Role of Entanglement in Two-Photon Imaging

VOLUME 87, NUMBER 12 PHYSICAL REVIEW LETTERS 17SEPTEMBER 2001

Ayman F. Abouraddy, Bahaa E. A. Saleh,* Alexander V. Sergienko, and Malvin C. Teich Quantum Imaging Laboratory, Departments of Electrical & Computer Engineering and Physics, Boston University, 8 Saint Mary’s Street, Boston, Massachusetts 02215-2421 (Received 9 April 2001; published 30 August 2001) The use of entangled photons in an imaging system can exhibit effects that cannot be mimicked by any other two-photon source, whatever the strength of the correlations between the two photons. We consider a two-photon imaging system in which one photon is used to probe a remote (transmissive or scattering) object, while the other serves as a reference. We discuss the role of entanglement versus correlation in such a setting, and demonstrate that entanglement is a prerequisite for achieving distributed quantum imaging.

DOI: 10.1103/PhysRevLett.87.123602 PACS numbers: 42.50.Dv, 42.65.Ky R A quantum two-particle system in an entangled state ex- where dx1 p͑x1͒ ෇ 1. Equation (2) is the familiar rela- hibits effects that cannot be attained with any classically tion describing a coherent optical system [6]. correlated system, no matter how strong the correlation. If the single-photon source is, instead, in a mixed state The difference between entanglement and classical corre- described by a density operator lation lies at the heart of Bell-type experiments. In this ZZ paper, we consider general conﬁgurations for two-photon 0 0 rˆ ෇ dx dx g͑x, x ͒ j1 ͗͘1 0 j , (3) imaging, with the goal of discerning the true role of entan- x x glement in contradistinction to correlation. We identify the R x0, x͒, then the͑ءunique advantage a source of entangled photon pairs has where dx g͑x, x͒ ෇ 1 and g͑x, x0͒ ෇ g over a source of classically correlated photon pairs in this photon probability density is context. Although imaging with entangled photons gener- ZZ ated by spontaneous parametric down-conversion has been (x , x0͒ , (4͑ءp͑x ͒ ~ dx dx0 g͑x, x0͒h͑x , x͒h proposed [1] and examined in some simple settings [2,3], it 1 1 1 has not been clear whether such experiments may, in prin- ciple, be reproduced using a classically correlated source. which is the familiar equation describing a partially co- Single-photon imaging.—To set the stage for our analy- herent optical system [6]. It therefore follows that the be- sis of two-photon imaging systems, we begin with a brief havior of an optical system with a single-photon source in summary of the basic equations governing single-photon an arbitrary mixed state is analogous to that of the same imaging. It is well known [4] that a single photon travel- system illuminated with partially coherent light. The in- ing through an optical system exhibits all the phenomena coherent limit is attained when g͑x, x0͒ ෇ g͑x͒d͑x 2 x0͒ of diffraction, interference, and imaging that are familiar whereas the coherent limit emerges when g is factorizable (x0͒, whereupon Eq. (2͑ءin classical optics. One needs only to repeat the single- in the form g͑x, x0͒ ෇ w͑x͒w photon experiment and accumulate observations over a is recovered. sufﬁciently large ensemble. Consider, for example, a thin planar single-photon source described by the pure state Z jC͘ ෇ dx w͑x͒ j1x͘ , (1) R ͘ ෇ 1 ik?x ͘ where j1x ͑2p͒2 dk e j1k is a position represen- tation of the single-photon state in terms of the familiar Fock state j1k͘ of the mode k, and theR state probability amplitude w͑x͒ is normalized such that dxjw͑x͒j2 ෇ 1. Assume now that this state is transmitted through a linear optical system, described by an impulse response function h͑x1, x͒, where x and x1 are the transverse coordinates on the input and output planes, respectively, as illustrated in FIG. 1. Single-photon imaging. A single-photon source S Fig. 1. If a photon arrives in the output plane, the proba- emits a photon from position x and sends it through an optical system described by its impulse response function, h͑x , x͒. bility density of its registrationÇ at a positionÇ x1 is [5] 1 Z 2 A scanning single-photon detector D detects the photons and ͑ ͒ p͑x ͒ ~ dx w͑x͒h͑x , x͒ , (2) thus measures the probability density of photon arrivals, p x1 . 1 1 An unknown object (shaded region) is imbedded in the system.

Two-photon imaging.—We now consider a planar at the output of system 1 and a photon at any location source that emits light in the pure two-photon state ͑2` , x2 ,`͒ at the output of system 2. We call this ͑ ͒ ZZ the marginal probability density p1 x1 [8]. We similarly 0 0 ͑ ͒ ͑ ͒ jC͘ ෇ dx dx w͑x, x ͒ j1x,1x0 ͘ , (5) define p2 x2 and p2 x2 . We proceed to demonstrate that, not RR remarkably, these two probability densities are always 0 0 2 identical. where dx dx jw͑x, x ͒j ෇ 1 and j1 ,1 0 ͘ is the two- x x ͑ ͒ ෇ photon state at transverse coordinates x and x0. For full The single-photon probability density, pj xj , j 1, 2, generality, we assume that the emitted photons are trans- is obtained by taking the trace over the subspace of the mitted through different linear optical systems with im- other photon. Each photon, considered separately from the 0 other, is in fact in a mixed state described by the density pulse response functions h1͑x1, x͒ and h2͑x2, x ͒, where x and x are transverse coordinates on the output planes operator provided in Eq. (3): 1 2 ZZ of the two systems, as depicted in Fig. 2. There are now 0 0 ෇ ͑ ͒ j ͗͘ 0 j ෇ many options for the placement of objects as well as many rˆ j dx dx gj x, x 1x 1x , j 1, 2 , (7) R 00 0 ء observation schemes. A single object may be placed in 0 00 00 where gR1͑x, x ͒ ෇ dx w͑x, x ͒w ͑x , x ͒ and 0͒ 00 ͑ء ͒ either system, or objects may be placed in both systems. ͑ 0͒ ෇ 00 ͑ 00 The photon coincidences at x and x may be recorded, g2 x, x dx w x , x w x , x . The probabil- 1 2 ity density of a photon registration at the output of system yielding the joint probability density p͑x1, x2͒ [5] Ç Ç j is then given by ZZ 2 ZZ ͑ ͒ 0 ͑ 0͒ ͑ ͒ ͑ 0͒ 0͒ ͑ء ͒ ͑ p x1, x2 ~ dx dx w x, x h1 x1, x h2 x2, x . ͑ ͒ 0 ͑ 0͒ pj xj ~ dx dx gj x, x hj xj, x hj xj, x , (6) j ෇ 1, 2 . (8) This function is the fourth-order correlation function (also ͑2͒ called the coincidence rate) G ͑x1,x2͒ [5,7]. This expression is similar to Eq. (4) for a single photon in Two distinct single-photon probability densities.—We a mixed state. It follows that the optics of a single photon now consider two distinct single-photon probability den- from a two-photon source that is in a pure state exhibits sity functions associated with this two-photon system. The the behavior of a partially coherent system. first is the probability density of observing a photon at In contrast to Eq. (8), the marginal probability density x1, regardless of whether the other photon is detected (or of detecting a photon at detector D1 is given by even whether there is another photon). This is called Z ͑ ͒ ͑ ͒ ෇ ͑ ͒ the single-photon probability density p1 x1 . The sec- p1 x1 dx2 p x1, x2 . (9) ond is the probability density of observing a photon at x1 This is the probability of observing one photon at x1, at the output of system 1, and another at the output of system 2 at any location. The detector at system 2 may then be termed a “bucket” detector since it does not reg- ister the arrival location of the photon. In this conception, the bucket detector can serve as a “gating” signal for a scanning detector at the output of system 1. The marginal ͑ ͒ probability density p2 x2 may be similarly defined and measured. Based on classical probability theory one would intu- ͑ ͒ ͑ ͒ itively expect that pj xj would be equal to pj xj . This is not always the case, however. Indeed, in some situa- ͑ ͒ tions, measurement of pj xj at the output of one system can be used to extract information about an object placed in the other system. Probability densities for a nonentangled source.—Let us examine a few special cases. We consider a nonentangled two-photon source in which the state probabil- 0 0 FIG. 2. Two-photon imaging. A two-photon source S emits ity amplitude is factorizable, i.e., w͑x, x ͒ ෇ w1͑x͒w2͑x ͒. one photon from point x and sends it through the system The joint probability density in this case is also factor- ͑ ͒ h1 x1, x with an unknown object imbedded in the system, and izable. There is nothing to be gained by measuring the joint emits the other from x0 and sends it through h ͑x , x0͒.Two 2 2 probability density (coincidence rate) since all informa- scanning detectors D1 and D2 record the singles, p1͑x1͒ and p2͑x2͒, and the coincidence rate p͑x1, x2͒. The shaded region tion is contained in the single-photon probability densities is an imbedded object. (singles rates). The photon arrivals are independent, and

123602-2 123602-2 VOLUME 87, NUMBER 12 PHYSICAL REVIEW LETTERS 17SEPTEMBER 2001 each is governed by a coherent imaging system [5,9]. the crystal thickness eventually leads to the limit of the Moreover, factorizable state. Distributed quantum imaging.—To demonstrate the p ͑x ͒ ෇ p ͑x ͒, j ෇ 1, 2 , (10) j j j j utility of measuring the marginal probability density ͑ ͒ so that this factorizable state, therefore, does not permit p2 x2 , consider the following scenario. Let system 1 the transfer of information in one system to the other. comprise a scattering object (see Fig. 2). The scattered Probability densities for an entangled source.— radiation impinges on detector D1. Such a system cannot Consider now an entangled [10] two-photon source by itself yield an image of the spatial distribution of the 0 described by the state probability amplitude w͑x, x ͒ ෇ scattering object. However, if D1 is a bucket detector that 0 w͑x͒d͑x 2 x ͒, in which case the two-photon state is gates the photon arrival registered by scanning D2, one is Z able to form a high quality image using this two-photon jC͘ ෇ dx w͑x͒ j1x,1x͘ , (11) scheme, as long as the scattering object is illuminated by one photon at a time. Using SPDC, simple experiments where the reduced densityR operators of the individual pho- along these lines have been carried out [2,3]. tons are rˆ ෇ rˆ ෇ dxjw͑x͒j2j1 ͗͘1 j, so that However, the origin of this distributed quantum- 1 Ç2 x x Ç Z 2 imaging phenomenon has not been adequately set forth p͑x1, x2͒ ~ dx w͑x͒h1͑x1, x͒h2͑x2, x͒ , (12a) heretofore. Indeed, it was stated in Ref. [3] that “it is possible to imagine some type of classical source that Z could partially emulate this behavior.” To obtain a better 2 2 pj͑xj͒ ~ dxjw͑x͒j jhj͑xj, x͒j , j ෇ 1, 2 , (12b) understanding of this effect we consider a two-photon source in an arbitrary mixed state. Can similar results be ZZ obtained by employing a two-photon source that exhibits ?classical statistical correlations but not entanglement 0͒ ͑ء ͒ ͑ 0 ͒ ͑ pj xj ~ dx dx w x w x That is, can distributed quantum imaging be achieved without entanglement? 0͒ ͑ء ͒ ͑ 0͒ ͑ 3 gk x, x hj xj, x hj xj, x , To answer this question we take a mixed state that exhibits the strongest possible classical correlations, i.e., one j, k ෇ 1, 2, j fi k , (12c) for which R Z ෇ ͑ ͒ ͗͘ 0 00 ء 00 00 0 with gk͑x, x ͒ ෇ dx hk͑x , x͒hk͑x , x ͒, k ෇ 1, 2. Evi- rˆ dx g x j1x,1x 1x,1xj , (14) ͑ ͒ ͑ ͒ ෇ dently pj xj fi pj xj , j 1, 2. Whereas the expression R in Eq. (12b) is similar to that for an incoherent optical sys- with dx g͑x͒ ෇ 1. This state represents a superposi- tem, Eq. (12c) is similar to that for a partially coherent tion of photon-pair emission probabilities from various ͑ ͒ locations x. [In contrast, the entangled state in Eq. (11) optical system. Moreover, measurement of p1 x1 con- tains information about the system function h2͑x2, x͒ via represents a superposition of probability amplitudes.] The 0 the function g2͑x, x ͒. density operatorsR of each photon taken individually are The most accessible entangled two-photon source is rˆ 1 ෇ rˆ 2 ෇ dx g͑x͒ j1x͗͘1xj, which are identical to based on the process of spontaneous parametric down- those of the entangled source if g͑x͒ ෇ jw͑x͒j2. In this conversion (SPDC) from a nonlinear crystal [11]. For case, then, Z a monochromatic pump, assuming the down-converted ͑ ͒ ͑ ͒ ͑ ͒ 2 ͑ ͒ 2 beams are filtered with narrow band spectral filters, the p x1, x2 ~ dx g x jh1 x1, x j jh2 x2, x j , (15a) state probability amplitude for SPDC is given by [5] Z Z 2 0 00 00 00 0 00 pj͑xj͒ ~ dx g͑x͒ jhj ͑xj, x͒j , j ෇ 1, 2, (15b) w͑x, x ͒ ~ dx Ep͑x ͒j͑x 2 x , x 2 x ͒ , (13) Z where E ͑x͒ is the pump field at the input of the crystal ͑ ͒ ͑ ͒ ͑ ͒ 2 p pj xj ~ dx gk x jhj xj, x j , and j͑x, x0͒ is a phase-matching function that depends on (15c) the crystal parameters. In the limit j͑x, x0͒ ! d͑x͒d͑x0͒, j, k ෇ 1, 2, j fi k , 0 0 the state function reduces to w͑x, x ͒ ෇ Ep͑x͒d͑x 2 x ͒, R ͑ ͒ ෇ ͑ ͒ 0 ͑ 0 ͒ 2 ෇ which corresponds to an entangled state. This can be where gk x g x dx jhk x ,x j , k 1, 2. The re- achieved by reducing the thickness of the crystal. In this sult in Eq. (15b) is similar to that in Eq. (12b), i.e., that of limit, the joint probability density, the single-photon proba- an incoherent optical system, since the reduced density op- bility density, and marginal probability density, which erators are identical. The distinction is that Eq. (15c) has are given by Eqs. (6), (8), and (9), respectively, yield the form of an incoherent optical system whereas Eq. (12c) Eqs. (12a)–(12c). The spatial coherence properties of has the form of a partially coherent optical system. For this source have been studied extensively [12]. Increasing the special case of a shift-invariant (isoplanatic) system,

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in which case hj͑xj, x͒ is a function of xj 2 x, then tion, by the David & Lucile Packard Foundation, and by ͑ ͒ ͑ ͒ ෇ gk x ~gx , k 1, 2, and Eq. (15c) becomes identical the Center for Subsurface Sensing and Imaging Systems to Eq. (15b). As a result of Eqs. (15a)–(15c), the dis- (CenSSIS), an NSF Engineering Research Center. tributed quantum-imaging scheme truly requires entanglement in the source and cannot be achieved using a classical source with correlations but without entanglement. There is, of course, a continuous transition between these two extremes, so that partial distributed quantum imaging is *Electronic address: [email protected] possible as entanglement enters the mixed state. [1] B. E. A. Saleh, S. Popescu, and M. C. Teich, in Proceed- ings of the Ninth Annual Meeting of the IEEE Lasers and To appreciate the consequences of the imaging formulas Electro-Optics Society, edited by P. Zory (IEEE, Pis- given in Eqs. (4), (12a)–(12c), (15a)–(15c), consider N cataway, NJ, 1996), Vol. 1, pp. 362–363, Catalog weak scatterers embedded in one of the optical systems (h No. 96CH35895. in the single-photon case or h1 in the two-photon case). [2] D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and The impulse response function of the system including the Y. H. Shih, Phys. Rev. Lett. 74, 3600–3603 (1995); scatterers is then given by T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y.H. Shih, Phys. Rev. A 53, XN ͑ ͒ ෇ ͑ ͒ ͑ j͒͑ ͒ 2804–2815 (1996). h x1, x h0 x1, x 1 ´jh x1, xj , (16) [3] T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. ෇ j 1 Sergienko, Phys. Rev. A 52, R3429–R3432 (1995). [4] G. I. Taylor, Proc. Cambridge Philos. Soc. 15, 114–115 where h0͑x1, x͒ represents the system in the absence of the scatterers, h͑ j͒͑x , x ͒ is the impulse response function (1909); P. A. M. Dirac, The Principles of Quantum Mechan- 1 j ics (Oxford University, Oxford, 1958). of the system following a scatterer at location x , and ´ j j [5] B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. is the strength of scatterer j. We assume that the system in Teich, Phys. Rev. A 62, 043816 (2000). which the scatterers are contained is inaccessible. By sub- [6] M. Born and E. Wolf, Principles of Optics (Cambridge stituting Eq. (16) in Eqs. (4), (12a)–(12c), (15a)–(15c), University, New York, 1999), 7th ed.; B. E. A. Saleh and it turns out that in the one-photon case the image of the M. C. Teich, Fundamentals of Photonics (Wiley, New York, scatterers is in general blurred by a distribution that de- 1991). ͑ j͒ pends on h ͑x1, xj͒, as expected. In the entangled two- [7] R. J. Glauber, Phys. Rev. 130, 2529–2539 (1963). 0 photon case, however, one can always select an h2͑x2, x ͒ [8] A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. ͑j͒ Sergienko, and M. C. Teich, Phys. Rev. A 63, 063803 (see Fig. 2) such that the combination with h ͑x1, xj͒ yields a diffraction limited imaging system for each scat- (2001). tering plane. The correlated two-photon case, on the other [9] L. Mandel, Opt. Lett. 16, 1882–1883 (1991). [10] E. Schrödinger, Naturwissenschaften 23, 807–812 (1935); hand, offers no such beneﬁt. 23, 823–828 (1935); 23, 844–849 (1935) [translation: J. D. Conclusion.—We have considered a distributed Trimmer, Proc. Am. Philos. Soc. 124, 323–338 (1980); quantum-imaging system in which one photon is used to reprinted in Quantum Theory and Measurement, edited by probe a remote transmissive or scattering object, while J. A. Wheeler and W. H. Zurek (Princeton University Press, the other serves as a reference. A high-spatial-resolution Princeton, NJ, 1983)]. detector scans the arrival position of the reference photon, [11] D. N. Klyshko, Photons and Nonlinear Optics (Nauka, while a bucket detector (which need have no spatial Moscow, 1980) (translation: Gordon and Breach, New resolution) registers the photon scattered by the object. York, 1988). The scanned reference detector is gated by the photoevent [12] A. Joobeur, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A registered by the bucket detector, and an image is formed. 50, 3349–3361 (1994); A. Joobeur, B. E. A. Saleh, T. S. We have shown that if the two-photon source is in an Larchuk, and M. C. Teich, Phys. Rev. A 53, 4360–4371 (1996); M. H. Rubin, Phys. Rev. A 54, 5349–5360 (1996); entangled state, the imaging is, in general, partially B. E. A. Saleh, A. Joobeur, and M. C. Teich, Phys. Rev. A coherent, and can possibly be fully coherent. On the other 57, 3991–4003 (1998); B. M. Jost, A. V. Sergienko, A. F. hand, if the source emits unentangled, but classically Abouraddy, B. E. A. Saleh, and M. C. Teich, Opt. Exp. correlated photon pairs, then the imaging is incoherent. 3, 81–88 (1998); A. F. Abouraddy, B. E. A. Saleh, A. V. We thank Zachary Walton for valuable discussions. Sergienko, and M. C. Teich, J. Opt. B: Quantum Semiclass. This work was supported by the National Science Founda- Opt. 3, S50–S54 (2001).

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