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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 12 — Jun. 17, 2013
  • pp: 13975–13985
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An optimized photon pair source for quantum circuits

Georg Harder, Vahid Ansari, Benjamin Brecht, Thomas Dirmeier, Christoph Marquardt, and Christine Silberhorn  »View Author Affiliations


Optics Express, Vol. 21, Issue 12, pp. 13975-13985 (2013)
http://dx.doi.org/10.1364/OE.21.013975


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Abstract

We implement an ultrafast pulsed type-II parametric down conversion source in a periodically poled KTP waveguide at telecommunication wavelengths with almost identical properties between signal and idler. As such, our source resembles closely a pure, genuine single mode photon pair source with indistinguishable modes. We measure the joint spectral intensity distribution and second order correlation functions of the marginal beams and find with both methods very low effective mode numbers corresponding to a Schmidt number below 1.16. We further demonstrate the indistinguishability as well as the purity of signal and idler photons by Hong-Ou-Mandel interferences between signal and idler and between signal/idler and a coherent field, respectively. Without using narrowband spectral filtering, we achieve a visibility for the interference between signal and idler of 94.8% and determine a purity of more than 80% for the heralded single photon states. Moreover, we measure raw heralding efficiencies of 20.5% and 15.5% for the signal and idler beams corresponding to detector-loss corrected values of 80% and 70%.

© 2013 OSA

1. Introduction

Complex photonic systems for quantum communication or quantum simulation applications rely on the interference between different quantum states of light at a beam splitter [1

1. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001) [CrossRef] [PubMed] .

4

4. M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 797–798 (2012).

]. The indistinguishability and purity of these states is crucial for the performance of such applications as dissimilarities and mixedness reduce the visibility and hence the quantum character of the interference. Quantum state sources for these applications should therefore not only meet demands of brightness and efficiency, but also generate pure and indistinguishable states.

Parametric down-conversion (PDC) is a versatile source of quantum states which can be used either as direct input states or for heralding single photons [5

5. A. B. U’Ren, C. Silberhorn, K. Banaszek, and I. A. Walmsley, “Efficient conditional preparation of high-fidelity single photon states for fiber-optic quantum networks,” Phys. Rev. Lett. 93, 093601 (2004) [CrossRef] .

8

8. K. Laiho, K. N. Cassemiro, and C. Silberhorn, “Producing high fidelity single photons with optimal brightness via waveguided parametric down-conversion,” Opt. Express 17, 22823–22837 (2009) [CrossRef] .

] as well as other non-Gaussian states [9

9. A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004) [CrossRef] [PubMed] .

11

11. E. Bimbard, N. Jain, A. MacRae, and A. I. Lvovsky, “Quantum-optical state engineering up to the two-photon level,” Nature Photon. 4, 243–247 (2010) [CrossRef] .

]. However, generic PDC sources feature spectral correlations between signal and idler [12

12. 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] .

]. When single photons are heralded from these sources, the correlations introduce mixedness in the heralded states [13

13. A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146 (2005).

, 14

14. 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] .

]. Similarly, when interfering states from independent sources, spectral and spatial correlations decrease the visibility of the interference. It is therefore desirable to have separable PDC sources with no correlations between signal and idler at ones disposal. This implies that it must be possible to define for each photon an ultrafast wavepacket, which also occupies exactly one single spatial mode.

The most simple and widely used approach to achieve spectral decorrelation is to use spectral filtering [5

5. A. B. U’Ren, C. Silberhorn, K. Banaszek, and I. A. Walmsley, “Efficient conditional preparation of high-fidelity single photon states for fiber-optic quantum networks,” Phys. Rev. Lett. 93, 093601 (2004) [CrossRef] .

9

9. A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004) [CrossRef] [PubMed] .

, 11

11. E. Bimbard, N. Jain, A. MacRae, and A. I. Lvovsky, “Quantum-optical state engineering up to the two-photon level,” Nature Photon. 4, 243–247 (2010) [CrossRef] .

, 15

15. T. B. Pittman and J. D. Franson, “Violation of bells inequality with photons from independent sources,” Phys. Rev. Lett. 90, 240401 (2003) [CrossRef] [PubMed] .

, 16

16. P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45, 595–604 (1998) [CrossRef] .

]. However, perfect decorrelation can only be achieved in the limit of a small bandwidth, reducing the brightness of the source significantly [17

17. A. M. Brańczyk, T. C. Ralph, W. Helwig, and C. Silberhorn, “Optimized generation of heralded fock states using parametric down-conversion,” New J. Phys. 12, 063001 (2010) [CrossRef] .

]. This tradeoff can be avoided by tailoring the dispersion properties of the nonlinear medium as well as the spectrum of the pump beam in such a way that the PDC process produces decorrelated states directly [12

12. 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] .

, 13

13. A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146 (2005).

]. This approach has first been realized in bulk PDC at wavelengths around 800 nm [18

18. 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] .

]. For telecom wavelengths, a decorrelated source in a waveguide structure was realized in periodically poled potassium titanyl phosphate (ppKTP) [19

19. A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett. 106, 013603 (2011) [CrossRef] [PubMed] .

]. Alternatively to PDC, four-wave mixing in microstructured fibers [20

20. O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett. 102, 123603 (2009) [CrossRef] [PubMed] .

] or silica waveguides [21

21. J. B. Spring, P. S. Salter, B. J. Metcalf, P. C. Humphreys, M. Moore, N. Thomas-Peter, M. Barbieri, X.-M. Jin, N. K. Langford, W. S. Kolthammer, M. J. Booth, and I. A. Walmsley, “On-chip low loss heralded source of pure single photons,” Opt. Express21, 13522–13532 (2011) [CrossRef] .

] can also be used to produced decorrelated photon pair states. The additional condition of indistinguishability between signal and idler has been realized in bulk KTP at telecom wavelengths [22

22. T. Gerrits, M. J. Stevens, B. Baek, B. Calkins, A. Lita, S. Glancy, E. Knill, S. W. Nam, R. P. Mirin, R. H. Hadfield, R. S. Bennink, W. P. Grice, S. Dorenbos, T. Zijlstra, T. Klapwijk, and V. Zwiller, “Generation of degenerate, factorizable, pulsed squeezed light at telecom wavelengths,” Opt. Express 19, 24434–24447 (2011) [CrossRef] [PubMed] .

] and in bulk BBO at around 800 nm [23

23. M. Tanida, R. Okamoto, and S. Takeuchi, “Highly indistinguishable heralded single-photon sources using parametric down conversion,” Opt. Express 20, 15275 (2012) [CrossRef] [PubMed] .

].

In this paper we report on an integrated source that combines decorrelation, indistinguishability and telecom wavelengths without relying on narrowband filtering. Being a waveguide source, it is efficient, exhibits decorrelated spatial Gaussian mode profiles and thus features high coupling efficiencies into single mode fibers allowing for easy integration with existing telecommunication networks.

Our source is also capable of producing bright two mode squeezed states with a mean photon number well above one. However, in this paper we mainly focus on the characterization as a photon pair source, i.e. we work in the low power regime with a mean photon number far below one.

2. Theory

2.1. Decorrelated parametric down-conversion

The spectral properties of a PDC process in waveguides are governed by the energy and momentum conservation laws
ωs+ωi=ωp,ks+ki+2πΛ=kp,
(1)
where ω is the frequency of the signal, idler or pump photons, respectively, k their propagation constant and Λ the period of a periodic poling of the waveguide. Energy and momentum are linked by the dispersion relation ω(k) of the waveguide, mainly influenced by the material properties. For short pump pulses, a distribution of frequencies has to be taken into account, the pump function α(ωp). Similarly, for waveguides of finite lengths, the momentum conservation is described by a phasematching function ϕ(ωs, ωi). The full process is then described by the joint spectral amplitude (JSA)
f(ωs,ωi)=ϕ(ωs,ωi)α(ωs,ωi)
(2)
and the Hamiltonian is given by H=ζdωsdωif(ωs,ωi)as(ωs)ai(ωi)+h.c., where a(ω) is the standard creation operator at frequency ω. The amount of spectral correlations can be quantified by the Schmidt number K = 1/∑k |ck|4, where the coefficients ck result from the Schmidt decomposition of the JSA: f(ωs, ωi) = ∑kckϕk(ωs)ψk(ωi), with orthonormal functions ϕk and ψk[24

24. C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: Effective finite hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304–5307 (2000) [CrossRef] [PubMed] .

]. In particular for K = 1, signal and idler are decorrelated and the state is separable.

Most PDC sources exhibit a negatively correlated phasematching function ϕ(ωs, ωi). Since the pump function α(ωs, ωi) is always oriented exactly at −45° due to energy conservation, spectral correlations cannot be avoided in these sources. With a positively oriented phasematching, however, spectral decorrelation is possible. By dispersion engineering, i.e. choosing the right material, and adjustment of the pump width, spectral decorrelation can be achieved [13

13. A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146 (2005).

]. KTP waveguides have a phasematching correlation angle of approximately 59° [25

25. A. Eckstein, “Mastering quantum light pulses with nonlinear waveguide interactions,” Ph.D. thesis, Naturwis-senschaftliche Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg (2012).

] at telecom wavelengths around 1550nm. Assuming a Gaussian pump spectrum and a Gaussian phasematching function, wwhich is a good approximation for realistic, slightly inhomogeneous waveguides, the joint spectral amplitude becomes perfectly separable with a shape that is close to round, see Fig. 1.

Fig. 1 Separable joint spectral amplitude resulting from a phasematching function ϕ with a positive slope multiplied by a pump function α of appropriate width.

2.2. Source characterization

Hong Ou Mandel interference between signal and idler

The interference of twin photons at a beamsplitter followed by a coincidence measurement with single photon detectors is a well established method to deduce information about a biphoton quantum state. First demonstrated by Hong, Ou and Mandel [26

26. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987) [CrossRef] [PubMed] .

] to measure the duration of photon wavepackets, it can be used as a measure of indistinguishability [6

6. T. Pittman, B. Jacobs, and J. Franson, “Heralding single photons from pulsed parametric down-conversion,” Optics Communications 246, 545–550 (2005) [CrossRef] .

, 27

27. P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric downconversion: a recipe for purity,” New J. Phys. 10, 093011 (2008) [CrossRef] .

, 28

28. F. W. Sun and C. W. Wong, “Indistinguishability of independent single photons,” Phys. Rev. A 79, 013824 (2009) [CrossRef] .

] and even for reconstruction of the spectral properties of a state [29

29. M. Avenhaus, M. V. Chekhova, L. A. Krivitsky, G. Leuchs, and C. Silberhorn, “Experimental verification of high spectral entanglement for pulsed waveguided spontaneous parametric down-conversion,” Phys. Rev. A 79, 043836 (2009) [CrossRef] .

31

31. P. Kolenderski and W. Wasilewski, “Derivation of the density matrix of a single photon produced in parametric down-conversion,” Phys. Rev. A 80, 015801 (2009) [CrossRef] .

]. The Hong-Ou-Mandel (HOM) interference is unaffected by loss and has been widely applied for the characterization of single photon states from PDC.

Assuming that signal and idler originate from a perfect, weakly pumped PDC source with a not necessarily decorrelated JSA f(ωs, ωi), the probability of a coincidence click after the interference at a 50/50 beam splitter for a time delay τ between signal and idler is given by
p(τ)=1212Redωsdωif*(ωs,ωi)f(ωi,ωs)eiτ(ωsωi).
(3)
For τ = 0, the integral term is the overlap between the JSA and its mirrored counterpart along the 45° line. A HOM measurement hence probes the symmetry of the state under the exchange of signal and idler, as illustrated in Fig. 2.

Fig. 2 The visibility of the HOM interference is proportional to the overlap (shaded area) of the JSA with itself under the exchange of signal and idler (contour plots). a) Positive correlation, K = 2 and visibility 70%. b) Decorrelation, K = 1 and visibility 96.7%. c) Negative correlation, K = 2 and visibility 99.8%. Panel b) corresponds to the situation presented in this paper.

Hong Ou Mandel interference with a reference field

For the phasematching correlation angle of our source, high visibilities in a HOM interference between signal and idler can be obtained in the decorrelated as well as in the negatively correlated case. Our goal, however, is to produce decorrelated states as in the central plot of Fig. 2. To distinguish between the decorrelated and the correlated case, it is necessary to interfere the state with an independent reference beam. This can be understood as a purity measurement of the marginal beams.

If signal and idler are correlated, tracing out the idler mode results in a mixed signal state, described in frequency space by a density matrix ρs(ωs, ωs). Being hermitian, this matrix must be symmetric with respect to the principal diagonal. In the following we assume that it is a 2-dimensional Gaussian function, which is well justified for our source as discussed in Sec. 3.2. It follows from straight forward calculations that the purity of the state is given by Tr(ρs2)=σ2/σ1, where σ1 and σ2 are the major and minor axes of the Gaussian function ρs(ωs,ωs'). The major axis is the spectral width of the signal beam and can be measured directly with a spectrometer. The minor axis cannot be accessed directly with a spectral measurement but has an influence on the coherence length of the state and hence the interference pattern, as depicted in Fig. 3. By a HOM interference measurement with a known reference field, the minor axis can be deduced from the temporal width of the HOM dip, which is given by
δ2=12σ22+12σβ2
(4)
where σβ is the width of the reference field. A detailed analysis of this technique can be found in [32

32. K. N. Cassemiro, K. Laiho, and C. Silberhorn, “Accessing the purity of a single photon by the width of the HongOuMandel interference,” New J. Phys. 12, 113052 (2010) [CrossRef] .

]. Note that a measurement of the widths is much more robust compared to a measurement of the visibility because it is not affected by higher photon number components and imperfect overlaps of the state in the spectral and polarization degree of freedom. For comparison with the JSA, the relation between the purity and the Schmidt number is simply given by P = 1/K.

Fig. 3 Comparison of an uncorrelated PDC state (dotted black lines) to a correlated PDC state (green, purple or solid black lines). Left: JSA. Center: reduced density matrix of signal. Right: HOM dip between signal and reference. Both cases, correlated and uncorrelated, have the same marginal spectral widths. Nevertheless, the width of the HOM dip reveals the amount of correlations.

Joint spectral intensity and second order correlation function

Alternatively to HOM interference measurements, information about the separability of the state can be inferred from the joint spectral intensity (JSI) |f(ωs, ωi)|2 or the second order Glauber correlation function g(2)(τ) [33

33. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963) [CrossRef] .

] of the signal or idler field without heralding on a photon in the opposite mode. The g(2)(0) value gives information about the photon number statistics of the marginal beams. For a decorrelated PDC state, the marginal statistics are thermal with g(2)(0) = 2 and for a strongly correlated state, the marginal statistics are Poissonian with g(2)(0) = 1 [16

16. P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45, 595–604 (1998) [CrossRef] .

]. The relation between the Schmidt number and the marginal Glauber correlation function is given by g(2)(0) = 1 + 1/K[19

19. A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett. 106, 013603 (2011) [CrossRef] [PubMed] .

, 34

34. A. Christ, K. Laiho, A. Eckstein, K. N. Cassemiro, and C. Silberhorn, “Probing multimode squeezing with correlation functions,” New J. Phys. 13, 033027 (2011) [CrossRef] .

].

All three methods for source characterization have already been applied in the literature. However, there is no ideal measurement: A JSI measurement is affected by a low spectral resolution and cannot resolve correlations hidden in the phase of the JSA. Furthermore, it is blind to any non spectral correlations. A g(2)(0) measurement is sensitive to correlations in all degrees of freedom, but is, unfortunately, strongly affected by detector dark counts and non-PDC background signal. The HOM interference measurement requires the exact knowledge of spectral widths of signal and reference fields. Therefore it makes sense to apply all three techniques and carefully compare their results as we do in this paper.

3. Experimental results

3.1. Experimental setup

We employ type-II PDC in ppKTP waveguides. At telecom wavelength it offers a positively correlated phasematching function with a correlation angle of 59 degrees, such that a decorrelated JSA can be produced using the correct pump width [19

19. A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett. 106, 013603 (2011) [CrossRef] [PubMed] .

]. Such waveguides can be routinely produced and are commercially available. Our chip is purchased from ADVR and has the following parameters: a length of 8mm, waveguide dimensions of about 4μm×6μm and a poling period of 117μm.

A scheme of the setup is shown in Fig. 4. It consists of a Ti:Sapph pumped OPO (Chameleon Compact OPO) running at 80MHz and producing short pulses of 250fs at 1536nm. A small part of the light is separated for the reference beam while the major part is frequency doubled in a BBO crystal. The cascade of Ti:Sapph, OPO and SHG is necessary to achieve a good synchronization between the coherent reference beam and the PDC light. A crucial part of the experiment is the spectral shaping of the pump. We employ a 4f -spectrometer, consisting of two gratings, two lenses and one slit in the center, all separated by the focal length of the lenses. The width of the spectrum can be narrowed by reducing the width of the slit. Additionally, the slit is tilted, effectively reducing the resolution of the spectrometer, to produce Gaussian spectral shapes rather than square-like shapes. The spectrum of both, pump and reference beams, is measured and monitored by an optical spectrum analyzer (OSA).

Fig. 4 Scheme of the setup. Pulsed light at telecom wavelengths is generated in an Ti:Sapph pumped optical parametric oscillator (OPO) and frequency doubled by second harmonic generation (SHG). Part of the light is separated for later use as a reference field. The repetition rate is lowered by an acousto-optical modulator (AOM) to match the repetition rate of the gated APDs. The 4-f spectrometer tailors the spectral width of the pump beam to achieve spectral decorrelation. The PDC state is generated inside the periodically poled KTP waveguide. The pump is separated by a long pass filter (LPF). A bandpass filter (BPF) is used to suppress background outside the PDC spectrum. Finally signal and idler are separated at a polarizing beam splitter (PBS).

We use aspheric lenses for in- and outcoupling of the waveguide and the fibers. Behind the waveguide, we employ a longpass filter (Semrock LP02-808RU-25) to block out the pump beam and a bandpass filter (Semrock NIR01-1535/3-25) with an FWHM of 8nm. This bandpass filter is the second crucial part of the experiment because the PDC generates background signal over a wide spectral range [35

35. A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Realistic g(2) measurement of a PDC source with single photon detectors in the presence of background,” Phys. Status Solidi 8, 1216–1219 (2011) [CrossRef] .

]. The bandpass filter is chosen such that it blocks as much background signal as possible without affecting the PDC signal itself. All beams are coupled into single mode fibers (SMF). For the detection at the single photon level, we use avalanche photo diodes (APDs) (either Id Quantique id201 or NuCrypt CPDS-1000-4). The repetition rate of the id201 is 1 MHz and of the CPDS-1000-4 40 MHz. We measure raw coincidence vs single ratios, i.e. Klyshko efficiencies [36

36. D. N. Klyshko, “Utilization of vacuum fluctuations as an optical brightness standard,” Sov. J. Quantum Electron. 7, 591–595 (1977) [CrossRef] .

38

38. D. Achilles, C. Silberhorn, and I. A. Walmsley, “Direct, loss-tolerant characterization of nonclassical photon statistics,” Phys. Rev. Lett. 97, 043602 (2006) [CrossRef] [PubMed] .

] of up to 20.5% for the signal beam and 15.5% for the idler beam. Corrected for the detector (id201) efficiencies of approximately 25% and 22%, these values correspond to coupling efficiencies into the SMFs of roughly 80% in one arm and 70% in the other arm, making our source highly compatible with existing fiber based telecommunication technologies.

3.2. Spectral measurements

We characterize the spectral properties of our source with a fiber spectrometer [39

39. M. Avenhaus, A. Eckstein, P. J. Mosley, and C. Silberhorn, “Fiber-assisted single-photon spectrograph,” Opt. Lett. 34, 2873 (2009) [CrossRef] [PubMed] .

] (see first frame of Fig. 5). Signal and idler travel through dispersive fibers before impinging onto two APDs. Different wavelengths arrive at different times. By scanning the gating times of the APDs, a joint spectral intensity (JSI) distribution is obtained from coincidence click rates. Similarly, the marginal spectral distribution of signal and idler are obtained from the single click rates. The resolution of the fiber spectrometer is limited by the gate width of the APDs and the length of the fiber. We use the id201 with a gate width of approximately 1.5 ns, resulting in a spectral resolution of 1.8 nm for the JSI and 0.9 nm for the marginal spectra.

Fig. 5 Scheme of different measurement settings. a) JSI measurement; b) g(2)(0) measurement; c) HOM interference between signal and idler; d) 2-fold or 3-fold HOM interference with a reference field. The Id Quantique detectors(1MHz) are used for the JSI measurement and the NuCrypt detectors (40MHz) for the rest.

The JSI and the marginal spectral distributions with and without the bandpass filter are shown in fig. 6. The bandpass filter has a width of 8 nm and one can see from the comparison of the unfiltered with the filtered spectrum in Fig. 6 that background signal is suppressed directly outside of the PDC range while the PDC spectrum itself is mostly unaffected. The spectra of signal and idler have Gaussian shapes and their widths at FWHM, obtained from Gaussian fits, are 5.2 nm and 4.0 nm, respectively. Assuming that the finite resolution is effectively a convolution with an 0.9 nm wide Gaussian, the true widths can be estimated to be 5.1 nm and 3.9 nm. The theoretical sinc-shape of the spectrum is smeared out by experimental imperfections of the waveguide structure.. The congruence of the data points with the fit curve supports the assumption of a Gaussian JSA in Sec. 2.2.

Fig. 6 a) Marginal idler spectra for the decorrelated case with and without the bandpass filter (BF). One can see that the spectral filter fits nicely the idler spectrum, touching it only slightly at the very edges. b) Signal and idler spectra with the bandpass filter. Bottom: Measured joint spectral intensity for anticorrelation (c) and decorrelation (d). The pump widths are 0.5nm and 2.1nm, respectively.

The JSI measurement in the decorrelated case shows a smooth, nearly round shape. This also supports the assumption of Gaussian phasematching and pump spectral profiles. Being an intensity distribution, the JSI does not provide full information about the state as additional phase or temporal information would be required to reconstruct the JSA [40

40. B. Brecht and C. Silberhorn, “Characterizing entanglement in pulsed parametric down-conversion using chrono-cyclic wigner functions,” Phys. Rev. A 87, 053810 (2013) [CrossRef] .

]. A singular value decomposition of the square root of the JSI, yielding an effective mode number of 1.001 in our case, can therefore be only regarded as a lower bound for the true Schmidt number.

To obtain a stronger statement about the correlations present in the source, we measure the second order Glauber correlation function g(2)(0) with a 50/50 fiber coupler [16

16. P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45, 595–604 (1998) [CrossRef] .

], as depicted in the second frame of Fig. 5. We find raw values of 1.83 ± 0.02 and 1.86 ± 0.02 for signal and idler, respectively. These values pose an upper bound on the correlations present in the PDC source. The corresponding Schmidt numbers are 1.20 and 1.16. Part of the deviation from the perfectly decorrelated case could be caused by background events. From comparison of the filtered with the unfiltered spectra, we conclude that the background still remaining under the PDC spectrum makes up for approximately 1.9% of the total count rates. Since background signal has a Poissonian photon number distribution, it strongly degrades the g(2)(0) value. Correcting for these background events [35

35. A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Realistic g(2) measurement of a PDC source with single photon detectors in the presence of background,” Phys. Status Solidi 8, 1216–1219 (2011) [CrossRef] .

], we get values of 1.90 and 1.94. However, we would like to note that the raw value of g(2)(0) = 1.86 is among the highest compared to other PDC sources [16

16. P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45, 595–604 (1998) [CrossRef] .

, 19

19. A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett. 106, 013603 (2011) [CrossRef] [PubMed] .

, 41

41. X. Li, L. Yang, L. Cui, Z. Y. Ou, and D. Yu, “Fiber-based source of photon pairs at telecom band with high temporal coherence and brightness for quantum information processing,” Opt. Lett. 33, 593 (2008) [CrossRef] [PubMed] .

, 42

42. H. Takesue and B. Miquel, “Entanglement swapping using telecom-band photons generated in fibers,” Opt. Express 17, 10748–10756 (2009) [CrossRef] [PubMed] .

] indicating that the amount of background events is relatively low.

3.3. Interference measurements

To demonstrate the indistinguishability between signal and idler, we interfere them at a 50/50 fiber coupler, as sketched in the third frame of Fig. 5. The coincidence rate versus a delay of one of the beams is shown in Fig. 7. The measurement is done with a pump energy as low as 0.6pJ per pulse leading to a mean photon number of 0.002. We obtain a visibility of (94.8 ± 0.6)%, which is to our knowledge the highest value reported in the literature for PDC sources without narrowband filtering. It is close to the theoretical value of 96.7% for the decorrelated case. Part of that small deviation from the theoretical value is caused by our fiber coupler which has a slightly uneven coupling ratio of 49.1/50.9. The high visibility of the interference shows that the indistinguishability between signal and idler is indeed very high.

Fig. 7 HOM interference between signal and idler.

As discussed in the theory section, a robust method for verifying the decorrelation of the PDC state is to measure the purity of the marginal beams by HOM interference with a reference field. As the reference field, we use part of the original laser beam and attenuate it to the single photon level. All three beams, signal, idler and reference are coupled into single mode fibers and the count rates are recorded with the APDs, as sketched in Fig. 5. We record two-fold and three-fold coincidences, where by two-fold we mean coincidence events behind the beam splitter disregarding the third APD and by three-fold we mean triple coincidences between all three output ports. The heralding with the second PDC beam in the three-fold case increases the visibility of the interference [8

8. K. Laiho, K. N. Cassemiro, and C. Silberhorn, “Producing high fidelity single photons with optimal brightness via waveguided parametric down-conversion,” Opt. Express 17, 22823–22837 (2009) [CrossRef] .

]. The results are shown in Fig. 8. As expected, the visibility in the three-fold case is higher than in the two-fold case. It depends on the spectral overlap between the two beams as well as the mean photon numbers which for this measurement are 0.006 for the marginal beam and 0.08 for the reference beam. For calculating the purity of the state, we use the width of the dip rather than the visibility. Both two-fold and three-fold curves have similar widths of 1.33±0.02ps and 1.28±0.04ps. Taking into account the spectral width of the Gaussian reference field of 4.5nm and the signal spectral width of 3.9nm, we calculate a purity of 82.1 ± 1.7% and 86.7 ± 4.3%. These values are in good agreement with the raw g(2)(0) values. Compared to other PDC sources without narrowband filtering, Mosley et. al [18

18. 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] .

] have shown higher purities around 95% by measuring the HOM interference between two independent PDC sources. Our measured purities are slightly below this value but still in the same range, demonstrating excellent source performance. Similar values have been obtained in [43

43. R.-B. Jin, J. Zhang, R. Shimizu, N. Matsuda, Y. Mitsumori, H. Kosaka, and K. Edamatsu, “High-visibility non-classical interference between intrinsically pure heralded single photons and photons from a weak coherent field,” Phys. Rev. A 83, 031805 (2011) [CrossRef] .

], who also utilized HOM interference with a coherent field.

Fig. 8 two-fold (left) and three-fold (right) HOM interference between signal and reference.

3.4. Source brightness

In the low power regime at which the above interference measurements are performed, we measure a photon pair generation efficiency of 3 × 109 (photon pairs)/J or 3.8 × 1011 (photon pairs)/(J × m), if taken per unit crystal length and pump energy. Taking into account that all photons are generated in a single mode, the source brightness is extremely high, even for low pump powers. For a mean photon number per pulse of 0.1 in the signal beam, the required cw equivalent power at a repetition rate of 1MHz is only 33μW. By using the full available power in our setup of 2.5mW (2.5nJ per pulse), we are able to reach a mean photon number of approximately 80 photon pairs per pulse. This high number is due to the single mode character of our source and much higher than is expected for a multimode source with the same photon pair generation efficiency.

4. Conclusion

In conclusion, we have implemented a source with remarkable properties in terms of brightness, purity and symmetry. The HOM interference between signal and idler showed, to our knowledge, an unprecedented visibility of 94.8%. The purity values for signal and idler obtained from g(2), JSI and interference with a reference beam reveal a purity of above 80%. In order to reach a desirable mean photon number around 0.1 photon pairs per pulse, pump energies as low as 33pJ are required, which significantly simplifies the simultaneous operation of several sources. Our source can be easily combined with identical sources or other optical fields and constitutes a step towards the feasibility of more complex quantum circuits or networks.

Acknowledgment

We thank Kaisa Laiho, Andreas Eckstein and Malte Avenhaus for valuable and helpful discussions. The research leading to these results has received funding from the European Community’s Seventh Framework Programme FP7/2001-2013 under grant agreement no. 248095 through the Integrated Project Q-ESSENCE.

References and links

1.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001) [CrossRef] [PubMed] .

2.

T. C. Ralph, “Quantum optical systems for the implementation of quantum information processing,” Rep. Prog. Phys. 69, 853–898 (2006) [CrossRef] .

3.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2012) [CrossRef] [PubMed] .

4.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 797–798 (2012).

5.

A. B. U’Ren, C. Silberhorn, K. Banaszek, and I. A. Walmsley, “Efficient conditional preparation of high-fidelity single photon states for fiber-optic quantum networks,” Phys. Rev. Lett. 93, 093601 (2004) [CrossRef] .

6.

T. Pittman, B. Jacobs, and J. Franson, “Heralding single photons from pulsed parametric down-conversion,” Optics Communications 246, 545–550 (2005) [CrossRef] .

7.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon fock state,” Phys. Rev. Lett. 87, 050402 (2001) [CrossRef] [PubMed] .

8.

K. Laiho, K. N. Cassemiro, and C. Silberhorn, “Producing high fidelity single photons with optimal brightness via waveguided parametric down-conversion,” Opt. Express 17, 22823–22837 (2009) [CrossRef] .

9.

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004) [CrossRef] [PubMed] .

10.

K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Photon subtracted squeezed states generated with periodically poled KTiOPO4,” Opt. Express 15, 3568–3574 (2007) [CrossRef] [PubMed] .

11.

E. Bimbard, N. Jain, A. MacRae, and A. I. Lvovsky, “Quantum-optical state engineering up to the two-photon level,” Nature Photon. 4, 243–247 (2010) [CrossRef] .

12.

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] .

13.

A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146 (2005).

14.

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] .

15.

T. B. Pittman and J. D. Franson, “Violation of bells inequality with photons from independent sources,” Phys. Rev. Lett. 90, 240401 (2003) [CrossRef] [PubMed] .

16.

P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt. 45, 595–604 (1998) [CrossRef] .

17.

A. M. Brańczyk, T. C. Ralph, W. Helwig, and C. Silberhorn, “Optimized generation of heralded fock states using parametric down-conversion,” New J. Phys. 12, 063001 (2010) [CrossRef] .

18.

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] .

19.

A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett. 106, 013603 (2011) [CrossRef] [PubMed] .

20.

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett. 102, 123603 (2009) [CrossRef] [PubMed] .

21.

J. B. Spring, P. S. Salter, B. J. Metcalf, P. C. Humphreys, M. Moore, N. Thomas-Peter, M. Barbieri, X.-M. Jin, N. K. Langford, W. S. Kolthammer, M. J. Booth, and I. A. Walmsley, “On-chip low loss heralded source of pure single photons,” Opt. Express21, 13522–13532 (2011) [CrossRef] .

22.

T. Gerrits, M. J. Stevens, B. Baek, B. Calkins, A. Lita, S. Glancy, E. Knill, S. W. Nam, R. P. Mirin, R. H. Hadfield, R. S. Bennink, W. P. Grice, S. Dorenbos, T. Zijlstra, T. Klapwijk, and V. Zwiller, “Generation of degenerate, factorizable, pulsed squeezed light at telecom wavelengths,” Opt. Express 19, 24434–24447 (2011) [CrossRef] [PubMed] .

23.

M. Tanida, R. Okamoto, and S. Takeuchi, “Highly indistinguishable heralded single-photon sources using parametric down conversion,” Opt. Express 20, 15275 (2012) [CrossRef] [PubMed] .

24.

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: Effective finite hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304–5307 (2000) [CrossRef] [PubMed] .

25.

A. Eckstein, “Mastering quantum light pulses with nonlinear waveguide interactions,” Ph.D. thesis, Naturwis-senschaftliche Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg (2012).

26.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987) [CrossRef] [PubMed] .

27.

P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric downconversion: a recipe for purity,” New J. Phys. 10, 093011 (2008) [CrossRef] .

28.

F. W. Sun and C. W. Wong, “Indistinguishability of independent single photons,” Phys. Rev. A 79, 013824 (2009) [CrossRef] .

29.

M. Avenhaus, M. V. Chekhova, L. A. Krivitsky, G. Leuchs, and C. Silberhorn, “Experimental verification of high spectral entanglement for pulsed waveguided spontaneous parametric down-conversion,” Phys. Rev. A 79, 043836 (2009) [CrossRef] .

30.

W. Wasilewski, P. Kolenderski, and R. Frankowski, “Spectral density matrix of a single photon measured,” Phys. Rev. Lett. 99, 123601 (2007) [CrossRef] [PubMed] .

31.

P. Kolenderski and W. Wasilewski, “Derivation of the density matrix of a single photon produced in parametric down-conversion,” Phys. Rev. A 80, 015801 (2009) [CrossRef] .

32.

K. N. Cassemiro, K. Laiho, and C. Silberhorn, “Accessing the purity of a single photon by the width of the HongOuMandel interference,” New J. Phys. 12, 113052 (2010) [CrossRef] .

33.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963) [CrossRef] .

34.

A. Christ, K. Laiho, A. Eckstein, K. N. Cassemiro, and C. Silberhorn, “Probing multimode squeezing with correlation functions,” New J. Phys. 13, 033027 (2011) [CrossRef] .

35.

A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Realistic g(2) measurement of a PDC source with single photon detectors in the presence of background,” Phys. Status Solidi 8, 1216–1219 (2011) [CrossRef] .

36.

D. N. Klyshko, “Utilization of vacuum fluctuations as an optical brightness standard,” Sov. J. Quantum Electron. 7, 591–595 (1977) [CrossRef] .

37.

J. G. Rarity, K. D. Ridley, and P. R. Tapster, “Absolute measurement of detector quantum efficiency using parametric downconversion,” Appl. Opt. 26, 4616–4619 (1987) [CrossRef] [PubMed] .

38.

D. Achilles, C. Silberhorn, and I. A. Walmsley, “Direct, loss-tolerant characterization of nonclassical photon statistics,” Phys. Rev. Lett. 97, 043602 (2006) [CrossRef] [PubMed] .

39.

M. Avenhaus, A. Eckstein, P. J. Mosley, and C. Silberhorn, “Fiber-assisted single-photon spectrograph,” Opt. Lett. 34, 2873 (2009) [CrossRef] [PubMed] .

40.

B. Brecht and C. Silberhorn, “Characterizing entanglement in pulsed parametric down-conversion using chrono-cyclic wigner functions,” Phys. Rev. A 87, 053810 (2013) [CrossRef] .

41.

X. Li, L. Yang, L. Cui, Z. Y. Ou, and D. Yu, “Fiber-based source of photon pairs at telecom band with high temporal coherence and brightness for quantum information processing,” Opt. Lett. 33, 593 (2008) [CrossRef] [PubMed] .

42.

H. Takesue and B. Miquel, “Entanglement swapping using telecom-band photons generated in fibers,” Opt. Express 17, 10748–10756 (2009) [CrossRef] [PubMed] .

43.

R.-B. Jin, J. Zhang, R. Shimizu, N. Matsuda, Y. Mitsumori, H. Kosaka, and K. Edamatsu, “High-visibility non-classical interference between intrinsically pure heralded single photons and photons from a weak coherent field,” Phys. Rev. A 83, 031805 (2011) [CrossRef] .

OCIS Codes
(190.4390) Nonlinear optics : Nonlinear optics, integrated optics
(230.7380) Optical devices : Waveguides, channeled
(270.0270) Quantum optics : Quantum optics

ToC Category:
Quantum Optics

History
Original Manuscript: April 26, 2013
Revised Manuscript: May 22, 2013
Manuscript Accepted: May 23, 2013
Published: June 3, 2013

Citation
Georg Harder, Vahid Ansari, Benjamin Brecht, Thomas Dirmeier, Christoph Marquardt, and Christine Silberhorn, "An optimized photon pair source for quantum circuits," Opt. Express 21, 13975-13985 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-12-13975


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References

  1. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature409, 46–52 (2001). [CrossRef] [PubMed]
  2. T. C. Ralph, “Quantum optical systems for the implementation of quantum information processing,” Rep. Prog. Phys.69, 853–898 (2006). [CrossRef]
  3. J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science339, 798–801 (2012). [CrossRef] [PubMed]
  4. M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science339, 797–798 (2012).
  5. A. B. U’Ren, C. Silberhorn, K. Banaszek, and I. A. Walmsley, “Efficient conditional preparation of high-fidelity single photon states for fiber-optic quantum networks,” Phys. Rev. Lett.93, 093601 (2004). [CrossRef]
  6. T. Pittman, B. Jacobs, and J. Franson, “Heralding single photons from pulsed parametric down-conversion,” Optics Communications246, 545–550 (2005). [CrossRef]
  7. A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon fock state,” Phys. Rev. Lett.87, 050402 (2001). [CrossRef] [PubMed]
  8. K. Laiho, K. N. Cassemiro, and C. Silberhorn, “Producing high fidelity single photons with optimal brightness via waveguided parametric down-conversion,” Opt. Express17, 22823–22837 (2009). [CrossRef]
  9. A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science306, 660–662 (2004). [CrossRef] [PubMed]
  10. K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Photon subtracted squeezed states generated with periodically poled KTiOPO4,” Opt. Express15, 3568–3574 (2007). [CrossRef] [PubMed]
  11. E. Bimbard, N. Jain, A. MacRae, and A. I. Lvovsky, “Quantum-optical state engineering up to the two-photon level,” Nature Photon.4, 243–247 (2010). [CrossRef]
  12. W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A64, 063815 (2001). [CrossRef]
  13. A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys.15, 146 (2005).
  14. 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]
  15. T. B. Pittman and J. D. Franson, “Violation of bells inequality with photons from independent sources,” Phys. Rev. Lett.90, 240401 (2003). [CrossRef] [PubMed]
  16. P. R. Tapster and J. G. Rarity, “Photon statistics of pulsed parametric light,” J. Mod. Opt.45, 595–604 (1998). [CrossRef]
  17. A. M. Brańczyk, T. C. Ralph, W. Helwig, and C. Silberhorn, “Optimized generation of heralded fock states using parametric down-conversion,” New J. Phys.12, 063001 (2010). [CrossRef]
  18. 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]
  19. A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Highly efficient single-pass source of pulsed single-mode twin beams of light,” Phys. Rev. Lett.106, 013603 (2011). [CrossRef] [PubMed]
  20. O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett.102, 123603 (2009). [CrossRef] [PubMed]
  21. J. B. Spring, P. S. Salter, B. J. Metcalf, P. C. Humphreys, M. Moore, N. Thomas-Peter, M. Barbieri, X.-M. Jin, N. K. Langford, W. S. Kolthammer, M. J. Booth, and I. A. Walmsley, “On-chip low loss heralded source of pure single photons,” Opt. Express 21, 13522–13532 (2011). [CrossRef]
  22. T. Gerrits, M. J. Stevens, B. Baek, B. Calkins, A. Lita, S. Glancy, E. Knill, S. W. Nam, R. P. Mirin, R. H. Hadfield, R. S. Bennink, W. P. Grice, S. Dorenbos, T. Zijlstra, T. Klapwijk, and V. Zwiller, “Generation of degenerate, factorizable, pulsed squeezed light at telecom wavelengths,” Opt. Express19, 24434–24447 (2011). [CrossRef] [PubMed]
  23. M. Tanida, R. Okamoto, and S. Takeuchi, “Highly indistinguishable heralded single-photon sources using parametric down conversion,” Opt. Express20, 15275 (2012). [CrossRef] [PubMed]
  24. C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: Effective finite hilbert space and entropy control,” Phys. Rev. Lett.84, 5304–5307 (2000). [CrossRef] [PubMed]
  25. A. Eckstein, “Mastering quantum light pulses with nonlinear waveguide interactions,” Ph.D. thesis, Naturwis-senschaftliche Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg (2012).
  26. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett.59, 2044–2046 (1987). [CrossRef] [PubMed]
  27. P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric downconversion: a recipe for purity,” New J. Phys.10, 093011 (2008). [CrossRef]
  28. F. W. Sun and C. W. Wong, “Indistinguishability of independent single photons,” Phys. Rev. A79, 013824 (2009). [CrossRef]
  29. M. Avenhaus, M. V. Chekhova, L. A. Krivitsky, G. Leuchs, and C. Silberhorn, “Experimental verification of high spectral entanglement for pulsed waveguided spontaneous parametric down-conversion,” Phys. Rev. A79, 043836 (2009). [CrossRef]
  30. W. Wasilewski, P. Kolenderski, and R. Frankowski, “Spectral density matrix of a single photon measured,” Phys. Rev. Lett.99, 123601 (2007). [CrossRef] [PubMed]
  31. P. Kolenderski and W. Wasilewski, “Derivation of the density matrix of a single photon produced in parametric down-conversion,” Phys. Rev. A80, 015801 (2009). [CrossRef]
  32. K. N. Cassemiro, K. Laiho, and C. Silberhorn, “Accessing the purity of a single photon by the width of the HongOuMandel interference,” New J. Phys.12, 113052 (2010). [CrossRef]
  33. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev.130, 2529–2539 (1963). [CrossRef]
  34. A. Christ, K. Laiho, A. Eckstein, K. N. Cassemiro, and C. Silberhorn, “Probing multimode squeezing with correlation functions,” New J. Phys.13, 033027 (2011). [CrossRef]
  35. A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, “Realistic g(2) measurement of a PDC source with single photon detectors in the presence of background,” Phys. Status Solidi8, 1216–1219 (2011). [CrossRef]
  36. D. N. Klyshko, “Utilization of vacuum fluctuations as an optical brightness standard,” Sov. J. Quantum Electron.7, 591–595 (1977). [CrossRef]
  37. J. G. Rarity, K. D. Ridley, and P. R. Tapster, “Absolute measurement of detector quantum efficiency using parametric downconversion,” Appl. Opt.26, 4616–4619 (1987). [CrossRef] [PubMed]
  38. D. Achilles, C. Silberhorn, and I. A. Walmsley, “Direct, loss-tolerant characterization of nonclassical photon statistics,” Phys. Rev. Lett.97, 043602 (2006). [CrossRef] [PubMed]
  39. M. Avenhaus, A. Eckstein, P. J. Mosley, and C. Silberhorn, “Fiber-assisted single-photon spectrograph,” Opt. Lett.34, 2873 (2009). [CrossRef] [PubMed]
  40. B. Brecht and C. Silberhorn, “Characterizing entanglement in pulsed parametric down-conversion using chrono-cyclic wigner functions,” Phys. Rev. A87, 053810 (2013). [CrossRef]
  41. X. Li, L. Yang, L. Cui, Z. Y. Ou, and D. Yu, “Fiber-based source of photon pairs at telecom band with high temporal coherence and brightness for quantum information processing,” Opt. Lett.33, 593 (2008). [CrossRef] [PubMed]
  42. H. Takesue and B. Miquel, “Entanglement swapping using telecom-band photons generated in fibers,” Opt. Express17, 10748–10756 (2009). [CrossRef] [PubMed]
  43. R.-B. Jin, J. Zhang, R. Shimizu, N. Matsuda, Y. Mitsumori, H. Kosaka, and K. Edamatsu, “High-visibility non-classical interference between intrinsically pure heralded single photons and photons from a weak coherent field,” Phys. Rev. A83, 031805 (2011). [CrossRef]

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