Analysis of photon flux distribution of type-Ⅱ SPDC for highly efficient entangled source adjustment

Transcription

1 Analysis of photon flux distribution of type-Ⅱ SPDC for highly efficient entangled source adjustment Ziyi Wang 1, Heying Wang 1, Wenbo Sun 1,* 1Physics Department, Tsinghua University, Beijing , P. R. China. * Corresponding author: swb@mail.tsinghua.edu.cn. Type-Ⅱspontaneous parametric down conversion (SPDC) of a beta-barium borate (BBO) crystal is a nonlinear optical process widely used as a polarization-entangled twin-photon source. Analysis of photon flux distribution of type-Ⅱspdc is needed for quickly and precisely determining the position of entangled twin photons, which is the key step to highly efficient adjustment for entangled source. In this article the photon flux distribution of the signal light ring and intersecting points of light cones generated via the type-Ⅱspdc from a BBO crystal is analyzed and experimental results exhibit good agreement with the theoretical analysis. Also it is experimentally observed and theoretically validated that for type-Ⅱspdc, the emergent direction of frequency degenerate photons deviates from the direction with the maximum photon flux. OCIS codes: ( ) Parametric processes; ( ) Quantum optics 1. INTRODUCTION The entanglement of particles is a special and important state in quantum mechanics. The concept of the entanglement of particles was firstly introduced into quantum mechanics by Schrödinger and the famous Einstein-Podolsky-Rosen experiment [1-3]. The entangled state has various significant applications such as quantum computing, secure cryptography, and teleportation of a quantum state [4-12]. A commonly used source of entangled photon pairs, the spontaneous parametric down-conversion (SPDC) is a nonlinear optical process in which a pump photon scatters into a signal photon and an idler photon in a nonlinear crystal [13-17]. The signal and idler photons emit in form of light cones or emit in a single mode. SPDC can be classified into type-Ⅰand type-Ⅱ according to its polarization pattern. Type-ⅠSPDC generates one light cone and Type-Ⅱ SPDC generates two light cones. Photon pairs emitted from the intersecting points of light cones generated via type- Ⅱ SPDC are polarization entangled. The spatial structure, wave vectors and polarization patterns of type-Ⅱspdc, along with the emergent angle of entangled twin-photons, are illustrated in Fig. 1. Non-frequency-degenerate light are emitted in forms of two light cones similar to the frequencydegenerate light, but one of the light cones has larger radius while the other has smaller radius. And the phase mismatching also contribute to the width of light cones like Fig. 1 Illustrates. Fig. 1. The schematic diagram of type-Ⅱ SPDC. The green and blue light rings are the cross sections of frequency-degenerate light cones and nonfrequency-degenerate light cones respectively. The width of these light cones is caused by the mismatching light with the same frequency. A typical polarization entangled twin-photon source involves a SPDC light field and two single photon detectors. Because there isn t precise study about the photon flux distribution of type-Ⅱspdc, it is impossible to adjust due to the counts of one single photon detector channel. Thus, in normal condition, the adjustment of such source depends on the coincident counts of two channels of single photon detectors, which represent the flux of entangled photon pairs. However, this kind of adjustment is time-consuming because from coincident counts of two detectors one cannot determine which detector gets a deviation. Thus the precise analysis of type-Ⅱ SPDC photon flux distribution is the key for highly efficient adjustment. With such analysis, in adjustment one can infer the detectors deviation from the proper location by the counts of one single detector. The complete theoretical analysis of SPDC power distribution was first presented by K. Koch et al along with the experimental result of type-Ⅰspdc.[18] Other studies for the photon flux distribution of type-Ⅱspdc were also presented, including different focusing of pump light and the thickness of crystal.[19] However, studies for photon flux distribution of type-Ⅱspdc have not been well built because the special structure of type- Ⅱ SPDC is more complex. Furthermore, highly precise analysis for type-Ⅱ SPDC with ultra-low photon fluxes, which is necessary to generated entangled states, has not been reported. ÖzgünSüzer and Theodore G. Goodson III used an aperture to scan type- Ⅱ SPDC and obtained a rough result.[20] O.Jedrkiewicz et al used CCD camera for the mapping of type- Ⅱ SPDC power distribution but the result wasn t precise enough for the analysis focusing on small portion of the light field such as the area near entangled photons.[21] Moreover, due to the different mechanism of CCD camera and single photon detector, the photon flux distribution measured by CCD camera cannot be straightly applied to help the adjustment of single photon detector. Futher, the entangled twin-photon source is based on single photon

2 detectors. Thus, to guide the adjustment of entangled twinphoton sourece based on only one single photon detector, it is necessary to measure the photon flux distribution using it. In this article the changing trend of photon flux distribution of signal light ring especially near the intersecting point from type-Ⅱspdc with very low photon flux is analyzed. And the spatial structure of 810nm type-Ⅱ SPDC is measured experimentally and compared with theoretical results. Finally, the photon flux near the entangled photons emergent angle is analyzed. 2. THEORETICAL ANALYSIS AND NUMERAL CALCULATION 2.1 The angular distribution of photon pairs The down conversion process is calculated according the conservations of energy and momentum, which is always referred as phase matching. The conservation of energy is expressed as ω s + ω i = ω p, (1) where ω p is the frequency of pump light, and the ω s and ω i are the frequencies of the signal and idler lights. The phase matching condition can be expressed as k s + k i = k p, (2) in which k p, k s and k i represent the wave vectors of pump, signal and idler lights respectively. For type-Ⅱspdc in a negative uniaxial BBO crystal, the polarizations of the lights involved can be expressed as e e + o, which indicates that the pump and signal light is the extraordinary light and the idler light is ordinary lights. Thus the vectorsk e andk o in Fig. 1 representsk s and k i. Consequently the refraction index of pump and signal light maybe replaced by effective refraction index, which is expressed as[22] n eff (λ e, φ e ) = [ cos2 φ e n o 2 (λ e ) + sin2 φ e n e 2 (λ e ) ] λ e represents the wavelength of extraordinary light andφ e represents the angle formed by extraordinary light s wave vector and optical axis as shown in Fig. 2. Combining the Eq. (1), Eq. (2), Eq. (3) and the geometrical relationship of optical axis and wave vectors of pump, signal and idler lights shown in Fig. 2, along with known parameters including the refraction index, the cutting angle of BBO crystal and wavelength of pump light, the emergent angle of frequency-degenerate signal and idler lights can be calculated in the case of frequency-degenerate down conversion, where ω s = ω i. Because the signal light is e-polarized and idler light is o-polarized, the photons at the intersecting point of signal light cone and idler light cone are polarization entangled. Its state can be expressed as H V + V H. 1 2 (3) Fig. 2. The geometry of wave vectors in type-Ⅱspdc. k p, k s and OA represent the direction of pump light, signal light and optical axis respectively. To simplify the calculation, k s is decomposed into k sz that is parallel to k p, and K s that is vertical to k p.

3 where K s, K i, k sz and k iz are the transverse and longitudinal components of wave vectors of signal and idler lights respectively. From the research by K.Koch et al, the photon flux generated from SPDC per unit frequency is[18] N s (ω s,k s )= ħd eff 2 ω s ω i ω p L 2 N p 2π 4 c 3 dω ε 0 n s n i n s d 2 K s d 2 δ exp ( 1 p 2 δ2 ) sinc 2 ( 1 2 L k z) (4) in which n p, n s and n i are the refraction indexes of pump, signal and idler lights in BBO crystal, d eff is the effective second-order nonlinear coefficient, L is the interacting crystal length, δ = K w is the dimensionless transverse momentum of signal and idler waves, and here w is the pump beam radius divided by e 2. From the derivation of Eq. (4) by F. Hsu and C. Lai, the photon flux per unit frequency can be expressed as[23] Fig. 3. The calculated angular distribution of type-Ⅱspdc. The wavelength of pump light is 405nm and the cut angle of BBO crystal is (a) The angular distribution of signal (upper) and idler (lower) light cones. Each point corresponds to equally distribute azimuthal angle. Thus the two rings intersecting positions correspond to the emergent angles of entangled twinphotons. (b) Angular distribution of signal light with wavelength changing from 796nm to 816nm. According to the calculation result, the emergent angle is positively correlated to the wavelength of signal light and consequently the wavelength and the emergent angle of signal lights can be approximately expressed as functions of each other. Such relationship is calculated and used in the latter calculation. The calculation result also illustrated that in a wide range of wavelength, signal and idler photon pairs can satisfy phase-matching function. And the receptor receives both non-frequency-degenerate and frequency degenerate signal photons. 2.2 The photon flux distribution of one SPDC ring The photon flux distribution away from the emergent angle of frequency-degenerate photon pairs has little contribution to the adjustment of entanglement. Thus the photon flux of signal lights with emergent angle in the range of ±0.5 is measured. In addition, the phase-mismatching photon pairs can still be generated with a lower intensity. The distribution of phase-mismatching photon pairs can be theoretically calculated [18]. The generated photon flux is strongly related to the mismatch vector k = k p k s k i. As illustrated in Fig. 2, the direction of pump light is the positive direction of z axis, the mismatching vector kis decomposed into the longitudinal part k z and the transversal part K, defining k z = k p k sz k iz, K = k x2 + k y 2 = K s + K i, N s (λ s,θ s, φ s )= ħ d eff 2 ω p ω i ω p L 2 N p 8π 4 c 3 2πdω ε 0 n s n i n s K s dk s dφ s p dδ i exp (- 1 2 (δ s δ i ) 2 ) sinc 2 ( 1 2 L k z) (5) In which ϕ s is the azimuthal angle of signal light illustrated in Fig. 2. Considering the top of the signal light ring, which is the furthest part from pump light, has the least interference from idler light ring, this part is therefore measured and analyzed to verify the theory and deduce the photon flux distribution near the entanglement photons emergent angle. The Eq. (5) includes the phase mis-matching factor sinc 2 ( 1 k L 2 L k z) = sin2 ( 2 ) k L ( 2 )2 which indicates that the range of emergent angle correspondeds to the phase-mismatching signal photons with certain wavelength is negatively related to the interacting crystal length. For a signal light from type-Ⅱ SPDC with certain wavelength, the effective range in wavelength and in emergent angle of phase-mismatching photon near it is rather small because of the greater thickness of crystal used for type-Ⅱspdc. Thus the effect of phase-mismatching is very small for type-Ⅱspdc from a thick bulk BBO crystal. For this reason and the calculation based on Eq. (5), the influence of phase-mismatching on photon flux distribution can be ignored in the case that the emergent angle of signal light is large enough, such as the top of signal light ring. However, if the emergent angle of signal light is relatively small, the effect of phasemismatching cannot be ignored. For the top of signal light ring, as previous discussion suggested, only signal lights accords to phase-matching function is necessary to be considered. In this case, it is obtained that δ i = δ s

4 sinc 2 ( 1 2 L k z) = 1 Therefore Eq. (5) is rewrote as N s (λ s, θ s, φ s ) = 2π 2πħd 2 eff ωp L 2 N p ε 0 n s n i n p λ5 sin (2θ s λi s )dλ s dθ s dδ i, (6) In the real situation of entangled twin-photon generation and the photon flux detection, the receptor composed by a fiber and a collimator is correlated to a constant solid angle. Therefore the receptor detects signal lights with different range of azimuthal angle φ s, making an additional corresponding factor necessary. Because in this case only the top of signal ring is studied, in which the angle φ s is approximately 0, the range of φ s received by the receptor is directly related to the emergent angle θ s out. In addition, as previously discussed, in a small range of signal light wavelength, the emergent angle of signal light is positively correlated with the signal light s wavelength, and thus the range of φ s correspond to the receptor is also related to signal light s wavelength. Defining factor ξ(λ s ) is relative value of the range of the azimuthal angle correlated to the signal light emergent angle, the equation for the photon flux per unit frequency is N s (λ s,θ s, φ s )= 2π 2πħd eff 2 ω p L 2 N p ε 0 n s n i n p λ s 5 λi sin(2θ s )dλ s dθ s dδ i ξ(λ s ). For the signal light near the entangled twin-photon s emergent angle, the influence of phase-mismatching cannot be ignored because the emergent angles of the signal lights are relatively small. The factor ξ(λ s ) is also necessary according to previous discussion. Therefore, adding this factor for the calculation of photon flux distribution near the entangled twin-photons, Eq. (5) is rewrote as N s (λ s,θ s, φ s )= ħ d eff 2 ω p ω i ω p L 2 N p 2πdω 8π 4 c 3 ε 0 n s n i n s K s dk s dφ s p dδ i e -1 2 (δ s δ i ) 2 sinc 2 ( 1 2 L k z) ξ(λ s ) (7) 3. EXPERIMENTAL METHODS The spatial structure near the intersecting point of the 810nm rings, the photon flux distribution of the top of type-Ⅱ SPDC and the photon flux distribution at the intersecting position two light cones from type-Ⅱ SPDC were measured, as schematically shown in Fig.4. Fig. 4. The schematic diagrams of experimental set-up and measurement. (a)the apparatus for 810nm rings spatial structure measurement. The Pol is a : 1 polarization plate, R1 is the receptor combined by fiber and a collimator, RM is the rotary mount used to fix the fiber and collimator, and SD is the single photon detector, S is the spectrometer. (b)the flux measurement at top of type-Ⅱspdc. (c)the photon flux measurement for intersecting point of type-Ⅱspdc. The type-Ⅱspdc measured is generated via a 405nm pump laser passing through a 2mm thick BBO crystal. The pump was a dioxide laser with a line width of 0.7nm and the output power of 18mw. The pump was focused on BBO crystal by a lens (L1) with focal length of 500mm. In order to adjust its position and direction, two mirrors (M1 and M2) were used before L1.

5 The BBO crystal was cut at the angles of θ m = 42.6 and φ = 30, which were selected for type-Ⅱ SPDC (e e + o) at 810nm from a 405nm pump. The crystal was fixed on a three-dimensional rotary mounting, with its crystal surface orienting right towards the pump laser. The angle formed by Optical Axis(OA) and pump s propagation wave vector satisfied the phase-matching condition for the max efficiency of parametric down conversion. With OA and pump s wave vector being in the same vertical plane, the light field was generated upright with the entangled photon pairs on the horizontal plane. The BBO crystal could be rotated in the transverse plane and thus the light field generated via SPDC could be rotated. The receptor used in experiments consisted of a collimator, a fiber and a single photon detector or spectrometer. The collimator focused the light to the fiber, and from which the light was received by the single photon detector or spectrometer. The collimator and one end of the fiber were fixed on a rotary mount that could move vertically and horizontally and rotate in both coronal plane and transverse plane. The collimator and end of the fiber on the mount were adjusted to be facing the same direction. Thus the collimator and the fiber pigtail could be moved on transverse plane and in a direction vertical to the direction of light it received. Also, the mount could be rotated to make the collimator and the fiber pigtail aligned on the direction of the measured light. In this way, the light generated via type-Ⅱspdc could be measured with respect to different emission angle of light generate via SPDC. The schematic diagram of this part of experiment is shown in Fig.4. To test the spatial structure of 810nm SPDC rings, the cross section of type- Ⅱ SPDC at 400mm from the BBO crystal was measured using the spectrometer. The receptor was aligned to the emission direction of entangled photons as the starting point. In the scanning the receptor were moved horizontally at the step size of 1mm. At each step the vertical coordinate of the receptor when it received the highest counts of 810nm photons after adjusting the angle of receptor. The experimental setup is shown in Fig.4 (a). The photon flux distribution was measured by a single photon detector (PerkinElmer, spcm-aqrh-13). The singlephoton detector used was a silicon avalanche diode and its quantum efficiency for different wavelength could be considered as a constant in this experiment. The measurement covered a relatively small portion of the light field, which was the top of SPDC as the illustration in Fig.3 (a). To reduce the error from mechanical stability, the SPDC was rotated 90 by rotating the BBO crystal and the polarization of pump laser by a half wave plate to place this portion of SPDC on the same horizontal plane pump laser. So that in the measurement the collimator and the fiber pigtail could be scanned in horizontal direction. The reason was that the mount moving horizontally has a more stable structure. In the experiment the receptor was firstly aligned to the emergent angle of frequency-degenerate photons at the top of SPDC. Then the receptor was horizontally moved in the direction vertical to the emission direction of signal photons. At each step the horizontal angle of the receptor was adjusted to attain the maximum counts of photon flux and the maximum counts was measured before and after 90 rotation of BBO crystal. The receptor moved in the range of ±2mm, which corresponds to ±0.5 for the signal photons emergent angle. The position of receptor as the starting point was adjusted due to reverse-light system. The schematic diagram is shown in Fig.4(b). To reduce the error from stray light, the BBO crystal was rotated 90 at every point to measure stray light from pump s diffuse reflection and background noise. Due to the calculation result, the receptor receives no considerable flux from type-Ⅱspdc after the rotation. At every point the crystal was rotated to the original state afterwards and compare the photon flux measured before the rotation to validate this method. Similar method is also used in other paper [23]. Using the same method, the photon flux distribution near the intersecting point was measured. The schematic diagram for this experiment is shown in Fig. 4(c). 4. RESULTS nm Light rings The experimental results of the spatial structure of 810nm type- Ⅱ SPDC are shown in Fig.5 with dots along with the theoretical calculation result with solid lines. This results reflect the basic structure of type-Ⅱ SPDC and can be used as reference for entangled twin-photon source adjustment. This analysis illustrates the overall structure of frequency-degenerate light cones and, along with the analysis of photon flux distribution, can help the adjustment of polarization-entangled twin-photon source. Fig. 5. The spatial structure of 810nm type-Ⅱspdc near the intersecting points. The cross section of signal and idler light cones at 400mm from BBO crystal. The green dots are the experimental results and the orange and blue curves are the theoretical results of signal and idler light rings respectively. Every point represents the vertical position at which the receptor receives the maximum counts of 810nm light. 4.2 Photon flux distribution Top of type-Ⅱ SPDC The experimental and theoretical results of photon flux distribution at the top of signal light ring of type-Ⅱspdc from BBO crystal are illustrated in Fig.6.

6 Fig. 6. Photon flux distribution at the top of signal light ring. At the zero point of lateral axis the receptor was correlated to the emergent angle of frequency-degenerate signal photon. The blue curve is the theoretical results and the green points are the experimental results. As the Fig. 6 represents, the changing trend of experimental results approximately fits that of theoretical result. Because the photon fluxes are very low, the error in the result of measurement can be relatively apparent. Such deviation also exists in other studies of SPDC photon flux distribution. The feature of the experimental results agree with the results in other photon flux distribution study.[18,23] Both the experimental result and theoretical result indicate that, because of the influence of phasemismatching light and the contribution of non-frequencydegenerate lights, the photon flux at the emergent direction of frequency degenerate photons isn t the greatest. The photon flux at the top of SPDC includes only the contribution of signal light ring and thus by testifying the theory of photon flux on one single ring, it can help analyzing of photon flux distribution at the intersecting point, which includes the photon fluxes from both two signal and idler light rings Intersecting point of type-Ⅱ SPDC The experimental results of photon flux distribution near the intersecting point are shown with the theoretical results in Fig.7. symmetric to each other. Therefore the photon flux near the entangled twin-photons emergent angle is relatively complex. The theoretical results of photon flux distribution near the intersecting point considering the phasemismatching effect illustrated in Fig. 7 is calculated using the Eq. (7). In this case, because of the relatively small emergent angle, the effect of phase-mismatch is considered. In this part of signal light ring, the emergent angle of frequency-degenerate photons is approximately The changing trend of experimental result nearly fits that of the theoretical results. Apparently, the experimental result has some deviation comparing to the theoretical result. Some deviation in elevation angle and height of the receptor caused the deviation of experimental result. Both theoretical and experimental results illustrate that, for the intersecting point of type-Ⅱspdc, the photon flux at the emergent direction of frequency degenerate photons isn t the greatest. Similar to previous experiment, the contributions of phase-mismatching and the non-frequencydegenerate lights together formed this deviation. Combining the theoretical and experimental result, it is demonstrated that the emergent angle of maximum photon flux is approximately 0.1 away from the emergent angle of entangled photons. From the photon flux distribution measured in this article and the counts of single photon detector in entangled twin photon source, the relative position of the receptor and the emergent direction of entangled photons can be inferred. Thus using method and results in this article, it is convenient to adjust the position of the receptor to receive entangled twin-photons and consequently established the polarization entangled twinphoton source.[24] 4.3 The polarization correlations of entangled twin-photon source Using the results and the analyzing methods shown above, a polarization entangled twin-photon source was established. The results of experimental polarization correlations is shown in Fig.8. Based on the result and the CHSH inequality, which is an extended form of Bell s inequality, it was testified that the experimental result violates the Bell inequality by S=2.3 with 50 standard deviation. This results verified that using method and results above, it can be efficient to adjust and establish the polarization entangled twin-photon source. Fig. 7. Photon flux distribution near the intersecting point of signal and idler rings in type-Ⅱspdc. The lateral axis is the position of receptor correspond to emergent angle of signal photons. The orange points in this chart is experimental results and the blue curve is the theoretical results. The entangled twin-photons emergent at the intersecting point of signal and idler ring, which are

7 Fig. 8. The result of experimental polarization correlations of the entangled twin-photon source. For each curve in the result, one polarization plate in front of one receptor is set at 0, 90, 45 or +45 while the other polarization plate in front of the other receptor rotates from 0 to 360. By applying the result to CHSH inequality, it is verified to have violated the Bell s inequality and consequently verifies the establishment of polarization entangled twin-photon source. 5. CONCLUSION The photon flux distribution at the top of signal light ring and near the intersecting point in type-Ⅱspdc at ultra low fluxes pumped by a 405-nm diode laser were measured using single photon counting system. The spatial structure of 810nm rings of type-Ⅱ SPDC was measured by a spectrometer. The results agree with the theoretical prediction. Both theoretical simulation and experimental result validate a property of type-Ⅱspdc that the emergent angle correlated to the maximum photon may deviate from the emergent angle of frequency-degenerate photons, which contradicts previous assumption. The results of the photon flux distribution contributes to the adjustment of this entangled twin-photon source by transforming the adjustment based on the coincident count of two channels to the adjustment based on the emission rate on one single channel. Combining the results of all three experiments, the photon flux distribution in the area important to the adjustment for polarization entangled twin-photon source is analyzed. Consequently, such analysis can guide and help this adjustment and increase its efficiency. With the help of experimental results and methods described in this article, a polarization entangled twin-photon source was established. ACKNOWLEDGEMENTS We would like to thank Xiangbin Wang from Tsinghua University and Yong Li from Beijing Computational Science Research Center for the warm hospitality and helpful discussions. This work is supported by National Natural Science Foundation of China (No. J ) and Tsinghua University Fund for Laboratory Innovation( No ) REFERENCES 1. A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev.47, (1935) 2. Schrödingeretal., Diegegenwärtige situation in der quantenmechanik, Naturwissenschaften 23, (1935) 3. Schrödinger et al, Discussion of probability relations between separated systems, Proceedings of the Cambridge Philosophical Society 31, (1935) 4. R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys.21(6-7), (1982) 5. D. Deutsch and R. Jozsa, Rapid solution of problems by quantum computation, Proc. R. Soc. London, Ser. A 439(1907), (1992) 6. P. W. 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