Quantum Mirrors and Crossing Symmetry as Heart of Ghost Imaging

Quantum Mirrors and Crossing Symmetry as Heart of Ghost Imaging D. B. Ion 1), M. L. Ion 2) and L. Rusu 1) 1) Institute for Physics and Nuclear Engineering, Deartment of Fundamental Physics, POB MG 6, Bucharest, Romania 2) Bucharest University, Deartment for Nuclear Physics, POB MG 11, Bucharest, Romania Abstract: In this aer it is roved that the key to understanding the ghost imaging mystery are the crossing symmetric hoton reactions in the nonlinear media. Hence, the laws of the lane quantum mirror (QM) and that of sherical quantum mirror, observed in the ghost imaging exeriments, are obtained as natural consequences of the energymomentum conservation laws. So, it is shown that the ghost imaging laws deend only on the energy-momentum conservation and not on the hotons entanglement. The extension of these results to the ghost imaging with other kind of light is discussed. Some fundamental exeriments for a decisive tests of the [SPDC-DFG]-quantum mirror are suggested. PACS: 11.10.-z; 42.50.-; 03.65.Ud 1. Introduction It is well known that the crossing symmetry of S-matrix is a fundamental theorem of quantum field theory [1,2]. So, the crossing symmetry is one of the most imortant ractical elements of dynamics which makes use of the analytical roerties of the scattering amlitudes [3]. It relates various amlitudes, for examle helicity amlitudes, in one channel to those in other channels, or more generally to other rocesses in which one or more or all incoming and outgoing articles have been interchanged. Roughly seaking, the rocesses in crossed channels rovide forces for the rocess in the original channel. For examle, forces resonsible for the binding of two articles may be attributed to the exchange of the other articles in crossed channels. As is well known, the crossing relations generalize Pauli exchange rincile. Quantum electrodynamics (QED) being one of the working examle of the field theory is indeed crossing symmetric. This roerty of the QED is exerimentally verified [3] with very high accuracy. The S-matrix crossing symmetry [2] is a symmetry that relates S-matrix elements. Interaction rocesses involving different kinds of articles can be obtained from each other by relacing incoming articles with outgoing antiarticles after taking the analytic continuation. The crossing symmetry alies to all known articles, including the hoton which is its own antiarticle. For examle, the annihilation of an electron with a ositron into two hotons is related to an elastic scattering of an electron with a hoton by crossing symmetry. This relation allows to calculate the scattering amlitude of one rocess from the amlitude for the other rocess if negative values of energy of some articles are substituted γ + e - e - + γ (1) e - + e + γ + γ (2) By examination, it can be seen that these two interactions are related by crossing symmetry. It could then be said that the observation of Comton scattering imlies the existence of air annihilation and redicts that it will roduce a air of hotons. Here some remarks are necessary in connection with the quantum entanglement. If the quantum entanglement is a quantum mechanical henomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be satially searated, then the crossing symmetry of an interaction can be interreted as a very secial kind of entanglement (see M. Pitkanen in Ref. [4]). On the other hand, in 1995, at Baltimore University, rofessor (Dr.) Yanhua Shih initiated ghostimaging research [5], by using entangled hotons. In that exeriment, one hoton assed through stenciled atterns in a mask to trigger a detector, and another hoton was catured by a second detector. Surrisingly, an image of the attern between the two detectors aeared which the hysics community called ghost-imaging. Some definitions of ghost imaging (see refs. [6-8]) are as follows: (i) Ghost-imaging is a visual image of an object created by means of light that has never interacted with the object. (ii) Ghost imaging is an unusual effect by which an image is formed using light 1

atterns that do not emanate from the target object. (iii) Ghost imaging, is a novel technique in which the object and the image system are on searate otical aths. (iv) Ghost-imaging is similar to taking a flash-lit hoto of an object using a normal camera. The image is formed by the hotons that come out of the flash, bounce off the object, and then are focused through the lens onto hoto-reactive film or a charge-couled array. But, in this case, the image is not formed from light that hits the object and bounces back, Dr. Shih said. The camera collects hotons from the light sources that did not hit the object, but are aired through a quantum effect with others that did. Here, we must underline that ten years ago, based on the crossing symmetry of the SPDC-hoton reactions, the authors of the aers [9-11] resented a new exlanation of all the ghost henomena. Then, they introduced the concet of SPDC-quantum mirror (QM) and on this basis they roved some imortant Quantum Mirror hysical laws (see also our aer [12]) which can be of great hel for a more dee understanding of the ghost imaging henomena. This article is a direct continuation of our aer [12]. So, in Sect. 2 we resent the crossing symmetric hoton reactions as well as the connection between crossing symmetry and difference frequency generation (DFG). Moreover, in Sect. 2, some theoretical results for the DFG-conversion rate, in three wave mixing aroximation, are resented. In Sect. 3 ghost imaging via [SPDC, DFG]- quantum mirrors are described and exerimentally verified. Sect. 4 is reserved for discussions and conclusions. 2. Crossing symmetric [SPDC-DFG]-hoton reactions The first exeriments on ghost imaging were erformed using a air of entangled hotons roduced by sontaneous arametric down conversion (SPDC). In this rocess, a rimary um () hoton is incident on a nonlinear crystal and roduces the hotons idler (i) and signal (s) by the reaction: s + i. These hotons are correlated in energy, momentum, olarizations and time of birth. Some of these features, such as energy and momentum conservations: ω = ωs + ω, i k = k s + k are i exloited to match in diverse exeriments. (e.g., Momentum conservation in the degenerate case when the idler and the signal hotons acquire the same frequency leads to the roduction of a air of simultaneous hotons that are emitted at equal angles relative to the incident beam). Now, if the S- matrix crossing symmetry [1-3] of the electromagnetic interaction in the sontaneous arametric down conversion (SPDC) crystals is taken into account, then the existence of the direct SPDC rocess (see Fig.1) γ + will imly the existence of the following crossing symmetric rocesses γ γ s γ i (3) + γs γ i (4) γ + γi γ s (5) as real rocesses which can be described by the same transition amlitude. Here, by s and i we denoted the time reversed hotons relative to the original hotons s and i, resectively. In fact, the SPDC-reactions (4)-(5) can be identified as being directly connected with the χ (2) second-order nonlinear effects called in general three waves mixing. So, the rocess (3) can be considered as being just the inverse of second-harmonic generation, while, the effects (4)-(5) corresonds to the difference-frequency generation (DFG) in the resence of um laser via three wave mixing (see Fig.1). Here we have the generation of a hoton at frequency ω i when hotons at frequencies ω and ω s are incident on a crystal with an areciable second-order suscetibility χ (2), such that ω i = ω ω s (assuming ω > ω s). In order for energy conservation to hold, this additionally imlies that, for every hoton generated at the difference frequency ω i, a hoton at ω s must also be created, while a hoton at the higher frequency ω must be annihilated. In addition, for the rocess to occur with an areciable efficiency of frequency conversion, hase-matching must occur, so that: k i = k k s. In many situations, the field at ω is an intense um field, while the field at ω s is a weak signal field. Difference-frequency generation yields amlification of the ω s field (along with generation of another 2

field at ω i, commonly called the idler field). Thus, this rocess is termed arametric amlification; when ω s = ω i, the device created is called Degenerate Parametric Amlifier. In fact, the SPDCreactions (4)-(5) can be identified as being directly connected with the χ (2) second-order nonlinear effects called in general three waves mixing. So, the rocess (3) can be considered as being just the inverse of second-harmonic generation, while, the effects (4)-(5) corresonds to the differencefrequency generation (DFG) in the resence of um laser via three wave mixing (see Fig. 1 ). Difference-frequency generation as a three-wave nonlinear interaction rocess has been theoretically analyzed by Armstrong et al. [13] in 1962. The difference-frequency conversion efficiency has been investigated by Boyd and Kleinman [14], based on the electric field generated by two focused Gaussian beams, studying the deendence of the generation ower on the focusing condition and the roerties of the nonlinear mixing material. Therefore, let us consider two collinear Gaussian beams [called um () and signal (s), with the owers P and P s at frequencies ω and ω s, for the um and signal beam, resectively] having identical confocal arameters b = k w 2 = k s w s2 (here w is the beam waist). In this case, the DFG outut ower P i, at the difference-frequency ωi = ω ω s, can be written as follows [15,16]: where c is the seed of light in vacuum, n is the index of refraction, L is the crystal length, is the effective nonlinear coefficient, and α is the absortion coefficient of the nonlinear otical medium at the DFG frequency. The subscrits s,, i refer to the signal, um, and idler (infrared) hotons, resectively. The focusing function h(μ, ξ,α) involving walk-off and focused beam effects is given as (focusing oint is assumed at the center of the crystal): h μ, ξ, α = 1 ξ ξ ex[ b. α 4. ξ. dτ dτ ' 4 ] τ τ ' ξ ξ 1 j 2 [. 1 μ 1 μ 1 μ (7). τ τ ' τ. τ ' 1 μ ] 3 d eff (6)

where µ = k / k s and the focusing arameter ξ = L/b which relates the crystal length to the beam size of the um and signal. Difference frequency generation rovides mid-infrared laser radiation by means of interaction of two near-infrared lasers in a non-linear medium. In connection with DFG-lasers, recently Stry et al. [18] develoed a continuously mode-ho free tunable difference DFG-laser system suitable for highresolution sectroscoy in the 3 micron region (see Fig.2 for a short descrition) 3. Ghost imaging via [SPDC-DFG] crossing symmetric hoton reactions The main urose here is to obtain an answer to the basic question: Is ghost imaging mystery solved via the quantum mirror (QM) mechanism introduced in refs. [9-12]? So, we start with the definition of the quantum mirror and some of its hysical laws [9-12]. [SPDC,DFG]-Quantum Mirror (QM) (see Fig.3). A quantum mirrors is called [SPDC,DFG]-QM if is based on the quantum SPDC and DFG henomena (3)-(5) in order to transform signal hotons, characterized by ω s, k s, e s, μ s, into idler hotons with * ( ω ωs, k k s, e s, µ s ) ( ωi, k i, ei, µ i ). Therefore, according to the schematic descrition from Fig. 3, a [SPDC,DFG]-QM is comosed from: a high quality laser um (), a transarent crystal in which all the three hoton reactions (3)-(5) can be roduced, all satisfying the same energymomentum conservation laws: ω = ωs + ωi, k = k s + ki. In these conditions a new geometric otics can be develoed on the basis of the concet of quantum mirrors (QM) as shown in refs. [9-11]. Hence, the laws of the lane quantum mirror (see Fig.4) and that of sherical quantum mirror (SQM) (see Fig. 5), observed in the ghost imaging exeriments [6-7], are roved as natural consequences of the energy-momentum conservation laws. By the quantum mirroring mechanism (see Fig.3) the objects and their images can be considered as being on the same otical aths. Therefore, the key of ghost imaging mystery can be given by the electromagnetic crossing symmetric hoton reactions (3)-(5) or more general by DFG-henomena (see Fig.1). Indeed, in the case of ghost imaging observed in the aers [6,7] the ghost image can be roduced as follows 4

(see Fig.3): The image forms indirectly from the signal hotons that come out of the flash, bounce off the aerture, and then are focused through the lens onto QM where they are transformed in idler hotons i s which are collected in the idler detector. So, in this case, the image is not formed directly from signal hotons that hits the aerture and bounces back. The image is formed only by idler i s - hotons from QM-sources that did not hit the object, but are obtained via crossing symmetric hoton reaction (4) [or equivalently, as frequency-difference generation (DFG)] and not via hoton reaction (3). Therefore, the crossing symmetry is the heart of ghost imaging, ghost diffraction and ghost interference, henomena. 5

Clearly, a SPDC crystal illuminated by a high quality laser beam can acts as real quantum mirrors since by the crossing rocesses (4) (or (5)) a signal hoton (or idler hoton) is transformed in an idler hoton i s (or signal hoton s i ), resectively. 6

The quantum mirrors can be ''lane''[11] (PQM) and ''sherical quantum mirrors'' (SQM) (see figs.3-4) according with the character of incoming laser waves (lane waves or sherical waves). In order to avoid many comlications, in this resentation we will work only in the thin crystal aroximation and only for SPDC-QM. Now, it is imortant to note that using the QM-concets [9-12] the results PQM and SQM from Fig.4-5 are obtained only as a consequence of the energy-momentum conservation (or hase matching conditions) without any kind of hoton entanglement. In order to illustrate this we resent in Figs. 5 a roof of SQM-law using only energy-momentum conservation law. Also, it is imortant to remark that, the very high selectivity of the SPDC-QM is given by the fact that the energy-momentum conservation laws acts as a daemon which selects only the imaging hotons i s which are roduced by the crossing symmetric hoton reaction (4) [or more general, in the (DFG), in electromagnetic transitions which are crossing symmetric]. 7

Eq. (1) from Fig.8 shows tyical features for a three-wave arametric mixing rocess: ID 1. So, the i-ower P i received by ID-detector is roortional to the nonlinear otical figure of merit, 2 / n i n s n. d eff 2. The outut idler ower varies linearly with the roduct of the inut owers P ob s. P ID 3. The i- ower P i is roortional to the square of the idler frequency ω i. 4. The above idler ower varies with the crystal length L in the case of Gaussian beam couling, and reaches a maximum value with an otimum focusing arameter of ξ 1.3. The h-function reduces to h =ξ when using loose focusing arameter ξ <<1, which makes the DFG ower roortional to L 2, as in the case of the lane-wave aroximation where the nonlinear generation ower at the resultant idler frequency ω i, in the non absortive medium, can be written as: P d 2 ID eff L 2 2.ω 2 i n i. n s.n i. P. P s [ ob. sin Δk L/2 (8) Δk L/2 ] Therefore, the image visibility in all ghost imaging henomena can be controlled by varying the above essential arameters Ps ob. P and d 2 eff / n i n s n, from Eq. (8). On the other hand is well known that Souto Riberiro et al. [23] erformed the Young s double-slit exeriment with light roduced in stimulated down conversion. They studied the degree of visibility of the interference atterns as a function of the mean occuation number er mode N, or the inducing laser intensity I s, and they demonstrated that the satial coherence in the signal beam can be controlled by means of its conjugate air. (See Figs. 9a,b for the exerimental results). We rove that QM-mechanism can exlain comletely all these very imortant results. The results resented in Fig. 9a,b strongly suggest that similar results can be obtained for ghost image. So, on the base of the results resented in Fig.8 visibility of a ghost image ID can be controlled with the intensity I s of the signal laser. So, with such an exeriment we can finally rove that the coincidences are not necessary for ghost henomena (imaging, diffraction and interferences) in agreement with the conclusion that the entanglement is not necessary for ghost imaging, ghost diffraction and ghost interferences henomena [see exeriments quoted in Fig.6]. 4. Discussions and Conclusions The results and conclusions obtained in this aer can be summarized as follows: (i) Quantum electrodynamics (QED) is one of the working examle of the field theory which is indeed crossing symmetric [1,2]. This roerty of the QED is exerimentally verified [3] with very high accuracy. (ii) The quantum mirror (QM) roerty of the SPDC-source (laser um+crystal) is clearly roduced by the crossing symmetric hoton reactions (4)-(5) [or DFG-crossing symmetric transitions in the source of thermal light, etc.] These QM-instruments ossesses some eculiar roerties, such as: (a) The angle of incidence θ s ω s is not equal with the angle of reflection θ i ω i, exceting the degenerate case ω s =ω i. These angles must obey the transverse momentum conservation law ω s sinθ s =ω i sin θ i at crystal surface; (b) A nondegenerate QM, roducing DFG, change the color of incident s-beam in the color of the reflected i s-hoton beam; (c) A QM-otical instrument ossess a high hoton-selectivity since only the s-hotons which satisfy the energy-momentum conservation laws ω ω s =ω i, k k s = k i are reflected as i s hotons; (d) The QM as an indeendent otical instrument can be combined with other otical classical devices. Hence, the laws of the lane quantum mirror (see Fig.4) and that of sherical quantum mirror (SQM) (see Fig.5), observed exerimentally in the ghost imaging exeriments [6-7], are obtained as natural consequences of the energy-momentum conservation laws; 8

(e) Substantial imrovements of the otical devices (e.g. telescoes, microscoes, etc.), are exected to be obtained by using quantum mirrors (see [9]-[12]) instead of the usual mirrors. Some geometrical advantages (e.g., the magnifications and distances control by varying the um wavelength, etc.) as well as signal rocessing advantages (e.g., the high resolution, the high fidelity and amlification of the incoming beam intensity, distortion undoing for the signal rays, coherence reserving, etc.) are exected to be obtained by using the quantum mirror instead of the usual mirrors. (iii) In the ghost-imaging exeriments (see Figs.3-5) the image forms indirectly from the signal hotons that come from the object, and are focused through the lens onto QM where they are transformed in idler hotons i s which are collected in the image detector [or on a hoto-reactive film, etc.]. So, in this case, the image is formed only by idler i s -hotons from QM-sources that did not hit the object (see again Fig.3), but are obtained via crossing symmetric hoton reaction (4). (iv) The recent results [19]-[22] definitely roved exerimentally that the entanglement is not necessary in ghost imaging. These imortant results are summarized in Fig. 6. Consequently, we susected that measurements in coincidence counting regime, can be used only for the background subtractions, just as in nuclear and elementary article hysics. An exerimental test of the SPDC- QM is suggested in Fig.7. (v) An fundamental more efficient exerimental test of the DFG-QM is roosed in Fig.8. The DFG tyical features exressed by Eq. (1), in Fig. 8, allowed us to conclude that image visibility of all ghost henomena can be controlled by varying the following arameters: Ps ob. P and d 2 eff / n i n s n, from Eq.(1) in Fig.8. This conclusion is in excellent agreement with the exerimental results of Souto Ribeiro et al. [23] resented in Figs.9a,b. We note of course that a high quality of a ghost image will be obtained by illuminating the object with a high intensity s-laser. Finally, we believe that the results obtained here are encouraging for new theoretical and exerimental investigations since the crossing symmetry as heart of ghost (imaging, interferencediffraction) henomena can be of great imortance not only from fundamental oint of view but also for ractical alications. 9

Acknowledgments This research was suorted by CNCSIS under contract ID-52-283/2007 and also by CNMP roject 71-nr.131/2007. 5. References [1] M. Gell-Mann and M.L. Goldberger, Phys. Rev. 96, 1433 (1954); For a general discussion of the role of crossing summetry, see [2] G.F. Chew, S-matrix Theory of Strong Interactions (W.A. Benjamin, Inc.,New York, 1962); A.D. Martin and T.D. Searman, Elementary Particle Theory, Nord Holland Publishing Co. Amsterdam, 1970. [3] See for examle: V. Alles Borelli, et al., Exerimental check of crossing symmetry in the electromagnetic interaction of letons. Lettre al Nuovo Cimento, 2, 376 (1971). [4] M. Pitkanen, Toological Geometrodynamics, ag 112, Beckington, Frome, U.K., Luniver-Press, 2006. [5] D.V. Strekalov, A.V. Sergienko, D.N. Klyshko, and Y.H. Shih, Phys. Rev.Lett., 74, 3600 (1995). [6] T.B. Pittman, Y.H. Shih., D.V. Strekalov, and A.V. Sergienko Phys. Rev. A, 52, R3429 (1995). [7] T.B. Pittman, D.V. Strekalov, D.N. Klyshko, M.H. Rubin, A.V. Sergienko, and Y.H. Shih, Phys. Rev. A, 53, 2804 2815 (1996). [8] For a review, see L.A. Lugiato, A. Gatti and E. Brambilla, Quantum Imaging, J.Ot. B: Quant. Semicl. Ot 4, S183 (2002); A. Gatti, E. Brambilla, and L.A. Lugiato, Quantum imaging, Progress in Otics, Vol. 51, E. Wolf, Ed., Elsevier, New York (2008). [9] D.B. Ion, P. Constantin and M.L. Ion, Rom. Journal of Phys. 43,3 (1998) ArXiv-0810.3401. [10] D.B. Ion, P. Constantin and M.L. Ion, Rom. Journal of Phys. 45, 3 (2000) ArXiv-0810.3986; [11] D.B. Ion, P. Constantin and M.L. Ion, Rom. Journal of Phys. 45, 15 (2000). [12] D. B. Ion, M.L.D. Ion, L. Rusu, Rom. Re. Phys. 60, 1151-1158 (2008). [13] J.A. Armstrong, N. Bloembergen, J. Ducuing, P.S. Pershan, Phys. Rev. 127, 1918 (1962). [14] G.D. Boyd, D.A. Kleinman, J. Al. Phys. 39, 3597 (1968). [15] T. Chu, M. Broyer, J. Phys. 4, 523 (1985). [16] J.-J. Zondy, Ot. Commun. 149, 181 (1998). 10

[17] W. Chen et al., C. R. Physique 8, 1129 (2007) [18] S. Stry, P. Hering, M. Mürtz, Al. Phys. B 75, 297 (2002). [19] A. Valencia, G. Scarcelli, M. D Angelo and Y. Shih, Phys. Rev. Lett. 94, 063601 (2005). [20] F. Ferri, D.Magatti, A. Gatti, M.Bache, E. Brambilla, and L. A. Lugiato, Phys.Rev.Lett. 94, 183602 (2005). [21] I.P.Degiovanni, M.Bondani, E.Puddu, A. Andreoni,M.G.A.Paris, Phys. Rev. 76, 062309 (2007) [22] G. Scarcelli et al., Al. Phys. Lett. 88, 061106 (2006). [23] P.H. Souto Ribeiro et al., Phys Rev. 51, 1631 (1995). 11