Two-photon double-slit interference experiment

Transcription

1 1192 J. Opt. Soc. Am. B/Vol. 15, No. 3/March 1998 C. K. Hong and T. G. Noh Two-photon double-slit interference experiment C. K. Hong and T. G. Noh Department of Physics, Pohang University of Science and Technology, Pohang , Korea Received August 5, 1997; revised manuscript received November 12, 1997 We report a two-photon double-slit interference experiment. Two correlated photons, generated by spontaneous parametric downconversion and traveling in different directions, are made to pass through a double slit in such a way that the path of one photon can be identified with that of the other photon. No single-photon interference pattern is observed even if the path is not actually identified, but a spatial interference pattern appears in the joint detection counting rates of the photon pairs. It can easily be shown that these nonlocal phenomena cannot be exhibited by any kind of classical field Optical Society of America [S (98) ] OCIS codes: , , INTRODUCTION Young s double-slit interference experiment is a textbook example of the manifestation of complementarity in quantum mechanics. 1,2 Both wave and particle aspects of light or matter are needed for an explanation of the phenomena, yet they are mutually exclusive. Light or material particles can pass through two slits simultaneously and exhibit an interference pattern. But if the path (particle aspect) is identified, the interference pattern (wave nature) disappears. So far, one cannot identify the path without destroying the light particle, i.e., the photon, and even in case of material particles path identification has been considered only hypothetically. 1 4 In this paper we report the first double-slit experiment in which one can identify the paths of photons without destroying the photons. In the process of spontaneous parametric downconversion, a pump photon incident upon a nonlinear crystal splits into a pair of signal and idler photons. 5,6 Because of the conservation of momentum, the emerging direction of the signal photon is highly correlated with that of the idler photon. If a double slit is inserted into the beam of signal photons, that correlation can be used to identify which slit a signal photon passes through from its conjugate idler photon s detection position, and this can be done even before the signal photon approaches the double slit. This distinguishability of the paths through the signal and idler correlation prohibits the appearance of the signal interference pattern, and complementarity holds. On the other hand, if idler photons are also made to pass through another double slit, the distinguishability is removed and the joint detection probability density of the signal and idler photon pairs shows a spatial interference pattern, which is observed in our experiment. We show in a simple way that this nonlocal interference effect originates purely by a photon photon correlation that cannot be described by classical wave theory. There have been numerous interesting two-photon interference experiments performed with the photon pairs generated by a spontaneous parametric downconversion process Of these experiments, those of Refs. 16, 18 21, 23, 24, 27, 29, and 31 demonstrate the nonlocal aspect of two-photon interference effects especially well because the paths of the signal and the idler photons are completely separated, as in our experiment. The experiments of Refs. 16, 18, 21, 23, 24, 27, 29, and 31 on the one hand and Refs. 19 and 20 on the other can be differentiated, even though in all of them temporal interference patterns were observed. In the former group, so-called Franson experiments, a Michelson or Mach Zehnder interferometer is inserted into both the signal and the idler beams. Two paths of the signal-side interferometer are independent of those of the idler-side interferometer, and therefore the path of the signal photon can be identified only after the photon has passed through the interferometer with precise timing. In the research reported in Refs. 19 and 20, two signal beams diverging from the downconversion crystal are combined with a beam splitter, and the same is done to the idler beams. These experiments are similar to ours in that the paths of the signal and the idler photons have a one-to-one correlation. There are some two-photon interference experiments in which spatial interference patterns are observed. Ghosh and Mandel directed a signal and an idler beam toward a same area and observed the dependence of the joint detection probability on the detector separation over that area. 7 Our experiment is a spread-out version of theirs: Two signal beams and two idler beams are combined at two different positions, which clearly illustrates the nonlocal nature. Likewise, the experiments of Refs. 19 and 20 can be considered spread-out versions of the experiment of Ref. 8 in which a signal and an idler beam are combined with a beam splitter. In other experiments two-photon spatial interference patterns are observed, with only one double slit inserted into the signal or the idler side for which the detector position is either scanned 36 or fixed. 37 The photon s path through the double slit cannot be identified in these experiments, and the observed phenomena were analyzed from different /98/ $ Optical Society of America

2 C. K. Hong and T. G. Noh Vol. 15, No. 3/March 1998/J. Opt. Soc. Am. B 1193 viewpoints. So far, ours is the only double-slit setup of which we are aware that permits path identification. 2. EXPERIMENT The experimental setup is depicted schematically in Fig. 1. The nm light of an argon-ion laser is used to pump a 20-mm long lithium iodate crystal. The FWHM of the pump beam is 2 mm. Horizontally polarized by type I phase matching, nm signal and idler photons emerge in a cone with 14 vertex angle. A double slit composed of two slits of 2-mm width with their centers separated by 5 mm is placed 90 mm from the center of the crystal. Two signal beams that have passed through the double slit are made to converge with a cylindrical lens of 100-mm focal length. Another cylindrical lens of the same focal length focuses the beams horizontally onto the input end of a 97- m-diameter optical fiber. The distance from the double slit to the first lens is 25 mm, and that from the lens to the input end of the optical fiber is 2.7 m. The output end of the optical fiber is coupled to a detector through a nm interference filter of 10-nm FWHM bandwidth. The same is done to the opposite idler beams. The vertical position of the input end of the signal fiber, y, is scanned and that of the idler fiber, y, is fixed. The output pulses of the signal and the idler detectors and their coincidences are counted simultaneously. Coincidence countings are corrected for the accidental coincidence of uncorrelated photons within an 86-ns resolving time, which is much longer than the 160-fs coherence time of 10-nm-bandwidth light that is due to the limitation of the electronics. The separation and position of the double slit are chosen such that no coincidences occur when the entire area of either the upper or the lower slit is blocked. Therefore coincidences originate only from signal and idler photons passing through opposite slits, which makes it possible to identify the path of the signal photon without disturbing it by the detection of the idler photon. This can be done even before the signal photon reaches the double slit, and we are aware of no other double-slit setup with such a capability. When the lower (upper) slit on the signal side and the upper (lower) one on the idler side are blocked, the coincidence-counting rate is on the average counts per second (c/s) ( c/s) without showing any significant variation Fig. 1. Experimental setup: DS, double slit; L s, cylindrical lenses; M s, mirrors; OF s, optical fibers; IF s, interference filters; D s detectors. Fig. 2. Measured signal photon-counting ( ) and coincidencecounting ( ) rates as functions of (y y ). Error bars for the signal photon-counting rate are smaller than the dot size. The solid curve is the least-squares fit to relation (7). with y. Similar results are obtained when one slit on the idler side is blocked with both slits on the signal side open. Figure 2 is a typical plot of the signal photon and the coincidence-counting rates measured when all four slit areas are open as functions of (y y ). The counting rate of the idler photons, whose detection position is fixed, is c/s. No interference pattern can be seen with the signal photon-counting rates, which means that the two signal beams passing through the double slit are completely incoherent with each other. On the other hand, the interference pattern that appears in the coincidencecounting rates implies that some kind of hidden coherence between them has existed and that it is uncovered by the detection of the idler photons in an appropriate arrangement. It is shown in Section 4 below that these phenomena cannot be described by classical wave theory. Therefore they should be discussed in the light of the Einstein Podolsky Rosen effect QUANTUM-MECHANICAL DESCRIPTION We now analyze the observed phenomena quantum mechanically. The experimental arrangement illustrated schematically in Fig. 3 is similar to those in Refs. 39 and 40. The signal and idler photons can pass through a double slit, whose center coincides with the pump beam, in four areas j ( s, i and j 1, 2) as in Fig. 3(a). If the slits are sufficiently narrow and their separation sufficiently wide, the downconverted fields passing through them can be considered four different single modes. Then the wave vectors of the signal, the idler, and the pump beams satisfy the phase-matching condition k s1 k i1 k s2 k i2 k p, and the downconverted field is prepared in an entangled initial state 41 : 1 2 1, 1 s1, i1 1, 1 s2, i2 ), (1) where 1, 1 sj, ik is a state with one photon in the sj mode and another in the ik mode.

3 1194 J. Opt. Soc. Am. B/Vol. 15, No. 3/March 1998 C. K. Hong and T. G. Noh the idler photon, as within the gray box in Fig. 3(b), only one initial state survives the detection, and the possibility of any interference effect disappears. In this case the probability density p (id) s (y) is given by p (id) s y K â i1 â i1 Ê s y Ê s y â i2 â i2 Ê s y Ê s y K 2 exp ik s1 y vac 2 exp ik s2 y vac 2 ] K, (4) which is the same as p s (y) calculated by Eq. (3). In Eq. (4), â ij is the photon-creation operator of the ij mode and vac is the vacuum state. In the experimental configuration of Fig. 3(c) the idler photons are also detected after they have passed through an identical double slit. The positive frequency part of the idler field operator at y is similarly given by Ê i y Ê i1 y Ê i2 y exp ik i1 y â i1 exp ik i2 y â i2. (5) The joint probability density of detecting a signal photon at y and an idler photon at y is then obtained as p si y, y K Ê s y Ê i y Ê i y Ê s y K 2 exp ik s1 y exp ik i1 y vac Fig. 3. Schematics of the experimental setup. (a) Four beams of the signal and the idler fields selected by a double slit. (b) Signal photon-detection scheme with a double slit. The box outlines the arrangement to identify the path of the signal photon with the idler photon. (c) Two-photon joint detection scheme with two identical double slits. At position y on the detection plane in Fig. 3(b) with the idler side ignored, the positive frequency part of the signal field operator can be expressed as Ê s y Ê s1 y Ê s2 y exp ik s1 y â s1 exp ik s2 y â s2, (2) where â sj is the photon-annihilation operator and k sj is now the y component of the wave vector of the sj mode. The probability density p s (y) of detecting a signal photon at y is given by 42 p s y K Ê s y Ê s y, K 2 exp ik s1 y 1 i1 exp ik s2 y 1 i2 ] 2, K, (3) where K is a normalization constant. No interference term appears in Eq. (3), consistent with the experimental observation. The attachment of the signal photon s path to the two orthogonal states of the idler mode prohibits the appearance of the interference pattern even though the path of the signal photon is not identified. On the other hand, if it is actually identified by the detection of exp ik s2 y exp ik i2 y vac ] 2 K 1 cos k s1 k s2 y y, (6) where K is another normalization constant and k s1 k i1 and k s2 k i2 have been assumed. Now Eq. (6) exhibits an interference pattern. It should be noted that if the initial state of any mode is other than a singlephoton state no interference pattern will appear. One such state is the coherent state that is equivalent to a classical wave. This is another proof in addition to the one given in Section 4 below that the observed phenomena originate from Einstein Podolsky Rosen-type photon photon correlation and that they cannot be described by classical wave theory. Although the probability density given by Eq. (6) is of the correct form, it cannot be discussed in terms of Bell s inequality violation because the inequality is composed of probabilities instead of probability densities. 43 It still remains to be shown that no theory based on local realism can explain such a nonlocal interference pattern. The slits used in our experiment are made wide to allow more photons to pass through them. Therefore the signal and the idler fields are mixtures of many incoherent modes, and the joint probability density given by Eq. (6) should be integrated over k sj ( j 1, 2) with some weighting functions determined by the geometry of the experimental setup. We simply consider them constant and integrate Eq. (6) over k. In addition to this, the finite cross section of the optical fibers should be considered. Then the joint probability density can be approximated as

4 C. K. Hong and T. G. Noh Vol. 15, No. 3/March 1998/J. Opt. Soc. Am. B sin k y y /2 is made and placed such that the correlated signal and p si y, y K 1 V idler photons pass through only the opposite slits. k y y /2 Therefore the fields that pass through the slits on the same side are uncorrelated and cos k 1 k 2 y y, (7) where K is a new normalization constant and k j is the central value of k sj. k is the range of k sj determined by the geometry of the setup, and V reflects the effect of the finite cross section of the optical fibers. The solid curve in Fig. 2 is the least-square fitting of Eq. (7). Its spatial period and V are mm and , and their theoretical values are 0.38 mm and 0.83, respectively. The difference between the coincidence-counting rates of the signal and idler photon pairs passing through two different combinations of opposite slits ( and c/s) also has been accounted for in the estimation of V. The distance between the first zeros of the sinc function depends on the width of the slits through k. It is 2.38 mm (corresponding to Fig. 2) or 4.08 mm when a 2- or a 1-mm-wide slit is used, and its theoretical value is 1.90 or 3.79 mm. 4. CLASSICAL DESCRIPTION We now show why the observed phenomena cannot be described with classical field theory. The classical quantity that corresponds to the joint probability density of detecting photon pairs is the intensity correlation function I(y)I(y ). By substituting complex field v j for annihilation operator â j and ensemble averages for the expectation values we obtain I y I y exp ik s1 y v s1 exp ik s2 y v s2 2 exp ik i1 y v i1 exp ik i2 y v i2 2 v s1 2 v i1 2 v s1 2 v i2 2 v s2 2 v i1 2 v s2 2 v i2 2 exp i i v s1 2 v i1 v i2 exp i i v s1 2 v i1 v i2 exp i i v s2 2 v i1 v i2 exp i i v s2 2 v i1 v i2 exp i s v i1 2 v s1 v s2 exp i s v i1 2 v s1 v s2 exp i s v i2 2 v s1 v s2 exp i s v i2 2 v s1 v s2 exp i s i v s1 v s2 v i1 v i2 exp i s i v s1 v s2 v i1 v i2 exp i s i v s1 v s2 v i1 v i2 exp i s i v s1 v s2 v i1 v i2, (8) where s (k s1 y k s2 y) and i (k i1 y k i2 y ). The fifth to twelfth terms of Eq. (8) should be zero. Otherwise Eq. (8) would exhibit a cosine modulation even if any one of the four fields were zero, which contradicts the fact that no interference effect can be observed with any one of the four slit areas blocked. We now consider the other eight terms. In the experiment the double slit v s1 2 v i2 2 v s2 2 v i1 2 v 2 2, (9) where we have assumed that the field intensities at the four slit areas are all same and equal to v. Each of the last four terms of Eq. (8) cannot be larger than Eq. (9), which can be seen from the Schwarz inequality u w 2 u 2 w 2, (10) and Eq. (8) can be expressed as I y I y v s1 2 v i1 2 v s2 2 v i2 2 O v 2 2. (11) When the pump intensity of the parametric downconversion process is low, the first two terms are proportional to the intensity v 2, which is proved in Ref. 44, and the last term can be ignored. In other words, the counting rate of the genuine coincidences between correlated signal and idler photon pairs is proportional to the intensity and can be much larger than that of the accidental coincidences between uncorrelated signal and idler photons at low intensity. If the coincidence gate time had been equal to the 160-fs coherence time instead of 86 ns while other experimental conditions remained the same, the genuine coincidence-counting rate would have been 140,000 times larger than the accidental coincidence-counting rate in our experiment. Therefore the interference effects that we observed cannot be exhibited by classical fields. In the case of other two-photon interference effects observed with a Mach Zehnder or a Michelson interferometer, 16,18,21,23,24,27,29,31 which were discussed in Section 1, it had been a question whether the interference fringe visibility smaller than 50% could be obtained with classical fields (The interference effects observed in Refs. 19 and 20 were never questioned in this regard.) To answer the question in the negative it is required that the classical fields be ergodic, 45 short pulses, 46 or Gaussian 48 in addition to the condition of low intensity. This is so because those temporal interference effects originate from the energy time correlation of the signal and the idler photons. On the other hand, we use the space momentum correlation in our experiment to see a spatial interference pattern, and any requirement other than low intensity is not necessary for proof of the nonclassicality of small visibility. 5. CONCLUSION We have described a new two-photon interference experiment. The uniqueness of our experiment is that each of two photons is made to pass through a double slit and that the path of one can be identified with that of the other. This makes ours, together with Young s original double-slit experiment, an ideal example of the manifestation of the complementarity principle of quantum mechanics. Unlike for other two-photon interference effects, disproving that a classical field can exhibit an interference pattern of less than 50% visibility requires

5 1196 J. Opt. Soc. Am. B/Vol. 15, No. 3/March 1998 C. K. Hong and T. G. Noh no special conditions, such as ergodicity, other than low intensity. This is so because our experiment is based on spatial correlation rather than on temporal correlation. Although what we have observed cannot be explained by classical field theory, it still remains to be seen that the phenomena are purely quantum-mechanical phenomena that cannot be explained by any local, realistic theory such as hidden-variable theories. The spatial interference pattern is a probability density, and it is not easy to formulate a criterion such as Bell s inequality. ACKNOWLEDGMENTS We thank T. S. Kim, Y. Ha, K. S. Kim, J. Noh, and M. H. Kang for helpful discussions. This study was supported by the Korea Science and Engineering Foundation under project REFERENCES 1. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1965), Vol. III, Chaps C. Cohen-Tannoudji, B. 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