Temporal Loop Multiplexing: A resource efficient scheme for multiplexed photon-pair sources
aa r X i v : . [ qu a n t - ph ] M a r Temporal Loop Multiplexing: A resource efficient scheme for multiplexed photon-pairsources
Robert J.A. Francis-Jones ∗ and Peter J. Mosley Centre for Photonics and Photonic Materials, Department of Physics,University of Bath, Bath, BA2 7AY, United Kingdom. (Dated: February 27, 2018)Single photons are a vital resource for photonic quantum information processing. However, evenstate-of-the-art single photon sources based on photon-pair generation and heralding detection haveonly a low probability of delivering a single photon when one is requested. We analyse a scheme thatuses a switched fibre delay loop to increase the delivery probability per time bin of single photonsfrom heralded sources. We show that, for realistic experimental parameters, combining the outputof up to 15 pulses can yield a performance improvement of a factor of 10. We consider the futureperformance of this scheme with likely component improvements.
Heralded single photon sources based on photon-pairgeneration in nonlinear media are at the forefront of de-velopments in photonic quantum technologies [1]. Pho-tons are created in pairs, usually by a pulsed pump laser,allowing the delivery of a single photon in a well-definedtime bin to be conditioned on the detection of its twin [2].The simplicity of this approach is attractive, but, becausepair generation is spontaneous, the generation probabil-ity must be kept low in order to limit contaminationfrom higher-order photon-pair components [3]. Com-bined with the effects of loss, the probability of deliveringa heralded photon from any individual pump laser pulseis typically less than 1%.Realistic quantum information protocols require manysingle photon sources to be operated simultaneously [1].The probability of this occurring, P , for N sources be-comes vanishingly small if the individual probability ofeach source producing a photon p ≪ P = p N .One method of increasing p is by multiplexing – activelycombining the output from several generation modes us-ing delay, feed-forward, and a fast switch [4–7]. Com-bining the output of pair generation in several separatesources – spatial multiplexing – brings significant perfor-mance benefits, but the resource overhead in nonlinearmedia and switches is high [8, 9].Alternatively, one can combine the output from dif-ferent temporal modes of the same source, bringing anoverhead cost only in time and not in additional physicalresources [10–14]. In this letter, we present a novel mul-tiplexing scheme in the temporal domain based on a fibreloop that uses only a single optical switch and delay line.We numerically evaluate the performance of the schemein the presence of imperfect detection, switch loss andattenuation in the delay line by extending the methodswe presented in [15] and consider the realistic limits tosource performance.Fig. 1 shows the proposed setup. A non-linear medium( χ (2) crystal or χ (3) optical fibre) is pumped by a pulsedlaser. The generated signal-idler pair is first split in wave-length, the signal photon is sent to a detector to heraldthe presence of the remaining idler photon. In this work, PulsedPump Pair Generation&HeraldingN-LMedium HeraldingSignalStorageLoop t = 1t = m ...
OpticalOutput FIG. 1. Schematic of temporal multiplexing scheme. See textfor details. we consider a detector that has photon number resolving(PNR) capability, though the analysis could be equallywell applied to a binary detector. The heralded idler pho-ton is coupled into fibre and subjected to a fixed delay togive sufficient time for detection and feed-forward. Fol-lowing the fixed delay a 2 × m laser pulses, labelled from t = 1 to t = m inorder of arrival, enters the nonlinear photon-pair gener-ation medium. If a pair is generated on the first pulseand the heralding detector fires, the switch is set to thecrossed state so that the corresponding idler photon isrouted into the the storage loop. On the next pulse if theheralding detector fires again the switch is once again setto the crossed state, simultaneously rejecting the photonfrom the previous pulse and storing the new photon inthe loop. If there is no heralding event, then the switchremains closed and photon in the storage loop completesanother pass through the switch. This process is repeatedup to and including the m ’th pulse and we refer to m asthe multiplexing depth .On the m ’th pulse, if there is no successful heraldingevent then the switch routes the stored photon amplitudeinto the optical output. However, if there is a successfulheralding event on the final pulse the switch allows thenewly-generated photon straight through to the output.Hence the photon leaving the output following the m ’thpulse may have passed through the switch and storageloop anywhere from t = 1 to t = m times and thereforeaccrued a vastly different amount of loss. At low valuesof t a photon must make more passes through the loopto be used after the m ’th pulse and so its contribution tothe overall probability of successfully delivering a singlephoton from the scheme is lower compared to photonsgenerated on later pulses where t → m . Furthermore, theloss of the switch and storage loop have a large effect onthe probability of successfully delivering a single photon.To evaluate the performance of the proposed systemwe follow a similar approach as outlined in [15]. We firstassume that the photon-pair source has been engineeredto generate signal and idler photons in only two spatio-temporal modes such that the modes are only correlatedin photon number [2]. This allows us to describe theprobability amplitudes of the state-vector, | Ψ i = a | s , i i + a | s , i i + a | s , i i + · · · , (1)with thermal statistics: | a n | = p th ( n ) = 1(¯ n + 1) (cid:18) ¯ n ¯ n + 1 (cid:19) n , (2)where ¯ n is the mean photon number per pulse.A simple result can be obtained by only consideringthe first non-zero term in the state-vector expansion inEq. 1. The overall probability of successfully deliveringa heralded single photon on the m ’th pulse, p (success),is given by p (success) = 1 − m Y t =1 (cid:0) − p th (1) η d η tL (cid:1) , (3)where η d and η L are the detector efficiency and lumpedefficiency of one pass of the switch and storage loop.However, this expression is only valid at low values of¯ n . In order to describe the behaviour of the sourcemore faithfully at higher mean photon numbers we haveanalysed the effects of higher photon-number terms fromthe state vector. To do so we applied the formalismdeveloped in [15] in which we determined the effect ofconcatenated loss in spatially multiplexed single photonsources. We used this to calculate the reduced densitymatrix, ˆ ρ i ( n s , t ), describing the heralded idler state givena heralding detection of n s photons on the t ’th pulse anddelayed until the m ’th pulse. We modelled the effects ofloss by applying a beam splitter transform in which thetransmission coefficient is given by the lumped efficiencyof t passes through the switch and delay line.For each pulse t there exists a set of normalised densitymatrices { ˆ ρ i ( n s , t ) : t, n s ∈ N } with Tr { ˆ ρ i ( n s , t ) } = 1, which are dependent on the number of photons n s de-tected by the heralding detector. By working with aPNR detector all detection results where n s = 1 can beignored. To extract the probability of successfully deliv-ering a single photon from the t ’th pulse we multipliedˆ ρ i ( n s = 1 , t ) by the probability of achieving a heraldingdetection result of n s = 1 and calculated the overlap witha pure single photon Fock state: p (success , t ) = p ( n s = 1) h | ˆ ρ i ( n s = 1 , t ) | i Tr { ˆ ρ i ( n s = 1 , t ) } . (4)The overall probability of successfully delivering a singlephoton from a pulse train divided into different tempo-ral bins, each containing m pulses is found through theproduct p (success , m ) = 1 − m Y t =1 (1 − p (success , t )) . (5)The contribution to noise, defined as a successful herald-ing detection followed by the delivery of an idler statecontaining more than one photon, was found through, p (noise , m ) = 1 − m Y t =1 − ∞ X n =2 h n | ˆ ρ i ( n s = 1 , t ) | n i Tr { ˆ ρ i ( n s = 1 , t ) } ! , (6)When ˆ ρ i is properly normalized such that Tr { ˆ ρ i } = 1,this reduces to: p (noise , m ) = 1 − m Y t =1 X n =0 h n | ˆ ρ i ( n s = 1 , t ) | n i ! , (7)it is nevertheless important to note that the accuracyof both p (success , m ) and p (noise , m ) will depend on thevalue of n at which the calculation is truncated due tothe effect this has on the normalization of ˆ ρ i . We definea corresponding signal-to-noise ratio,SNR = p (success , m ) p (noise , m ) , (8)that allows us to make direct comparisons of systemswith disparate characteristics. Using these techniques,we have simulated numerically the effects of higher-orderphoton-pair components on a temporally-multiplexedsource to enable the optimization of real-world systemsin the presence of loss.Figure 2 shows the performance of the proposed mul-tiplexing scheme with η d = 0 . η L = 0 . m . As the multiplexing depthis increased the resulting probability of successfully de-livering a single photon from the output increases. Thecorresponding SNR is shown in Fig. 2(b). To maintaina high SNR the mean photon number is constrained tobe low in order to reduce the contribution from multi-photon components in the heralded density matrix. By p ( s u cc e ss ) (a) S i gna l − t o − N o i s e ¯ n ¯ n (b) ∆ p ¯ n n = 2 n = 4n = 5n = 3 (c) p ( s u cc e ss ) (d) FIG. 2. a.) Variation in p (success) with increasing ¯ n for mul-tiplexing depths of m = 1 (blue), 3 (green), 5 (red) and 15(purple) pulses. b.) Corresponding signal-to-noise ratio. c.)Difference (∆ p ) between p (success) calculated using Eq 3 andthe full numerical calculation with up to 5 photon numbercomponents. Need to label the lines somehow d.) The in-crease in p (success) with increasing multiplexing depth at afixed ¯ n (green squares) and fixed SNR = 100 (blue circles).Triangles show the comparison with a spatial multiplexingscheme for fixed ¯ n and fixed SNR in teal and red respectively.Simulations carried out with a detector efficiency η d = 0 . η L = 0 . multiplexing, the overall probability of successfully deliv-ering a single photon at the output is increased at fixedSNR, as seen in Fig. 2(d).It is insightful to find the limits of validity to the sim-ple analytic result of eq. 3 and thence to set the pointat which our numerical calculations can safely be trun-cated. The difference between the analytic result and thenumerical density matrix method truncated at differentnumbers of photon pairs is plotted in Fig. 2(c). We cansee that for larger mean photon numbers it is essentialto include higher-order terms in the density matrix, how-ever, the benefit in using any amplitudes above 5 photonpairs quickly diminishes. As a result, all subsequent cal-culations were made by truncating the state vector at n = 5.Figure 2(d) shows the increase in p (success) with mul-tiplexing depth m . Results from our previous simulationsof a spatial multiplexing scheme are shown for compari-son [15]. Our numerical approach allows a direct compar-ison between multiplexed devices at a fixed SNR of 100. It can be seen that the temporal multiplexing loop yieldsan increase in p (success) commensurate with spatial mul-tiplexing schemes of approximately the same depth, butwith a large reduction in the resource overhead requiredfor implementation. In the case of temporal multiplex-ing the exponentially diminishing gain in p (success) withincreasing passes through the storage loop leads to theimprovement saturating at larger values of ¯ n as shown inFig. 2(d). For small values of ¯ n < .
01, as the multiplex-ing depth increases, p (success) tends to a constant valuegiven bylim m →∞ { p (success , m ) } = (cid:18) η L − η L (cid:19) p (success , . (9)This is in contrast to spatially-multiplexed sources, inwhich the probability of success continues to increasewith multiplexing depth over the range studied here;nevertheless, as shown in [15], even in the spatial case p (success) will begin to decrease at a value dependent onthe concatenated switch loss.At a fixed SNR of 100, a storage loop and switch withtotal loss η L = 0 . m = 8 pulses. Thisrequires only one photon-pair source, one detector, andone switch. To achieve a similar increase in performancefrom a spatially-multiplexed scheme would require vastlymore resources: 8 sources, 8 detectors, and 7 switches anddelay lines. Not only is building such a network compli-cated and expensive, but the sources and delays mustbe carefully matched to ensure that the photons outputare in identical pure states. The resource scaling for theschemes discussed is outlined in Table I.We note that after temporal multiplexing has takenplace the number of time bins in which a photon can bedelivered has been reduced by a factor of m . Therefore,in terms of number of heralded single photons delivered inone mode per second, this scheme may be outperformedby non-switched photon-pair sources pumped by a high-repetition-rate laser [16]. Nevertheless, if our goal is todeliver several single photons from independent sourcessimultaneously the scheme will yield huge improvementsover current source architectures. This is summarizedin terms of the waiting time to deliver N single photonsfrom N independent sources shown in Fig. 3.It is clear that to construct the highest performancesingle photon sources the loss of all the components be-tween the point of generation and the output must beminimised. Switch loss is critical in this endeavour.Notwithstanding this, our analysis shows that, in thelight of the inevitable imperfections in currently-availablecomponents, meaningful gains in source performance canbe readily achieved by implementing this scheme.Our method of analysis at a fixed SNR enables theeffects of detector inefficiency and switch loss to bestraightforwardly compared. Fig. 4 shows the behaviourof p (success) with varying detector and switch efficiencies TABLE I. Multiplexing scheme performance comparison. Improvement calculated for multiplexing depth m = 8, relative tosingle source with a heralding detector of efficiency η d = 0 . η L = 0 . n ImprovementTemporal 1 1 1 R p /d
100 - 6 . d d d − R p
100 - 7 . R p /d - 0.01 3 . d d d − R p - 0.01 4 . −5 No. of Independent Photons W a i t i ng T i m e ( s ) FIG. 3. The waiting time to deliver N single photons from N independent sources at fixed SNR = 100. Blue (solid) -Shows the waiting time for a standard single-photon sourcepumped at 80MHz. Green (circles) - A multiplexing schemewith detector efficiency η d = 0 .
7, switch efficiency η L = 0 . η d = 0 .
98, switchefficiency η L = 0 .
95 and a multiplexing depth of 15. For largenumbers of independent photons, these temporal multiplexingschemes exhibit a distinct advantage over traditional singlephoton sources. at a fixed SNR = 100 at a fixed multiplexing depth of10.We see in Fig. 4(a) that at low values of η d the de-tector has limited ability to discriminate between pulseson which a single pair or multi-pair generation event hasoccurred. The source must then be operated at very lowmean photon number in order to maintain the signal tonoise, severely limiting the success probability. As thedetector efficiency increases, the probability of successgrows rapidly due to the larger value of mean photonnumber that can be accessed; as η d → p ( s u cc e ss ) (a) p ( s u cc e ss ) (b) ¯ n p ( s u cc e ss ) (c) S w i t c h E ff i c i en cy D e t e c t o r E ff i c i en cy p ( s u cc e ss ) (d) FIG. 4. a.) Variation of p (success) with detector efficiency, η d ,for a fixed switch efficiency ( η L = 0 .
8) at fixed SNR = 100.b.) Variation of p (success) with switch efficiency, η s , for afixed detector efficiency ( η d = 0 .
7) at fixed SNR = 100. Allpoints correspond to multiplexing depth of 10. c.) Realisticfuture device with η d = 0 .
98 and η L = 0 .
95, multiplexingdepth 1 (solid), 5 (dashed), 10 (dotted) and 15 (dot-dash)which approaches deterministic operation. d.) Probability ofsuccess surface plot showing the overall effect of componentefficiency for a multiplexing depth of 10 at a fixed SNR = 100. of the fibre in the storage loop. In the case of Fig. 4(b),provided the switch has an efficiency greater than 0 . ACKNOWLEDGEMENTS
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