High-performance source of spectrally pure, polarization entangled photon pairs based on hybrid integrated-bulk optics

Entangled photon pair sources based on bulk optics are approaching optimal design and implementation, with high state fidelities, spectral purities and heralding efficiencies, but generally low brightness. Integrated entanglement sources, while providing higher brightness and low-power operation, often sacrifice performance in output state quality and coupling efficiency. Here we present a polarization-entangled pair source based on a hybrid approach of waveguiding and bulk optics, addressing every metric simultaneously. We show 96% fidelity to the singlet state, 82% Hong-Ou-Mandel interference visibility, 43% average Klyshko efficiency, and a high brightness of $2.9\times10^6$ pairs/(mode$\cdot$s$\cdot$mW), while requiring only microwatts of pump power.


I. INTRODUCTION
Modern entangled photon pair sources based on parametric down-conversion (PDC) are approaching the ideal: high state fidelity, spectral purity, and heralding efficiency are commonly demonstrated, enabling applications such as tests of Bell's inequality [1,2], probing the boundaries of quantum physics [3,4], quantum communication over long distance [5][6][7], and quantum metrology beyond classical limits [8][9][10]. However, sources using bulk nonlinear crystals suffer an intrinsic three-way tradeoff between brightness, fiber-coupling efficiency, and spectral purity [11]. This deficiency is now becoming critical, as many new experiments and applications rely on the interference of multiple photons. For high rate, high quality multi-photon experiments, all three of the aforementioned parameters must be maximized simultaneously [12][13][14][15][16][17][18]. This is because the overall rate in multiphoton experiments with N pairs scales with probability p of producing and detecting a single pair as p N , requiring both high brightness and coupling efficiency. The quality of multi-photon interference is determined by spectral purity and indistinguishably as only pure, indistinguishable photons are able to interfere with high visibility.
In contrast to bulk sources, integrated sources provide high brightness due to strong confinement in waveguides and long interaction lengths, and can be designed to be spatially and spectrally single-mode, enabling simultaneously high fiber-coupling efficiency, spectral purity, and brightness. Many examples exist of highbrightness integrated sources, for example based on PDC in waveguides [19][20][21], or four-wave mixing in optical fibers [22,23]. However, these sources have not yet demonstrated simultaneous high performance in all other parameters comparable to their bulk-optical counterparts. Entangled pairs from quantum dots, while promising [24,25], also do not yet reach the performance of pairs from nonlinear optical sources. * evan.meyer-scott@upb.de Here we solve the performance problem of integrated optics while retaining the coupling and brightness benefits by employing a hybrid bulk-waveguide solution: photon pairs are produced in a single-mode waveguide, then made to interfere and coupled to optical fiber using bulk optics. Our source combines for the first time excellent performance in all parameters simultaneously.

II. INTEGRATED SINGLE-MODE PHOTON PAIR SOURCES
Over the last two decades, efforts in improving entangled photon-pair sources based on bulk crystals and bulk optics have resulted in impressive performance in many measures (see end of document for comparison table). Entanglement fidelities above 99 % are readily achieved [1,2,17,26], and even above 99.9 % is possible [4]. Klyshko (heralding) efficiency [27], defined as the ratio of coincidence to singles counts, can reach 75 % [1,2,26,28,29]. The spectral purity, required to interfere photons from separate sources for multi-photon experiments, has been shown above 99 % [30].
Unfortunately, bulk sources suffer from an intrinsic tradeoff between the brightness, or emitted photon rate per pump power, and the Klyshko efficiency [11]; for example setting the pump focus to enable coupling photon pairs to single mode fiber with 95 % efficiency necessarily reduces the brightness by a factor of ten from the maximum [31]. This tradeoff arises due to conflicting requirements on the focusing conditions: high brightness requires a tight focus which decreases the effective area of the interaction and therefore increases its strength. High coupling efficiency, however, requires a weak focus which more strongly correlates the spatial modes of signal and idler photons such that if one photon is coupled into fiber, the other is likely to be coupled too. This tradeoff means the fundamental performance limits of bulk sources have largely been saturated. Furthermore, sources at telecommunications wavelengths are much less bright than those with visible-range photons, due to the wavelength depen-dence of the down-conversion efficiency [32].
Integrated photon sources can surpass these limits, as the waveguide, rather than spatial phasematching, defines the allowed modes into which photons are emitted [33]. Single-spatial-mode waveguides in particular completely decouple the brightness from the focusing conditions [34], and can be produced with appropriate choice of the waveguide width and height. Then the the maximum coupling efficiency depends only on the mode overlap between the waveguide and fiber modes. While the brightness of bulk sources with optimal focusing scales with increasing nonlinear crystal length L as √ L [35] or constant [11], the brightness of waveguide sources increases proportionally to L, as well as inversely with the effective area [32]. This allows waveguide sources, beyond removing the brightness-efficiency tradeoff, to be significantly brighter overall [36], requiring much less pump power for reasonable photon pair probabilities (e.g. 2 mW average power for 0.1 pairs generated per pulse [37]).
It is also desirable to have spectrally single-mode photons, where a frequency measurement of the signal provides no information on the properties of the idler, meaning each is in a pure spectral state. This is required for interference between independent sources, essential for quantum networking [15,16,38,39], boson sampling [12,13] or linear optic quantum computing [12][13][14]. This high spectral purity can be asymptotically accomplished by narrowband filtering, but filtering both photons unavoidably lowers the Klyshko efficiency [40,41]. Engineering the group velocities of the pump, signal, and idler avoids this problem by producing intrinsically single-spectral-mode photons [42,43]. However, for bulk crystals, even a spectrally-engineered source has only a certain range of focusing parameters where the spectral purity is maximized [11,44,45]. In waveguides, this spectral-spatial coupling is eliminated thanks to the single-spatial-mode propagation, allowing the spectral purity to be independently optimized.
Yet to date the most advanced experiments do not use waveguide sources, which can be understood in light of the difficulty in optimizing performance in integrated optics (see end of document for comparison table). The brightness of integrated sources is orders of magnitude larger than possible in bulk, reaching above 10 8 pairs/(s·mW) [21,23]. Entanglement fidelity over 95 % has been achieved in a few integrated systems [20,23,46,47], and while source engineering has allowed high spectral purity in waveguided photon pair sources [34,48,49], so far it has not been combined with qubit entanglement. The biggest drawback of current implementations however is the Klyshko efficiency, which is often very low due to lossy integrated components and poor coupling between elements. Though high Klyshko efficiencies from unentangled waveguide sources have been demonstrated [37], only two examples of an entangled pair source with efficiency >3 % exist so far [20,50].
By combining both integrated and bulk approaches we benefit from the advantages of waveguide photon pair sources -single mode operation, high brightness, independent optimization of parameters -and the flexibility of efficient free-space coupling to fiber. We describe the experimental setup and results below.

III. EXPERIMENT
Our hybrid source of entangled photon pairs is based on a free-space-coupled waveguide in a periodically poled potassium titanyl phosphate (KTP) crystal. In fact we test two such chips, and find one has better entanglement properties and the other better coupling efficiency. We present a full data set for the latter, and discuss the differences between the samples in Section III D. The periodic poling is designed for type-II phasematching at 1550 nm, and the material KTP is chosen such that the group velocity of the pump is between that of the signal and idler. Matching the pump and phasematching bandwidths then provides intrinsic spectral purity of the photons [34,51].

A. Setup
The KTP waveguide is placed in a Sagnac loop [52] as shown in Fig. 1(a). The pump (Coherent Mira 900f) at 770 nm is coupled through a single-mode fiber for spatial mode cleaning, then its polarization is set with a halfwave plate (HWP), and it passes through a dichroic mirror (Thorlabs DMSP1180). The pump is split equally at a dual-wavelength polarization beam splitter (PBS, Op-toSigma), and propagates both ways around the Sagnac loop. In the counter-clockwise direction the pump polarization is rotated to horizontal with a superachromatic HWP (B. Halle), balanced with an identical HWP at 0 • on the clockwise path to reduce distinguishably from dispersion. Then the pump is coupled using achromatic lenses of focal length 6 mm (Edmund Optics ACH-NIR 6 X 9 NIR-II) into the KTP waveguide (ADVR, Inc.) with length 9 mm. For producing a single spectral mode, this waveguide length requires a pump bandwidth of 1.8 nm, which we set using a 4f-line in the pump path. The same lenses couple out the photon pairs, where now the clockwise-propagating pairs are rotated 90 • in polarization by the HWP. These interfere with the un-rotated counter-clockwise pairs at the dualwavelength PBS, which divides signal and idler to the two output ports while creating the polarization entangledstate |ψ = 1 √ 2 |HV + e iφ |V H . The phase φ of the state is set to π in the signal arm using a Soleil-Babinet compensator (Thorlabs, SBC-IR), and both photons pass filters to remove sinc lobes of the PDC spectrum and fluorescence (signal: Semrock NIR01-1550/3, idler: Thorlabs FBH1550-12). Finally the entangled state is analyzed with half-and quarter-wave plates and PBSs (not shown), then coupled into optical fiber for detection with superconducting nanowire single photon detectors (Photon Spot, Inc.).

B. Differences to bulk sources
The single-mode nature of the PDC in our waveguide brings significant advantages over bulk optics. Since the clockwise and counterclockwise photons are necessarily in the same spatial mode, interference at the PBS is simple, and alignment is straightforward. This also relaxes much of the strict symmetry needed in the crystal position in bulk Sagnac sources with respect to the focusing lenses and PBS [53]. The high brightness of the source allows for low pump power, which means the pump spatial mode can be cleaned in standard single mode fiber without spectral broadening due to nonlinearities. One drawback is the lenses used to couple light in and out of the waveguide are the same for pump and photon pairs, meaning the focus is optimized only for the photons. Nonetheless, with achromatic lenses we can reach >40 % coupling of the pump through the waveguide.
There is one point in waveguides that requires special attention compared to bulk sources: the antireflection coating on the crystal surface. In bulk sources, photons reflected internally at the end facets have a different focal position when they reach the coupling fibers, and thus couple poorly. In waveguides by contrast, photons reflected at the end facets remain in the single spatial mode and couple well to the fibers. Unfortunately these photons end up in exactly the wrong polarization compared to their non-reflected partners, directly lowering the entanglement fidelity as in Fig. 1(b). We solve this using an ion-assisted coating technique to deposit anti-reflection coatings for both wavelengths on both end facets. These coatings reduce the end facet reflectivity to around 2 % for the first chip and below the measurement uncertainty of 0.06 % for the second chip, giving a maximum achievable fidelities of 96 % and 99.9 % respectively.

C. Distinguishability in time and frequency
For any polarization-entangled photon pair source, it is essential that the two polarization paths are completely indistinguishable in all other degrees of freedom. The single-mode waveguide and output fiber coupling ensure this indistinguishability in the spatial degree of freedom, but extra care must be taken to ensure time-frequency overlap, particularly as spectrally pure photons require relatively broadband pump pulses, especially compared to continuous-wave sources. The Sagnac scheme does not require degenerate signal and idler emission, but does require that the clockwise (c) and counter-clockwise (cc) paths remain indistinguishable. Even though both paths encounter the exact same optical components, they encounter them at different wavelengths and polarizations (e.g. pump vs photon wavelength, signal vs idler polarization). Any uncompensated dispersive or birefringent materials or coatings thereby reduce the polarization entanglement generated by coupling polarization information to the time-frequency degree of freedom.
This coupling can be modelled in the joint polarization and time-frequency space for signal and idler photons in modes defined by creation operatorsâ andb, respectively, as This model state always has perfect anti-correlations in the rectilinear (H/V) basis. The projection probability in the diagonal basis is If the joint spectral amplitudes for the clockwise and counter-clockwise paths f c (ω s , ω i ) and f cc (ω s , ω i ) are exactly identical, the polarization and time-frequency degrees of freedom are separable, and we obtain a "perfect" polarization entangled state, with anti-correlations in the diagonal (D/A) basis. For distinguishable paths with f c (ω s , ω i ) = f cc (ω s , ω i ), undesired coincidence counts will be measured when projecting on |DD , as described by Eq. (2). Notably, this projection is sensitive to spectral phase differences between the two paths.    To diagnose sources of distinguishability, we spectrally resolve the polarization correlations [54,55]. In Fig. 2, we show the joint spectral intensities reconstructed using time-of-flight spectrometers [56,57] when projecting on polarization combinations in the rectilinear and diagonal polarization bases. The results can be explained by relative time delays τ ( = p, s, i) between the two directions for the signal, idler, and pump, defined as

x U s 4 T Q C R m x v p G S R E z 7 2 f z y H F 8 Y Z 4 j D W J k n A c / d v x M Z i b S e R Y H p j A i M 9 W p W m P 9 l / R T C W z / j M k m B S b p Y F K Y C Q 4 w L D H j I F a M g Z k Y Q q r i 5 F d M x U Y S C g b W 0 x X A x i P K a A e O u Y l g X 3 a u G 6 z T c h + t 6 8 6 5 E V E V n 6 B x d I h f d o C Z q o T b q I I q m 6 A W 9 o j f r 2 X q 3 P q z P R W v F K m d O 0 V J Z X 7 8 1 r p w I < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " g 4 O N D C S f S 3 c y 9 2 8 d v W e T i L t w l q g = " > A A A C D X i c b Z D N S s N A F I U n 9 a / W v 6 h L N 4 N F c F U S E R R X B T d d V r C t 0 I Q y m U 7 a o Z N J m L k p l J B n 8 A X c 6 h u 4 E 7 c + g y / g c z h p s 7 C t F 4 Y 5 n H M v 9 / I F i e A a H O f b q m x s b m 3 v V H d r e / s H h 0 f 2 8 U l X x 6 m i r E N j E a u n g G g m u G Q d 4 C D Y U 6 I Y i Q L B e s H k v s h 7 U 6 Y 0 j + U j z B L m R 2 Q k e c g p A W M N b N u b M M h a 3 d w L F C n + g V 1 3 G s 6 8 8 L p w S 1 F H Z b U H 9 o 8 3 j G k a M Q l U E K 3 7 r p O A n x E F n A q W 1 7 x U s 4 T Q C R m x v p G S R E z 7 2 f z y H F 8 Y Z 4 j D W J k n A c / d v x M Z i b S e R Y H p j A i M 9 W p W m P 9 l / R T C W z / j M k m B S b p Y F K Y C Q 4 w L D H j I F a M g Z k Y Q q r i 5 F d M x U Y S C g b W 0 x X A x i P K a A e O u Y l g X 3 a u G 6 z T c h + t 6 8 6 5 E V E V n 6 B x d I h f d o C Z q o T b q I I q m 6 A W 9 o j f r 2 X q 3 P q z P R W v F K m d O 0 V J Z X 7 8 1 r p w I < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " g 4 O N D C S f S 3 c y 9 2 8 d v W e T i L t w l q g = " > A A A C D X i c b Z D N S s N A F I U n 9 a / W v 6 h L N 4 N F c F U S E R R X B T d d V r C t 0 I Q y m U 7 a o Z N J m L k p l J B n 8 A X c 6 h u 4 E 7 c + g y / g c z h p s 7 C t F 4 Y 5 n H M v 9 / I F i e A a H O f b q m x s b m 3 v V H d r e / s H h 0 f 2 8 U l X x 6 m i r E N j E a u n g G g m u G Q d 4 C D Y U 6 I Y i Q L B e s H k v s h 7 U 6 Y 0 j + U j z B L m R 2 Q k e c g p A W M N b N u b M M h a 3 d w L F C n + g V 1 3 G s 6 8 8 L p w S 1 F H Z b U H 9 o 8 3 j G k a M Q l U E K 3 7 r p O A n x E F n A q W 1 7 x U s 4 T Q C R m x v p G S R E z 7 2 f z y H F 8 Y Z 4 j D W J k n A c / d v x M Z i b S e R Y H p j A i M 9 W p W m P 9 l / R T C W z / j M k m B S b p Y F K Y C Q 4 w L D H j I F a M g Z k Y Q q r i 5 F d M x U Y S C g b W 0 x X A x i P K a A e O u Y l g X 3 a u G 6 z T c h + t 6 8 6 5 E V E V n 6 B x d I h f d o C Z q o T b q I I q m 6
which could arise due to spectral phase profiles specific to the vertical ports or facet coatings of the dualwavelength PBS, or due to unpoled regions on one end of the waveguide. By projecting onto the diagonal basis, these time delays will manifest as fringing across the signal-idler joint spectral intensity oriented at an angle θ = arctan τp+τi τp+τs . If a relative chirp exists between the two paths, this will appear as a frequency-dependent time delay, τ = τ 0 + 2A δω , and the angle of the fringes will change as the central frequencies are shifted, which we observe when the pump wavelength is shifted in Fig. 2. The theoretical simulations in Fig. 2 correspond to a relative time delay of 600 fs between horizontal and vertical components (τ s = −τ i = 300 fs), equivalent to the birefringent delay of approximately 2.2 mm of KTP. To describe the dependence on pump wavelength, a chirp on the pump for the counter-clockwise process of A p = 5800 fs 2 is sufficient. To optimize the indistinguishably, we use this polarization-resolved joint spectral characterization to identify waveguides with more uniform poling, and obtain the results presented below.

D. Results
We measure the spectral purity of our source via the joint spectral intensity (JSI) and Hong-Ou-Mandel (HOM) interference [58] between independent photons. The joint spectral intensity of Fig. 3 is reconstructed using a time-of-flight spectrometer [56,57], and returns an upper bound to the purity of 98 %.
To measure HOM interference we use the Sagnac loop to create two photon pairs without polarization entanglement, one in each direction around the loop. This is accomplished by detecting two signal photons simultaneously, one |V s and one |H s , making use of both output ports of the signal photon's PBS. This heralds two idler photons |1H, 1V i in the same spatial mode heading toward the idler's PBS. To make these photons interfere we rotate the idler HWP, which at 22.5 • leads to the state (in a single spatial mode) 1 √ 2 (|2H i − |2V i ) due to the indistinguishability of the two heralded photons [59]. Thus we should be able to tune the coincidence probability between the horizontal and vertical output ports of the idler's PBS between 0 and 1 by rotating the HWP. This is in contrast to typical HOM interference, where distinguishability in the photons is introduced via a time delay, and they impinge from separate ports on a 50:50 beam splitter. In that case the coincidence probability varies between 0 and 1/2. Changing the HWP in our case is like changing the splitting ratio of the beam splitter from 100:0 to 50:50 to 0:100. Mapping the HOM visibility from the temporal to the polarization case gives V HOM = Nmax/2−Nmin Nmax/2 , where N max and N min are the maximum and minimum number of fourfold coincidences we measure, respectively, and the factor one half comes from the maximum probability being 1 compared to 1/2 in the temporal case. This visibility depends on the spectral purity and indistinguishably of the photons [60], and can also be degraded by higher-order down-conversion events. Our measured value is V HOM = (82 ± 2) %, without background subtraction. That the HOM visibility is lower than the upper bound given by the JSI's purity is probably due to chirp on the pump pulses, backreflections as in Fig. 1(b), and residual distinguishability in the photons due to being produced in different directions in the waveguide.
We plot the measured count rates and Klyshko efficiencies in Fig. 4. The rates scale linearly with pump power (at low power), allowing us to plot also versus the mean pair number per pulse produced in the crystal. We are limited to around 2 mW pump power and 1 million coincidences per second by saturation and latching of our detectors. From Fig. 4(a) we extract a brightness of (3.5 ± 0.1) × 10 6 pairs/(s·mW), competitive with state-of-the-art waveguide processes. The Klyshko efficiencies, as expected, drop at low power due to dark counts, and at high power due to detector saturation. The average Klyshko efficiencies (excluding the first two and last five points) are (38.0 ± 0.5) % and (46.8 ± 1.3) % respectively for signal and idler. We can compare these values to those estimated from classical measurements: component transmission from waveguide to fiber for the signal (85 ± 3) % (idler (79 ± 3) %), fiber coupling efficiency for the signal (66 ± 5) % (idler (84 ± 5) %), fiber transmission to detectors (95 ± 5) %, detector efficiency (90 ± 5) %, giving a total expected efficiency of (48 ± 4) % for the signal and (57 ± 6) % for the idler. Waveguide losses partially account for the difference to the measured efficiencies, as do losses due to the gentle filtering used to remove sinc lobes [41]. None of these losses are fundamental: better coatings on our optics and lower loss waveguides would boost the efficiency dramatically. Additionally, the heralded g (2) h (0), an indication of noise photons in the system, is consistent with zero extra noise. For spectrally pure photons and low pump power, g (2) h (0) = 2(2 − η h )µ, where µ is the mean pair number per pulse and η h is the Klyshko efficiency of the heralding signal photon. This gives g  Finally we present the entanglement visibility curves and reconstructed two-qubit density matrix in Fig. 5. We find a maximum visibility of (96.0 ± 0.1) % in the rectilin-ear basis and (94.3 ± 0.1) % in the diagonal basis, where the error bars come from Poissonian statistics. We then perform overcomplete quantum state tomography [61] with an average coincidence rate of 59 000 pairs/s, finding a fidelity of F = ψ − | ρ |ψ − = (95.78 ± 0.04) %. The fidelity and visibilities are limited by the small end facet reflectivity as in Fig. 3(b), some residual temporal-spectral distinguishability as in Section III C, and 30 accidental coincidences per second due to multi-pair emissions. We also measured a different waveguide chip with the same design, but from a different batch. Due to better end coatings, the entanglement fidelity of this chip was (98.82 ± 0.05) %, and the Hong-Ou-Mandel visibility was (89 ± 2) %. Unfortunately the waveguide and coupling losses were higher, resulting in Klyshko efficiencies of (26.3 ± 0.3) % and (28.2 ± 0.5) %. Thus we chose to focus on the chip with best Klyshko efficiency, at the cost in this case of entanglement and Hong-Ou-Mandel visibility. However both the reduced visibility of the first chip and the reduced efficiency of the second chip are technical, rather than fundamental problems, and neither prevents these sources from being used in cuttingedge experiments.

E. Comparison with other sources
It is instructive to compare the properties and performance of our source to previous work. Compared to bulk sources (see Table I), our waveguide source provides 1−3 orders of magnitude higher brightness, as well as higher Klyshko efficiency than the bright bulk sources due to the waveguide decoupling the brightness and efficiency. Our entanglement fidelity and HOM interference visibility are comparable to many of the best bulk sources.
Compared to integrated sources (see Table II), our source has by far the highest Klyshko efficiency, in some cases by nearly two orders of magnitude. Our high spectral purity leads to an interesting brightness comparison: instead of the typical pairs/(s·mW), we can compare the brightness additionally per spectral mode. Our brightness by this metric is still very high (2.9 × 10 6 pairs/(mode·s·mW)), while CW pumped sources and those without spectral engineering drop by orders of magnitude. Our source is the brightest to emit entangled pairs into a single spectral-temporal mode, suitable for multi-photon interference.

IV. CONCLUSION
We have demonstrated a source of entangled photon pairs that has nearly ideal polarization entanglement, spectral purity, and brightness, and shown for the first time high Klyshko efficiency in a waveguided entangled pair source. Further optimization of optical coatings and beam reshaping to maximize the overlap between waveguide and fiber modes [62] will allow coupling efficiencies approaching 100 %, independent of the source brightness. Our use of spectral engineering to produce pure, indistinguishable photons is a great advantage over spectral filtering for quantum networking, and could be further enhanced by apodization of the nonlinearity as shown in bulk sources [63][64][65]. Overall we have taken a significant step towards the ideal integrated source of entangled photons, and laid out the straightforward optimizations necessary for maximum performance.
Looking forward, the high brightness, fidelity and purity of our source make it an excellent candidate for multiplexing to create multi-photon entangled states [17,18], and for time-multiplexed multi-photon experiments [66]. The wavelength is also compatible with telecommunications infrastructure, making the source suitable for quantum communications in optical fiber, in particular teleportation and entanglement swapping. Another exciting application is polarization squeezing [67], which requires simultaneously high brightness and coupling efficiency, to produce squeezed and entangled continuousvariable states that can be detected without a local oscillator [68,69].  I. Comparison of high-performance entangled-pair sources based on bulk optics. † denotes photons at telecommunications wavelengths; * denotes values inferred from published data. In particular, entanglement fidelity is approximated from reported visibilities as F = 1 − (1 − V )/2, where V is the highest reported average visibility in rectilinear and diagonal bases without subtracting accidental coincidences, and HOM visibility between two individually heralded photons is upper bounded by the spectral purity. The brightness encompasses the filtered source bandwidth, and all losses are removed to give the brightness inside the crystal. The Klyshko efficiency is the average of the signal and idler photons, evaluated as the coincidence rate divided by the singles rate. Values that could not be estimated are indicated with -. Reference