Target Detection aided by Quantum Temporal Correlations: Theoretical Analysis and Experimental Validation
11 Target Detection aided by Quantum TemporalCorrelations: Theoretical Analysis and ExperimentalValidation
Han Liu, Bhashyam Balaji,
Senior Member, IEEE,
Amr S. Helmy,
Fellow, OSA, Senior Member, IEEE
Abstract —The detection of objects in the presence of significantbackground noise is a problem of fundamental interest in sensing.In this work, we theoretically analyze a prototype target detectionprotocol, the quantum temporal correlation (QTC) detectionprotocol, which is implemented in this work utilizing sponta-neous parametric down-converted photon-pair sources. The QTCdetection protocol only requires time-resolved photon-countingdetection, which is phase-insensitive and therefore suitable foroptical target detection. As a comparison to the QTC detectionprotocol, we also consider a classical phase-insensitive targetdetection protocol based on intensity detection that is practical inthe optical regime. We formulated the target detection problem asa total probe photon transmission estimation problem and obtainan analytical expression of the receiver operating characteristic(ROC) curves. We carry out experiments using a semiconductorwaveguide source, which we developed and previously reported.The experimental results agree very well with the theoreticalprediction. In particular, we find that in a high-level environ-ment noise and loss, the QTC detection protocol can achieveperformance comparable to that of the classical protocol (thatis practical in the optical regime) but with (cid:39) times lowerdetection time in terms of ROC curve metric. The performanceof the QTC detection protocol experiment setup could be furtherimproved with a higher transmission of the reference photon andbetter detector time uncertainty. Furthermore, the probe photonsin the QTC detection protocol are completely indistinguishablefrom the background noise and therefore useful for covertranging applications. Finally, our technological platform is highlyscalable as well as tunable and thus amenable to large scaleintegration, which is necessary for practical applications. Index Terms —Quantum Lidar, Quantum Radar, QuantumTemporal Correlation, Covert Ranging N OTATION
I. I
NTRODUCTION
The fundamental problem in sensing is to detect an objectin the presence of interference and noise. Different regions ofthe electromagnetic spectrum are exploited for such sensingapplications; those include, radar in the radiofrequency regimeand LIDAR in the optical regime. Current radars/LIDARsare based on signals that can be described by classicalproperties of electromagnetic radiation. These propertiespose some limitations on the sensing performance includingthermal and shot noise. A novel fashion that can be devisedto overcome such performance limits is to utilize non-classical sources of electromagnetic radiation. This is possiblebecause quantum physics allows for other types of statesof electromagnetic radiation that exploit two key propertiesin quantum mechanics, namely the quantization of theelectromagnetic field [1] (where the quantized excitations ν Photon pair generation rate ν b Noise photon detection rate ∆ t Detector time uncertainty ∆ t Intrinsic correlation time ∆ t eff effective time uncertainty τ Observation time η r Total transmission of reference photons η p Total transmission of probe photons T c Length of the temporal detection window t p Probe photon detection time t r Reference photon detection time P p Probe photon detection rate P r Reference photon detection rate P c Coincidence detection rate P d Probability of target detection P fa Probability of false alarmROC Receiver operator characteristicSPDC Spontaneous Parametric DownconversionQTI Quantum Temporal CorrelationCTI Classical Temporal Intensityare referred to as photons ) and quantum entanglement andcorrelations [2] . It has been experimentally demonstratedthat quantum entanglement [3] and correlations[4], [5], [6]that exist within entangled photon pairs could be utilizedto enhance the accuracy of target detection in lossy andnoisy environments. The basic idea of these protocols canbe described as follows: A pair of nonclassical photons ineither the radio or optical frequency regimes are used asthe source, which consists of the probe and the reference photons (also referred to in the literature as signal and idler photons, respectively). The probe photons are sent towards thetarget and the back-reflected ones are collected for detection.The reference photons are stored or immediately detectedlocally. Through analyzing the quantum entanglement, orcorrelation, between the probe and the reference photonwith appropriate measurement scheme, it is possible toreduce the environmental noise effect on the target detectionperformance. This is because environmental noise photonsare not correlated or entangled with the reference photons. Correlation between two particles means that a certain degree of freedomof the two particles always takes correlated values. Quantum entanglementbetween two particles means that the two particles are correlated in differentdegrees of freedom, in such a way that it is not possible to describe each ofthe two particles separately. a r X i v : . [ phy s i c s . op ti c s ] A p r An example of quantum-enhanced target detection protocolsis quantum illumination , which can effectively distinguish thepresence and absence of the target object through analyzingthe entanglement between the reference and the probe light. Ithas been shown that the sensitivity of the quantum illuminationprotocol can surpass the classical limit that is achieved byphase-sensitive homodyne detection [7]. However, the im-plementation of the quantum illumination protocol requiresa significant level of system complexity, including phase-sensitive joint detection . For example, the optical phasesbetween the probe and reference light have to be stabilizeddown to the sub-wavelength level [7], which is not practicalif the target distance is unknown or fluctuating. As such,formidable challenges lie in implementing the quantum illu-mination protocol to benefit practical target detection applica-tions. An alternative approach to enhancing the performance oftarget detection, while mitigating the complexity of quantumillumination, is to only use correlations that exist in the non-classical states of light. The correlation enhanced protocolsonly require independent, phase-insensitive measurements toanalyze the correlation between the probe and reference light.The performance of correlation enhanced phase-insensitivetarget detection protocols may be inferior to the optimalclassical target detection protocol based on phase-sensitive de-tection, it is nevertheless useful for practical phase-insensitivetarget detection, such as radar and LIDAR. The key aspect ofcorrelation enhanced target detection protocols is the type ofcorrelation that is utilized. One type of correlation that existsin entangled photon pairs is the temporal correlation , whichmeans the detection time of the probe and reference photonsare completely random but always ‘simultaneous’. In otherwords, the probe and the reference photons are always detectedwith the detection time difference smaller than the intrinsiccorrelation time ∆ t (assuming zero detector time uncertainty),which could be as short as tens of femtoseconds[8]. To mea-sure the temporal correlation, it suffices to conduct separatephase-insensitive time-resolving photon counting detection forthe reference and probe photons. Recent years have seenrapid progress in single photon detection technology with ultrashort detection time uncertainty down to picosecond level[9],which has made possible the utilization of the strong temporalcorrelation of non-classical photon pairs.In this paper, we discuss an approach to target detection (theQTC detection protocol) that utilizes the temporal correlationof non-classical photon pairs that are generated in a monolithicsemiconductor waveguide. The rest of this paper is organizedas follows: In Section II the basic formalism of the QTCdetection protocol is explained and is compared to a similarprotocol in the radio frequency domain. The QTC detectionprotocol is also compared to a classical intensity detectionbased (CTI) protocol that is commonly used in the opticaldomain. In Section III a simple theoretical model for the QTCdetection protocol is developed. In particular, the parameterestimation theory is applied to construct the optimal detector a joint measurement requires bringing the two photons to the same physicallocation at the same time. function for the QTC detection protocol, based on which thetheoretical prediction of the ROC curve is obtained. In SectionIV the experiment setup is described. Section V shows theexperimentally measured time series of the detector functionsof both the QTC and CTI detection protocol, from which theROC curve is calculated. The experiment result is comparedto the theoretical prediction which is obtained in section IV. Insection VI the limiting factors of the QTC detection protocolperformance are discussed and the QTC detection protocolis compared to the CTI detection protocol with pulsed probelight and other correlation enhanced protocols. The conclusionis discussed in Section VII.II. B ACKGROUND AND O VERVIEW
The goal of this paper is to develop and analyze a phase-insensitive target detection protocol (the QTC detection pro-tocol) that utilizes the strong temporal correlations of non-classical photon pairs. This enhanced protocol will be com-pared to a classical and practical optical target detection proto-col (the CTI detection protocol) based on time-resolved inten-sity (photon-counting) detection. Both the enhanced and thebaseline classical target detection experiments are conducted ina lossy and noisy environment, where the environment noiseis assumed to be overlapping with the probe signal in bothtemporal and spectral domain (i.e., background photons are inthe same frequency range as the signal and could also arriveat the same time as a signal photon).
A. Temporal Correlations
In the QTC detection protocol, the temporal correlationbetween the probe and the reference photon is utilized.Temporally correlated photons could be generated throughcontinuous-wave (CW) pumped spontaneous parametricdown-conversion (SPDC)[10], [11]. In the SPDC process,a pump photon with a short wavelength ( (cid:39) nm) isannihilated in the nonlinear medium and subsequentlygenerates the probe and reference photon pair with longercentral wavelength ( (cid:39) nm). While the bandwidth of theCW pump light is narrow, the generated probe and referencephotons could be broadband, depending on the structureof the semiconductor waveguide [8]. This is because theannihilation of the pump photon could generate a photonpair with different possible frequency combinations, as longas the sum of the frequencies of the photon pairs equals thefrequency of the pump. The probe and reference photonsare in different polarization states (i.e., Type II SPDC) andare separated by a polarization beamsplitter. Note that thephotons in the SPDC photon pair are entangled in polarizationbefore the polarization splitting. However, in this paper, weare only considering temporal correlations between the probeand the reference photon after the polarization splitting.Since the pump light is CW, the probe and reference photonsare generated at completely random times. The temporalcorrelation of the probe and reference photon could beresolved via time-resolving photon detection: the detectiontime difference of the probe and reference photon is alwayssmaller than the intrinsic correlation time , ∆ t , of the Pump Laser
Semiconductor waveguideProbe detectorReference detector Target ObjectEnvironmental Noise Source
Source & detector Transreceiver
Signal Analysis
Fig. 1. Top: the schematic of the SPDC process in the semiconductorwaveguide. The red wave λ is the pump light at 783nm, the green λ p and blue λ r wave is the down-converted probe and reference photon thatis in horizontal (vertical) polarization. Middle: joint probability distributionof photon detection time. t p : detection time of the probe photon; t r detectiontime of the reference photon. The standard deviation of the detection timedifference is ∆( t p − t s ) = 100 ps for this plot. Bottom: Block diagram ofthe experiment setup. photon pair, if both photons have traveled equal optical pathlength and the detectors have the perfect temporal resolution.This corresponds to the heuristic idea that the probe andthe reference photons are always generated at the ‘sametime’. The typical joint probability distribution of probe andreference photon detection time is shown in Fig. 1. B. Experimental details of the QTC detection protocol
The block diagram of the QTC detection experiment setupis shown in Fig. 1. After the generation of the probe and thereference photon in the nonlinear-waveguide, the referencephoton is immediately detected on a single photon detector(the reference detector) and the probe photon is sent towardthe target object. The probe photons which are back-reflectedfrom the target object are collected and detected on anothersingle-photon detector (the probe detector). Regardless of thepresence or absence of the target object, the strong environ-mental noise power is always detected on the probe detector.The strong background noise is assumed to be completelyoverlapping with the probe photons in terms of both spectral and temporal domain distribution, i.e. CW and broadband.Therefore for the cases discussed here, it is not possible toreduce the noise power through temporal or spectral filtering.The presence and absence of the target object could bedetermined by analyzing the time-resolved photon detectionstatistics on both detectors. Since the background noise pho-tons are uncorrelated with the reference photon, the effect ofthe background noise on the target detection accuracy couldbe reduced by analyzing the temporal correlation informationthat is extracted from the photon detection statistics. The CTIdetection protocol is a simple intensity-based target detectionprotocol with experiment setup identical to that of the QTCdetection protocol, except that the reference photons are notdetected (nor is the information used in the processing). Inother words, the CTI protocol simply detect the intensity ofthe reflected probe light to determine the presence of the targetobject, without using any correlation information. It is true thatthe CTI protocol is not the optimal classical target detectionprotocol and it can be much improved using many otherclassical techniques such as coherent detection and matchedfiltering. However, since intensity detection has already bewidely adopted for optical target detection, the CTI protocolcan still be chosen as a baseline for comparison for the QTCdetection protocol, at least in the optical regime.
C. Experimental Parameters
There are some important characteristic constants of theQTC detection protocol and the CTI detection protocol thatare closely related to their performance. These constants willbe used in both the theoretical analysis and the experimentaldata analysis. • The source pair rate , ν , is defined as the number ofSPDC photon pairs generated inside the waveguide persecond. • The measured temporal correlation of the probe and thereference photon depends on two factors: – Detector time uncertainty ( ∆ t ): assuming thereference and probe single-photon detector haveidentical performance, ∆ t is defined as twice themaximal difference between the recorded photondetection time and the actual photon detection time. – Intrinsic Temporal Correlation ( ∆ t ): it isdefined as the maximal time difference between thedetection of the probe and the reference photons,assuming zero detector time uncertainty.The effective time uncertainty , ∆ t eff , of photon detec-tion is defined as the sum of detector time uncertainty ∆ t and intrinsic correlation time ∆ t : ∆ t eff = ∆ t + ∆ t . (1) • The total reference photon transmission efficiency η r is defined as the ratio between the number of photon detection rate on the reference detector and the sourcepair rate ν , i.e., η r = Reference photon detection ratePhoton source pair rate . (2)The total reference photon transmission efficiency η r includes the effect of both the optical loss of thereference photons and the reference detector efficiency. • The total probe photon transmission efficiency , η p , issimilarly defined as (assuming no background noise) η p = Probe photon detection ratePhoton source pair rate . (3)Note that η p is affected by the detection efficiency andthe coupling losses as well. In addition, η p is affectedby the distance of the target object, the reflectivity ofthe object and the collection efficiency of back-reflectedprobe photons. • The power of background noise ν b is characterized bythe number of photon detection events per second thatis due to the noise source (i.e. with probe beam blocked). • Another important parameter is the coincidencedetection window , T c , relating to the definition of coincidence detections , which will be made clear in thefollowing section.III. T HEORETICAL ANALYSIS
A. Temporally correlated photon pairs
In order to exploit the temporal correlation, it sufficesto model the correlated photon pair states with the jointprobability density P ( t p , t r ) of generating a probe photon attime t p and a reference photon at time t r . The average numberof the photon pairs generated per second ν equals the averagenumber of reference photons generated per second, i.e., ν = (cid:90) + ∞−∞ dt p P ( t p , t r ) (4)For CW pumped SPDC process, P ( t p , t r ) only depends onthe time difference t p − t r . Therefore ν does not depend on t r (despite the explicit appearance of t r in the above expres-sion). The intrinsic correlation time ∆ t , which is definedas the maximal detection time difference t p − t r , could beapproximated as 3 times the standard deviation of the timedifference t p − t r : ∆ t = 3 × (cid:115) ν (cid:90) + ∞−∞ dt r P ( t p , t r )( t p − t r ) (5)The intrinsic correlation time ∆ t is determined by the spectralproperty of the SPDC photon pair and could be tuned fromaround 20 fs to around 10 ps through different waveguidedesign[8]. The joint probability density distribution of thephoton detection time is a convolution between the jointprobability density of the photon generation time P ( t p , t r ) and the detector response function (parametrized by the detectortime uncertainty ∆ t ), aside from a constant time shift of t p or t r that is due to the unbalanced optical path lengths ofthe probe and the reference photon. Such unbalance could becompensated for through post data processing and is irrelevantfor target detection. Therefore in the rest of the paper, weshall assume that the optical path length difference betweenthe probe photon and the reference photon is zero. However, itshould be noted that this time shift contains the target distanceinformation and is useful for ranging-detection. The ability ofranging of the QTC detection protocol is discussed in sectionVI.C. B. Photon detection statistics of the QTC detection protocol
In the QTC detection protocol experiment, the referencephotons are detected on the reference detector and the back-reflected probe photon along with background noise photonsare detected on the probe detector. The time of each photondetection event is recorded. The absolute time of each photondetection event is not useful since the target detection systemis time-invariant: the nonclassical photon pair source is CWpumped and the target object is stationary. For a QTC detectionexperiment that last for a fixed period of time τ , there are onlythree useful quantities that can be extracted from the photondetection statistics: • Number of single channel photon detection events N p on the probe detector • Number of single channel photon detection events N r on the reference detector • Number of coincidence detection events N c that dependson the time difference between the photon detectionevents on different detectors.A coincidence detection event is defined as detecting twophotons on the different detectors ‘almost simultaneously’,with the detection time t p and t r satisfying: t p − t r ∈ (cid:20) − T c , − T c (cid:21) (6)where T c is the length of the coincidence window. Thesingle channel detection events on the probe or referencedetector are defined as the photon detection events that donot contribute to the coincidence detection events.It can be shown (see the Appendix) the N p , N c and N r each satisfy independent Poisson distribution: P ( N p , N r , N c ) = f ( N p , P p τ ) f ( N r , P p τ ) f ( N c , P p τ ) (7)where f ( k, λ ) = e − λ λ k k ! (8)is the Poisson distribution function and the average number ofphoton detections event N p , N r , N c per unit time P p , P r and P c are given by: P p = νη p + ν b − P c , (9) P r = νη r − P c ,P c = νη p η r + νη r ν b T c . ( T c ≥ t eff ) The above expressions for the photon detection eventrates are rigorously derived through quantum mechanicalanalysis[12]. A simplified derivation that is only based on thejoint temporal probability density P ( t s , t i ) could be found inthe Appendix. The condition for (9) to hold is that the temporaldetection window T c is larger than twice the effective timeuncertainty t eff . This is to ensure that probe-reference photonpairs that are detected will always contribute to a coincidencedetection event (see the Appendix). Equation (9) could be usedto predict the experimentally measured photon detection ratesas well as to calculate experiment parameters ν, η p , η r fromexperimental photon detection statistics. C. Transmission estimation and detector functions
For the QTC detection protocol, it is in principle possible toinfer the presence or absence of the target object directly fromthe experimental photon detection statistics (7). However, thefact that (7) is a multidimensional probability distributionprevents the direct application of standard target detectionanalysis such as ROC analysis, which requires a single-valuedtarget detection signal as the input. Therefore, in order toquantify the performance of the QTC detection protocol, onemust first construct a suitable single-valued detector function,which is a real function of the experimentally measuredphoton counting statistics N p , N r and N c .Our approach to constructing the optimal detector functionfor the QTC detection protocol is to formulate the target de-tection problem as a probe photon transmission η p estimationproblem[12]. An unbiased estimator S of η p is a function ofexperimental photon detection statistics N p , N r , N c and haveits mean value equal the probe transmission η p , which couldbe formally expressed as: target present : S = η p + n ( η p ) (10) target absent : S = n (0) where n ( η p ) and n (0) are corresponding zero-mean noisevariance when the object is present and absent (note that η p = 0 when the target object is absent). The unbiasedestimator S could be used as the detector function. This isbecause the target object is present if and only if the meanvalue of S is positive. The performance of the detector function S is given by the estimation variance, which characterizes theminimal probe photon transmission efficiency η p that couldbe distinguished from zero transmission. The performance ofthe detector function is also affected by the value of totalprobe transmission η p itself, which is the same for all detectorfunctions. In this article, we will consider three different types ofestimators (detector functions): S QTC1 = argmax η p P ( N p , N r , N c ) (11) S QTC2 = argmax η p f ( N c , P c τ ) S CTI = argmax η p f ( N p , P p τ ) ( η r = 0) where N p , N r , N c are the experimentalymeasured photon detection statistics and f ( N p , P p τ ) , f ( N c , P c τ ) , P ( N p , N r , N c ) are thecorresponding probability distribution as given in (7)and (8). Note that S CTI does not use the information aboutthe reference photon ( η r = 0 ), and therefore corresponds tothe CTI detection protocol.In the limit of long detection time τ , the minimal estimationvariance could be achieved by QTC1, which is the maximallikelihood estimation (MLE) of η p from the experimentalphoton counting statistics N p , N r and N c [13]: (cid:104) ∆ S QTC1 (cid:105) = 1 I QTC1 , (12)where I QTC1 = + ∞ (cid:88) N p ,N r ,N c =0 P ( N p , N r , N c ) (13) × (cid:20) ∂∂η p log P ( N p , N r , N c ) (cid:21) . It therefore follows that (see the Appendix for the derivation) I QTC1 = (cid:20) η r ν P c + (1 − η r ) ν P p + η r ν P r (cid:21) τ. (14)The three terms form left to right in the above expressionrepresent the contribution to the total Fisher information fromcoincidence detection events N c , single channel detectionevents on the probe and reference detector N p , N r , asindicated by their respective denominator P c , P p and P r .When the target detection environment is noisy ( ν b (cid:29) νη p ),the contribution of coincidence detection N c dominates inexpression (14), since the denominator in the first term P c ismuch lower than the denominator in the second term P p . Thisis because the environmental noise will directly increase thesingle-channel detection rate P p but insignificantly affect thecoincidence detection rate P c . From (14) it can be seen thatthe effect of the environmental noise ν b to the coincidencedetection rate P c decreases as the coincidence window size T c decreases. The minimal coincidence window T c is limitedby twice the effective time uncertainty ∆ t eff . This impliesthat the performance of the QTC1 detector function dependson the temporal correlation of non-classical photon pairsas well as the detector time uncertainty. In most cases, thedetector time uncertainty ∆ t is orders of magnitude largerthan the intrinsic correlation time ∆ t . Therefore the effectivetime uncertainty ∆ t eff is effectively just the detector timeuncertainty, i.e., ∆ t eff ≈ ∆ t . In the limit of zero coincidencewindow (zero effective time uncertainty T c = 2∆ t eff = 0 ), the contribution to the total Fisher information from thecoincidence detection events N c is immune to environmentnoise.The detector function S QTC2 could be considered as theMLE of η p when only the marginal probability distributionof coincidence detections N c is considered. S QTC2 = argmax η p f ( N c , τ P c ) , (15) = argmax η p { exp( − ( η p η r ν + νη r ν b T c )) × ( η p η r ν + νν b η r T c ) N c N c ! } = N c − ν b νη r T c νη r , . (16)The corresponding Fisher information of S QTC2 is given by I QTC2 (see the Appendix for the detailed derivation), whichequals the first term of I QTC1 in (14): I QTC2 = η r ν P c τ. (17)The CTI detector function S CT I is the MLE of the probetransmission efficiency η p from the marginal distribution ofprobe channel counts N p (when η r = 0 ), and the correspond-ing detector function S CTI and Fisher information ( I CTI ) aregiven by S CTI = argmax η p f ( N p , η p ν + ν b ) , (18) = N p − ν b ν , (19) I CTI = ν P p τ. (20)The derivation of S CTI and I CTI is similar to the derivation of S QTC2 and I QTC2 . The detector function S QTC2 and S CTI areequivalent to more straight forward definitions S (cid:48) QTC2 = N c and S (cid:48) CTI = N p in terms of ROC analysis as will be shownlater, since they are merely re-parametrizations of each other.The detector function S (cid:48) QTC2 has already been used in aprevious study [5].
D. Gaussian approximation and the ROC curve
The central limit theorem suggests that if the measurementtime τ of a target detection experiment is sufficiently long, thedetector functions S QTC1 , S QTC2 and S CTI approximately obeysnormal distribution. Therefore, under the large τ limit, thenoise variance n ( η p ) of the detector functions S QTC1 , S QTC2 and S CTI could be approximated as zero-mean Gaussian noiserandom variables as follows: n QTC1 ( η p ) ∼ normal (0 , /I QTC1 ) (21) n QTC2 ( η p ) ∼ normal (0 , /I QTC2 ) n CTI ( η p ) ∼ normal (0 , /I CTI ) It is important to remember that the Fisher information ( I QTC1 , I QTC2 and I CTI ) are functions of η p . In particular, η p = 0 when the target object is absent. For a fixed target detectiontime τ , the validity of this Gaussian signal approximationshould be verified with the experimental photon detectionstatistics.For target detection, the presence and the absence of thetarget could be determined by comparing the detected signal(the value of the detector function for a particular experiment)to a signal intensity threshold V : when the detected signalis stronger than the threshold the presence of the targetobject could be asserted and vice versa. However, even ifthe object is absent ( η p = 0 ), there could be non-vanishingprobability (false alarm possibility P fa ) of the detected signalbeing higher than the detection threshold. Similarly, if thetarget object is present ( η p > ), the probability (detectionprobability P d ) of the detector function being higher than thethreshold could be lower than unity too. The false alarm rate( P fa ) and detection rate ( P d )both depend on the threshold V .By varying the threshold V , the relationship between P fa and P d could be calculated, which is defined as the ROC curve.The Gaussian signal approximation could be used to obtainan analytic expression of the ROC curve. Consider any one ofthe detector functions in (11). Denote the Fisher information I QTC1 (or I QTC2 , I CTI ) by I ( η p ) as the function of total probetransmission η p (note that I (0) correspond to the absence ofthe target). Then the expression of the ROC curve is givenby[14]: P d = Φ (cid:32) η p (cid:113) I ( η p ) + (cid:115) I ( η p ) I (0) Φ − ( P fa ) (cid:33) (22)where P d is the detection rate and P fa is the false alarm rateand Φ is the cumulative Gaussian distribution function: Φ( x ) = 1 √ π (cid:90) x −∞ exp( − z / dz (23)The ROC curve is independent of the re-parametrization of de-tector functions[14]: if a detector function could be expressedin terms of another detector function and constants, then theircorresponding ROC curve will be identical. TABLE IT
ABLE OF SOME EXPERIMENT PARAMETERS
Transmit Power 0.41 pWRange to Target 8 inSource ( ν ) 3.19 MHzBackground ( ν b ) 0.85 MHzTransmission ( η p ) 0.24%Transmission ( η r ) 17%Window ( T c ) 486 psDetector time uncertainty ( ∆ t ) 243 ps Intrinsic correlation time ( ∆ t ) 0.02 psEffective time uncertainty ( ∆ t eff ) (cid:39)
243 psObservation time ( τ ) 0.01 s Fig. 2. The schematic of the experimental setup is divided into two parts: the source and detection part and the transceiver part. The source and detection part(green background) includes the photon pair source and detectors. Pump laser: Ti-Sapphire CW laser at 783nm. PBS: polarization beam splitter. LPF: longpass ( > IV. E
XPERIMENTAL SETUP
A. Layout
As shown in the block diagram in Fig. 1 and theexperiment setup Fig. 2, the QTC detection protocol setupconsists of two parts: the source and detector part andthe transceiver part . The source and detector part consistsof the semiconductor SPDC source and both single-photondetectors. In the transceiver part, the probe photons are senttoward the target and the back-reflected probe photons arecollected. The two parts are connected via single-mode fiber,which enables a flexible deployment of the target detectionsystem. In the source and detection part, the photon pairs aregenerated through the SPDC process inside the semiconductorwaveguide, which is externally pumped by a CW Ti-Sapphirelaser at 783 nm. The broadband, CW probe (with horizontalpolarization) and reference(with vertical polarization) photonsare separated upon a polarization beam-splitter. The referencephotons are coupled into a single-mode fiber and are detectedon the reference detector. The probe photons are coupled intoanother single-mode fiber and sent to the transceiver part. Inthe transceiver part, the probe photons are emitted throughan optical collimator towards the target object. The probephotons back-scattered from the target object are collectedby a telescope and are directed to the probe detector fordetection. To simulate strong environment noise, collimated broadband CW light shines towards the collection optics,leading to a large number of noise photons getting detectedon the probe detector. The noise light source is a light-emitting diode (BeST-SLQTCD ® ,LUMUX) with the outputpower attenuated by a tunable optical attenuator (HP ® B. The target object and the collection optics
The target object used in the experiment is aluminum foil,which has a high reflectivity but diffusive reflection at thefrequency of the probe photons. The target object is placedaround 8 inches away from the first lens of the collectionoptics as shown in Fig. 3. The presence and the absence ofthe target object are simulated by turning on and off of thecoupling of the probe photon into the single mode fiber in thesource and detection part, as shown in Fig. 2. This is carriedout using a beam chopper to periodically block and unblockthe probe photons at 2 Hz frequency with a 50:50 duty cycle.The reason for not directly modulating the transmission of theprobe photon in front of the target object is that doing so willaffect the coupling of the noise photons into the collectionoptics as well, which could be due to the multiple reflectionsof the noise photons between the collection optics and theoptical chopper. Although physically different, turning on (off)the coupling of the probe photons within the source & detectorpart is equivalent to the presence (absence) of the target object,both leading to the detection (non-detection) of the back-reflected probe photon at the probe detector.The collection optics is a home-made telescope using threelenses. The target object is placed around 8 inches away fromthe first lens of the telescope. The collection efficiency, whichdetermines the limit in our lab for target detection range,could be easily enhanced with an improved optical designof the telescope. The total transmission of the probe photons η p is estimated to be around 0.24%, which includes variouslosses from the source to the detection, such as collectionefficiency, detection efficiency, and the target reflectivity. Thisvalue of η p is obtained by conducting a trial experiment withlong detection time and then the value of η p could be calcu-lated from experimentally measured photon detection statistics N r , N p , N c according to (9). The value of other experimentalalparameters ν, η r , ν b could be calculated similarly. C. Single photon detectors
The probe and reference detector are two superconductingnanowire single-photon detectors (Quantum Opus)[15]. Bothdetectors are placed inside the same cryogenic system cooleddown to 2.7K. Both detectors have similar detection efficiency( (cid:39) at 1566 nm) and time uncertainty. The time of eachphoton detection event on both detectors is recorded by a time-digital converter(ID800, IDQuantique). The detector temporaluncertainty ∆ t (including the effect of electronic jitteringof the voltage adapter between the detector and the time-digital converter and the time uncertainty of the time-digitalconverter) for both the reference and the probe detectors are243 ps. Fig. 4. the scanning electron microscope image of the semiconductorwaveguide
D. Semiconductor waveguide source of non-classical photonpairs
The semiconductor SPDC waveguide used is based onaluminum gallium arsenide (AlGaAs) material platform. Thedimension of the waveguide is (cid:39) mm long and (cid:39) µm wide.The ridge and the substrate of the waveguide are based onthe Bragg structure, consisting of multiple layers of AlGaAsalloy with different compositions ( Al x Ga − x As, ≤ x ≤ )as could be seen in Fig. 4 [16]. This semi-periodical structureof the waveguide provides simultaneous confinement ofthe 783nm pump light and the generated photon pairsaround 1566 nm. In addition, the waveguide structure is alsodesigned to satisfy the momentum matching condition[17] ofthe SPDC process. The conversion efficiency is estimated tobe 2.1 × − probe-reference photon pairs per 783nm pumpphoton. In the experiment setup, the nonlinear waveguide ismounted on a two-dimensional translation stage. The input783nm pump light is coupled into the rear facet of thewaveguide through an objective lens. The probe-referencephoton pairs are coupled out from the front facet waveguide through another objective lens.The advantages of the semiconductor photon-pair source fortarget detection as compared to conventional SPDC sourcebased on bulk nonlinear crystals are many-fold. • The significantly smaller form factor of thesemiconductor waveguide enables large scale integrationof photon pairs sources, which is crucial for realistictarget detection applications where a large flux ofnon-classical photon pairs is needed. • For each of the waveguides, the generation of theprobe-reference photon pairs is also efficient because ofthe strong nonlinearity of the semiconductor material. • The AlGaAs platform also allows the possibility ofthe active waveguide with electrically pumping (i.e.generating the correlated photon pairs by applying avoltage across the waveguide), which is also favorablein terms of large scale integration. • The spectrum of the SPDC photon pairs could beengineered through different designs of the waveguidestructure[8]. For example, the central frequency of theprobe could be shifted to suit the need in different targetdetection scenarios without shifting the central frequencyof the reference photons.V. E
XPERIMENTAL RESULTS
We conduct repeated ( (cid:39) × ) target detection exper-iments for the QTC detection protocol to characterize itsperformance. Each of the experiments lasts for a short period τ = 0 . s . The target object is absent (probe photons blocked)for half of the total number of the experiments. The sameexperimentally recorded photon counting data is used for theCTI detection protocol too, but the photon detection events onthe reference detector are neglected. By doing so the drift ofthe experiment condition between the CTI detection protocolexperiment and the QTC detection protocol experiment iseliminated. A table of experimental parameters could be foundin the table(I)(note that the ROC curves are also plotted fordifferent values of measurement time τ apart from the listedvalue). A qualitative sense of the experiment is provided by theratio of the number of probe photons and the number of noisephotons, i.e.,
10 log ( νη p /ν b ) ≈ − dB. The performanceof the three detector functions S QTC2 , S QTC1 and S CTI arequantified and compared through the ROC analysis of theexperimental photon counting data.
A. Time series, histogram and detector performances
The experimental photon counting data is recorded by theID800 time-digital converter in a ‘detection time - channelindex’ format, where the ‘channel index’ stands for either theprobe or the reference detector. A coincidence detection eventis registered if two consecutive photon detection events are ondifferent detectors and have time difference shorter than half index of experiments d e t e c t o r o u t p u t S Q T C
1e 3 detector S QTC output n o r m a li z e d p r o b a b ili t y d e n s i t y index of experiments d e t e c t o r o u t p u t S Q T C
1e 3 detector S QTC output n o r m a li z e d p r o b a b ili t y d e n s i t y index of experiments d e t e c t o r o u t p u t S C T I
1e 2 detector S CTI output n o r m a li z e d p r o b a b ili t y d e n s i t y Fig. 5. Left:the time series of different detector functions S QTC1 , S QTC2 and S CTI over 1400 independent target experiment. X-axis: the index of independentexperiment that last time τ = 0 . s. Blue (red) bar: probability density when the target is absent (present). Blue (red) solid line, the theoretically calculatedprobability distribution using the Gaussian signal approximation (21). The histograms are calculated from a larger portion of data ( . × independentexperiments). the coincidence window T c (after compensating the differenceof the optical path lengths of the probe and reference photon).The number of single-channel detection events N p and N r are obtained by subtracting the total number of coincidencedetection events N c from the total number of photon detectionevents on the probe and the reference detector. From thesestatistics different detector functions S QTC1 , S QTC2 and S CTI could be evaluated. Fig. 5 shows the time series of thethree detector functions from 1400 repetitions of independenttarget detection experiments (700 experiments with the targetobject present and 700 experiments with the target objectabsent). As could be seen, both detector functions for theQTC detection protocol ( S QTC1 and S QTC2 ) have much lowerfluctuation compared to the detector function of the CTIdetection protocol. The histogram of the values of the differentdetector functions is also shown in Fig. 5. The experimentally measured histograms agree well with the theoretical theorycurve that is obtained from the Gaussian signal approximation,which validates such approximation. For the detector function S QTC2 there exists ’gaps’ within the histogram. This is becausethe number of coincidence detection events is relatively lowand quantized in each experiment.
B. ROC curve
The ROC curves of the different detector functions areplotted in Fig. 6 with different value of experiment time τ = 0 . , . , . s. For the CTI detection protocol, theexperimental ROC curve perfectly matches the theoreticalROC curve. The theoretical ROC curve of detector function S QTC2 is only slightly lower than that of the S QTC1 detector,suggesting that coincidence detection is close to optimal forthe QTC detection protocol. The experimental ROC curve P fp False Alarm Probability P d D e t e c t i o n P r o b a b ili t y S CTI theory S CTI theory,42.62 times longer time S QTC theory S QTC theory S CTI experiment S QTC experiment S QTC experiment 10 P fp False Alarm Probability P d D e t e c t i o n P r o b a b ili t y S CTI theory S CTI theory,42.62 times longer time S QTC theory S QTC theory S CTI experiment S QTC experiment S QTC experiment10 P fp False Alarm Probability P d D e t e c t i o n P r o b a b ili t y S CTI theory S CTI theory,56.75 times longer time S QTC theory S QTC theory S CTI experiment S QTC experiment S QTC experiment 10 P fp False Alarm Probability P d D e t e c t i o n P r o b a b ili t y S CTI theory S CTI theory,56.75 times longer time S QTC theory S QTC theory S CTI experiment S QTC experiment S QTC experiment10 P fp False Alarm Probability P d D e t e c t i o n P r o b a b ili t y S CTI theory S CTI theory,80.55 times longer time S QTC theory S QTC theory S CTI experiment S QTC experiment S QTC experiment 10 P fp False Alarm Probability P d D e t e c t i o n P r o b a b ili t y S CTI theory S CTI theory,80.55 times longer time S QTC theory S QTC theory S CTI experiment S QTC experiment S QTC experiment Fig. 6. Left:ROC curves for the detector function S QTC1 , S QTC2 and S CTI for τ = 0 . s, τ = 0 . s and τ = . s (from top to bottom). Red curve: thetheoretical ROC curve for S QTC1 calculated based on the Gaussian signal approximation. Red triangles:the experimental ROC curve for S QTC1 . The bluecurve, blue triangles, and black curve, black triangles are similarly plotted for detector functions S QTC2 and S CTI , respectively. Dashed black curve: theoreticalROC curve for the detector function S CTI with longer integration time τ (cid:48) = k τ τ such that the detection rate P d same as that of S QTC1 is achieved at falsealarm rate P fa = 10 − . The ratios are k τ = 42 . , . , . for τ = 0 . s, . s, . s . For the same integration time τ , the CTI detection protocolcan also achieve the same performance with . , . , . times more source power or total probe photon transmission. Right: the same ROC curve plottedwith log scaled P d axis. for S QTC1 and S QTC2 detector functions are considerablylower than the theoretical prediction. We attribute this to thedrift of the SPDC pump power and the probe transmission η p over time. This is confirmed as the drift of the SPDCpump power and the probe transmission is experimentallyobserved. Moreover, when the ROC curves are plotted for asmaller sample of independent experiments that last a shorterduration, the theoretical and experimental curve will have abetter matching. The fact that the experimental ROC curvefor the detection function S QTC1 is considerably higher thanthat of S QTC2 may suggest the better robustness of the S QTC1 against the drift of the experiment condition.Both detector functions S QTC1 and S QTC2 have significant performance advantages over the CTI detection protocol de-tector function. For measurement time τ = 0 . s, when thefalse alarm rate is set around P fa = 10 − , the experimentaldetection rate of S QTC1 and S QTC2 are around . To achievethe same theoretical detection rate P d of the detector function S QTC1 , the CTI detection protocol must last for around (cid:39) times longer, as shown in Fig. 6. Such a time reduction factorimplies the much higher target detection speed of the QTCdetection protocol. Alternatively, for the same measurementtime τ = 0 . , the CTI achieve can achieve the samedetection rate P d with (cid:39) τ . It couldbe seen that the QTC detection protocol could have even higher performance advantages compared to the CTI detectionprotocol when the detection time τ is short and when the falsealarm rate P fa is low. P fp False Alarm Probability P d D e t e c t i o n P r o b a b ili t y S QTC without improvement S QTC , t =25ps S QTC , r =100% S CTI S CTI ,80.55 times longer time
Fig. 7. Theoretical ROC curves with measurement time τ = 0 . s andimproved effective time uncertainty ∆ t eff or reference photon transmission η r for the QTC detection protocol. Black: the CTI detection protocol forcomparison; Red: the QTC detection protocol ( S QTC1 ) without any improve-ment; Black dashed: the CTI detection protocol with (cid:39) . times longermeasurement time; Red dashed: the QTC detection protocol with improvedeffective time uncertainty ∆ t eff = 25 ps ; Red dotted: the QTC detectionprotocol with perfect reference photon transmission η r = 100% VI. D
ISCUSSION
A. Factors impacting the target detection performance
The factors influencing the target detection performancecould be classified into two sets: • Factors that only affect the performance of the QTCdetection protocol, namely, reference photon transmissionefficiency ( η r ) and effective time uncertainty ( ∆ t eff ); • Factors that affect both the QTC detection and the CTIdetection protocols, particularly, the source pair rate ( ν ),background noise photon detection rate ( ν b ), and probephoton transmission efficiency ( η p ).The most important limiting factor that only affects the QTCdetection protocol is the transmission efficiency, η r (cid:39) in our experiment, of the reference photon. In the limit ofzero reference photon transmission η r = 0 , the QTC detectionprotocol will be identical to the CTI detection protocol andhence provide no advantage. In the experiment, η r is mainlylimited by the finite efficiency of the coupling of the referencephoton into the single-mode fiber that is connected to thereference detector. This could be due to the low modal overlapbetween the waveguide mode and the fiber mode for thereference and probe photons. Such inefficiency could be alle-viated with an improved design of the coupling optics, or thesemiconductor waveguide to achieve better modal overlap. Inthe limit of perfect reference photon transmission ( η r = 100% )the ROC curve of the QTC detection protocol ( with thedetector function S QTC1 ) is plotted in Fig. 7.Another important limiting factor for the QTC detectionprotocol only is the effective time uncertainty ∆ t eff , whichdetermines the temporal coincidence window T c = 2∆ t eff .From (17) and (9) it could be seen that for the detector function S QTC2 (whose performance is close to the detectorfunction S QTC1 ), the effect of background noise could bereduced by a factor of x when the effective time uncertainty ∆ t is reduced by a factor of x . The effective time uncertainty ∆ t eff is limited by the intrinsic correlation time ∆ t and the the detector time uncertainty ∆ t . In our currentimplementation of the QTC detection protocol the effectivetemporal uncertainty ∆ t eff is mainly limited by the detectortemporal uncertainty ∆ t eff (cid:39) ∆ t = 243 ps. This valuecould be improved with single-photon detectors that have afaster response or with improved photon detection techniques[12]. Fig. 7 also shows the ROC curve of the QTC detectionprotocol for the detection function S QTC1 when the effectivetime uncertainty is reduced to ∆ t eff (cid:39) ps.Factors affecting both the QTC detection protocol and theCTI detection protocol performance include the source pairrate ν , background noise photon detection rate ν b and theprobe photon transmission efficiency η p . The performanceof both the QTC detection protocol and the CTI detectionprotocol increase when the source pair rate ν is high and thebackground noise power ν b is low. This corresponds to theintuitive idea that better target detection performance couldbe achieved with higher source power and lower backgroundnoise. However, it is worth noting that ν and ν b does notaffect the QTC and the CTI detection protocols equally. Theperformance advantage of the QTC detection protocol overthe CTI detection protocol is larger in high noise ν b andlow source power ν regime[12], suggesting that the QTCdetection protocol is more favorable for target detection withlow power and high noise background.The total probe transmission efficiency, η p , impacts theperformance of both the QTC detection protocol and the CTIdetection protocol. The total transmission η p is affected bymany factors including the target distance, the reflectivityof the target, propagation medium, and the probe photoncollection efficiency. This corresponds to the intuitive idea thattarget within a short distance and high reflectivity is easy todetect. The experiment result and theoretical analysis showthe significant performance advantages of the QTC detectionprotocol over the CTI detection protocol with the same η p . Inparticular, the variance of the estimation can be considerablylower for the QTC detection protocol. This suggests thatfor the same level of target detection performance, the QTCdetection protocol is capable of detecting a target object withlower reflectivity and longer distance than the CTI detectionprotocol. B. QTC and CTI detection protocols with pulsed or narrow-band probe
It is important to note that in the theoretical model andthe experiment of both the QTC and CTI detection protocol,the background noise is assumed to be overlapping with theprobe photon in both temporal and spectral domains, i.e.,broadband and CW. Therefore is not possible to reduce thein-band noise power through filtering. However, if a short pulse or narrowband probe light is utilized, then it is possiblefor the CTI detection protocol to reduce the in-band noisepower through spectral filtering or temporal gating. In suchcases, the performance of the CTI detection protocol couldbe increased and is also ultimately limited by the detectortime or frequency uncertainty. Nevertheless, the utilization ofshort-pulse or narrow-band probe light will not only increasethe system complexity by using a pulsed or narrowband laserbut also reduced the stealth of the target detection: suchnoise reduction comes at the price of concentrated opticalpower in either the temporal or spectral domain, and henceincreasing the visibility of the target detection channel. If theadversary party is able to distinguish the probe photons fromthe background noise by their different spectral-temporalproperties, then they could selectively jam the target detectionchannel with noise photons that are indistinguishable fromthe probe photons. Although there exist classical scramblingtechniques such as frequency scrambling to increase theindistinguishability between the probe and noise photons,such indistinguishability is not guaranteed by fundamentalphysical principles. On the other hand, the QTC detectionprotocol provides unconditional indistinguishability betweenthe probe and noise photons. This is because each of theprobe photons is generated at a fundamentally random timeand frequency, albeit its strong temporal correlation to thereference photon. When the environment noise power is veryhigh compared to the probe power, the probe photon will bealmost invisible to the adversary party, but the QTC detectionprotocol could still achieve meaningful target detectionperformance, as shown in the results and analysis above.Another important advantage of the QTC detection protocolis jamming resilience. Note that in the case of pulsed source,if the background photons and probe photons arrive at thesame time, it would be totally and perfectly jammed. However,intuitively, the QTC detection protocol based on CW pumpingis more resilient to jamming noise as the probe photons arriveat random times. This intuition is confirmed by the theoreticalanalysis of the Fisher information. C. Covert ranging
Covert ranging is an application of the QTC detectionprotocol that could take advantage of its stealth property. Clas-sical ranging protocol typically utilizes pulsed electromagneticradiation to be probe the target, and the target distance couldbe calculated from the time-of-flight of the probe photon.However, time-variant probe radiation is distinguishable fromthe CW background noise and is therefore visible to theunauthorized receivers. Ranging with classical time-invariantradiation is not possible since the time-invariant back-reflectedsignal does not contain much information about the targetdistance. On the other hand, ranging with time-invariant probelight is possible for the QTC detection protocol. Since eachprobe photon is temporally correlated with a reference photon,the travel distance of the probe photon could be calculatedfrom the detection time difference of the probe and referencephoton. Meanwhile, the probe photon in the QTC detection protocol is generated at a fundamentally random time andfrequency and is therefore indistinguishable from the CWbroadband environment noise. A preliminary ranging exper-iment with the QTC detection protocol was reported in [12],achieving distance resolution of (cid:39)
D. Other enhanced target detection protocols
The major difference between the QTC detection protocol,discussed here, and the previously reported quantum two-mode squeezing (QTMS) detection protocol [6] is the differenttype of correlation of the photon pairs that is utilized. In theQTMS protocol, which is operating in radio frequency, thephase-sensitive complex amplitude correlation between theprobe and reference light is utilized. Correspondingly, thedetection in the QTMS protocol consists of phase-sensitiveheterodyne detection of both the reference and the probephotons. In the optical frequency, the generation of complexamplitude correlated state (squeezed state) is also possiblethrough spontaneous parametric down-conversion process[18]. However, the homodyne measurement that is requiredto measure such complex amplitude correlation is much morechallenging. In optical frequencies, homodyne measurementrequires precise temporal matching of the waveform of thelocal oscillator and the probe/reference photon, which meansthe spatial overlap between the local oscillator and the probelight have to be stabled to sub-wavelength level. Such stringentrequirements make phase-sensitive homodyne measurementimpractical for optical target detection and ranging sincethe exact position of the target is usually unknown or unstable.Compared to the phase-sensitive heterodyne measurementthat is used in QTMS radar, time-resolved photon detectionin the QTC detection protocol is phase-insensitive and easy toimplement in the optical domain. This is because single-photondetection technologies are relatively mature in optics, whilequadrature (I and Q) measurements are very challenging. It isworth emphasizing that this does not mean the QTC detectionprotocol is restricted to the optical frequencies. In fact, therelatively higher noise background in the microwave regimecould potentially make the advantages of the QTC detectionprotocol over the classical intensity detection based protocoleven more evident. However, it must be noted that in themicrowave regime, one can do better using coherent signalprocessing (e.g., match filtering), and that current radars donot use match filtering in the single-photon level. So a moredetailed analysis needs to be carried out to investigate utilityof QTC in the microwave regime.In the optical domain, there are other correlation-enhancedtarget detection protocols utilizing the photon-numbercorrelation in discrete timebins[4], [5]. These protocols arenot suitable for covert operations since they utilized pulsedphoton pairs as the sources that are distinguishable from theCW noise background. Moreover, the bulk crystal photon pairsources that are utilized in these protocols are not suitable forlarge scale integration. In general, the correlation enhanced target detectionprotocols (including QTMS) are inferior to the optimalentanglement-based protocols in terms of absoluteperformance. However, the correlation enhanced protocolonly entails independent measurements instead of jointmeasurement of the probe and reference photons, and is,therefore, not suitable for radar and ranging applicationswhere the target distance is unknown.In this paper, we have made a comparison with practicallyrelevant CTI detection protocol that is in the optical domain,and not the optimal classical protocol. We believe it is still afair comparison in the sense that the QTC detection protocolis also a practically realizable protocol. The proposed QTCdetection protocol can be viewed as a separate practicallyrealizable sensing technology made possible due to advancesin single-photon detector technologies. This new technologyneeds to be compared with current practical sensing technolo-gies, particularly in the microwave. The theoretical results inthis paper enable us to carry out such an analysis, which weplan to carry out in the future.
E. Practical applications
A QTC source based on the proposed semiconductorwaveguide approach is compact and practical. For instance,a moderate-size array could have the form factor of a laserpointer and be battery operable. Unlike many quantum tech-nologies, no cryogenics is required. The accompanying de-tectors would be much bulkier and with significant powerrequirements, as such detectors need to be cooled. Applicationspace could include covert ranging as well as covert imaging.However, significant signal processing work is required to fullyexploit this capability, and the theory in this paper can beviewed as the first step of such an endeavor.VII. C
ONCLUSION AND F UTURE W ORK
We theoretically analyzed and experimentally demonstrateda prototype target detection protocol (the QTC detectionprotocol) with SPDC photon-pair sources. This protocolis similar to the previously reported QTMS radar, but itutilizes the temporal correlation instead of the complexamplitude correlation between the probe and the referencephoton and is in the optical regime (not microwave). TheQTC detection protocol only requires time-resolved photon-counting detection, which is phase-insensitive and thereforesuitable for optical target detection. As a comparison tothe QTC detection protocol, we also consider a classicalphase-insensitive target detection protocol based on intensitydetection which is the basis of current technologies in theoptical regime.The experimental and theoretical results of this paper issummarized as follows: • The performance of both the CTI (and practical in theoptical regime) and the QTC detection protocol arequantified with the standard ROC analysis that is widelyadopted in radar/signal processing research. We present simple models and derive analytical expressions forthe ROC curves, and demonstrate excellent agreementwith experimental data. In particular, we applied theparameter estimation theory to construct the detectionfunctions for target detection. • We considered two different detector functions toquantify the performance of the QTC detection protocol.The QTC detection protocol is compared to the CTIdetection protocol with intensity (photon counting)detection that is practically relevant in the opticalregime. Experimental results show that in a lossy andnoisy environment, the QTC detection protocol couldachieve a probability of detection ( P d ) of (cid:39) . for afalse alarm probability around − , while the currentclassical/practical CTI detection protocol yields a P d ≈ . • This performance of the QTC detection protocol iscomparable to the CTI detection protocol with the sameoutput power but with detection time that is (cid:39)
57 timeslonger. • The QTC detection protocol is also capable of covertranging, with a low flux CW signal that is completelyindistinguishable from the background noise. • We implemented a free-space target detection systemwith a semiconductor waveguide photon pair source. Thisdemonstration provides an already proven, technologicalroute to realizing a practical sensor in the optical regime.There are many directions for further development of theQTC detection protocol. It would be important to model theprobe photon transmission in greater detail for a specificQTC detection protocol, including the effect of target objectreflectivity, range to target, the design of collection opticsand the turbulence of the target detection channel. It is alsoimportant to explore the possibilities of array variants ofthe QTC detection protocol, i.e., with multiple sources anddetectors to obtain a stronger probe photon flux. It is also im-portant to investigate the possibility of overcoming the detectortime uncertainty limit with novel photon detection techniques.These areas will be explored in future publications.R
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Liu Han received his B.E. degree in opto-electronics informa-tion science and engineering from Tianjin university in 2016.He is now pursuing his Ph. D. degree in the university ofToronto. His research interests include quantum metrology andsensing with time-frequency entangled photon pairs.
Bhashyam Balaji (SMIEEE, FIET) graduated from KendriyaVidyalaya (Central School), Sector VIII, Ram Krishna Puram,New Delhi, India, in 1987. He received his B.Sc. (Honors)degree in physics from St. Stephens College, University ofDelhi, India, and his Ph.D. degree in theoretical particlephysics from Boston University, Boston, MA, in 1997. Since1998, he has been a scientist at the Defence Research andDevelopment Canada, Ottawa, Canada. His research interestsinclude all aspects of radar sensor outputs, including space-time adaptive processing, multi-target tracking, and meta-leveltracking, as well as multisource data fusion. His theoreticalresearch also includes the application of Feynman path integraland quantum field theory methods to the problems of nonlinearfiltering and stochastic control. Most recently, his researchinterests include quantum sensing, in particular, quantum radarand quantum imaging. He is an IET Fellow and a recipient ofthe IEEE Canada Outstanding Engineer Award in 2018. Amr S. Helmy is a Professor in the department of electricaland computer engineering at the University of Toronto.Prior to his academic career, he held a position at AgilentTechnologies photonic devices, R&D division, in the UKbetween 2000 and 2004. He received his Ph.D. and M.Sc.from the University of Glasgow with a focus on photonicdevices and fabrication technologies, in 1999 and 1995respectively. He received his B.Sc. from Cairo University in1993, in electronics and telecommunications engineering.Amr is a senior member of the IEEE and a Fellow of the Opti-cal Society of America. His research interests include photonicdevice physics and characterization techniques, with emphasison nonlinear and quantum optics in III-V semiconductors;applied optical spectroscopy in III-V optoelectronic devicesand materials; III-V fabrication and monolithic integrationtechniques. A
PPENDIX
A. Derivation of the photon counting statistics
To calculate the coincidence detection rate P c , it suffices toconsider a very small time interval ∆ τ such that there couldbe at most one photon detection event on each detector. Thenthe probability of both the probe and the reference photonsare generated and detected is given by νη p η r ∆ τ . In suchcases, consider the temporal distribution of recorded probeand reference photon detection times t p and t r respectively.According to the definition of the detector time uncertainty ∆ t , if the recorded detection time of the reference photon isgiven by t r , then the actual reference photon arriving time onthe detector is within the range [ t r − ∆ t , t r + ∆ t ] . And by thedefinition of the intrinsic correlation time ∆ t , the actual probephoton arriving time on the probe detector is within the rangeof [ t r − ∆ t − ∆ t , t r + ∆ t + ∆ t ] . Again by the definition ofdetector time uncertainty, the recorded probe photon detectiontime is within the range [ t r − ∆ t − ∆ t , t r + ∆ t + ∆ t ] .Therefore the coincidence detection window width T c mustbe set to twice the effective temporal uncertainty ∆ t eff =∆ t +∆ t to ensure that every reference-probe photon pair thatis detected will contribute to a coincidence detection event.Due to the non-zero width of the coincidence window T c ,the noise photons and the reference photons will contribute tocoincidence detection events too. The probability of detectinga reference photon and a noise photon that is within thecoincidence window T c is given by νη r ν b T c ∆ τ . The total rate of coincidence detection events is given by: P c = η p η r ν ∆ τ + η r νν b T c ∆ τ ∆ τ (24) = η p η r ν + η r νν b T c (25)The derivation of single channel detection event rates P p , P r is directly from the definition: P p = η p ν + ν b − P c (26) P r = η r ν − P c (27)For a target detection experiment that last time τ , wemake an important approximation that N p , N r and N c each obey independent Poisson distribution, with mean value P p τ , P r τ and P c τ . The independent Poisson distributionapproximation is valid for the following reasons. The singlechannel detection events and coincidence detection events arecompletely uncorrelated if the time separation is larger thanthe effective time uncertainty ∆ t eff (assuming the opticalpath length difference between reference and probe photonis zero). This is because the probe photon is only correlatedto the reference photon from the same SPDC photon pair.Secondly, the probability of having more than one singlechannel or coincidence detection events within an effectivetime uncertainty ∆ t eff is negligible. This is because thephoton detection rate is much lower than the inverse ofthe effective time uncertainty. Based on these two reasons,single-channel detection events and coincidence detectionevents could be treated as if completely uncorrelated andtherefore each obeys independent Poisson distribution. B. Derivation of the Fisher information expressions
For a discrete random variable X whose probability massfunction P ( X = x | η ) is parametrized by a parameter η , theFisher information of X about η is given by: I X ( η ) = (cid:88) x ∈ χ (cid:20) ddη log P ( X = x | η ) (cid:21) P ( X = x | η ) (28)where χ is the space of different outcome of X and for theQTC detection experiment χ is given by: χ = { ( N p , N r , N c ) | N p , N r , N c ∈ N } (29)To simplify the derivation of the Fisher information for theQTC detection protocol, we show first the additive propertyof Fisher information I xy . Consider a joint probability dis-tribution of two independent random variables f xy ( x, y ) = f x ( x ) f y ( y ) : I xy = (cid:88) x,y (cid:20) ddη log f xy ( x, y ) (cid:21) f xy ( x, y ) (30) = (cid:88) x,y f x ( x ) f y ( y )[ f x ( x ) f y ( y )] (cid:20) df x ( x ) dη f y ( y ) + df y ( y ) dη f x ( x ) (cid:21) = (cid:88) x,y f x ( x ) f y ( y ) (cid:20) df x ( x ) dη f y ( y ) + df y ( y ) dη f x ( x ) (cid:21) = (cid:88) x,y { [ df x ( x ) dη ] f y ( y ) f x ( x ) + [ df y ( y ) dη ] f x ( x ) f y ( y )+ 2 df x ( x ) dη df y ( y ) dη } = (cid:88) x (cid:40) f x ( x ) (cid:20) df x ( x ) dη (cid:21) + (cid:88) y f y ( y ) (cid:20) df y ( y ) dη (cid:21) (cid:41) = (cid:88) x (cid:20) ddη log f x ( x ) (cid:21) f x ( x )+ (cid:88) y (cid:20) ddη log f y ( y ) (cid:21) f y ( y )= I x + I y (31)In (31) the normalization property of probability distribution( f z ( z ) , z ∈ { x, y } ) is used: (cid:88) z f z ( z ) = 1 (32) (cid:88) z df z ( z ) dη = 0 (33)Therefore, to calculate the Fisher information I QTC1 for thejoint probability distribution P ( N p , N r , N c ) , which is a prod-uct of independent Poisson distributions, it suffices to calculatethe sum of Fisher information of the probability distributionsof N p , N r and N c : I QTC1 = + ∞ (cid:88) N p ,N r ,N c =0 (34) P ( N p = N p , N r = N r , N c = N c ) × (cid:20) ∂∂η p log P ( N p = N p , N r = N r , N c = N c (cid:21) = + ∞ (cid:88) N p =0 (cid:20) ∂∂η p log f ( N p , P p τ ) (cid:21) f ( N p , P p τ )+ + ∞ (cid:88) N r =0 (cid:20) ∂∂η p log f ( N r , P r τ ) (cid:21) f ( N r , P r τ )+ + ∞ (cid:88) N c =0 (cid:20) ∂∂η p log f ( N c , P c τ ) (cid:21) f ( N c , P c τ ) It can be shown that for Poisson distribution f ( k, λ ) = e − λ λ k k ! (35)whose mean value λ is parametrized by η , the Fisher infor- mation is given by: I k = + ∞ (cid:88) k =0 (cid:20) ∂∂η p log f ( k, λ ) (cid:21) f ( k, λ ) (36) = + ∞ (cid:88) k =0 f ( k, λ ) (cid:20) df ( k, λ ) dλ dλdη (cid:21) = 1 λ (cid:20) dλdη (cid:21) ∞ (cid:88) k =0 e − λ λ k k ! ( k − λ )= 1 λ (cid:20) dλdη (cid:21) Then from (35) we obtain I QTC1 = 1 P p τ (cid:20) dP p dη p τ (cid:21) + 1 P r τ (cid:20) dP r dη p τ (cid:21) + 1 P c τ (cid:20) dP c dη p τ (cid:21) , (37) = τ ν (cid:20) (1 − η r ) P p + η r P r + η r P c (cid:21) The Fisher information I QTC2 and I CTI is obtained fromPoisson distribution P ( N c = N c ) = f ( N c , P c τ ) and P ( N p = N p ) = f ( N p , P p τ ) . Then it directly follows from (37) that: I QTC2 = η r P c τ ν (38)and I CTI = (1 − η r ) P p τ ν (cid:12)(cid:12)(cid:12) η r =0 (39) = ν P p τ. FIGURE CAPTIONS
Fig. 1
Top: the schematic of the SPDC process in thesemiconductor waveguide. The red wave λ is the pumplight at 783nm, the green λ p and blue λ r wave is thedown-converted probe and reference photon that is inhorizontal (vertical) polarization. Middle: joint probabilitydistribution of photon detection time. t p : detection time ofthe probe photon; t r detection time of the reference photon.The standard deviation of the detection time difference is ∆( t p − t s ) = 100 ps for this plot. Bottom: Block diagram ofthe experiment setup. Fig. 2
The schematic of the experimental setup isdivided into two parts: the source and detection partand the transceiver part. The source and detection part(green background) includes the photon pair source anddetectors. Pump laser: Ti-Sapphire CW laser at 783nm. PBS:polarization beam splitter. LPF: long pass ( > Fig. 3
Photograph of the transceiver part of the experimentsetup. Different optical elements are marked with red boxes.The green line marks the optical path of the probe photonbefore it hits on the target object. The reflective mirror on thebottom-right corner is purely for optical alignment purposes.
Fig. 4 the scanning electron microscope image of thesemiconductor waveguide
Fig. 5
Left:the time series of different detector functions S QTC1 , S QTC2 and S CTI over 1400 independent targetexperiment. X-axis: the index of independent experimentthat last time τ = 0 . s. Blue (red) bar: probability densitywhen the target is absent (present). Blue (red) solid line,the theoretically calculated probability distribution usingthe Gaussian signal approximation (21). The histogramsare calculated from a larger portion of data ( . × independent experiments). Fig. 6
Left:ROC curves for the detector function S QTC1 , S QTC2 and S CTI for τ = 0 . s, τ = 0 . s and τ = . s (fromtop to bottom). Red curve: the theoretical ROC curve for S QTC1 calculated based on the Gaussian signal approximation.Red triangles:the experimental ROC curve for S QTC1 . Theblue curve, blue triangles, and black curve, black trianglesare similarly plotted for detector functions S QTC2 and S CTI ,respectively. Dashed black curve: theoretical ROC curvefor the detector function S CTI with longer integration time τ (cid:48) = k τ τ such that the detection rate P d same as that of S QTC1 is achieved at false alarm rate P fa = 10 − . The ratiosare k τ = 42 . , . , . for τ = 0 . s, . s, . s .For the same integration time τ , the CTI detection protocolcan also achieve the same performance with . , . , . times more source power or total probe photon transmission.Right: the same ROC curve plotted with log scaled P d axis. Fig. 7
Theoretical ROC curves with measurement time τ = 0 . s and improved effective time uncertainty ∆ t eff or reference photon transmission η r for the QTC detectionprotocol. Black: the CTI detection protocol for comparison;Red: the QTC detection protocol ( S QTC1 ) without any im-provement; Black dashed: the CTI detection protocol with (cid:39) . times longer measurement time; Red dashed: the QTCdetection protocol with improved effective time uncertainty ∆ t eff = 25 ps ; Red dotted: the QTC detection protocol withperfect reference photon transmission η r = 100%= 100%