bg,I}}$) detectors,
detection efficiency of the signal ($\eta_{S}$) and idler detectors
($\eta_{S}$). The detection efficiencies account for all system loss and
finite quantum efficiency in the limit of a perfect target reflectivity. Below
we extend the analysis framework presented in Ref. [33] to the regime of
detection in the presence of a Poissonian background source, as used in this
experiment setup.

To model the CI case, the probabilities of signal detector firing with the
target present $p^{\mathrm{CI}}_{\mathrm{H1}}$ (or absent
$p^{\mathrm{CI}}_{\mathrm{H0}}$) are given by

$\displaystyle p^{\mathrm{CI}}_{\mathrm{H0}}$
$\displaystyle=1-\exp\left(-\bar{n}_{\mathrm{bg}}\eta_{\mathrm{S}}\right),$
(2a) $\displaystyle P^{\mathrm{CI}}_{\mathrm{H1}}$
$\displaystyle=1-\frac{1}{1+\gamma\eta_{\mathrm{S}}\xi\bar{n}}\exp\left(-\frac{\bar{n}_{\mathrm{bg}}\eta_{\mathrm{S}}}{(1+\eta_{\mathrm{S}}\gamma\xi\bar{n})}\right).$
(2b)

In the QI case, the single shot probabilities reflect the probability of a
signal event within the coincidence time $\tau_{c}$ conditioned on the idler
firing. When the object is absent, the detector only fires due to noise and
the probability of a signal event is equivalent to the CI case with no target.
With target present, it is necessary to derive the click probability by
considering the idler conditioned signal state as a thermal-difference
state[40], resulting in the QI click probabilities given by

$\displaystyle p^{\mathrm{QI}}_{\mathrm{H0}}$
$\displaystyle=p^{\mathrm{CI}}_{\mathrm{H0}},$ (3a) $\displaystyle
p^{\mathrm{QI}}_{\mathrm{H1}}$
$\displaystyle=1-\frac{1}{p^{\mathrm{QI}}_{\mathrm{I}}}\biggl{(}\frac{1}{1+\bar{n}\xi\eta_{\mathrm{S}}\beta}\text{exp}\bigl{(}\eta_{\mathrm{S}}\bar{n}_{\mathrm{bg}}(\frac{\bar{n}\xi\eta_{\mathrm{S}}\beta}{1+\bar{n}\xi\eta_{\mathrm{S}}\beta}-1)\bigr{)}$
$\displaystyle-$
$\displaystyle\frac{(1-p^{\mathrm{QI}}_{\mathrm{I}})}{1+\bar{n}_{\mathrm{I:X}}\xi\eta_{\mathrm{S}}\beta}\text{exp}\bigl{(}\eta_{\mathrm{S}}\bar{n}_{\mathrm{bg}}(\frac{\bar{n}_{\mathrm{I:X}}\xi\eta_{\mathrm{S}}\beta}{1+\bar{n}_{\mathrm{I:X}}\xi\eta_{\mathrm{S}}\beta}-1)\bigr{)}\biggr{)},$
(3b)

where
$\bar{n}_{\mathrm{I:X}}=\bar{n}(1+\eta_{I}\bar{n}_{\mathrm{bg,I}}-\eta_{\mathrm{I}})/(1+\eta_{\mathrm{I}}\bar{n}_{\mathrm{bg,I}}+\bar{n}\eta_{\mathrm{I}})$
is the signal state mean photon number after conditioning from a no click
event at the idler and $p^{\mathrm{QI}}_{\mathrm{I}}$ is the idler firing
probability equal to

$p^{\mathrm{QI}}_{\mathrm{I}}=1-\frac{1}{1+\eta_{\mathrm{I}}\bar{n}+\eta_{\mathrm{I}}\bar{n}_{\mathrm{bg,I}}}$
(4)

Furthermore, in these equations we introduce two additional parameters,
$\gamma$ and $\beta$. These parameters allow for adjustments to match the data
obtained for CI and QI respectively, to account for detector non-linearities
and variations in the heralding efficiency due to changes in pump power and
coincidence window duration.

####  III.2.2 Log-Likelihood ratio

In the limit of $k$ trials it is possible to express the LLV defined in Eq. 1
in a linear form [33] dependent only upon the single-shot probabilities
$p_{\mathrm{H0}}$ and $p_{\mathrm{H1}}$ using $\Lambda(x,k)=Mx+Ck$, where
$M=\log((p_{\mathrm{H1}}*(1-p_{\mathrm{H0}}))/(p_{\mathrm{H0}}*(1-p_{\mathrm{H1}})))$
and $C=\log((1-p_{\mathrm{H1}})/(1-p_{\mathrm{H1}}))$.

For data acquired over an integration time $T$, for the CI LLV is calculated
using $x$ as the number of detected signal events and $k=T/\tau_{c}$
corresponding to the number of trials of duration $\tau_{c}$ within the
integration window. For the QI LLV, $x$ is the number of measured coincidence
counts and $k$ is the number of idler firing events.

####  III.2.3 Analytical Distinguishability

To evaluate the analytical distinguishability it is necessary to calculate the
mean and standard deviation of the underlying LLV distributions corresponding
to target present and absent, and then integrate over this distribution to
find the detection and false alarm probabilities from which $\phi$ is then
calculated.

In the limit where the underlying Poisson count distributions can be modelled
as Gaussian [33] (valid for the data shown in the paper), the LLV
distributions for H1 and H0 also become Gaussian, with mean value
$\mu_{Hi:\Lambda}=M\bar{x}_{Hi}+Ck$ and a standard deviation equal to
$\sigma_{Hi:\Lambda}=M\sigma_{Hi}/\sqrt{N_{\mathrm{av}}}$, where
$N_{\mathrm{av}}$ corresponds to the average window length being considered.

For the case of CI, as above $k=T/\tau_{c}$, with mean signal levels given by
$\bar{x}^{\mathrm{CI}}_{Hi}=kp_{Hi}^{\mathrm{CI}}$ and standard deviation
$\sigma^{\mathrm{CI}}_{Hi}=\sqrt{kp^{\mathrm{CI}}_{Hi}(1-p^{\mathrm{CI}}_{Hi})}$.
For the QI case, we use $k=(T/\tau_{c})p^{\mathrm{QI}}_{I}$ corresponding to
the average number of times the idler fires. The mean coincidence rate is then
$\bar{x}^{\mathrm{QI}}_{Hi}=kp_{Hi}^{\mathrm{QI}}$ with standard deviation
$\sigma^{\mathrm{QI}}_{Hi}=\sqrt{kp^{\mathrm{QI}}_{Hi}(1-p^{\mathrm{QI}}_{Hi})}$.

Defining a threshold LLV value $\Lambda_{c}$, the probabilities of detection
and false alarm can then be evaluated as

$\displaystyle P_{\mathrm{D}}$
$\displaystyle=\displaystyle\int_{\Lambda_{c}}^{\infty}\exp\left(-\frac{(x-\mu_{\mathrm{H1}:\Lambda})^{2}}{2\sigma_{\mathrm{H1}:\Lambda}^{2}}\right)dx,$
(5a) $\displaystyle P_{\mathrm{FA}}$
$\displaystyle=\displaystyle\int_{\Lambda_{c}}^{\infty}\exp\left(-\frac{(x-\mu_{\mathrm{H0}:\Lambda})^{2}}{2\sigma_{\mathrm{H0}:\Lambda}^{2}}\right)dx.$
(5b)

For the distinguishabilities quoted above we use threshold $\Lambda_{c}=0$
throughout the paper, with the ROC curve in Fig. 3b calculated by plotting
$P_{\mathrm{FA}}$ versus $P_{\mathrm{D}}$ for different values of
$\Lambda_{c}$.

###  III.3 Estimating Experimental System Parameters

To obtain parameters relevant for calculating single-shot probabilities using
Eqs. 2\- 3, the detection efficiencies $\eta_{S,I}$ are obtained using the
known source brightness and comparing measured count rates at each detector in
the absence of a simulated target, giving $\eta_{S,I}=0.2329,0.1958$
respectively for $\tau_{c}=2$ ns. For each scenario, background rates
$\bar{n}_{\mathrm{bg,S}}$ and $\bar{n}_{\mathrm{bg,I}}$ are measured with the
SPDC source blocked and only the noise source active. The emitted signal
strength $\bar{n}$ is extracted from the idler detector with the pump active.
This, combined with the calibrated target reflectivity $\xi$, provides the
full set of system parameters. In the analysis above, a subset of data points
(typically the first 100) for the target absent and present cases are compared
against the theoretical model, using free parameters $\beta$ and $\gamma$ to
fit against the measured single shot probabilities. Typically both parameters
yield values close to unity corresponding to correcting only for the slight
change of filter attenuation when replaced in the beam.

###  III.4 Dynamic Background Tracking

For the slow classical jamming data shown in Fig. 4a the system parameters are
initially obtained by performing analysis of the first 200 measurements with
static background $\bar{n}_{\mathrm{bg,S}}$. We then define 25 discrete
background levels in the range 2.1-2.7 MHz, and for each value recalculate the
single shot click probabilities in Eqs. 2 and 3 using a re-scaled value of
$\bar{n}_{\mathrm{bg,S}}^{\prime}$. Subsequently the experimental data
obtained with jamming engaged was analysed by calculating the LLV for each
point in time by using the relevant click probabilities associated with the
level that closest matches the instantaneous signal counts to dynamically
track slow changes in background.

##  IV Data Availability

The datasets used in this work are available at [DOI to be added].

##  V References

## References

  * McManamon [2012] P. McManamon, Opt. Eng. 51, 060901 (2012).
  * Li  _et al._ [2022] N. Li, C. P. Ho, J. Xue, L. W. Lim, G. Chen, Y. H. Fu, and L. Y. T. Lee, Laser & Photonics Reviews 16, 2100511 (2022).
  * Amann  _et al._ [2001] M.-C. Amann, T. M. Bosch, M. Lescure, R. A. Myllylae, and M. Rioux, Optical Engineering 40, 10 (2001).
  * Hooijer and Vernimmen [2021] A. Hooijer and R. Vernimmen, Nature Communications 12, 3592 (2021).
  * Royo and Ballesta-Garcia [2019] S. Royo and M. Ballesta-Garcia, Applied Sciences 9, 4093 (2019).
  * Massa  _et al._ [1998] J. S. Massa, G. S. Buller, A. C. Walker, S. Cova, M. Umasuthan, and A. M. Wallace, Appl. Opt. 37, 7298 (1998).
  * Sun  _et al._ [2016] M.-J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, Nature Communications 7, 12010 (2016).
  * Tachella  _et al._ [2019] J. Tachella, Y. Altmann, N. Mellado, A. McCarthy, R. Tobin, G. S. Buller, J.-Y. Tourneret, and S. McLaughlin, Nature Communications 10, 4984 (2019).
  * Wallace  _et al._ [2020] A. M. Wallace, A. Halimi, and G. S. Buller, IEEE Transactions on Vehicular Technology 69, 7064 (2020).
  * Tobin  _et al._ [2021] R. Tobin, A. Halimi, A. McCarthy, P. J. Soan, and G. S. Buller, Scientific Reports 11, 11236 (2021).
  * Pirandola  _et al._ [2018] S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, Nature Photonics 12, 724 (2018).
  * England  _et al._ [2019] D. G. England, B. Balaji, and B. J. Sussman, Phys. Rev. A 99, 023828 (2019).
  * Liu  _et al._ [2019] H. Liu, D. Giovannini, H. He, D. England, B. J. Sussman, B. Balaji, and A. S. Helmy, Optica 6, 1349 (2019).
  * Liu  _et al._ [2020] H. Liu, B. Balaji, and A. S. Helmy, IEEE Transactions on Aerospace and Electronic Systems 56, 3529 (2020).
  * Lloyd [2008] S. Lloyd, Science 321, 1463 (2008).
  * Tan  _et al._ [2008] S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, Phys. Rev. Lett. 101, 253601 (2008).
  * Shapiro [2020] J. H. Shapiro, IEEE Aerosp. Electron. Syst. Mag. 35, 8 (2020).
  * Guha and Erkmen [2009] S. Guha and B. I. Erkmen, Phys. Rev. A 80, 052310 (2009).
  * Zhang  _et al._ [2015] Z. Zhang, S. Mouradian, F. N. C. Wong, and J. H. Shapiro, Phys. Rev. Lett. 114, 110506 (2015).
  * Zhuang  _et al._ [2017a] Q. Zhuang, Z. Zhang, and J. H. Shapiro, Phys. Rev. Lett. 118, 040801 (2017a).
  * Zhuang  _et al._ [2017b] Q. Zhuang, Z. Zhang, and J. H. Shapiro, Phys. Rev. A 96, 020302 (2017b).
  * Luong  _et al._ [2020] D. Luong, C. W. S. Chang, A. M. Vadiraj, A. Damini, C. M. Wilson, and B. Balaji, IEEE Trans. Aerosp. Electron. Syst. 56, 2041 (2020).
  * Chang  _et al._ [2019] C. W. S. Chang, A. M. Vadiraj, J. Bourassa, B. Balaji, and C. M. Wilson, Appl. Phys. Lett. 114, 10.1063/1.5085002 (2019).
  * Barzanjeh  _et al._ [2020] S. Barzanjeh, S. Pirandola, D. Vitali, and J. M. Fink, Science Adv. 6, eabb0451 (2020).
  * Yang  _et al._ [2021] H. Yang, W. Roga, J. D. Pritchard, and J. Jeffers, Opt. Express 29, 8199 (2021).
  * Lopaeva  _et al._ [2013] E. D. Lopaeva, I. Ruo Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, Phys. Rev. Lett. 110, 153603 (2013).
  * Balaji and England [2018] B. Balaji and D. England, in  _ICCST 2018_ (2018) pp. 1–4.
  * Rarity  _et al._ [1990] J. G. Rarity, P. R. Tapster, J. G. Walker, and S. Seward, Appl. Opt. 29, 2939 (1990).
  * Frick  _et al._ [2020] S. Frick, A. McMillan, and J. Rarity, Opt. Express 28, 37118 (2020).
  * Blakey  _et al._ [2022] P. S. Blakey, H. Liu, G. Papangelakis, Y. Zhang, Z. M. Léger, M. L. Iu, and A. S. Helmy, Nature Comms. 13, 5633 (2022).
  * Yang  _et al._ [2022] H. Yang, N. Samantaray, and J. Jeffers, Phys. Rev. Appl. 18, 034021 (2022).
  * Murchie  _et al._ [2021] R. J. Murchie, J. D. Pritchard, and J. Jeffers, in  _Quantum Communications and Quantum Imaging XIX_, Vol. 11835, edited by K. S. Deacon and R. E. Meyers, International Society for Optics and Photonics (SPIE, 2021) p. 118350G.
  * Murchie  _et al._ [2023] R. J. Murchie, J. D. Pritchard, and J. Jeffers, arXiv:2307.10785 [quant-ph] (2023).
  * Rapp  _et al._ [2020] J. Rapp, J. Tachella, Y. Altmann, S. McLaughlin, and V. K. Goyal, IEEE Signal Processing Magazine 37, 62 (2020).
  * Taylor  _et al._ [2020] G. G. Taylor, A. McCarthy, B. Korzh, A. D. Beyer, D. Morozov, R. M. Briggs, J. P. Allmaras, B. Bumble, M. D. Shaw, R. H. Hadfield, and G. S. Buller, in  _Conference on Lasers and Electro-Optics_ (Optica Publishing Group, 2020) p. SM2M.6.
  * McCarthy  _et al._ [2013] A. McCarthy, N. J. Krichel, N. R. Gemmell, X. Ren, M. G. Tanner, S. N. Dorenbos, V. Zwiller, R. H. Hadfield, and G. S. Buller, Opt. Express 21, 8904 (2013).
  * Buller  _et al._ [2011] G. S. Buller, N. J. Krichel, A. McCarthy, N. R. Gemmell, M. G. Tanner, C. M. Natarajan, X. Ren, and R. H. Hadfield, in  _Infrared Sensors, Devices, and Applications; and Single Photon Imaging II_, Vol. 8155, edited by M. Razeghi, P. D. LeVan, A. K. Sood, and P. S. Wijewarnasuriya, International Society for Optics and Photonics (SPIE, 2011) p. 81551I.
  * Lee  _et al._ [2016] S. M. Lee, H. Kim, M. Cha, and H. S. Moon, Opt. Express 24, 2941 (2016).
  * Beck [2007] M. Beck, J. Opt. Soc. Am. B 24, 2972 (2007).
  * Horoshko  _et al._ [2019] D. B. Horoshko, S. De Bièvre, G. Patera, and M. I. Kolobov, Phys. Rev. A 100, 053831 (2019).

##  VI Acknowledgements

This project was funded by the UK Ministry of Defence.

##  VII Author Contributions

M.M. and J.P. devised the experiment; M.M. conducted the experiment, R.M. and
J.J. developed the underpinning theoretical framework; J.P., R.M. and M.M.
performed analysis of the data. All authors contributed to the discussion of
results and preparation of the manuscript.

##  VIII Competing Interests

The authors declare no competing interests.

##  IX Additional information

###  IX.1 Correspondence

Correspondence and requests for materials should be addressed to Jonathan D.
Pritchard.