Abstract

The continuous-variable (CV) orbital angular momentum (OAM) entanglement is very different to the traditional quadrature entanglement. The Stokes-operators directly reflect the character of OAM light. Here, we report the first direct experimental demonstration the Stokes-operator entanglement of continuous-variable OAM entanglement. Generated by transforming quadrature entanglement in the HG01 mode onto the orbital Stokes-operator basis, the entanglement is measured in the Stokes-operator basis using a self-designed detection scheme. An inseparability of I(O^2,O^3)<1 is achieved over a wide analyzing frequency of 1–10 MHz. Moreover, experimental fluctuations at 5.0 MHz are visualized using the quantum orbital Poincaré sphere representation. The OAM entanglement with Stokes-operators measurement has a promising application in certain nonlocal quantum information protocols and rotational optomechanics by interacting with nanoparticle or atoms.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (6)

M. P. J. Lavery, Ch. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, Ch. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, 1700552 (2017).
[Crossref]

A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “High-dimensional intracity quantum cryptography with structured photons,” Optica 4(9), 1006–1010 (2017).
[Crossref]

M. J. Padgett, “Orbital angular momentum 25 years on,” Opt. Express 25(10), 11265–11274 (2017).
[Crossref] [PubMed]

J. Guo, C. Cai, L. Ma, K. Liu, H. Sun, and J. Gao, “Measurement of Stokes-operator squeezing for continuous-variable orbital angular momentum,” Sci. Rep. 7, 4434 (2017).
[Crossref] [PubMed]

J. Guo, C. Cai, L. Ma, K. Liu, H. Sun, and J. Gao, “Higher order mode entanglement in a type ii optical parametric oscillator,” Opt. Express 25, 4985–4993 (2017).
[Crossref] [PubMed]

Z. Yin and T. Li, “Bringing quantum mechanics to life: from Schrödinger’s cat to Schrödinger’s microbe,” Contemp. Phys. 58(2), 119–139 (2017).
[Crossref]

2016 (2)

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10,010,” PNAS 113(48), 13642–13647 (2016).
[Crossref] [PubMed]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10(4), 248–252 (2016).
[Crossref]

2015 (3)

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7(1), 66–106 (2015).
[Crossref]

X. L. Wang, X. D. Cai, Z. E. Su, M. C. Chen, D. Wu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref] [PubMed]

H. Shi and M. Bhattacharya, “Optomechanics based on angular momentum exchange between light and matter,” J. Phys. B 49, 153001 (2015).
[Crossref]

2014 (5)

N. Behbood, F. Martin Ciurana, G. Colangelo, M. Napolitano, G. Tóth, R. J. Sewell, and M. W. Mitchell, “Generation of macroscopic singlet states in a cold atomic ensemble,” Phys. Rev. Lett. 113, 093601 (2014).
[Crossref] [PubMed]

C. Peuntinger, B. Heim, C. R. Müller, C. Gabriel, Ch. Marquardt, and G. Leuchs, “Distribution of squeezed states through an atmospheric channel,” Phys. Rev. Lett. 113, 060502 (2014).
[Crossref] [PubMed]

K. Liu, J. Guo, C. Cai, S. Guo, and J. Gao, “Experimental generation of continuous-variable hyperentanglement in an optical parametric oscillator,” Phys. Rev. Lett. 113, 170501 (2014).
[Crossref] [PubMed]

A Nicolas, L Veissier, L Giner, E Giacobino, D Maxein, and J Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8, 234–238 (2014).
[Crossref]

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. USA 111, 6243–6247 (2014).
[Crossref]

2013 (2)

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545 (2013).
[Crossref] [PubMed]

2012 (2)

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[Crossref] [PubMed]

2011 (1)

2009 (4)

D. Elser, T. Bartley, B. Heim, C. Wittmann, D. Sych, and G. Leuchs, “Feasibility of free space quantum key distribution with coherent polarization states,” New J. Phys. 11(4), 045014 (2009).
[Crossref]

M. T. L. Hsu, W. P. Bowen, and P. K. Lam, “Spatial-state stokes-operator squeezing and entanglement for optical beams,” Phys. Rev. A 79, 043825 (2009).
[Crossref]

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous variable entanglement and squeezing of orbital angular momentum states,” Phys. Rev. Lett. 102, 163602 (2009).
[Crossref] [PubMed]

B dos Santos, K. Dechoum, and A. Khoury, “Continuous-variable hyperentanglement in a parametric oscillator with orbital angular momentum,” Phys. Rev. Lett. 103, 230503 (2009).
[Crossref]

2008 (1)

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282 (2008).
[Crossref]

2007 (2)

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. A. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
[Crossref] [PubMed]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[Crossref]

2006 (1)

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H. A. Bachor, “Generation of squeezing in higher order Hermite–Gaussian modes with an optical parametric amplifier,” J. Eur. Opt. Soc. Rapid Publ. 1, 06003 (2006).
[Crossref]

2004 (1)

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

2002 (2)

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and C. Silberhorn, “Polarization squeezing and continous-variable polarization entanglement,” Phys. Rev. A 65(2), 052306 (2002).
[Crossref]

W. P. Bowen, N. Treps, R. Schnabel, and P. K. Lam, “Experimental demonstration of continuous variable polarization entanglement,” Phys. Rev. Lett 89, 253601 (2002).
[Crossref] [PubMed]

2001 (1)

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

1999 (1)

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular-momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Ahmed, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7(1), 66–106 (2015).
[Crossref]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular-momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Andersen, U. L.

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous variable entanglement and squeezing of orbital angular momentum states,” Phys. Rev. Lett. 102, 163602 (2009).
[Crossref] [PubMed]

Aolita, L.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Ashrafi, N.

Ashrafi, S.

Bachor, H. A.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. A. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
[Crossref] [PubMed]

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H. A. Bachor, “Generation of squeezing in higher order Hermite–Gaussian modes with an optical parametric amplifier,” J. Eur. Opt. Soc. Rapid Publ. 1, 06003 (2006).
[Crossref]

Banzer, P.

M. P. J. Lavery, Ch. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, Ch. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, 1700552 (2017).
[Crossref]

Bao, C.

Barreiro, J. T.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282 (2008).
[Crossref]

Bartlett, S. D.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Bartley, T.

D. Elser, T. Bartley, B. Heim, C. Wittmann, D. Sych, and G. Leuchs, “Feasibility of free space quantum key distribution with coherent polarization states,” New J. Phys. 11(4), 045014 (2009).
[Crossref]

Behbood, N.

N. Behbood, F. Martin Ciurana, G. Colangelo, M. Napolitano, G. Tóth, R. J. Sewell, and M. W. Mitchell, “Generation of macroscopic singlet states in a cold atomic ensemble,” Phys. Rev. Lett. 113, 093601 (2014).
[Crossref] [PubMed]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular-momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Bhattacharya, M.

H. Shi and M. Bhattacharya, “Optomechanics based on angular momentum exchange between light and matter,” J. Phys. B 49, 153001 (2015).
[Crossref]

Bouchard, F.

Bowen, W. P.

M. T. L. Hsu, W. P. Bowen, and P. K. Lam, “Spatial-state stokes-operator squeezing and entanglement for optical beams,” Phys. Rev. A 79, 043825 (2009).
[Crossref]

W. P. Bowen, N. Treps, R. Schnabel, and P. K. Lam, “Experimental demonstration of continuous variable polarization entanglement,” Phys. Rev. Lett 89, 253601 (2002).
[Crossref] [PubMed]

Boyd, R. W.

A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “High-dimensional intracity quantum cryptography with structured photons,” Optica 4(9), 1006–1010 (2017).
[Crossref]

M. P. J. Lavery, Ch. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, Ch. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, 1700552 (2017).
[Crossref]

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545 (2013).
[Crossref] [PubMed]

Buchhave, P.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. A. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
[Crossref] [PubMed]

Buchler, B.

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10,010,” PNAS 113(48), 13642–13647 (2016).
[Crossref] [PubMed]

Cai, C.

J. Guo, C. Cai, L. Ma, K. Liu, H. Sun, and J. Gao, “Measurement of Stokes-operator squeezing for continuous-variable orbital angular momentum,” Sci. Rep. 7, 4434 (2017).
[Crossref] [PubMed]

J. Guo, C. Cai, L. Ma, K. Liu, H. Sun, and J. Gao, “Higher order mode entanglement in a type ii optical parametric oscillator,” Opt. Express 25, 4985–4993 (2017).
[Crossref] [PubMed]

K. Liu, J. Guo, C. Cai, S. Guo, and J. Gao, “Experimental generation of continuous-variable hyperentanglement in an optical parametric oscillator,” Phys. Rev. Lett. 113, 170501 (2014).
[Crossref] [PubMed]

C. Cai, K. Liu, and J. Gao, “Rotation angular measurement beyond quantum noise limit with an orbital angular position squeezed state,” in Quantum Information and Measurement (Optical Society of America, 2017), pp. QF3A-5.

Cai, X. D.

X. L. Wang, X. D. Cai, Z. E. Su, M. C. Chen, D. Wu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref] [PubMed]

Campbell, G.

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10,010,” PNAS 113(48), 13642–13647 (2016).
[Crossref] [PubMed]

Cao, Y.

Chen, M. C.

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Khoury, A.

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M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10(4), 248–252 (2016).
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R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10,010,” PNAS 113(48), 13642–13647 (2016).
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M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H. A. Bachor, “Generation of squeezing in higher order Hermite–Gaussian modes with an optical parametric amplifier,” J. Eur. Opt. Soc. Rapid Publ. 1, 06003 (2006).
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M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. USA 111, 6243–6247 (2014).
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R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
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M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous variable entanglement and squeezing of orbital angular momentum states,” Phys. Rev. Lett. 102, 163602 (2009).
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J. Guo, C. Cai, L. Ma, K. Liu, H. Sun, and J. Gao, “Measurement of Stokes-operator squeezing for continuous-variable orbital angular momentum,” Sci. Rep. 7, 4434 (2017).
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X. L. Wang, X. D. Cai, Z. E. Su, M. C. Chen, D. Wu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
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[Crossref]

Molisch, A. F.

Müller, C. R.

C. Peuntinger, B. Heim, C. R. Müller, C. Gabriel, Ch. Marquardt, and G. Leuchs, “Distribution of squeezed states through an atmospheric channel,” Phys. Rev. Lett. 113, 060502 (2014).
[Crossref] [PubMed]

Napolitano, M.

N. Behbood, F. Martin Ciurana, G. Colangelo, M. Napolitano, G. Tóth, R. J. Sewell, and M. W. Mitchell, “Generation of macroscopic singlet states in a cold atomic ensemble,” Phys. Rev. Lett. 113, 093601 (2014).
[Crossref] [PubMed]

Nicolas, A

A Nicolas, L Veissier, L Giner, E Giacobino, D Maxein, and J Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8, 234–238 (2014).
[Crossref]

O’Brien, J. L.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Padgett, M.

Padgett, M. J.

M. J. Padgett, “Orbital angular momentum 25 years on,” Opt. Express 25(10), 11265–11274 (2017).
[Crossref] [PubMed]

M. P. J. Lavery, Ch. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, Ch. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, 1700552 (2017).
[Crossref]

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
[Crossref]

Pan, J. W.

X. L. Wang, X. D. Cai, Z. E. Su, M. C. Chen, D. Wu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref] [PubMed]

Peuntinger, C.

Peuntinger, Ch.

M. P. J. Lavery, Ch. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, Ch. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, 1700552 (2017).
[Crossref]

Plick, W. N.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[Crossref] [PubMed]

Pryde, G. J.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Ralph, T. C.

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and C. Silberhorn, “Polarization squeezing and continous-variable polarization entanglement,” Phys. Rev. A 65(2), 052306 (2002).
[Crossref]

Ramachandran, S.

Ramelow, S.

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. USA 111, 6243–6247 (2014).
[Crossref]

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[Crossref] [PubMed]

Ren, Y.

Ren, Y. X.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Schaeff, C.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[Crossref] [PubMed]

Schnabel, R.

W. P. Bowen, N. Treps, R. Schnabel, and P. K. Lam, “Experimental demonstration of continuous variable polarization entanglement,” Phys. Rev. Lett 89, 253601 (2002).
[Crossref] [PubMed]

Sciarrino, F.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Sewell, R. J.

N. Behbood, F. Martin Ciurana, G. Colangelo, M. Napolitano, G. Tóth, R. J. Sewell, and M. W. Mitchell, “Generation of macroscopic singlet states in a cold atomic ensemble,” Phys. Rev. Lett. 113, 093601 (2014).
[Crossref] [PubMed]

Shi, H.

H. Shi and M. Bhattacharya, “Optomechanics based on angular momentum exchange between light and matter,” J. Phys. B 49, 153001 (2015).
[Crossref]

Silberhorn, C.

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and C. Silberhorn, “Polarization squeezing and continous-variable polarization entanglement,” Phys. Rev. A 65(2), 052306 (2002).
[Crossref]

Sit, A.

Slussarenko, S.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Spagnolo, N.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular-momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Su, Z. E.

X. L. Wang, X. D. Cai, Z. E. Su, M. C. Chen, D. Wu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref] [PubMed]

Sun, H.

J. Guo, C. Cai, L. Ma, K. Liu, H. Sun, and J. Gao, “Measurement of Stokes-operator squeezing for continuous-variable orbital angular momentum,” Sci. Rep. 7, 4434 (2017).
[Crossref] [PubMed]

J. Guo, C. Cai, L. Ma, K. Liu, H. Sun, and J. Gao, “Higher order mode entanglement in a type ii optical parametric oscillator,” Opt. Express 25, 4985–4993 (2017).
[Crossref] [PubMed]

Sych, D.

D. Elser, T. Bartley, B. Heim, C. Wittmann, D. Sych, and G. Leuchs, “Feasibility of free space quantum key distribution with coherent polarization states,” New J. Phys. 11(4), 045014 (2009).
[Crossref]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[Crossref]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[Crossref]

Tóth, G.

N. Behbood, F. Martin Ciurana, G. Colangelo, M. Napolitano, G. Tóth, R. J. Sewell, and M. W. Mitchell, “Generation of macroscopic singlet states in a cold atomic ensemble,” Phys. Rev. Lett. 113, 093601 (2014).
[Crossref] [PubMed]

Treps, N.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. A. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
[Crossref] [PubMed]

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H. A. Bachor, “Generation of squeezing in higher order Hermite–Gaussian modes with an optical parametric amplifier,” J. Eur. Opt. Soc. Rapid Publ. 1, 06003 (2006).
[Crossref]

W. P. Bowen, N. Treps, R. Schnabel, and P. K. Lam, “Experimental demonstration of continuous variable polarization entanglement,” Phys. Rev. Lett 89, 253601 (2002).
[Crossref] [PubMed]

Tur, M.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7(1), 66–106 (2015).
[Crossref]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545 (2013).
[Crossref] [PubMed]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Vaziri, A.

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

Veissier, L

A Nicolas, L Veissier, L Giner, E Giacobino, D Maxein, and J Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8, 234–238 (2014).
[Crossref]

Wagner, K.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. A. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
[Crossref] [PubMed]

Walborn, S. P.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Wang, J.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7(1), 66–106 (2015).
[Crossref]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Wang, X. L.

X. L. Wang, X. D. Cai, Z. E. Su, M. C. Chen, D. Wu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref] [PubMed]

Wei, T. C.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282 (2008).
[Crossref]

Weihs, G.

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

White, A. G.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Willner, A. E.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7(1), 66–106 (2015).
[Crossref]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545 (2013).
[Crossref] [PubMed]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Wittmann, C.

D. Elser, T. Bartley, B. Heim, C. Wittmann, D. Sych, and G. Leuchs, “Feasibility of free space quantum key distribution with coherent polarization states,” New J. Phys. 11(4), 045014 (2009).
[Crossref]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular-momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Wu, D.

X. L. Wang, X. D. Cai, Z. E. Su, M. C. Chen, D. Wu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref] [PubMed]

Xie, G.

Yan, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7(1), 66–106 (2015).
[Crossref]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Yang, J. Y.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Yao, A. M.

Yin, Z.

Z. Yin and T. Li, “Bringing quantum mechanics to life: from Schrödinger’s cat to Schrödinger’s microbe,” Contemp. Phys. 58(2), 119–139 (2017).
[Crossref]

Yue, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545 (2013).
[Crossref] [PubMed]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Zeilinger, A.

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10,010,” PNAS 113(48), 13642–13647 (2016).
[Crossref] [PubMed]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10(4), 248–252 (2016).
[Crossref]

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. USA 111, 6243–6247 (2014).
[Crossref]

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[Crossref] [PubMed]

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

Zhao, Z.

Adv. Opt. Photon. (2)

Contemp. Phys. (1)

Z. Yin and T. Li, “Bringing quantum mechanics to life: from Schrödinger’s cat to Schrödinger’s microbe,” Contemp. Phys. 58(2), 119–139 (2017).
[Crossref]

J. Eur. Opt. Soc. Rapid Publ. (1)

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H. A. Bachor, “Generation of squeezing in higher order Hermite–Gaussian modes with an optical parametric amplifier,” J. Eur. Opt. Soc. Rapid Publ. 1, 06003 (2006).
[Crossref]

J. Phys. B (1)

H. Shi and M. Bhattacharya, “Optomechanics based on angular momentum exchange between light and matter,” J. Phys. B 49, 153001 (2015).
[Crossref]

Nat. Commun. (1)

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Nat. Photonics (3)

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10(4), 248–252 (2016).
[Crossref]

A Nicolas, L Veissier, L Giner, E Giacobino, D Maxein, and J Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8, 234–238 (2014).
[Crossref]

Nat. Phys. (2)

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282 (2008).
[Crossref]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[Crossref]

Nature (2)

X. L. Wang, X. D. Cai, Z. E. Su, M. C. Chen, D. Wu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref] [PubMed]

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

New J. Phys. (1)

D. Elser, T. Bartley, B. Heim, C. Wittmann, D. Sych, and G. Leuchs, “Feasibility of free space quantum key distribution with coherent polarization states,” New J. Phys. 11(4), 045014 (2009).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Optica (1)

Phys. Rev. A (3)

M. T. L. Hsu, W. P. Bowen, and P. K. Lam, “Spatial-state stokes-operator squeezing and entanglement for optical beams,” Phys. Rev. A 79, 043825 (2009).
[Crossref]

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and C. Silberhorn, “Polarization squeezing and continous-variable polarization entanglement,” Phys. Rev. A 65(2), 052306 (2002).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular-momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Phys. Rev. Lett (1)

W. P. Bowen, N. Treps, R. Schnabel, and P. K. Lam, “Experimental demonstration of continuous variable polarization entanglement,” Phys. Rev. Lett 89, 253601 (2002).
[Crossref] [PubMed]

Phys. Rev. Lett. (7)

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H. A. Bachor, P. Lam, N. Treps, P. Buchhave, C. Fabre, and C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
[Crossref] [PubMed]

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous variable entanglement and squeezing of orbital angular momentum states,” Phys. Rev. Lett. 102, 163602 (2009).
[Crossref] [PubMed]

B dos Santos, K. Dechoum, and A. Khoury, “Continuous-variable hyperentanglement in a parametric oscillator with orbital angular momentum,” Phys. Rev. Lett. 103, 230503 (2009).
[Crossref]

K. Liu, J. Guo, C. Cai, S. Guo, and J. Gao, “Experimental generation of continuous-variable hyperentanglement in an optical parametric oscillator,” Phys. Rev. Lett. 113, 170501 (2014).
[Crossref] [PubMed]

N. Behbood, F. Martin Ciurana, G. Colangelo, M. Napolitano, G. Tóth, R. J. Sewell, and M. W. Mitchell, “Generation of macroscopic singlet states in a cold atomic ensemble,” Phys. Rev. Lett. 113, 093601 (2014).
[Crossref] [PubMed]

C. Peuntinger, B. Heim, C. R. Müller, C. Gabriel, Ch. Marquardt, and G. Leuchs, “Distribution of squeezed states through an atmospheric channel,” Phys. Rev. Lett. 113, 060502 (2014).
[Crossref] [PubMed]

PNAS (1)

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10,010,” PNAS 113(48), 13642–13647 (2016).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. USA 111, 6243–6247 (2014).
[Crossref]

Sci. Adv. (1)

M. P. J. Lavery, Ch. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, Ch. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, 1700552 (2017).
[Crossref]

Sci. Rep. (1)

J. Guo, C. Cai, L. Ma, K. Liu, H. Sun, and J. Gao, “Measurement of Stokes-operator squeezing for continuous-variable orbital angular momentum,” Sci. Rep. 7, 4434 (2017).
[Crossref] [PubMed]

Science (2)

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[Crossref] [PubMed]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545 (2013).
[Crossref] [PubMed]

Other (3)

C. Cai, K. Liu, and J. Gao, “Rotation angular measurement beyond quantum noise limit with an orbital angular position squeezed state,” in Quantum Information and Measurement (Optical Society of America, 2017), pp. QF3A-5.

L. Kong, R. Liu, Z. Wang, Y. Si, W. Qi, S. Huang, C. Tu, Y. Li, W. Hu, F. Xu, Y. Lu, and H. Wang, “Complete orbital angular momentum Bell-state measurement and superdense coding,” https://arXiv:1709.03770 (2017).

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” https://arXiv:1708.06101 (2017).

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Figures (6)

Fig. 1
Fig. 1 The degree of inseparability for (a) I ( O ^ 1 , O ^ 2 ) , (b) I ( O ^ 3 , O ^ 1 ) and (c) I ( O ^ 2 , O ^ 3 ) as a function of the amplitude ratio (α10/α01) and the relative phase θ between HG01 and HG10 modes.
Fig. 2
Fig. 2 Experimental generation and characterization of entanglement of first-order OAM modes. The entanglement source, specifically, (a) HG01 mode quadrature and (b) OAM state entanglement, are measured in quadrature detection and Stokes detection, respectively. DBS: dichroic beam splitter; F1: flip mirrors; PBS: polarizing beam splitter; MS: measuring system; DP: Dove prism; SA: spectrum analyzer.
Fig. 3
Fig. 3 Correlation variance measurements of the HG01-mode entangled beam for (a) quadrature amplitude sum Δ A + B 2 X ^ 01 , (b) phase difference Δ A B 2 Y ^ 01 , and (c) degree of inseparability. In (a) and (b), trace (i) is the SNL and trace (ii) the correlation variance normalized to SNL. In (3), trace (i) is the bound of inseparability (unity), and trace (ii) degree of inseparability. All data are normalized with respect to shot-noise level. The measurement parameters of spectrum analyzer: RBW=300 kHz, VBW=200 Hz.
Fig. 4
Fig. 4 Measured variance spectra of quantum noise on (a) Δ A ± B 2 O ^ 2 and (b) Δ A ± B 2 O ^ 3 for OAM entangled beams. In (a) and (b), trace (i) is the SNL, trace (ii) the correlation variance and trace (iii) the anti-correlation variance normalized to SNL. All data are normalized with respect to shot noise level. The measurement parameters of the spectral analyzer: RBW=300 kHz, VBW=200 Hz.
Fig. 5
Fig. 5 Experimental measurement of I ( O ^ 2 , O ^ 3 ) at Fourier frequencies from 1 to 10 MHz. Trace (i) is the bound of inseparability (unity), and trace (ii) degree of inseparability. Values below unity indicate entanglement.
Fig. 6
Fig. 6 Entanglement descriptions mapped onto quantum orbital Poincaré spheres at 5.0 MHz. (a) Location of output beams on the orbital Poincaré sphere. (b) Knowledge of beam A before any measurement of beam B. (c) Conditional knowledge of beam A after measurements of Ô 2 on beam B. (d) Conditional knowledge of beam A after measurements of Ô 3 on beam B. The dashed circles define the classical correlation limit, and conditional knowledge outside the dashed circles indicates entanglement.

Equations (18)

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O ^ 0 = a ^ 10 a ^ 10 + a ^ 01 a ^ 01
O ^ 1 = a ^ 10 a ^ 10 a ^ 01 a ^ 01
O ^ 2 = a ^ 10 a ^ 01 e i θ + a ^ 01 a ^ 10 e i θ
O ^ 3 = i a ^ 10 a ^ 10 e i θ i a ^ 01 a ^ 01 e i θ
O ^ 0 = α 10 2 + α 01 2
O ^ 1 = | α 10 2 α 01 2 |
O ^ 2 = 2 α 10 α 10 cos θ
O ^ 3 = 2 α 10 α 10 sin θ
Δ O ^ 0 = α 10 Δ X ^ 10 + α 10 Δ X ^ 01
Δ O ^ 1 = α 10 Δ X ^ 10 + α 10 Δ X ^ 01
Δ O ^ 2 = cos θ ( α 01 Δ X ^ 10 + α 10 Δ X ^ 01 ) + sin θ ( α 01 Δ Y ^ 10 + α 10 Δ Y ^ 01 )
Δ O ^ 2 = sin θ ( α 01 Δ X ^ 10 + α 10 Δ X ^ 01 ) + cos θ ( α 01 Δ Y ^ 10 + α 10 Δ Y ^ 01 )
I ( O ^ 1 , O ^ 2 ) = Δ A ± B 2 O ^ 1 + Δ A ± B 2 O ^ 2 8 | α 10 α 01 cos θ |
I ( O ^ 3 , O ^ 1 ) = Δ A ± B 2 O ^ 1 + Δ A ± B 2 O ^ 3 8 | α 10 α 01 cos θ |
I ( O ^ 2 , O ^ 3 ) = Δ A ± B 2 O ^ 2 + Δ A ± B 2 O ^ 3 4 | α 10 2 α 01 2 |
I ( O ^ 1 , O ^ 2 ) = α 10 8 α 01 | sin θ | ( Δ A ± B 2 X ^ 10 + Δ A ± B 2 X ^ 01 cos 2 θ + Δ A B 2 Y ^ 01 sin 2 θ ) + α 01 8 α 01 | sin θ | ( Δ A ± B 2 X ^ 01 + Δ A ± B 2 X ^ 01 cos 2 θ + Δ A B 2 Y ^ 10 sin 2 θ )
I ( O ^ 3 , O ^ 1 ) = α 10 8 α 01 | cos θ | ( Δ A ± B 2 X ^ 10 + Δ A ± B 2 X ^ 01 sin 2 θ + Δ A B 2 Y ^ 01 cos 2 θ ) + α 01 8 α 01 | cos θ | ( Δ A ± B 2 X ^ 01 + Δ A ± B 2 X ^ 10 sin 2 θ + Δ A B 2 Y ^ 10 cos 2 θ )
I ( O ^ 2 , O ^ 3 ) = α 10 2 ( Δ A ± B 2 X ^ 01 + Δ A B 2 Y ^ 01 ) + α 01 2 ( Δ A ± B 2 X ^ 10 + Δ A B 2 Y ^ 10 ) 8 | α 10 2 α 01 2 |

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