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Demonstration of spectral correlation control in a source of polarization-entangled photon pairs at telecom wavelength Thomas Lutz,1,2 Piotr Kolenderski,3,* and Thomas Jennewein1 1
Institute for Quantum Computing, University of Waterloo, 200 University Ave. West, Waterloo, Ontario N2L 3G1, Canada 2 3
Institut für Quantenmaterie, Universität Ulm, 89069 Ulm, Germany
Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5, 87-100 Torun, Poland *Corresponding author: [email protected] Received November 28, 2013; revised February 7, 2014; accepted February 7, 2014; posted February 10, 2014 (Doc. ID 202038); published March 11, 2014 Spectrally correlated photon pairs can be used to improve the performance of long-range fiber-based quantum communication protocols. We present a source based on spontaneous parametric downconversion, which allows one to control spectral correlations within the entangled photon pair without spectral filtering by changing the pump-pulse duration or the characteristics of the coupled spatial modes. The spectral correlations and polarization entanglement are characterized. We find that the generated photon pairs can feature both positive spectral correlations, decorrelation, or negative correlations at the same time as polarization entanglement with a high fidelity of 0.97 (no background subtraction) with the expected Bell state. © 2014 Optical Society of America OCIS codes: (190.4410) Nonlinear optics, parametric processes; (300.6190) Spectrometers; (270.4180) Multiphoton processes; (270.5565) Quantum communications. http://dx.doi.org/10.1364/OL.39.001481
Controlling the spectral correlation of each polarizationentangled photon pair produced via spontaneous parametric downconversion (SPDC) could have important benefits in optical quantum information. Photonic quantum gates require pure single photon states, which can be created by sources producing pairs of spectrally decorrelated photons [1–11]. On the other hand, long-distance fiber-based quantum communication and quantum metrology [12,13] suffer from chromatic dispersion. Polarization mode dispersion [14] is a major source of decoherence [15,16] in fiber-based entanglement distribution protocols. These detrimental effects could potentially be mitigated [17] with positive spectral correlations in photon pairs [18–20]. Here we demonstrate a SPDC photon pair source that enables the control of the spectral correlation within the photon pair. The source is based on a β-barium borate (BBO) crystal, which was characterized previously in [21], where we experimentally measured the phasematching function using a CW laser and concluded that this particular setting for a BBO crystal at 1550 nm allows for control of spectral correlation. This can be done by changing the pump pulse durations and/or collection optics. In the previous configuration, we could produce only spectrally anticorrelated pairs in a separable polarization state. The source presented here produces polarization entangled pairs in the telecom band and can be tuned to create negative, none, or positive spectral correlations. The high-quality polarization entanglement is achieved without using spectral filtering. Also, in the typical scenario for type II SPDC, the walk-off effects have to be compensated by putting respective birefringent crystals in the optical paths of the generated photons. In our design of the experimental setup, no compensation crystals are required. This is due to the mixing of the photons at the polarizing beam splitter (PBS), which removes any distinguishability between the photons and compensates for the walk-off effect in the crystal [22,23]. 0146-9592/14/061481-04$15.00/0
Let us now discuss the main features of the source in detail. In the SPDC process, one photon of the pump downconverts into a photon pair. Energy and momentum conservation relations, jointly described as phase matching, and the properties of the pump photons govern the characteristics of the generated photons. The probability amplitude for a photon pair emission in a given direction and at a given frequency can be described by the product of the pump spatiotemporal amplitude and the phasematching function [5,22]. The phase matching, which depends on the properties of the nonlinear media, specifies the allowed emissions for a given pump beam. Typically, the output photons are coupled into optical fibers, which correspond to collecting photons from a specific range of directions that are defined by the fiber and the optics. From this point of view, coupling can be understood as an additional condition to phase-matching. Therefore, one can introduce an effective phase-matching function (EPMF) [5,21], which fully describes the joint effect of the crystal and coupling into fibers. It consists of a Gaussian function with the characteristic width σ. The spectral part of the pump can also be approximated by a Gaussian function with the characteristic spectral width 1∕2τp . The spectral wave function then reads: νs νi 2 τ2p ν − ν 2 : (1) exp − ψνs ; νi N exp − s 2 i 4 σ Here N is a normalization factor, and νs∕i are the frequency detunings of signal and idler photon from their central frequencies, which correspond to 1550 nm in our case. τp is the pump pulse duration. Note that the first exponent in Eq. (1) is an approximation of the EPMF for our particular nonlinear crystal, which allows for spectral correlation control [21]. It has already been shown experimentally [21] that the particular EPMF of the type-II BBO crystal cut for © 2014 Optical Society of America
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telecom wavelength enables spectral correlation control. Tuning capabilities arise due to the variable pulse duration τp of the pump laser and the variable collection optics, which influences the characteristic width σ of the EPMF. We introduce spectral qa correlation parameter
defined as r hνs νi i∕ hν2s ihν2i i. The brackets stand for expectation values, which we compute using the wave function given by Eq. (1). As a result, one obtains r
1∕4τ2p − σ 2 : 1∕4τ2p σ 2
(2)
This parameter, together with the cooperativity parameter K, introduced in [24], which gives information regarding the number of spectral modes involved, is very useful to characterize the performance of the source. Our source allows us to generate polarizationentangled pairs with the desired spectral correlation. In order to ensure the quality of polarization entanglement, we use a technique previously described in [22,25]. Using this approach, no compensating crystals are required. In addition, this allows us to avoid problems arising due to the spectral distinguishability of the extraordinary and ordinary photons as those are coupled into two distinct fibers, and no spectral filtering is required to improve the polarization entanglement visibility. The schematic setup of the experiment is shown in Fig. 1. The BBO crystal used in this source is cut at
Fig. 1. Experimental setup. The BBO crystal is pumped using a Ti:Sapphire laser. The beam is focused onto the crystal by a lens (L3). Downconverted photons are collimated using lenses (L2, f 25 cm). The retroreflectors are used to adjust the distances the extraordinary and ordinary photon travel. A half-wave plate (HWP) rotates the polarization by 90 deg, which results in splitting both photons at the PBS. After they pass polarizers (P), they are coupled into single-mode fiber (SMF-28e) by aspheric lenses (L1, f 15.4 mm). The stray light is filtered out by a long-pass filter (Semrock BLP01-1319R-25). Photons are detected using two InGaAs/InP detectors D2 (Micro-Photon-Devices, InGaAs/InP single-photon avalanche diode [26]) and D3 (idQuantique, id201). Timing analysis is performed using FPGA electronics. A small percentage of the pump light is directed to a photodiode (DET10A) D1 using a glass plate (GP), a coupling lens (L4), and a SMF at 800 nm. D1 is used to measure pump pulse time.
29.14 deg for type-II phase matching. A pump photon at wavelength 775 nm is downconverted into a photon pair at wavelength 1550 nm. The crystal is pumped using either a CW (MIRA-900, Coherent) or a femtosecond (TiF-50M, Atseva) tunable Ti:Sapphire laser. First we investigate the spectral correlation between the two photons of a pair. Based on our theoretical predictions [5], the source is expected to generate photons featuring spectral correlations that are (1) positive when τp < 110 fs; (2) reduced for τp ≈ 110 fs; and (3) negative when τp > 110 fs. Experimentally we are limited to measurements for three pump settings: τp 70 fs, 98 fs, and CW. For correlation analysis, we used a method similar to the one already used to reconstruct the EPMF in [21]. The results are presented in Fig. 2 and Table 1. It can be seen in Fig. 2 that the joint spectrum is (1) spectrally positively correlated; (2) shows very little positive correlations; and (3) is spectrally negatively correlated in case of CW pumping. In order to directly relate the experimental results with our theory, we compare the correlation parameter defined by Eq. (2) and show the theoretical prediction for the joint spectrum as contours in Fig. 2. The results for the measured/theoretical correlation parameter r ex ∕r th , and the measured/theoretical co-operativity K ex ∕K th can be seen in Table 1. For the first two measurements, we use a fast photodiode D1 to measure the pump-pulse arrival time. This, in combination with the relative detection time of a generated photon pair, allows us to reconstruct the spectral correlation characteristics [21]. Because the measurement was based on timing statistics, the spectral resolution is limited by the timing jitter of the single photon detectors and photodiode. In order to improve the final results, we use a deconvolution technique based on the overall timing resolution. We measured the following time jitters: 43 ps for the photodiode, 127 ps for D2, and 400 ps for D3. The finite resolution and experimental uncertainties are reasons for the slight difference between predicted and measured correlation parameters and cooperativities for pulsed pump settings (see Table 1). For the CW setting, our method of postprocessing assumes a perfect, monochromatic pump laser, which, in combination with energy conservation, results in perfect anticorrelation. Next we characterized the quality of the polarization entanglement. We measured polarization interference fringes by fixing the signal polarizer orientation in the arm monitored by D1 (see Fig. 1) to horizontal (H), vertical (V), diagonal (D), or antidiagonal (A) polarization and measured coincidences in the arm
Fig. 2. Measured joint spectral functions for pump durations of (a) 70 fs, acquired for 71 min, and (b) 98 fs, acquired for 43 min. (c) Measurement taken with a continuous pump laser (CW) pump, acquired for 28 min. The contour lines correspond to the theoretical prediction.
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Table 1. Measured Spectral and Polarization Characteristics of Photon Pairs Generated by the Sourcea τp fs 70 98 CW
r ex jr th
K ex jK th
S
Vis.
Fid.
Conc.
0.55j0.63 −0.05j0.10 −1j − 1
1.81j1.14 1.32j1.02 >26j∞
2.2(1) 2.37(5) 2.4(2)
99j87 99j86 98j84
0.97 0.94 n/a
0.75 0.76 n/a
a See text for a description of correlation parameter r, cooperativity K, Bell parameter S, visibility, fidelity, and concurrence. In the visibility column, the first number corresponds to the H/V basis, the second one to the D/A basis. Polarization tomography was not performed for CW pumping.
monitored by D2 for the full range of idler polarizer orientations. Figure 3 shows the resulting interference fringes in the (a) H/V and (b) A/D basis for τp 98 fs. The interference visibilities (vis) and Bell parameters (S) for all our measurements are displayed in Table 1. From the analysis of the measured interference fringes, we see that the Bell inequality is clearly violated because we measured S > 2 for all the three different kinds of pump settings. This shows that the source in each case generates entangled photon pairs. Note that we did not subtract the H/V background counts in the A/D basis. This is often done to remove experimental imperfections such as dark counts and after-pulsing in order to obtain polarization interference visibility curves and Bell parameters limited only by the distinguishability of the photons. Finally, we analyzed the polarization-entangled state produced by the source using quantum state tomography methods. To do this, we placed polarization analyzers consisting of a polarizer, quarter-wave plate, and halfwave plate in front of each fiber coupler. We performed 36 coincidence measurements for analyzers set to all combinations from the set of polarizations: H, V, D, A, left (L), right (R) [27]. An example of a reconstructed polarization state for a pump-pulse duration of τp 98 fs is shown in Figs. 3(c) and 3(d). The pillars in Fig. 3(c) have different heights, and the imaginary part in Fig. 3(d) doesn’t vanish completely, which shows that the state is imperfect. We quantify the quality of the state by computing its fidelity [28] compared to the expected p state HH V V ∕ 2. We also calculated the entanglement monotone (concurrence [29]). The computed values for our measurements are displayed in Table 1. The fidelity of the measured state and also its entanglement quality are lowered by experimental imperfections such as detector dark counts and after-pulsing but also the imperfect pump beam mode. In summary, we have shown that the source characterized previously in [21] produces both any desired kind of spectral correlations and high-quality polarization entanglement at the same time. This could be useful for longdistance quantum communication where photons sent through long single-mode fibers suffer from chromatic dispersion. The same positive spectral correlation feature has a potential to improve two photon absorption experiments, where two photon coherence effects are important. Future work will include increasing the relatively low count rates, which are a result of low coupling into single-mode fibers originating from nonoptimal
Fig. 3. Measured polarization entanglement interference fringes in (a) HV and (b) AD basis for a pump pulse duration of τp 98 fs. (c) Real and (d) imaginary part of tomographically reconstructed polarization state produced by the source when pumped with τp 98 fs pulses. The fidelity between the measured state and the expected entangled state HH p V V ∕ 2 is 0.94 (without background subtraction). The entanglement monotone (concurrence) is found to be 0.75.
spatial modes. This is due to the transverse walk-off in the crystal and the nonoptimal pumping condition. Our numerical simulations suggest that it is possible to increase the coupling efficiency by engineering the spatial mode of the pump and the collected modes. The authors acknowledge funding from NSERC (CGS, QuantumWorks, Discovery, USRA), the Ontario Ministry of Research and Innovation (ERA program and research infrastructure program), CIFAR, Industry Canada and the CFI, and support by OEC, which provided InGaAs/InP SPAD module produced by Micro-Photon-Devices. P. K. acknowledges support by Mobility Plus project no. 602/MOB/2011/0 financed by Polish Ministry of Science and Higher Education, by Foundation for Polish Science under Homing Plus no. 2013-7/9 program supported by European Union under PO IG project, and NCU internal grant no. 1625-F. We also thank Carmelo Scarcela and Alberto Tosi from Politecnico di Milano for insightful discussions about detection techniques. References 1. A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, Laser Phys. 15, 146 (2005). 2. A. B. U’Ren, Y. Jeronimo-Moreno, and H. Garcia-Gracia, Phys. Rev. A 75, 023810 (2007). 3. C. I. Osorio, A. Valencia, and J. P. Torres, New J. Phys. 10, 113012 (2008). 4. P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. URen, C. Silberhorn, and I. A. Walmsley, Phys. Rev. Lett. 100, 133601 (2008). 5. P. Kolenderski, W. Wasilewski, and K. Banaszek, Phys. Rev. A 80, 013811 (2009). 6. A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, Phys. Rev. Lett. 106, 013603 (2011). 7. P. G. Evans, R. S. Bennink, W. P. Grice, T. S. Humble, and J. Schaake, Phys. Rev. Lett. 105, 253601 (2010).
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8. 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, Opt. Express 19, 24434 (2011). 9. M. Hendrych, M. Micuda, and J. P. Torres, Opt. Lett. 32, 2339 (2007). 10. A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, Phys. Rev. Lett. 99, 243601 (2007). 11. R.-B. Jin, R. Shimizu, K. Wakui, H. Benichi, and M. Sasaki, Opt. Express 21, 10659 (2013). 12. B. Dayan, A. Pe’Er, A. A. Friesem, and Y. Silberberg, Phys. Rev. Lett. 94, 043602 (2005). 13. B. Dayan, Phys. Rev. A 76, 043813 (2007). 14. J. Gordon and H. Kogelnik, Proc. Natl. Acad. Sci. USA 97, 4541 (2000). 15. C. Antonelli, M. Shtaif, and M. Brodsky, Phys. Rev. Lett. 106, 080404 (2011). 16. M. Brodsky, E. C. George, C. Antonelli, and M. Shtaif, Opt. Lett. 36, 43 (2011). 17. T. Lutz, “Entangled photon sources for ultra-long distance quantum entanglement distribution,” Master’s thesis (University of Waterloo, Ulm University, 2013).
18. O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. Wong, and F. X. Kärtner, Phys. Rev. Lett. 94, 083601 (2005). 19. O. Kuzucu, F. N. C. Wong, S. Kurimura, and S. Tovstonog, Phys. Rev. Lett. 101, 153602 (2008). 20. G. Harder, V. Ansari, B. Brecht, T. Dirmeier, C. Marquardt, and C. Silberhorn, Opt. Express 21, 13975 (2013). 21. T. Lutz, P. Kolenderski, and T. Jennewein, Opt. Lett. 38, 697 (2013). 22. Y.-H. Kim and W. P. Grice, J. Mod. Opt. 49, 2309 (2002). 23. H. S. Poh, J. Lim, I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, Phys. Rev. A 80, 043815 (2009). 24. H. Huang and J. Eberly, J. Mod. Opt. 40, 915 (1993). 25. H. S. Poh, C. Y. Lum, I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, Phys. Rev. A 75, 043816 (2007). 26. A. Tosi, A. Della Frera, A. Bahgat Shehata, and C. Scarcella, Rev. Sci. Instrum. 83, 013104 (2012). 27. R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, Phys. Rev. A 66, 012303 (2002). 28. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 1st ed. (Cambridge University, 2000). 29. W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
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Demonstration of spectral correlation control in a source of polarization-entangled photon pairs at telecom wavelength Thomas Lutz,1,2 Piotr Kolenderski,3,* and Thomas Jennewein1 1
Institute for Quantum Computing, University of Waterloo, 200 University Ave. West, Waterloo, Ontario N2L 3G1, Canada 2 3
Institut für Quantenmaterie, Universität Ulm, 89069 Ulm, Germany
Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5, 87-100 Torun, Poland *Corresponding author: [email protected] Received November 28, 2013; revised February 7, 2014; accepted February 7, 2014; posted February 10, 2014 (Doc. ID 202038); published March 11, 2014 Spectrally correlated photon pairs can be used to improve the performance of long-range fiber-based quantum communication protocols. We present a source based on spontaneous parametric downconversion, which allows one to control spectral correlations within the entangled photon pair without spectral filtering by changing the pump-pulse duration or the characteristics of the coupled spatial modes. The spectral correlations and polarization entanglement are characterized. We find that the generated photon pairs can feature both positive spectral correlations, decorrelation, or negative correlations at the same time as polarization entanglement with a high fidelity of 0.97 (no background subtraction) with the expected Bell state. © 2014 Optical Society of America OCIS codes: (190.4410) Nonlinear optics, parametric processes; (300.6190) Spectrometers; (270.4180) Multiphoton processes; (270.5565) Quantum communications. http://dx.doi.org/10.1364/OL.39.001481
Controlling the spectral correlation of each polarizationentangled photon pair produced via spontaneous parametric downconversion (SPDC) could have important benefits in optical quantum information. Photonic quantum gates require pure single photon states, which can be created by sources producing pairs of spectrally decorrelated photons [1–11]. On the other hand, long-distance fiber-based quantum communication and quantum metrology [12,13] suffer from chromatic dispersion. Polarization mode dispersion [14] is a major source of decoherence [15,16] in fiber-based entanglement distribution protocols. These detrimental effects could potentially be mitigated [17] with positive spectral correlations in photon pairs [18–20]. Here we demonstrate a SPDC photon pair source that enables the control of the spectral correlation within the photon pair. The source is based on a β-barium borate (BBO) crystal, which was characterized previously in [21], where we experimentally measured the phasematching function using a CW laser and concluded that this particular setting for a BBO crystal at 1550 nm allows for control of spectral correlation. This can be done by changing the pump pulse durations and/or collection optics. In the previous configuration, we could produce only spectrally anticorrelated pairs in a separable polarization state. The source presented here produces polarization entangled pairs in the telecom band and can be tuned to create negative, none, or positive spectral correlations. The high-quality polarization entanglement is achieved without using spectral filtering. Also, in the typical scenario for type II SPDC, the walk-off effects have to be compensated by putting respective birefringent crystals in the optical paths of the generated photons. In our design of the experimental setup, no compensation crystals are required. This is due to the mixing of the photons at the polarizing beam splitter (PBS), which removes any distinguishability between the photons and compensates for the walk-off effect in the crystal [22,23]. 0146-9592/14/061481-04$15.00/0
Let us now discuss the main features of the source in detail. In the SPDC process, one photon of the pump downconverts into a photon pair. Energy and momentum conservation relations, jointly described as phase matching, and the properties of the pump photons govern the characteristics of the generated photons. The probability amplitude for a photon pair emission in a given direction and at a given frequency can be described by the product of the pump spatiotemporal amplitude and the phasematching function [5,22]. The phase matching, which depends on the properties of the nonlinear media, specifies the allowed emissions for a given pump beam. Typically, the output photons are coupled into optical fibers, which correspond to collecting photons from a specific range of directions that are defined by the fiber and the optics. From this point of view, coupling can be understood as an additional condition to phase-matching. Therefore, one can introduce an effective phase-matching function (EPMF) [5,21], which fully describes the joint effect of the crystal and coupling into fibers. It consists of a Gaussian function with the characteristic width σ. The spectral part of the pump can also be approximated by a Gaussian function with the characteristic spectral width 1∕2τp . The spectral wave function then reads: νs νi 2 τ2p ν − ν 2 : (1) exp − ψνs ; νi N exp − s 2 i 4 σ Here N is a normalization factor, and νs∕i are the frequency detunings of signal and idler photon from their central frequencies, which correspond to 1550 nm in our case. τp is the pump pulse duration. Note that the first exponent in Eq. (1) is an approximation of the EPMF for our particular nonlinear crystal, which allows for spectral correlation control [21]. It has already been shown experimentally [21] that the particular EPMF of the type-II BBO crystal cut for © 2014 Optical Society of America
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telecom wavelength enables spectral correlation control. Tuning capabilities arise due to the variable pulse duration τp of the pump laser and the variable collection optics, which influences the characteristic width σ of the EPMF. We introduce spectral qa correlation parameter
defined as r hνs νi i∕ hν2s ihν2i i. The brackets stand for expectation values, which we compute using the wave function given by Eq. (1). As a result, one obtains r
1∕4τ2p − σ 2 : 1∕4τ2p σ 2
(2)
This parameter, together with the cooperativity parameter K, introduced in [24], which gives information regarding the number of spectral modes involved, is very useful to characterize the performance of the source. Our source allows us to generate polarizationentangled pairs with the desired spectral correlation. In order to ensure the quality of polarization entanglement, we use a technique previously described in [22,25]. Using this approach, no compensating crystals are required. In addition, this allows us to avoid problems arising due to the spectral distinguishability of the extraordinary and ordinary photons as those are coupled into two distinct fibers, and no spectral filtering is required to improve the polarization entanglement visibility. The schematic setup of the experiment is shown in Fig. 1. The BBO crystal used in this source is cut at
Fig. 1. Experimental setup. The BBO crystal is pumped using a Ti:Sapphire laser. The beam is focused onto the crystal by a lens (L3). Downconverted photons are collimated using lenses (L2, f 25 cm). The retroreflectors are used to adjust the distances the extraordinary and ordinary photon travel. A half-wave plate (HWP) rotates the polarization by 90 deg, which results in splitting both photons at the PBS. After they pass polarizers (P), they are coupled into single-mode fiber (SMF-28e) by aspheric lenses (L1, f 15.4 mm). The stray light is filtered out by a long-pass filter (Semrock BLP01-1319R-25). Photons are detected using two InGaAs/InP detectors D2 (Micro-Photon-Devices, InGaAs/InP single-photon avalanche diode [26]) and D3 (idQuantique, id201). Timing analysis is performed using FPGA electronics. A small percentage of the pump light is directed to a photodiode (DET10A) D1 using a glass plate (GP), a coupling lens (L4), and a SMF at 800 nm. D1 is used to measure pump pulse time.
29.14 deg for type-II phase matching. A pump photon at wavelength 775 nm is downconverted into a photon pair at wavelength 1550 nm. The crystal is pumped using either a CW (MIRA-900, Coherent) or a femtosecond (TiF-50M, Atseva) tunable Ti:Sapphire laser. First we investigate the spectral correlation between the two photons of a pair. Based on our theoretical predictions [5], the source is expected to generate photons featuring spectral correlations that are (1) positive when τp < 110 fs; (2) reduced for τp ≈ 110 fs; and (3) negative when τp > 110 fs. Experimentally we are limited to measurements for three pump settings: τp 70 fs, 98 fs, and CW. For correlation analysis, we used a method similar to the one already used to reconstruct the EPMF in [21]. The results are presented in Fig. 2 and Table 1. It can be seen in Fig. 2 that the joint spectrum is (1) spectrally positively correlated; (2) shows very little positive correlations; and (3) is spectrally negatively correlated in case of CW pumping. In order to directly relate the experimental results with our theory, we compare the correlation parameter defined by Eq. (2) and show the theoretical prediction for the joint spectrum as contours in Fig. 2. The results for the measured/theoretical correlation parameter r ex ∕r th , and the measured/theoretical co-operativity K ex ∕K th can be seen in Table 1. For the first two measurements, we use a fast photodiode D1 to measure the pump-pulse arrival time. This, in combination with the relative detection time of a generated photon pair, allows us to reconstruct the spectral correlation characteristics [21]. Because the measurement was based on timing statistics, the spectral resolution is limited by the timing jitter of the single photon detectors and photodiode. In order to improve the final results, we use a deconvolution technique based on the overall timing resolution. We measured the following time jitters: 43 ps for the photodiode, 127 ps for D2, and 400 ps for D3. The finite resolution and experimental uncertainties are reasons for the slight difference between predicted and measured correlation parameters and cooperativities for pulsed pump settings (see Table 1). For the CW setting, our method of postprocessing assumes a perfect, monochromatic pump laser, which, in combination with energy conservation, results in perfect anticorrelation. Next we characterized the quality of the polarization entanglement. We measured polarization interference fringes by fixing the signal polarizer orientation in the arm monitored by D1 (see Fig. 1) to horizontal (H), vertical (V), diagonal (D), or antidiagonal (A) polarization and measured coincidences in the arm
Fig. 2. Measured joint spectral functions for pump durations of (a) 70 fs, acquired for 71 min, and (b) 98 fs, acquired for 43 min. (c) Measurement taken with a continuous pump laser (CW) pump, acquired for 28 min. The contour lines correspond to the theoretical prediction.
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Table 1. Measured Spectral and Polarization Characteristics of Photon Pairs Generated by the Sourcea τp fs 70 98 CW
r ex jr th
K ex jK th
S
Vis.
Fid.
Conc.
0.55j0.63 −0.05j0.10 −1j − 1
1.81j1.14 1.32j1.02 >26j∞
2.2(1) 2.37(5) 2.4(2)
99j87 99j86 98j84
0.97 0.94 n/a
0.75 0.76 n/a
a See text for a description of correlation parameter r, cooperativity K, Bell parameter S, visibility, fidelity, and concurrence. In the visibility column, the first number corresponds to the H/V basis, the second one to the D/A basis. Polarization tomography was not performed for CW pumping.
monitored by D2 for the full range of idler polarizer orientations. Figure 3 shows the resulting interference fringes in the (a) H/V and (b) A/D basis for τp 98 fs. The interference visibilities (vis) and Bell parameters (S) for all our measurements are displayed in Table 1. From the analysis of the measured interference fringes, we see that the Bell inequality is clearly violated because we measured S > 2 for all the three different kinds of pump settings. This shows that the source in each case generates entangled photon pairs. Note that we did not subtract the H/V background counts in the A/D basis. This is often done to remove experimental imperfections such as dark counts and after-pulsing in order to obtain polarization interference visibility curves and Bell parameters limited only by the distinguishability of the photons. Finally, we analyzed the polarization-entangled state produced by the source using quantum state tomography methods. To do this, we placed polarization analyzers consisting of a polarizer, quarter-wave plate, and halfwave plate in front of each fiber coupler. We performed 36 coincidence measurements for analyzers set to all combinations from the set of polarizations: H, V, D, A, left (L), right (R) [27]. An example of a reconstructed polarization state for a pump-pulse duration of τp 98 fs is shown in Figs. 3(c) and 3(d). The pillars in Fig. 3(c) have different heights, and the imaginary part in Fig. 3(d) doesn’t vanish completely, which shows that the state is imperfect. We quantify the quality of the state by computing its fidelity [28] compared to the expected p state HH V V ∕ 2. We also calculated the entanglement monotone (concurrence [29]). The computed values for our measurements are displayed in Table 1. The fidelity of the measured state and also its entanglement quality are lowered by experimental imperfections such as detector dark counts and after-pulsing but also the imperfect pump beam mode. In summary, we have shown that the source characterized previously in [21] produces both any desired kind of spectral correlations and high-quality polarization entanglement at the same time. This could be useful for longdistance quantum communication where photons sent through long single-mode fibers suffer from chromatic dispersion. The same positive spectral correlation feature has a potential to improve two photon absorption experiments, where two photon coherence effects are important. Future work will include increasing the relatively low count rates, which are a result of low coupling into single-mode fibers originating from nonoptimal
Fig. 3. Measured polarization entanglement interference fringes in (a) HV and (b) AD basis for a pump pulse duration of τp 98 fs. (c) Real and (d) imaginary part of tomographically reconstructed polarization state produced by the source when pumped with τp 98 fs pulses. The fidelity between the measured state and the expected entangled state HH p V V ∕ 2 is 0.94 (without background subtraction). The entanglement monotone (concurrence) is found to be 0.75.
spatial modes. This is due to the transverse walk-off in the crystal and the nonoptimal pumping condition. Our numerical simulations suggest that it is possible to increase the coupling efficiency by engineering the spatial mode of the pump and the collected modes. The authors acknowledge funding from NSERC (CGS, QuantumWorks, Discovery, USRA), the Ontario Ministry of Research and Innovation (ERA program and research infrastructure program), CIFAR, Industry Canada and the CFI, and support by OEC, which provided InGaAs/InP SPAD module produced by Micro-Photon-Devices. P. K. acknowledges support by Mobility Plus project no. 602/MOB/2011/0 financed by Polish Ministry of Science and Higher Education, by Foundation for Polish Science under Homing Plus no. 2013-7/9 program supported by European Union under PO IG project, and NCU internal grant no. 1625-F. We also thank Carmelo Scarcela and Alberto Tosi from Politecnico di Milano for insightful discussions about detection techniques. References 1. A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, Laser Phys. 15, 146 (2005). 2. A. B. U’Ren, Y. Jeronimo-Moreno, and H. Garcia-Gracia, Phys. Rev. A 75, 023810 (2007). 3. C. I. Osorio, A. Valencia, and J. P. Torres, New J. Phys. 10, 113012 (2008). 4. P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. URen, C. Silberhorn, and I. A. Walmsley, Phys. Rev. Lett. 100, 133601 (2008). 5. P. Kolenderski, W. Wasilewski, and K. Banaszek, Phys. Rev. A 80, 013811 (2009). 6. A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, Phys. Rev. Lett. 106, 013603 (2011). 7. P. G. Evans, R. S. Bennink, W. P. Grice, T. S. Humble, and J. Schaake, Phys. Rev. Lett. 105, 253601 (2010).
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