Partially reflecting jamming objects in correlation-enhanced target detection with entangled photons

Quantum illumination (QI)^1–3 is a general name for a class of target detection protocols where an entangled photon state is used to illuminate a region of space where a reflecting target may be present. In the case of interest, a weakly reflecting target is detected against a strong background. A source feeds entangled photons to two channels called signal (“s”) and idler (“i”), with the signal radiation being sent to the target. The receiver collects both the background radiation and the signal reflected from the target and recombines them with the retained idler radiation. The entanglement itself, as a property of the radiation state, is generally hard to detect,⁴ but under QI, it manifests via correlations in frequency, phase, and arrival time between the photons in signal and idler channels, increasing the sensitivity of the target detection protocol.^5,6 These correlations have a greater impact the larger the number of radiation modes filled by the source, provided there is no correlation between different noise modes. A striking feature of quantum illumination is that the benefit of entanglement survives in a lossy communication channel, and a performance enhancement of 6 dB (four times) over any classical radar of the same transmitted power is possible.⁷ Initially applied to entangled photon pairs,² QI has been further expanded^7,8 to the more general case of Gaussian states,⁹ which are easier to prepare and manipulate.

The impact of thermal background radiation is especially pronounced in the microwave range, where the benefits of quantum illumination have been experimentally demonstrated for radars—first with the entanglement between a microwave photon and an optical one⁶ and then solely with the microwave entangled photons.¹⁰ In the optical range, demonstrated applications of quantum illumination include increasing the robustness of quantum key distribution (QKD) toward eavesdropping,⁵ improving radar sensitivity for target detection,¹¹ and correlation-enhanced imaging,^12,13 including light-sensitive applications. Another important approach is quantum optical coherence tomography,¹⁴ where a Hong–Ou–Mandel interference is used to map the object under study. However, in all the previous studies, the background radiation has been considered independently from the illumination state.

Any particular implementation of the QI protocol defines a specific set of properties of entangled states available for correlation analysis.¹⁵ For radar applications, the phase information can be omitted, as the phase of the target return state is typically a random variable.¹⁶ While having a resolution in frequency unfolds the full advantage of the QI protocol,⁵ in many works, only the classical time correlations between photocounts are utilized.^17,18 When only time information is used, a purely classical time-of-flight system is a strong alternative to QI. Classical systems based on attenuated infrared femtosecond lasers demonstrate centimeter resolution depth imaging of low reflectivity objects in daylight at stand-off distances of the order of 1 km.¹⁹ Practical applications of QI in photonics are mainly obstructed by the low efficiency of entangled photon sources. One known solution is to perform time-correlation-based detection with classical pseudo-thermal sources, sacrificing maximum achievable contrast in favor of the brightness of the source. A thorough discussion on the topic is available in the context of ghost imaging.²⁰ 

Here, we study the time-correlation-based target detection with entangled photons and investigate the impact of a partially reflecting jamming object such as a beam-splitter, located between the illumination source and the main target, on the QI protocol. We show that if a part of the signal radiation is sent back to the receiver and preserves the correlation with the stored idler channel, this reflected radiation becomes a source of noise to which QI is vulnerable. This jamming reflectance can come from back reflections from optical elements or any semi-transparent objects in the beam path such as a cover glass in microscopy, etc. We compare the double beam (DB) detection, where the coincidence counting is performed between signal and idler channels, with the single beam (SB) detection, where only the photons in the signal channel are used.

The manuscript is organized as follows: In Sec. II, we introduce the density matrices describing the radiation and evaluate the asymptotic minimal error probability p_err(N) as a function of the number of trials N using the quantum Chernoff bound^8,21 for SB and DB protocols. For a quantitative analysis, we introduce the average minimal number of trials (N_min) necessary to prove the target’s presence or absence. We first consider the SB protocol in detail and then reproduce the results for the DB one. The section is concluded with a simplified description of the photon counting detection that is applicable to the experimental part. In Sec. III, we perform correlation-enhanced target detection with photon pairs in the optical range and determine the experimental value of N_min. In Sec. IV, we compare analytical and experimental results for SB and DB protocols and outline their potential applications. The  Appendix contains additional details on the experimental data processing.

Following the original work by Lloyd,² let us consider a target detection protocol where a source feeds entangled photons in a state given by a density matrix ρ_IN to the d pairs of conjugated signal (“s”) and idler (“i”) radiation modes, and the signal photon is used to illuminate a region of space where a reflecting target can be present. In addition to the signal radiation, thermal noise is admixed at the detection stage. In contrast to the previous studies, a part of the radiation in the signal channel sent to the target is reflected back to the receiver by the jamming object. Our goal is to investigate the influence of the jamming reflectance on differentiating with statistical significance between the two hypotheses: target present (H₁) and target absent (H₀), as illustrated in Figs. 1(a) and 1(b), respectively. The radiation incident on the target and on the jammer is described by the density matrices ρ_T and ρ_J, respectively, which can differ due to decoherence, losses, and re-distribution of the signal and idler modes. In this work, we focus on temporal modes, when ρ_T and ρ_J are disparate matrices due to different distances to the target and to the jammer.

We thus restrict our consideration to (a) a low number of photons, ignoring all the twin beam components except the photon pairs; (b) low reflectances of both the target $η$ and the jammer $ζ$⁠; and (c) the case of uncorrelated background noise, with thermal radiation as a key example. For SB, the density matrix is obtained from Eq. (1) by taking the partial trace over the idler channel: $ρINSB=TriρINDB$⁠.

Here, we have introduced a real-valued indistinguishability parameter μ quantifying the orthogonality between the matrices $ρT′≡ρT−ρT00|0〉〈0|$ and $ρJ′≡ρJ−ρJ00|0〉〈0|$⁠; 0 ≤ μ ≤ 1 from a Cauchy–Schwartz inequality. If μ = 0, the target and jammer return states are mutually orthogonal, the observer can distinguish between them, and the jammer does not obstruct the target detection. The larger the value of μ, the more impact the jammer has on the detection protocol. At μ = 1, the target and the jammer become indistinguishable: $ρJ′=ρT′$⁠. For pure states, μ is simply the squared norm of the scalar product between the state vectors. Note that for SB and DB protocols, the values of μ are generally different due to the presence or absence of information about the radiation in the idler channel: μ^SB ≠ μ^DB. In many applications, SB inherently implies the indistinguishability (μ^SB = 1) between radiation reflected from the target and from the jammer.

The low-noise regime for SB is achieved when η ≫ μ^SBζ + (bd/Trρ′), and the detected photon is most probably the one reflected from the target. It is thus sufficient to detect just one photon, and $NminSB$ is equal to $NminSB=1/pH1SB≈1/(ηTrρ′)$⁠. In the high-noise regime [η ≪ μ^SBζ + (bd/Trρ′)], both H₀ and H₁ hypotheses result in comparable numbers of photon detection events, and $NminSB$ scales as $NminSB∝(μSBζ+(bd/Trρ′))/(η2Trρ′)$⁠.

The results of this section are summarized in Table I and in Sec. IV below.

Our experimental set-up for measuring the value of N_min is shown in Fig. 2. We use type-I frequency-degenerate spontaneous parametric down-conversion (SPDC)²² in a 0.25 mm-thick bismuth triborate $[Bi(BO2)3]$ crystal. The pump source is a continuous-wave laser diode (403 nm) with 10 mW of power at the crystal. The beam diameter of the crystal is 2 mm. After the crystal, pump radiation is removed with a dichroic mirror and a long-pass filter. In the idler (reference) arm, the radiation is coupled into a multimode 105 μm fiber. In the signal arm, we have a beam splitter, which provides a jamming signal, and a target (a metal mirror) in a Sagnac interferometer. For photon detection and timing, we use a time-correlated single-photon counting (TCSPC) system based on silicon avalanche photodiodes (Aurea LynXea; 40% quantum efficiency at 800 nm, 360 ps jitter, 25 ns dead time). The signal coming from the jammer is centered at ∼6.5 ns and can be clearly separated from that from the target, located at ∼13 ns. A set of neutral density filters can be introduced before (T₁) and after (T₂) the beam splitter to individually tune target (η) and jammer (ζ) effective reflectances. Reflectance (R) and transmittance (T) of the beam splitter were obtained by fitting the number of single counts as a function of T₂, resulting in T = (52 ± 5)%. We also used the idler channel as a reference for monitoring the laser power to compensate for long-term drifts in the signal channel. We do not perform phase-sensitive detection, as the phase of the target return state at optical wavelengths is typically a random variable for atmospheric target detection tasks.¹⁶ 

Experiments have been performed in two different configurations. For jammed detection (“J”), target on/off states are obtained by opening or blocking the interferometer path. The beam splitter acts as a jammer with an effective reflectance of ζ = RT₁, and the jamming signal is present in both target-on and target-off states. The first target return signal from the interferometer corresponds to a target with an effective reflectance of η = T₁T₂T²T_cav. Here, T_cav is the transmittance of the interferometer cavity, excluding the beam splitter. For non-jammed detection (“NJ”), the interferometer path is always blocked, and the beam splitter acts as a target with an effective reflectance of η = RT₁. The “Target off” state corresponds to the detector path being completely blocked.

Here, we assume a high event generation rate and neglect the uncertainty in the number of generated events—this assumption is valid when the number of trials is large: N ≫ 1. Event rate f₀ was measured in the “NJ” configuration with no filters installed, and the correction for the beam-splitter reflectance was applied. For the single-beam protocol, the event rate was the number of photons registered in the signal channel per unit time: $f0SB=3708±40$ Hz. For the double beam protocol, it was the coincidence counting rate $f0DB=202±10$ Hz in a coincidence window of Δt = 3.0 ns full width around the target return peak at ∼6.5 ns.

In our experiment, μ^SB = 1, as no information is available on the arrival time when only the signal channel is used, and μ^DB = 0, as photons reflected from the target and from the jammer have different time delays with respect to the idler channel [illustrated in Fig. 4(a)]. The value of μ^DB can be changed by altering the size of the coincidence window during the post-processing, as is shown at the end of this section and in Fig. 3(c). The number of distinct modes being detected (d) is determined by the histogramming step in the data processing. We estimate d ≈ 10² given the width of the coincidence window of $≈2$ ns and the total number of bins equivalent to 200 ns. The effective value of b is mostly determined by the dark noise of the detectors (∼150 Hz). We should note the moderate stability of our laser diode, as illustrated in Fig. 4(a) of the  Appendix in the SB case by the spread of the total counts. The minimal observation time interval τ_min was then determined by measuring the detection rates of single (SB) and coincidence (DB) photons and fitting the statistical errors as described in  Appendix and Fig. 4.

In Sec. II, we have shown that quantum illumination illustrated by the DB protocol softens the low-noise regime boundary by a factor of (Trρ′/d) ≪ 1, where d is the number of modes uniformly filled by the source and registered by the detector. This stems from the availability (under DB) of the noise-free idler channel, which increases the dimensionality of the full Hilbert space available to the system. The performance of both protocols is dependent on the types of states (ρ_J) and (ρ_T) used (similar to Ref. 15) and on the emission rate of the source described with the parameter Trρ′ ≡ (1 − ρ₀₀) ≪ 1. The ratio (Trρ′)/Tr(ρ′²) affects the crossover between the low-noise and the high-noise regimes, taking values from 1/2 for pure states to (d^q/Trρ′) ≫ 1 for fully mixed states, where q = 1 for SB and q = 2 for DB. In the low-noise regime, the minimal number of trials, N_min, is the same for all the protocols and scales as η⁻¹. In the high-noise regime, N_min scales as η⁻² with target reflectance.

The presence of a jamming object (ζ) strongly affects both protocols. It is particularly noticeable in the high-noise DB regime, where for highly reflective jammers (μ^DBζ ≫ b), the increase in N_min equals $μDBζTr(ρ′2)b(Trρ′)2/d≫1$⁠. The performance of the protocols is strongly dependent upon whether or not the detector can distinguish between the radiation returned from the jammer (ρ_J) and from the target (ρ_T). Quantitatively, this is described with the indistinguishability parameter μ [defined in Eq. (15)], which generally differs for SB and DB protocols due to the presence or absence of information about the radiation in the idler channel: μ^SB ≠ μ^DB. When ρ_T and ρ_J can be fully distinguished (μ = 0), the influence of the jammer can be fully avoided, and the number of trials is unaffected by ζ. Otherwise, a partially reflecting target acts as an effective background even under the DB detection protocol, and for low background radiation levels [b Trρ′/Tr(ρ′²) ≪ 1], DB has the same performance as SB.

In the non-jammed case, both SB and DB techniques were operating in the low-noise regime, as illustrated in Fig. 3(a) by the linear behavior of N_min as a function of the inverse target reflectance (1/η). Without attenuation, when T₁ = 1 (and thus η = R = 0.48), the corresponding minimal numbers of trials were equal to 2.11 ± 0.02 (SB) and 2.27 ± 0.07 (DB), which simply showed that the photon is reflected to the detector in 50% of the cases. Figure 3(a) is a good example of the inaccuracy of approximate Eqs. (10), (33), and (35). The deviation from the straight line predicted by Eq. (33) for SB protocol is due to both the non-zero noise level and the difference between Eqs. (9) and (10). The non-zero noise contribution to SB is due to the detector’s dark counts. The use of exact Eqs. (9), (19), and (26) results in a better fit, as shown by the solid black lines in Figs. 3(a) and 3(b). For the DB protocol, the observed noise level is low, and both fitting procedures yield identical results. Using the fit results, we estimate the number of thermal photons per mode to be equal to b ∼ 10⁻⁴ and (b/d) · Trρ′ < 10⁻⁹.

Changing to the jammed mode [Fig. 3(b)], under DB detection, the N_min does not depend on the jamming reflectance ζ, showing that DB still operates in the low-noise mode. SB detection gives a linear dependence of N_min on ζ, corresponding to the high-noise detection regime. When the jammer reflectance ζ is low, both protocols give similar values of N_min, which are close to the case of non-jammed detection: $NminSB(η=0.005,ζ=0.01)=158±11$⁠, $NminDB(η=0.005,ζ=0.01)=144±44$⁠. We thus demonstrate that the jammer increases the effective noise level if the detector cannot differentiate between the radiation from the target and from the jammer, i.e., when μ^SB = 0.

The different performance of protocols SB and DB illustrates the impact of the jammer on correlation-enhanced target detection. This impact can be mitigated by increasing the distinguishability between the radiation coming from the jammer and the target. In our current set-up, the advantage of the DB protocol over the SB protocol reaches $NminSB/NminDB=20$ for η = 0.005 and ζ = 0.52 and can be greatly increased by increasing the ζ/η ratio, e.g., for dimmer targets.

We have demonstrated that a partially reflecting jamming object introduces noise to correlation-enhanced target detection. In contrast to the background noise, the radiation reflected from the jammer is correlated with the stored idler beam. We have introduced the indistinguishability parameter μ (0 ≤ μ ≤ 1) between the states of radiation incident on the target and on the jammer. The value of μ is given by Eq. (15) and is, in a general case, different for SB and DB protocols: μ^SB ≠ μ^DB. To mitigate the impact of the jammer, the value of μ needs to be minimized.

We have established a measuring routine for the expected value of the minimal number of trials (N_min) required to differentiate with statistical significance between target absent and target present hypotheses. Results on N_min are summarized in Eqs. (9), (19), and (26), and the asymptotic behavior is given in Table I. If the detection protocol is incapable of differentiating between the jammer and target reflected radiation, the performance of the protocol degrades. It is particularly noticeable in the high-noise DB regime, where for highly reflective jammers (μ^DBζ ≫ b), the increase in N_min equals $μDBζTr(ρ′2)b(Trρ′)2/d≫1$⁠. Experimentally, we obtained an improvement of $NminSB/NminDB=20$ between SB and DB protocols for η = 0.01 and ζ = 0.5. This should increase further for dimmer targets. We have shown that a partially reflecting object can change the detection mode from low-noise to high-noise and the asymptotic behavior of N_min. Our consideration also holds for a full quantum illumination protocol, where correlations in frequency and phase between signal and idler channels are used.

Further research will be directed toward investigating the impact of jamming objects on correlation-based detection with more practical Gaussian states, following Refs. 7, 8, and 15. Another promising route lies in the analysis of multimode detection, where the numbers of radiation modes filled by the source, the background noise, and the detector are different. The asymptotic behavior of N_min by simultaneously processing the data from both SB and DB protocols can be used in the design of target detection protocols and for applications such as imaging or scanning microscopy, similar to quantum optical coherence tomography,¹⁴ especially under low illumination levels such as in photosensitive materials.

This work has been supported by the Finnish Scientific Advisory Board for Defense (MATINE) under the project “Covert optical detection under adverse weather conditions” and by the Flagship on Photonics Research and Innovation (PREIN) program managed by the Academy of Finland. The present work has been conducted at the Micronova Nanofabrication Center at Aalto University. This work has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 882135-BrightNano-vdW. We acknowledge the reviewers for their valuable comments during the peer-review stage.

The authors have no conflicts to disclose.

V. V. Kornienko: Conceptualization (supporting); Data curation (lead); Formal analysis (lead); Funding acquisition (supporting); Investigation (lead); Methodology (lead); Project administration (equal); Resources (supporting); Software (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). C. Vidal: Methodology (supporting); Validation (equal); Writing – original draft (supporting); Writing – review & editing (supporting). A. Pönni: Methodology (supporting); Validation (equal); Writing – original draft (supporting). M. Raasakka: Conceptualization (supporting); Formal analysis (supporting); Methodology (equal); Validation (supporting); Writing – original draft (supporting). I. Tittonen: Conceptualization (lead); Formal analysis (supporting); Funding acquisition (lead); Methodology (lead); Project administration (lead); Resources (lead); Supervision (lead); Validation (supporting); Writing – original draft (supporting); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

We then calculated the mean $fdiff̄$ and standard deviation σ_f values for the differential rate over all the time intervals for a given τ [Fig. 4(b)]. According to the distinguishability criterion (32), we defined the minimal observation time interval τ_min as the value of τ when $σfτ=τmin$ reaches the mean value of differential rate $fdiff̄τ→∞$⁠, which was calculated for the whole τ range available (300 s). To obtain τ_min values, we fitted the values of the standard deviation σ_f according to Eq. (31): $σf∝1/τ$ [Fig. 4(c)].