Mitigating indistinguishability issues in photon pair sources by delayed-pump Intermodal Four Wave Mixing

Large arrays of independent, pure and identical heralded single photon sources are ubiquitous in today's Noise Intermediate Scale Quantum devices (NISQ). In the race towards the development of increasingly ideal sources, delayed-pump Intermodal Four Wave Mixing (IFWM) in multimode waveguides has recently demonstrated record performances in all these metrics, becoming a benchmark for spontaneous sources in integrated optics. Despite this, fabrication imperfections still spoil the spectral indistinguishability of photon pairs from independent sources. Here we show that by tapering the width of the waveguide and by controlling the delay between the pump pulses, we add spectral tunability to the source while still inheriting all the record metrics of the IFWM scheme. This feature is used to recover spectral indistinuishability in presence of fabrication errors. Under realistic tolerances on the waveguide dimensions, we predict>99.5% indistinguishability between independent sources on the same chip, and a maximum degradation of the Heralded Hong Ou Mandel visibility<0.35 %.


I. INTRODUCTION
Spontaneous sources of photon pairs are primary resources in emerging large scale NISQ architectures, especially those based on integrated optics [1,2].Through repeated application of heralding, large arrays of sources can be used to deterministically prepare many independent photons, which constitutes an important substrate for quantum information processing [3,4].Their quality influences the ultimate computational power of the hardware, and limits the effective size of resources which are available for quantum algorithms [5][6][7].Two of the most relevant metrics are the purity and the indistinguishability of the heralded states [8].In essence, they bound the visibility of multiphoton interference, which lies at the heart of protocols, algorithms and building blocks for quantum computation and quantum information.Examples include scattershot [9] and gaussian boson sampling [10], preparation of cluster states [11], realization of entangling gates [12,13] and state teleportation [14].Several devices and methods have been developed to herald photons in pure states, characterized by a single and well defined spectral-temporal mode.These span from phase matching engineering [15], pump manipulation [16], selective control of the quality factor in microresonators [17] and spectral filtering [18].Even if the purity can be improved from a clever design of the device, the indistinguishability relies exclusively on the fabrication uniformity of the array of sources.To date, even state of the art lithographic techniques can not guarantee sufficient uniformity levels, and errors must be compensated in post-fabrication.Indeed, the thickness uniformity of the silicon waveguide layer (long range disorder) has a rms value of 3 − 4 nm, while at die level (∼ cm 2 size, short range disorder), the uniformity in the waveguide width has an rms value < 10 nm [19].
Independent sources based on microresonators can be made indistinguishable by aligning and locking their resonance wavelengths through thermo optic tuning [10,20].However, this method does not compensate slightly differences in the Free Spectral Range (FSR) or in the cavity linewidth, which are especially relevant for resonators of high quality factor.
Waveguide sources without phase matching engineering emit photons in a broad spectral interval, and off or on-chip filters are used to increase their purity at the expense of reducing the heralding efficiency [21].Therefore, the indistinguishability depends on the fabrication uniformity of the filters.In general, waveguide sources of spectrally uncorrelated photon pairs are not easily reconfigurable.Small tuning ranges can be obtained by heating the whole chip [22], while wider variations require to modify the pump wavelength [23].Other techniques aim to erase the spectral distinguishability only after that the pair is generated.
This can be achieved in materials with a strong second order nonlinearity by electro-optic frequency shearing [24], or in third order materials by Four Wave Mixing Bragg Scattering [25].
In this work, we propose and validate the design of a waveguide source which emits highly pure and spectrally tunable photons without spectral filtering.This is achieved through delayed-pump Intermodal Four Wave Mixing, a scheme recently reported on the SOI platform and which showed a record heralded Hong Ou Mandel (HHOM) visibility of 96% between independent sources [26].In contrast to the original work, we introduce an adiabatic change of the waveguide width along the propagation direction, and we tune the relative delay between the two pumps to reconfigure the phase matching wavelength of the emitted photons.The delay determines the point where the pump pulses overlap, which in turn selects the segment of the waveguide where pair generation occurs.Since the Signal/Idler frequencies depend on the waveguide cross-section, the delay reconfigures the generation wavelengths of the photon pair.We numerically investigate how this feature can be used to mitigate the distinguishability issues between different sources which arise from fabrication imperfections.We consider errors on both the waveguide width and height, focusing on realistic ranges provided by commercial foundries.We show that an indistinguishability level > 95% can be guaranteed up to height differences of 4.3 nm, and for width differences greater than 100 nm.The HHOM visibility is shown to degrade by less than 0.35% from its value in two identical sources for devices on the same chip.In all the considered cases, the spectral tunability allows to dramatically improve the visibility of both Reverse (RHOM) and Heralded Hong Ou Mandel interference.We also prove that the principal source metrics and the spectral tunability are not degraded by the Self and the Cross Phase Modulation induced by the pump on the Signal and the Idler photon.

II. PRINCIPLE OF OPERATION AND THEORY
In spontaneous IFWM, photons from two bright Pump fields (labeled as p 1 and p 2 ) annihilate to produce Signal (s) and Idler (i) photon pairs propagating in the different transverse mode orders of a multimode waveguide.The generation process occurs within narrow frequency ranges located at large spectral distances from the pump wavelength, where phase matching is satisfied [27].By denoting the wavevectors of the fields as (k p 1 , k p 2 , k s , k i ), and their central wavelengths as ( λp1 , λp2 , λs , λi ), this condition implies that k p1 ( λp1 ) + k p2 ( λp2 ) = k s ( λs ) + k i ( λi ).The great flexibility offered by the choice of the modal combination and by the waveguide cross section has been exploited to tune the emission wavelengths from the Near Infrared to the Mid-Infrared [27,28] range.At the same time, the narrow generation bandwidth and the different group velocities of the modes can be exploited to engineer the emission of spectrally uncorrelated photon pairs.Within this framework, we revisit the configuration described in [26], where IFWM is demonstrated on a 220 nm thick SOI waveguide. .

A. Tuning the phase matching wavelengths with the waveguide width
We use a Pump pulse of gaussian shape with a Full Width at Half Maximum (FWHM) duration of T 0 = 0.8 ps, a repetition rate of 50 MHz and a wavelength of 1550 nm.This is coupled in a coherent superposition of the two lowest order Transverse Magnetic (TM) modes (TM0 and TM1 mode), with a relative delay τ between them.From now on, we will refer to the faster and delayed pulse in the TM0 mode as the pump 1, while the pulse in the TM1 mode as the pump 2. The multimode waveguide has a width of w = 2.25 µm and a length of L = 1.5 cm.Signal and Idler photons are generated in the TM1 and TM0 modes respectively at the wavelengths λs = 1581.4nm and λi = 1519.9nm.Geometrical variations with respect to this reference configuration lead to a shift ∆λ s(i) of their phase matching wavelengths.This is shown in Fig. 1(a), in which ∆λ s is plotted as a function of the deviation ∆w and ∆h in the waveguide width and height.Due to the remarked sensitivity of TM modes with the latter, we have that dλs d∆h ∼ 1 while dλs d∆w ∼ −0.015.Despite this, changing the waveguide width is easier than locally varying the thickness of the silicon device layer, so we can adjust ∆w to tailor the emission wavelength of the source.We exploit two key characteristics of IFWM to realize a single device which can be reconfigured.The first is that due to the temporal walk-off between the pump pulses, the position z = L match along the waveguide where the pair generation probability is maximum depends on the delay τ .This is given by is the group velocity of pump 1(2)), which is the coordinate where the two pump pulses overlap (see Appendix C).The effective width which determines the phase matching wavelengths corresponds to the local waveguide width w(z) at position z = L match .The second feature which we exploit is the fact that by letting w to vary along the propagation direction, the effective width where pair generation occurs can be controlled with τ .As a consequence, the generation wavelengths can be continuously tuned, as shown in Fig. 1(a).We focus on the configuration shown in Fig. 1(b), where the width of the waveguide is linearly tapered from w(z = 0) = w + ∆w to w(z = L) = w − ∆w, with ∆w ≥ 0. We define τ max = 4.9 ps as the delay which makes the two pump pulses to overlap at the end of the waveguide.In Fig. 1(b) we analyze three extremal cases.When τ = 0, the maximum pump overlap occurs at w + ∆w, and according to Fig. 1(a), ∆λ s < 0, i.e., pairs are generated at wavelengths closer to the one of the pump.When τ = τmax 2 , the overlap is maximum at w , and the phase matching wavelengths are not changed with respect to the case ∆w = 0.When τ = τ max , the pump pulses catch at the narrower end of the waveguide, and photon pairs are generated at larger spectral detunings with respect to the pump wavelength.As long as L exceeds the walk-off length between the Pump pulses, and that the choice of τ allows a complete progression of one pulse over the other, the generation bandwidth and the efficiency remains constant.In the next section, we quantitatively evaluate ∆λ s(i) as a function of τ and ∆w, focusing on how the pair generation probability and the purity of the heralded single photon states are affected.

B. Theory of photon pair generation in the tapered source
The spectral (temporal) properties of photon pairs are characterized their Joint Spectral (Temporal) Amplitude (JSA/JTA), and most of the source metrics can be derived from this function [29].We then focus on the derivation of the JSA/JTA, taking into account the multiple spatial modes in the FWM process, the delayed pump configuration, the varying waveguide width along the propagation direction and the effects of SPM and XPM between the pumps and the Signal/Idler photons.The electric fields of the two pumps are treated classically and are expressed as [30]: where e is the unit vector of polarization, F p is the transverse mode profile (normalized such that |F p (x, y)| 2 dxdy = 1), n is the refractive index of the waveguide core, βp1(2) and ωp1(2) the central wavevector and frequency of the fields, and Ãp1(2) a slowly varying envelope function.The power carried by the field in Eq.( 1) is where ãs(i) (z, ω) represents the Fourier Transform of the slowly varying annihilation operator ãs(i) (z, t) for the Signal(Idler) photon.It is possible to formally derive the propagation equation for E p1(2) in a fully quantum mechanical framework by treating Ãp1(2) as an operator and by using the Heisenberg equation of motion.However, we anticipate the result of the classical regime, in line with the fact that the field in Eq.( 1) is not quantized.This is given by the well known set of coupled Nonlinear Schrodinger equations (NLSE) [31]: where the dimensionless time T = t − z v g1 /T 0 refers to a reference frame moving at the group velocity of pump 1.The definition of the parameters can be found in Appendix A.
The second term on the right hand side of Eqs.(3,4) is defined as ∆β p = β p (z, ωp ) − βp , and accounts for the varying waveguide width along the propagation direction.We numerically integrated this set of equations using a third order, symmetrized Split-Step Fourier method (SSFM) [32].To obtain a similar set of equations for the Signal and the Idler field operators, we use the Heisenberg equation of motion generated by the momentum operator M (z, t), where O is any operator in the Heisenberg picture.The total momentum can be written as M = M L + M SPM + M XPM + M FWM , which is the sum of the linear, the SPM, the XPM and the FWM induced momentum [34], and whose expressions can be found in Appendix A. We then move in the interaction picture and split the total momentum into M = M 0 + M FWM , where all the trivial evolution is generated by The pair generation process is described by the interaction momentum M FWM .Using the expressions for M L , M SPM and M XPM provided in Appendix A, and the equal position commutation relation [ã we get [35]: where we have neglected the XPM and the SPM of the Signal and the Idler fields.It is worth to note that losses have been phenomenologically introduced by the linear loss coefficients α s(i) .Losses spoil the photon number correlation between the Signal and the Idler photon in the two-mode squeezed state generated by M FWM , which could be accounted by introducing a reservoir of loss modes that is coupled to the Signal/Idler fields [36].Beside that, the simultaneous presence of squeezing and loss differs from the case where the two effects separately act [36].However, the latter well approximates the case of IFWM, since the interaction length is small compared to the one of the waveguide, and losses can be assumed to be all lumped after pair generation.Provided that we restrict our attention to the low squeezing regime of single pair generation, the loss term in Eq.( 5) simply scales the pair generation probability by a factor α s α i , and does not contribute to modify the shape of the JSA.The state |Ψ(z) of the Signal and the Idler photon, lying in vacuum at z = 0, evolves as −i d|Ψ dz = M FWM |Ψ [34], and its solution can be formally written in terms of a space propagator |Ψ(z) = U (z, 0) |Ψ(0) [30].In the regime of single pair generation, this is given by U (z, 0) = I + i z 0 M FWM (z )dz , where I denotes the identity operator.From the two-photon state, we can define the joint amplitude probability Φ(T s , T i , z) of detecting, at position z, the Signal photon at time T s and the Idler photon at time T i , as Φ(T s , T i , z) = ãs (T s , z)ã i (T s , z)U (z, 0) , where the expectation value is evaluated on vacuum.When Φ is normalized such that |Φ| 2 dT s dT i = 1, this coincides with the definiton of the JTA [35].In the rest of the paper, we will refer to Φ(T s , T i , z) as the JTA without distinction.The JSA Φ(ω s , ω i , z), expressed in the dimensionless frequencies ω s(i) = (ω s(i) − ωs(i) )T 0 , is related to Φ(T s , T i , z) by a two-dimensional Fourier Transform [34].Following the derivation detailed in Appendix B, and similarly reported in [37], we can write a propagation equation for the JTA.
By expressing the latter as Φ = Φ(T s , T i , z)e iΘ si (z) , where Θ si = z 0 (∆β s (z ) + ∆β i (z ))dz , the function Φ obeys the equation: where the operators L s(i) , N s(i) and the driving term S are defined as: where we wrote the Fourier Transform of Ãp1(2) (z, T ) as Āp1(2) (z) exp i z 0 ∆β p1(2) (z )dz to factor out the accumulated phase due to the tapering.Equation 6has the same structure of a two-dimensional NLSE in the dimensionless time variables (T s , T i ), with the inclusion of an external driving term S.

III. ANALYSIS OF THE SOURCE PERFORMANCE
Using the third order SSFM developed in [30,37], we numerically integrated Eq.6 to calculate the JTA and the JSA for different tapering amplitudes ∆w and for different delays τ .The average pump power is set to 1 mW, and is equally distributed between the TM0 and the TM1 modes.The mean wavelength shift ∆λ s of the Signal photon, calculated from the JSA, is shown in Fig. 2 One of the strengths of Eq.( 6) is to provide a natural framework to track evolution of any metric along the waveguide.As an example, the accumulated pair generation probability arrival times of the Signal (black) and of the Idler (blue) photon at the end of the waveguide, calculated using Eq.( 9).These values are relative to the arrival time of the pump pulse which lies in the TM0 mode.The inset shows an example of the JTA for τ τmax = 0.5, from which the mean T s(i) and the standard deviation σ T s(i) on the arrival times are extracted.These are plotted as shaded regions ( T s(i) ± σ T s(i) ).ξ(z) = |Φ(T s , T i , z)| 2 dT s dT i , from the beginning of the waveguide to position z, can be computed starting from Eq.( 6) as: where R denotes the real part and we have used the fact that, from Eq.7, In Fig. 3(a), we plot ξ as a function of z for τ τmax = {0.25,0.5, 0.75} and ∆w = 0.1 µm.The essence of IFWM emerges from these curves.The generation probability is approximately zero until z L ∼ τ τmax , which is the point where the two pump pulses match.Then, the value of ξ smoothly grows from the 5% to the 95% of its maximum in a length of L grow ∼ 0.36L.Then, the cumulative generation probability saturates since the pump pulses lose their spatial overlap, and after that it exponentially decays due to the propagation losses.
As shown in Appendix C, the function ξ(z) can be approximated by an erf function, which implies that its derivative, representing the pair generation probability per unit length, is a gaussian peaked at z L = τ τmax .Its FWHM ∆ z can be assessed from ∆ z = ln(2) 2 L grow ∼ 0.21L.
This value is very close to the the approximated analytic result found in Appendix C, which is ∆ z = √ 2L wp = 0.20L.For ∆w = 0.1 µm, the width of the waveguide changes by 2∆w∆z L ∼ 0.06 µm along ∆ z , which is the ∼ 2.5% of the waveguide width.
Since the pair generation process is well localized in space, so they have to be the arrival times of the Signal and the Idler photon at the end of the waveguide.As shown from the JTA in the inset of Fig. 3(b), photons are generated in a well defined gaussian temporal wavepacket, whose size is of the same order of the pump duration (0.8 ps).From the JTA, the mean and the standard deviation on the Signal/Idler arrival times are calculated, which are shown in Fig. 3(b) (shaded regions) as a function of τ and for ∆w = 0.These values are relative to the arrival time of the faster pump pulse, in accordance to the fact that Eq.( 7) is expressed in a moving reference frame.The arrival times can be analytically predicted by assuming that the pair is generated at the position where the two pump pulses have their maximum overlap, which for a delay τ occurs at z = L match = τ T 0 Lw p .From z = L match , the time required for the Signal (Idler) photon to reach the end of the waveguide is , from which is easy to show that the arrival times T s(i) are given by: where the + sign is used for the Signal.The solid lines in Fig. 3(b), obained from Eq. ( 9), show a good agreement with the arrival times calculated from the JTA.

IV. MITIGATING INDISTINGUISHABILITY ISSUES IN TWO PHOTON IN-TERFERENCE
We now exploit the tunability of the source to mitigate the indistinguishability issues which arise from fabrication imperfections in indepedent devices.Suppose to have two sources, labelled 1 and 2, which can either lie on the same die or on two different chips.In general, due to fabrication imperfections, they will have a different cross-section and JSA, which will compromise their capability to interfere.We can try to recover their spectral indistinguishability by respectively applying pump delays τ 1 and τ 2 to the two sources in order to overlap their Signal/Idler spectra.Unfortunately, as shown in Fig. 3(b), whenever τ 1 = τ 2 , the Signal(Idler) photons will arrive at the end of the waveguide at the different times τ s1 (τ i1 ) and τ s2 (τ i2 ).In other terms, they will be spectrally indistinguishable but temporarily dis- tinguishable.In order to erase the temporal information, additional delay stages have to be placed at the end of the waveguide, which make τ s1 (τ i1 ) = τ s2 (τ i2 ).To this purpose, the same component used to delay the pump pulses can be implemented, as shown in Fig. 1(b).
We numerically investigated the maximum visibility of two photon interference that can be obtained for increasing amounts of fabrication error.We focused on two key experiments, which are respectively based on the RHOM and on the HHOM effect.In RHOM, the two-photon states In the case of HHOM, in each source we use one photon of the pair, say the Idler, to herald its partner.Among the heralded Signals, one is delayed with respect to the other, after that the two are interfered at the input ports of a 50/50 beasmplitter.A dip in the coincidences between the photons emerging at the output ports is observed at zero delay, with visibility [38]: This quantity depends on both the indistinguishability and the purity of the heralded photons.Figure 4 ) of each photon from the JTAs Φ 1 (2) .Second, the JTA of source 2 is shifted in )) to mimic the presence of a delay stage on the path of each photon.Using the JTA of source 1 and the delayed JTA of source 2, the visibilities are computed according to Eq. (10)(11).From Fig. 4(a), we see that V RHOM is higher than 0.95 for |h 2 − h 1 | ≤ 4.3 nm, while in the same range V HHOM ≥ 0.9, which is only 3% less than its value at |h 2 − h 1 | = 0 (V HHOM = 0.93).On the contrary, both V RHOM and V HHOM rapidly decrease to zero if the delays are not optimized.As shown in the inset of Fig. 4(a), we have that 1 nm of error in the waveguide height is sufficient to drop V RHOM to 0.89 and V HHOM to 0.81, while their values are almost unaffected (V RHOM > 0.995, This range is especially relevant for sources which lie on the same die, for which the thickness uniformity of the silicon device layer is sub-nm [19].When the fabrication error is considered on the average waveguide width w (assuming the same height for both sources), a similar result is found.This is shown in Fig. 4(b) for ∆w = 0.1 µm.As already discussed in Section II, the phase matching wavelengths are less affected by small variations in the waveguide width, reason why for this configuration we choose a smaller tapering amplitude.
Without delay optimization, for | w 1 − w 2 | = 60 nm their value drop to V RHOM = 0.92 and V RHOM = 0.88.The inset in Fig. 4(b) shows that for errors in the waveguide width below 15 nm, which is a meaningfull range for sources lying on the same die [19,39], the optimized values of V RHOM and V HHOM are almost equal to the case of identical waveguides.
In Fig. 4(c) we report the values of τ 1 and τ 2 which maximize the fringe visibility as a function of the error on the average waveguide width.As the latter increases, τ 1 and τ 2 show opposite trends.From the trivial case τ 1 = τ 2 = τmax 2 , which occurs at w 1 = w 2 , we have that by increasing the difference in the waveguide width, τ 1 monotonically decreases while τ 2 increases (we arbitrarily choose w 1 ≤ w 2 to fix the sign of ∆λ s , the behaviour will be inverted in the opposite case).To intuitively understand this trend, suppose that due to an error on the waveguide width, sources 1 and 2 emit pairs with a wavelength difference ∆λ s(i) = λ s(i)1 − λ s(i)2 .We could recover the spectral indistinguishability by acting exclusively on the delay of source 1, i.e., τ 1 → τmax 2 ± ∆τ , where the choice of the sign depends on the one of ∆λ s (equivalently ∆λ i ).However, Fig. 2(b) indicates that at both large and small delays, the shape of the JSA is asymmetric, and the purity of the heralded single photon states decreases with respect to τ = τmax 2 .It is then more convenient to modify the delay of both sources, choosing τ 1 ∼ τmax 2 ± ∆τ 2 and τ 2 ∼ τmax 2 ∓ ∆τ 2 , rather than imparting the whole delay ∆τ on source 1.In this way, the JSA of both sources will have less distortions.
We evaluated that in order to compensate for silicon device thickness inhomogeneties ≤ 1 nm the delay must be tunable in the range [0.35, 0.65]τ max , which corresponds to [1.71, 3.18] ps.
With reference to the device sketched in Fig. 1(b), this could be achieved by placing a delay line in the lower arm after the input beamsplitter, which is reconfigurable in the range ∆τ = τ 0 ± ∆T 2 , where τ 0 is a bias delay and ∆T = (3.18− 1.71) = 1.47 ps.When the delay line is set into its rest state (∆τ = τ 0 ), the length difference between the lower and the upper arm after the input beamsplitter must be ∆L = v p1 ( τmax 2 − τ 0 ).Among the different devices which can physically implement the delay line, a good candidate is the one based on cascaded asymmetric Mach Zendher Interferometers (aMZI) reported in [40].This device is attractive since it can be easily reconfigured using thermo optic phase shifters, it is built using standard and robust optical components, and has a broadband spectral response.While an in-depth discussion lies out of the scope of this work, we only comment on the feasibility of the method.Following the results found in [40], the maximum delay ∆T is linked to the FSR of the aMZI as FSR λ = 2λ 2 c∆T .Using ∆T = 1.47 ps, we have that FSR λ ∼ 11 nm, and the minimum 3 dB-bandwidth of the device transmittance is ∼ 1.27 × FSR λ 2 ∼ 7 nm [40].
This should be sufficiently large to transmit the pump, the Signal and the Idler photons without significant distortions of their temporal wavepackets.

V. INFLUENCE OF SPM AND XPM ON THE SOURCE TUNABILITY
In our scheme, we use sub-ps pulses of high peak power (∼ 10 W for 1 mW of average power) to compensate the large effective area of the FWM interaction.It is well known that SPM and XPM, triggered by the high power intensities, influence the shape of the JSA [34,35,41].In our case, the accumulated SPM of the pumps and their XPM on the Signal/Idler photons both depend on τ , because the delay determines the position along the waveguide where the two pump pulses overlap and the pair is generated.When τ ∼ 0, the SPM accumulated by the pumps is minimum, but the XPM induced on the photons is maximum.The opposite holds when τ ∼ τ max .We numerically simulated these regimes, focusing in particular on how the pump power influences the maximum tuning range ∆λ max s(i) .In Fig. 5(a) we plot this quantity as a function of the average pump power and a tapering amplitude of ∆w = 0.25 µm.The tuning range increases with the pump power, with the Idler photon being slightly more sensitive (∼ 0.33 nm mW ) than the Signal (∼ 0.26 nm mW ) to power variations.Figure 5(b) shows ∆λ s(i) as a function of τ τmax for different input powers.Nonlinear effects alter the wavelengths of the Signal and the Idler especially at small delays, which suggests that they originate from XPM.As the pump power increases, the Signal blue shifts from the low power condition, while the Idler red shifts.To better understand the origin of this phenomenon, we plot in Fig. 5(c) the spectrally-resolved cumulative probability to generate the Idler along the waveguide, which is obtained by marginalizing Φ(ω s , ω i , z) over ω s .This is shown for τ τmax = 0.15 in both the low power (0.5 mW) and the high power regime (3 mW).It is evident that, in both cases, Idlers are generated at approximately the same wavelength.We then observe a red shift and a spectral broadening of the Idler spectra only at high power.This is a clear signature that XPM and SPM are not affecting the phase matching condition, but rather that the spectral shift arises from XPM after that the pair is generated.This phenomenon, called XPM induced asymmetric spectral broadeding, is well known to occur in optical fibers in presence of a temporal walk-off between an intense pump and a weak probe beam [32].Since ∆λ s(i) are modified by XPM after that the pair is generated, they do not obey the energy conservation relation ∆λ i,EC = − λ i0 λ s0 2 ∆λ s,EC .This implies that any spectral distinguishability arising from XPM can not be recovered by changing the delay τ .In Fig. 5(d) we plot the discrepancy of the Idler wavelength λ i from the one λ i,EC expected by energy conservation.To determine λ i,EC , the pump wavelength is fixed and we use the average wavelength of the Signal extracted from the JSA.At low power, the deviation is zero at τ = τmax 2 , while the small discrepancies at τ → 0 and τ → τ max have exclusively be attributed to the asymmetric marginal spectra of the Idler which arise from border effects (see Fig. 2(b)).Up to 1 mW, nonlinear effects still have a limited impact, with |λ i − λ i,EC | ≤ 0.1 nm.At 2 mW, deviations from energy conservation can be as high as 1 nm at τ = 0.However, as shown in Fig. 4(c), the delays which are used to correct the fabrication errors lie in the range τ ∈ (0.25, 0.7), and within this interval |λ i − λ i,EC | ≤ 0.3 nm, which is less than 10% of the spectral linewidth of each photon.We then conclude that, up to 2 mW, XPM and SPM effects do not severely compromise the spectral indistinguishability.
It is worth to note that the source is conceived to work in the low (e.g., ξ < 0.1) squeezing regime to limit multiphoton contamination in the heralded photon states [3].Therefore, it is very unlikely that we will use input powers higher than 1 mW, since this level already corresponds to ξ = 0.1 (see Fig. 2(d)).As a comparison, we have that ξ = 0.25 at 2 mW of input power.

VI. CONCLUSIONS
We proposed a scheme to generate highly pure and spectrally tunable photon pairs using delayed-pump Intermodal Four Wave Mixing.The high purity is inherited from the engineering of the phase matching relation and from the adiabatic switching of the nonlinear interaction.The tunability of the emission wavelength is added by tapering the width of the waveguide, and by changing the delay between the pump pulses.We demonstrate that the tunability range can be extended by increasing the tapering amplitude, with only a modest reduction in the purity of the heralded single photon states and with almost no impact on the pair generation probability.We show that, by optimizing the pump delay, we can drastically reduce the distinguishability among independent sources which arise from fabrication errors.Under realistic fabrication tolerances, an indistinguishability level > 95% can be guaranteed up to a difference in the waveguide height of 4 nm, and for errors in the waveguide width larger than 100 nm.Under these circumstances, we predicted a degradation of the HHOM visibility of less than 3% of its value compared to the case of two identical sources.In comparison, the visibility and the indistinguishability will be both below 20% without delay optimization.We also show that, in the regime of low pair generation, XPM and SPM effects are not affecting the device performance.The proposed device can be built using standard integrated optical components provided by commercial photonic design kits and could be reconfigured using thermo optical phase shifters.Its implementation can mitigate indistinguishability issues either in large scale quantum photonic circuits encompassing arrays of sources, or in distant devices for quantum communication which are manufactured on different chips.

MENTUM
The momentum flux M governing the spatial evolution along the waveguide length of each operator is defined as [33]: where D − and E + denote respectively the negative and the positive frequency part of the displacement and the electric field operator.The first involves only photon creation operators, while the second only annihilation operators (see the field expansion in Eq.( 2)).In Eq.( 12), we assumed that the E field is entirely polarized along the y direction (TM modes), and that B can be expressed as B = 1 c (E y , 0, 0) (plane wave approximation).Then, one writes D y = 0 n 2 E y + P NL y , where n is the material refractive index and P NL y is the nonlinear polarization, which in our case consists only in the term yyyy E 3 y , where χ yyyy is the isotropic contribution to the third order nonlinear susceptibility.In the next steps, one finds suitable expressions for E(x, y, z, t), as the ones in Eq.( 2), insert them into Eq.(12), and separates the linear terms from the ones generated by the nonlinear polarization.This standard procedure can be found, e.g., in [42], hence we will only report the final result.
Since the pump, the Signal and the Idler fields are narrowband and centered into three non-overlapping frequency ranges, the integral in Eq.( 13) can be split into M L = M p L + M s L + M i L .Each term has the same form of Eq.( 13), but with the integral restricted to the frequency range of the corresponding beam.Within these intervals, one can define the slowly varying operators (see Eq.( 2)) ãq (z, ω q ) = a(ω q + ωq , z)e −i βqz , where q = {p, s, i}.By Taylor expanding the wavevector β up to the second order in the frequency detuning (ω − ω) in Eq.( 13), we have: q={p,s,i} ( βq + v q (ω q − ωq ) + β (2) q 2 (ω q − ωq ) 2 )ã † q (ω q − ωq , z)ã q (ω q − ωq , z)dω q , (14 where β (2) q = d 2 βq dω 2 .The SPM, XPM and FWM terms are more easily expressed in the time domain, where they have the following form [34]: The definitions in Eq.( 15) make use of the nonlinear parameter γ ijkl = ωn 2 cA ijkl , where n 2 is the nonlinear refractive index of silicon and A ijkl is the nonlinear effective area, defined as: wg F i (x, y)F j (x, y)F k (x, y) * F l (x, y) * dxdy . ( The values of the parameters introduced so far and the ones appearing in Eqs.(3-7) are calculated using the commercial Lumerical MODE package [43], and are listed in Table I.
Losses are taken from the measured values in [26].

JTA
We start from the definition of the JTA given in Section II, that we rewrite here for clarity: Φ(T s , T i , z) = ãs (T s , z)ã i (T i , z)U (z, 0) .
By performing the derivative in z of both members in Eq.( 17) we get: We now use Eq.( 5) to express ∂ã s(i) ∂z = (L s(i) + N s(i) )ã s(i) , where L s(i) = L s(i) + i∆β s(i) (z) and the definition of the operators L s(i) and N s(i) are given in Eq. (7).Moreover, from Section II we have that ∂U ∂z = i M FWM , so as Eq.( 18) becomes: The first two terms on the right hand side have exactly the same form, so we will only treat the case of the Signal and apply the same result to the Idler.Writing ãs (T s , z) using its  Fourier Transform: ãs (T s , z) = ãs (ω s , z)e −iTsω s dω s , time integral, the result is a gaussian function centered at z = L match = τ 1 v 2 − 1 v 1 = τ τmax L and with standard deviation σ z : which is the result of the main text.Therefore:

2 FIG. 1 .
FIG. 1.(a) Variation of the phase matching wavelength of the Signal ∆λ s with respect to changes in the width (∆w) and height (∆h) of the reference waveguide cross-section 2.25 ×0.22 µm.The white dashed line follows ∆λ s = 0. (b) Top: sketch of the source, with indicated the relevant components and parameters (MC = Mode Converter).Bottom: principle of operation of the tunable source.In order to tune the phase matching wavelengths of the Signal (λ s ) and of the Idler (λ i ) (right sketch), the two pump pulses are delayed by a variable amount of time τ , making them to overlap in different positions (magenta color) along the waveguide.At each delay, a different waveguide width is sensed, and the phase matching wavelengths change according to panel (a).The delays τ 1 and τ 2 are respectively applied to the Signal and the Idler photon to control their arrival time.
as can be verified by integrating the Poyinting vector S = E×B µ 0 across the waveguide cross section.The two pump envelopes are temporally delayed gaussians, and are defined in Appendix C. The Signal and the Idler fields are quantized as:

FIG. 2 .
FIG. 2. (a) Relative shift of the average Signal wavelength as a function of the time delay τ between the pumps.Different curves refer to different tapering amplitudes ∆w.(b) JSA of the photon pair source for different choices of the delay τ .The tapering amplitude is fixed to ∆w = 0.25 µm.(c) Purity of the heralded photon states as a function of the delay τ and for different values of ∆w.(d) Same as in (c), but relative to the pair generation probability ξ.

FIG. 3 .
FIG. 3. (a) Cumulative photon pair generation probability ξ as a function of the position z along the waveguide.Different curves refer to different choices of the relative pump delay τ .(b) Average

FIG. 4 .
FIG. 4. (a) RHOM and HHOM visibilities as a function of the height difference of the two waveguide sources in the delay-optimized (RHOM opt , HHOM opt ) and not optimized (RHOM raw ,HHOM raw ) cases.The inset shows a detail of the region |h 1 − h 2 | ≤ 1. Visibilities V are calculated in the two dimensional grid spanned by h 1 and h 2 , which are stored in the matrix V ij .From this matrix, we calculated the mean and the standard deviation on V as a function of the absolute height difference |h 1 − h 2 | by tracing along the anti-diagonal lines.The shaded regions cover one standard deviation on V .(b) Same as in (a), but the visibilities are shown for a fixed thickness of the waveguide (220 nm) and as a function of the difference on the average waveguide width between the sources.(c) Values of the delays τ 1 (black, relative to source 1) and τ 2 (green, relative to source 2) which maximize the visibility in panel (b).

1 √ 2
by source 1 and 2 are sent at the input ports a and b of a balanced beamsplitter, in the coherent superposition |Ψ = |II 1,a + e iθ |II 2,b .Coincidences are monitored between the output ports as a function of θ.It can be demonstrated that the visibility V RHOM of the two-photon fringe coincides with the indistinguishability [26], i.e.: (a)  shows the maximum values of V RHOM and V HHOM which can be achieved after optimization of τ 1 and τ 2 .The quantities are shown as a function of the height difference |h 1 − h 2 | of the two waveguides sources, which are assumed to have the same average width w = 2.25 µm and tapering ∆w = 0.25 µm.In the optimization procedure, the temporal distinguishability is erased in two steps.First, we compute the mean arrival times

FIG. 5 .
FIG. 5. (a) Maximum tuning range of the Signal and of the Idler wavelength as a function of the input pump power.(b) Shift of the Signal and Idler wavelength as a function of τ for different values of the input pump power.In both panels (a) and (b), the tapering amplitude is fixed to ∆w = 0.25 µm.(c) Spectrally-resolved cumulative generation probability of the Idler photon as a function of the position along the waveguide.This quantity has been normalized to its maximum for clarity, so it has not to be interpreted as a true probability.The two plots refer to an input power of 3 mW (left) and 0.5 mW (right), while τ = 0.15τ max .(d) Deviation of the average Idler wavelength λ i with respect to the one predicted by energy conservation λ i,EC .This quantity is shown as a function of τ and for different values of the input power.

TABLE I .
Values of the parameters used in the simulation of the JSA and JTA through this work.