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.

© 2011 Optical Society of America

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  42. 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.
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2010 (1)

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]

2009 (1)

2008 (9)

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

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]

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]

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]

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]

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]

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]

2007 (5)

C. Gopaul and R. Andrews, "The effect of atmospheric turbulence on entangled orbital angular momentum states," N. J. Phys. 9, 94 (2007).
[CrossRef]

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]

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]

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]

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]

2006 (2)

2005 (4)

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]

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

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

2004 (4)

2003 (3)

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]

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

2002 (1)

H. de Riedmatten, I. Marcikic, H. Zbinden, and N. Gisin, "Creating high dimensional entanglement using modelocked lasers," Quant. Inf. Comput. 2, 425-433 (2002).

2001 (1)

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]

1998 (2)

1997 (1)

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]

1996 (1)

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

1991 (1)

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

1982 (1)

1975 (1)

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

1966 (1)

’t Hooft, G. W.

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]

Aiello, A.

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]

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]

Alexandrescu, A.

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

Ali-Khan, I.

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]

Andrews, R.

C. Gopaul and R. Andrews, "The effect of atmospheric turbulence on entangled orbital angular momentum states," N. J. Phys. 9, 94 (2007).
[CrossRef]

Anguita, J. A.

Baas, A.

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]

Baek, B.

Bao, X.-H.

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]

Barbieri, C.

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]

Barnett, S. M.

Bernhardt, M.

Blauensteiner, B.

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]

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]

Böhm, H. R.

Boyd, R. W.

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]

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]

Bradley, C.

Branciard, C.

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]

Broadbent, C. J.

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]

Brukner, ?. C.

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]

Bruno, T. L.

Cai, W.

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

Caro, A.

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

Chen, J.-L.

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]

Courtial, J.

Dainty, C.

de Riedmatten, H.

H. de Riedmatten, I. Marcikic, H. Zbinden, and N. Gisin, "Creating high dimensional entanglement using modelocked lasers," Quant. Inf. Comput. 2, 425-433 (2002).

Doesburg, S.

Durt, T.

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]

Ekert, A. K.

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

Eliel, E. R.

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]

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]

Fante, R. L.

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

Fedrizzi, A.

Fejer, M. M.

Feng, F.-Y.

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Other (8)

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.

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

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

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

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).

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

Supplementary Material (3)

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Figures (8)

Fig. 1
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.

Fig. 2
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.

Fig. 4
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 αβ = π/2. Typically, the measurement values reproduce to within a few percent.

Fig. 3
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).

Fig. 5
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).

Fig. 6
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.

Fig. 7
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
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.

Equations (7)

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| Ψ = l , p c l p | l , p | l , p .
e i ϕ ( r 1 ) i ϕ ( r 2 ) t = e 1 2 6.88 [ r 1 r 2 r 0 ] 5 / 3 ,
w 0 r 0 = ( w l e / w d l ) 2 1 3.0 ,
P ( α β ) = | A ( α ) | B ( β ) | S ^ A | Ψ | 2 t
S ^ A | A ( α ) A ( α ) | S ^ A t = 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 .
D = 1 Tr [ ( ρ A α ) 2 ] .
D ˜ = Tr [ ( ρ A t ) 2 ] Tr [ ( ρ A t , α ) 2 ] .

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