Entanglement of orbital angular momentum for the

journal of modern optics, 2002, vol. 49, no. 5/6, 777 ±785

Entanglement of orbital angular momentum for the signal and idler beams in parametric down-conversion MILES PADGETT, JOHANNES COURTIAL, L. ALLEN, Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK SONJA FRANKE-ARNOLD and STEPHEN M. BARNETT Department of Physics and Applied Physics, University of Strathclyde, Glasgow G4 0NG, UK (Received 5 September 2001 ) Abstract. We calculate the anticipated correlation between measurements of the orbital angular momentum of the signal and idler beams for parametric down-conversion. These calculations apply to the experiments where the orbital angular momentum state is measured by the use of computer-generated holograms. Displacement of these holograms with respect to the beam axis allows the measurement of superpositions of Laguerre±Gaussian modes. The correlations between such superposition modes of the signal and idler beams show their entanglement and could be used for Bell-type tests of nonlocality.

Over the past 20 years many experiments have been performed to study polarization entanglement of photon pairs. The experiments of Aspect and coworkers [1] in the early 1980s are generally regarded as having provided the ®rst experimental support for nonlocal interpretations of quantum mechanics. Since that time, other experiments on photon pairs have successfully shown entanglement in their polarization states [2], their arrival times [3, 4] and their transverse position [5]. In addition to investigating the interpretation of quantum mechanics, entanglement also plays important roles in quantum cryptography [6] and teleportation [7]. Recently, Mair et al. [8] reported the ®rst experiments to investigate the entanglement of the orbital angular momentum of photon pairs using computergenerated holograms to measure the orbital angular momentum of individual photons. While the polarization of light, related to the spin angular momentum, can be characterized by two orthogonal states, the orbital angular momentum is higher dimensional [9]. The multi-dimensional entanglement of orbital angular momentum states could ®nd interesting applications in quantum cryptography and communication. This paper seeks to apply our recently derived theoretical treatment [10] of the correlation between orbital angular momentum states to experiments of their type. We also suggest more detailed experiments to observe the entanglement of the Journal of Modern Optics ISSN 0950±0340 print/ISSN 1362±3044 online # 2002 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/09500340110108594

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orbital angular momentum and to analyse the possibility to test the non-locality of multi-dimensional variables via Bell-type measurements. Although early experiments on entanglement used two-photon decay processes to produce the photon pairs [11], most recent experiments [12] have favoured parametric down-conversion as this results in a greater photon ¯ux and, subsequently, more accurate measurements [13, 14]. In type-I and type-II degenerate down-conversion, the signal and idler photons are emitted with parallel and orthogonal polarizations, respectively. In such systems the polarization direction of one photon is completely correlated with that of the other. However, the polarization states are generally not entangled, as the tensor nature of the nonlinear susceptibility dictates both signal and idler polarizations with respect to the crystal orientation. In order to entangle the polarization states of the signal and idler beams, they have to be subsequently combined at a beam splitter [15] or, in case of type II down-conversion, collected at the overlapping positions of the emission cones [2]. Each of the two output beams then contains a superposition of horizontally and vertically polarized states. It is now well established that, in addition to the spin angular momentum, light beams can also possess an orbital angular momentum arising from helical phase fronts [16]. Such beams, of which Laguerre±Gaussian modes are one example, are characterized by an azimuthal phase term exp …il¿†. The helical phase fronts give an azimuthal component to the wave vector, k· ¢ ¿^ ˆ l·h=r;

…1†

where ¿^ denotes the azimuthal unit vector. This azimuthal component to the wave vector results in an orbital angular momentum of l· h per photon. In addition to l, which is referred to as the azimuthal mode index, Laguerre±Gaussian modes are characterized by the radial mode index, p, which describes the number of concentric rings in the intensity distribution. Beams with helical phase fronts can readily be produced using computergenerated holograms [17], which, in their simplest form, consist of the interference pattern between a plane-wave input beam and a beam with the chosen azimuthal phase dependence. The result is a distorted diÄraction grating with an l-fold dislocation on the beam axis (see ®gure 1). When illuminated with a fundamental Gaussian mode, the ®rst-order diÄracted beam has l intertwined phase fronts.

Figure 1.

Blazed hologram designs for producing/detecting helically phased beams with l ˆ 2 and l ˆ 1 respectively.

Entanglement of orbital angular momentum for signal and idler beams

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When the hologram is fabricated as a phase grating with the appropriate blaze conversion, eÅciencies of over 50% are readily achievable. Such holograms can also be used to detect helical phase fronts. When a hologram with an lholo -fold dislocation is illuminated by a beam with azimuthal index lbeam , the ®rst-order diÄracted beam has an azimuthal mode index lholo ‡ lbeam . If lholo ˆ ¡lbeam , then the resulting beam has planar wavefronts and a large on-axis intensity, which can be deduced from the transmission of the beam through a pinhole or ®bre aperture. We have recently shown theoretically, in parametric down-conversion that the orbital angular momentum of the pump signal and idler beams are related by [10]. l3 ˆ l2 ‡ l1 ;

…2†

where the signal, idler and pump ®elds are ®elds 1, 2 and 3, respectively. Indeed, for second harmonic generation, where a single input beam of azimuthal index l provides ®elds 1 and 2, the output beam, 3, is observed to possess an azimuthal index 2l [18]. For upconversion, equation (2) can be understood to arise from stringent phase matching. The phase-matching condition, if rigorously applied, ensures that the three wave vectors are related by k·3 ˆ k·2 ‡ k·1 :

…3†

By considering the azimuthal component we can write k·3 ¢ ¿^ ˆ k·2 ¢ ¿^ ‡ k·1 ¢ ¿^:

…4†

As this holds at all radii within the beam, we can readily substitute equation (1) to again obtain equation (2). In the case of down-conversion, it is insightful to consider the phase relationship between the three ®elds that, under conditions of perfect phase matching, is [19] p …5† ’3 ¡ ’2 ¡ ’1 ˆ ¡ 2 For down-conversion, the pump has an azimuthal phase structure of ’3 …¿† ˆ lpump ¿ ‡ Constpump :

…6†

Hence, although the azimuthal phase structures of the signal and idler ®eld are not uniquely de®ned, their phase relation is ®xed. That is, measurement of the phase structure of either signal or idler determines the phase structure of the other. Note that this entanglement of the azimuthal phase structure and resulting orbital angular momentum follows directly from the phase-matching condition. There is no dependence on the tensor form of the susceptibility, nor do the signal and idler beams need to be combined at a beam splitter. In their recent experiment [7], Mair et al. used appropriate holograms to measure simultaneously the orbital angular momentum of the signal and idler ®elds. A schematic of this type of experiment is shown in ®gure 2. They observed a high coincidence when and only when equation (2) is satis®ed. Although conservation of orbital angular momentum shows a strong correlation between the signal and idler photons, it does not establish their entanglement. Indeed, correlations of this kind can also occur in the classical world. Entanglement, however, is a feature of quantum mechanics and can be tested via Bell’s inequality. Applied to polarization entanglement, Bell’s inequality is not violated for the

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Figure 2.

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Experimental con®guration for the simultaneous measurement of the orbital angular momentum state of the signal and idler beams.

coincidence measurement of parallel or orthogonal polarization directions of the signal and idler mode, but shows the strongest violation for measurements performed with an angle between the polarizers of 22.58 and 67.58. In the case of orbital angular momentum, correlation measurements of `pure l modes’ are equivalent to polarization coincidence rates measured for parallel and crossed polarizers only. Clearly, for measurements of orbital angular momentum, rotational alignment of the hologram plays no role. Instead, it is useful to think of the parallel and orthogonal settings of the polarizers as producing measurements between two parallel or orthogonal states. Intermediate settings of the polarizers give measurements between superpositions of orthogonal states. For a hologram, this corresponds to one designed to generate/detect a superposition of two or more helically phased beams with diÄering mode indices. Although speci®c mode combinations of modes can be obtained by using appropriately designed holograms [20] some degree of adjustment is desirable. A variable phase mode converter [21], which is analogous to a Babinet compensator, can achieve this, but only by means of superpositions of modes of the same mode order. An alternative approach, identi®ed by Mair [22], for obtaining superpositions of modes with diÄering indices and diÄering mode order is to displace the hologram with respect to the optical axis. For example, ®gure 3 shows the calculated beam pro®les and mode composition resulting from a hologram designed to produce an l ˆ 2 beam as a function of hologram displacement from the beam axis. When the hologram is aligned on the beam axis, the beam can be decomposed into a series of Laguerre±Gaussian modes with the same l index. The range of p indices depends upon the size of the beam waist chosen for the beam composition. If the beam waist is chosen correctly, over 90% of the energy will be


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Figure 3. Mode composition resulting from displacement of an l ˆ 2 hologram away from the beam axis (note the contribution is summed over all p, but in all cases p ˆ 0 is the dominant component).

contained in the single ringed p ˆ 0 mode. As the hologram is displaced from the beam axis the contribution from modes with lower l indices increases and in the extreme, for displacements large compared with the illumination beam size, the l ˆ 0, p ˆ 0 mode dominates. In a recent work [10], we considered theoretically the correlation between orbital angular momentum states. We have shown that the normalized coincidence probability for detection of the signal and idler modes with complex amplitudes FS …r; ¿† and Fi …r; ¿†, if the phase matching condition is assumed to be rigorously applied, is P…FS …r; ¿†; Fi …r; ¿††

≠… … ≠2 ≠ ≠ ≠ F¤ …r; ¿†F¤ …r; ¿†Fp …r; ¿†r dr d¿≠ S i ≠ ≠ ˆ ≥… … ´1=2 ≥… … ´1=2 : 2 2 ¤ ¤ jFS …r; ¿†Fp …r; ¿†j r dr d¿ jFi …r; ¿†Fp …r; ¿†j r dr d¿

…7†

For the purpose of this present work, it is informative to examine the numerator of this equation in isolation. For modes which are cylindrically symmetric in intensity, we can separate out the amplitude and phase terms to obtain ≠… ≠2 ≠ ≠ P / ≠≠ exp …¡ilS ¿† exp …¡ili ¿† exp …ilp ¿† d¿≠≠ : …8†

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For integer values of l, we see reassuringly that when lp 6ˆ lS ‡ li , equation (8) integrates to zero. In the remainder of this work we numerically evaluate equation (7) to derive the coincidence probability for the signal and idler beams as measured using various holograms. Speci®cally we examine how the normalized coincidence probability changes as a function of hologram displacement from the beam axis. Our numerical model calculates the mode functions FS …r; ¿† and Fi …r; ¿† by `back propagating’ the light detected through the pinhole, the focusing lens and displaced hologram to the plane of the crystal. The free space propagation algorithm is based on a Fourier decomposition of the beam into its constituent plane waves [23, 24] where the eÄect of phase elements such as holograms and lenses can be calculated by a position-dependent multiplication of the complex beam amplitude. Figure 4 shows the calculated normalized coincidence probability given in equation (7) for various combinations of signal and idler holograms as a function of the hologram oÄset. Figure 4 (a) is calculated for lsignal holo: ˆ 2, lidler holo: ˆ ¡2 and lpump ˆ 0. With both holograms aligned along the beam axis, the correlation between the pure signal and idler modes are measured. As for these modes the orbital angular momentum is conserved according to equation (2), the resulting coincidence rate is high. A displaced signal hologram, instead, detects superpositions of Laguerre±Gaussian modes. In order to obtain an equally high coincidence rate, the idler hologram has to be displaced by the same amount. This follows from the fact that the same displacement for both holograms results in the same mode decomposition for both beams, each common component of which still satis®es equation (2). When the holograms are displaced in opposite directions, the modes are weighted in the same way but acquire diÄerent phases. This results in a low coincidence rate. Finally we note that for large displacements of both holograms in either direction, the coincidence rate remains ®nite since the dominant mode component of both signal and idler is then l ˆ 0. Figure 4 (b) is calculated for lsignal holo: ˆ 1, lidler holo: ˆ ¡2 and lpump ˆ 0. In this case, holograms that are perfectly aligned along the beam axis will measure the correlation between states that do not conserve the orbital angular momentum and the coincidence rate is zero. A displacement of the idler hologram, in either direction from the beam axis, increases the l ˆ ¡1 component of the idler beam and thus increases the coincidence count rate. Again, for large displacements of both holograms, the large l ˆ 0 component of both beams leads to a higher coincidence rate. Figure 4 (c) shows an example of a non-Gaussian pump beam, with lsignal holo: ˆ 1, lidler holo: ˆ ¡2 and lpump ˆ ¡1. As in ®gure 4 (a), an alignment of the holograms along the beam axis measures the correlation between modes that conserve the orbital angular momentum and the coincidence rate is high. Note that in this case for large displacements of both holograms the large l ˆ 0 components of both beams do not satisfy equation (2) and the coincidence rate is now low. In conclusion, we can say that high coincidence rates are found between pure l states that conserve the orbital angular momentum as well as between superpositions of such states. As already pointed out by Mair et al. [8], this is a sign of quantum entanglement. If instead the signal and idler modes were classically correlated, the emitted signal and idler would contain mixtures of photon pairs that individually conserve orbital angular momentum. This would still allow high


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Figure 4. Calculated coincidence rates for diÄerent combinations of signal and idler holograms as a function of displacement with respect to the beam axis.

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coincidence rates between pure l states. By displacing the idler hologram and detecting a superposition of diÄerent l modes, the signal beam would then be projected into a mixture of l modes that together with the signal modes conserve orbital angular momentum. The coincidence rate should then depend only on the mode decomposition but not on the phases, and the direction of the signalhologram displacement should not matter. This is in contrast to our predictions based on a numerical model of the two photon wavefunction described in this paper. The fact that the coincidence rate between superpositions of l states is as high as between pure l states shows that, instead, coherent states are emitted. These predictions are similar to the results obtained experimentally by Mair et al. Instead of directly detecting the coincidence rates between superposition modes of signal and idler, as we suggest here, they concentrated on the determination of the beam pro®le and its dependence on the displacement of the hologram in the other beam. Their work showed that if the photons were detected in coincidence, displacing the hologram in one beam displaces the axis of the other beam by the same amount. Such a result is consistent with the behaviour predicted in ®gure 4 (a). In general, our simulations relate to experiments that have not yet been performed. Such experiments are potentially attractive because the predicted correlations have clearly identi®ed zeros and maxima, enabling a quantitative analysis of orbital angular momentum entanglement. The most rigorous proof of entanglement would be to perform measurements that test a Bell-type inequality for orbital angular momentum states. Measurements of the kind described in this paper could form the basis of such a test.

Acknowledgments This work was supported by the Leverhulme Trust. LA also wishes to acknowledge individual support. MJP thanks the Royal Society for the award of a University Research Fellowship. SMB thanks the Royal Society of Edinburgh and the Scottish Executive Education and Lifelong Learning Department for the award of a Support Research Fellowship.

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