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OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014
Quantum correlation of fiber-based telecom-band photon pairs through standard loss and random media Yong Meng Sua,1 John Malowicki,2 and Kim Fook Lee3,* 1
3
Department of Physics, Michigan Technological University, Houghton, Michigan 49931, USA 2 Air Force Research Laboratory, Rome, New York 13441, USA
Center for Photonic Communication and Computing, Department of Electrical Engineering and Computer Science, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3118, USA *Corresponding author: [email protected] Received April 24, 2014; revised July 9, 2014; accepted July 9, 2014; posted July 11, 2014 (Doc. ID 210556); published August 11, 2014 We study quantum correlation and interference of fiber-based telecom-band photon pairs with one photon of the pair experiencing multiple scattering in a random medium. We measure joint probability of two-photon detection for signal photon in a normal channel and idler photon in a channel, which is subjected to two independent conditions: standard loss (neutral density filter) and random media. We observe that both conditions degrade the correlation of signal and idler photons, and depolarization of the idler photon in random medium can enhance twophoton interference at certain relative polarization angles. Our theoretical calculation on two-photon polarization correlation and interference as a function of mean free path is in agreement with our experiment data. We conclude that quantum correlation of a polarization-entangled photon pair is better preserved than a polarization-correlated photon pair as one photon of the pair scatters through a random medium. © 2014 Optical Society of America OCIS codes: (270.0270) Quantum optics; (190.4370) Nonlinear optics, fibers; (060.1660) Coherent communications; (270.5585) Quantum information and processing. http://dx.doi.org/10.1364/OL.39.004808
Despite the success of demonstrating entanglementbased quantum key distribution (QKD) in the Tokyo Network and 144 km of free-space channel by Anton Zeilinger’s group, the practicality of using entangled photon pairs for long distance or eventually global QKD remains in doubt. For the free space channel, questions arise due to major limitations such as a single decoherence-free state of two photons to collective noise [1], atmospheric scattering, turbulence, and propagation losses [2–7]. On the other hand, in addition to linear propagation and connection loss with current optical fiber technology, a recent experiment has demonstrated that polarization mode dispersion [8] is a major source of decoherence limiting the fiber network distance for QKD. In short, it requires that the quantum correlations be preserved over the disturbances in the transmission channel when separating the photon pair over a large scale. Hence, explicit investigation on the propagation of the photon pairs through standard loss and scattering medium [9,10] is of great interest for the implementation of various entanglement-based QKDs in free space, fiber networks, and under the sea. In this work, we measure the joint probability of twophoton detection, which is a normally ordered correlation function given by Pθ1 ; θ2 h∶aˆ †1 aˆ †20 aˆ 20 aˆ 1 ∶i, where aˆ 1 is the annihilation operator of the signal (s) photon, propagating through a normal channel (free space and fiber) and experiencing negligible loss, and aˆ 20 is the annihilation operator of the idler (i) photon, propagating through a random medium and experiencing depolarization and the noncollective decoherence. The θ1 (θ2 ) is the projection angle of the polarization analyzer in the signal (idler) channel. The aˆ 20 of the idler photon consists of coherent and incoherent components. Its coherent part is associated with transmission amplitude T i , corresponding to the ballistic photon that undergoes no decoherence. Its incoherent part is associated with the 0146-9592/14/164808-04$15.00/0
depolarization amplitude Li ∝ 1∕l, where the l is the scattering mean free path. This means that the large mean free path corresponds to small amplitude Li , i.e., the idler photon will experience fewer scattering events. The output annihilation operator of the idler (after the random medium) at the x–y basis is given by aˆ 20 T i;x aˆ i;x xˆ T i;y aˆ i;y yˆ Li;x Li;y ˆc eˆ ;
(1)
where aˆ i;x and aˆ i;y are the x and y components of the input annihilation operator of the idler (before the random medium), cˆ is the annihilation operator of the depolarized idler photon, and eˆ is a polarization projection of an analyzer. For a homogenous random medium, the depolarization p amplitude at the x–y plane is Li;x Li;y 1∕ 2Li , and the transmission amplitude at the p x–y plane is T i;x T i;y 1∕ 2T i . A polarizer is used to project the scattered idler photon to a polarization state eˆ cos θ2 xˆ sin θ2 yˆ . After the polarizer, the annihilation operator of the idler is given by aˆ 20 aˆ 20 · eˆ T i;x aˆ i;x cos θ2 T i;y aˆ i;y sin θ2 Li;x Li;y ˆc: (2) Note that cˆ can operate at any polarization axis, i.e., (ˆc eˆ ·ˆe → cˆ ), and so it is not associated with terms cos θ2 and sin θ2 because the depolarization effect is not dependent on the orientation of a polarizer. As for the signal photon that is propagating in a normal channel with negligible loss (T s 1, Ls 0), the annihilation operator of the signal after a polarizer is aˆ 1 aˆ s;x cos θ1 aˆ s;y sin θ1 , where aˆ s;x and aˆ s;y are the x and y components of the annihilation operator of the signal. Then, the joint probability detections h∶aˆ †1 aˆ †20 aˆ 20 aˆ 1 ∶i for the horizontal polarization-correlated two-photon state © 2014 Optical Society of America
August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
jΨicor j1s;x ; 0s;y ; 1i;x ; 0i;y i andppolarization-entangled two-photon state jΨient 1∕ 2j1s;x ; 0s;y ; 1i;x ; 0i;y i j0s;x ; 1s;y ; 0i;x ; 1i;y i are given by P cor
P ent
respectively. We take the average of these two maximum points fθ1 − θ2 0°; 180°g and calculate the visibility, as given by
T 2i cos2 θ2 cos2 θ1 2L2i 2T i Li cos θ2 cos θ1 2 (3)
and 1 T 2i 2 2 cos θ2 − θ1 8Li 4T i Li cosθ2 − θ1 ; 2 2 (4)
corresponds to respectively. The term L2i † † haˆ s;x;y cˆ cˆ aˆ s;x;y i; while the term T i Li corresponds to haˆ †s;x;y aˆ †i;x;y cˆ aˆ s;x;y i and haˆ †s;x;y cˆ † aˆ i;x;y aˆ s;x;y i. Both Li 2 and T i Li are attributed to completely and partially depolarized idler photons. In our detection system, a coincidence count is recorded when both avalanche photodiodes (APDs) detect a photon at the same gated time interval, while an accidental coincidence count is recorded when both APDs detect a photon at the adjacent gated time interval. From Eq. (3) with the θ1 0° and θ2 0°, we have coincidence P cor T 2i ∕2 2L2i 2T i Li , which is contributed from photon pairs and depolarized idler photons. The term 2L2i 2T i Li corresponds to accidental, which is contributed from the depolarized idler photons. One can also set θ1 0° and θ2 90° for measuring the component 2L2i alone. The coincidence (Cc) to accidental (Ac) ratio (CAR) is given by Cc T 2i 4L2i 4T i Li : Ac 4L2i 4T i Li
(5)
One can calculate the visibility of two-photon interference (TPI) V cor Cc − Ac∕Cc Ac as given by V cor
T 2i : T 2i 8L2i 8T i Li
(6)
The CAR estimate visibility V cor is usually used to predict the visibility of TPI of an entangled state generated from the correlated photon pairs. The joint probability of two-photon detection for polarization-entangled photon pairs P ent in Eq. (4) is a TPI fringe as a function of relative polarization angle (θ2 − θ1 ) of the signal and idler photons. There are two different maximum points and one minimum point from the function P ent . First, the maximum fθ2 − θ1 0°g and minimum fθ2 − θ1 90°g interference points are given by 1∕2T 2i ∕2 8L2i 4T i Li and 4L2i , respectively. The incoherent term 4T i Li , which is partially due to depolarization of the idler photon, does contribute to the maximum interference but not to the minimum. This term exhibits enhancement of TPI at certain relative polarization angles. Second, the maximum fθ2 − θ1 180°g and minimum fθ2 − θ1 90°g interference points are given by 1∕2T 2i ∕2 8L2i − 4T i Li and 4L2i ,
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V ent
T 2i : T 32L2i
(7)
2 i
As shown in Fig. 1, we use a compact counterpropagating scheme (CPS) [11–13] for preparing the polarizationentangled photon pairs (signal and idler) through a spontaneous four-wave mixing process in 10 m of highly nonlinear fiber (HNLF) [14] at room temperature. For preparing the polarization-correlated photon pairs in the similar setup, we use only horizontally polarized pump to propagate through the HNLF. The photon pair generated from the CPS is separated by dual-channel wavelength division multiplexing filters with 1 nm bandwidth at 1560.6 nm (idler) and 1547.7 nm (signal). The signal/idler photons are guided through their polarization analyzers; each consists of a quarter-wave plate, a halfwave plate, and a polarizing beam splitter (PBS). Both signal and idler photons are detected by fiber-coupled InGaAs/InP APDs operated in gated Geiger mode at room temperature. They are counted and correlated by using a dual-channel gated photon counter (SR400) and multichannel scaler (SR430). We first investigate the effect of standard loss on quantum correlation and interference of photon pairs by placing a neutral density filter (not a random medium) in the idler channel. We measure the CAR of horizontal polarization-correlated photon pairs with the attenuations of 1, 3, and 5 dB in the idler channel. For each attenuation, we measure the CAR at different average pump powers, as shown in the inset of Fig. 2. The maximum CAR values for attenuations of 1, 3, and 5 dB at the idler channel are 26, 23, and 16 at the average pump powers of 0.450, 0.530, and 0.600 mW, respectively. The observation of decreasing maximum CAR value in line with attenuation proves that standard losses in the transmission channel indeed degrade the correlation of signal-idler photons. p We then prepare the polarization-entangled state 1∕ 2jH i H s i jV i V s i and measure the TPI as a function of the relative polarization angle of signal-idler photons. The measured TPI’s visibilities V ent (CAR estimate visibilities V cor ) for attenuations of 1, 3, and 5 dB are 93.3%(92.6%), 91.8% (91.4%) and 89.1% (88.2%), respectively, as shown in Fig. 2. The V ent is in agreement with V cor , which implies that quantum correlation and interference for both polarization-correlated and polarization-entangled phoAPD1
Correlated and entangled photon source
Signal
SR400 DWDM filters
QWP FC
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FC
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Attenuator (Neutral Density Filter)
Random media
Coincidence Counting
Fig. 1. Experiment setup for measuring CAR and two-photon interference of the signal photon in a normal channel and the idler photon experiencing multiple scattering events.
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OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014 ( : 93.3%) (x: 92.6%)
( : 91.8%) (x: 91.4%)
24 ( : 89.1%) 30
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Fig. 2. Measured visibility of two-photon interference (blue boxes), CAR estimate visibility (crosses), and maximum CAR (solid circles) versus standard losses. Inset: coincidence to accidental ratio (CAR) versus pump power with different attenuations. Green squares = 1 dB, blue diamonds = 3 dB, red dots = 5 dB.
ton pairs are equally sensitive to the standard losses in the transmission channel. We then replace the neutral density filter with a random medium in the idler channel. We consider that the scatterers are isotropic and uncorrelated. The multiple scattering sample is prepared by dispersing uniform polystyrene microspheres (Duke Standards) in oil suspension and kept in a quartz cuvette with the length of 10 mm. The samples prepared in this work are in a weak scattering regime, where the scattering mean free path l is much larger than the wavelength of the photon. The refractive indices of oil and polystyrene microspheres are 1.47 and 1.59, respectively. The mean diameters (NIST traceable) of polystyrene microspheres used in this work are 0.5, 0.8, 1.6, and 5.0 μm. The scattered idler photon transmitted from a sample is collected by using a free space-to-fiber collimator (NA 0.25), which is placed right after the PBS. In order to achieve a 3 dB loss for all samples, we prepare concentrations of 1.22× 1014 m−3 , 1.13 × 1014 m−3 , 0.58 × 1014 m−3 , and 0.08 × 1014 m−3 for scatterer diameters (ϕ1;2;3;4 ) of 0.5, 0.8, 1.6, and 5.0 μm with the scattering mean free path l1;2;3;4 of 0.019, 0.010, 0.004, and 0.003 m, respectively. The main purpose for us to use different scatterer diameters with different concentrations for changing the mean free path while keeping the total loss of 3 dB is to explore and compare the results obtained from the standard loss of 3 dB (neutral density). We measure CAR as a function of scattering mean free path l, as shown in Fig. 3 for the horizontal polarizationcorrelated photon pair. We obtain a CAR of 16.9, 18.3, 19.8, and 20.3 for the mean free path of l4 0.003 m (ϕ4 5.0 μm), l3 0.004 m (ϕ3 1.6 μm), l2 0.010 m (ϕ2 0.8 μm), and l1 0.019 m (ϕ1 0.5 μm) at different average pump powers of 0.600, 0.570, 0.530, and 0.500 mW, respectively. The maximum CAR value decreases as the idler photon experiences more scattering events or short scattering mean free path. Also shown is the CAR value 23 for a standard loss of 3 dB, which is obviously higher than the CAR values obtained with multiple scattering random media.
0
0.005
0.6
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Pump Power (mW)
0.01
1.4
0.015
0.02
Mean free path (m)
Fig. 3. Maximum coincidence to accidental ratio (CAR) versus scattering mean free path. The CAR of 23 is for the standard loss of 3 dB (solid red line). Inset: coincidence to accidental ratio (CAR) versus pump power with different scattering mean free path (l). Blue dots, l1 0.019 m; green diamonds, l2 0.010 m; black circles, l3 0.004 m; red squares, l4 0.003 m.
We then measure TPI of the polarization-entangled state with the idler photon scattered through a sample, as shown in the inset of Fig. 4. We fit the TPI fringe with Eq. (4) (shown as red dotted line) in the inset of the figure. The two maximum conditions fθ2 − θ1 0°; 180°g fall within in the error bars of our experimental data. We then obtain the TPI’s visibility, which is the average visibility as discussed in Eq. (7). We repeat the measurement of TPI for all samples and plot the visibility in Fig. 4 as a function of mean free path (l). We also plot the visibility V cor obtained from the CAR measurement. Also shown is TPI’s visibility of 91.8% for a standard loss of 3 dB, which is obviously higher than both V cor and V ent obtained with multiple scattering medium. The TPI’s visibility V ent is better than the visibility V cor obtained from the CAR measurement. This is not observed in 0.915
0.9
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5
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0
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Fig. 4. V ent (blue squares) and V cor (red dots) versus scattering mean free path, the solid blue and red lines are fitting curves for V ent and V cor . The dashed line is the visibility measured with 3 dB standard loss. Inset: two-photon interference fringes (blue squares) versus relative angle for scattering random medium, l3 0.004 m with HNLF at 300 K. The red dotted line is the theoretical plot of two-photon interference fringe.
August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
standard loss as shown in Fig. 2, where the CAR estimate visibility agrees with the measured TPI’s visibility. This observation directly attests that quantum correlation of the polarization-entangled photon pair is better preserved than polarization-correlated photon pair as one photon of the pair experiences random scattering process in the random medium. The fitting curves of the V cor and V ent are obtained from Eqs. (6) and (7), respectively. Our experiment/theory results show that the visibility of TPI (V ent or V cor ) is asymptotically approaching a constant value (>91%) provided by the standard loss (3 dB), i.e., without the random medium. The V ent is approaching the asymptotic value faster than the V cor . The V ent is 10% higher than the V cor for the mean free path in the range of 0.005–0.02 m. The V ent is for the entangled state prepared before the random medium. The V cor is for the entangled state prepared after the random medium that is for the situation one needs to prepare the entangled state from the correlated photon transmitted through the random medium. From the fittings of Eqs. (6) and (7), we obtain the average of T i 0.77, i.e., T 2i 0.6. The FWHM of the scattering angles for all samples is around 6°–7.4°. The acceptance angle of the fiber collimator is about 14°. With the approximation of 80% for the scattering light coupled into the fiber collimator, we have 0.8 × 0.6 0.48, which is close to the 3 dB loss that we claimed for all samples. In particular, our experiment setup is modeled in analogues to the entanglement-based QKD demonstrated in the Tokyo Network and 144 km of free-space channel by Anton Zeilinger’s group, where one photon from the entangled pair is measured locally and the second photon is sent to a distant observer via a transmission channel with losses or disturbances. In conclusion, we demonstrate that both standard loss (neutral density) and random scattering processes in the transmission channel are detrimental to the quantum correlation and interference of polarization-correlated and polarization-entangled photon pairs. We observe that polarization-entangled photon pairs are better preserved as one photon of the pair propagates through multiple
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scattering medium, hence it will be a better candidate for free-space long-distance QKD compared to correlated photon pairs. The Contractor acknowledges the Government’s support in the publication of this paper. This material is based upon research sponsored by AFRL under agreement number FA8750-12-1-0136. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of Air Force Research Laboratory or the U.S. Government. References 1. P. G. Kwiat, A. J. Berglund, J. B. Altepeter, and A. G. White, Science 290, 498 (2000). 2. P. W. Milonni, J. H. Carter, C. G. Peterson, and R. J. Hughes, J. Opt. B 6, S742 (2004). 3. A. A. Semenov and W. Vogel, Phys. Rev. A 80, 021802(R) (2009). 4. A. A. Semenov and W. Vogel, Phys. Rev. A 81, 023835 (2010). 5. I. Capraro, A. Tomaello, A. Dall’Arche, F. Gerlin, R. Ursin, G. Vallone, and P. Villoresi, Phys. Rev. Lett. 109, 200502 (2012). 6. D. Yu. Vasylyev, A. A. Semenov, and W. Vogel, Phys. Rev. Lett. 108, 220501 (2012). 7. M. V. da Cunha Pereira, L. A. P. Filpi, and C. H. Monken, Phys. Rev. A 88, 053836 (2013). 8. M. Brodsky, K. E. George, C. Antonelli, and M. Shtaif, Opt. Lett. 36, 43 (2011). 9. A. Aiello and J. P. Woerdman, Phys. Rev. Lett. 94, 090406 (2005). 10. G. Puentes, A. Aiello, D. Voigt, and J. P. Woerdman, Phys. Rev. A 75, 032319 (2007). 11. K. F. Lee, J. Chen, C. Liang, X. Li, P. L. Voss, and P. Kumar, Opt. Lett. 31, 1905 (2006). 12. X. Li, C. Liang, K. F. Lee, J. Chen, P. L. Voss, and P. Kumar, Phys. Rev. A 73, 052301 (2006). 13. H. Takesue and K. Inoue, Phys. Rev. A 70, 031802(R) (2004). 14. Y. M. Sua, J. Malowicki, M. Hirano, and K. F. Lee, Opt. Lett. 38, 73 (2013).
OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014
Quantum correlation of fiber-based telecom-band photon pairs through standard loss and random media Yong Meng Sua,1 John Malowicki,2 and Kim Fook Lee3,* 1
3
Department of Physics, Michigan Technological University, Houghton, Michigan 49931, USA 2 Air Force Research Laboratory, Rome, New York 13441, USA
Center for Photonic Communication and Computing, Department of Electrical Engineering and Computer Science, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3118, USA *Corresponding author: [email protected] Received April 24, 2014; revised July 9, 2014; accepted July 9, 2014; posted July 11, 2014 (Doc. ID 210556); published August 11, 2014 We study quantum correlation and interference of fiber-based telecom-band photon pairs with one photon of the pair experiencing multiple scattering in a random medium. We measure joint probability of two-photon detection for signal photon in a normal channel and idler photon in a channel, which is subjected to two independent conditions: standard loss (neutral density filter) and random media. We observe that both conditions degrade the correlation of signal and idler photons, and depolarization of the idler photon in random medium can enhance twophoton interference at certain relative polarization angles. Our theoretical calculation on two-photon polarization correlation and interference as a function of mean free path is in agreement with our experiment data. We conclude that quantum correlation of a polarization-entangled photon pair is better preserved than a polarization-correlated photon pair as one photon of the pair scatters through a random medium. © 2014 Optical Society of America OCIS codes: (270.0270) Quantum optics; (190.4370) Nonlinear optics, fibers; (060.1660) Coherent communications; (270.5585) Quantum information and processing. http://dx.doi.org/10.1364/OL.39.004808
Despite the success of demonstrating entanglementbased quantum key distribution (QKD) in the Tokyo Network and 144 km of free-space channel by Anton Zeilinger’s group, the practicality of using entangled photon pairs for long distance or eventually global QKD remains in doubt. For the free space channel, questions arise due to major limitations such as a single decoherence-free state of two photons to collective noise [1], atmospheric scattering, turbulence, and propagation losses [2–7]. On the other hand, in addition to linear propagation and connection loss with current optical fiber technology, a recent experiment has demonstrated that polarization mode dispersion [8] is a major source of decoherence limiting the fiber network distance for QKD. In short, it requires that the quantum correlations be preserved over the disturbances in the transmission channel when separating the photon pair over a large scale. Hence, explicit investigation on the propagation of the photon pairs through standard loss and scattering medium [9,10] is of great interest for the implementation of various entanglement-based QKDs in free space, fiber networks, and under the sea. In this work, we measure the joint probability of twophoton detection, which is a normally ordered correlation function given by Pθ1 ; θ2 h∶aˆ †1 aˆ †20 aˆ 20 aˆ 1 ∶i, where aˆ 1 is the annihilation operator of the signal (s) photon, propagating through a normal channel (free space and fiber) and experiencing negligible loss, and aˆ 20 is the annihilation operator of the idler (i) photon, propagating through a random medium and experiencing depolarization and the noncollective decoherence. The θ1 (θ2 ) is the projection angle of the polarization analyzer in the signal (idler) channel. The aˆ 20 of the idler photon consists of coherent and incoherent components. Its coherent part is associated with transmission amplitude T i , corresponding to the ballistic photon that undergoes no decoherence. Its incoherent part is associated with the 0146-9592/14/164808-04$15.00/0
depolarization amplitude Li ∝ 1∕l, where the l is the scattering mean free path. This means that the large mean free path corresponds to small amplitude Li , i.e., the idler photon will experience fewer scattering events. The output annihilation operator of the idler (after the random medium) at the x–y basis is given by aˆ 20 T i;x aˆ i;x xˆ T i;y aˆ i;y yˆ Li;x Li;y ˆc eˆ ;
(1)
where aˆ i;x and aˆ i;y are the x and y components of the input annihilation operator of the idler (before the random medium), cˆ is the annihilation operator of the depolarized idler photon, and eˆ is a polarization projection of an analyzer. For a homogenous random medium, the depolarization p amplitude at the x–y plane is Li;x Li;y 1∕ 2Li , and the transmission amplitude at the p x–y plane is T i;x T i;y 1∕ 2T i . A polarizer is used to project the scattered idler photon to a polarization state eˆ cos θ2 xˆ sin θ2 yˆ . After the polarizer, the annihilation operator of the idler is given by aˆ 20 aˆ 20 · eˆ T i;x aˆ i;x cos θ2 T i;y aˆ i;y sin θ2 Li;x Li;y ˆc: (2) Note that cˆ can operate at any polarization axis, i.e., (ˆc eˆ ·ˆe → cˆ ), and so it is not associated with terms cos θ2 and sin θ2 because the depolarization effect is not dependent on the orientation of a polarizer. As for the signal photon that is propagating in a normal channel with negligible loss (T s 1, Ls 0), the annihilation operator of the signal after a polarizer is aˆ 1 aˆ s;x cos θ1 aˆ s;y sin θ1 , where aˆ s;x and aˆ s;y are the x and y components of the annihilation operator of the signal. Then, the joint probability detections h∶aˆ †1 aˆ †20 aˆ 20 aˆ 1 ∶i for the horizontal polarization-correlated two-photon state © 2014 Optical Society of America
August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
jΨicor j1s;x ; 0s;y ; 1i;x ; 0i;y i andppolarization-entangled two-photon state jΨient 1∕ 2j1s;x ; 0s;y ; 1i;x ; 0i;y i j0s;x ; 1s;y ; 0i;x ; 1i;y i are given by P cor
P ent
respectively. We take the average of these two maximum points fθ1 − θ2 0°; 180°g and calculate the visibility, as given by
T 2i cos2 θ2 cos2 θ1 2L2i 2T i Li cos θ2 cos θ1 2 (3)
and 1 T 2i 2 2 cos θ2 − θ1 8Li 4T i Li cosθ2 − θ1 ; 2 2 (4)
corresponds to respectively. The term L2i † † haˆ s;x;y cˆ cˆ aˆ s;x;y i; while the term T i Li corresponds to haˆ †s;x;y aˆ †i;x;y cˆ aˆ s;x;y i and haˆ †s;x;y cˆ † aˆ i;x;y aˆ s;x;y i. Both Li 2 and T i Li are attributed to completely and partially depolarized idler photons. In our detection system, a coincidence count is recorded when both avalanche photodiodes (APDs) detect a photon at the same gated time interval, while an accidental coincidence count is recorded when both APDs detect a photon at the adjacent gated time interval. From Eq. (3) with the θ1 0° and θ2 0°, we have coincidence P cor T 2i ∕2 2L2i 2T i Li , which is contributed from photon pairs and depolarized idler photons. The term 2L2i 2T i Li corresponds to accidental, which is contributed from the depolarized idler photons. One can also set θ1 0° and θ2 90° for measuring the component 2L2i alone. The coincidence (Cc) to accidental (Ac) ratio (CAR) is given by Cc T 2i 4L2i 4T i Li : Ac 4L2i 4T i Li
(5)
One can calculate the visibility of two-photon interference (TPI) V cor Cc − Ac∕Cc Ac as given by V cor
T 2i : T 2i 8L2i 8T i Li
(6)
The CAR estimate visibility V cor is usually used to predict the visibility of TPI of an entangled state generated from the correlated photon pairs. The joint probability of two-photon detection for polarization-entangled photon pairs P ent in Eq. (4) is a TPI fringe as a function of relative polarization angle (θ2 − θ1 ) of the signal and idler photons. There are two different maximum points and one minimum point from the function P ent . First, the maximum fθ2 − θ1 0°g and minimum fθ2 − θ1 90°g interference points are given by 1∕2T 2i ∕2 8L2i 4T i Li and 4L2i , respectively. The incoherent term 4T i Li , which is partially due to depolarization of the idler photon, does contribute to the maximum interference but not to the minimum. This term exhibits enhancement of TPI at certain relative polarization angles. Second, the maximum fθ2 − θ1 180°g and minimum fθ2 − θ1 90°g interference points are given by 1∕2T 2i ∕2 8L2i − 4T i Li and 4L2i ,
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V ent
T 2i : T 32L2i
(7)
2 i
As shown in Fig. 1, we use a compact counterpropagating scheme (CPS) [11–13] for preparing the polarizationentangled photon pairs (signal and idler) through a spontaneous four-wave mixing process in 10 m of highly nonlinear fiber (HNLF) [14] at room temperature. For preparing the polarization-correlated photon pairs in the similar setup, we use only horizontally polarized pump to propagate through the HNLF. The photon pair generated from the CPS is separated by dual-channel wavelength division multiplexing filters with 1 nm bandwidth at 1560.6 nm (idler) and 1547.7 nm (signal). The signal/idler photons are guided through their polarization analyzers; each consists of a quarter-wave plate, a halfwave plate, and a polarizing beam splitter (PBS). Both signal and idler photons are detected by fiber-coupled InGaAs/InP APDs operated in gated Geiger mode at room temperature. They are counted and correlated by using a dual-channel gated photon counter (SR400) and multichannel scaler (SR430). We first investigate the effect of standard loss on quantum correlation and interference of photon pairs by placing a neutral density filter (not a random medium) in the idler channel. We measure the CAR of horizontal polarization-correlated photon pairs with the attenuations of 1, 3, and 5 dB in the idler channel. For each attenuation, we measure the CAR at different average pump powers, as shown in the inset of Fig. 2. The maximum CAR values for attenuations of 1, 3, and 5 dB at the idler channel are 26, 23, and 16 at the average pump powers of 0.450, 0.530, and 0.600 mW, respectively. The observation of decreasing maximum CAR value in line with attenuation proves that standard losses in the transmission channel indeed degrade the correlation of signal-idler photons. p We then prepare the polarization-entangled state 1∕ 2jH i H s i jV i V s i and measure the TPI as a function of the relative polarization angle of signal-idler photons. The measured TPI’s visibilities V ent (CAR estimate visibilities V cor ) for attenuations of 1, 3, and 5 dB are 93.3%(92.6%), 91.8% (91.4%) and 89.1% (88.2%), respectively, as shown in Fig. 2. The V ent is in agreement with V cor , which implies that quantum correlation and interference for both polarization-correlated and polarization-entangled phoAPD1
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Fig. 1. Experiment setup for measuring CAR and two-photon interference of the signal photon in a normal channel and the idler photon experiencing multiple scattering events.
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OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014 ( : 93.3%) (x: 92.6%)
( : 91.8%) (x: 91.4%)
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Fig. 2. Measured visibility of two-photon interference (blue boxes), CAR estimate visibility (crosses), and maximum CAR (solid circles) versus standard losses. Inset: coincidence to accidental ratio (CAR) versus pump power with different attenuations. Green squares = 1 dB, blue diamonds = 3 dB, red dots = 5 dB.
ton pairs are equally sensitive to the standard losses in the transmission channel. We then replace the neutral density filter with a random medium in the idler channel. We consider that the scatterers are isotropic and uncorrelated. The multiple scattering sample is prepared by dispersing uniform polystyrene microspheres (Duke Standards) in oil suspension and kept in a quartz cuvette with the length of 10 mm. The samples prepared in this work are in a weak scattering regime, where the scattering mean free path l is much larger than the wavelength of the photon. The refractive indices of oil and polystyrene microspheres are 1.47 and 1.59, respectively. The mean diameters (NIST traceable) of polystyrene microspheres used in this work are 0.5, 0.8, 1.6, and 5.0 μm. The scattered idler photon transmitted from a sample is collected by using a free space-to-fiber collimator (NA 0.25), which is placed right after the PBS. In order to achieve a 3 dB loss for all samples, we prepare concentrations of 1.22× 1014 m−3 , 1.13 × 1014 m−3 , 0.58 × 1014 m−3 , and 0.08 × 1014 m−3 for scatterer diameters (ϕ1;2;3;4 ) of 0.5, 0.8, 1.6, and 5.0 μm with the scattering mean free path l1;2;3;4 of 0.019, 0.010, 0.004, and 0.003 m, respectively. The main purpose for us to use different scatterer diameters with different concentrations for changing the mean free path while keeping the total loss of 3 dB is to explore and compare the results obtained from the standard loss of 3 dB (neutral density). We measure CAR as a function of scattering mean free path l, as shown in Fig. 3 for the horizontal polarizationcorrelated photon pair. We obtain a CAR of 16.9, 18.3, 19.8, and 20.3 for the mean free path of l4 0.003 m (ϕ4 5.0 μm), l3 0.004 m (ϕ3 1.6 μm), l2 0.010 m (ϕ2 0.8 μm), and l1 0.019 m (ϕ1 0.5 μm) at different average pump powers of 0.600, 0.570, 0.530, and 0.500 mW, respectively. The maximum CAR value decreases as the idler photon experiences more scattering events or short scattering mean free path. Also shown is the CAR value 23 for a standard loss of 3 dB, which is obviously higher than the CAR values obtained with multiple scattering random media.
0
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Fig. 3. Maximum coincidence to accidental ratio (CAR) versus scattering mean free path. The CAR of 23 is for the standard loss of 3 dB (solid red line). Inset: coincidence to accidental ratio (CAR) versus pump power with different scattering mean free path (l). Blue dots, l1 0.019 m; green diamonds, l2 0.010 m; black circles, l3 0.004 m; red squares, l4 0.003 m.
We then measure TPI of the polarization-entangled state with the idler photon scattered through a sample, as shown in the inset of Fig. 4. We fit the TPI fringe with Eq. (4) (shown as red dotted line) in the inset of the figure. The two maximum conditions fθ2 − θ1 0°; 180°g fall within in the error bars of our experimental data. We then obtain the TPI’s visibility, which is the average visibility as discussed in Eq. (7). We repeat the measurement of TPI for all samples and plot the visibility in Fig. 4 as a function of mean free path (l). We also plot the visibility V cor obtained from the CAR measurement. Also shown is TPI’s visibility of 91.8% for a standard loss of 3 dB, which is obviously higher than both V cor and V ent obtained with multiple scattering medium. The TPI’s visibility V ent is better than the visibility V cor obtained from the CAR measurement. This is not observed in 0.915
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Fig. 4. V ent (blue squares) and V cor (red dots) versus scattering mean free path, the solid blue and red lines are fitting curves for V ent and V cor . The dashed line is the visibility measured with 3 dB standard loss. Inset: two-photon interference fringes (blue squares) versus relative angle for scattering random medium, l3 0.004 m with HNLF at 300 K. The red dotted line is the theoretical plot of two-photon interference fringe.
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standard loss as shown in Fig. 2, where the CAR estimate visibility agrees with the measured TPI’s visibility. This observation directly attests that quantum correlation of the polarization-entangled photon pair is better preserved than polarization-correlated photon pair as one photon of the pair experiences random scattering process in the random medium. The fitting curves of the V cor and V ent are obtained from Eqs. (6) and (7), respectively. Our experiment/theory results show that the visibility of TPI (V ent or V cor ) is asymptotically approaching a constant value (>91%) provided by the standard loss (3 dB), i.e., without the random medium. The V ent is approaching the asymptotic value faster than the V cor . The V ent is 10% higher than the V cor for the mean free path in the range of 0.005–0.02 m. The V ent is for the entangled state prepared before the random medium. The V cor is for the entangled state prepared after the random medium that is for the situation one needs to prepare the entangled state from the correlated photon transmitted through the random medium. From the fittings of Eqs. (6) and (7), we obtain the average of T i 0.77, i.e., T 2i 0.6. The FWHM of the scattering angles for all samples is around 6°–7.4°. The acceptance angle of the fiber collimator is about 14°. With the approximation of 80% for the scattering light coupled into the fiber collimator, we have 0.8 × 0.6 0.48, which is close to the 3 dB loss that we claimed for all samples. In particular, our experiment setup is modeled in analogues to the entanglement-based QKD demonstrated in the Tokyo Network and 144 km of free-space channel by Anton Zeilinger’s group, where one photon from the entangled pair is measured locally and the second photon is sent to a distant observer via a transmission channel with losses or disturbances. In conclusion, we demonstrate that both standard loss (neutral density) and random scattering processes in the transmission channel are detrimental to the quantum correlation and interference of polarization-correlated and polarization-entangled photon pairs. We observe that polarization-entangled photon pairs are better preserved as one photon of the pair propagates through multiple
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scattering medium, hence it will be a better candidate for free-space long-distance QKD compared to correlated photon pairs. The Contractor acknowledges the Government’s support in the publication of this paper. This material is based upon research sponsored by AFRL under agreement number FA8750-12-1-0136. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of Air Force Research Laboratory or the U.S. Government. References 1. P. G. Kwiat, A. J. Berglund, J. B. Altepeter, and A. G. White, Science 290, 498 (2000). 2. P. W. Milonni, J. H. Carter, C. G. Peterson, and R. J. Hughes, J. Opt. B 6, S742 (2004). 3. A. A. Semenov and W. Vogel, Phys. Rev. A 80, 021802(R) (2009). 4. A. A. Semenov and W. Vogel, Phys. Rev. A 81, 023835 (2010). 5. I. Capraro, A. Tomaello, A. Dall’Arche, F. Gerlin, R. Ursin, G. Vallone, and P. Villoresi, Phys. Rev. Lett. 109, 200502 (2012). 6. D. Yu. Vasylyev, A. A. Semenov, and W. Vogel, Phys. Rev. Lett. 108, 220501 (2012). 7. M. V. da Cunha Pereira, L. A. P. Filpi, and C. H. Monken, Phys. Rev. A 88, 053836 (2013). 8. M. Brodsky, K. E. George, C. Antonelli, and M. Shtaif, Opt. Lett. 36, 43 (2011). 9. A. Aiello and J. P. Woerdman, Phys. Rev. Lett. 94, 090406 (2005). 10. G. Puentes, A. Aiello, D. Voigt, and J. P. Woerdman, Phys. Rev. A 75, 032319 (2007). 11. K. F. Lee, J. Chen, C. Liang, X. Li, P. L. Voss, and P. Kumar, Opt. Lett. 31, 1905 (2006). 12. X. Li, C. Liang, K. F. Lee, J. Chen, P. L. Voss, and P. Kumar, Phys. Rev. A 73, 052301 (2006). 13. H. Takesue and K. Inoue, Phys. Rev. A 70, 031802(R) (2004). 14. Y. M. Sua, J. Malowicki, M. Hirano, and K. F. Lee, Opt. Lett. 38, 73 (2013).
