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Role of entanglement in two-photon imaging

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.

attained with any classically correlated system, no matter how strong the correlation. The difference between entanglement and classical correlation lies at the heart of Bell-type experiments. In this paper, we consider general configurations for two-photon imaging, with the goal of discerning the true role of entanglement in contradistinction to correlation. We identify the unique advantage a source of entangled photon pairs has over a source of classically correlated photon pairs in this context. Although imaging with entangled photons generated by spontaneous parametric down-conversion has been proposed [1] and examined in some simple settings [2,3], it has not been clear whether such experiments may, in principle, be reproduced using a classically correlated source.
Single-photon imaging.-To set the stage for our analysis of two-photon imaging systems, we begin with a brief summary of the basic equations governing single-photon imaging. It is well known [4] that a single photon traveling through an optical system exhibits all the phenomena of diffraction, interference, and imaging that are familiar in classical optics. One needs only to repeat the single-photon experiment and accumulate observations over a sufficiently large ensemble. Consider, for example, a thin planar single-photon source described by the pure state where is a position representation of the single-photon state in terms of the familiar Fock state k 1 of the mode k , and the state probability amplitude is normalized such that ( ) where and are transverse coordinates on the output planes of the two systems, as depicted in Fig. 2. There are now many options for the placement of objects as well as many observation schemes. A single object may be placed in either system, or objects may be placed in both systems. The photon coincidence at and may be recorded, yielding the joint probability density This function is the fourth-order correlation function (also called the coincidence rate) [5,7].
Two distinct single-photon probability densities.-We now consider two distinct single-photon probability density functions associated with this two-photon system. The first is the probability density of observing a photon at , regardless of whether the other photon is detected (or even whether there is another photon). This is called the singlephoton probability density . The second is the probability density of observing a photon at at the output of system 1 and a photon at any location ( ) at the output of system 2. We call this the marginal probability density ( ) where The probability density of a photon registration at the output of system is then given by This expression is similar to Eq. (4) for a single photon in a mixed state. It follows that the optics of a single photon from a two-photon source that is in a pure state exhibits the behavior of a partially coherent system.
In contrast to Eq. (8), the marginal probability density of detecting a photon at detector D 1 is given by This is the probability of observing one photon at , 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 register 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  Probability densities for a non-entangled source.-Let us examine a few special cases. We consider a non-entangled two-photon source in which the state probability amplitude is factorizable, i.e., . The joint probability density in this case is also factorizable. There is nothing to be gained by measuring the joint probability density (coincidence rate) since all information is contained in the single-photon probability densities (singles rates). The photon arrivals are independent, and each is governed by a coherent imaging system [5,9]. Moreover, so that this factorizable state, therefore, does not permit the transfer of information in one system to the other.
Probability densities for an entangled source.-Consider now an entangled [10] two-photon source described by the state probability amplitude ( ) ( ) ( x , in which case the two-photon state is where the reduced density operators of the individual photons are Whereas the expression in Eq. (12b) is similar to that for an incoherent optical system, Eq. (12c) is similar to that for a partially coherent optical system. Moreover, measurement of ( )  The most accessible entangled two-photon source is based on the process of spontaneous parametric down-conversion (SPDC) from a nonlinear crystal [11]. For a monochromatic pump, assuming the down-converted beams are filtered with narrowband spectral filters, the state probability amplitude for SPDC is given by [5] ( ) ( ) ( where is the pump field at the input of the crystal and ( ) is a phase-matching function that depends on the crystal parameters. In the limit ( ) ( ) (x , which corresponds to an entangled state. This can be achieved by reducing the thickness of the crystal. In this limit, the joint probability density, the single-photon probability density, and marginal probability density, which are given by Eqs. (6), (8), and (9) respectively, yield Eqs. (12a) -(12c).
The spatial coherence properties of this source have been studied extensively [12].
Increasing the crystal thickness eventually leads to the limit of the factorizable state. , consider the following scenario. Let system 1 comprise a scattering object (see Fig. 2). The scattered radiation impinges on detector D 1 .
Such a system cannot by itself yield an image of the spatial distribution of the scattering 10/13/01 object. However, if D 1 is a bucket detector that gates the photon arrival registered by scanning D 2 , one is able to form a high quality image using this two-photon scheme, as long as the scattering object is illuminated by one photon at a time. Using SPDC, simple experiments along these lines have been carried out [2,3].
However, the origin of this distributed quantum-imaging phenomenon has not been adequately set forth heretofore. Indeed, it was stated in Ref. [3] that "it is possible to imagine some type of classical source that could partially emulate this behavior". To obtain a better understanding of this effect we consider a two-photon source in an arbitrary mixed state. Can similar results be obtained by employing a two-photon source that exhibits classical statistical correlations but not entanglement? That is, can distributed quantum-imaging be achieved without entanglement?
To answer this question we take a mixed state that exhibits the strongest possible classical correlations, i.e., one for which with . This state represents a superposition of photon-pair emission probabilities from various locations . (In contrast, the entangled state in Eq. (11) represents a superposition of probability amplitudes.) The density operators of each photon taken individually are , which are identical to those of the entangled source if ( ) . In this case, then, 10/13/01 . The result in Eq. (15b) is similar to that in Eq. (12b), i.e., that of an incoherent optical system, since the reduced density operators are identical. The distinction is that Eq. (15c) has the form of an incoherent optical system whereas Eq. (12c) has the form of a partially coherent optical system. For the special case of a shift-invariant (isoplanatic) system, in which case ( ) , and Eq. (15c) becomes identical to Eq. (15b). As a result of Eqs. (15a-c), the distributed 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 possible as entanglement enters the mixed state.
To appreciate the consequences of the imaging formulas given in Eqs. (4), (12ac), and (15a-c), consider weak scatterers embedded in one of the optical systems ( in the single-photon case or in the two-photon case). The impulse response function of the system including the scatterers is then given by: (see Fig. 2) such that the combination with yields a diffraction limited imaging system for each scattering plane. The correlated-two-photon case, on the other hand, offers no such benefit.
Conclusion.-We have considered a distributed quantum imaging system in which one photon is used to probe a remote transmissive or scattering object, while the other serves as a reference. A high-spatial-resolution detector scans the arrival position of the reference photon, while a bucket detector (which need have no spatial resolution) registers the photon scattered by the object. The scanned reference detector is gated by the photoevent registered by the bucket detector, and an image is formed. We have shown that if the two-photon source is in an entangled state, the imaging is, in general, partially coherent, and can possibly be fully coherent. On the other hand, if the source emits unentangled, but classically-correlated photon pairs, then the imaging is incoherent.