# Spin-orbit hybrid entanglement of photons and quantum contextuality

###### Abstract

We demonstrate electromagnetic quantum states of single photons and of correlated photon pairs exhibiting “hybrid” entanglement between spin and orbital angular momentum. These states are obtained from entangled photon pairs emitted by spontaneous parametric down conversion, by employing a -plate for coupling the spin and orbital degrees of freedom of a photon. Entanglement and contextual quantum behavior (that is also non-local, in the case of photon pairs) is demonstrated by the reported violation of the Clauser-Horne-Shimony-Holt inequality. In addition a classical analog of the hybrid spin-orbit photonic entanglement is reported and discussed.

## I Introduction

Entangled states are at the heart of most quantum paradoxes and provide the main tool for quantum information processing, including applications such as teleportation, cryptography, superdense coding, etc. Entangled quantum states are also the basis of Bell’s inequality violations, which ruled out classical hidden-variable theories in favor of quantum mechanics Bell (1966). Bell’s inequalities were originally derived for two particles, as a consequence of locality and realism. In almost all experimental demonstrations of these inequalities to date, the same degree of freedom of two particles has been used, e.g. the spin of a photon. Very recently, however, the case of so-called “hybrid entanglement”, occurring when the involved degrees of freedom of the two particles are not the same, has attracted a certain interest, and the first experimental demonstrations with spin and spatial-mode degrees of freedom have been reported s. Ma et al. (2009); Neves et al. (2009). Using different degrees of freedom also opens up another opportunity, i.e. that of realizing entanglement between different degrees of freedom of a single particle. In this case, no role is played by non-locality, but Bell-type inequalities can still be formulated by assuming realism and the so-called “non-contextuality” of the two involved commuting observables, i.e. the assumption that the result of a particular measurement of one observable is determined independently of any simultaneous measurement of the other one Roy and Singh (1993); mermin93 (); Liu et al. (2009). Non-contextual hidden variable models have been excluded by recent experiments where the violation of suitable inequalities was observed using neutrons Hasegawa et al. (2003), ions Kirchmair et al. (2009), and single photons prepared in entangled spin-path states Gadway et al. (2009). Finally, single-particle entanglement, in the case of bosons such as photons, has a classical analog that is obtained by replacing single-photon states with multi-photon coherent states realized within the same field mode spreeuw98 (). Such a classical analog helps visualizing the nature of the single-particle entanglement.

A particularly convenient framework in which to explore these concepts is provided by photons carrying both spin angular momentum (SAM) and orbital angular momentum (OAM). While the former is the most widely employed internal degree of freedom of photons for quantum manipulations, the latter is becoming an interesting additional resource for quantum applications (see, e.g., Molina-Terriza et al. (2007); Franke-Arnold et al. (2008); Barreiro et al. (2008); Nagali et al. (2009a)). In this work, we study three conceptually related experimental situations. Firstly, heralded single photons are prepared in a state where SAM and OAM are entangled (as proposed in Chen and She (2009)), and are then used for testing the contextuality of different degrees of freedom of the same particle. Secondly, correlated photon pairs, where the SAM of one photon is entangled with the OAM of the other, i.e. photon pairs exhibiting SAM-OAM hybrid entanglement, are generated and used for testing the contextuality and non-locality of these degrees of freedom when they are spatially separated. Finally, optical coherent states involving many photons, are used to demonstrate a classical analog of SAM-OAM hybrid entanglement.

## Ii Experimental Setup

The experimental layout we used in the quantum regime (the first two experiments) is presented in Fig. 1. Our down-conversion source generates photon pairs that are entangled in the OAM degree of freedom Jack et al. (2009); Leach et al. (2009), each photon being horizontally polarized, as described by

(1) |

Here and denote the signal and idler photons traveling along the two corresponding arms of the setup shown in Fig. 1 and , denote SAM and OAM degrees of freedom, respectively. The integer is the photon OAM in units of and denotes horizontal linear polarization.

### ii.1 Single-photon experiment

. In this case we use photon to herald a single photon which we convert into an OAM-SAM maximally entangled state. Starting from state given in Eq. (1), we post-select photon pairs having , i.e. in state , by coupling photon into a single-mode optical fiber. Photon is thus also projected into . Spatial light modulator SLM B in this case is patterned as a uniform grating, deflecting the beam but not affecting its transverse spatial mode (see upper-right inset of Fig. 1). Photon is sent first through a -plate Marrucci et al. (2006); marrucci06b () so as to generate the maximally entangled SAM-OAM state Nagali et al. (2009b)

(2) |

where , denote left-circular and right-circular polarization states, respectively. The polarization state of photon emerging from the -plate is then measured by a half-wave plate (HWP) oriented at a variable angle and a fixed linear polarizer, restoring the horizontal polarization. This HWP-polarizer combination filters incoming photons having linear polarization at angle with respect to the horizontal direction. In the circular polarization basis, the state of the filtered photons is written as . The SAM measurement does not affect the OAM degree of freedom. Non-contextuality can be assumed between the -component of photon SAM and OAM, because, in the paraxial approximation, the SAM operator commutes with the OAM operator . After SAM filtering, the photon’s OAM is also measured by a suitable computer-generated hologram, displayed on SLM A, followed by coupling into a single-mode fiber. The hologram pattern is defined by the four-sector alternated -shift phase structure shown in the upper-right inset of Fig. 1, with the four sectors rotated at a variable angle (the grating fringes are not rotated). On diffraction, this hologram transforms the photons arriving in the OAM superposition state back into the state, which is then filtered by coupling in fiber. The OAM superposition state is the spatial mode analog of the linear polarization, and we may refer to its angle as to its “orientation” footnote1 (). The overall effect of our apparatus is therefore to perform a joint measurement of the polarization and spatial mode orientations of photons at angles and , respectively. When photon is in the entangled Bell state described by Eq. (2), we expect that the final probability to detect it (in coincidence with the trigger photon) is given by

(3) |

To test entanglement we adopt the Clauser-Horne-Shimony-Holt (CHSH) inequality, given by CHSH ()

(4) |

where is calculated from the - photon coincidence counts according to

(5) |

Whilst the CHSH inequality is commonly applied to non-local measurements on two spatially separated entangled photons, testing for hidden variable theories, here we apply it to single-photon entanglement to test for contextuality. In Fig. 2a the coincidence counts are shown as a function of spatial mode orientation for different values of polarization angles .

The occurrence of high-visibility fringes indicates (single-particle) entanglement in the SAM-OAM spaces. The CHSH value calculated from this data is shown in Fig. 3 (green dots). A violation of the CHSH inequality is clearly obtained, in good agreement with quantum theory predictions, confirming the entanglement and providing a demonstration of quantum SAM-OAM contextuality for single photons.

### ii.2 Two-photon experiment

. In this case, we generate and verify entanglement between the SAM of one photon and the OAM of the other, i.e. we demonstrate non-local hybrid entanglement in these two degrees of freedom. To this purpose, the four-sector and uniform holograms of arms A and B were swapped, as displayed in the top-right inset of Fig. 1. The -plate in arm A and the sector hologram in arm B of the apparatus, together with subsequent coupling into the single-mode fiber before detection, act so as to post-select the photons with in Eq. (1), i.e. the post-selected initial two-photon state is . The photon passes through the -plate, acting in this case as a OAM-to-SAM transferrer Nagali et al. (2009b), so that the OAM eigenstates are mapped into and polarized photons with , respectively. After this process, the photon pair is projected into the nonlocal state

(6) |

where the SAM of one photon is maximally entangled with the OAM of the other. Next, the polarization of the photon is measured by the HWP rotated at angle followed by the polarizer, and the spatial mode of the photon by the sector hologram rotated at angle followed by coupling in fiber. Well-defined coincidence fringes with visibility up to 90% are obtained, as shown in Fig. 2b. Repeating the measurements for different angles and , the quantity was evaluated from Eqs. (4) and (5) and the violation of the CHSH inequality was verified, as shown in Fig. 3 (blue dots). This violation provides a demonstration of SAM-OAM hybrid entanglement and non-locality, for separated photon pairs.

### ii.3 Classical light experiment

. In our final experiment, we move to a classical regime of non-separable optical modes occupied by many photons, corresponding to coherent quantum states. A 100 mW frequency-doubled linearly-polarized continuous wave Nd:YVO laser beam is sent in an optical line equal to arm A of our quantum apparatus, so as to obtain, after the -plate, a coherent state in the SAM-OAM non-separable mode given by Eq. (2) footnote2 (). The calculated structure of this mode is shown in Fig. 2d, for a given input polarization. The mode non-separability is evident, as the polarization is spatially non-uniform Borges et al. (2010). The beam polarization is then filtered by the combination of the HWP at angle and polarizer and its spatial mode by the sector hologram rotated at angle , as in the single-photon experiment (a). In this case, no trigger is used and the count rates in Eq. (5) are replaced by average power measurements, corresponding to photon fluxes. When the angles and are changed, high contrast sinusoidal fringes proportional to were observed in the overall transmitted power fraction, as shown in Fig. 2c. As shown in Fig. 3 (red dots) we note that the classical experiment mimics the results of the single photon experiment. However, the experiment can of course also be interpreted without assuming the existence of photons. In this case, SAM and OAM measurements can be understood just as wave filtering procedures, and no conclusion can be drawn about discrepancies between classical-realistic and quantum behaviour. Nevertheless, providing a classical analog of single-particle entanglement is interesting in itself and may offer the basis for some entirely classical implementations of quantum computational tasks Spreeuw (2001).

## Iii Conclusions

In conclusion, we have demonstrated hybrid entanglement between the spin and the orbital angular momentum of light in two different regimes: single photons and entangled photon pairs. We have reported an additional classical experiment which mimics the quantum result and although the experimental results appear very similar in the three cases, they provide different and complementary insight into the contextual quantum nature of light.

## Iv Acknowledgment

The project PHORBITECH acknowledges the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under FET-Open grant number 255914.

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