Transport of orbital-angular-momentum entanglement through a turbulent atmosphere

doi:10.1364/OE.19.006671

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Transport of orbital-angular-momentum entanglement through a turbulent atmosphere

Bart-Jan Pors, C. H. Monken, Eric R. Eliel, and J. P. Woerdman »View Author Affiliations

Optics Express, Vol. 19, Issue 7, pp. 6671-6683 (2011)
http://dx.doi.org/10.1364/OE.19.006671

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Abstract

We demonstrate experimentally how orbital-angular-momentum entanglement of two photons evolves under the influence of atmospheric turbulence. Experimental results are in excellent agreement with our theoretical model, which combines the formalism of two-photon coincidence detection with a Kolmogorov description of atmospheric turbulence. We express the robustness to turbulence in terms of the dimensionality of the measured correlations. This dimensionality is surprisingly robust: scaling up our system to real-life dimensions, a horizontal propagation distance of 2 km seems viable.

1. Introduction

Quantum communication by means of entangled photon pairs allows for an intrinsically secure transmission of data, by distributing the pairs via a free-space or fiber channel to distant parties [1

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991). [CrossRef] [PubMed]

]. Most popular is polarization entanglement, which has dimensionality 2. Higher dimensionalities can be achieved using orbital-angular-momentum (OAM) entanglement [2

A. Mair, G. W. A. Vaziri, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). [CrossRef] [PubMed]

–4

J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008). [CrossRef] [PubMed]

] or energy-time entanglement [5

H. de Riedmatten, I. Marcikic, H. Zbinden, and N. Gisin, “Creating high dimensional entanglement using mode-locked lasers,” Quant. Inf. Comput. 2, 425–433 (2002).

, 6

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007). [CrossRef] [PubMed]

]; this route provides for a larger channel capacity and an increased security against eavesdroppers [7

T. Durt, D. Kaszlikowski, J.-L. Chen, and L. C. Kwek, “Security of quantum key distributions with entangled qudits,” Phys. Rev. A 69, 032313 (2004). [CrossRef]

Č. Brukner, T. Paternek, and M. Żukowski, “Quantum communication complexity protocols based on higher-dimensional entangled systems,” Int. J. Quantum Inf. 1, 519–525 (2003). [CrossRef]

]. However, the performance of a real-world high-dimensional quantum channel is an open issue. Here, we address this issue for the case of OAM entanglement distribution via a free-space channel [9

B.-J. Pors, C. H. Monken, E. R. Eliel, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” arXiv:0909.3750v1 [quant-ph] (2010).

For quantum communication to be of practical relevance, it is imperative that the entanglement between the photons carrying the information survives over a reasonably long propagation distance. Entanglement distribution over fiber-based transmission lines has proven to be feasible over distances over a hundred kilometers [10

A. Poppe, A. Fedrizzi, R. Ursin, H. R. Böhm, T. Lorünser, O. Maurhardt, M. Peev, M. Suda, C. Kurtsiefer, H. Weinfurter, T. Jennewein, and A. Zeilinger, “Practical quantum key distribution with polarization entangled photons,” Opt. Express 12, 3865–3871 (2004). [CrossRef] [PubMed]

–12

D. Salart, A. Baas, C. Branciard, N. Gisin, and H. Zbinden, “Testing the speed of ’spooky action at a distance’,” Nature 454, 861–864 (2008). [CrossRef] [PubMed]

]. However, the use of free-space links is needed when considering such purposes as airplane and satellite quantum links or hand-held communication devices [13

K. J. Resch, M. Lindenthal, B. Blauensteiner, H. R. Böhm, A. Fedrizzi, C. Kurtsiefer, A. Poppe, T. Schmitt-Manderbach, M. Taraba, R. Ursin, P. Walther, H. Weier, H. Weinfurter, and A. Zeilinger, “Distributing entanglement and single photons through an intra-city, free-space quantum channel,” Opt. Express 13, 202–209 (2005). [CrossRef] [PubMed]

–15

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007). [CrossRef]

The increased quantum-channel capacity that is available when encoding the information in the OAM of the entangled photons was argued to be severely limited in a practical free-space link, due to atmospheric turbulence that causes wavefront distortions. Several studies have addressed this aspect [16

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005). [CrossRef] [PubMed]

–24

F. S. Roux, “Decoherence of orbital angular momentum entanglement in a turbulent atmosphere,” arXiv:1009.1956v2 [physics.optics] (2010).

], but there is no unanimity on exactly how sensitive OAM entanglement is to atmospheric perturbations. So far, no experimental verdict has been obtained to clarify this issue.

In this paper, we present the first such experiment. We start with bipartite OAM entanglement of effective dimensionality 6, and demonstrate how the corresponding correlations evolve when one of the photons traverses a turbulent atmosphere, emulated by controlled mixing of cold and hot air. Our experimental results are in excellent agreement with our theoretical model, which combines a Kolmogorov description of atmospheric turbulence with our formalism of bi-photon correlation detection.

2. Experimental setting

Our experimental setup is depicted in Fig. 1. A PPKTP down-conversion crystal is pumped by the single transverse mode output of a Kr⁺ laser operating at 413 nm. It emits correlated photon pairs with complementary OAM at 826 nm, in a state of the form [25

J. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301 (2003). [CrossRef]

, 26

Within the mode space our analyzers have access to, we can safely approximate c_lp to be independent of l.

]

| Ψ 〉 = \sum_{l, p} c_{l p} | l, p 〉 | - l, p 〉 .

(1)

Here, |l, p〉 indicates the Schmidt mode containing one photon with orbital angular momentum lh̄, with p the radial mode index, and we can write

〈 r | l, p 〉 = 〈 r | l, p 〉 e^{i l θ} / \sqrt{2 π}

. For our source, the total number of entangled azimuthal modes, the so-called angular Schmidt number K, is of order 30 [27

M. P. van Exter, P. S. K. Lee, S. Doesburg, and J. P. Woerdman, “Mode counting in high-dimensional orbital angular momentum entanglement,” Opt. Express 15, 6431–6438 (2007). [CrossRef] [PubMed]

]. The correlated photons are spatially separated by a 50/50 beam splitter.

Fig. 1 Experimental setup. A type-I PPKTP crystal emits two frequency-degenerate photons (λ = 826 nm) that are entangled in their OAM degree of freedom. A beam splitter serves to separate the twin photons spatially. The entanglement is analyzed by two angular-phase-plate projectors, variably oriented at α and β, respectively, which are linked to a coincidence circuit. Each phase plate has one elevated quadrant sector with optical thickness λ/2 (inset). In one of the beam lines we place a turbulence cell.

The entanglement is analyzed by means of two state projectors, which are composed of an angular phase plate that is lens-coupled to a single-mode fiber, and a single-photon counter. The two phase plates are identical and carry a purely azimuthal variation of their optical thickness: they have one elevated quadrant sector with an optical thickness that is λ/2 larger than that of the remainder of the plate (see inset Fig. 1). The two phase plates can be rotated around their normals over an angle α and β, respectively. The detection state |A(α)〉 (or |B(β)〉) of one such analyzer is a high-dimensional superposition of OAM modes, the relative phases of which depend on the orientation of the plate. It can be written as

〈 r | A (α) 〉 = (2 / w_{0}) exp (- r^{2} / w_{0}^{2}) \sum_{l} λ_{l} e^{i l (θ + α)}

, where the Gaussian factor describes the fiber mode profile with field radius w₀, and the summation over the orbital-angular-momentum states describes the phase imprint imparted by the phase plate. When rotating the quadrant phase plate over 2π rad, the analyzer scans a mode space of dimensionality D = 6 [28

J. B. Pors, A. Aiello, S. S. R. Oemrawsingh, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, “Angular phase-plate analyzers for measuring the dimensionality of multi-mode fields,” Phys. Rev. A 77, 033845 (2008). [CrossRef]

3. Turbulence cell

In one of the beam lines, we place a turbulence cell where cold and hot air are mixed to bring about random variations of the refractive index that vary over time (Fig. 2(a)). We can tune the strength of the turbulence by varying the heating power and air flow through the cell. Similar cells have been used as a realistic emulation of atmospheric turbulence [29

O. Keskin, L. Jolissaint, and C. Bradley, “Hot-air optical turbulence generator for the testing of adaptive optics systems: principles and characterization,” Appl. Opt. 45, 4888–4897 (2006). [CrossRef] [PubMed]

]. Figure 2(b) gives an impression of the cell’s functioning: We inject one of the analyzers backwards with diode laser light and monitor the beam, which traverses the turbulence cell, in the far field. We do this for two cases; the analyzer is equipped with no phase plate (top row), or with the quadrant phase plate (bottom row). We observe that the input beams (left column) become deformed by the refractive index fluctuations, as can be seen when taking a 10 ms snapshot (middle column). Time averaging these fluctuations over 10 s reveals a beam broadening that is spatially isotropic (right column).

Fig. 2 Beam corruption after passage through the turbulence cell. (a) Impression of how an OAM eigenmode, having a helical wavefront, gets distorted when transiting the turbulence cell. The cell consists of a 7 cm long, 26 mm diameter glass tube, containing several resistors that produce up to 60 W of heat. A gentle flow of room temperature air is driven through the tube. (b) ( Media 1, Media 2, Media 3) Far-field intensity patterns of the analyzer, which is fed backwards with diode laser light at 826 nm. The analyzer is equipped with no phase plate (top row), or quadrant phase plate (with its sector aligned along the Cartesian axes) (bottom row). The diffraction limited patterns (left column) get perturbed when turbulence is switched on (middle column): for mild turbulence, the dominant effect is a randomly evolving beam deflection; for the more severe turbulence conditions used here (w₀/r₀ = 0.65), the beam profile can get significantly distorted. Taking a 10 s time average reveals an isotropic beam broadening (right column). The apparent asymmetry along the diagonal in the bottom left and right windows is due to the 3% discrepancy of the quadrant phase step from the ideal value of π.

It has been noted that misalignment of OAM beams with respect to the receiver (as can be caused by turbulence) could compromise quantum communication applications [17

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004). [CrossRef] [PubMed]

]. As shown in Fig. 2(b), however, the long-term average beam pointing does not get displaced, so that on this time scale misalignments do not occur.

We describe our cell by the Kolmogorov theory of turbulence [30

V. I. Tatarski, Wave propagation in a turbulent medium , 2nd ed. (Dover Publications Inc., 1961),

]. This standard model treats the optical effects of the atmosphere at any moment as a random phase operation e^iϕ(r), the time evolution of which follows a Gaussian distribution. It is conveniently described in terms of its coherence function, given by

{〈 e^{i ϕ (r_{1}) - i ϕ (r_{2})} 〉}_{t} = e^{- \frac{1}{2} 6.88 {[\frac{r_{1} - r_{2}}{r_{0}}]}^{5 / 3}},

(2)

where 〈...〉_t denotes averaging over time [31

D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966). [CrossRef]

]. The relevant parameter in this model is the Fried parameter r₀, being the transverse distance over which the beam profile gets distorted by approximately 1 rad of root-mean-square phase aberration [31

D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966). [CrossRef]

]. In the absence of turbulence r₀ → ∞, but when turbulence becomes stronger, the spatial coherence is reduced and hence r₀ shortens. From the Gaussian beam broadening in Fig. 2(b) (top row) we can determine the relation between the Fried parameter and the 1/e beam size w₀,

\frac{w_{0}}{r_{0}} = \frac{\sqrt{{(w_{l e} / w_{d l})}^{2} - 1}}{3.0},

(3)

with w_dl and w_le the 1/e far-field radius of the diffraction limited beam and long-exposure broadened beam, respectively [32

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975). [CrossRef]

The model presented here constitutes the canonical Kolmogorov model of turbulence, which behaves similarly at all length scales. Several refinements of the model exist, but in many cases Eq. (2) suffices as a proper description for the outside atmosphere [33

R. L. Lucke and C. Y. Young, “Theoretical wave structure function when the effect of the outer scale is significant,” Appl. Opt. 46, 559–569 (2007). [CrossRef] [PubMed]

]. It is fair to question, however, how well our experimental turbulence cell emulates the outside atmosphere [29

]. For this purpose, we have compared our experimental results that will be presented in Fig. 4 to advanced models that allow the outer scale L₀ (at which large wind flows turn unstable and turbulent) and the inner scale ℓ₀ (at which energy associated with turbulent motion is dissipated) to be finite [33

R. L. Lucke and C. Y. Young, “Theoretical wave structure function when the effect of the outer scale is significant,” Appl. Opt. 46, 559–569 (2007). [CrossRef] [PubMed]

]. It is known that the effect of a finite inner scale is modest. A finite outer scale, on the contrary, may have considerable influence, and is experimentally constrained by the size of our turbulence cell. A conservative estimate would therefore be that L₀ = 26 mm, being the tube diameter. (In fact, the air is blown out of the tube and remains turbulent at some distance from it, suggesting that the outer scale is probably larger). Using advanced Kolmogorov modeling for a finite outer scale L₀ = 26 mm, we found that the turbulence strength is underestimated by as much as a factor of 2. Instead, the derivation of the theoretical curves in Fig. 4 is based on Eq. (2), and thus L₀ → ∞. We see that for large w₀/r₀ this canonical Kolmogorov theory indeed slightly overestimates the turbulence in our system. However, the agreement between experimental and theoretical curves is sufficiently good to warrant our use of Eq. (2)).

Fig. 4 Survival of OAM coincidence curves under influence of turbulence. Experimental coincidence rates (data points) and theoretical predictions (curves) obtained with two quadrant-sector phase plates for: no turbulence (blue), w₀/r₀ = 0.30 (green) and w₀/r₀ = 0.65 (red). The inset shows a blow-up of the wiggles around α–β = π/2. Typically, the measurement values reproduce to within a few percent.

We calculated the effect of Kolmogorov turbulence on a single beam with a mode profile described by |A(α)〉. The blue curve in Fig. 3(a) shows the survival probability of an OAM eigenmode l = l₀ upon passing through a turbulent atmosphere as described by Eq. (2). The survival probability degrades gradually for increasing turbulence strength. We note that this decay depends on the ratio w₀/r₀ only and not on the specific OAM eigenvalue l₀, provided that the propagation distance L is small compared to the diffraction length

z_{R} = π w_{0}^{2} / λ

. Furthermore, the turbulence produces a coupling between the orthogonal OAM modes, leading to a non-vanishing mode overlap between the l₀ eigenmode and its neighbors Δl = ±1 (red) and Δl = ±2 (green). A different perspective on this mode mixing is presented in histogram Fig. 3(b), which shows how an OAM eigenmode (blue bar) spreads out over its neighboring azimuthal modes for w₀/r₀ = 0.65 (red bars). We note that normalization is not preserved, because some intensity is scattered to radial modes that are not sustained by the single-mode fiber. This illustrates the importance of taking into account the radial content of the generated two-photon state and the analyzers’ detection states when dealing with OAM modes in the presence of turbulence.

Fig. 3 Mode scattering due to turbulence. (a) Time-averaged survival probability of an analyzer’s OAM eigenmode l = l₀ as a function of turbulence strength (blue). The red and green curves denote turbulence-induced coupling probabilities to neighboring modes for Δl = ±1 and Δl = ±2, respectively. (b) Time-averaged spreading of the l₀ OAM eigenmode (blue bar) over its neighbors for w₀/r₀ = 0.65 (red bars).

Although turbulence acts as a decohering process when time averages are observed, in real time it simply imprints a phase perturbation on the beam. We note that it is possible to fully undo these perturbations if one could monitor and unwrap the wavefront deformations in real time by means of a phase corrector. This can be done using modern adaptive-optics techniques [34

B. M. Levine, E. A. Martinsen, A. Wirth, A. Jankevics, M. Toledo-Quinones, F. Landers, and T. L. Bruno, “Horizontal line-of-sight turbulence over near-ground paths and implications for adaptive optics corrections in laser communications,” Appl. Opt. 37, 4553–4560 (1998). [CrossRef]

4. Results

In the experiment, the phase plates are rotated around their normals, and the time-averaged photon coincidence probability

P (α - β) = {〈 {| 〈 A (α) | 〈 B (β) | {\hat{S}}_{A} | Ψ 〉 |}^{2} 〉}_{t}

(4)

is recorded as a function of their independent orientations. Here, the time-averaged behavior of the turbulence scattering operator Ŝ_A working on channel A can be described in terms of its coherence function Eq. (2),

{〈 {\hat{S}}_{A} | A (α) 〉 〈 A (α) | {\hat{S}}_{A}^{†} 〉}_{t} = \int d r_{1} d r_{2} | r_{1} 〉 〈 r_{1} | A (α) 〉 〈 A (α) | r_{2} 〉 〈 r_{2} | {〈 e^{i ϕ (r_{1}) - i ϕ (r_{2})} 〉}_{t} .

(5)

Figure 4 shows our main experimental results. In the absence of turbulence, we observe a piecewise-parabolic coincidence curve (blue circles), i.e., the coincidence rate follows a parabolic dependence for |α – β| ≤ π/2 and is zero elsewhere [4

]. The coincidence rate depends on the relative orientation of the phase plates only. We have investigated how the coincidence rates evolve for 6 different turbulence strengths, two of them shown in Fig. 4: w₀/r₀ = 0.30 (green triangles) and w₀/r₀ = 0.65 (red stars). The latter strength was also used for Fig. 2(b) and 3(b). Note that the 20 s integration time used in the experiment assures isotropic sampling of the wavefront fluctuations (see Fig. 2(b)). We observe a partial “smoothening” of the coincidence curve, which is excellently described by our theoretical predictions based on Eqs. (2) and (4), without any fit parameter. The turbulence-induced wiggles at |α – β| = π/2 are reproduced remarkably well (see inset). The symmetry of the experimental data around α–β = π attests to the stability of the experimental setup.

We stress that, while the coincidence count rates exhibit this rich behavior, the single count rates on both detectors are independent of the orientations of the phase plates and independent of the turbulence strength.

Note that arrival-time differences between the two photons of a pair, that could arise due to longitudinal phase distortions, are negligible compared to the 3 ns coincidence window of our detectors. Unlike transverse phase fluctuations, which lead to modal scattering, longitudinal phase fluctuations are largely washed out along the path of propagation; the time of flight is dominated by the average refractive index n, not Δn. To put things in perspective, even for light propagation over a 1 km distance through the outside atmosphere, the timing difference is only of the order of 1 ps [35

L. Kral, I. Prochazka, and K. Hamal, “Optical signal path delay fluctuations caused by atmospheric turbulence,” Opt. Lett. 30, 1767–1769 (2005). [CrossRef] [PubMed]

, 36

T. Schmitt-Manderbach, “Long distance free-space quantum key distribution,” Ph.D. thesis, Ludwig-Maximilians-Universität München (2007).

5. Discussion

Figure 4 shows that the coherence of the two-photon state is partly conserved even in the presence of rather strong turbulence. However, the turbulence inevitably damages the purity of the quantum state to some degree. Naturally, this damage, or equivalently the mixedness of the state, increases with increasing turbulence strength.

Quantifying entanglement for mixed states is a notoriously hard problem, especially if the entanglement is high dimensional [37

B. Schumacher, “Sending entanglement through noisy quantum channels,” Phys. Rev. A 54, 2614–2628 (1996). [CrossRef] [PubMed]

–39

S. Ryu, W. Cai, and A. Caro, “Quantum entanglement of formation between qudits,” Phys. Rev. A 77, 052312 (2008). [CrossRef]

]. Nevertheless, for mixed states of two qubits, some mathematical techniques exist to quantify the entanglement [40

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998). [CrossRef]

]. Two recent theoretical studies used this approach to investigate the robustness of entanglement between two spatial (OAM) modes against transmission through a turbulent noise channel [24

F. S. Roux, “Decoherence of orbital angular momentum entanglement in a turbulent atmosphere,” arXiv:1009.1956v2 [physics.optics] (2010).

,41

A. K. Jha, G. A. Tyler, and R. W. Boyd, “Effects of atmospheric turbulence on the entanglement of spatial two-qubit states,” Phys. Rev. A 81, 053832 (2010). [CrossRef]

]. In our experiment, however, we are explicitly in the regime of high-dimensional OAM entanglement. Consequently, it is near impossible to extract a proper entanglement measure from the experimental data at hand. This notwithstanding, the prospect of a high dimensionality is the very motivation to study OAM entanglement in the first place.

In the following, we therefore take a first step towards analyzing mixed high-dimensional entanglement, and apply the techniques developed in Ref. [4

] for pure entangled states to the partially mixed states we are dealing with here. We attempt to quantify the robustness of the correlations in terms of the Shannon dimensionality D, as introduced in Ref. [4

]. It is an operationally defined measure and gives the effective number of modes the combined analyzers have access to when scanning over their possible settings, viz. the phase-plate orientations. For two identical analyzers in the absence of turbulence, we can express D in terms of the pure detection state operator ρ_A (or ρ_B), where ρ_A = |A(α)〉〈A(α)|, as [4

]

D = \frac{1}{Tr [{({〈 ρ_{A} 〉}_{α})}^{2}]} .

(6)

Here, 〈ρ_A〉_α is the density operator obtained by averaging ρ_A over all phase-plate orientations α.

In the presence of turbulent scattering, however, the detection state becomes randomly time dependent:

ρ_{A} = {\hat{S}}_{A} | A (α) 〉 〈 A (α) | {\hat{S}}_{A}^{†} = ρ_{A} (t)

. The relevant detection state operator is therefore not ρ_A, but rather 〈ρ_A〉_t, i.e., the density operator averaged over time. In general, 〈ρ_A〉_t is no longer a single-mode projector, but just a positive operator. In other words, when averaging over the random fluctuations, the detection state becomes multimode. We can attach a - non-utilizable - dimensionality D⁻¹ = Tr[(〈ρ_A〉_t)²] to this detection operator, which gives the effective number of modes captured by the analyzer for fixed orientation α. The mixed nature of the detection operator blurs the analyzer’s modal resolution when scanning its orientation setting α.

In analogy to Eq. (6), the total number of modes captured by the analyzer when scanning its orientation setting α is given by Tr[(〈ρ_A〉_t,α)²], where 〈ρ_A〉_t,α denotes the average of the detection state operator ρ_A over time t and orientation α. However, we need to compensate for the contribution that arises from the degradation of resolution due to turbulence. Therefore, in the presence of turbulence, the Shannon dimensionality for mixed detection states is written as [42

We note that in our experiment we have turbulence in one arm only. For this case, the Shannon dimensionality can be generalized as D̃ = Tr(〈ρ_A〉_tρ_B)/Tr(〈ρ_A〉_t,α 〈ρ_B〈_β). It can be shown that this reduces to Eq. (7) when one has similar but weaker turbulence in both arms.

]

\tilde{D} = \frac{Tr [{({〈 ρ_{A} 〉}_{t})}^{2}]}{Tr [{({〈 ρ_{A} 〉}_{t, α})}^{2}]} .

(7)

This expression can be understood as being the ratio between the dimensionality of the operator 〈ρ_A〉_t,α that is averaged over α and the dimensionality of 〈ρ_A〈_t for fixed α. Note that in the limit of no turbulence, this result reduces to Eq. (6).

It is worth noting that the numerator in Eq. (7) is independent of the specific phase plate in use. It can be shown that its evolution under the action of turbulence follows the survival probability discussed in Fig. 3(a). The denominator in Eq. (7), on the other hand, does depend on the specific phase plate in use.

Continuing with our naive approach, we can extract D̃ straightforwardly from the experimental coincidence curves in Fig. 4. Working in the regime K ≫ D and using identical phase-plate analyzers in both arms, it can be shown that the numerator in Eq. (7) is associated to the maximum coincidence probability, and the denominator in Eq. (7) is associated to the average coincidence probability. It then follows that D̃ = 2πN_max/A, where N_max is the maximum coincidence rate and A is the area underneath the curve.

Figure 5 shows how D̃ evolves for increasing turbulence strength according to theory and experiment. In the absence of turbulence, we find an experimental value D = 5.7 vs. a theoretical prediction D = 6 [4

]. As the turbulence strength increases, the modal resolution of the analyzers degrades, constraining the dimensionality to smaller values, ultimately to D̃ = 1 [43

In this limit for extreme turbulence, the azimuthal fingerprint of the analyzer mode is fully wiped out. The detection state thus becomes circularly isotropic, leading to D̃ = 1.

]. The number of modes is reduced by ∼ 50% to D̃ = 3.1 when w₀/r₀ = 0.65. Considering the severity of the wavefront distortions (see Fig. 2(b)), we conclude that our dimensionality D̃ is surprisingly robust. For comparison, we also plotted our experimental results obtained with two half-sector phase plates, having one semicircle phase shifted by π (see inset in Fig. 5). For this case we observe that the dimensionality, initially at a value D = 3, decays considerably more slowly. This indicates that the resilience to atmospheric turbulence is quite sensitive to the nature of the OAM superposition state, an aspect also noted in Ref. [21

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47, 2414–2429 (2008). [CrossRef] [PubMed]

Fig. 5 Decay of the Shannon dimensionality. Experimental (data points) and theoretical (curves) dimensionality as a function of turbulence strength, for two quadrant-sector phase plates (circles) and two half-sector phase plates (triangles). The turbulence strength is expressed in the ratio w₀/r₀ (lower abscissa) and in the corresponding real-life propagation distance L (upper abscissa).

To further substantiate this observation, we consider two distinct OAM superposition states that have the same zero-turbulence dimensionality, and compute how their dimensionality decays for increasing turbulence strength (see Fig. 6). For instance, we compare the decay of the dimensionality for the case D = 6, using analyzers equipped with (i) quadrant phase plates and (ii) phase plates having two opposing octants of optical thickness λ/2 (see inset Fig. 6 for the plate profile). These two types of phase plates bear a large similarity; the OAM superposition state corresponding to the double-octant phase plate has an identical OAM eigenvalue distribution as the superposition of the quadrant phase plate, albeit with twice the mode spacing. This double mode spacing reflects the two-fold symmetry of the double-octant plate as compared to the quadrant phase plate.

Fig. 6 Robustness of the Shannon dimensionality for various OAM superposition states. Decay of the dimensionality for two quadrant phase plates (blue solid) and two double-octant plates (blue dashed), both of initial dimensionality D = 6. Similarly, decay of the dimensionality for half-sector phase plates (red solid) and half-integer spiral phase plates (red dashed), both of initial dimensionality D = 3.

Figure 6 shows that the dimensionality D̃ decays considerably more slowly for the double-octant plates (dashed blue curve) as compared to the quadrant phase plates (solid blue curve). A qualitative understanding of this difference in robustness can be obtained from the data represented in Fig. 3(b). First of all, the dominant effect of turbulence is a loss of modal strength (Δl = 0 scattering). The non-conserving OAM scattering probability decays rapidly as a function of |Δl|. This spreading of a mode applies to any initial value of l. Therefore, neighboring modes in the OAM superposition associated with a particular phase-plate analyzer affect each other strongly when separated by |Δl| = 1 (as for the quadrant phase plates) and much weaker when separated by |Δl| = 2 (as for the double-octant phase plates). This explains the robustness of the latter phase plates as compared to the standard quadrant phase plates.

As a second example, we compare OAM superposition states for the case D = 3, associated to (i) our half-sector phase plates (solid red curve) and (ii) half-integer spiral phase plates having a helical phase ramp of optical height λ/2 (dashed red curve). Since the OAM eigenvalue spectrum of the half-sector phase plate has twice the modal spacing of the half-integer spiral phase plate, the argumentation given above applies also here. Naturally, the argument can be extended, suggesting that OAM superposition states can be designed that have an optimal robustness against atmospheric perturbations.

So far, we have expressed the turbulence strength in terms of the ratio w₀/r₀. However, this quantity allows us to estimate the propagation distance L that can be reached outside the laboratory, since the Kolmogorov theory (see Eq. (2)) used to describe our data is also a fair description of a real-life atmosphere [30

V. I. Tatarski, Wave propagation in a turbulent medium , 2nd ed. (Dover Publications Inc., 1961),

]. For horizontal propagation, the Fried parameter can be expressed as

r_{0} = 3.02 {(k^{2} L C_{n}^{2})}^{- 3 / 5}

, with k = 2π/λ the wavenumber of the light, L the propagation length and

C_{n}^{2}

the structure constant quantifying the phase perturbations [44

D. L. Fried, “Anisoplanism in adaptive optics,” J. Opt. Soc. Am. 72, 52–61 (1982). [CrossRef]

]. To put this correspondence in perspective, we consider a wavelength λ = 1550 nm in the transmission window of the atmosphere and assume moderate ground-level perturbations

C_{n}^{2} = 10^{- 14} m^{- 2 / 3}

[45

R. A. Johnston, N. J. Wooder, F. C. Reavell, M. Bernhardt, and C. Dainty, “Horizontal scintillation detection and ranging $C_{n}^{2}$ estimation,” Appl. Opt. 42, 3451–3459 (2003). [CrossRef] [PubMed]

] and a beam size w₀ = 6 cm. At w₀/r₀ = 0.65, where D̃ has decayed to 50% of its initial level, we find a propagation length of 2 km (satisfying the requirement L < z_R). This distance would suffice for use in a metropolitan environment.

It is interesting to compare this result to values published in the literature for single-photon horizontal transport, such as given by Ref. [17

] (15 m), Ref. [16

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005). [CrossRef] [PubMed]

] (200 m), and Refs. [20

G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25, 225–230 (2008). [CrossRef]

–22

S. P. Walborn, D. S. Lemelle, D. S. Tasca, and P. H. Souto Ribeiro, “Schemes for quantum key distribution with higher-order alphabets using single-photon fractional Fourier optics,” Phys. Rev. A 77, 062323 (2008). [CrossRef]

] (1 km or further). However, one should be careful in comparing these numbers, since all these experiments have their own system characteristics and measurement criteria.

For vertical propagation from ground level through the entire column of the atmosphere, the Fried parameter is typically of the order of 5–15 cm, depending on the elevation and weather conditions [46

M. C. Roggemann, B. M. Welsh, and R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997). [CrossRef]

, 47

S.-X. Li, Y.-F. Fu, Y.-L. Huang, J.-G. Li, and J.-T. Mao, “Calculation and statistical analysis of the Fried parameter r₀ of astronomical seeing in China,” Chin. Astron. Astrophys. 28, 222–237 (2004). [CrossRef]

]. For the turbulence strength and beam size mentioned above, the Fried parameter equals 9 cm, suggesting that also a satellite communication link may be viable.

6. Conclusions

We have presented the first experimental data on the transmission of OAM-entangled photons through a turbulent atmosphere. We have found that the shape of the coincidence curve is quite robust under the action of the turbulence, and that the robustness can be enhanced by judiciously designing the OAM superposition that acts as an information carrier. Present-day adaptive-optical techniques are sufficiently developed that OAM entanglement for free-space distribution is viable.

Appendices

7. Appendix

In this Appendix, we show the full measurement set of the coincidence count rates used for Figs. 4 and 5. Figures 7 and 8 show data obtained with half-sector phase plates and quadrant-sector phase plates, respectively. The agreement between experimental data (open circles) and theoretical predictions (solid curves) is seen to be excellent. Note that the theoretical curves require just a single fit parameter; a trivial vertical scaling factor, which is determined for the initial case of no turbulence (see Figs. 7(a) and 8(a)) and kept fixed for increasing turbulence strengths (see Figs. 7(b)–7(g) and 8(b)–8(g)).

Fig. 7 Coincidence curves for two half-sector phase plates. Circles denote experimental data points measured during a 20 s collection time. Curves denote theoretical predictions, using no fit parameter other than a trivial scaling factor that is determined in the absence of turbulence and kept fixed for all other graphs. (a) w₀/r₀ = 0 (no turbulence), (b) w₀/r₀ = 0.24, (c) w₀/r₀ = 0.30, (d) w₀/r₀ = 0.36, (e) w₀/r₀ = 0.42, (f) w₀/r₀ = 0.53, (g) w₀/r₀ = 0.65.

Fig. 8 Coincidence curves for two quadrant-sector phase plates. Circles denote experimental data points measured during a 20 s collection time. Curves denote theoretical predictions, using no fit parameter other than a trivial scaling factor that is determined in the absence of turbulence and kept fixed for all other graphs. (a) w₀/r₀ = 0 (no turbulence), (b) w₀/r₀ = 0.24, (c) w₀/r₀ = 0.30, (d) w₀/r₀ = 0.36, (e) w₀/r₀ = 0.42, (f) w₀/r₀ = 0.53, (g) w₀/r₀ = 0.65.

Acknowledgments

We acknowledge valuable discussions with Steven Habraken, Laurent Jolissaint and Remko Stuik. CHM acknowledges financial support from the Brazilian agencies CNPq and CAPES. This project received funding from the Stichting voor Fundamenteel Onderzoek der Materie (FOM) and from the EU Seventh Framework Programme HIDEAS (grant agreement no. 221906).

References and links

1.	A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991). [CrossRef] [PubMed]
2.	A. Mair, G. W. A. Vaziri, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). [CrossRef] [PubMed]
3.	D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Observing quantum correlation of photons in Laguerre-Gauss modes using the Gouy phase,” Phys. Rev. Lett. 101, 050501 (2008). [CrossRef] [PubMed]
4.	J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008). [CrossRef] [PubMed]
5.	H. de Riedmatten, I. Marcikic, H. Zbinden, and N. Gisin, “Creating high dimensional entanglement using mode-locked lasers,” Quant. Inf. Comput. 2, 425–433 (2002).
6.	I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007). [CrossRef] [PubMed]
7.	T. Durt, D. Kaszlikowski, J.-L. Chen, and L. C. Kwek, “Security of quantum key distributions with entangled qudits,” Phys. Rev. A 69, 032313 (2004). [CrossRef]
8.	Č. Brukner, T. Paternek, and M. Żukowski, “Quantum communication complexity protocols based on higher-dimensional entangled systems,” Int. J. Quantum Inf. 1, 519–525 (2003). [CrossRef]
9.	B.-J. Pors, C. H. Monken, E. R. Eliel, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” arXiv:0909.3750v1 [quant-ph] (2010).
10.	A. Poppe, A. Fedrizzi, R. Ursin, H. R. Böhm, T. Lorünser, O. Maurhardt, M. Peev, M. Suda, C. Kurtsiefer, H. Weinfurter, T. Jennewein, and A. Zeilinger, “Practical quantum key distribution with polarization entangled photons,” Opt. Express 12, 3865–3871 (2004). [CrossRef] [PubMed]
11.	Q. Zhang, H. Takesue, S. W. Nam, C. Langrock, X. Xie, B. Baek, M. M. Fejer, and Y. Yamamoto, “Distribution of time-energy entanglement over 100 km fiber using superconducting single-photon detectors,” Opt. Express 16, 5776–5781 (2008). [CrossRef] [PubMed]
12.	D. Salart, A. Baas, C. Branciard, N. Gisin, and H. Zbinden, “Testing the speed of ’spooky action at a distance’,” Nature 454, 861–864 (2008). [CrossRef] [PubMed]
13.	K. J. Resch, M. Lindenthal, B. Blauensteiner, H. R. Böhm, A. Fedrizzi, C. Kurtsiefer, A. Poppe, T. Schmitt-Manderbach, M. Taraba, R. Ursin, P. Walther, H. Weier, H. Weinfurter, and A. Zeilinger, “Distributing entanglement and single photons through an intra-city, free-space quantum channel,” Opt. Express 13, 202–209 (2005). [CrossRef] [PubMed]
14.	C.-Z. Peng, T. Yang, X.-H. Bao, J. Zhang, X.-M. Jin, F.-Y. Feng, B. Yang, J. Yang, J. Yin, Q. Zhang, N. Li, B.-L. Tian, and J.-W. Pan, “Experimental free-space distribution of entangled photon pairs over 13 km: towards satellite-based global quantum communication,” Phys. Rev. Lett. 94, 150501 (2005). [CrossRef] [PubMed]
15.	R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007). [CrossRef]
16.	C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005). [CrossRef] [PubMed]
17.	G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004). [CrossRef] [PubMed]
18.	B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006). [CrossRef]
19.	C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” N. J. Phys. 9, 94 (2007). [CrossRef]
20.	G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25, 225–230 (2008). [CrossRef]
21.	J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47, 2414–2429 (2008). [CrossRef] [PubMed]
22.	S. P. Walborn, D. S. Lemelle, D. S. Tasca, and P. H. Souto Ribeiro, “Schemes for quantum key distribution with higher-order alphabets using single-photon fractional Fourier optics,” Phys. Rev. A 77, 062323 (2008). [CrossRef]
23.	G. A. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34, 142–144 (2009). [CrossRef] [PubMed]
24.	F. S. Roux, “Decoherence of orbital angular momentum entanglement in a turbulent atmosphere,” arXiv:1009.1956v2 [physics.optics] (2010).
25.	J. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301 (2003). [CrossRef]
26.	Within the mode space our analyzers have access to, we can safely approximate c_lp to be independent of l.
27.	M. P. van Exter, P. S. K. Lee, S. Doesburg, and J. P. Woerdman, “Mode counting in high-dimensional orbital angular momentum entanglement,” Opt. Express 15, 6431–6438 (2007). [CrossRef] [PubMed]
28.	J. B. Pors, A. Aiello, S. S. R. Oemrawsingh, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, “Angular phase-plate analyzers for measuring the dimensionality of multi-mode fields,” Phys. Rev. A 77, 033845 (2008). [CrossRef]
29.	O. Keskin, L. Jolissaint, and C. Bradley, “Hot-air optical turbulence generator for the testing of adaptive optics systems: principles and characterization,” Appl. Opt. 45, 4888–4897 (2006). [CrossRef] [PubMed]
30.	V. I. Tatarski, Wave propagation in a turbulent medium , 2nd ed. (Dover Publications Inc., 1961),
31.	D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966). [CrossRef]
32.	R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975). [CrossRef]
33.	R. L. Lucke and C. Y. Young, “Theoretical wave structure function when the effect of the outer scale is significant,” Appl. Opt. 46, 559–569 (2007). [CrossRef] [PubMed]
34.	B. M. Levine, E. A. Martinsen, A. Wirth, A. Jankevics, M. Toledo-Quinones, F. Landers, and T. L. Bruno, “Horizontal line-of-sight turbulence over near-ground paths and implications for adaptive optics corrections in laser communications,” Appl. Opt. 37, 4553–4560 (1998). [CrossRef]
35.	L. Kral, I. Prochazka, and K. Hamal, “Optical signal path delay fluctuations caused by atmospheric turbulence,” Opt. Lett. 30, 1767–1769 (2005). [CrossRef] [PubMed]
36.	T. Schmitt-Manderbach, “Long distance free-space quantum key distribution,” Ph.D. thesis, Ludwig-Maximilians-Universität München (2007).
37.	B. Schumacher, “Sending entanglement through noisy quantum channels,” Phys. Rev. A 54, 2614–2628 (1996). [CrossRef] [PubMed]
38.	M. B. Plenio and S. Virmani, “An introduction to entanglement measures,” arXiv:quant-ph/0504163v3 (2006).
39.	S. Ryu, W. Cai, and A. Caro, “Quantum entanglement of formation between qudits,” Phys. Rev. A 77, 052312 (2008). [CrossRef]
40.	W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998). [CrossRef]
41.	A. K. Jha, G. A. Tyler, and R. W. Boyd, “Effects of atmospheric turbulence on the entanglement of spatial two-qubit states,” Phys. Rev. A 81, 053832 (2010). [CrossRef]
42.	We note that in our experiment we have turbulence in one arm only. For this case, the Shannon dimensionality can be generalized as D̃ = Tr(〈ρ_A〉_tρ_B)/Tr(〈ρ_A〉_t,α 〈ρ_B〈_β). It can be shown that this reduces to Eq. (7) when one has similar but weaker turbulence in both arms.
43.	In this limit for extreme turbulence, the azimuthal fingerprint of the analyzer mode is fully wiped out. The detection state thus becomes circularly isotropic, leading to D̃ = 1.
44.	D. L. Fried, “Anisoplanism in adaptive optics,” J. Opt. Soc. Am. 72, 52–61 (1982). [CrossRef]
45.	R. A. Johnston, N. J. Wooder, F. C. Reavell, M. Bernhardt, and C. Dainty, “Horizontal scintillation detection and ranging $C_{n}^{2}$ estimation,” Appl. Opt. 42, 3451–3459 (2003). [CrossRef] [PubMed]
46.	M. C. Roggemann, B. M. Welsh, and R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997). [CrossRef]
47.	S.-X. Li, Y.-F. Fu, Y.-L. Huang, J.-G. Li, and J.-T. Mao, “Calculation and statistical analysis of the Fried parameter r₀ of astronomical seeing in China,” Chin. Astron. Astrophys. 28, 222–237 (2004). [CrossRef]

OCIS Codes
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(030.7060) Coherence and statistical optics : Turbulence
(270.5565) Quantum optics : Quantum communications

ToC Category:
Quantum Optics

History
Original Manuscript: December 1, 2010
Revised Manuscript: February 18, 2011
Manuscript Accepted: February 20, 2011
Published: March 24, 2011

Citation
Bart-Jan Pors, C. H. Monken, Eric R. Eliel, and J. P. Woerdman, "Transport of orbital-angular-momentum entanglement through a turbulent atmosphere," Opt. Express 19, 6671-6683 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6671

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References

A. K. Ekert, "Quantum cryptography based on Bell’s theorem," Phys. Rev. Lett. 67, 661-663 (1991). [CrossRef] [PubMed]
A. Mair, G. W. A. Vaziri, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001). [CrossRef] [PubMed]
D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, "Observing quantum correlation of photons in Laguerre-Gauss modes using the Gouy phase," Phys. Rev. Lett. 101, 050501 (2008). [CrossRef] [PubMed]
J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, "Shannon dimensionality of quantum channels and its application to photon entanglement," Phys. Rev. Lett. 101, 120502 (2008). [CrossRef] [PubMed]
H. de Riedmatten, I. Marcikic, H. Zbinden, and N. Gisin, "Creating high dimensional entanglement using modelocked lasers," Quant. Inf. Comput. 2, 425-433 (2002).
I. Ali-Khan, C. J. Broadbent, and J. C. Howell, "Large-alphabet quantum key distribution using energy-time entangled bipartite states," Phys. Rev. Lett. 98, 060503 (2007). [CrossRef] [PubMed]
T. Durt, D. Kaszlikowski, J.-L. Chen, and L. C. Kwek, "Security of quantum key distributions with entangled qudits," Phys. Rev. A 69, 032313 (2004). [CrossRef]
ˇ. C. Brukner, T. Paternek, and M. ˙. Zukowski, "Quantum communication complexity protocols based on higher dimensional entangled systems," Int. J. Quantum Inf. 1, 519-525 (2003). [CrossRef]
B.-J. Pors, C. H. Monken, E. R. Eliel, and J. P. Woerdman, "Transport of orbital-angular-momentum entanglement through a turbulent atmosphere," arXiv:0909.3750v1 [quant-ph] (2010).
A. Poppe, A. Fedrizzi, R. Ursin, H. R. Böhm, T. Lorünser, O. Maurhardt, M. Peev, M. Suda, C. Kurtsiefer, H. Weinfurter, T. Jennewein, and A. Zeilinger, "Practical quantum key distribution with polarization entangled photons," Opt. Express 12, 3865-3871 (2004). [CrossRef] [PubMed]
Q. Zhang, H. Takesue, S. W. Nam, C. Langrock, X. Xie, B. Baek, M. M. Fejer, and Y. Yamamoto, "Distribution of time-energy entanglement over 100 km fiber using superconducting single-photon detectors," Opt. Express 16, 5776-5781 (2008). [CrossRef] [PubMed]
D. Salart, A. Baas, C. Branciard, N. Gisin, and H. Zbinden, "Testing the speed of ’spooky action at a distance’," Nature 454, 861-864 (2008). [CrossRef] [PubMed]
K. J. Resch, M. Lindenthal, B. Blauensteiner, H. R. Böhm, A. Fedrizzi, C. Kurtsiefer, A. Poppe, T. Schmitt-Manderbach, M. Taraba, R. Ursin, P. Walther, H. Weier, H. Weinfurter, and A. Zeilinger, "Distributing entanglement and single photons through an intra-city, free-space quantum channel," Opt. Express 13, 202-209 (2005). [CrossRef] [PubMed]
C.-Z. Peng, T. Yang, X.-H. Bao, J. Zhang, X.-M. Jin, F.-Y. Feng, B. Yang, J. Yang, J. Yin, Q. Zhang, N. Li, B.-L. Tian, and J.-W. Pan, "Experimental free-space distribution of entangled photon pairs over 13 km: towards satellite-based global quantum communication," Phys. Rev. Lett. 94, 150501 (2005). [CrossRef] [PubMed]
R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, "Entanglement-based quantum communication over 144 km," Nat. Phys. 3, 481-486 (2007). [CrossRef]
C. Paterson, "Atmospheric turbulence and orbital angular momentum of single photons for optical communication," Phys. Rev. Lett. 94, 153901 (2005). [CrossRef] [PubMed]
G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, "Free-space information transfer using light beams carrying orbital angular momentum," Opt. Express 12, 5448-5456 (2004). [CrossRef] [PubMed]
B. J. Smith and M. G. Raymer, "Two-photon wave mechanics," Phys. Rev. A 74, 062104 (2006). [CrossRef]
C. Gopaul and R. Andrews, "The effect of atmospheric turbulence on entangled orbital angular momentum states," N. J. Phys. 9, 94 (2007). [CrossRef]
G. Gbur and R. K. Tyson, "Vortex beam propagation through atmospheric turbulence and topological charge conservation," J. Opt. Soc. Am. A 25, 225-230 (2008). [CrossRef]
J. A. Anguita, M. A. Neifeld, and B. V. Vasic, "Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link," Appl. Opt. 47, 2414-2429 (2008). [CrossRef] [PubMed]
S. P. Walborn, D. S. Lemelle, D. S. Tasca, and P. H. Souto Ribeiro, "Schemes for quantum key distribution with higher-order alphabets using single-photon fractional Fourier optics," Phys. Rev. A 77, 062323 (2008). [CrossRef]
G. A. Tyler and R. W. Boyd, "Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum," Opt. Lett. 34, 142-144 (2009). [CrossRef] [PubMed]
F. S. Roux, "Decoherence of orbital angular momentum entanglement in a turbulent atmosphere," arXiv:1009.1956v2 [physics.optics] (2010).
J. P. Torres, A. Alexandrescu, and L. Torner, "Quantum spiral bandwidth of entangled two-photon states," Phys. Rev. A 68, 050301 (2003). [CrossRef]
Within the mode space our analyzers have access to, we can safely approximate cl p to be independent of l.
M. P. van Exter, P. S. K. Lee, S. Doesburg, and J. P. Woerdman, "Mode counting in high-dimensional orbital angular momentum entanglement," Opt. Express 15, 6431-6438 (2007). [CrossRef] [PubMed]
J. B. Pors, A. Aiello, S. S. R. Oemrawsingh, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, "Angular phase plate analyzers for measuring the dimensionality of multi-mode fields," Phys. Rev. A 77, 033845 (2008). [CrossRef]
O. Keskin, L. Jolissaint, and C. Bradley, "Hot-air optical turbulence generator for the testing of adaptive optics systems: principles and characterization," Appl. Opt. 45, 4888-4897 (2006). [CrossRef] [PubMed]
V. I. Tatarski, Wave propagation in a turbulent medium, 2nd ed. (Dover Publications Inc., 1961)
D. L. Fried, "Optical resolution through a randomly inhomogeneous medium for very long and very short exposures," J. Opt. Soc. Am. 56, 1372-1379 (1966). [CrossRef]
R. L. Fante, "Electromagnetic beam propagation in turbulent media," Proc. IEEE 63, 1669-1692 (1975). [CrossRef]
R. L. Lucke and C. Y. Young, "Theoretical wave structure function when the effect of the outer scale is significant," Appl. Opt. 46, 559-569 (2007). [CrossRef] [PubMed]
B. M. Levine, E. A. Martinsen, A. Wirth, A. Jankevics, M. Toledo-Quinones, F. Landers, and T. L. Bruno, "Horizontal line-of-sight turbulence over near-ground paths and implications for adaptive optics corrections in laser communications," Appl. Opt. 37, 4553-4560 (1998). [CrossRef]
L. Kral, I. Prochazka, and K. Hamal, "Optical signal path delay fluctuations caused by atmospheric turbulence," Opt. Lett. 30, 1767-1769 (2005). [CrossRef] [PubMed]
T. Schmitt-Manderbach, "Long distance free-space quantum key distribution," Ph.D. thesis, Ludwig-Maximilians-Universität München (2007).
B. Schumacher, "Sending entanglement through noisy quantum channels," Phys. Rev. A 54, 2614-2628 (1996). [CrossRef] [PubMed]
M. B. Plenio and S. Virmani, "An introduction to entanglement measures," arXiv:quant-ph/0504163v3 (2006).
S. Ryu, W. Cai, and A. Caro, "Quantum entanglement of formation between qudits," Phys. Rev. A 77, 052312 (2008). [CrossRef]
W. K. Wootters, "Entanglement of formation of an arbitrary state of two qubits," Phys. Rev. Lett. 80, 2245-2248 (1998). [CrossRef]
A. K. Jha, G. A. Tyler, and R. W. Boyd, "Effects of atmospheric turbulence on the entanglement of spatial two-qubit states," Phys. Rev. A 81, 053832 (2010). [CrossRef]
We note that in our experiment we have turbulence in one arm only. For this case, the Shannon dimensionality can be generalized as D = Tr(〈ρA〉tρB)/Tr(〈ρA〉t,α 〈ρB〉β). It can be shown that this reduces to Eq. (7) when one has similar but weaker turbulence in both arms.
In this limit for extreme turbulence, the azimuthal fingerprint of the analyzer mode is fully wiped out. The detection state thus becomes circularly isotropic, leading to D = 1.
D. L. Fried, "Anisoplanism in adaptive optics," J. Opt. Soc. Am. 72, 52-61 (1982). [CrossRef]
R. A. Johnston, N. J. Wooder, F. C. Reavell, M. Bernhardt, and C. Dainty, "Horizontal scintillation detection and ranging C2 n estimation," Appl. Opt. 42, 3451-3459 (2003). [CrossRef] [PubMed]
M. C. Roggemann, B. M. Welsh, and R. Q. Fugate, "Improving the resolution of ground-based telescopes," Rev. Mod. Phys. 69, 437-505 (1997). [CrossRef]
S.-X. Li, Y.-F. Fu, Y.-L. Huang, J.-G. Li, and J.-T. Mao, "Calculation and statistical analysis of the Fried parameter r0 of astronomical seeing in China," Chin. Astron. Astrophys. 28, 222-237 (2004). [CrossRef]

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