1. Introduction

In 1935 Einstein, Podolsky, and Rosen (EPR) raised a famous argument concerning the completeness of quantum mechanics [1]. Since then, there have been a number of schemes proposed to test local realism, which are generally grouped into two sets, namely, with and without inequalities [2]. Those with inequalities can be traced back to Bell’s theorem formulated in 1964 [3]. In contrast, Hardy in 1993 demonstrated a logical proof of nonlocality not involving inequalities [4] (see also [5, 6]), which was considered “the best version of Bell’s theorem” [7]. Hardy’s proof was first demonstrated by measuring photon polarization [8, 9]. A significant progress was the generalization to a ladder proof to increase the maximum fraction of measurements that demonstrate nonlocality [10, 11]. In this type of experiments, a source of non-maximally entangled photons is necessary. These sources have been implemented by clever arrangements of wave plates and Fresnel rhomb rotators to vary the degree of entanglement, and the correlations were inferred from Hong-Ou-Mandel peaks (dips) [10]. Another simpler implementation is a bidirectional-pump scheme for producing non-maximally entangled polar-isation states [12]. Here we report an experiment showing Hardy’s ladder proof, relying only on a standard parametric down-conversion setup without any sophisticated state preparation. Instead, we measure a different degree of freedom, namely the orbital angular momentum of entangled pairs of twisted photons.

Twisted photons are so named because of their characteristic helical wave fronts. OAM eigenstates |ℓ〉 have a phase front described by exp(iℓϕ), where ϕ is the azimuthal angle and ℓ is an integer. A photon in the state |ℓ〉 has an OAM of ℓh̄. Because ℓ is an integer, the OAM state-space is theoretically unbounded. This provides a promising playground for exploring high-dimensional entanglement and deeper features of quantum mechanics, and also for increasing the information capacity of photons [13–15]. Our motivation for using OAM as another degree of freedom to test Hardy’s theory is twofold. First, there is a clear analogy between polarization and OAM, which allows for a straightforward implementation of the Hardy paradox in OAM. Second, spontaneous parametric downconversion (SPDC), which has proven to be a reliable source of entangled photons [16], result to a naturally nonmaximally entangled OAM state. The entanglement of OAM in these photon pairs has been established experimentally [17]. Since then, various tests of local realism, such as the EPR paradox [18] and Bell-type tests [19, 20] have been performed. While polarization is a two-dimensional state space, OAM is not [13]. We can choose to work on any two-dimensional subspace of the theoretically unbounded OAM state space. Choosing different subspaces effectively allows us to choose the degree of entanglement directly and without the introduction of any other component for state preparation. Using OAM states not only lead to convenient state preparation, but also to convenient state measurement when done with programmable spatial light modulators. The work we present here is the first time the Hardy paradox is demonstrated in the high-dimensional state-space of photon OAM.

2. Theory and experiment

Let us first summarize the Hardy paradox, in which quantum mechanics allows a set of probabilities that is logically inconsistent within a classical framework [4–7]. Assume there are two observers measuring dichotomic observables. Alice measures A₀ and A₁, while Bob measures B₀ and B₁. We define P(A_i, B_j) as the joint probability of obtaining A_i = 1 and B_j = 1, while P(Ā_i, B_j) is that of A_i = − 1 and B_j = 1. In the classical framework of local hidden-variable theory, if the three conditions: (I) P(A₀, B₀) = 0, (II) P(Ā₀, B₁) = 0 and (III) P(A₁, B̄₀) = 0 hold, then (IV) P(A₁, B₁) should be exactly zero. However, quantum mechanics allows suitable observables A₀, B₀ and A₁, B₁, satisfying (I), (II) and (III), but P₁ = P(A₁, B₁) > 0. This has been generalised to a ladder proof [10], where when considering K + 1 defined dichotomic observables A_k and B_k where k = 0, 1,···,K), the following chain of probabilities hold:

SPDC naturally results to entangled OAM states which are not maximally entangled– the different OAM states have different weightings, and these weightings consist the spiral spectrum [21]. This suggests many sets of nonmaximally entangled states within different OAM subspaces with varying degrees of entanglement. We exploit the analogy between polarization and OAM. On the conventional Poincaré sphere [22], the north and south poles are left- and right- circular polarizations, respectively. The points on the equator indicate linear polarizations, the rest are elliptical polarizations. Although OAM has an unbounded number of orthogonal states, we can choose to work within two-dimensional subspaces represented by Bloch spheres [19]. Unlike [19], we construct asymmetric Bloch spheres spanned by OAM modes with different |ℓ| values (e.g. Fig. 1). The north pole corresponds to |ℓ = +2〉 (or |ℓ = +1〉) and the south pole correspond to |ℓ = 0〉; the surface of each sphere encompasses all possible superpositions of these modes. The necessary measurements, {A_k} and {B_k}, to show the Hardy paradox reside on these spheres.

 

Fig. 1 The Bloch spheres for our Hardy test: (a) |+2〉 and |0〉; (b) |+1〉 and |0〉. We show the intensity of the modes on the poles and some representatives on the equator. We also show the holograms required for Alice’s measurements (Bob’s not shown).

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Our experiment is sketched in Fig. 2. A collimated 355 nm beam pumps a 5-mm long BBO crystal, where degenerate 710 nm signal and idler photons are produced in pairs via type-I collinear SPDC. These are separated by a non-polarizing beam splitter (BS). In each arm, we have a spatial light modulator (SLM) that allows us to program any phase profile. For the simple case of just measuring photons in OAM state |ℓ〉, the SLMs are encoded with a diffraction grating having ℓ dislocations. This simple hologram transforms light having the OAM state we intend to measure, into a fundamental mode in the direction of the first diffraction order. The SLMs are imaged onto single-mode fibers (SMF) and the output of the SMFs are connected to avalanche photodiodes whose outputs are fed to a coincidence counting circuit. The measurements needed to demonstrate the Hardy paradox consists of specific superpositions of OAM states which require modulation of both intensity and phase (Fig. 1). Nonetheless, we can employ the same concept to measure these states: we program our SLMs to holograms which consists of the specific superposition of OAM states, in addition to the grating. The SLM can only modulate the phase. However, we can incorporate a spatially dependent blazing function to the phase of the grating and the measurement state (Fig. 2), and this will allow us to also modulate the intensity [20]. This technique has been used previously for generating (in contrast to measuring) complex optical vortex topologies from the fundamental mode [23]. Intensity masking in this manner works, but there is a cost in efficiency. The holograms we use has a diffraction efficiency of about 60%, and the introduction of the blazing function will make this even smaller, translating to a decrease in coincidence counts.

 

Fig. 2 Experimental setup for showing Hardy’s paradox with entangled OAM. Left inset is spiral spectrum; top and bottom insets are examples of intensity and phase of measurement modes and the hologram displayed in SLM.

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We use the Laguerre-Gaussian modes characterised by the azimuthal index ℓ and radial index p, to express the state produced in SPDC [21, 24],

In our consideration, the Hilbert space of the signal photon is spanned by two arbitrary OAM eigenstates |m〉_A and |n〉_A. The idler space is then spanned by |− m〉_B and |− n〉_B, following the conservation law of angular momentum. Thus, the nonmaximally entangled state is naturally post-selected,

By substituting Eqs. (6)–(8) to Eqs. (1)–(4) and after some lengthy but straightforward algebra, we have tan θ_k = (−1)^k × ζ^{(2k+1)(|m|−|n|)/2}. Subsequently, we obtain the Hardy fraction,

In contrast to [10, 11], here the degree of entanglement is ε = 0.66^|m|−|n| and can be easily varied by simply choosing different OAM bases, |m〉 and |n〉. The required measurements can be conveniently implemented using SLMs, which act as computer reconfigurable refractive elements and can be utilized to specify any non-trivial superpositions of OAM states [20]. Without loss of generality, we take two states, |Ψ〉_2,0 and |Ψ〉_1,0, for examples to demonstrate the Hardy paradox with K = 1 and K = 2, respectively. The corresponding degrees of entanglement, ε(2,0) = 0.43 and ε(1,0) = 0.66, are known from the measured spiral bandwidth (left inset in Fig. 2). Based on Eqs. (7)–(9), we can calculate the states to be measured, and represent them on the asymmetric Bloch spheres, we show the holograms to measure the states |A_k〉 and $|A_k^{\perp }〉$ (Fig. 1). We illustrate the case of K = 1 more closely for which we choose m = 2 and n = 0. In Fig. 2, we show the intensity and phase of the states we want to measure (|A₀〉 in SLM_A and |B₀〉 in SLM_B) and the holograms we use to detect these.

Our results are shown in Fig. 3(a) and 3(b). We obtain P₁ = 0.0778 ± 0.0039 for |Ψ〉_2,0 and P₂ = 0.1389 ±0.0047 for |Ψ〉_1,0, which are slightly less than the theoretical predictions of 0.0890 and 0.1573 from Eq. (10), respectively. The difference can be attributed to slight misalignment (the small, but nonzero probabilities for the rest measurements suggest some misalignment in our setup) and non-uniformity in the diffraction efficiency of different holograms used (we have been stringent in calculating the probabilities and we did not correct for these differences). All other probabilities are low as anticipated, consistent with the Hardy paradox. The Hardy paradox can also be put in a more general framework in terms of an inequality [10],

 

Fig. 3 Maximum fraction allowed by quantum mechanics (blank bars, red) and experimental results (solid, green) for (a) |Ψ〉_2,0 with K = 1; (b) |Ψ〉_1,0 with K = 2.

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3. Conclusion

To conclude, we have shown experimentally Hardy’s nonlocality proof by measuring the entangled OAM of twisted photon pairs. This enables us to directly use the nonmaximally entangled state produced by SPDC. We note that apart from post-selecting suitable OAM bases, we can also modify the degree of entanglement by tuning phase-matching conditions to directly change the spiral spectrum [25]. The good agreement between theoretical and experimental results point, again, to an incompatibility between local realism and quantum mechanics.