Multi-photon absorption limits to heralded single photon sources

Transcription

1 Multi-photon absorption limits to heralded single photon sources Chad Husko, Alex S. Clark, Matthew J. Collins, Chunle Xiong, and Benjamin J. Eggleton Centre for Ultrahigh-bandwidth Devices for Optical Systems (CUDOS), Institute of Photonics and Optical Science (IPOS), School of Physics, University of Sydney, NSW 2006, Australia Alfredo De Rossi, Sylvain Combrié, and Gaëlle Lehoucq Thales Research and Technology, Route Départementale 128, Palaiseau, France Isabella H. Rey SUPA, School of Physics and Astronomy, University of St Andrews, Fife, KY16 9SS, UK Thomas F. Krauss Department of Physics, University of York, YORK, YO10 5DD, UK These authors contributed equally to this work. 1. Photon-pair generation in the presence of nonlinear losses We define µ as the number of generated photon-pairs per pulse. For a source of photon pairs consisting of the signal and idler photons of a spontaneous four wave mixing (SFWM) process, this is described by µ = ( ντ)g = ( ντ)φ 2 NLΘ 2, (1) where ν is the bandwidth of the filtering in the signal and idler arms (here 60 GHz), τ is the temporal full-width halfmaximum of the pulse, β 2 is the group-velocity dispersion (GVD) of the medium, L is the length of the waveguide, Φ NL is the nonlinear phase shift and Θ the phase matching terms required by the parametric SFWM process [1 3]. This is essentially the four-wave mixing conversion efficiency, G = Φ 2 NL Θ2, with the phase matching equation: [ ] sinh(gl) 2 Θ 2 =. (2) gl These earlier works employed a linear definition of Φ NL = γp 0 L e f f S 2, where γ = k o n 2 /A e f f is the nonlinear coefficient with k 0 = 2π/λ P at the pump wavelength λ P, n 2 is the nonlinear refractive index, A e f f the mode area, P 0 is the peak pump power, and L e f f = (1 e αl )/α is the effective length in the presence of propagation loss α and S = n g /n 0 is the slow-down factor. It is known that Φ NL scales nonlinearly in TPA-limited [4] and ThPA-limited materials [5]. A recent work considers the role of these different Φ NL scalings in slow-light for the FWM process [6]. While the linear definition of Φ NL = γp 0 L e f f S 2 is valid for the GaInP (TPA-free) sample in the low-power regime investigated here, in the silicon (TPA-limited) samples one must consider the appropriate description for Φ NL [4]: Φ NL = k 0n 2 α 2 ln [ 1 + α 2 I o L e f f S 2], (3) with α 2 the two-photon absorption coefficient. One must also consider the slow-light scaling of the nonlinear effects, as well as linear scattering as demonstrated earlier [3, 5, 7, 8]. Note that Θ is also a function of Φ NL through the parametric gain term g, which is defined as: g = (ΦNL L 2 ) ( klin 2 + Φ ) 2 NL, (4) L with k lin = 2k pump k signal k idler the linear phase mismatch. For our degenerate pump case we evaluate k lin as: k lin = ( Ω) 2 β 2 (ω pump ) ( Ω)4 β 4 (ω pump ), (5) 1

2 where Ω = ω pump ω signal is the frequency detuning of the single photons from the pump and β 2 (ω pump ) and β 4 (ω pump ) are the group-velocity dispersion (GVD) and fourth-order dispersion (FOD) terms, respectively. For a further description of these terms with particular attention to photonic crystals, slow-light, and dispersion see Refs. [3, 6, 9, 10]. 2. Impact of TPA on the pump beam intensity and estimate of free-carrier density In these experiments only two-photon absorption (TPA) plays a significant role at the intensity levels required to generate photon pairs by SFWM. First, we consider the TPA-limited pump in the generation process. The first important quantity is: 1 η T PA = 1 + α 2 I o L e f f S 2, (6) where η T PA <1 is the attenuation due to TPA. The TPA process creates free carriers, which we also described in the main text. Free carrier absorption (FCA) of single photons is described by: η FCA = 1, (7) [1 + σn c L e f f (2α)] 1/2 where σ is the absorption cross section, N c is the carrier density and L e f f (2α) = 1 exp( 2αL)/(2α) [4,5]. Finally, the pump experiences linear loss as η α = exp( αl). Fig. 1 shows η pump versus Φ for the experimental data η exp. along with the analytic equations for TPA - η T PA, FCA - η FCA, and linear loss - η al pha. The pump beam in the experiments (full model) is described by: η pump = η T PA η FCA η α. With the analytic model we are able to add in the various effects one at a time. The top (dotted) curve is TPA only, while the middle (dashed) curve corresponds to TPA-FCA, with the full model also including linear loss shown as the bottom (solid) line. These results are further confirmed with an independent numerical nonlinear Schrödinger equation (NLSE) including free-carriers (dot-dashed curves) [4, 5, 11]. We estimate the free-carrier density N c in the analytic formulations both here and in the main text from these NLSE numerical solutions. Most importantly, we plot our experimental transmission measurements as the red circles, showing solid agreement with our analytic formulation. Fig. 1. Role of nonlinear loss processes. This figure shows analytic modeling of the pump loss due to: (i) TPA, (ii) TPA and free-carriers generated from TPA (FCA), and (iii) TPA, FCA, and linear loss. The circles indicate the experimental parameters, while the black dot-dashed lines indicate independent numerical nonlinear Schrödinger equation modeling (black lines) confirming the analytic formulation. The top (dotted) curve is TPA only, while the middle (dashed) curve corresponds to TPA-FCA, with the full model also including linear loss shown as the bottom (solid) line. 2

3 Fig. 2. Schematic of the experimental setup for the CAR measurements. 3. CAR model - matrix formulation of experimental losses and detection for two detectors We formulated a two detector model for the coincidence-to-accidental ratio (CAR) based on the matrix representation for the detection statistics similar to Ref. [12]. We begin by constructing a vector which describes the joint state of the two detectors P, for the signal and idler detectors S and I as shown in Figure 2: P = ( P SI,P SI,P SI,P SI ). (8) Each element of the matrix corresponds to different sets of detectors switching from a no click state (P SI ) to both clicked (P SI ) during an elementary observation time which we take as one pump repetition. We then construct 4x4 matrices for the probability a generated photon-pair causes the detectors to transition to the clicked state (M η ), M η = and the probability that dark counts in the detectors cause such a transition (M D ): M D = ( (1 ηs )(1 η I ) 0 0 0) η S (1 η I ) (1 η I 0 0, (9) η I (1 η S ) 0 (1 η S 0 η S η I η I η S 1 ( (1 Ds )(1 D i ) 0 0 0) D s (1 D i ) (1 D i 0 0. (10) D i (1 D s ) 0 (1 D s 0 D s D i D i D s 1 Here we have defined the channel efficiencies and dark counts for the signal and idler channels as η S,I and D S,I. We define the state of the detectors after the emission of up to i photon pairs as: P = i=0 p i M D (M η ) i P 0, (11) where P 0 is an initial vector with all the detectors set to off (i.e. P 0 = (1,0,0,0)) and p i are the probability distributions for the generated states. In our case p i is a function of the likelihood to generate a pair per pulse, defined previously as µ, following a thermal distribution: p i = µ i. (12) (1 + µ) i+1 We take the different elements of the final vector P and calculate the total number of singles counts: N S = P SI + P SI, (13) N I = P SI + P SI (14) 3

4 with the coincidences C tot = P SI, (15) which includes both coincidence and accidental coincidence counts. Since we know the true coincidences C can only arise from the SFWM process C = η S η I µ, we then define the coincidence-to-accidental ratio (CAR) as: CAR = C A = η Sη I µ N S N I (16) We expand the transmission coefficients η S,I, which are also present in the final expansion of P SI, to account for coupling efficiencies and transmission losses η couples,i, propagation losses η α and nonlinear losses of single photons (signal or idler) from cross-two-photon absorption η NL with the pump such that: η S,I = η couples,i η α η NL (17) Using our measured experimental parameters from the device under test we calculate the propagation and nonlinear loss terms, η α and η NL = η XT PA η FCA, respectively, with η XT PA = 1 (1 + α 2 I o L e f f S 2 ) 2. (18) 4. g (2) (0) model - matrix formulation of experimental losses and detection for three detectors The model for the g (2) (0) correlation function involves a larger matrix representation for the detection statistics similar to Ref. [12]. We begin by constructing a vector which describes the joint state of the detectors P, noting that we now have three detectors, the heralding detector H and the two detectors A and B after a 50:50 fibre coupler as shown in Figure 3: P = ( P ABH,P ABH,P ABH,P ABH,P ABH,P ABH,P ABH,P ABH ). (19) Each element of the matrix corresponds to different sets of detectors switching from a no click state (P ABH ) to all clicked (P ABH ) during an elementary observation time, which we take as one pump repetition. We construct 8x8 matrices for the probability a generated photon-pair causes the detectors to transition to the clicked state (M η ) and the probability that dark counts in the detectors cause such a transition (M D ): and M η = M D = (1 η H +(η A +η B )(η H 1)) (η A (1 η H )) ((1 η B )(1 η H )) (η B (1 η H )) 0 ((1 η A )(1 η H )) (η H (1 (η A +η B ))) 0 0 (1 (η A +η B )) (η B (1 η H )) (η A (1 η H )) 0 (1 η H ) (η A η H ) (η H (1 η B )) 0 η A 0 (1 η B ) 0 0 (η B η H ) 0 (η H (1 η A )) η B 0 0 (1 η A ) 0 0 (η B η H ) (η A η H ) 0 η H η B η A 1 ((1 d A )(1 d B )(1 d H )) d A (1 d B ) (1 d H ) (1 d B ) (1 d H ) (1 d A )d B (1 d H ) 0 (1 d A ) (1 d H ) (1 d A )(1 d B )d H 0 0 (1 d A ) (1 d B ) d A d B (1 d H ) (d B (1 d H )) (d A (1 d H )) 0 (1 d H ) d A (1 d B ) d H (d H (1 d B )) 0 (d A (1 d B )) 0 (1 d B ) 0 0 (1 d A )d B d H 0 (d H (1 d A )) (d B (1 d A )) 0 0 (1 d A ) 0 d A d B d H (d B d H ) d A d H d A d B d H d B d A 1, (20). (21) Here we have defined the channel efficiencies and dark counts for the three channels as η A,B,H and d A,B,H. We then define the state of the detectors after the emission of up to i photons as: P = i=0 p i M D (M η ) i P 0, (22) where P 0 is an initial vector with all the detectors set to off (i.e. P 0 = (1,0,0,0,0,0,0,0)) and p i are the probability distributions for the different generated states. Again, p i describes the likelihood of generating a pair per pulse by, defined previously as µ, in a thermal distribution: 4

5 Fig. 3. Schematic of the experimental setup for the g (2) (x) correlation measurements. p i = µ i. (23) (1 + µ) i+1 Taking the appropriate elements of the final vector P, we calculate the g (2) (x) correlation function using the formula g (2) (0) = Π AB H Π H Π A H Π B H. (24) The elements corresponding to the above counts are the Π AB H = P ABH, Π A H = P ABH +P ABH, Π B H = P ABH +P ABH and Π H = P ABH +P ABH +P ABH +P ABH. We once again expand the transmission coefficients η A,B,H to account for coupling efficiencies and transmission losses η couplea,b,h, propagation losses η α and nonlinear losses η NL, η A,B,H = η couplea,b,h η α η NL, (25) where the subscripts A,B correspond to the channels at the output of the 50:50 beamsplitter and H is the heralding channel. We note this is the counting formulation for the more general description: with the quantum intensity operators ˆ I S,Î I [13, 14]. g (2) (τ) = Iˆ S (t + τ)î I (t) Iˆ S (t + τ) Î I (t), (26) 5

6 5. Generation of multiple pairs-per-pulse: experiment and model We previously defined the state of the detectors after the emission of up to i photon pairs as: P = i=0 p i M D (M η ) i P 0. (27) Explicitly i > 1 corresponds to the probability of generating multiple photon pairs-per-pulse. This is a well known source of noise in spontaneous photon pair sources [15, 16]. Our flexible matrix model allows us to quickly unveil the role of multiple pairs in our experiments. Figure 4 shows the role of multi-pair generation in our three-photon measurement of g (2) (0) for both silicon and GaInP. We immediately notice that increasing the value of i from the single pair regime (i = 1) in our model increases the agreement of the model to the experimental output. We define the edge of the single pair regime in the main text as the point at which i = 2 deviates from the i = 1 line. This occurs at approximately Φ 0.1 radians for both materials. It takes the inclusion of approximately 5 pairs (i = 5) for silicon and 3 pairs (i = 3) to accurately model the experimental range. We also note that nonlinear losses cause the silicon device to have a lower g (2) (0) value, as the losses push the source more into the single photon regime, but at a detrimental cost to the photon coincidence rate. Fig. 4. Role of multiple pairs-per-pulse. Experimental g (2) (0) correlation measurements versus Φ with our theoretical model as the number of photon pairs i are increased for (a) silicon and (b) GaInP. 6. Coincidence count, CAR and g (2) (0) contributions to the enhanced QMU To optimise probabilistic heralded single photon sources one must take care to include counting rates as well as quantum signal-to-noise metrics including the CAR and g (2) (0). We do so by formulating the quantum utility, or QMU, defined as QMU = µ SNR [1 g (2) (0)]. (28) For a single probabilistic source, µ = µη 2, where η 2 = η 2 αη 2 NL with with η NL the nonlinear loss efficiency given above, as well as in the main text, and η α the propagation loss efficiency of single photons. The SNR is the signal to noise ratio of desirable photons to accidental photons, here the CAR. Figure 5 shows the QMU with respect to nonlinear phase shift Φ for both the silicon and GaInP experimental results, as well as plotting the results of the devices with similar lengths and slow-down factors, and more favourable dispersion β 2 = -0.1 ps 2 /mm repeated from the main text. The QMU curves at can be separated into their constituent parts, namely the achievable experimental coincidence count rates [Fig. 5(b)-(c)], the CAR [Fig. 5(d)-(e)], and the g (2) (0) [Fig. 5(f)-(g)], all of which we compare to the original data (open circles/squares). We note the nonlinear loss limits the silicon device, but the GaInP device shows a 5-fold increase in count rate whilst achieving a higher CAR value and a lower g (2) (0). 6

7 Fig. 5. Plots of the QMU and constituent parts corresponding to Fig. 6 in the main text. (a) A plot of the theoretical QMU with the current experimental data for silicon (red line and circles) and GaInP (blue line and squares) with respect to nonlinear phase shift Φ. Predicted curves at a more favourable β 2 for devices of the same length and slow-down factor are shown as dashed lines for silicon (red) and GaInP (blue), (b) and (c) show the improved counts compared to the original data. (d) and (e) show the improved CAR plots whilst (f) and (g) are g (2) (0) values for silicon and GaInP respectively. 7

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