Observation of interference pattern in the intensity correlation of a non-local object using a Hanbury Brown and Twiss-type experiment

I. Vidal, D. P. Caetano, E. J. S. Fonseca and J. M. Hickmann

Optics and Materials Group - Optma Instituto de Física, Universidade Federal de Alagoas Caixa Postal 2051, 57061-970, Maceió, AL, Brazil

E-mail: jmh@optma.org

Received 20 December 2007, accepted for publication 19 March 2008
Published 22 April 2008

Classically, image and interference patterns of objects and apertures are usually observed by intensity distribution measurements of scattered electromagnetic fields where their phases play a fundamental role [1]. Recently, new ways to form images exploiting two-photon or intensity correlations have been demonstrated using parametric downconverted photons [2–5] and pseudothermal light [6–9] as radiation sources. Many of these results can be interpreted with the partial coherence theory, like the original Hanbury Brown and Twiss (HBT) experiment [10, 11]. Likewise, these experiments can seem counterintuitive when we consider that we do not need to invoke the quantum nature of light and sometimes they even apparently contradict it.

Measurements of intensity correlations of two light beams allow us to understand fundamental properties and observe fascinating phenomena of electromagnetic fields and its interactions. In particular, second-order intensity correlations of twin-photons from parametric downconversion (PDC) give rise to an unusual way to observe interference patterns and images. Commonly, these two-photon patterns are referred to as quantum imaging [5]. Such effect was originally attributed to the quantum correlations which are present in the quantum description of the twin-photons state. Basically, the quantum imaging configuration is based on splitting of the down-converted photons into two separated optical paths where an object is placed in one of the paths (test arm). When coincidence detection is performed scanning the detector placed in the other path (reference arm) the image or the interference pattern of the object located at the test arm is observed. Different schemes have been developed [12] and many interesting applications have been proposed and implemented, for example, quantum lithography [13] and quantum holography [14]. Non-local aspects of the quantum correlations of the twin photons have also been exploited for quantum imaging of a distributed object [15]. In this case, different parts of an object are placed in different paths of the photon pairs. The peculiar feature of this non-local configuration is that the spatial superposition of each part of the object forms the entire object and its spatial information is recovered when photon pairs are detected in coincidence.

It is important to distinguish between non-local and local configurations for quantum imaging. In the local configuration, interference pattern formation can be interpreted as a local filtering process of the wave vectors. We can regard the one-photon pattern as the result of superposed random phase interference patterns, which smear out the fringes, as illustrated in fig. 1a. However, a particular localized interference pattern is obtained when coincidence measurements are performed. In this case, the reference photon works as a trigger and the coincidence detection filters out only the wave vectors that are transversely correlated with those wave vectors scattered by the object, in this case a localized double-slit. On the other hand, this local filtering process interpretation may not be applied to a distributed object. Figure 1b illustrates different one-photon interference patterns for each part of the object, none of them being a typical double-slit interference pattern. In this non-local configuration, there is no reference detector which could be used to filter out a localized interference pattern. Actually, each detector works as a trigger to each other. However, the same result of the local configuration is obtained when coincidence measurements are performed.

Apart from the success of the experimental observations and some agreement with theoretical descriptions, a number of papers has been published showing that some of the quantum imaging aspects do not require quantum correlation in the wave vectors of the twin photons. Recently, it was experimentally demonstrated that coincidence image (local configuration) can also be obtained with the same configuration of fig. 1a, using a pseudothermal light source [6–9]. The same filtering explanation works fine for this situation.

A further step in the understanding classical and quantum aspects of these phenomena is to investigate the non-local aspect of interference pattern formation, which was explored only by using entangled photons from PDC [15]. Based on fig. 1b, the different parts of a double-slit, slit and wire, are illuminated by each beam and arranged in such way that if they were spatially superimposed over each other they would form a double-slit. In this work we show that performing a HBT-like experiment with pseudothermal light we can emulate a non-local behavior without quantum correlations, when different objects are placed in front of each detector [16]. The non-local signature appears on the intensity correlation where we observe the interference pattern of the combined objects. Interesting enough, if our experiment were performed with an electron source, we should obtain a negative image of the interference pattern in a similar fashion as in the fermionic HBT experiment [17].

The experimental setup is sketched in fig. 2a and its implementation followed the setup shown in fig. 2b. The pseudothermal light source [18] was built using an argon ion laser, operating at 514 nm, passing through a slow rotating ground glass followed by a 2.5 mm diameter pinhole. The rotating ground glass acts as a spatially incoherent object reducing heavily the transverse coherence length of the laser beam in such a way that it is not possible to identify any visible diffraction pattern due to any object place at the beam path. The light beam is split in two beams by a beam splitter (BS1) and then each beam illuminates two different parts of the distributed object, a 0.2 mm diameter wire and a 0.4 mm × 50 mm single-slit. These objects are placed at the same distance from the beam splitter. The alignment of the objects must be carefully done in order to guarantee that if the wire and the single-slit were spatially superposed a localized Young's double-slit should come up. The scattered beams are combined in a different spot of the second beam splitter (BS2) which is used to guide the beams to the charge-coupled device (CCD) camera. A 190 mm focal length lens L is used to focus the wire and single-slit scattered beams in a CCD camera. The measurement consists in acquiring an image corresponding to the intensity of each scattered beam incident in complete different areas of the CCD, resulting in two 200 × 200 pixels matrices. The intensity correlation between the beams is numerically calculated using these matrices.

As the original HBT experiment [10, 11] we have measured the intensity distributions in each arm and calculated the intensity correlations. The intensity distribution of each scattered beam is shown in fig. 3. These measurements do not produce any visible information of the presence of the single-slit (fig. 3a) and the wire (fig. 3b) in the path of the beams. It means that both beams are incoherent enough to erase any interference pattern due to the objects. When the intensity correlation of these images is calculated, an interference pattern corresponding to a double-slit is obtained, as shown in fig. 4a. In fig. 4b there is shown the intensity correlation corresponding to a vertical line at the center of the interference pattern shown in fig. 4a. Even though the interference pattern does not present a large visibility, it clearly is a double-slit interference pattern. In fig. 4c there is shown the interference pattern, obtained with a coherent light beam, but with the wire and slit in the same beam, forming a real "local" double-slit. This measurement confirms that the result of fig. 4b corresponds to the interference pattern of the entire object.

It is important to note that all measurements were performed detecting the intensity of the scattered beams by each part of the double-slit, losing any direct information about the phase of the field. However, when we calculate the intensity correlation, an interference pattern of a double-slit is obtained. This counterintuitive effect leads us to stress that classical wave vector correlations play a central role in intensity correlation imaging using pseudothermal light beams. Although we agree that some aspects of the quantum correlation of twin photons cannot be mimicked by classical correlation, we have demonstrated that the classically correlated wave vectors can also connect non-local realities. It is very important to point out that the non-local aspect which we are discussing here is only due to the way that the object is distributed and it is not related with any Bell's inequality violation.

To confirm that classical wave vectors correlations can also reproduce non-local aspects of quantum images, we have calculated the intensity correlation I₁(x₁)I₂(x₂), following the usual formalism which has been applied to the local configuration [7, 19]. The fields intensity I₁(x₁) and I₂(x₂) are evaluated in the coordinates x₁ and x₂, respectively. The objects are described by impulse response functions h₁(x₁', x₁) and h₂(x₂', x₂). Based on the experimental situation, let us consider an initial pseudothermal field passing through a 50/50 beam splitter. Two copies E₀(x₁) and E₀(x₂) are produced, each one evaluated in different positions x₁ and x₂. These fields propagate passing through two different objects, whose transmission function is given by the impulse response functions h₁(x₁', x₁) and h₂(x₂', x₂). Between each object and CCD camera we have inserted a lens to produce the far-field distribution of scattered fields in the detection plane. The intensity correlation of these fields is given by

where E₁(x₁) and E₂(x₂) are the fields at the detection plane. In the paraxial approximation the expression for the fields in the detection plane can be written as

where q_j is the transverse wave vector component, f is the focal length of the lenses, is the field after the object in the wave vector domain, and is the Fourier transform of the impulse response functions of each lens. The transmission function for a thin lens is given by

In terms of the impulse response of the objects, the fields immediately after the objects are given by

which, in the wave vector domain, takes the simple form , where is the field immediately before the objects in wave vector domain. Using these expressions and taking into acount that the fields obey the Gaussian statistics [20], eq. (1) takes the form

is the Fourier transform of the function

where k is the wave number of the fields and f is the focal length of the lenses used to produce the far-field distribution at x₁ and x₂. Analyzing the result from eq. (5), we can see that the first term is a constant which gives rise to a background that changes the contrast of the resultant pattern. This term explains the low visibility of the experimental result presented in fig. 4. The second term deserves a special attention. It corresponds to the Fourier transform of the function g(x₁, x₂) (eq. (6)), which is a product of the functions describing the objects and the square modulus of the initial field. Note that these functions depend on the difference of the coordinates evaluated in different spatial regions x₁ and x₂. It means that the last term of intensity correlation (eq. (5)) links the distributed object, due to wave vector correlations of the fields. Therefore, this term is responsible for the so-called non-local aspect of the interference pattern that we have experimentally obtained. It is worth emphasizing that although we are dealing with classical correlations, it is not necessary to invoke quantum correlation to obtain interference pattern in intensity correlation experiments even for the case of a non-local object.

The classical interpretation used for the local configuration [7, 19] can be used for our results. However, a quantum interpretation can be given in terms of interference between two-photon probability amplitudes [21, 22], as pointed out by Fano [23]. Let us consider two light source points S1 and S2 in the ground glass which are detected by two detectors D1 and D2. In our case, the non-local interference pattern comes from of the indistinguishability between two two-photons events. One photon is emitted by S1 and detected by D1 (slit path) and one photon is emitted by S2 and detected by D2 (wire path) or one photon is emitted by S1 and detected by D2 (wire path) and one photon is emitted by S2 and detected by D1 (slit path).

An interesting experiment using fermionic particles can be envisaged based on the quantum description of our experiement. It has been experimentally demonstrated [24] that a fermionic version of the HBT experiment presents anti-bunching in its intensity correlation (anticorrelation) instead of bunching (correlation), which is present in the bosonic version. By performing our experiment with an independent electrons source, a negative image of our results could be obtained and easily understood by the Fano argument.

In conclusion, we have investigated a subtle aspect of the HBT experiment related with quantum imaging. We have experimentally demonstrated an explicit non-local interference phenomenon using a pseudothermal light source. Our results show that transverse entanglement is not necessary even in the case of the so-called non-local quantum imaging. We also presented a quantum explanation for our results based on two-photon probability amplitudes. The interpretations presented in this work belong to a recent discussion about the nature of two-photon interference effects [22, 25, 26]. For the case of second-order interference of a non-local object, both classical and quantum interpretations can be used to describe the interference pattern formation. However, the non-local aspect of our experiment cannot be related with the non-locality present in Bell's inequality experiments, which has no classical analogy.

Acknowledgments

We acknowledge with thanks the support from Instituto do Milênio de Informação Quântica, CAPES, MCT/CNPq, PRONEX/Fapeal, Nanofoton research network, Finep/CTInfra/Proinfra, and ANP-CTPETRO.

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