Abstract

Weak measurement has been shown to play important roles in the investigation of both fundamental and practical problems. Anomalous weak values are generally believed to be observed only when post-selection is performed, i.e., only a particular subset of the data is considered. Here, we experimentally demonstrate that an anomalous weak value can be obtained without discarding any data by performing a sequential weak measurement on a single-qubit system. By controlling the blazing density of the hologram on a spatial light modulator, the measurement strength can be conveniently controlled. Such an anomalous phenomenon disappears when the measurement strength of the first observable becomes strong. Moreover, we find that the anomalous weak value cannot be observed without post-selection when the sequential measurement is performed on each of the components of a two-qubit system, which confirms that the observed anomalous weak value is based on sequential weak measurement of two noncommutative operators.

© 2020 Chinese Laser Press

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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  42. E. Cohen and M. Nowakowski, “Comment on ‘Measurements without probabilities in the final state proposal’,” Phys. Rev. D 97, 088501 (2018).
    [Crossref]
  43. R. Bousso and D. Stanford, “Reply to “Comment on ‘Measurements without probabilities in the final state proposal’”,” Phys. Rev. D 97, 088502 (2018).
    [Crossref]
  44. Y. Aharonov, S. Popescu, and J. Tollaksen, “Each instant of time a new universe,” in Quantum Theory: A Two-Time Success Story (Springer, 2014), pp. 21–36.
  45. Y.-W. Cho, Y. Kim, Y.-H. Choi, Y.-S. Kim, S.-W. Han, S.-Y. Lee, S. Moon, and Y.-H. Kim, “Emergence of the geometric phase from quantum measurement back-action,” Nat. Phys. 15, 665–670 (2019).
    [Crossref]

2020 (1)

M. Yang, Y. Xiao, Y. W. Liao, Z. H. Liu, X. Y. Xu, J. S. Xu, C. F. Li, and G. C. Guo, “Zonal reconstruction of photonic wavefunction via momentum weak measurement,” Laser Photon. Rev. 14, 1900251 (2020).
[Crossref]

2019 (5)

Y.-W. Cho, Y. Kim, Y.-H. Choi, Y.-S. Kim, S.-W. Han, S.-Y. Lee, S. Moon, and Y.-H. Kim, “Emergence of the geometric phase from quantum measurement back-action,” Nat. Phys. 15, 665–670 (2019).
[Crossref]

A. A. Abbott, R. Silva, J. Wechs, N. Brunner, and C. Branciard, “Anomalous weak values without post-selection,” Quantum 3, 194–208 (2019).
[Crossref]

E. Cohen, “Quantum measurements-yet another surprise,” Quantum Views 3, 27 (2019).
[Crossref]

J. S. Chen, M. J. Hu, X. M. Hu, B. H. Liu, Y. F. Huang, C. F. Li, C. G. Guo, and Y. S. Zhang, “Experimental realization of sequential weak measurements of non-commuting Pauli observables,” Opt. Express 27, 6089–6097 (2019).
[Crossref]

Q. Li, C.-J. Zhang, Z.-D. Cheng, W.-Z. Liu, J.-F. Wang, F.-F. Yan, Z.-H. Lin, Y. Xiao, K. Sun, Y.-T. Wang, J.-S. Tang, J.-S. Xu, C.-F. Li, and G.-C. Guo, “Experimental simulation of anti-parity-time symmetric Lorentz dynamics,” Optica 6, 67–71 (2019).
[Crossref]

2018 (2)

E. Cohen and M. Nowakowski, “Comment on ‘Measurements without probabilities in the final state proposal’,” Phys. Rev. D 97, 088501 (2018).
[Crossref]

R. Bousso and D. Stanford, “Reply to “Comment on ‘Measurements without probabilities in the final state proposal’”,” Phys. Rev. D 97, 088502 (2018).
[Crossref]

2017 (2)

L. Vaidman, “Weak value controversy,” Philos. Trans. R. Soc. London, Ser. A 375, 20160395 (2017).
[Crossref]

L. P. García-Pintos and J. Dressel, “Past observable dynamics of a continuously monitored qubit,” Phys. Rev. A 96, 062110 (2017).
[Crossref]

2016 (1)

J. Martínez-Rincón, W.-T. Liu, G. I. Viza, and J. C. Howell, “Can anomalous amplification be attained without postselection?” Phys. Rev. Lett. 116, 100803 (2016).
[Crossref]

2015 (2)

Y. Turek, H. Kobayashi, T. Akutsu, C. P. Sun, and Y. Shikano, “Post-selected von Neumann measurement with Hermite–Gaussian and Laguerre–Gaussian pointer states,” New J. Phys. 17, 083029 (2015).
[Crossref]

G. I. Viza, J. Martinez-Rincon, G. B. Alves, A. N. Jordan, and J. C. Howell, “Experimentally quantifying the advantages of weak-value-based metrology,” Phys. Rev. A 92, 032127 (2015).
[Crossref]

2014 (6)

C. Emary, N. Lambert, and F. Nori, “Leggett-Garg inequalities,” Rep. Prog. Phys. 77, 016001 (2014).
[Crossref]

A. N. Jordan, J. Martinez-Rincon, and J. C. Howell, “Technical advantages for weak-value amplification: when less is more,” Phys. Rev. X 4, 011031 (2014).
[Crossref]

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

J. Dressel, K. Y. Bliokh, and F. Nori, “Classical field approach to quantum weak measurements,” Phys. Rev. Lett. 112, 110407 (2014).
[Crossref]

C. Ferrie and J. Combes, “Weak value amplification is suboptimal for estimation and detection,” Phys. Rev. Lett. 112, 040406 (2014).
[Crossref]

J. Combes, C. Ferrie, Z. Jiang, and C. M. Caves, “Quantum limits on postselected, probabilistic quantum metrology,” Phys. Rev. A 89, 052117 (2014).
[Crossref]

2013 (2)

X.-Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.-F. Li, and G.-C. Guo, “Phase estimation with weak measurement using a white light source,” Phys. Rev. Lett. 111, 033604 (2013).
[Crossref]

G. Strubi and C. Bruder, “Measuring ultrasmall time delays of light by joint weak measurements,” Phys. Rev. Lett. 110, 083605 (2013).
[Crossref]

2012 (3)

H. Kobayashi, G. Puentes, and Y. Shikano, “Extracting joint weak values from two-dimensional spatial displacements,” Phys. Rev. A 86, 053805 (2012).
[Crossref]

A. G. Kofman, S. Ashkab, and F. Nori, “Nonperturbative theory of weak pre-and post-selected measurements,” Phys. Rep. 520, 43–133 (2012).
[Crossref]

Y. Suzuki, M. Iinuma, and H. F. Hofmann, “Violation of Leggett–Garg inequalities in quantum measurements with variable resolution and back-action,” New J. Phys. 14, 103022 (2012).
[Crossref]

2011 (3)

J. Dressel, C. J. Broadbent, J. C. Howell, and A. N. Jordan, “Experimental violation of two-party Leggett-Garg inequalities with semiweak measurements,” Phys. Rev. Lett. 106, 040402 (2011).
[Crossref]

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

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

2010 (2)

N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: weak measurements or standard interferometry?” Phys. Rev. Lett. 105, 010405 (2010).
[Crossref]

A. Palacios-Laloy, F. Mallet, F. Nguyen, P. Bertet, D. Vion, D. Esteve, and A. N. Korotkov, “Experimental violation of a Bell’s inequality in time with weak measurement,” Nat. Phys. 6, 442–447 (2010).
[Crossref]

2009 (3)

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

J. S. Lundeen and A. M. Steinberg, “Experimental joint weak measurement on a photon pair as a probe of Hardy’s paradox,” Phys. Rev. Lett. 102, 020404 (2009).
[Crossref]

K. Yokota, T. Yamamoto, M. Koashi, and N. Imoto, “Direct observation of Hardy’s paradox by joint weak measurement with an entangled photon pair,” New J. Phys. 11, 033011 (2009).
[Crossref]

2008 (1)

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

2007 (3)

H. M. Wiseman, “Grounding Bohmian mechanics in weak values and bayesianism,” New J. Phys. 9, 165–175 (2007).

R. Mir, J. S. Lundeen, M. W. Mitchell, A. M. Steinberg, J. L. Garretson, and H. M. Wiseman, “A double-slit which-way experiment on the complementarity uncertainty debate,” New J. Phys. 9, 287 (2007).
[Crossref]

G. Mitchison, R. Jozsa, and S. Popescu, “Sequential weak measurement,” Phys. Rev. A 76, 062105 (2007).
[Crossref]

2004 (2)

K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Experimental realization of the quantum box problem,” Phys. Lett. A 324, 125–131 (2004).
[Crossref]

G. Horowitz and J. Maldacena, “The black hole final state,” J. High Energy Phys. 2004, 008 (2004).
[Crossref]

2002 (2)

Y. Aharonov, A. Botero, S. Popescu, B. Reznik, and J. Tollaksen, “Revisiting Hardy’s paradox: counterfactual statements, real measurements, entanglement and weak values,” Phys. Lett. A 301, 130–138 (2002).
[Crossref]

H. M. Wiseman, “Weak values, quantum trajectories, and the cavity-QED experiment on wave-particle correlation,” Phys. Rev. A 65, 032111 (2002).
[Crossref]

1989 (1)

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a “weak measurement” of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).
[Crossref]

1988 (1)

Y. Aharonov, D. Z. Albert, and 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]

Abbott, A. A.

A. A. Abbott, R. Silva, J. Wechs, N. Brunner, and C. Branciard, “Anomalous weak values without post-selection,” Quantum 3, 194–208 (2019).
[Crossref]

Aharonov, Y.

Y. Aharonov, A. Botero, S. Popescu, B. Reznik, and J. Tollaksen, “Revisiting Hardy’s paradox: counterfactual statements, real measurements, entanglement and weak values,” Phys. Lett. A 301, 130–138 (2002).
[Crossref]

Y. Aharonov, D. Z. Albert, and 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]

Y. Aharonov and E. Cohen, “Weak values and quantum nonlocality,” arXiv:1504.03797 (2015).

Y. Aharonov, S. Popescu, and J. Tollaksen, “Each instant of time a new universe,” in Quantum Theory: A Two-Time Success Story (Springer, 2014), pp. 21–36.

Akutsu, T.

Y. Turek, H. Kobayashi, T. Akutsu, C. P. Sun, and Y. Shikano, “Post-selected von Neumann measurement with Hermite–Gaussian and Laguerre–Gaussian pointer states,” New J. Phys. 17, 083029 (2015).
[Crossref]

Albert, D. Z.

Y. Aharonov, D. Z. Albert, and 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]

Alves, G. B.

G. I. Viza, J. Martinez-Rincon, G. B. Alves, A. N. Jordan, and J. C. Howell, “Experimentally quantifying the advantages of weak-value-based metrology,” Phys. Rev. A 92, 032127 (2015).
[Crossref]

Ashkab, S.

A. G. Kofman, S. Ashkab, and F. Nori, “Nonperturbative theory of weak pre-and post-selected measurements,” Phys. Rep. 520, 43–133 (2012).
[Crossref]

Bertet, P.

A. Palacios-Laloy, F. Mallet, F. Nguyen, P. Bertet, D. Vion, D. Esteve, and A. N. Korotkov, “Experimental violation of a Bell’s inequality in time with weak measurement,” Nat. Phys. 6, 442–447 (2010).
[Crossref]

Bliokh, K. Y.

J. Dressel, K. Y. Bliokh, and F. Nori, “Classical field approach to quantum weak measurements,” Phys. Rev. Lett. 112, 110407 (2014).
[Crossref]

Botero, A.

Y. Aharonov, A. Botero, S. Popescu, B. Reznik, and J. Tollaksen, “Revisiting Hardy’s paradox: counterfactual statements, real measurements, entanglement and weak values,” Phys. Lett. A 301, 130–138 (2002).
[Crossref]

Bousso, R.

R. Bousso and D. Stanford, “Reply to “Comment on ‘Measurements without probabilities in the final state proposal’”,” Phys. Rev. D 97, 088502 (2018).
[Crossref]

Boyd, R. W.

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

Branciard, C.

A. A. Abbott, R. Silva, J. Wechs, N. Brunner, and C. Branciard, “Anomalous weak values without post-selection,” Quantum 3, 194–208 (2019).
[Crossref]

Braverman, B.

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

Broadbent, C. J.

J. Dressel, C. J. Broadbent, J. C. Howell, and A. N. Jordan, “Experimental violation of two-party Leggett-Garg inequalities with semiweak measurements,” Phys. Rev. Lett. 106, 040402 (2011).
[Crossref]

Bruder, C.

G. Strubi and C. Bruder, “Measuring ultrasmall time delays of light by joint weak measurements,” Phys. Rev. Lett. 110, 083605 (2013).
[Crossref]

Brunner, N.

A. A. Abbott, R. Silva, J. Wechs, N. Brunner, and C. Branciard, “Anomalous weak values without post-selection,” Quantum 3, 194–208 (2019).
[Crossref]

N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: weak measurements or standard interferometry?” Phys. Rev. Lett. 105, 010405 (2010).
[Crossref]

Caves, C. M.

J. Combes, C. Ferrie, Z. Jiang, and C. M. Caves, “Quantum limits on postselected, probabilistic quantum metrology,” Phys. Rev. A 89, 052117 (2014).
[Crossref]

Chen, J. S.

Cheng, Z.-D.

Cho, Y.-W.

Y.-W. Cho, Y. Kim, Y.-H. Choi, Y.-S. Kim, S.-W. Han, S.-Y. Lee, S. Moon, and Y.-H. Kim, “Emergence of the geometric phase from quantum measurement back-action,” Nat. Phys. 15, 665–670 (2019).
[Crossref]

Choi, Y.-H.

Y.-W. Cho, Y. Kim, Y.-H. Choi, Y.-S. Kim, S.-W. Han, S.-Y. Lee, S. Moon, and Y.-H. Kim, “Emergence of the geometric phase from quantum measurement back-action,” Nat. Phys. 15, 665–670 (2019).
[Crossref]

Cohen, E.

E. Cohen, “Quantum measurements-yet another surprise,” Quantum Views 3, 27 (2019).
[Crossref]

E. Cohen and M. Nowakowski, “Comment on ‘Measurements without probabilities in the final state proposal’,” Phys. Rev. D 97, 088501 (2018).
[Crossref]

Y. Aharonov and E. Cohen, “Weak values and quantum nonlocality,” arXiv:1504.03797 (2015).

Combes, J.

C. Ferrie and J. Combes, “Weak value amplification is suboptimal for estimation and detection,” Phys. Rev. Lett. 112, 040406 (2014).
[Crossref]

J. Combes, C. Ferrie, Z. Jiang, and C. M. Caves, “Quantum limits on postselected, probabilistic quantum metrology,” Phys. Rev. A 89, 052117 (2014).
[Crossref]

Dixon, P. B.

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

Dressel, J.

L. P. García-Pintos and J. Dressel, “Past observable dynamics of a continuously monitored qubit,” Phys. Rev. A 96, 062110 (2017).
[Crossref]

J. Dressel, K. Y. Bliokh, and F. Nori, “Classical field approach to quantum weak measurements,” Phys. Rev. Lett. 112, 110407 (2014).
[Crossref]

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

J. Dressel, C. J. Broadbent, J. C. Howell, and A. N. Jordan, “Experimental violation of two-party Leggett-Garg inequalities with semiweak measurements,” Phys. Rev. Lett. 106, 040402 (2011).
[Crossref]

Duck, I. M.

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a “weak measurement” of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).
[Crossref]

Emary, C.

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A. Palacios-Laloy, F. Mallet, F. Nguyen, P. Bertet, D. Vion, D. Esteve, and A. N. Korotkov, “Experimental violation of a Bell’s inequality in time with weak measurement,” Nat. Phys. 6, 442–447 (2010).
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A. Feizpour, X. Xing, and A. M. Steinberg, “Amplifying single-photon nonlinearity using weak measurements,” Phys. Rev. Lett. 107, 133603 (2011).
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Guo, G. C.

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G. I. Viza, J. Martinez-Rincon, G. B. Alves, A. N. Jordan, and J. C. Howell, “Experimentally quantifying the advantages of weak-value-based metrology,” Phys. Rev. A 92, 032127 (2015).
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G. Mitchison, R. Jozsa, and S. Popescu, “Sequential weak measurement,” Phys. Rev. A 76, 062105 (2007).
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Liu, B. H.

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J. Martínez-Rincón, W.-T. Liu, G. I. Viza, and J. C. Howell, “Can anomalous amplification be attained without postselection?” Phys. Rev. Lett. 116, 100803 (2016).
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Liu, Z. H.

M. Yang, Y. Xiao, Y. W. Liao, Z. H. Liu, X. Y. Xu, J. S. Xu, C. F. Li, and G. C. Guo, “Zonal reconstruction of photonic wavefunction via momentum weak measurement,” Laser Photon. Rev. 14, 1900251 (2020).
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G. I. Viza, J. Martinez-Rincon, G. B. Alves, A. N. Jordan, and J. C. Howell, “Experimentally quantifying the advantages of weak-value-based metrology,” Phys. Rev. A 92, 032127 (2015).
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J. Martínez-Rincón, W.-T. Liu, G. I. Viza, and J. C. Howell, “Can anomalous amplification be attained without postselection?” Phys. Rev. Lett. 116, 100803 (2016).
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J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: understanding quantum weak values: basics and applications,” Rev. Mod. Phys. 86, 307–316 (2014).
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S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
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R. Mir, J. S. Lundeen, M. W. Mitchell, A. M. Steinberg, J. L. Garretson, and H. M. Wiseman, “A double-slit which-way experiment on the complementarity uncertainty debate,” New J. Phys. 9, 287 (2007).
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G. Mitchison, R. Jozsa, and S. Popescu, “Sequential weak measurement,” Phys. Rev. A 76, 062105 (2007).
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H. Kobayashi, G. Puentes, and Y. Shikano, “Extracting joint weak values from two-dimensional spatial displacements,” Phys. Rev. A 86, 053805 (2012).
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S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
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K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Experimental realization of the quantum box problem,” Phys. Lett. A 324, 125–131 (2004).
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Y. Aharonov, A. Botero, S. Popescu, B. Reznik, and J. Tollaksen, “Revisiting Hardy’s paradox: counterfactual statements, real measurements, entanglement and weak values,” Phys. Lett. A 301, 130–138 (2002).
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S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
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Y. Turek, H. Kobayashi, T. Akutsu, C. P. Sun, and Y. Shikano, “Post-selected von Neumann measurement with Hermite–Gaussian and Laguerre–Gaussian pointer states,” New J. Phys. 17, 083029 (2015).
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A. Feizpour, X. Xing, and A. M. Steinberg, “Amplifying single-photon nonlinearity using weak measurements,” Phys. Rev. Lett. 107, 133603 (2011).
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S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
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S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
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Y. Suzuki, M. Iinuma, and H. F. Hofmann, “Violation of Leggett–Garg inequalities in quantum measurements with variable resolution and back-action,” New J. Phys. 14, 103022 (2012).
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Y. Turek, H. Kobayashi, T. Akutsu, C. P. Sun, and Y. Shikano, “Post-selected von Neumann measurement with Hermite–Gaussian and Laguerre–Gaussian pointer states,” New J. Phys. 17, 083029 (2015).
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J. Martínez-Rincón, W.-T. Liu, G. I. Viza, and J. C. Howell, “Can anomalous amplification be attained without postselection?” Phys. Rev. Lett. 116, 100803 (2016).
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Wang, Y.-T.

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M. Yang, Y. Xiao, Y. W. Liao, Z. H. Liu, X. Y. Xu, J. S. Xu, C. F. Li, and G. C. Guo, “Zonal reconstruction of photonic wavefunction via momentum weak measurement,” Laser Photon. Rev. 14, 1900251 (2020).
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Xu, X. Y.

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Yang, M.

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Corr(A,B)=∑i,j(Ai,j−A¯)(Bi,j−B¯)∑i,j(Ai,j−A¯)2∑i,j(Bi,j−B¯)2, where Ai,j(Bi,j) represents the gray level of image A(B) at pixel (i,j), and A¯(B¯) means the average value.

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Figures (6)

Fig. 1.
Fig. 1. Theoretical protocol. The system initially in the state |ψ(0) is sequentially weak coupled with two pointers in the initial states of |ϕ1(0) and |ϕ2(0), respectively. The time sequence is denoted as t1, t2, and t3. The pointers are measured individually or jointly.
Fig. 2.
Fig. 2. Experimental setup and deflection images. (a) Single photons from a single photon emitter (SPE) are sent to the sequential weak measurement setup. The single photon property is characterized by the second order autocorrelation function, in which the dip at the zero delay time is fitted to be g2(0)=0.025. The polarization of the single photons is set by a half-wave plate (HWP1). A lens (f=150  mm) is used to focus the photon to the right screen of the spatial light modulator (SLM) for the horizontal weak coupling, where the hologram loaded is a vertical grating. The coupling strength is adjusted by changing the density of the grating. The photons are then refocused on the left screen of the SLM by a lens with f=75  mm for the vertical coupling, where the hologram loaded is a horizontal grating with the same density. The HWP2 is used to rotate the polarization of the photon before the screen to adjust the direction of the pointer. The photons are then finally detected by an intensified charge coupled device (ICCD) in the focus plane of a lens with f=150  mm. (b) The images of photon distributions detected by the ICCD with different coupling strengths γexp. The inserts with blue background are the theoretical predictions of the corresponding experimental images when γexp>0.2366, and the Corr represents the correlation value between experimental and theoretical images.
Fig. 3.
Fig. 3. Deflections of the pointer’s position and the normalized result of sequential weak measurements in the one-qubit system. (a) The brown and blue dots represent the experimental results of the pointer positions x^ and y^, respectively. The brown and blue lines represent the corresponding theoretical predictions. (b) The green dots represent the experimental results of the joint pointer position x^y^ with the green solid line representing the corresponding theoretical prediction. The red data represents the anomalous joint pointer position. (c) The black dots represent the inferred values of M=x^y^/γexp2, while the theoretical prediction is shown as a black line.
Fig. 4.
Fig. 4. Deflections of pointer positions via sequential weak measurements in the two-qubit system. (a) The brown and blue dots represent the experimental results with the brown and blue lines representing the corresponding theoretical predictions, respectively. (b) The green dots represent the joint average pointer positions x^y^ with the green line representing the corresponding theoretical prediction.
Fig. 5.
Fig. 5. Weak measurement based on the liquid crystal spatial light modulator (SLM). (a) The input photons are transformed from the coordinate space to the momentum space by a Fourier lens and focused on the screen of SLM. A phase that changes linearly along the x direction is applied on photons by the SLM, and then photons are re-transformed from the momentum space to the coordinate space by another Fourier lens. The photon wave packet will be transversely shifted slightly, which is known as weak measurement. (b) The relation between grating densities α on SLM and coupling strength γexp.
Fig. 6.
Fig. 6. Experimental setup of the single photon emitter (SPE).

Equations (11)

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AψfW:=f|A^|ψf|ψ,
AψIW:=ψ|A^|ψ.
(A1A2)ψIW:=ψ|A^1A^2|ψ.
x^1x^2=116(13eγ128σ12)γ1γ2,
U^r=exp(iγexp|HH|p^r),
K^=x,yK·|ϕ(x,y)|2/x,y|ϕ(x,y)|2,
|Ψ=|ϕ(x,y)(a|H+b|V),
F[|ϕ(x,y)](a|H+b|V)=|U(η,ξ)(a|H+b|V),
a|U(η,ξ)eiγexpη|H+b|U(η,ξ)|V),
aF1[|U(η,ξ)eiγexpη]|H+b|F1[U(η,ξ)]|V=a|ϕ(xγexp,y)|H+b|ϕ(x,y)|V.
U^|ϕ(x,y)(a|H+b|V)=a|ϕ(xγexp,y)|H+b|ϕ(x,y)|V,

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