Quantum Rangefinding

Quantum light generated in non-degenerate squeezers has many applications such as sub-shot-noise transmission measurements to maximise the information extracted by one photon or quantum illumination to increase the probability in target detection. However, any application thus far fails to consider the thermal characteristics of one half of the bipartite down-converted photon state often used in these experiments. We show here that a maximally mixed state, normally viewed as nuisance, can indeed be used to extract information about the position of an object while at the same time providing efficient camouflaging against other thermal or background light.


I. INTRODUCTION
Successfully harnessing the properties of quantum states of light has become one of the greatest promises of quantum optics to revolutionise computation [1][2][3], sensing [4], and metrology [5]. This promise has been realised for many applications in metrology [6,7], sensing [8,9], and partially for computation [10]. However, all applications to date fail to realise any quantum advantage when, as in every realistic system, loss and background are introduced.
It was realised in the early 90's that the energy time correlations of pair photon sources could be used to establish optical communication [11] and rangefinding [12] (LIDAR) in high background situations. These early works exploited the time correlations of photon pairs allwoing heralding of sent photons thus suppressing uncorrelated background, and outperforming weak coherent state pulses when adjusted to equivalent levels of sent photons per pulse, an advantage recently quantified in [13]. However this approach is limited to a maximum of one photon per heralded time gate, constraining us to low loss (< −50dB) scenarios not typically encountered. This was again highlighted by Lloyd in 2008, who extended this to a theoretical framework where light entangled over n modes is used to illuminate a target [14]. This entangled illumination promised suppression of false detection probabilities by a factor of n even if loss is present in the system. Later, this prediction of quantum illumination was softened when comparing the performance of entangled photons to Gaussian [15] and coherent [16] states.
Of course the simple advantage of using multiple entangled modes immediately helps with high loss scenarios as now each mode can contain one heralded photon. However, even with hundreds of modes source brightness is limited to the sub-microwatt region. Classical pulsed sources with thousands (or millions) of photons per pulse thus easily outperform quantum illumination in most scenarios and being single-moded, narrowband filtering can * Now at University of Innsbruck; stefan.frick@uibk.ac.at be used to reduce background. Hence the use of multimoded entangled sources (quantum illumination) needs to be motivated by means other than enhanced signalto-noise ratios (SNRs) or increased contrasts.
For LIDAR this justification can be found in the application of covert ranging or quantum rangefinding. In our protocol, entanglement is not used to improve the SNR compared to single-moded illumination schemes. However, one half of a bipartite entangled or two-mode squeezed state is always in a maximally mixed state, indistinguishable from the state of a single mode of thermal background radiation. If a single spatial mode of spectrally multi-moded background can be replaced with one half of a state produced in spontaneous parametric down-conversion (SPDC), efficient camouflaging can be achieved, if the occupied spectral modes are identical to the ones replaced. Such a broadband state can be tailored using quasi-phase matching [17][18][19] in non-linear crystals such as periodically poled potassium titanyl phosphate (ppKTP). Careful engineering of the poling structure of these crystals can be used to emulate similar spectral behaviour as a single spatial mode from thermal background radiation emitted from the surroundings. This means that a single spectral mode (K = 1, with mode number K) will show the exact photon statistics of a single mode of a truly thermal source in terms of their second order correlation function g (2) . While also K > 1 spectral modes show the same behaviour as thermal background light A proof of this behaviour is given in [20], with experimental verifications presented in [21,22]. However, using a spectrally broadband state for illumination would typically result in high background pollution of the signal, because a wider bandwidth of background has to be accepted by the detectors. Here the energy anti-correlation between signal and idler photons can be harnessed to achieve background suppression (see figure 1). If the locally kept photon is probed with respect to its colour, conservation of energy can be used to predict the colour of its partner, if the frequency of the pump beam is known where ω {p,s,i} denotes the energy of pump, signal and idler photons, respectively. Using this fundamental principle in SPDC the background rate can be filtered by categorising photons and their partner into n colour/frequency channels and only accepting events that are constrained by energy conservation.

II. NON-LINEAR CRYSTAL DESIGN
Quasi phase-matching allows tailoring the spectral properties of down-converted photons through implementation of special poling periods Λ, engineered for specific applications. This makes quasi phase-matching and the non-linear processes using it an extremely powerful tool for quantum optics. Engineering a poling period to emulate the behaviour of thermal background light, as is necessary for efficient camouflaging, is a non trivial task that involves sophisticated poling structures with different periods. For this purpose we developed a specialised software tool capable of predicting joint spectral amplitudes (JSAs) of signal and idler photons from arbitrary poling structures. We want to acknowledge that helpful tools for this task now exist [18,23], however, during the development phase of the crystals used in this work no such tools were available. These computer aided design methods for crystal poling help the experimenter to easily engineer the spectral properties of the down-conversion crystals and are an important contribution to harvesting the full potential of these customisable sources. Resulting joint spectral intensity of a crystal designed with our specialised software. The custom poling of the crystal, comprising a linear chirp from Λ = 9 µm to Λ = 13 µm, results in a phase-matching condition allowing for broadband photons between 700 nm and 950 nm for both, signal and idler. Figure 2 shows the simulation of a poling structure designed with our software. Introducing a chirp in the poling period Λ from 9 µm to 13 µm allows broadband type-II phase-matching which generates photons between 700 nm and 950 nm wavelength from a 405 nm pump laser. These parameters where chosen since high power lasers at 405 nm are ubiquitously available while the longest down-converted wavelength of 950 nm is still detectable with off-the-shelf silicon avalanche photo diodes (APDs). However, the non-linear material would allow for spectrally broader photons.

III. CLASSICAL SIGNAL-TO-NOISE RATIO MODEL
While the "quantum advantage" in [14] was found by comparing the quantum Chernoff bound [24] of a bipartite state entangled over n modes with the state of a single photon. We want to show here that we can infer a similar advantage by simple estimations of background and coincidence rates as well as attenuation (optical gain Q < 1) in the quantum channel of our rangefinder. This is still true when applying a linear detection scheme, where correlations between n frequency channels of signal and idler modes are considered.
Our model sets out to estimate the resulting number of coincident photon pairs S in a time-correlated histogram with bin width ∆t after an integration time T . All the coincidences stemming from signal and idler modes occur in one time bin corresponding to the distance of the target. Hence, the time correlated histogram of photon events corresponds directly to a RADAR/LIDAR waveform in classical systems. We compare S to the number of accidental coincidences N appearing in such a histogram. Events contributing to N can be from a variety of dif-ferent sources. Besides background light our model also considers detector dark counts and imperfect heralding of single photons. Table I shows the different combinations of detector events that are considered in our model and how they are denoted throughout this article. The signal-to-noise ratio (SNR) is then defined by the ratio of the number of photons in the LIDAR waveform's coincidence peaks over the standard deviation of the noise and signal contributions combined where N comprises all noise terms from table I and Poissonian statistics are assumed due to the discrete nature of photon counting. For a full definition of all noise and signal terms please refer to the supplement material. Importantly, it is possible to categorise the noise terms into three different categories. A first category N c ( ) includes all noise terms that are constant with the number of frequency channels n. It comprises all combinations of background light and dark counts from table I. A second category N p ( ), which is proportional to the number of frequency channels n, only considers dark counts in both arms. Lastly, the category N i ( ) is inversely proportional to the number of channels. Since this category contains all combinations of background and unpaired single photons it is typically the largest contribution to the SNR and thereby guarantees the advantage from using the frequency correlations inherent to the down-conversion process. Our overall SNR model can thus be written as From equation (4) the advantage gained by higher channel numbers is immediately visible and is true while n ≤ n opt , with optimal channel number n opt = Ni Np . This behaviour can also be observed in figure 3, where our model is plotted for (a) different attenuations and (b) different background rates. A clear advantage is gained by adding more channels while an effect of "diminishing returns" becomes apparent with increasing channel numbers: The increase of SNR between n = 1 and n = 5 is similar to the one between n = 25 and n = 125 reaching the optimal value at n = n opt . Typical numbers for n opt are > 10 k and occurs when the dark count rate of the detectors surpasses the background rate per channel.

IV. EXPERIMENTAL VERIFICATION
The predictions made by our model are using Poissonian statistics only. Thus, quantum mechanics is not needed to explain the advantage of increased SNR, instead this advantage can be achieved by classical correlations. However, quantum mechanics guarantees the covertness of the rangefinder.
Any practical implementation of our protocol will suffer from technical imperfections causing additional noise in the system. To prove the technical feasibility of our protocol, we here demonstrate it in a lab experiment.

A. Experiment Setup
To verify our model for quantum rangefinding, we devised an experiment that can, at the same time, examine multiple channel numbers under different background and loss conditions.
For this purpose the channel number n = 2 was implemented using a frequency splitting setup realised with a dichroic mirror for both signal and idler photons (see figure 4).
With this setup we can simulate one frequency channel by combining all coincidences between the four detectors or two frequency channels by combining only channels that should contain coincidences constrained by energy conservation All coincidence rates are calculated from a LIDAR waveform generated from a time-correlated histogram of arrival times at the respective single photon detectors. The time tagger used in this experiment is developed in house at the University of Bristol with a resolution of ≈ 50 ps and a maximal event rate of ≈ 1 MHz [25].
Additionally custom electronics were engineered containing the current and temperature control for the pump laser diode and temperature control for the crystal oven stabilised at 40 • C. See supplementary material for details.

B. Methods & Results
First we verify the spectral properties of the custom poling structure of our down-crystal by separately measuring the signal and idler spectra on a single photon spectrometer (see supplement) and by using the two channel frequency splitting setup. By measuring the coincidence rates between all four detectors we can confirm that all correlations are contained within two of the four possible combinations of detectors ( figure 5). This then justifies disregarding coincidences from the other two detector pairings, which arise primarily from background light.
To compare cases of two and one frequency channel the software of our time tagger is capable of calculating time-correlated histograms between all four detectors in real-time. These histograms are then summed up bin by bin, where all four histograms are considered to emulate a one frequency channel solution (5) and the off-diagonal   pairings represent the two channel setup (6). The resulting time-correlated histograms correspond to a LIDAR waveform as can be found in conventional systems. The peak position corresponds to the time-of-flight of the second photon to the target and back and thus can be used to estimate the target distance. By recording 600 of these waveforms we can estimate a distribution of signal peak heights above noise floor ( figure 6). These statistics are then used to calculate the SNR in our system. Since our estimate of the SNR calculated here depends on both the mean and the variance of the measured distributions, we are not allowed to directly infer measurement errors from the collected data. In order to give an accuracy to our measurements, we employ a fitting algorithm (LevenbergMarquardt [26,27]) to approximate the measured distributions with a normal distributions. We have chosen a normal distribution in this case, instead of a Poissonian distribution, to allow for standard deviations that are independent of the mean value. With this we can guarantee a fair comparison between model and experiment, in which technical noise might lead to reduced SNRs. The residual fitting error of mean and standard deviation is then used to give an uncertainty of our measurement. Figure 7 depicts a comparison between our model and the data measured in the experiment. The absolute increase of SNR is predicted with good agreement while the absolute value deviates from the model, especially for higher background rates. This deviation is still well within a < 10% error margin, or even within the error bars of our experiment. For high background rates the deviation is explained by saturating detectors/time tagging electronics.
Most importantly the advantage of using frequency, anti-correlations which occur naturally within the SPDC process, shows a clear advantage by enabling the removal of unwanted background radiation. This is possible despite the target being illuminated with a perfectly mixed/noisy state which is not carrying any information in absence of its partner mode and is consequently undetectable.

V. CONCLUSION & OUTLOOK
In this paper we have shown that multi-mode rangefinding can be used to develop a covert rangefinding system operating at light levels significantly below daylight background and spectrally and statistically indistinguishable from that background.
Our source exploits the energy anti-correlations between the broadband states produced in signal and idler modes and we have constructed and SNR model equivalent to classical narrowband illumination. Although we only demonstrate a two-channel system here, our model einvsages a high number of channels, such as 10 4 would with an aspheric lens (f) into a non-linear crystal (X) after its polarization is conditioned for the correct pump orientation for phase-matching. Subsequently removing the pump light with a long-pass filter (F) the signal and idler photons are split on a polarizing beam-splitter (PBS) and collected into two single-mode fibres using the doublet couplers C1 and C2. A channel number n = 2 is implemented by using a dichroic mirror DM after coupling the locally kept photons back into free space. Depending on their wavelength these photons are then either guided to detector A or B. The other half of the photon pair is injected to a laser range consisting of two mirrors and a corner cube mirror (CC). Finally, the returning photon is coupled into an identical frequency splitting setup using a dichroic mirror and detectors C and D. Attenuations can be simulated by inserting a neutral density filter (ND) into the laser range, while back illumination of the corner cube with an LED can be used to simulate different background rates. allow us to reach emission rates > 10 12 Hz (assuming 0.1 pairs per ∆t ≈ 1ns bin width) leading to return loss tolerances of order 100 dB enabling rangefinding of passive targets in high background. Such a source would still re-tain covertness even at higher pair generation rates due to the thermal nature of the transmitted quantum state.
In this paper we present our work on covert or quantum rangefinding. The use of frequency anti-correlations in imaging systems has lately been proposed in [28], however, the protocol there aims to distinguish the absence or presence of a target and not to measure its distance [29]. This enables to use both temporal and frequency (anti-)correlations to remove background light, which is not possible when no a priori information on the target position is available. If we compare our protocol to classical sources without any temporal correlations, we of course can claim a much higher "quantum" advantage as was done in [13].
At the same time we want to point out that our measurement of frequency correlations are an artefact of a quantum process still observable after tremendous amounts of loss and background pollution. The possibility to use maximally mixed light, in a quantum state that carries no information (or modulation), to illuminate a target and then to reconstruct position information using a measurement realised with a linear detection scheme is only possible using the spectro-temporal entanglement of the photon pair state.  The signal-to-noise ratio is increased between the two tested channel numbers and is accurately predicted by our model.

Signal
The signal term S in our model consists of photon pairs detected in one time bin after integration time T .
with photon pair rate c p and optical loss (gain Q < 1).
Meaning that photons sent towards the target get lost with probability Q. This reduces the detected coincidence rate to c P Q leaving S photon pair events after integration time T .

Proportional Terms
Table I lists only one term that is proportional to the number of detector pairs/frequency bins used in our protocol. Accidental coincidences caused by the detection of dark counts within a bin width ∆t are the only contributors to this noise term. Hence, where c d is the dark count rate of the detectors. Note how the term is dependant on the bin width ∆t since, like all other noise terms, dark counts are a random process. Consequently they have a finite probability to coincide within this bin width. Photon pairs in the signal term S are not dependant on this bin width (as long as the detector jitter is low enough), since they will always coincide. Accidental coincidences between dark counts are proportional to the number of detectors since every detector introduces the same dark count rate to the system.

Constant Terms
We identified six terms in total that contribute a constant amount of noise to our system, independently of the frequency bin number n. These terms are caused by event combinations where one detector click is caused by a dark count event. For example, the combination of a single photon of a detected photon pair in the idler mode and a dark count in the signal mode occurs in a single frequency bin and in n frequency bins.
No physical implementation of a photon pair source will ever reach a unity heralding efficiency η < 1. Hence, every photon pair source will also produce photons who's partner was lost between the photon creation and the detection. These terms are then, similar like background light from the surrounding, contributing to the noise as with unpaired photon rate c s defined at the pair source by the heralding efficiency η = cp cp+cs . Lastly combinations between background light from the environment an dark counts can occur times, where B 0 is the backgrounds spectral density in Hz nm −1 and ∆λ denotes the combined spectral bandwidth of all frequency bins.
Of course these terms also exist for the case where a dark count event was registers in the idler mode. However, the magnitudes of the different terms vary greatly. This is mostly so because it is much less likely to detect a photon from the source (either paired or unpaired) after it has travelled to the target. On the other hand background light is typically much higher on the target facing detectors.

Inversely Proportional Terms
Quantum rangefinding can reduce the typically high background rates associated with broadband single photon detection by only correlating frequency bins that are constraint under energy conservation. Events between detectors violating this condition can not have been produced in the down-conversion process and can hence be omitted. While this mechanic greatly helps with removing environmental background it is also beneficial towards other events emerging from imperfect implementations.
The terms most relevant to the advantage gained by omitting non energy constraint correlations include the typically high environmental background in the idler mode.
denotes contributions from photon pairs in the signal mode and background light in its partner mode. Quantum rangefinding also helps to reduce noise emerging due to a non-unity heralding efficiency η < 1 .
and N s,s = Q c 2 s · ∆t T (A12) account for accidental coincidences between photon pairs and unpaired photons, unpaired photons and photon pairs as well as unpaired photons in both modes. The inversely proportional relation of these terms can be easily seen. For example in N s,s , c s is the rate of unpaired photons in the source. Consequently, cs n unpaired photon rate remains per frequency bin and yields for each detector pair constraint under energy conservation Q cs n 2 · ∆t T events, introducing n photon pairs leaves n · Q c s n 2 · ∆t T = 1 n N s,s . (A13)

Appendix B: Electronics and Temperature Stabilisation
To control the pump laser diode temperature and current as well as the temperature of the down-conversion crystal, we employed custom build electronics integrated in a small form factor of (160 × 220 × 52) mm 3 . These electronics were capable of stabilising the crystal temperature to ± 0.1 • C as shown in figure 8. Stabilisation of both, the down-conversion crystal and the laser diode, are necessary to avoid spectral drifts in the downconverted photons and to guarantee consistent energy correlations between different frequency bins.

Appendix C: Signal & Idler Spectra
To verify the broadband spectral properties of the down-converted signal and idler beams, they were measured independently from each other on a single photon resolving spectrometer. Both spectra are shown in fig-ure 9. The solid backdrops show the calculations performed with our software while the lines in the foreground show the measured spectral density. Originally both spectra were designed to be identical. However, a mismatch between the laser diode wavelength and the pump wavelength the phase-matching was designed for causes a red and blue shift for signal and idler photons, respectively.