Abstract

Retrieving the vast amount of information carried by a photon is an enduring challenge in quantum metrology science and quantum photonics research. The transverse spatial state of a photon is a convenient high-dimensional quantum system for study, as it has a well-understood classical analog as the transverse complex field profile of an optical beam. One severe drawback of all currently available quantum metrology techniques is the need for a time-consuming characterization process, which scales very unfavorably with the dimensionality of the quantum system. Here we demonstrate a technique that directly measures a million-dimensional photonic spatial state with a single setting of the measurement apparatus. Through the arrangement of a weak measurement of momentum and parallel strong measurements of position, the complex values of the entire photon state vector become measurable directly. The dimension of our measured state is approximately four orders of magnitude larger than previously measured. Our work opens up a practical route for characterizing high-dimensional quantum systems in real time. Furthermore, our demonstration also serves as a high-speed, extremely high-resolution unambiguous complex field measurement technique for diverse classical applications.

© 2015 Optical Society of America

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Feiran Wang, Manuel Erhard, Amin Babazadeh, Mehul Malik, Mario Krenn, and Anton Zeilinger
Optica 4(12) 1462-1467 (2017)

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2014 (3)

M. Mirhosseini, O. S. Magaña Loaiza, S. M. Hashemi Rafsanjani, R. W. Boyd, “Compressive direct measurement of the quantum wave function,” Phys. Rev. Lett. 113, 090402 (2014).
[Crossref]

M. Malik, M. Mirhosseini, M. P. J. Lavery, J. Leach, M. J. Padgett, R. W. Boyd, “Direct measurement of a 27-dimensional orbital angular momentum state vector,” Nat. Commun. 4, 3115 (2014).

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, R. W. Boyd, “Colloquium: understanding quantum weak values: basics and applications,” Rev. Mod. Phys. 86, 307–316 (2014).
[Crossref]

2013 (4)

R. Fickler, M. Krenn, R. Lapkiewicz, S. Ramelow, A. Zeilinger, “Real-time imaging of quantum entanglement,” Sci. Rep. 3, 1914 (2013).
[Crossref]

M. Mirhosseini, M. Malik, Z. Shi, R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref]

W. Shengjun, “State tomography via weak measurements,” Sci. Rep. 3, 1193 (2013).

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photonics 7, 316–321 (2013).
[Crossref]

2012 (5)

J. Söderholm, G. Björk, A. B. Klimov, L. L. Sánchez-Soto, G. Leuchs, “Quantum polarization characterization and tomography,” New J. Phys. 14, 115014 (2012).
[Crossref]

D. Sych, J. Řeháček, Z. Hradil, G. Leuchs, L. L. Sánchez-Soto, “Informational completeness of continuous-variable measurements,” Phys. Rev. A 86, 052123 (2012).
[Crossref]

M. Edgar, D. Tasca, F. Izdebski, R. Warburton, J. Leach, M. Agnew, G. Buller, R. Boyd, M. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 1038 (2012).
[Crossref]

J. S. Lundeen, C. Bamber, “Procedure for direct measurement of general quantum states using weak measurement,” Phys. Rev. Lett. 108, 070402 (2012).
[Crossref]

M. Malik, M. N. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195–13200 (2012).
[Crossref]

2011 (5)

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[Crossref]

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

A. Feizpour, X. Xing, A. M. Steinberg, “Amplifying single-photon nonlinearity using weak measurements,” Phys. Rev. Lett. 107, 133603 (2011).
[Crossref]

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
[Crossref]

2010 (1)

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[Crossref]

2009 (1)

P. B. Dixon, D. J. Starling, A. N. Jordan, J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[Crossref]

2008 (1)

O. Hosten, P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[Crossref]

2007 (2)

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

V. Arrizón, U. Ruiz, R. Carrada, L. A. González, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A 24, 3500–3507 (2007).
[Crossref]

2005 (3)

B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30, 3365–3367 (2005).
[Crossref]

M. N. O’Sullivan-Hale, I. Ali Khan, R. W. Boyd, J. C. Howell, “Pixel entanglement: experimental realization of optically entangled d = 3 and d = 6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).
[Crossref]

K. J. Resch, P. Walther, A. Zeilinger, “Full characterization of a three-photon Greenberger-Horne-Zeilinger state using quantum state tomography,” Phys. Rev. Lett. 94, 070402 (2005).
[Crossref]

2004 (4)

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C. Kieffer, P. B. Corkum, D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867–871 (2004).
[Crossref]

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

D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, J. M. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett. 92, 043601 (2004).
[Crossref]

L. M. Johansen, “Weak measurements with arbitrary probe states,” Phys. Rev. Lett. 93, 120402 (2004).
[Crossref]

2003 (2)

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref]

A. M. Dawes, M. Beck, K. Banaszek, “Mode optimization for quantum-state tomography with array detectors,” Phys. Rev. A 67, 032102 (2003).
[Crossref]

2002 (1)

V. V. Dodonov, “‘Nonclassical’ states in quantum optics: a ‘squeezed’ review of the first 75 years,” J. Opt. B 4, R1–R33 (2002).
[Crossref]

2001 (3)

M. Beck, C. Dorrer, I. A. Walmsley, “Joint quantum measurement using unbalanced array detection,” Phys. Rev. Lett. 87, 253601 (2001).
[Crossref]

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

A. G. White, D. F. V. James, W. J. Munro, P. G. Kwiat, “Exploring Hilbert space: accurate characterization of quantum information,” Phys. Rev. A 65, 012301 (2001).
[Crossref]

2000 (1)

M. Beck, “Quantum state tomography with array detectors,” Phys. Rev. Lett. 84, 5748–5751 (2000).
[Crossref]

1999 (1)

1991 (1)

N. W. M. Ritchie, J. G. Story, R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1991).
[Crossref]

1989 (1)

I. M. Duck, P. M. Stevenson, E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin-½ particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).
[Crossref]

1988 (1)

Y. Aharonov, D. Z. Albert, L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref]

1972 (1)

R. N. Smartt, J. Strong, “Point-diffraction interferometer (abstract),” J. Opt. Soc. Am. 62, 737 (1972).

Agnew, M.

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photonics 7, 316–321 (2013).
[Crossref]

M. Edgar, D. Tasca, F. Izdebski, R. Warburton, J. Leach, M. Agnew, G. Buller, R. Boyd, M. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 1038 (2012).
[Crossref]

Aharonov, Y.

Y. Aharonov, D. Z. Albert, L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref]

Albert, D. Z.

Y. Aharonov, D. Z. Albert, L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref]

Ali Khan, I.

M. N. O’Sullivan-Hale, I. Ali Khan, R. W. Boyd, J. C. Howell, “Pixel entanglement: experimental realization of optically entangled d = 3 and d = 6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).
[Crossref]

Andersson, E.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[Crossref]

Arrizón, V.

Bachor, H.-A.

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

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref]

Bamber, C.

J. S. Lundeen, C. Bamber, “Procedure for direct measurement of general quantum states using weak measurement,” Phys. Rev. Lett. 108, 070402 (2012).
[Crossref]

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
[Crossref]

Banaszek, K.

Barnett, S. M.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[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, A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref]

Beck, M.

A. M. Dawes, M. Beck, K. Banaszek, “Mode optimization for quantum-state tomography with array detectors,” Phys. Rev. A 67, 032102 (2003).
[Crossref]

M. Beck, C. Dorrer, I. A. Walmsley, “Joint quantum measurement using unbalanced array detection,” Phys. Rev. Lett. 87, 253601 (2001).
[Crossref]

M. Beck, “Quantum state tomography with array detectors,” Phys. Rev. Lett. 84, 5748–5751 (2000).
[Crossref]

Björk, G.

J. Söderholm, G. Björk, A. B. Klimov, L. L. Sánchez-Soto, G. Leuchs, “Quantum polarization characterization and tomography,” New J. Phys. 14, 115014 (2012).
[Crossref]

Bolduc, E.

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photonics 7, 316–321 (2013).
[Crossref]

Bowen, W. P.

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref]

Boyd, R.

M. Edgar, D. Tasca, F. Izdebski, R. Warburton, J. Leach, M. Agnew, G. Buller, R. Boyd, M. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 1038 (2012).
[Crossref]

Boyd, R. W.

M. Mirhosseini, O. S. Magaña Loaiza, S. M. Hashemi Rafsanjani, R. W. Boyd, “Compressive direct measurement of the quantum wave function,” Phys. Rev. Lett. 113, 090402 (2014).
[Crossref]

M. Malik, M. Mirhosseini, M. P. J. Lavery, J. Leach, M. J. Padgett, R. W. Boyd, “Direct measurement of a 27-dimensional orbital angular momentum state vector,” Nat. Commun. 4, 3115 (2014).

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, R. W. Boyd, “Colloquium: understanding quantum weak values: basics and applications,” Rev. Mod. Phys. 86, 307–316 (2014).
[Crossref]

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photonics 7, 316–321 (2013).
[Crossref]

M. Mirhosseini, M. Malik, Z. Shi, R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref]

M. Malik, M. N. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195–13200 (2012).
[Crossref]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[Crossref]

M. N. O’Sullivan-Hale, I. Ali Khan, R. W. Boyd, J. C. Howell, “Pixel entanglement: experimental realization of optically entangled d = 3 and d = 6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).
[Crossref]

Braverman, B.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

Buchhave, P.

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

Buller, G.

M. Edgar, D. Tasca, F. Izdebski, R. Warburton, J. Leach, M. Agnew, G. Buller, R. Boyd, M. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 1038 (2012).
[Crossref]

Buller, G. S.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[Crossref]

Campos, J.

Carrada, R.

Chiao, R. Y.

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Treps, N.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, C. C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
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N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
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Vaidman, L.

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Wagner, K.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, C. C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
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A. M. Yao, M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).

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Zeilinger, A.

R. Fickler, M. Krenn, R. Lapkiewicz, S. Ramelow, A. Zeilinger, “Real-time imaging of quantum entanglement,” Sci. Rep. 3, 1914 (2013).
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K. J. Resch, P. Walther, A. Zeilinger, “Full characterization of a three-photon Greenberger-Horne-Zeilinger state using quantum state tomography,” Phys. Rev. Lett. 94, 070402 (2005).
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Adv. Opt. Photon. (1)

Appl. Opt. (1)

J. Opt. B (1)

V. V. Dodonov, “‘Nonclassical’ states in quantum optics: a ‘squeezed’ review of the first 75 years,” J. Opt. B 4, R1–R33 (2002).
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J. Opt. Soc. Am. (1)

R. N. Smartt, J. Strong, “Point-diffraction interferometer (abstract),” J. Opt. Soc. Am. 62, 737 (1972).

J. Opt. Soc. Am. A (1)

Nat. Commun. (3)

M. Edgar, D. Tasca, F. Izdebski, R. Warburton, J. Leach, M. Agnew, G. Buller, R. Boyd, M. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 1038 (2012).
[Crossref]

M. Mirhosseini, M. Malik, Z. Shi, R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref]

M. Malik, M. Mirhosseini, M. P. J. Lavery, J. Leach, M. J. Padgett, R. W. Boyd, “Direct measurement of a 27-dimensional orbital angular momentum state vector,” Nat. Commun. 4, 3115 (2014).

Nat. Photonics (1)

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photonics 7, 316–321 (2013).
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Nat. Phys. (1)

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
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Nature (3)

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C. Kieffer, P. B. Corkum, D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867–871 (2004).
[Crossref]

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
[Crossref]

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

New J. Phys. (1)

J. Söderholm, G. Björk, A. B. Klimov, L. L. Sánchez-Soto, G. Leuchs, “Quantum polarization characterization and tomography,” New J. Phys. 14, 115014 (2012).
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Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (3)

D. Sych, J. Řeháček, Z. Hradil, G. Leuchs, L. L. Sánchez-Soto, “Informational completeness of continuous-variable measurements,” Phys. Rev. A 86, 052123 (2012).
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A. M. Dawes, M. Beck, K. Banaszek, “Mode optimization for quantum-state tomography with array detectors,” Phys. Rev. A 67, 032102 (2003).
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Phys. Rev. D (1)

I. M. Duck, P. M. Stevenson, E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin-½ particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).
[Crossref]

Phys. Rev. Lett. (14)

N. W. M. Ritchie, J. G. Story, R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1991).
[Crossref]

L. M. Johansen, “Weak measurements with arbitrary probe states,” Phys. Rev. Lett. 93, 120402 (2004).
[Crossref]

J. S. Lundeen, C. Bamber, “Procedure for direct measurement of general quantum states using weak measurement,” Phys. Rev. Lett. 108, 070402 (2012).
[Crossref]

K. J. Resch, P. Walther, A. Zeilinger, “Full characterization of a three-photon Greenberger-Horne-Zeilinger state using quantum state tomography,” Phys. Rev. Lett. 94, 070402 (2005).
[Crossref]

Y. Aharonov, D. Z. Albert, L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref]

D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, J. M. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett. 92, 043601 (2004).
[Crossref]

P. B. Dixon, D. J. Starling, A. N. Jordan, J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[Crossref]

A. Feizpour, X. Xing, A. M. Steinberg, “Amplifying single-photon nonlinearity using weak measurements,” Phys. Rev. Lett. 107, 133603 (2011).
[Crossref]

M. Mirhosseini, O. S. Magaña Loaiza, S. M. Hashemi Rafsanjani, R. W. Boyd, “Compressive direct measurement of the quantum wave function,” Phys. Rev. Lett. 113, 090402 (2014).
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N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
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M. N. O’Sullivan-Hale, I. Ali Khan, R. W. Boyd, J. C. Howell, “Pixel entanglement: experimental realization of optically entangled d = 3 and d = 6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).
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M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, C. C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98, 083602 (2007).
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Rev. Mod. Phys. (1)

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, R. W. Boyd, “Colloquium: understanding quantum weak values: basics and applications,” Rev. Mod. Phys. 86, 307–316 (2014).
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Sci. Rep. (2)

W. Shengjun, “State tomography via weak measurements,” Sci. Rep. 3, 1193 (2013).

R. Fickler, M. Krenn, R. Lapkiewicz, S. Ramelow, A. Zeilinger, “Real-time imaging of quantum entanglement,” Sci. Rep. 3, 1914 (2013).
[Crossref]

Science (4)

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[Crossref]

O. Hosten, P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[Crossref]

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref]

Other (3)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Plenum, 1995).

S. Kilin, Quanta and Information, E. Wolf, ed., Vol. 42 of Progress in Optics (Elsevier Science, 2001), pp. 1–90.

M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2010).

Supplementary Material (2)

» Media 1: PDF (736 KB)     
» Supplement Media 2: MP4 (7983 KB)     

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Figures (3)

Fig. 1.
Fig. 1. Experimental implementation of a scan-free direct measurement on the transverse spatial state of photons. The photons prepared using a phase-only spatial light modulator (SLM) pass through a 4 f imaging system. The weak measurement is performed in the momentum space, i.e., the common focal plane of the 4 f system, where the linear polarization state of the photons in the zero-momentum state is rotated by a small angle α . The strong measurement is performed in the position basis, i.e., the image plane, using a detector array in combination with some polarization optics, where the change in polarization for all position states is measured simultaneously. Specifically, the real and imaginary parts of the weak values are measured in terms of the rotation of the photons’ polarization in the diagonal (D)–anti-diagonal (A) linear and left (L)–right (R) handed circular bases, respectively, as labeled in the detector array plane. BS, beam splitter; PBS, polarizing beam splitter; HWP, half-wave plate; QWP, quarter-wave plate.
Fig. 2.
Fig. 2. Measured (a) real and (b) imaginary parts of the two-dimensional weak values and the corresponding (c) phase and (d) amplitude profiles of photons carrying orbital angular momentum (OAM) with quantum number l = 3 . The measured weak values have very large magnitude toward the center of the mode and therefore are truncated for better visualization purposes. (e)–(h) Extracted phase profile of photons carrying OAM with quantum number l ranging from 2 to 2.
Fig. 3.
Fig. 3. Upper row: measured magnitude of the probability amplitude | ψ ( x ) | of photons with an amplitude profile incorporating a University of South Florida “Bull” logo using (a) our direct measurement approach and (b) conventional strong measurement. Lower row: (c) measured magnitude of the probability amplitude | ψ ( x ) | of photons with a truncated Gaussian amplitude profile using our scan-free direct measurement approach and (d) one cross section of the directly measured result (thick red line) in comparison with result of the conventional strong (intensity) measurement (thin blue line). The actual | ψ ( x ) | profile is the result of diffraction and propagation of the photons through our nonideal imaging system. (See also Media 1.)

Equations (3)

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π p x w = x | p 0 p 0 | ψ x | ψ = e i p 0 x / ψ ˜ ( p 0 ) ψ ( x ) ,
π p 0 x w = ν ψ ( x ) ,
ψ ( x ) = x | ψ = ν π p x w = ν [ s f ( x ) | σ ^ 1 | s f ( x ) i s f ( x ) | σ ^ 2 | s f ( x ) ] ,

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