## Abstract

We study the high-dimensional orbital angular momentum (OAM) entanglement contained in the spatial profiles of two quantum-correlated photons. For this purpose, we use a multi-mode two-photon interferometer with an image rotator in one of the interferometer arms. By measuring the two-photon visibility as a function of the image rotation angle we measure the azimuthal Schmidt number, i.e., we count the number of OAM modes involved in the entanglement; in our setup this number is tunable from 1 to 8.

© 2007 Optical Society of America

## 1. Introduction

The most popular variety of quantum entanglement involves the *polarization* degree of freedom of two photons; in this case we deal obviously with two (polarization) modes per photon [1, 2, 3]. Recently, there has been a lot of interest in *spatial* entanglement of two photons; in this case the number of modes per photon can be much larger than two so that entanglement is correspondingly richer [4, 5, 6, 7, 8, 9, 10, 11]. This interest is motivated, fundamentally, by the desire to understand the nature of quantum entanglement in a high-dimensional Hilbert space. From the point of view of applications the high-dimensional case is important since it holds promise for implementing high-dimensional alphabets for quantum information, e.g. for quantum key distribution [12]. A popular basis for the spatial modes is the basis in which the modes are distinguished on account of their orbital angular momentum (OAM) [5, 6, 7]. An issue of much discussion in high-dimensional entanglement, OAM or otherwise, is how many modes are involved, beyond the statement that this number can be much larger than 2 [4, 5, 6, 7, 13, 14, 15]. In this article we demonstrate a practical method to quantify the number of OAM spatial modes involved in bi-photon entanglement; in our experiment this number has been varied in a controlled way from 1 to 8. This result has been achieved by using a special two-photon interferometer.

Our two-photon interferometer contains an image rotator in one of its arms (see Fig. 1). Similar interferometers with built-in rotation have only been tested at the *one-photon* level, where the rotation has been linked to a topological (Berry) phase [16]. A one-photon interferometer with an image reversal has been shown to act as a sorter between even and odd spatial modes [17, 18]. We will instead consider *two-photon* interference in an interferometer with built-in rotation.

In two-photon interference experiments, two photons are combined on a beam splitter, before being detected. These experiments, which have been pioneered by Hong, Ou and Mandel (HOM) [19], demonstrate an effective bunching between the photons in each pair, but only if the optical beams have good spatial and temporal overlap. More recent versions of these “HOM” experiments study the generation of spatial anti-bunching [8], and the effect of a modified pump profile (TEM_{01} versus TEM_{00}) on the interference pattern (bunching versus anti-bunching) [9, 10].

The key question that we will address is what the two-photon interference in our interferometer-with-built-in-rotation tells us about the spatial entanglement between the two multi-mode beams. As our geometry leads to an effective separation of the radial and azimuthal degrees of freedom, where only the latter are manipulated, the experiment only provides information on the entanglement between the orbital angular momenta (OAM) of the two photons [5, 6, 7]. It allows to measure the azimuthal Schmidt number, i.e., to count the number of entangled OAM modes.

## 2. Experimental results

Figure 1 is a schematic view of our two-photon interferometer. We mildly focus light from a krypton ion laser (γ =407 nm, θ_{P} = 0.50 mrad divergence) onto a 1-mm-thick β-barium borate (BBO) crystal to generate quantum-entangled photon pairs at 814 nm via (type-I) spontaneous parametric down-conversion. These twin photons travel along individual interferometer arms, one of them through an image rotator *R*(θ), before they are combined at a beam splitter. The experimental results shown in Figs. 2–4 are obtained with an interferometer that contains an odd number of mirrors (as in Fig. 1). Two-photon interference is observed by recording the number of coincidences as a function of the delay ∆*t* between the two arms with single-photon counters (SPC). An adjustable circular aperture, positioned in one detection arm at *L* = 1.5 m from the crystal (≈ far field), allows us to control the detected number of entangled spatial modes. The limited detection bandwidth (5 nm) and detection angle (< 7 mrad) assure operation in the so-called thin-crystal limit [20], where phase matching is automatically fulfilled. In this limit, the spatial properties of the detected two-photon field are solely determined by the TEM_{00} pump profile.

The image rotator *R*(*θ*) consists of a fixed mirror M_{1} and a rotatable “open Dove prism”, comprising three separate mirrors, arranged in an equilateral triangle, i.e., operating at reflection angles of 60°,30° and 60°. This construction has several advantages over alternative arrangements. By working with a unit of 3 mirrors instead of a glass Dove prism [21], we avoid any detrimental effects of wavelength dispersion, which could lead to temporal labelling and reduced interference. Furthermore, the adjustability of the mirrors allows us to reduce unwanted beam deflection to angles ≤ 0.1 mrad, whereas glass Dove prisms have typical wedge angles of ≈ 1 mrad. Finally, whereas image rotations are generally accompanied by polarization rotations [16], our rotator hardly changes the polarization. This convenient property is obtained by using silver mirrors (protected by a thin SiO_{2} cover layer) instead of dielectric mirrors. The measured phase difference *ϕ _{s-p}* = 0.81

*π*between the (three-fold) reflected s- and p-polarized light, is in fact close enough to the ideal value of

*ϕ*=

_{s-p}*π*needed for a polarization-insensitive rotator, to limit the measured power in the orthogonal polarization to at most 8% (a precise quantitative check of these values based upon the optical constants of silver is hampered by the fact that we do not know the precise thickness and porosity of the protective SiO

_{2}coating). As both polarization components have the same spatial profile, we simply remove this small unwanted orthogonal component with a fixed polarizer (

*P*

_{1}in Fig. 1). The topological phase [22] that originates from the mentioned polarization changes goes unnoticed, as two-photon interferometers are insensitive to the optical phase.

Figure 2 shows the measured two-photon coincidence rate as a function of the time delay Δ*t* at a fixed rotation angle of *θ* = -30° and for two aperture sizes. The reduced coincidence rate around Δ*t* = 0 demonstrates how two-photon interference produces an effective bunching of the two incident photons in either of the two output channels [19]. The shape of the interference pattern is the same for both apertures: its width of ≈ 260 fs (FWHM) is Fourier-related to the transmission spectrum of our filters (not shown in Fig. 1) and agrees within a few percent with the value expected for a Δλ = 5 nm bandwidth. The modulation depth or so-called HOM visibility, however, is quite different, being 89.5±0.5 % for the 1 mm aperture and only 15.5±0.5 % for the 10 mm aperture.

Coincidence measurements like those presented in Fig. 2 were repeated for various aperture sizes. Combining these results lead to Fig. 3, which shows the HOM visibility at a fixed rotation angle of *θ* = -30° as a function of the aperture diameter. The drop in visibility at larger apertures results from spatial labeling; the two-photon interference disappears if we are able to decide, even only in principle, which of the two photons exiting the beam splitter travelled which path in the interferometer. The diffraction due to the smaller apertures frustrates the observation of this labeling and thus increases the visibility. Experiments with one extra mirror in the interferometer always yielded visibilities close to 100% irrespective of aperture size and rotation angle; labeling does not occur when the number of mirrors in the interferometer is even. This strong dependence on the parity of the number of mirrors is related to the sign change in OAM experienced at each reflection, which in turn affects the symmetry of the detected two-photon wavefunction.

By repeating the measurements shown in Fig. 3 for a series of *fixed rotation angles* we obtain a two-dimensional table of visibilities *V*(*a*, *θ*) and from that the visibility *V*(*θ*) as a function of rotation angle *θ* for various *fixed detection geometries*. Figure 4 shows these results for four different geometries, which are specified by their azimuthal Schmidt number *K _{az}* (see below). All curves are symmetric under the operation

*θ*↔

*θ*(

*θ*= 0° corresponds to no image rotation) and periodic in

*θ*↔

*θ*+ 180°. This last observation, that a rotation over 180° instead of 360° already produces identical physics, reflects the two-photon character of the interference.

For detection behind single-mode fibers (labeled as *K _{az}* = 1) we obtained visibilities of at least 98%, independent of

*θ*. As the fundamental fiber mode is rotationally symmetric, spatial labeling and loss of interference under image rotation will not occur. For free-space detection behind small apertures (small

*K*) the effect of image rotation on the two-photon interference is relatively mild. For larger apertures, this effect is much more drastic and leads to a visibility as low as 4% at

_{az}*θ*= 90° for

*K*= 8. The reason for this reduction is that free-space detectors also monitor linear combinations of higher-order modes, which are no longer invariant under rotation and thus provide labeling information.

_{az}The theoretical curves in Figs. 3 and 4 are based on the following analytic expression [23]

where ξ = 2(*a*/*w _{d}*)

^{2}sin

^{2}

*θ*and

*a*is the aperture radius. The diffraction waist

*w*= 2

_{d}*Lθ*, or angular spread of one photon at a fixed position of the other, is twice the size of the pump in the (far-field) detection plane [24]. The solid curve in Fig. 3 is a fit based on

_{p}*w*= 1.4 mm, in agreement with the mentioned values of

_{d}*L*and

*θ*. The three dashed curves in Fig. 4 are based on the same value and contain no adjustable parameter, apart from a small uniform scaling of the vertical axis. We attribute the (small) deviations between theory and experiment to imperfect beam alignment. These deviations show up most prominently at small

_{p}*K*(

_{az}*K*= 1.13 in Fig. 4), where the two-photon visibility should remain high over a large angular range. At large

_{az}*K*the visibility drop upon rotation is fast enough to dominate spurious misalignment effects.

_{az}## 3. Mode counting

We now come to the essence of our paper, being the question “How can we count the number of orbital angular momentum (OAM) modes involved in the high-dimensional entanglement?”. In the appendix we will show that there exists a natural Fourier relation between the OAM modal distribution (or OAM spectrum) and the angular dependence of the two-photon interference, quantified by our visibility function *V*(*θ*). More specifically, we find

where the spiral weight *P _{l}* (with -∞ <

*l*< ∞ and ∑

_{l}

*P*= 1) is the probability to detected a photon pair with orbital angular momenta (

_{l}*l*, -

*l*) [14]. Equation (2) expresses the observed visibility

*V*(

*θ*) as a weighted sum over contributions from groups of

*l*-modes, each contribution oscillating between

*V*= 1 (HOM dip) and

_{l}*V*= - 1 (HOM peak), with its own angular dependence. It shows the power of our experiment, where a simple Fourier transformation of the measured

_{l}*V*(

*θ*) yields the complete OAM distribution {

*P*}.

_{l}In order to convert the modal distribution {*P _{l}*} into a single number that counts the effective number of entangled OAM modes, we use the azimuthal Schmidt number as

*K*≡ 1/∑

_{az}_{l}

*P*

_{l}^{2}, in analogy with the general form for modal decompositions. The azimuthal Schmidt number

*K*that we introduce is closely related to the quantum spiral bandwidth introduced in ref. [13]; both single out the azimuthal behavior by summing over all radial mode numbers. The relation between the azimuthal Schmidt number

_{az}*K*and the full 2D Schmidt number

_{az}*K*

_{2D}, where the summation runs over both azimuthal and radial mode numbers, depends somewhat on the shape of the detecting apertures. For a geometry with one or two Gaussian apertures, which allows for a complete analytic Schmidt decomposition [27], we find

*K*= 2√

_{az}*K*

_{2D}/(1 + 1/

*K*

_{2d}).

Based on the above description, we count the number of entangled OAM modes in our experiment as follows: For the three lower curves in Fig. 4 we first performed a Fourier analysis of the normalized *V*(*θ*)/*V*(0) to obtain the probability distribution *P _{l}* for each theoretical curve. The azimuthal Schmidt numbers that we calculated from these distributions are

*K*= 1.13 for the 1 mm aperture,

_{az}*K*= 2.9 for the 4 mm aperture, and

_{az}*K*= 8 for the 10 mm aperture. The aperture clearly allows us to tune the effective number of entangled modes.

_{az}One might wonder whether, and if so, in what sense, our experiment proofs the existence of spatial entanglement and the conservation of OAM in the SPDC pair production. Suppose we would have based our analysis on a more general two-photon input state that is not restricted to *l*
_{1} + *l*
_{2} = 0. The calculated visibility *V*(*θ*) for an interferometer with an odd number of mirrors would then contain terms of the form *P*
_{l1,l2} cos[(*l*
_{1} -*l*
_{2})*θ*], again using the mirror symmetry *P*
_{l1,l2} = *P*
_{l2,l1}. We can not exclude this possibility a priori. For an interferometer with an even number of mirrors, however, *V*(*θ*) would then contain terms of the form *P*
_{l1,l2} cos[(*l*
_{1} + *l*
_{2})*θ*] instead. Our observation that *V*(*θ*) ≈ 1 at any angle *θ* in the “even-mirror geometry”, can thus be interpreted as a real proof of the existence of OAM entanglement; any photon pair with *l*
_{1} ≠ -*l*
_{2} would make *V*(*θ*) angular dependent.

From a theoretical perspective, it is instructive to also consider a configuration with a *Gaussian* instead of a hard-edged transmission profile, having an aperture with an intensity transmission profile *T*(*r*) = exp(-*r*
^{2}/*ã*
^{2}). This combination allows for a complete (radial and azimuthal) analytic Schmidt decomposition of the detected field[27], and yields[23]

where *K*
_{2D} = 1 +(*ã*/*w _{d}*)

^{2}is the 2D Schmidt number. The Airy profile of Eq. (3) has almost the same shape as the function described by Eq. (1). It again allows for a Fourier decomposition of the form (2), providing analytic expressions for

*P*as a power series of the form

_{l}*P*∝

_{l}*α*(with

^{l}*α*< 1).

## 4. Summary

In summary, we have demonstrated how the high-dimensional entanglement of orbital angular momentum (OAM) can be characterized with a two-photon interferometer that contains an odd number of mirrors and an image rotator in one of its interferometer arms. We have shown how a Fourier analysis of the observed angle-dependent visibility *V*(θ) profile yields the full probability distribution over the OAM modes involved in the entanglement. Finally, we have calculated the azimuthal Schmidt number *K _{az}* corresponding to the effective number of entangled OAM modes. At

*K*= 1 (fiber-coupled detection) the detected two-photon field is a direct product state that contains no spatial entanglement; at

_{az}*K*= 2 the two-photon field acts as a pair of entangled qubits; at

_{az}*K*= 8 we deal with entangled quNits (with

_{az}*N*= 8) with a much richer internal structure.

## Appendix

The derivation of the Fourier relation between the visibility *V*(θ) and the OAM spectrum {*P _{l}*}, as presented in Eq. (2), starts with a (Schmidt) decomposition of the two-photon field in a sum over discrete spatial modes, instead of an integral over a plane-wave continuum. The two-photon field is thus represented by the pure state:

where ∣*u _{i}*〉 and ȣ

*v*〉 are two sets of orthonormal transverse modes. The Schmidt number

_{i}*K*= 1/(Σ

_{i}λ

_{i}

^{2}), with Σλ

_{i}= 1, quantifies the effective number of participating modes.

Two ingredients are added to this description. First of all, we perform a modal decomposition of the *detected* two-photon field instead of the full *generated* field, to incorporate any spatial filtering of apertures from the start and avoid unnecessary complications related to phase matching. The Schmidt decomposition of the generated field is generally very difficult to calculate, as its spatial extent depends both on the pump geometry and on phase matching [13, 15]. The Schmidt decomposition of the *detected* field [27], being the two-photon field behind the detection aperture(s), is quite different and doable if the apertures are small enough to satisfy phase matching, as is the case in our experiment.

As a second ingredient, we use the rotation symmetry of the pump beam and the detection apertures, in combination with the small-angle approximation, to single out the azimuthal dependence (OAM) of the two-photon field. Our Schmidt decomposition thus factorizes to

where *l* and *p* are the azimuthal and radial quantum numbers and ∣*l,p*〉′ and ∣ - *l,p*〉″ are the Schmidt eigenmodes of the detected field. The rotation symmetry restricts these modes to “Laguerre-Gaussian-like” field profiles of which the precise radial distribution is co-determined by the detection apertures. As our amplitude coefficients √γ_{l,p} already contain the effects of aperture filtering, they will decrease rapidly both for high *p* and high *l* values (high *l*-states are quite extended even for *p* = 0). A summation over the radial mode number *p* yields the OAM probability *P _{l}* = Σ

_{p}λ

_{l,p}.

As a last step, we propagate the two-photon field of Eq. (5) through our interferometer and calculate the expected two-photon visibility *V*(θ). This propagation will modify the two-photon field in the following ways: every mirror reflection changes the handedness by inverting the OAM of each *l*-state from *l* to −*l*. The image rotation *R*(θ) adds a phase factor exp(*ilθ*) to each *l*-state. The relevant beam splitter operations are the double transmission, which leaves the *l*-states unaffected, and the double reflection, which swaps the labels and changes the handedness. Application of these operations to the state of Eq. (5), in combination with the “signal ↔ idler” symmetry (λ_{-l,p} = λ_{l,p} and φ_{-i,p} = φ_{l,p}) and the orthogonality of the (*l,p*) states, finally yields

for the combined field behind a (50/50) beam splitter, where ∆π is the frequency difference between the two detected photons and τ is the time delay difference in the interferometer. By comparing the expected coincidence rate *R _{cc}*(θ, ∆τ) ∝ 〈Ψ(θ)∣Ψ(θ)〉

_{det}with the definition of the two-photon visibility, we obtain Eq. (2) as final result.

## Acknowledgements

This work is supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM).

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