Fractional Fourier processing of quantum light Yifan Sun,1,2 Ran Tao,3 and Xiangdong Zhang2,* 1
Department of Physics, Beijing Normal University, Beijing 100875, China School of Physics, Beijing Institute of Technology, 100081, Beijing, China 3 Fundamental Science on Multiple Information Systems Laboratory, Department of Electronic Engineering, Beijing Institute of Technology, Beijing, China *[email protected] 2
Abstract: We have extended Fourier transform of quantum light to a fractional Fourier processing, and demonstrated that a classical optical fractional Fourier processor can be used for the shaping of quantum correlations between two or more photons. Comparing the present method with that of Fourier processing, we find that fractional Fourier processing for quantum light possesses many advantages. Based on such a method, not only quantum correlations can be shaped more rich, but also the initial states can be easily identified. Moreover, the twisted phase information can be recovered and quantum states are easily controlled in performing quantum information experiments. Our findings open up new avenues for the manipulation of correlations between photons in optical quantum information processing. ©2014 Optical Society of America OCIS codes: (070.2590) ABCD transforms; (270.5585) Quantum information and processing.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25(3), 241–265 (1980). A. C. McBride and F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. Appl. Math. 39(2), 159–175 (1987). A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10(10), 2181–2186 (1993). D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10(9), 1875–1881 (1993). H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. A 10(12), 2522–2531 (1993). D. Mendlovic, Z. Zalevsky, R. G. Dorsch, Y. Bitran, A. W. Lohmann, and H. Ozaktas, “New signal representation based on the fractional Fourier transform: definitions,” J. Opt. Soc. Am. A 12(11), 2424–2431 (1995). B. Hennelly and J. T. Sheridan, “Fractional Fourier transform-based image encryption: phase retrieval algorithm,” Opt. Commun. 226(1-6), 61–80 (2003). Y. Zhang, B. Dong, B. Gu, and G. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A 15(5), 1114–1120 (1998). H. E. Hwang and P. Han, “Fractional Fourier transform optimization approach for analyzing optical beam propagation between two spherical surfaces,” Opt. Commun. 245(1-6), 11–19 (2005). Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. 86(2), 021112 (2005). Y. Cai and S. Y. Zhu, “Coincidence fractional Fourier transform implemented with partially coherent light radiation,” J. Opt. Soc. Am. A 22(9), 1798–1804 (2005). F. Wang, Y. J. Cai, and S. L. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express 14(16), 6999–7004 (2006). J. Liu, A. Tan, and Z. Hong, “Experimental observation of coincidence fractional Fourier transform with entanglement photon pairs,” Opt. Commun. 282(17), 3524–3526 (2009). D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78(1), 010304 (2008). D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009). D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer, 2000). M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000). C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University, 2005).
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Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 727
19. J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012). 20. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19(5), 1174–1184 (2002). 21. M. Atatüre, G. Di Giuseppe, M. D. Shaw, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Multiparameter entanglement in quantum interferometry,” Phys. Rev. A 66(2), 023822 (2002). 22. B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94(22), 223601 (2005). 23. R. Shimizu, K. Edamatsu, and T. Itoh, “Quantum diffraction and interference of spatially correlated photon pairs and its Fourier-optical analysis,” Phys. Rev. A 74(1), 013801 (2006). 24. A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78(4), 043810 (2008). 25. A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. W. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104(1), 010501 (2010). 26. E. Poem, Y. Gilead, Y. Lahini, and Y. Silberberg, “Fourier processing of quantum light,” Phys. Rev. A 86(2), 023836 (2012). 27. H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
1. Introduction Fractional Fourier transform (FrFT) was proposed by Namias in 1980 as generalization of conventional Fourier technique [1, 2]. It was introduced into optics in 1993 [3–5]. It has been found wide applications in information processing, optical image encryption, beam analysis and shaping, etc [6–9]. Subsequently, coincidence FrFT with partially coherent beam and entangled photon pairs have also been analyzed theoretically, and demonstrated experimentally [10–15]. In recent years, there is a growing interest in the control and manipulation of quantum light due to its exciting applications in quantum information processing [16–19]. The optical Fourier transform (FT) has been used for the manipulation of quantum light [20–25]. Very recent investigations have shown that a classical optical Fourier processor can be used for the shaping of quantum correlations between two or more photons, and the class of filter functions applicable in the multiphoton Fourier space is identified [26]. As we know, using the FrFT in classical information processing possesses many advantages in comparison with using the FT. For example, the FrFT can show comprehensive signal information of space domain and frequency domain at the same time, we can use it to realize multichannel and multistage filters and so on [3–9]. The problem is whether or not the classical optical FrFT can also be used for the shaping of quantum correlations? if it can, whether or not such a method possesses some advantages in the processing of optical quantum information? Motivated by these problems, in this work we discuss an analogy between the fractional Fourier processing of classical light to that of quantum light. We explore the possibility to realize FrFT of quantum light by borrowing a classical optical fractional Fourier processor. 2. Theory and method It has been demonstrated that the transition between photon quantum states can be realized by using linear optical devices [16–19]. For example, we introduce at the input plane of the optical device a two-photon state [26] Ψ = dx1dx2 AS ( x1 , x2 ) aˆ † ( x2 ) aˆ † ( x1 ) 0 ,
(1)
where aˆ † ( x ) is photon creation operator at the coordinate x of the input plane of the device. The coordinates can be either one-(1D) or two-dimensional(2D), depending on the studied system, and the operators obey bosonic commutation relations. Here As ( x1 , x2 ) = As ( x2 , x1 ) is an exchange-symmetric complex probability amplitude. The spatial intensity correlation
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Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 728
function of this state is given by its modulus square. For two noninteracting photons passing together through the device, the output two-photon state is expressed as [26] Ψ out = dxo1dxo 2 BS ( xo1 , xo 2 ) bˆ† ( xo 2 ) bˆ† ( xo1 ) 0
(2)
BS ( xo1 , xo 2 ) = dxi1dxi 2 AS ( xi1 , xi 2 ) U ( xo 2 , xi 2 ) U ( xo1 , xi1 ) ,
(3)
with
where bˆ† ( xi ) is photon creation operator at the coordinate xi of the output plane of the device, U ( xo1 , xi1 ) is the (complex) amplitude of that transition from the input(i) to output(o), BS ( xo1 , xo 2 ) represents the output complex probability amplitude. Then, the spatial intensity
correlation function of the output two-photon state can be obtained by calculating BS ( xo1 , xo 2 )
2
[26].
Recently, based on the above theoretical description, Poem et. al. has demonstrated the Fourier processing of quantum light [26]. In the following, we explore to realize the fractional Fourier processing of quantum light. The scheme design is described in Fig. 1. In classical optics, it is well known that 1D fractional Fourier processing can be realized by lenses [3]. In our scheme, we also use lens-mask-lens structure. The first lens is used to modulate the spatial distribution of two-photon states, and there is no difference whether FT or FrFT is applied. Here, we use FT in the transform preceding. The transfer function can be expressed as U F ( xo , xi ) = exp ( −i 2π xo xi / (λ f ) ) , where xi and xo are the coordinates in the input and output plane, respectively, λ is the wavelength of the incident light and f is the focal length of the lens. After a phase mask, we use the second lens to do FrFT, which the transfer function for the FrFT by lenses can be expressed as iπ U Fr ( xo , xi ) = exp xo2 cot φ − 2 xo xi csc φ + xi 2 cot φ ) . ( λ f1
(4)
Here f1 = Qf is called standard focal length. In general, there are two kinds of method to realize the fractional Fourier processing of classical light by using lenses as shown in the line-box of Fig. 1 [3]. One is by using one lens. In such a case, the related FrFT parameter Q is taken as Q = tan( φ /2). The other method is by using two lenses, Q is taken as Q = sin( φ ). Here φ = pπ / 2 , p represents fractional degree with period 4, which can be taken continuously within the period. It is easy to find from Eq. (4) that, U Fr ( x0 , xi ) = δ ( x0 − xi ) for p = 0 , which represents the identity transformation. When p = 1 , U Fr ( x0 , xi ) goes back to the Fourier processor. For p = 2 , U Fr ( x0 , xi ) = δ ( x0 + xi ) represents the spatial reflection transformation, and for p = 3 , it is the compound of two cases with p = 1 and p = 2 . As for the mask in the fractional Fourier processing, we use a 4-f filter with a sinusoidal phase mask M ( x ) = exp iAp cos ( 2πυ x ) , where Ap is the phase mask amplitude and ν is the spatial frequency of modulation [26]. It has been pointed out that this filter implements the tight-binding model [18] and emulates the quantum walk [27] of the two entangled photons on a periodic lattice.
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Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 729
Fig. 1. Schematic picture depicting Fractional Fourier processing of quantum light. The path-entangled photon-pairs are taken as source, and an initial phase factor can be added to one of the paths by phase control part. The photon-pairs go through a 4-f filter equipped with a spatial light modulator (SLM), in which a classic FrFT process is applied, and finally detected by two fiber-coupled detectors (D1, D2). Coincidence detection is manifested by a coincidence counting unit (CCU). The classic FrFT process, which could be realized by lenses shown in the line-box. Here f represents the focal length of the lens and f1 = Qf is called standard focal length. R and Q are related FrFT parameters, R = tan( φ /2) and Q = sin( φ ) for upper case, and R = sin( φ ) and Q = tan( φ /2) for lower case. Here
φ
= pπ/2,
p
represents fractional degree.
We consider two-photon path-entangled state at the input plane, which can be written as 1 ( 2a ,0b + eiϕ 0a , 2b ) in Fock representation [26], where the subscripts a(b) represent the 2 path mark and φ is the phase difference between the two paths. The complex probability amplitude AS for such a case can be written as 1 g ( xi1 − xa ) g ( xi 2 − xa ) + eiϕ g ( xi1 − xb ) g ( xi 2 − xb ) , where xa ( xb ) is the AS ( xi1 , xi 2 ) = 2 central lateral position of the first (second) path, g ( x ) is its lateral profile. The subscript 1 and 2 are marked for the distinction of two photons. The profile of g ( x ) is a Gaussian-like distribution, which can be measured experimentally. Here, we set g ( x ) to be a Dirac-δ function, which is a limit form of Gaussian distribution with zero width. Then, the output complex probability amplitude can be obtained as BS ( xo1 , xo 2 ) = dxi1dxi 2 AS ( xi1 , xi 2 ) U T ( xo 2 , xi 2 ) U T ( xo1 , xi1 ) ∝
1 U T ( xo1 , xa ) U T ( xo 2 , xa ) + eiϕU T ( xo1 , xb )U T ( xo 2 , xb ) , 2
(5)
where U T ( xo , xi ) is the transfer function for the FrFT, and can be calculated by the following relation U T ( xo , xi ) = dx f U Fr ( xo , x f ) M ( x f ) U F ( x f , xi ) iπ 2 = exp xo cot φ × (1 + i ) × λ f1
f1λ tan φ ×
(6)
2 xo xi csc φ + − nυ λ f1 tan φ . i J n ( Ap ) exp −iπ λf n =−∞ λ f1 n =∞
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n
Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 730
Here Jn(x) represents the Bessel function of the first kind and U F ( x f , xi ) is the transfer function for the FT that is related with the first lens. Thus, the information of correlation functions can be obtained by calculating BS ( xo1 , xo 2 ) . In the same manner the theory can be extended to the case for an arbitrary number of noninteracting photons. Similar to the FT, the FrFT of N non-interacting bosons in a D-dimensional space can be regarded as a transformation in ND-dimensional space. 3. Numerical results and discussion We begin to study correlation manipulation similar to fractional Fourier image processing as shown in Fig. 2(a). We use a 4-f filter with a sinusoidal phase mask similar to the previous investigation on the Fourier processing in [26]. In our design, the output for the Fourier processing can be regenerated exactly as p = 1 . The FrFT processing and its difference from the FT processing can be understood clearly by analyzing the spatial focus mode of the output intensity as a function of the parameter Ap . The output intensity patterns for two input beams in the Fourier processor are shown in Fig. 2(b). The incident light, originally concentrated in two input spatial modes, spreads among a number of output modes increasing linearly with Ap [26]. However, when p deviates from 1, which can be realized by changing the position of lens, the situation becomes different. This is because spectral lines of output focus modes are broaden with such a deviation and the interactions among modes occur. When the deviation is small, only the nearest neighbor interaction between modes appears. This results in the spectral line splitting and more new modes produce as shown in Fig. 2(c). With the increase of the deviation, the stretches of the spectral lines increase gradually, the interactions among the modes go beyond the nearest neighbor case. Off to a certain extent for the p, strong overlap among the modes can occur. In such a case, a handful of output modes, even single mode, can appear as shown in Fig. 2(d). This can be seen more clearly from Fig. 3. Figures 3(a) and 3(b) show output intensity patterns of two input modes as a function of fractional degree and position at Ap = 0.86π and 1.67π, respectively. We find that various kinds of output modes can be obtained by choosing suitable values of p. It is interesting that the FrFT could yield a large deviation from the FT for a small deviation from p = 1. This is because the output intensity depends on U T ( xo , xi ) . The factor tan φ in the expression of U T ( xo , xi ) plays a major role in such a deviation, and tan φ = tan ( pπ / 2 ) → ∞ as p → 1 . This means that a small deviation from p = 1 can cause a large change of tan φ , which leads to a large deviation of the output intensity.
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Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 731
Fig. 2. (a) Schematic of the beam shaping for FrFT process with a sinusoidal phase mask. For FrFT process, the transform part will be replaced by the lens system in the line-box of Fig. 1. (b) Output intensity patterns for different values of the phase amplitude Ap for two input beams when fractional degree p is 1 (FT case). The output intensity patterns as a function of Ap and position for p = 1-6.66 × 10−9 (c) and p = 0.8 (d).
Fig. 3. Output intensity patterns as a function of fractional degree p and position with two input modes for Ap = 0.86π (a) and Ap = 1.67π (b). The other parameters are taken identical with those in Fig. 2.
The above rich features of output modes can be used directly to shape quantum correlation. Figure 4 displays the calculated results for shaping of quantum states by using 2D mask. The incident state is the two-photon path-entangled state as has been described in Sec.2. The phase of the 2D mask is presented in Fig. 4(b). The calculated results for correlation maps are presented in Figs. 4(c)-4(f) for p = 0.95, 0.85, 0.75 and 0.65 at Ap = 1.67π, respectively. Here the initial phase factor is taken as φ = 0. For comparison, the correlation map of the initial state when no mask was applied, is presented in Fig. 4(a). It is seen clearly that the output quantum states strongly depend on the fractional degree p, which is corresponded to those in Fig. 2 and
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Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 732
Fig. 3. When p is near 1, many correlated states are found (see Fig. 4(c)). With the decrease of p, more overlaps among discrete quantum states occur. With the change of overlapping, various states of aggregation have been observed as shown in Figs. 4(d) and 4(e). Taking an appropriate value for p, the output of single quantum state of aggregation can be realized (see Fig. 4(f)). In all, we can control correlation properties of quantum states through the fractional Fourier process according to our requirements.
Fig. 4. Normalized spatial correlation functions for the incident two-photon path-entangled state with different p. (a) Spatial correlation function of the input state without mask. (b) A section of the phase pattern created by the applied sinusoidal phase mask in the two-photon fractional Fourier plane. (c)–(f) correspond to the spatial correlation patterns at Ap = 1.67π for p = 0.95, 0.85, 0.75 and 0.65, respectively. The initial phase factor is taken as φ = 0.
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Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 733
Fig. 5. Spatial correlation functions for the incident two-photon path-entangled state at Ap = 1.67π and p = 0.65. (a)–(d) correspond to the phase difference between two photons φ = 0, φ = π, φ = π/2 and φ = -π/2, respectively. The other parameters are taken identical with those in Fig. 4.
The output quantum states not only depend on the fractional degree p, they are also related to the input states. Figure 5 shows spatial correlation functions of two-photon path-entangled states at Ap = 1.67π and p = 0.65. Figures 5(a)–5(d) correspond to the phase difference between two photons φ = 0, π, π/2 and -π/2, respectively. In the previous investigations, it has been shown that the output states for φ = π/2 and φ = -π/2 are indistinguishable through the Fourier process without an extra Zernike filter [26]. However, our calculated results in Figs. 5(c) and 5(d) exhibit different features. This is because using the FrFT is equivalent to add a tunable parameter (dimension) in comparison with the FT. Some degeneracy states (indistinguishable sates) in the FT process become distinguishable in the FrFT process. This means that it is easier to distinguish the different input states by using the fractional Fourier process in comparison with using the Fourier processing. Another advantage to shape quantum states by using the fractional Fourier processing in comparison with using the Fourier processing is that it is easy to recover the twist phase information. In fact, deformation and distortion inevitably occur in the fabrication of the phase mask. Both thermal fluctuation and solid strain can also cause deformation and distortion of the phase mask. If the phase mask is twisted, the shaped quantum state is also deformed. The deformation of the phase mask is usually complex. In the microscopic version, it can be considered as a continuous function and has Taylor series. In Taylor-series, x 2 , which represents the one of the simplest bend form of a plane, can be usually considered as a approximate form, if the deformation is small enough. Here, we add x 2 bend to the phase iAp a ( 2πυ x 2 ) + cos( 2πυ x ) mask. So the expression of the phase mask changes to the form M ′ ( x ) = e . The mask is 1D and becomes 2D when acting in the two-photon probability amplitude. Our calculated results are shown in Fig. 6. The left column in Fig. 6 describes the phase mask pattern. The middle column is the normalized spatial correlation function of the output two-photon state corresponding to the phase pattern. The right column is the output intensity #201558 - $15.00 USD (C) 2014 OSA
Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 734
pattern for the case with a single photon incident state, which is deeply related to the two-photon correlation function. Figure 6(b) corresponds to the case with the twist phase mask pattern. For comparison, the perfect case is shown in Fig. 6(a). Comparing Figs. 6(a) with 6(b), we find that quantum correlation spectrum of the Fourier processing becomes blurry which may bring about a measure problem in experiment, for example, low contrast. However, such a problem can be overcome by setting the appropriate parameters (tuning the position of lens in experiment) in the fractional Fourier processing. For example, let p = 0.992, the twisted information gets the perfect resume as shown in Fig. 6(c). Although the above discussions are only for the theoretical results, we believe that the corresponding experiment can be performed according to the design as shown in Fig. 1.
Fig. 6. State information recovery under small perturbation by FrFT. (a) Sinusoidal phase mask pattern, spatial correlation function and output intensity patterns for the FT process. (b) 2D bend phase mask pattern, deformed spatial correlation function and output intensity patterns in the FT process. (c) State recovery by FrFT process. Here the fractional degree p = 0.992 is taken. The other parameters are taken identical with those in Fig. 4
4. Summary In summary, we have extended FT of quantum light to the fractional Fourier processing, and demonstrated that the spatial correlations of quantum light states can be manipulated by using the classical fractional Fourier processor. Comparing our method with that of the Fourier processing, we find that the fractional Fourier processing for quantum light possesses many advantages. Not only various kinds of quantum correlation properties can be constructed by using the fractional Fourier process, but also the initial states can be easily identified and the twisted phase information can be recovered by using such a method. In addition, there are quite a few standard fractional Fourier processing schemes used in classical signal processing [6–8]. In fact, some of these methods can also be theoretically adapted to manipulate quantum light. We believe that these findings are very beneficial for optical quantum information processing.
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Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 735
Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 11274042) and the National Key Basic Research Special Foundation of China under Grant 2013CB632704.
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Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 736
Department of Physics, Beijing Normal University, Beijing 100875, China School of Physics, Beijing Institute of Technology, 100081, Beijing, China 3 Fundamental Science on Multiple Information Systems Laboratory, Department of Electronic Engineering, Beijing Institute of Technology, Beijing, China *[email protected] 2
Abstract: We have extended Fourier transform of quantum light to a fractional Fourier processing, and demonstrated that a classical optical fractional Fourier processor can be used for the shaping of quantum correlations between two or more photons. Comparing the present method with that of Fourier processing, we find that fractional Fourier processing for quantum light possesses many advantages. Based on such a method, not only quantum correlations can be shaped more rich, but also the initial states can be easily identified. Moreover, the twisted phase information can be recovered and quantum states are easily controlled in performing quantum information experiments. Our findings open up new avenues for the manipulation of correlations between photons in optical quantum information processing. ©2014 Optical Society of America OCIS codes: (070.2590) ABCD transforms; (270.5585) Quantum information and processing.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25(3), 241–265 (1980). A. C. McBride and F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. Appl. Math. 39(2), 159–175 (1987). A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10(10), 2181–2186 (1993). D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10(9), 1875–1881 (1993). H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. A 10(12), 2522–2531 (1993). D. Mendlovic, Z. Zalevsky, R. G. Dorsch, Y. Bitran, A. W. Lohmann, and H. Ozaktas, “New signal representation based on the fractional Fourier transform: definitions,” J. Opt. Soc. Am. A 12(11), 2424–2431 (1995). B. Hennelly and J. T. Sheridan, “Fractional Fourier transform-based image encryption: phase retrieval algorithm,” Opt. Commun. 226(1-6), 61–80 (2003). Y. Zhang, B. Dong, B. Gu, and G. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A 15(5), 1114–1120 (1998). H. E. Hwang and P. Han, “Fractional Fourier transform optimization approach for analyzing optical beam propagation between two spherical surfaces,” Opt. Commun. 245(1-6), 11–19 (2005). Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. 86(2), 021112 (2005). Y. Cai and S. Y. Zhu, “Coincidence fractional Fourier transform implemented with partially coherent light radiation,” J. Opt. Soc. Am. A 22(9), 1798–1804 (2005). F. Wang, Y. J. Cai, and S. L. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express 14(16), 6999–7004 (2006). J. Liu, A. Tan, and Z. Hong, “Experimental observation of coincidence fractional Fourier transform with entanglement photon pairs,” Opt. Commun. 282(17), 3524–3526 (2009). D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78(1), 010304 (2008). D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009). D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer, 2000). M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000). C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University, 2005).
#201558 - $15.00 USD (C) 2014 OSA
Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 727
19. J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012). 20. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19(5), 1174–1184 (2002). 21. M. Atatüre, G. Di Giuseppe, M. D. Shaw, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Multiparameter entanglement in quantum interferometry,” Phys. Rev. A 66(2), 023822 (2002). 22. B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94(22), 223601 (2005). 23. R. Shimizu, K. Edamatsu, and T. Itoh, “Quantum diffraction and interference of spatially correlated photon pairs and its Fourier-optical analysis,” Phys. Rev. A 74(1), 013801 (2006). 24. A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78(4), 043810 (2008). 25. A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. W. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104(1), 010501 (2010). 26. E. Poem, Y. Gilead, Y. Lahini, and Y. Silberberg, “Fourier processing of quantum light,” Phys. Rev. A 86(2), 023836 (2012). 27. H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
1. Introduction Fractional Fourier transform (FrFT) was proposed by Namias in 1980 as generalization of conventional Fourier technique [1, 2]. It was introduced into optics in 1993 [3–5]. It has been found wide applications in information processing, optical image encryption, beam analysis and shaping, etc [6–9]. Subsequently, coincidence FrFT with partially coherent beam and entangled photon pairs have also been analyzed theoretically, and demonstrated experimentally [10–15]. In recent years, there is a growing interest in the control and manipulation of quantum light due to its exciting applications in quantum information processing [16–19]. The optical Fourier transform (FT) has been used for the manipulation of quantum light [20–25]. Very recent investigations have shown that a classical optical Fourier processor can be used for the shaping of quantum correlations between two or more photons, and the class of filter functions applicable in the multiphoton Fourier space is identified [26]. As we know, using the FrFT in classical information processing possesses many advantages in comparison with using the FT. For example, the FrFT can show comprehensive signal information of space domain and frequency domain at the same time, we can use it to realize multichannel and multistage filters and so on [3–9]. The problem is whether or not the classical optical FrFT can also be used for the shaping of quantum correlations? if it can, whether or not such a method possesses some advantages in the processing of optical quantum information? Motivated by these problems, in this work we discuss an analogy between the fractional Fourier processing of classical light to that of quantum light. We explore the possibility to realize FrFT of quantum light by borrowing a classical optical fractional Fourier processor. 2. Theory and method It has been demonstrated that the transition between photon quantum states can be realized by using linear optical devices [16–19]. For example, we introduce at the input plane of the optical device a two-photon state [26] Ψ = dx1dx2 AS ( x1 , x2 ) aˆ † ( x2 ) aˆ † ( x1 ) 0 ,
(1)
where aˆ † ( x ) is photon creation operator at the coordinate x of the input plane of the device. The coordinates can be either one-(1D) or two-dimensional(2D), depending on the studied system, and the operators obey bosonic commutation relations. Here As ( x1 , x2 ) = As ( x2 , x1 ) is an exchange-symmetric complex probability amplitude. The spatial intensity correlation
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Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 728
function of this state is given by its modulus square. For two noninteracting photons passing together through the device, the output two-photon state is expressed as [26] Ψ out = dxo1dxo 2 BS ( xo1 , xo 2 ) bˆ† ( xo 2 ) bˆ† ( xo1 ) 0
(2)
BS ( xo1 , xo 2 ) = dxi1dxi 2 AS ( xi1 , xi 2 ) U ( xo 2 , xi 2 ) U ( xo1 , xi1 ) ,
(3)
with
where bˆ† ( xi ) is photon creation operator at the coordinate xi of the output plane of the device, U ( xo1 , xi1 ) is the (complex) amplitude of that transition from the input(i) to output(o), BS ( xo1 , xo 2 ) represents the output complex probability amplitude. Then, the spatial intensity
correlation function of the output two-photon state can be obtained by calculating BS ( xo1 , xo 2 )
2
[26].
Recently, based on the above theoretical description, Poem et. al. has demonstrated the Fourier processing of quantum light [26]. In the following, we explore to realize the fractional Fourier processing of quantum light. The scheme design is described in Fig. 1. In classical optics, it is well known that 1D fractional Fourier processing can be realized by lenses [3]. In our scheme, we also use lens-mask-lens structure. The first lens is used to modulate the spatial distribution of two-photon states, and there is no difference whether FT or FrFT is applied. Here, we use FT in the transform preceding. The transfer function can be expressed as U F ( xo , xi ) = exp ( −i 2π xo xi / (λ f ) ) , where xi and xo are the coordinates in the input and output plane, respectively, λ is the wavelength of the incident light and f is the focal length of the lens. After a phase mask, we use the second lens to do FrFT, which the transfer function for the FrFT by lenses can be expressed as iπ U Fr ( xo , xi ) = exp xo2 cot φ − 2 xo xi csc φ + xi 2 cot φ ) . ( λ f1
(4)
Here f1 = Qf is called standard focal length. In general, there are two kinds of method to realize the fractional Fourier processing of classical light by using lenses as shown in the line-box of Fig. 1 [3]. One is by using one lens. In such a case, the related FrFT parameter Q is taken as Q = tan( φ /2). The other method is by using two lenses, Q is taken as Q = sin( φ ). Here φ = pπ / 2 , p represents fractional degree with period 4, which can be taken continuously within the period. It is easy to find from Eq. (4) that, U Fr ( x0 , xi ) = δ ( x0 − xi ) for p = 0 , which represents the identity transformation. When p = 1 , U Fr ( x0 , xi ) goes back to the Fourier processor. For p = 2 , U Fr ( x0 , xi ) = δ ( x0 + xi ) represents the spatial reflection transformation, and for p = 3 , it is the compound of two cases with p = 1 and p = 2 . As for the mask in the fractional Fourier processing, we use a 4-f filter with a sinusoidal phase mask M ( x ) = exp iAp cos ( 2πυ x ) , where Ap is the phase mask amplitude and ν is the spatial frequency of modulation [26]. It has been pointed out that this filter implements the tight-binding model [18] and emulates the quantum walk [27] of the two entangled photons on a periodic lattice.
#201558 - $15.00 USD (C) 2014 OSA
Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 729
Fig. 1. Schematic picture depicting Fractional Fourier processing of quantum light. The path-entangled photon-pairs are taken as source, and an initial phase factor can be added to one of the paths by phase control part. The photon-pairs go through a 4-f filter equipped with a spatial light modulator (SLM), in which a classic FrFT process is applied, and finally detected by two fiber-coupled detectors (D1, D2). Coincidence detection is manifested by a coincidence counting unit (CCU). The classic FrFT process, which could be realized by lenses shown in the line-box. Here f represents the focal length of the lens and f1 = Qf is called standard focal length. R and Q are related FrFT parameters, R = tan( φ /2) and Q = sin( φ ) for upper case, and R = sin( φ ) and Q = tan( φ /2) for lower case. Here
φ
= pπ/2,
p
represents fractional degree.
We consider two-photon path-entangled state at the input plane, which can be written as 1 ( 2a ,0b + eiϕ 0a , 2b ) in Fock representation [26], where the subscripts a(b) represent the 2 path mark and φ is the phase difference between the two paths. The complex probability amplitude AS for such a case can be written as 1 g ( xi1 − xa ) g ( xi 2 − xa ) + eiϕ g ( xi1 − xb ) g ( xi 2 − xb ) , where xa ( xb ) is the AS ( xi1 , xi 2 ) = 2 central lateral position of the first (second) path, g ( x ) is its lateral profile. The subscript 1 and 2 are marked for the distinction of two photons. The profile of g ( x ) is a Gaussian-like distribution, which can be measured experimentally. Here, we set g ( x ) to be a Dirac-δ function, which is a limit form of Gaussian distribution with zero width. Then, the output complex probability amplitude can be obtained as BS ( xo1 , xo 2 ) = dxi1dxi 2 AS ( xi1 , xi 2 ) U T ( xo 2 , xi 2 ) U T ( xo1 , xi1 ) ∝
1 U T ( xo1 , xa ) U T ( xo 2 , xa ) + eiϕU T ( xo1 , xb )U T ( xo 2 , xb ) , 2
(5)
where U T ( xo , xi ) is the transfer function for the FrFT, and can be calculated by the following relation U T ( xo , xi ) = dx f U Fr ( xo , x f ) M ( x f ) U F ( x f , xi ) iπ 2 = exp xo cot φ × (1 + i ) × λ f1
f1λ tan φ ×
(6)
2 xo xi csc φ + − nυ λ f1 tan φ . i J n ( Ap ) exp −iπ λf n =−∞ λ f1 n =∞
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n
Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 730
Here Jn(x) represents the Bessel function of the first kind and U F ( x f , xi ) is the transfer function for the FT that is related with the first lens. Thus, the information of correlation functions can be obtained by calculating BS ( xo1 , xo 2 ) . In the same manner the theory can be extended to the case for an arbitrary number of noninteracting photons. Similar to the FT, the FrFT of N non-interacting bosons in a D-dimensional space can be regarded as a transformation in ND-dimensional space. 3. Numerical results and discussion We begin to study correlation manipulation similar to fractional Fourier image processing as shown in Fig. 2(a). We use a 4-f filter with a sinusoidal phase mask similar to the previous investigation on the Fourier processing in [26]. In our design, the output for the Fourier processing can be regenerated exactly as p = 1 . The FrFT processing and its difference from the FT processing can be understood clearly by analyzing the spatial focus mode of the output intensity as a function of the parameter Ap . The output intensity patterns for two input beams in the Fourier processor are shown in Fig. 2(b). The incident light, originally concentrated in two input spatial modes, spreads among a number of output modes increasing linearly with Ap [26]. However, when p deviates from 1, which can be realized by changing the position of lens, the situation becomes different. This is because spectral lines of output focus modes are broaden with such a deviation and the interactions among modes occur. When the deviation is small, only the nearest neighbor interaction between modes appears. This results in the spectral line splitting and more new modes produce as shown in Fig. 2(c). With the increase of the deviation, the stretches of the spectral lines increase gradually, the interactions among the modes go beyond the nearest neighbor case. Off to a certain extent for the p, strong overlap among the modes can occur. In such a case, a handful of output modes, even single mode, can appear as shown in Fig. 2(d). This can be seen more clearly from Fig. 3. Figures 3(a) and 3(b) show output intensity patterns of two input modes as a function of fractional degree and position at Ap = 0.86π and 1.67π, respectively. We find that various kinds of output modes can be obtained by choosing suitable values of p. It is interesting that the FrFT could yield a large deviation from the FT for a small deviation from p = 1. This is because the output intensity depends on U T ( xo , xi ) . The factor tan φ in the expression of U T ( xo , xi ) plays a major role in such a deviation, and tan φ = tan ( pπ / 2 ) → ∞ as p → 1 . This means that a small deviation from p = 1 can cause a large change of tan φ , which leads to a large deviation of the output intensity.
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Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 731
Fig. 2. (a) Schematic of the beam shaping for FrFT process with a sinusoidal phase mask. For FrFT process, the transform part will be replaced by the lens system in the line-box of Fig. 1. (b) Output intensity patterns for different values of the phase amplitude Ap for two input beams when fractional degree p is 1 (FT case). The output intensity patterns as a function of Ap and position for p = 1-6.66 × 10−9 (c) and p = 0.8 (d).
Fig. 3. Output intensity patterns as a function of fractional degree p and position with two input modes for Ap = 0.86π (a) and Ap = 1.67π (b). The other parameters are taken identical with those in Fig. 2.
The above rich features of output modes can be used directly to shape quantum correlation. Figure 4 displays the calculated results for shaping of quantum states by using 2D mask. The incident state is the two-photon path-entangled state as has been described in Sec.2. The phase of the 2D mask is presented in Fig. 4(b). The calculated results for correlation maps are presented in Figs. 4(c)-4(f) for p = 0.95, 0.85, 0.75 and 0.65 at Ap = 1.67π, respectively. Here the initial phase factor is taken as φ = 0. For comparison, the correlation map of the initial state when no mask was applied, is presented in Fig. 4(a). It is seen clearly that the output quantum states strongly depend on the fractional degree p, which is corresponded to those in Fig. 2 and
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Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 732
Fig. 3. When p is near 1, many correlated states are found (see Fig. 4(c)). With the decrease of p, more overlaps among discrete quantum states occur. With the change of overlapping, various states of aggregation have been observed as shown in Figs. 4(d) and 4(e). Taking an appropriate value for p, the output of single quantum state of aggregation can be realized (see Fig. 4(f)). In all, we can control correlation properties of quantum states through the fractional Fourier process according to our requirements.
Fig. 4. Normalized spatial correlation functions for the incident two-photon path-entangled state with different p. (a) Spatial correlation function of the input state without mask. (b) A section of the phase pattern created by the applied sinusoidal phase mask in the two-photon fractional Fourier plane. (c)–(f) correspond to the spatial correlation patterns at Ap = 1.67π for p = 0.95, 0.85, 0.75 and 0.65, respectively. The initial phase factor is taken as φ = 0.
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Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 733
Fig. 5. Spatial correlation functions for the incident two-photon path-entangled state at Ap = 1.67π and p = 0.65. (a)–(d) correspond to the phase difference between two photons φ = 0, φ = π, φ = π/2 and φ = -π/2, respectively. The other parameters are taken identical with those in Fig. 4.
The output quantum states not only depend on the fractional degree p, they are also related to the input states. Figure 5 shows spatial correlation functions of two-photon path-entangled states at Ap = 1.67π and p = 0.65. Figures 5(a)–5(d) correspond to the phase difference between two photons φ = 0, π, π/2 and -π/2, respectively. In the previous investigations, it has been shown that the output states for φ = π/2 and φ = -π/2 are indistinguishable through the Fourier process without an extra Zernike filter [26]. However, our calculated results in Figs. 5(c) and 5(d) exhibit different features. This is because using the FrFT is equivalent to add a tunable parameter (dimension) in comparison with the FT. Some degeneracy states (indistinguishable sates) in the FT process become distinguishable in the FrFT process. This means that it is easier to distinguish the different input states by using the fractional Fourier process in comparison with using the Fourier processing. Another advantage to shape quantum states by using the fractional Fourier processing in comparison with using the Fourier processing is that it is easy to recover the twist phase information. In fact, deformation and distortion inevitably occur in the fabrication of the phase mask. Both thermal fluctuation and solid strain can also cause deformation and distortion of the phase mask. If the phase mask is twisted, the shaped quantum state is also deformed. The deformation of the phase mask is usually complex. In the microscopic version, it can be considered as a continuous function and has Taylor series. In Taylor-series, x 2 , which represents the one of the simplest bend form of a plane, can be usually considered as a approximate form, if the deformation is small enough. Here, we add x 2 bend to the phase iAp a ( 2πυ x 2 ) + cos( 2πυ x ) mask. So the expression of the phase mask changes to the form M ′ ( x ) = e . The mask is 1D and becomes 2D when acting in the two-photon probability amplitude. Our calculated results are shown in Fig. 6. The left column in Fig. 6 describes the phase mask pattern. The middle column is the normalized spatial correlation function of the output two-photon state corresponding to the phase pattern. The right column is the output intensity #201558 - $15.00 USD (C) 2014 OSA
Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 734
pattern for the case with a single photon incident state, which is deeply related to the two-photon correlation function. Figure 6(b) corresponds to the case with the twist phase mask pattern. For comparison, the perfect case is shown in Fig. 6(a). Comparing Figs. 6(a) with 6(b), we find that quantum correlation spectrum of the Fourier processing becomes blurry which may bring about a measure problem in experiment, for example, low contrast. However, such a problem can be overcome by setting the appropriate parameters (tuning the position of lens in experiment) in the fractional Fourier processing. For example, let p = 0.992, the twisted information gets the perfect resume as shown in Fig. 6(c). Although the above discussions are only for the theoretical results, we believe that the corresponding experiment can be performed according to the design as shown in Fig. 1.
Fig. 6. State information recovery under small perturbation by FrFT. (a) Sinusoidal phase mask pattern, spatial correlation function and output intensity patterns for the FT process. (b) 2D bend phase mask pattern, deformed spatial correlation function and output intensity patterns in the FT process. (c) State recovery by FrFT process. Here the fractional degree p = 0.992 is taken. The other parameters are taken identical with those in Fig. 4
4. Summary In summary, we have extended FT of quantum light to the fractional Fourier processing, and demonstrated that the spatial correlations of quantum light states can be manipulated by using the classical fractional Fourier processor. Comparing our method with that of the Fourier processing, we find that the fractional Fourier processing for quantum light possesses many advantages. Not only various kinds of quantum correlation properties can be constructed by using the fractional Fourier process, but also the initial states can be easily identified and the twisted phase information can be recovered by using such a method. In addition, there are quite a few standard fractional Fourier processing schemes used in classical signal processing [6–8]. In fact, some of these methods can also be theoretically adapted to manipulate quantum light. We believe that these findings are very beneficial for optical quantum information processing.
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Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 735
Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 11274042) and the National Key Basic Research Special Foundation of China under Grant 2013CB632704.
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Received 19 Nov 2013; revised 24 Dec 2013; accepted 24 Dec 2013; published 6 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000727 | OPTICS EXPRESS 736
