## Abstract

We use pulsed spontaneous parametric down-conversion in KTiOPO${}_{4}$, with a Gaussian phase-matching function and a transform-limited Gaussian pump, to achieve near-unity spectral purity in heralded single photons at telecommunication wavelength. Theory shows that these phase-matching and pump conditions are sufficient to ensure that a biphoton state with a circularly symmetric joint spectral intensity profile is transform limited and factorable. We verify the heralded-state spectral purity in a four-fold coincidence measurement by performing Hong-Ou-Mandel interference between two independently generated heralded photons. With a mild spectral filter we obtain an interference visibility of $98.4\pm 1.1\%$ which corresponds to a heralded-state purity of 99.2%. Our heralded photon source is potentially an essential resource for measurement-based quantum information processing and quantum network applications.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In a prototypical quantum network with multiple quantum nodes, single photons in a well-defined single-spatiotemporal mode are highly desirable for implementing qubit measurement-based quantum information processing and networking applications. High-purity single-mode photons are especially important for complex tasks that may involve multiple quantum interference measurements, such as quantum computation and simulation [1, 2], or realization of a scalable quantum network [3, 4]. It is quite simple to herald single photons from biphotons generated in pulsed spontaneous parametric down-conversion (SPDC). In general, however, SPDC biphotons are frequency entangled, as can be seen from their predicted joint spectral amplitude (JSA), or shown by analyzing their factorability using the Schmidt decomposition [5, 6]. Under such conditions, heralding a signal photon by detecting its idler companion results in the signal photon being left in a spectrally mixed state [7] that does not yield high visibility in quantum interference measurements.

Recent works in this area focus on spectrally engineered SPDC sources to create factorable biphotons for generating pure-state heralded single photons. One approach is to create an elongated JSA oriented along the idler (or signal) frequency axis with rectangular symmetry [8, 9], as demonstrated in [9], which showed 94.4% visibility in Hong-Ou-Mandel interference (HOMI) [10] between two independent heralded photons at 830 nm that was observed without spectral filters. The other approach, as employed in this work, is to create a circularly symmetric JSA at a more convenient telecommunication wavelength in periodically poled KTiOPO${}_{4}$ (PPKTP).

In a low-flux, perturbative treatment of SPDC, the biphoton output has the frequency-domain representation [11]

*ε*is anti-diagonally oriented in that space. Therefore

_{p}*ε*and Φ are orthogonal in signal-idler frequency space and serve as two independent control parameters to shape the JSA to be circularly symmetric.

_{p}Several attempts to generate a circularly symmetric JSA utilized standard phase matching that has a sinc function dependence whose side lobes degrade circular symmetry [12–15]. To recover the circular symmetry, a nonlinear crystal’s phase-matching function can be modified to possess a Gaussian shape by engineering its nonlinearity profile [16], poling pattern [17–19], or poling periods [20]. Recently, we custom-fabricated a PPKTP crystal with a Gaussian phase-matching function [20, 21] to generate biphotons with a circularly symmetric JSI, as shown in Fig. 1. The JSI of Fig. 1(a) shows residual side lobes that are at least 24 dB lower than the main lobe, yielding the Gaussian signal and idler spectra of Fig. 1(b) by tracing over the other photon’s spectrum in Fig. 1(a). We infer a heralded-state spectral purity of 99% if we assume the joint distribution of signal and idler amplitudes are transform limited in frequency and time [21]. Because the JSI contains no phase information, we could not verify that the generated biphotons were indeed factorable with near-unity heralded-state purity. Moreover, due to the limited spectral resolution and range in the JSI measurement [22], the inferred value can only serve as an upper bound of the spectral purity. In this work, we first show theoretically that, given a circularly symmetric JSI, it is both necessary and sufficient to use a transform-limited Gaussian pump with an appropriate bandwidth to obtain a factorable biphoton state with transform-limited signal and idler. We then experimentally verify the heralded-state purity by performing HOMI between two independent heralded photons, obtaining a HOMI visibility of $93.9\pm 1.8$% without filtering and $98.4\pm 1.1$% with mild filtering of the SPDC output. Without correcting for accidental coincidences due to multipair events and dark counts, we obtained a heralded-state purity of 99.2%.

## 2. Theory

To fully characterize the biphoton state requires measuring the spectral as well as the temporal correlations, as demonstrated recently [23, 24]. However, most measurements are performed in the spectral or temporal domain, but not both, rendering these measurements incomplete. We cannot infer from a circularly symmetric JSI the factorability of the biphoton without assumptions. Given the Gaussian, circularly symmetric JSI of Fig. 1 together with a Gaussian phase-matching function [20, 21], we derive the necessary and sufficient conditions for a factorable biphoton state. We take the pump field to be Gaussian

*E*is the pump field’s amplitude at its center frequency

_{p}*ω*,

_{p}*σ*is its root-mean-square (RMS) bandwidth, and

_{p}*β*is its second-order dispersion, in units of $f{s}^{2}$. We assume that the PPKTP crystal is phase matched at frequency degeneracy, satisfies the extended phase-matching condition [11], and has a Gaussian phase-matching function [20, 21]. We can then write the phase-matching function as

*ω*, and ${k}_{s\left(i\right)}^{\text{'}}=d{k}_{s\left(i\right)}/d\omega {|}_{\omega ={\omega}_{p}/2}$ is the inverse of the signal (idler) group velocity. The approximation in Eq. (3) is obtained by expanding the phase mismatch to second order at frequency degeneracy while imposing the extended phase-matching condition [11]. Inserting Eqs. (2) and (3) into Eq. (1), we find that a circularly symmetric JSI results if and only if the pump bandwidth is given by $1/4{\sigma}_{p}^{2}=K{({k}_{s}^{\text{'}}-{k}_{i}^{\text{'}})}^{2}$, in which case the signal and idler have RMS bandwidths ${\sigma}_{s}={\sigma}_{i}={\sigma}_{p}/\sqrt{2}$. To have a factorable biphoton state, i.e., a factorable JSA, we must also require a transform-limited (chirp-free) pump, viz., one whose second-order dispersion vanishes,

*β*= 0. The resulting factorable state then has transform-limited signal and idler photons:

Heralded photons from the factorable state in Eq. (4) are in a spectrally pure state with a heralded-state purity *P* = 1. If the Gaussian pump field satisfies the bandwidth condition for a circularly-symmetric JSI, but is not transform limited because of $\beta \ne 0$, the dispersive term $\mathrm{exp}\text{}[-i\beta \left(\omega -{\omega}_{p}{)}^{2}/4\right]$ in Eq. (2) introduces a JSA phase term that is proportional to the product of the signal and idler frequencies, which decreases their heralded-state purities. The degraded purity can be quantified by performing a Schmidt decomposition on the SPDC signal (idler) state [25, 26]: $P=1/\sqrt{1+{\beta}^{2}{\sigma}_{s\left(i\right)}^{4}}$. Our theory implies that the heralded-state purity depends on both the pump’s spectrum and its second-order dispersion, which is consistent with the theoretical and numerical analysis in [22, 27]. Furthermore, our analysis shows that measurements of the transform-limited Gaussian pump and the circularly symmetric JSI are sufficient to ensure a high-purity factorable SPDC ouptut state.

## 3. Experiment

To verify the high purity of the SPDC output state experimentally, we measured the HOMI visibility between two heralded signal photons generated by the same SPDC source at two different times. High HOMI visibility can only occur if the heralded photons are spectrally pure. Moreover, we examined the effects of the pump in three different cases: (i) transform-limited pump, *β* = 0, with a non-Gaussian profile, (ii) Gaussian pump with nonzero second-order dispersion $\beta \ne 0$, and (iii) transform-limited Gaussian pump.

Figure 2 shows the experimental setup for SPDC biphoton generation and HOMI measurement of two time-separated heralded signal photons. The SPDC pump was derived from an 80-MHz mode-locked Ti:Sapphire laser centered at 791 nm with a maximum full width at half-maximum (FWHM) bandwidth of 7.8 nm. To modify the pump bandwidth, we implemented a linear spectral filtering system using a pair of identical diffraction gratings in a 4*f* optical configuration [28, 29]. Two identical lenses with focal length *f* = 20 cm were placed 2*f* apart, and the two diffraction gratings were located a distance *f* from the lenses, as shown in Fig. 2(a). The first grating spatially dispersed the broadband pump’s spectral components that were then focused by the first lens at the Fourier plane located at a distance *f* from the lens. We placed an apodizing mask at the Fourier plane to shape the pump spectrum. The type of mask and its exact placement determined the pump’s spectrum and second-order dispersion. Characterization of the spectral filtering system is detailed in Appendix A.

After the pump spectral filter, we loosely focused the pump beam at the center of the PPKTP crystal that was temperature stabilized at $22.4\pm {0.1}^{\circ}$C for operating at wavelength degeneracy with signal and idler outputs at 1582 nm. Reference [21] provides details of the custom PPKTP crystal featuring a Gaussian phase-matching profile under extended phase matching. After propagation through the crystal, the pump was removed by a long-pass filter with a 1300-nm cutoff wavelength. The orthogonally polarized signal and idler were coupled into a single-mode polarization-maintaining (PM) fiber, and then separated by a fiber-based polarization beam splitter (PBS) into their respective channels. At 1 mW of pump power we detected signal and idler singles of ∼3,000/s and ∼3,200/s, respectively, and signal-idler coincidences of ∼1,200/s, which implies a system efficiency of $\sim 38\%$ and a correlated mode coupling efficiency of $\sim 80\%$ [30].

We configured the setup of Fig. 2(b) to measure the HOMI between two heralded signal photons separated by 8 mode-locked pulses or ∼100.7 ns, chosen to be greater than the ∼80-ns detector deadtime of our superconducting nanowire single-photon detectors (SNSPDs). After the fiber PBS, the idler photons were directed to SNSPD D3 for detection to herald the presence of the signal photons. The signal photons went through a 50:50 beam splitter (BS) that randomly sent them to the long-path delay or short-path delay. Given two idler detection events that are separated by ∼100.7 ns, there is a 25% chance that the first heralded signal photon went through the long path and the second heralded signal photon passed along the short path. The path difference was adjusted to match the time separation of 8 mode-locked pump pulses, so that the two heralded signal photons would interfere at the second BS before detection at SNSPDs D1 and D2. A successful HOMI data point is a four-fold coincidence event: two idler D3 detections separated by 100.7 ns and simultaneous signal detection at D1 and D2. All detection events were recorded using a time tagger. The long path was constructed of dispersion-shifted fiber so that the two paths introduced the same amount of dispersion, and the relative path delay (excluding the 100.7 ns path length difference) was adjusted using a movable air gap. Although the photon pairs are generated by different pulses from the same pump laser, the heralded photons have a random phase relationship because of phase diffusion of the SPDC process [31, 32]. Therefore, our results also apply to heralded photons generated from two independent sources.

We recorded the 4-fold coincidence events with a 2 ns coincidence gate as we varied the relative delay between the long and short paths. The zero delay position was assigned to the location where the two signal photons arrived at the BS at the same time. We use the standard HOMI visibility definition $V=\left({N}_{max}-{N}_{min}\right)/{N}_{max}$, where ${N}_{max\left(min\right)}$ represents the maximum (minimum) coincidence counts. For a heralded state with purity *P*, ${N}_{min}\approx {N}_{max}(1-{P}^{2}$), implying $V={P}^{2}$.

## 4. Results

Initial measurements were made with a hard-aperture mask. As detailed in Appendix A, adjusting the hard aperture location lets us optimize the pump’s second-order dispersion or its spectrum, but not both simultaneously. We obtained 78.1% HOMI visibility for a transform-limited non-Gaussian pump in Fig. 3(a) and 87.3% for a Gaussian pump that was not transform limited in Fig. 3(b), in agreement with our theory that the pump being Gaussian or transform limited, but not both, is not sufficient for obtaining high visibility HOMI.

We then used a custom Gaussian transmission mask [33] that shaped the transform-limited pump to have a Gaussian spectrum. A pump with 15 mW of power and 1.0 nm bandwidth produced a mean SPDC photon pair per pulse $\alpha =0.002$, and we measured HOMI visibility of $V=93.9\pm 1.8\%$ without spectral filtering of the SPDC output, as shown in Fig. 4(a). For $\alpha \approx 0.003$ (at 20 mW of pump), we obtained a lower visibility $V=91.6\pm 1.5\%$ because the higher pump power produced more multi-pair events and therefore increased accidental coincidences. At $\alpha =0.003$ we changed the pump bandwidth to 0.97 nm and 1.05 nm and obtained visibilities of $89.1\pm 2.1\%$ and $90.6\pm 1.9\%$, respectively, thus confirming that the pump bandwidth of 1.0 nm was optimal. We note that the reported visibilities are the Gaussian fitted results along with the fitted uncertainties. Compared with a similar experiment at 1550 nm with a domain-engineered crystal reporting HOMI visibility of $90.7\pm 0.3\%$ [19], our results show visibility improvements partly because we operated at the optimal wavelength for PPKTP’s extended phase-matching condition.

In order to reduce the weak residual side lobes in Fig. 1’s JSI, we sent the SPDC output through a 10-nm filter with near-unity transmission for the 6-nm center portion (compared with 2.62 nm bandwidth of heralded photons) so that the SPDC flux remained about the same and the Gaussian spectrum was not disturbed (see Appendix B). With this filter we measured $V=98.4\pm 1.1\%$ ($96.9\pm 1.2\%$) for $\alpha =0.002$ (0.003), as shown in Fig. 4(b). The measured HOMI visibility of 98.4% corresponds to 99.2% heralded-state spectral purity without background corrections, in agreement with our previous estimate of the heralded-state purity of mildly-filtered SPDC under the assumption of a transform-limited biphoton state [20].

## 5. Conclusion

We have generated heralded single photons with high intrinsic spectral purity of 99.2% by use of a custom Gaussian phase-matching profile in PPKTP and a transform-limited Gaussian pump. We verified the heralded-state purity by performing a four-fold coincidence measurement of HOMI between two time-separated heralded photons, obtaining an interference visibility of 98.4% without correcting for degradation due to dark counts and multi-pair effects. We also showed, theoretically, that given a circularly symmetric Gaussian JSI and a Gaussian phase-matching function, it is necessary and sufficient to achieve a factorable biphoton state if the pump has a transform-limited Gaussian spectrum. Our technique can be easily replicated and the generated single-spatiotemporal-mode heralded photons can be utilized in many measurement-based quantum information processing applications that involve interference between independent single photons.

## Appendix

## A. Modification of the pump’s spectrum

The pump at the input to its spectral filtering setup in the main text’s Fig. 2(a) had a FWHM bandwidth of 6.25 nm as determined by an optical spectrum analyzer. We note that the pump’s FWHM deviates from its maximum value of 7.8 nm because the laser was driven at a lower current at the time of this particular measurement. The direct output of the mode-locked pump laser should have approximately a sech${}^{2}$ pulse shape, and our aim was to modify its shape to be Gaussian with a transform-limited bandwidth of ∼1 nm. In order to check if the pump was transform limited we measured the pulse duration of the pump before and after the 4*f* setup without an apodizing mask. The pulse width measurement was done by autocorrelation based on second harmonic generation. We obtained a pulse broadening ratio of 1.2 which corresponds to a second-order dispersion of *β* = 8,578 $f{s}^{2}$. The pulse broadening observed can be caused by aberrations, spatial chirp, or imperfect alignment. This amount of dispersion is negligible for apump bandwidth of ∼1 nm that we used to produce the joint spectral intensity (JSI) in the main text’s Fig. 1.

At the Fourier plane we placed an adjustable slit to reduce the pump bandwidth to 1.0 nm and to obtain a Gaussian shape. That hard-aperture mask, however, has a rectangular transmission profile and does not result in a Gaussian spectral profile, while the pump remained transform limited. By trial and error, we moved the slit several mm away from the Fourier plane to obtain a Gaussian shape. This maneuver reduced the filtered pump power and caused the pump to acquire an appreciable amount of dispersion. In order to have both a Gaussian spectrum and negligible dispersion, we chose to use a Gaussian transmission mask [33]. We fabricated multiple transmission masks lithographically on a single 2-mm thick chromium mask. Our Gaussian transmission mask consists of 16 individual strips of 2000 × 200 opaque squares each measuring 4 *μ*m in size. Each strip had a Gaussian spatial distribution of the opaque squares spread over slightly different strip widths that effectively imposed a pump bandwidth range 0.61–1.15 nm in 0.003 nm increments. In the main text we show the HOMI results for the three mask choices: slit at the Fourier plane that produces a transform-limited non-Gaussian pump, slit away from the Fourier plane that produces a non-transform-limited Gaussian pump, and the Gaussian transmission mask at the Fourier plane that produces a transform-limited Gaussian pump. The pump beam is then focused onto the crystal with a beam waist of $110\text{}\mu $m. The generated SPDC photons are collected with a collection beam waist of ∼ 90 *μ*m.

## B. Spectral filter transmission profile

We applied a 10-nm spectral filter to the SPDC output in order to remove the residual side lobes of the signal-idler JSI that are clearly visible in the main text’s Fig. 1(a) and reproduced here in Fig. 5’s inset. For optimal filtering, it is ideal to have unity transmission over the central peak and sufficient absorption in the side lobes such that the SPDC flux remains about the same and the JSI becomes more circularly symmetric, thereby improving the heralded-state spectral purity. Figure 5 shows the measured transmission profile of the flat-top spectral filter and the signal spectrum obtained from the marginal distribution of the JSI. We see that the filter transmission profile fully covers the SPDC spectrum with littleattenuation of the central peak, suggesting that the spectral filter introduces negligible loss to the main lobe of the SPDC output. The mild spectral filtering allows us to improve the Hong-Ou-Mandel interference visibility from 93.9% without filteringto 98.4% with filtering as noted in the main text.

## Funding

Air Force Office of Scientific Research (FA9550-14-1-0052).

## Acknowledgment

The authors thank Zheshen Zhang and Feihu Xu for helpful discussions, and Di Zhu for help with the SNSPD system. The contribution of NIST, an agency of the U.S. government, is not subject to copyright.

## References

**1. **A. Politi, J. C. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science **325**, 1221 (2009). [CrossRef]

**2. **N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics **11**, 447 (2017). [CrossRef]

**3. **N. Sangouard, C. Simon, H. De Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. **83**, 33 (2011). [CrossRef]

**4. **K. Azuma, K. Tamaki, and H.-K. Lo, “All-photonic quantum repeaters,” Nat. Commun. **6**, 6787 (2015). [CrossRef] [PubMed]

**5. **C. Law, I. A. Walmsley, and J. Eberly, “Continuous frequency entanglement: effective finite hilbert space and entropy control,” Phys. Rev. Lett. **84**, 5304 (2000). [CrossRef] [PubMed]

**6. **C. Law and J. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. **92**, 127903 (2004). [CrossRef] [PubMed]

**7. **P. P. Rohde, W. Mauerer, and C. Silberhorn, “Spectral structure and decompositions of optical states, and their applications,” New J. Phys. **9**, 91 (2007). [CrossRef]

**8. **W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A **64**, 063815 (2001). [CrossRef]

**9. **P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. **100**, 133601 (2008). [CrossRef] [PubMed]

**10. **C.-K. Hong, Z.-Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. **59**, 2044 (1987). [CrossRef] [PubMed]

**11. **V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong, “Generating entangled two-photon states with coincident frequencies,” Phys. Rev. Lett. **88**, 183602 (2002). [CrossRef] [PubMed]

**12. **O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, “Joint temporal density measurements for two-photon state characterization,” Phys. Rev. Lett. **101**, 153602 (2008). [CrossRef] [PubMed]

**13. **G. Harder, V. Ansari, B. Brecht, T. Dirmeier, C. Marquardt, and C. Silberhorn, “An optimized photon pair source for quantum circuits,” Opt. Express **21**, 13975–13985 (2013). [CrossRef] [PubMed]

**14. **F. Kaneda, K. Garay-Palmett, A. B. U’Ren, and P. G. Kwiat, “Heralded single-photon source utilizing highly nondegenerate, spectrally factorable spontaneous parametric downconversion,” Opt. Express **24**, 10733–10747 (2016). [CrossRef] [PubMed]

**15. **M. M. Weston, H. M. Chrzanowski, S. Wollmann, A. Boston, J. Ho, L. K. Shalm, V. B. Verma, M. S. Allman, S. W. Nam, R. B. Patel, S. Slussarenko, and G. J. Pryde, “Efficient and pure femtosecond-pulse-length source of polarization-entangled photons,” Opt. Express **24**, 10869–10879 (2016). [CrossRef] [PubMed]

**16. **A. M. Brańczyk, A. Fedrizzi, T. M. Stace, T. C. Ralph, and A. G. White, “Engineered optical nonlinearity for quantum light sources,” Opt. Express **19**, 55–65 (2011). [CrossRef]

**17. **J. Tambasco, A. Boes, L. Helt, M. Steel, and A. Mitchell, “Domain engineering algorithm for practical and effective photon sources,” Opt. Express **24**, 19616–19626 (2016). [CrossRef] [PubMed]

**18. **F. Graffitti, D. Kundys, D. T. Reid, A. M. Brańczyk, and A. Fedrizzi, “Pure down-conversion photons through sub-coherence-length domain engineering,” Quantum Sci. Technol. **2**, 035001 (2017). [CrossRef]

**19. **F. Graffitti, P. Barrow, M. Proietti, D. Kundys, and A. Fedrizzi, “Independent high-purity photons created in domain-engineered crystals,” Optica **5**, 514–517 (2018). [CrossRef]

**20. **P. B. Dixon, J. H. Shapiro, and F. N. C. Wong, “Spectral engineering by gaussian phase-matching for quantum photonics,” Opt. Express **21**, 5879–5890 (2013). [CrossRef] [PubMed]

**21. **C. Chen, C. Bo, M. Y. Niu, F. Xu, Z. Zhang, J. H. Shapiro, and F. N. C. Wong, “Efficient generation and characterization of spectrally factorable biphotons,” Opt. Express **25**, 7300–7312 (2017). [CrossRef] [PubMed]

**22. **F. Graffitti, J. Kelly-Massicotte, A. Fedrizzi, and A. M. Brańczyk, “Design considerations for high-purity heralded single-photon sources,” Phys. Rev. A **98**, 053811 (2018). [CrossRef]

**23. **J.-P. W. MacLean, J. M. Donohue, and K. J. Resch, “Direct characterization of ultrafast energy-time entangled photon pairs,” Phys. Rev. Lett. **120**, 053601 (2018). [CrossRef] [PubMed]

**24. **A. O. Davis, V. Thiel, M. Karpiński, and B. J. Smith, “Measuring the single-photon temporal-spectral wave function,” Phys. Rev. Lett. **121**, 083602 (2018). [CrossRef] [PubMed]

**25. **L. Praxmeyer and K. Wodkiewicz, “Time and frequency description of optical pulses,” Laser Phys. **15**, 1477–1485 (2005).

**26. **V. Ansari, J. M. Donohue, M. Allgaier, L. Sansoni, B. Brecht, J. Roslund, N. Treps, G. Harder, and C. Silberhorn, “Tomography and purification of the temporal-mode structure of quantum light,” Phys. Rev. Lett. **120**, 213601 (2018). [CrossRef] [PubMed]

**27. **N. Quesada and A. M. Brańczyk, “Gaussian functions are optimal for waveguided nonlinear-quantum-optical processes,” Phys. Rev. A **98**, 043813 (2018). [CrossRef]

**28. **A. M. Weiner, J. P. Heritage, and E. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B **5**, 1563–1572 (1988). [CrossRef]

**29. **A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. **71**, 1929–1960 (2000). [CrossRef]

**30. **P. B. Dixon, D. Rosenberg, V. Stelmakh, M. E. Grein, R. S. Bennink, E. A. Dauler, A. J. Kerman, R. J. Molnar, and F. N. C. Wong, “Heralding efficiency and correlated-mode coupling of near-ir fiber-coupled photon pairs,” Phys. Rev.A **90**, 043804 (2014). [CrossRef]

**31. **R. Graham and H. Haken, “The quantum-fluctuations of the optical parametric oscillator. I,” Z. Physik **210**, 276–302 (1968). [CrossRef]

**32. **E. J. Mason and N. C. Wong, “Observation of two distinct phase states in a self-phase-locked type ii phase-matched optical parametric oscillator,” Opt. Lett. **23**, 1733–1735 (1998). [CrossRef]

**33. **T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A **82**, 031802 (2010). [CrossRef]