Non-classical Semiconductor Photon Sources Enhancing the Performance of Classical Target Detection Systems
Haoyu He, Daniel Giovannini, Han Liu, Eric Chen, Zhizhong Yan, Amr S. Helmy
JJOURNAL OF JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. XX, NO. X, JULY 2019 1
Non-classical Semiconductor Photon SourcesEnhancing the Performance of Classical TargetDetection Systems
Haoyu He, Daniel Giovannini, Han Liu, Eric Chen, Zhizhong Yan, Amr S. Helmy,
Senior Member, IEEE,
Abstract —We demonstrate and analyze how deploying non-classical intensity correlations obtained from a monolithic semi-conductor quantum photon source can enhance classical targetdetection systems. This is demonstrated by examining the advan-tages offered by the utilization of the non-classical correlations ina correlation based target detection protocol. We experimentallydemonstrate that under the same condition, the target contrastobtained from the protocol when non-classical correlations areutilized exhibits an improvement of up to .
79 dB over the bestclassical intensity correlation-based target detection protocol [1],under .
69 dB channel loss and excess noise .
40 dB strongerthan the probe signal. We also assessed how the strong frequencycorrelations within the non-classical photon pairs can be used tofurther enhance this protocol.
Index Terms —Radar target recognition
I. I
NTRODUCTION O PTICAL target detection has been receiving increasingattention owing to many emerging applications inthe domains of computing, human/machine interaction,LIDAR, and non-invasive biological imaging, amongst others.Conventionally, the sensitivity of optical target detection couldbe improved by increasing the source brightness, detectorsensitivity or improving the throughput of the optical setup.In addition, it has been shown that such sensitivity could alsobe significantly boosted through using quantum propertiesof entangled light [2], [3], [4], where one photon servesas a probe and the other as a reference. More recent workproposed that entanglement within non-classical photon pairscould be utilized to enhance the target detection sensitivityeven beyond conventional limits encountered in the classicalregime [5]. However, a significant level of complexityin the instrumentation involved including phase-sensitivejoint detection is essential to boost the target detectionsensitivity beyond the classical regime limits. As such,formidable challenges lie ahead on the route to harvesting theentanglement advantages because, amongst other issues, itrequires sub-wavelength-level stabilization of optical phasesbetween the probe and reference photon.The previous demonstrations that utilize non-classicalstate of light to enhance target detection systems require
The first three authors contributed equally to this work. Haoyu He, DanielGiovannini,Han Liu, Zhizhong Yan, Eric Chen and Amr Helmy was withthe Edward S. Rogers Department of Electrical and Computer Engineering,University of Toronto, 10 King’s College Road, Toronto, Ontario M5S 3G4,Canada table-top, mechanically unstable and poorly scalable setups.For example, previous work has relied entirely on bulk opticsand nonlinear crystals, such as BBO and PPLN [5], [1], forthe generation of the entangled photon pairs that illuminatethe target of interest. For practical target detection protocolsutilizing non-classical photon pairs, the source and theassociated setup, need to offer a form factor which enablesboth remote operation and quantum state generation in thedirect vicinity to the object under illumination.There has been astounding progress in the prowess ofnon-classical sources in the last decade [6], [7]. In particular,it has been shown that integrated monolithic semiconductordevices can be used to generate and tailor high-qualityquantum states of light in active semiconductor structures,such as AlGaAs devices [8], [9]. Such structures can directlyproduce entangled photon pairs without any additionaloff-chip interferometry, spectral filtering, compensation, orpost-selection and then be coupled effectively into optical fiberor integrated topic target detection systems. The flexibility inwaveguide structure design also allows for efficient dispersioncontrol and quantum state engineering.In this work, we exploit a monolithic quantum light sourcebased on a semiconductor device to enhance the performanceof the intensity correlation based target detection protocol,which is otherwise classical. We demonstrate a significantperformance enhancement compared to a similar detectionsystem using classical sources. The performance enhancementis also comparable with the previous comparable systems [1]that is based on bulk non-classical light source. In addition,we discussed a possible approach to further enhancing the per-formance of the intensity correlation based protocols withoutdecreasing the flux of the probe photon, through utilizing thestrong frequency correlation within the non-classical photonpairs.II. I
NTENSITY CORRELATION TARGET DETECTION WITHNON - CLASSICAL PHOTON PAIRS
A. Target detection with intensity correlation
The intensity correlation signal could be defined as thecovariance between the total photon number operator of theprobe mode N p and the reference mode N r [1]: S = N p N r − (cid:104) N p (cid:105)(cid:104) N r (cid:105) , (1) a r X i v : . [ phy s i c s . op ti c s ] A p r OURNAL OF JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. XX, NO. X, JULY 2019 2
PBSPump laser(80 MHz, 780 nm) 80 MHz trigger780 nmdetector CCD P B S H W P L P fi l t e r PBS SMFFiber BSEDFA BP filter Attenuator
NOISE MODULE LOSS MODULE
Fiber BSFiber BS
LOSS MODULENOISEMODULE (a) (b)
80 MHz trigger
HV HV
Gated SPADsTDC Gated SPADsTDC
Probe R e f e r e n c e Fig. 1. Schematics of the experimental setup for (a) the nonclassical source enhanced protocol(ICQ), and (b) the classical protocol (ICC). The left-hand side ofthe ICC setup is omitted for clarity. In ICQ, the H-polarized photon in each pair produced by type-II SPDC is used as a reference beam; the V-polarized photonis used as a probe beam. In ICC, the classically correlated probe and reference beams are obtained by sending the H-polarized photon through a balancedbeamsplitter. HWP: half-wave plate; PBS: polarizing beamsplitter; BS: beamsplitter; BRW: Bragg reflection waveguide; LP: long-pass; BP: band-pass; SMF:single-mode fiber; SPAD: single-photon avalanche diode; TDC: time-to-digital converter.
Note that the average value (cid:104) S (cid:105) is independent of environ-mental noise and equals zero when the target is absent. Thisproperty may be useful in practical applications where thenoise power is drifting, and a priori information about it is hardto obtain[1]. The contrast ε of the object could be defined asthe contrast between S in and S out (the subscript ‘in’ and ‘out’denote the presence and absence of the object, respectively)and normalized against its standard deviation : ε = (cid:104) S in (cid:105) − (cid:104) S out (cid:105) (cid:112) (cid:104) δ S in (cid:105) + (cid:104) δ S out (cid:105) (2)To quantify the sensitivity enhancement brought by thestrong intensity correlation of non-classical photon pairsources, we are comparing the intensity correlation targetdetection protocol with non-classical photon pair sources andthat with optimal classical sources, namely, correlated thermalbeams [1]. For brevity, these two protocols will be referredto as the ICQ and ICC protocol in the rest of this paper,respectively. However, it should be noted that in practicalapplications coherent light and intensity detection (will bereferred to as the INT protocol) are often used for classicaltarget detection and the ICC protocol may not be optimal.However, the reason why the ICC protocol is considered forcomparison is two-fold. First, the ICC protocol has a similarexperimental setup and is therefore directly comparable to theICQ protocol from an implementation point of view. Second,the ICQ protocol we demonstrate here suffers from the largetransmission loss of the reference light. This is an experimentalimperfection and could be alleviated with a better detectorand collection optics. However, since the performance of theICC protocol is also affected by the transmission loss of thereference photons, comparisons between these two protocolscan still reflect the quantum enhancement of the quantum in-tensity correlation regardless of the experimental imperfection.In the appendix, the ICQ protocol is also compared to theINT protocol. The result shows that the ICQ protocol cannot outperform the INT protocol if the INT protocol transmitsall the probe light in one single pulse. However, if the INTprotocol spreads the probe light in the same number of pulses that have equal average photon number as the ICQ protocol,then the ICQ protocol could possibly outperform the INTprotocol, provided that perfect transmission of the referencelight can be achieved. B. Experimental setup
The experimental approach for the ICQ protocol relieson the generation of correlated photon pairs from a type-IIspontaneous parametric down-conversion (SPDC) process.Our semiconductor quantum light source is capable ofgenerating high-quality entangled states with highly versatileand tunable properties [10], including non-degenerate,continuously tunable operation [11]. Its operation is basedon a dispersion engineered AlGaAs waveguide architecture[9].The semiconductor source is pumped using a femtosecondpulsed laser. For each pump pulse, the down-convertedphotons are always generated with different polarizations inpairs. Therefore the number of vertically polarized SPDCphotons and horizontally polarized SPDC photons arealways equal. The average number of photon pairs generatedby one pump pulse is denoted by µ , which is typicallymuch less than one. This joint state with correlated photonnumber in different polarizations could be used to detectthe presence or absence of a weakly reflecting object. Inthe experimental setup for ICQ (shown in Fig. 1(a)), thesignal and idler SPDC photons are deterministically separatedvia a polarization beamsplitter (PBS) into the reference andprobe beams. The reference photon is detected (with totaldetection efficiency η r , including both optical losses anddetection efficiency) immediately and the probe photon issent toward the reflecting object. The reflectivity of the objectis modeled by mixing the probe photons with environmentnoise upon a fiber beamsplitter with power transmission η o ( η o = η e , to simulate the additional lossof the probe channel, and including the detector efficiency OURNAL OF JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. XX, NO. X, JULY 2019 3 as well) and detected by a second single photon detector.The total efficiency is given by η p = η e η o . The probe andidler photons detected on both detectors are time tagged forcorrelation analysis. In the ICC setup (shown in Fig. 1(b)),the correlated probe and reference photons (with same meanphoton number per mode µ ) are obtained through splittingthe (locally) thermal state of the SPDC signal light on abalanced beamsplitter.It must be noted that in our experimentally achievableimplementation of the ICC protocol, the generated probeand reference photons are not optimal: the down-convertedsignal light are not in a single-mode thermal state, but rathera statistical mixture of thermal states of many orthogonalfrequency modes. This spectral multimodeness could becharacterized by the Schmidt number M of the down-converted states. As shown in Eq. (4), the ICC coincidencecounts is a function of the number of the Schmidt modes M of the down-converted photon pairs. The Schmidt number M ≥
13 for the photon pair source is obtained through thenumerical Schmidt decomposition of the experimentallymeasured joint spectral intensity (Fig. 4). By imposing M =1, we obtain the best theoretical single-mode ICC protocolachievable with single-mode thermal state sources. This idealcase is used to calculate the quoted ratios ε ICQ / ε ICC togetherwith the experimental results from the ICQ protocol, toquantify the enhancement of the ICQ protocol.For each experimental data point, the contrast ε ICQ ( ε ICC ) ofthe ICQ (ICC) protocol are directly calculated according to thedefinition (2), with experimentally measured photon countingstatistics. To obtain a theoretical plot of ε ICQ and ε ICC one canstill apply this definition, but with photon counting statisticsexpressed as a function of the probe and reference efficiency, η p and η r respectively, and the pair generation rate of thesource, µ (full derivation of the photon counting statisticscould be found in the appendix.) (cid:104) N p N r (cid:105) ICQ = η p η r µ ( µ ( + M ) + ) + η r µ (cid:104) N b (cid:105) (3) (cid:104) N p N r (cid:105) ICC = η p η r µ ( + M ) + η r µ (cid:104) N b (cid:105) (4) (cid:104) N p (cid:105) = η p µ + (cid:104) N b (cid:105) (5) (cid:104) N r (cid:105) = η r µ (6) (cid:104) δ S (cid:105) = (cid:104) δ ( N p N r ) (cid:105) (cid:39) (cid:104) N p N r (cid:105) − (cid:104) N p N r (cid:105) (7)where (cid:104) N b (cid:105) denotes the average noise photon number at thedetector that overlap with each induividual probe pulse. Notethat the expressions (3)-(7) apply to both the target present( η p = η o η e ) and absent ( η p =
0) case. The values of µ , η p , and η r used in the theoretical calculation are calculated from theaveraged photon counting statistics from different experiments.III. E XPERIMENTAL R ESULTS
Fig. 2 shows a comparison between the ICQ and ICCprotocol for different brightness levels, µ . In our experiment,the number of realizations is equal to the number of recordedpump pulses. The duration of each measurement is 40 s, (a) probe photon per pulse C o n t r a s t ( × ) ICQICC (b) probe photon per pulse C o n t r a s t ( × ) ICQ ICC (c) probe photon per pulse c o n t r a s t r a t i o I C Q I CC ( d B ) p = 3.8 × 10 , N b = 5.6 × 10 p = 1.1 × 10 , N b = 2.1 × 10 Fig. 2. (a) Normalized contrast ε as a function of the average photon pairnumber generated per pulse, µ , for the ICQ (solid black) protocol, with noadditional loss and noise( η r = . × − , η p = . × − , (cid:104) N b (cid:105) = . × − ).Black error bar: the experimentally measured contrast for the ICQ protocol.Dashed black curve: the bound of the maximum achievable contrast ε for thesingle-mode ICC implementation ( M =
1, with same channel transmissionand noise power). Dashed blue line: the noise floor for the contrast ofthe ICC experiment. Error bars are given by the standard deviation ofthree measurements. (b) Normalized contrast ε as a function of the averagephoton pair number generated per pulse, µ , with additional loss and noiseinjection( η r = . × − , η p = . × − , (cid:104) N b (cid:105) = . × − ). (c) Contrastratio ε ICQ / ε ICC as a function of the average probe photon number generatedper pulse µ for plots (a) (black line) and (b) (red line). The value of ε ICQ and ε ICC are the theoretical value of contrast for the best ICC protocol and theICQ protocol (the dashed and solid curve in (a) and (b)). The experimentallyprobed parameter region is marked by the shaded area for the two experiments.The circles on the left of the solid curve correspond to the maximal valuesof the ε ICQ / ε ICC ratio, being 18.57dB (without additional loss and noise) and17.89db(with additional loss and noise) respectively. with a number of valid realizations of around 2 × . Theaverage probe photon number generated per pump pulse, µ ,is calculated from the photon counting statistics((3),(5),(6)).The ICQ protocol performance is compared to only the besttheoretical (single-mode, M =
1) implementation of ICC(“ICC bound” in the plots). See the appendix for detailsof the experimental implementation and the effect of theSchmidt number of the photon pair source on the ICC contrast.Under 0 dB additional loss (introduced by the tunable lossmodule and the 50:50 beamsplitter) and 0 dB additional noise
OURNAL OF JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. XX, NO. X, JULY 2019 4 (a) noise photon per pulse log N b C o n t r a s t ( × ) ICQ ICC (b)
Probe channel efficiency p (×10 ) C o n t r a s t ( × ) ICQICC (c)
Probe channel efficiency p (×10 ) C o n t r a s t ( × ) ICQICC
Fig. 3. (a) Normalized contrast ε as a function of the average noise photonnumber per pulse, (cid:104) N b (cid:105) ( η r = . × − , η p = × − , µ = .
077 ). Blackerrorbar: the experimentally measured contrast for the ICQ protocol. Dashedblack curve: the bound of the maximum achievable contrast ε for the single-mode ICC implementation ( M =
1, with the same channel transmissionand source pair rate). Dashed blue line: the noise floor for the contrastof the ICC experiment. The maximum ε ICQ / ε ICC achieved is 9 . ε as a function of the probe channel transmissivitywith no additional noise ( η r = . × − , µ = . , (cid:104) N b (cid:105) = . × − ). Themaximum ε ICQ / ε ICC achieved is 11 . ε as afunction of the probe channel transmission with additional noise, η p ( η r = . × − , µ = . , (cid:104) N b (cid:105) = . × − ). The maximum ε ICQ / ε ICC achievedis 9 . (injected by the EDFA), the ICQ protocol shows an im-provement of up to 18 .
57 dB over the ICC contrast in low-brightness conditions for the region explored experimentally.This improvement is with respect to the theoretical ICC bound;the contrast ratio between the ICQ and ICC protocol becomeslarger at smaller photon flux (lower µ ), highlighting the betterperformance for the ICQ protocol in low-brightness condition.When introducing additional loss (3 dB beamsplitter loss) andnoise (13 .
40 dB compared to the detected probe signal), thisquantum advantage is still found to be considerable, as shownin Figs. 2(b) and (c). In both cases, the performance advantagesof the ICQ protocol over the ICC protocol decrease as thesource pair generation rate µ increases.The performance of the ICQ and ICC protocol underdifferent additional noise and loss levels are shown in Fig. 3.The loss is varied by adjusting the coupling efficiency of the adjustable fiber optic attenuator. In both the ICQ and ICCexperiment, the probe beam is mixed with thermal noise usinga 50:50 fiber beamsplitter, introducing a 3 dB loss into theprobe channel. The ICQ and ICC contrast ε as a function of thenoise injected into the probe path, is shown in Fig. 3(a). Whilethe theoretical value of ε for both ICQ and ICC decreases asmore noise is injected into the probe path, ICQ is shown to bemore resilient to noise compared to the ICC protocol. The ICQadvantage is also shown as a function of loss in the absenceand presence of further noise, in Figs. 3(b) and (c) respectively.This loss-tolerance property makes ICQ a suitable protocol todetect a low-reflection target in a high-loss environment.IV. D ISCUSSION
As evident from the results, a quantum enhancement hasbeen demonstrated in the ICQ protocol using a monolithic,on-chip quantum light source. A contrast enhancementpersists even in the presence of high levels of noise andadditional loss in the channel. Our experimental protocolproduces results comparable with previous experimental workin this area [1]. The ICQ protocol further shows its resilienceto noise and loss, especially in the low-brightness regime.Since the detection of each probe and reference photon aretime tagged, the photon counting statistics could also beused to calculate the traveled distance of the probe photonsfrom the time of flight of the probe photon. In addition tothe performance enhancement, the compact semiconductorwaveguide source of the ICQ protocol also enables largescale integration.A major limitation of the ICQ protocol is that the strength ofintensity correlation between the probe and reference photon(hence the enhancement of the target detection performance)is limited by the average number of probe photon per pulse µ . When the mean photon number per pulse µ decreases, theperformance advantage of the ICQ protocol as compared tothe ICC protocol increases, but at the price of sacrificing theabsolute performance of the ICQ protocol, as could be seenin Fig. 2(a) and Fig. 2(c). A possible approach to increasingthe performance of the ICQ protocol without decreasing µ is to utilize the frequency correlation that also exists withinnonclassical photon pairs: while the frequency of the probeand reference photon are broadband individually, the sum oftheir frequencies could be within a narrow frequency range,which implies strong frequency correlations. To see how thefrequency correlation could benefit target detection, it sufficesto consider a photon pair state | ψ (cid:105) that is correlated in thefrequency degree of freedom | ψ (cid:105) : | ψ (cid:105) = (cid:90)(cid:90) d ω p d ω r ψ ∗ ( ω p , ω r ) a † p ( ω p ) a † r ( ω r ) | (cid:105) (8)where a p ( ω p ) , a r ( ω r ) are the annihilation operators of theprobe and reference photon at frequency ω p and ω r , respec-tively and the function ψ ( ω r , ω r ) is the joint spectral ampli-tude. It could be further shown that | ψ (cid:105) can be decomposed OURNAL OF JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. XX, NO. X, JULY 2019 5 as a superposition of different photon pair states through theSchmidt decomposition: | ψ (cid:105) = ∑ n (cid:112) λ n A † p , n A † r , n | (cid:105) (9)where { A p , n } , { A r , n } are the different discrete frequency modeoperators for the probe and reference light and { λ n } are thecorresponding Schmidt eigenvalues. The number of the modepairs could be quantified as the Schmidt number M = / ∑ n λ n .The expression (9) suggests that, similar to the case of the ICQprotocol where probe and reference photons are always createdin pairs in different pulse pairs, the frequency correlated probeand reference photons are also created in pairs in different fre-quency mode pairs { ( A p , n , A r , n ) } . Therefore for each frequencymode pair ( A p , n , A r , n ) , the same intensity correlation analysisin the ICQ protocol applies. Since the number of differentfrequency mode pairs M could be very large, each of thefrequency mode pair ( A p , n , A r , n ) can have mean photon numbermuch less than one, which translates to high performanceenhancement of the ICQ protocol. In general, the Schmidtnumber of the non-classical photon pair could be approximatedby the ratio of the SPDC photon bandwidth and the SPDCpump bandwidth. For semiconductor waveguides with a spe-cific structure design[12], around 400nm of the SPDC photonbandwidth could be achieved. The detection of each frequencymode ( A p , n and A r , n ) could be done through frequency resolvedphoton counting or frequency to time mapping based onthe fast fiber spectrogram technique[13]. Fig. 4 shows theexperimental setup and result of the measurement of the jointspectral intensity. Through numerical Schmidt decompositionof the joint spectral amplitude (which is assumed to be thesquare root of the joint spectral intensity), the Schmidt numberof the pulsed SPDC photon pair source is estimated to bearound M =
13. Larger Schmidt number M could be achievedwith a narrowband pump.V. C ONCLUSIONS
We demonstrated an intensity correlation target detectionprotocol enhanced by non-classical light generated in a semi-conductor chip source. This is the first instance where aquantum enhancement in a target detection protocol over athermal background has been shown in an integrated platform.Our device can achieve up to 18 .
57 dB experimentally verifiedcontrast improvement over the classical intensity correlationtarget detection protocol, in the absence of additional loss andnoise. A high quantum contrast has also been measured evenunder both 29 .
69 dB loss, and noise 13 .
40 dB stronger than thedetected probe field. The ratio between the experimental valueof ε ICQ and the best theoretical ε ICC in equivalent conditions is17 .
79 dB. When separately analyzing the system performancein terms of noise and additional loss, we have experimentallydemonstrated a contrast enhancement up to 9 .
95 dB as afunction of noise, and 11 .
68 dB as a function of loss. We alsoproposed a method to further improve the performance of theICQ protocol by utilizing the strong frequency correlations ofSPDC photon pairs.
Pump laser(80 MHz, 780 nm) 80 MHz triggerSemiconductorchip source L P fi l t e r PBS HV Gated SPADsTDC5 km SMF1.5 1.55 1.60 1.651.621.601.581.561.541.521.501.48 Signal wavelength (nm) I d l e r w a v e l e n g t h ( n m ) (a)(b) Fig. 4. (a) A schematic of the experimental setup for joint spectral intensitymeasurements using the frequency to time mapping technique. LP: long-passfilter ( ≥ A PPENDIX AE XPERIMENT S ETUP D ETAILS
We refer to the photons in different polarization generated inthe semiconductor waveguide as signal and idler photons. Forthe ICQ experiment, the signal photon is used as a “local”reference and detected by an MPD InGaAs single-photondetection module, gated by the pulsed pump. The idler photonis used as the probe and mixed on a 50:50 fiber beamsplitterwith a broadband amplified spontaneous emission (ASE) noiseproduced by an erbium-doped fiber amplifier (EDFA). TheASE noise level is adjusted through a variable attenuator. Themixed output is detected by an id210 single-photon detector.To simulate the absence of a target, a beam block is placed inthe idler path. The id210 is externally triggered by the 80 MHzpump, open for 3 ns, with 20 µs dead time, and quantumefficiency set to 25 %. The time-to-digital converter used tocorrelate the signal is an id800 model.For the ICC experiment, only the signal path is used, withthe idler path blocked. The signal beam passes through a 50:50fiber beamsplitter to produce correlated thermal light. Onepath is used as a reference path and detected by the MPDdetector. The probe path is mixed with ASE noise by a 50:50fiber beamsplitter and detected by the id220 detector. Tunablelosses are introduced by adjusting the fiber-to-fiber couplingin the probe path. The no-target condition is simulated by
OURNAL OF JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. XX, NO. X, JULY 2019 6 disconnecting the probe path fiber from the fiber beamsplitterinput. A
PPENDIX BD ERIVATION OF PHOTON COUNTING STATISTICS
In the experiment, the light source used are Gaussianstates and the channel loss and noise could be modeled asGaussian operation upon Gaussian states[14]. As a result,the Gaussian state formalism could be used to compute theexpectation value (3)-(6). The advantages of using Gaussianstate formalism to compute (3)-(6) is two-fold:first it does notrequire complicated operator expansion (e.g. unitary transformbetween the input and output modes of the beam-splitter) sothe computation is much more scalable as the complexityof the optical setup increase; secondly the evolution of thequantum states could be modeled as the transform of theircovariance matrices and could be calculated symbolically with program. A Gaussian state is completely characterized by itscovariant matrix and first-order moment: σ kl = (cid:104) ˆ R k ˆ R l + ˆ R k ˆ R l (cid:105) − (cid:104) ˆ R k (cid:105)(cid:104) ˆ R l (cid:105) (10) d l = (cid:104) ˆ R l (cid:105) (11)Where ˆ R i − , ˆ R i is the x and p quadrature of the i th mode, re-spectively. The commutation relationship could be expressed: [ ˆ R k , ˆ R l ] = i Ω kl (12) Ω kl = ⊗ ni = ω (13) ω = (cid:20) − (cid:21) (14)Throughout the experiment setup, all the quantum stateshave zero first order moment (no coherent state component).The covariance matrices of the down-converted photon pairssource and the correlated thermal light source is given by: σ SPDC = µ + (cid:112) µ ( µ + ) µ + (cid:112) µ ( µ + ) (cid:112) µ ( µ + ) µ + (cid:112) µ ( µ + ) µ + (15) σ thermal = µ + µ µ + µµ µ + µ µ + (16)To model the loss on each channel, one could first append(direct sum) a vacuum covariance matrix: σ vac = I × η ) on the joint covariance matrix : S bs = √ η √ − η √ η √ − η −√ − η √ η −√ − η √ η (18)(note that only the relevant dimensions are shown in thesymplectic transform. The other dimensions of the symplectictransform corresponding to non-interacting modes are identi- ties and omitted)Then the covariant matrix after the loss isgiven by: σ loss = S bs σ source S Tbs (19)where σ source could be either σ SPDC for ICQ or σ thermal forICC. Mixing with thermal noise could be treated similarly,except that a thermal noise state with covariance matrix: σ noise = I × (cid:104) N b (cid:105) + σ vac (20)is appended(direct sum) to the source covariant matrixinstead and the beam-splitting is 50-50( η = σ SPDC , fin σ thermal , fin is given by: OURNAL OF JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. XX, NO. X, JULY 2019 7 σ SPDC , fin = (cid:104) N b (cid:105) + η p µ + (cid:112) η p η r µ ( µ + ) (cid:104) N b (cid:105) + η p µ + (cid:112) η p η r µ ( µ + ) (cid:112) η p η r µ ( µ + ) η r µ + (cid:112) η p η r µ ( µ + ) η r µ + (21) σ thermal , fin = (cid:104) N b (cid:105) + η p µ + (cid:112) η p η r µ (cid:104) N b (cid:105) + η p µ + (cid:112) η p η r µ (cid:112) η p η r µ η r µ + (cid:112) η p η r µ η r µ + (22)The final state is completely specified by the covariancematrices and the expectation value of the left-hand side ofequation (3)(4)(5)(6) could be calculated according to [15]and shown to be: (cid:104) N p N r (cid:105) ICQ = η p η r µ ( µ + ) + η r µ (cid:104) N b (cid:105) (23) (cid:104) N p N r (cid:105) ICC = η p η r µ + η r µ (cid:104) N b (cid:105) (24) (cid:104) N p (cid:105) = η p µ + (cid:104) N b (cid:105) (25) (cid:104) N r (cid:105) = η r µ , (26)To calculate the higher order moment (cid:104) δ S (cid:105) : (cid:104) δ S (cid:105) (27) = (cid:104) δ ( N r N p − (cid:104) N r (cid:105)(cid:104) N p (cid:105) ) (cid:105) (28) = (cid:104) δ ( N r N p ) (cid:105) (29) = (cid:104) N r N p N r N p − (cid:104) N r N p (cid:105) (cid:105) (30) (cid:39) (cid:104) N r N p (cid:105) − (cid:104) N r N p (cid:105) (31)The approximation in the last equation is valid becasue inthe intensity regime we are interested in, operator N r N p onlyhave eigenvalue 0 and 1. Thus N r N p N r N p = N r N p .In the analysis above single-mode SPDC/thermal source isassumed. However, in the actual experiment, the SPDC stategenerated by a single pump pulse is a tensor product of manysimultaneously squeezed states on different probe/referencemode pairs, and the state obtained by splitting the H polarizedSPDC photons(which are generated within a single pumppulse) is a tensor product of many correlated thermal statepairs: ρ multi_SPDC = ⊗ Mn = ρ SQZ (32) ρ multi_thermal = ⊗ Mn = ρ thermal_pair (33)Experimentally, the singles counting on each channel is tocount the total number of probe/reference photons in all theM probe/reference modes and could be model mathematicallyas: (cid:104) N p (cid:105) = M ∑ n = (cid:104) N p , n (cid:105) (34) (cid:104) N r (cid:105) = M ∑ n = (cid:104) N r , n (cid:105) (35)Where N p , n and N r , n are the photon number operator ofthe probe/reference mode of the n th probe-reference modepair.Similarly, the coincidence counting is modeled by the total number of probe photons in all the M probe modes times thetotal number of the reference photon in all the M referencemodes. (cid:104) N p N r (cid:105) = (cid:104) M ∑ h = N p , h M ∑ k = N r , k (cid:105) (36) = M ∑ h (cid:54) = k (cid:104) N p , h (cid:105)(cid:104) N r , k (cid:105) (37) + M ∑ n = (cid:104) N p , n N r , n (cid:105) (38)For simplicity, we assume the total µ probe/reference photonsand is evenly distributed among all M probe/reference modesand the total (cid:104) N b (cid:105) noise photon are evenly distributed amongthe M probe modes. Then for each of the M probe/referencepair, the signal mode result (3)-(6) applies with µ → µ M and (cid:104) N b (cid:105) → (cid:104) N b (cid:105) M : (cid:104) N p , n N r , n (cid:105) ICQ = η p η r µ M ( µ M + ) + η r µ M (cid:104) N b (cid:105) M (39) (cid:104) N p , n N r , n (cid:105) ICC = η p η r ( µ M ) + η r µ M (cid:104) N b (cid:105) M (40) (cid:104) N p , n (cid:105) = η p µ M + (cid:104) N b (cid:105) M (41) (cid:104) N r , n (cid:105) = η r µ M (42)Then the experimentally measured photon counting statisticsbecomes: (cid:104) N p N r (cid:105) ICQ = η p η r ( µ + µ + µ M ) + η r (cid:104) N b (cid:105) µ (43) (cid:104) N p N r (cid:105) ICC = η p η r µ ( + M ) + η r (cid:104) N b (cid:105) µ (44) (cid:104) N p (cid:105) = η p µ + (cid:104) N b (cid:105) (45) (cid:104) N r (cid:105) = η r µ (46)It could be seen that the coincidence counting in ICQ caseis only slightly affected by the multimodal nature of thesource because in typical SPDC regime µ (cid:29) µ M . Thus forsimplicity,(23) could still be used instead of (43) as a goodapproximation. A PPENDIX C COMPARISON WITH INTENSITY DETECTION PROTOCOL
To give a more complete assessment of the ICQ protocol,we also compare its performance to classical target detectionprotocols with coherent probe light and intensity detection.
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To be specific, we will consider the following two types ofclassical intensity detection protocol for comparison: • (a) the classical coherent state light source transmits asingle probe light pulse that contains a large number ofphotons. • (b) the classical coherent state light source transmits atrain of probe light pulses, each contains the same averagenumber of photons as the ICQ protocol.To quantify the performance of the protocol (a), it sufficesto calculate its error probability P e , COH : P e , COH (cid:39) exp ( − µ coh η p ) / µ coh is the total number of photons in the probe pulse.To reach the error probability level of 10 − , the number ofdetected photons η p µ coh needs to reach 8.5. Note that theeffect of background noise is negligible ( (cid:104) N b (cid:105) = . × − ),because of the small overlap of the single probe pulse and theenvironmental background noise. On the other hand, the errorprobability of the ICQ protocol is approximately given by: P e , ICQ = √ π (cid:90) ∞ √ K ε (cid:48) ICQ exp ( − t / ) dt (48)where K is the number of transmitted probe pulses and ε (cid:48) ICQ is the modified contrast defined as: ε (cid:48) ICQ = (cid:104) S in (cid:105) − (cid:104) S out (cid:105) (cid:112) (cid:104) δ S in (cid:105) + (cid:112) (cid:104) δ S out (cid:105) (49)Assuming µ = . , η p = . × − , (cid:104) N b (cid:105) = . × − , η r = K that is requiredto achieve P e , ICQ = − is 1 . × , which correspond todetection of η p µ K =
20 probe photons. The reason for theinferior performance of the ICQ protocol compared to theprotocol (a) is that the concentration of the probe light inone single pulse can effectively reduce the overlap betweenthe probe light with the continuous wave background noise.Therefore in the protocol (a) the effect of the noise backgroundon the target detection performance is minimized.Although sending a bright single probe pulse can reduce theeffect of environmental noise on the target detection perfor-mance, it is still desirable to spread the probe light into mul-tiple pulses in some target detection scenario, such as stealthoperation, where the distinguishability between the probephotons and the CW background noise is to be minimized.This corresponds to the protocol (b), whose performance canalso be quantified in the form of contrast similar to (2): ε INT = (cid:104) N p (cid:105) in − (cid:104) N p (cid:105) out (cid:112) (cid:104) N p (cid:105) in + (cid:104) N p (cid:105) out (50)Note that for coherent state probe light the variance of thephoton number equals the mean value of the photon number.The figure below compares ε ICQ , ε ICC and ε INT as functions ofthe environmental noise power (cid:104) N b (cid:105) as well as the transmissionof the reference photons η r . It could be seen that (1) in thelimit of perfect reference photon transmission η r =
1, the ICQprotocol has non-trivial performance advantage compared toprotocol (b) and (2) the performance of the ICQ protocol isclosely related to the transmission of the reference photon. average noise photon per pulse N b c o n t r a s t ( × ) ICQ r = 0.1%ICQ r = 4.0%ICQ r = 20.0%ICQ r = 100.0%ICC r = 100%COH Fig. 5. The contrast of the ICQ, ICC and intensity detection protocol asa function of the environmental noise power (cid:104) N b (cid:105) . The transmission of theprobe and the probe power is given by µ = . × − and η p = . × − . The authors thank OptoElectronic Components for the loanof an MPD InGaAs single-photon detection module.R
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