Abstract

Interpretations of quantum mechanics (QM), or proposals for underlying theories, that attempt to present a definite realist picture, such as Bohmian mechanics, require strong non-local effects. Naively, these effects would violate causality and contradict special relativity. However if the theory agrees with QM the violation cannot be observed directly. Here, we demonstrate experimentally such an effect: we steer the velocity and trajectory of a Bohmian particle using a remote measurement. We use a pair of photons and entangle the spatial transverse position of one with the polarization of the other. The first photon is sent to a double-slit-like apparatus, where its trajectory is measured using the technique of Weak Measurements. The other photon is projected to a linear polarization state. The choice of polarization state, and the result, steer the first photon in the most intuitive sense of the word. The effect is indeed shown to be dramatic, while being easy to visualize. We discuss its strength and what are the conditions for it to occur.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref]
  2. D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables. II,” Phys. Rev. 85, 180 (1952).
    [Crossref]
  3. 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 (1988).
    [Crossref] [PubMed]
  4. A. G. Kofman, S. Ashhab, and F. Nori, “Nonperturbative theory of weak pre-and post-selected measurements,” Phys. Rep. 520, 43 (2012).
    [Crossref]
  5. 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 (2014).
    [Crossref]
  6. B. Tamir and E. Cohen, “Introduction to weak measurements and weak values,” Quanta 2, 7 (2013).
    [Crossref]
  7. Y. Aharonov, E. Cohen, and A. C. Elitzur, “Foundations and applications of weak quantum measurements,” Phys. Rev. A 89, 052105 (2014).
    [Crossref]
  8. 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 photon in a two-slit Interferometer,” Science 332, 1170 (2011).
    [Crossref] [PubMed]
  9. B. Braverman and C. Simon, “Proposal to observe the nonlocality of Bohmian trajectories with entangled photons,” Phys. Rev. Lett. 110, 060406 (2013).
    [Crossref] [PubMed]
  10. D. H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K. J. Resch, H. M. Wiseman, and A. Steinberg, “Experimental nonlocal and surreal Bohmian trajectories,” Sci. Adv. 2, e1501466 (2016).
    [Crossref] [PubMed]
  11. E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Camb. Philos. Soc. 31, 555 (1935).
    [Crossref]
  12. E. Schrödinger, “Probability relations between separated systems,” Proc. Camb. Philos. Soc. 32, 446 (1936).
    [Crossref]
  13. H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
    [Crossref] [PubMed]
  14. J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein-Podolsky-Rosen Steering,” Phys. Rev. Lett. 112, 200402 (2014).
    [Crossref]
  15. M. T. Quintino, T. Vértesi, and N. Brunner, “Joint measurability, Einstein-Podolsky-Rosen steering, and Bell nonlocality,” Phys. Rev. Lett. 113, 160402 (2014).
    [Crossref] [PubMed]
  16. M. Piani and J. Watrous, “Necessary and sufficient quantum information characterization of Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 114, 060404 (2015).
    [Crossref] [PubMed]
  17. I. Kogias, A. R. Lee, S. Ragy, and G. Adesso, “Quantification of Gaussian quantum steering,” Phys. Rev. Lett. 114, 060403 (2015).
    [Crossref] [PubMed]
  18. D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845 (2010).
    [Crossref]
  19. K. Sun, J.-S. Xu, X.-J. Ye, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental demonstration of the Einstein-Podolsky-Rosen steering game based on the all-versus-nothing proof,” Phys. Rev. Lett. 113, 140402 (2014).
    [Crossref] [PubMed]
  20. K. Sun, X.-J. Ye, J.-S. Xu, X.-Y. Xu, J.-S. Tang, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental quantification of asymmetric Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
    [Crossref] [PubMed]
  21. S. Wollmann, N. Walk, A. J. Bennet, H. M. Wiseman, and G. J. Pryde, “Observation of Genuine One-Way Einstein-Podolsky-Rosen Steering,” Phys. Rev. Lett. 116, 160403 (2016).
    [Crossref] [PubMed]
  22. Y. Xiao, X.-J. Ye, K. Sun, J.-S. Xu, C.-F. Li, and G.-C. Guo, “Demonstration of multisetting one-way Einstein-Podolsky-Rosen steering in two-qubit systems,” Phys. Rev. Lett. 118, 140404 (2017).
    [Crossref] [PubMed]
  23. G. Naaman-Marom, N. Erez, and L. Vaidman, “Position measurements in the de Broglie-Bohm interpretation of quantum mechanics,” Ann. Phys.,  327, 2522 (2012).
    [Crossref]
  24. A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger, “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Opt. Express 15, 15377 (2007).
    [Crossref] [PubMed]
  25. W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 78, 5022 (1997).
  26. H. M. Wiseman, “Grounding Bohmian mechanics in weak values and bayesianism,” New J. Phys. 9, 165 (2007).
    [Crossref]
  27. D. Durr, S. Goldstein, and N. Zanghi, “On the weak measurement of velocity in Bohmian mechanics,” J. Stat. Phys. 134, 1023 (2009).
    [Crossref]
  28. H. M. Wiseman, “Bohmian analysis of momentum transfer in Welcher Weg measurements,” Phys. Rev. A 58, 1740 (1998).
    [Crossref]
  29. B.-G. Englert, M. O. Scully, G. Süssmann, and H. Walther, “Surrealistic Bohm trajectories,” Z. Naturforsch. 47a, 1175 (1992).
  30. B. J. Hiley and R. E. Callaghan, “Delayed-choice experiments and the Bohm approach,” Phys. Scr. 74, 336 (2006).
    [Crossref]

2017 (1)

Y. Xiao, X.-J. Ye, K. Sun, J.-S. Xu, C.-F. Li, and G.-C. Guo, “Demonstration of multisetting one-way Einstein-Podolsky-Rosen steering in two-qubit systems,” Phys. Rev. Lett. 118, 140404 (2017).
[Crossref] [PubMed]

2016 (3)

K. Sun, X.-J. Ye, J.-S. Xu, X.-Y. Xu, J.-S. Tang, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental quantification of asymmetric Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
[Crossref] [PubMed]

S. Wollmann, N. Walk, A. J. Bennet, H. M. Wiseman, and G. J. Pryde, “Observation of Genuine One-Way Einstein-Podolsky-Rosen Steering,” Phys. Rev. Lett. 116, 160403 (2016).
[Crossref] [PubMed]

D. H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K. J. Resch, H. M. Wiseman, and A. Steinberg, “Experimental nonlocal and surreal Bohmian trajectories,” Sci. Adv. 2, e1501466 (2016).
[Crossref] [PubMed]

2015 (2)

M. Piani and J. Watrous, “Necessary and sufficient quantum information characterization of Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 114, 060404 (2015).
[Crossref] [PubMed]

I. Kogias, A. R. Lee, S. Ragy, and G. Adesso, “Quantification of Gaussian quantum steering,” Phys. Rev. Lett. 114, 060403 (2015).
[Crossref] [PubMed]

2014 (5)

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein-Podolsky-Rosen Steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

M. T. Quintino, T. Vértesi, and N. Brunner, “Joint measurability, Einstein-Podolsky-Rosen steering, and Bell nonlocality,” Phys. Rev. Lett. 113, 160402 (2014).
[Crossref] [PubMed]

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 (2014).
[Crossref]

Y. Aharonov, E. Cohen, and A. C. Elitzur, “Foundations and applications of weak quantum measurements,” Phys. Rev. A 89, 052105 (2014).
[Crossref]

K. Sun, J.-S. Xu, X.-J. Ye, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental demonstration of the Einstein-Podolsky-Rosen steering game based on the all-versus-nothing proof,” Phys. Rev. Lett. 113, 140402 (2014).
[Crossref] [PubMed]

2013 (2)

B. Tamir and E. Cohen, “Introduction to weak measurements and weak values,” Quanta 2, 7 (2013).
[Crossref]

B. Braverman and C. Simon, “Proposal to observe the nonlocality of Bohmian trajectories with entangled photons,” Phys. Rev. Lett. 110, 060406 (2013).
[Crossref] [PubMed]

2012 (2)

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

G. Naaman-Marom, N. Erez, and L. Vaidman, “Position measurements in the de Broglie-Bohm interpretation of quantum mechanics,” Ann. Phys.,  327, 2522 (2012).
[Crossref]

2011 (1)

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 photon in a two-slit Interferometer,” Science 332, 1170 (2011).
[Crossref] [PubMed]

2010 (1)

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845 (2010).
[Crossref]

2009 (1)

D. Durr, S. Goldstein, and N. Zanghi, “On the weak measurement of velocity in Bohmian mechanics,” J. Stat. Phys. 134, 1023 (2009).
[Crossref]

2007 (3)

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

A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger, “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Opt. Express 15, 15377 (2007).
[Crossref] [PubMed]

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[Crossref] [PubMed]

2006 (1)

B. J. Hiley and R. E. Callaghan, “Delayed-choice experiments and the Bohm approach,” Phys. Scr. 74, 336 (2006).
[Crossref]

1998 (1)

H. M. Wiseman, “Bohmian analysis of momentum transfer in Welcher Weg measurements,” Phys. Rev. A 58, 1740 (1998).
[Crossref]

1997 (1)

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 78, 5022 (1997).

1992 (1)

B.-G. Englert, M. O. Scully, G. Süssmann, and H. Walther, “Surrealistic Bohm trajectories,” Z. Naturforsch. 47a, 1175 (1992).

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 (1988).
[Crossref] [PubMed]

1952 (2)

D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables. I,” Phys. Rev. 85, 166 (1952).
[Crossref]

D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables. II,” Phys. Rev. 85, 180 (1952).
[Crossref]

1936 (1)

E. Schrödinger, “Probability relations between separated systems,” Proc. Camb. Philos. Soc. 32, 446 (1936).
[Crossref]

1935 (1)

E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Camb. Philos. Soc. 31, 555 (1935).
[Crossref]

Adesso, G.

I. Kogias, A. R. Lee, S. Ragy, and G. Adesso, “Quantification of Gaussian quantum steering,” Phys. Rev. Lett. 114, 060403 (2015).
[Crossref] [PubMed]

Aharonov, Y.

Y. Aharonov, E. Cohen, and A. C. Elitzur, “Foundations and applications of weak quantum measurements,” Phys. Rev. A 89, 052105 (2014).
[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 (1988).
[Crossref] [PubMed]

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 (1988).
[Crossref] [PubMed]

Ashhab, S.

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

Bennet, A. J.

S. Wollmann, N. Walk, A. J. Bennet, H. M. Wiseman, and G. J. Pryde, “Observation of Genuine One-Way Einstein-Podolsky-Rosen Steering,” Phys. Rev. Lett. 116, 160403 (2016).
[Crossref] [PubMed]

Bohm, D.

D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables. I,” Phys. Rev. 85, 166 (1952).
[Crossref]

D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables. II,” Phys. Rev. 85, 180 (1952).
[Crossref]

Bowles, J.

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein-Podolsky-Rosen Steering,” Phys. Rev. Lett. 112, 200402 (2014).
[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 (2014).
[Crossref]

Braverman, B.

B. Braverman and C. Simon, “Proposal to observe the nonlocality of Bohmian trajectories with entangled photons,” Phys. Rev. Lett. 110, 060406 (2013).
[Crossref] [PubMed]

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 photon in a two-slit Interferometer,” Science 332, 1170 (2011).
[Crossref] [PubMed]

Brunner, N.

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein-Podolsky-Rosen Steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

M. T. Quintino, T. Vértesi, and N. Brunner, “Joint measurability, Einstein-Podolsky-Rosen steering, and Bell nonlocality,” Phys. Rev. Lett. 113, 160402 (2014).
[Crossref] [PubMed]

Callaghan, R. E.

B. J. Hiley and R. E. Callaghan, “Delayed-choice experiments and the Bohm approach,” Phys. Scr. 74, 336 (2006).
[Crossref]

Chen, J.-L.

K. Sun, X.-J. Ye, J.-S. Xu, X.-Y. Xu, J.-S. Tang, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental quantification of asymmetric Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
[Crossref] [PubMed]

K. Sun, J.-S. Xu, X.-J. Ye, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental demonstration of the Einstein-Podolsky-Rosen steering game based on the all-versus-nothing proof,” Phys. Rev. Lett. 113, 140402 (2014).
[Crossref] [PubMed]

Cohen, E.

Y. Aharonov, E. Cohen, and A. C. Elitzur, “Foundations and applications of weak quantum measurements,” Phys. Rev. A 89, 052105 (2014).
[Crossref]

B. Tamir and E. Cohen, “Introduction to weak measurements and weak values,” Quanta 2, 7 (2013).
[Crossref]

Doherty, A. C.

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[Crossref] [PubMed]

Dressel, J.

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 (2014).
[Crossref]

Durr, D.

D. Durr, S. Goldstein, and N. Zanghi, “On the weak measurement of velocity in Bohmian mechanics,” J. Stat. Phys. 134, 1023 (2009).
[Crossref]

Elitzur, A. C.

Y. Aharonov, E. Cohen, and A. C. Elitzur, “Foundations and applications of weak quantum measurements,” Phys. Rev. A 89, 052105 (2014).
[Crossref]

Englert, B.-G.

B.-G. Englert, M. O. Scully, G. Süssmann, and H. Walther, “Surrealistic Bohm trajectories,” Z. Naturforsch. 47a, 1175 (1992).

Erez, N.

G. Naaman-Marom, N. Erez, and L. Vaidman, “Position measurements in the de Broglie-Bohm interpretation of quantum mechanics,” Ann. Phys.,  327, 2522 (2012).
[Crossref]

Fedrizzi, A.

Fisher, K.

D. H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K. J. Resch, H. M. Wiseman, and A. Steinberg, “Experimental nonlocal and surreal Bohmian trajectories,” Sci. Adv. 2, e1501466 (2016).
[Crossref] [PubMed]

Goldstein, S.

D. Durr, S. Goldstein, and N. Zanghi, “On the weak measurement of velocity in Bohmian mechanics,” J. Stat. Phys. 134, 1023 (2009).
[Crossref]

Guo, G.-C.

Y. Xiao, X.-J. Ye, K. Sun, J.-S. Xu, C.-F. Li, and G.-C. Guo, “Demonstration of multisetting one-way Einstein-Podolsky-Rosen steering in two-qubit systems,” Phys. Rev. Lett. 118, 140404 (2017).
[Crossref] [PubMed]

K. Sun, X.-J. Ye, J.-S. Xu, X.-Y. Xu, J.-S. Tang, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental quantification of asymmetric Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
[Crossref] [PubMed]

K. Sun, J.-S. Xu, X.-J. Ye, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental demonstration of the Einstein-Podolsky-Rosen steering game based on the all-versus-nothing proof,” Phys. Rev. Lett. 113, 140402 (2014).
[Crossref] [PubMed]

Herbst, T.

Hiley, B. J.

B. J. Hiley and R. E. Callaghan, “Delayed-choice experiments and the Bohm approach,” Phys. Scr. 74, 336 (2006).
[Crossref]

Jennewein, T.

Jones, S. J.

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845 (2010).
[Crossref]

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[Crossref] [PubMed]

Jordan, A. N.

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 (2014).
[Crossref]

Kocsis, S.

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 photon in a two-slit Interferometer,” Science 332, 1170 (2011).
[Crossref] [PubMed]

Kofman, A. G.

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

Kogias, I.

I. Kogias, A. R. Lee, S. Ragy, and G. Adesso, “Quantification of Gaussian quantum steering,” Phys. Rev. Lett. 114, 060403 (2015).
[Crossref] [PubMed]

Lee, A. R.

I. Kogias, A. R. Lee, S. Ragy, and G. Adesso, “Quantification of Gaussian quantum steering,” Phys. Rev. Lett. 114, 060403 (2015).
[Crossref] [PubMed]

Li, C.-F.

Y. Xiao, X.-J. Ye, K. Sun, J.-S. Xu, C.-F. Li, and G.-C. Guo, “Demonstration of multisetting one-way Einstein-Podolsky-Rosen steering in two-qubit systems,” Phys. Rev. Lett. 118, 140404 (2017).
[Crossref] [PubMed]

K. Sun, X.-J. Ye, J.-S. Xu, X.-Y. Xu, J.-S. Tang, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental quantification of asymmetric Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
[Crossref] [PubMed]

K. Sun, J.-S. Xu, X.-J. Ye, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental demonstration of the Einstein-Podolsky-Rosen steering game based on the all-versus-nothing proof,” Phys. Rev. Lett. 113, 140402 (2014).
[Crossref] [PubMed]

Mahler, D. H.

D. H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K. J. Resch, H. M. Wiseman, and A. Steinberg, “Experimental nonlocal and surreal Bohmian trajectories,” Sci. Adv. 2, e1501466 (2016).
[Crossref] [PubMed]

Malik, M.

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 (2014).
[Crossref]

Miatto, F. M.

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 (2014).
[Crossref]

Mirin, R. P.

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 photon in a two-slit Interferometer,” Science 332, 1170 (2011).
[Crossref] [PubMed]

Naaman-Marom, G.

G. Naaman-Marom, N. Erez, and L. Vaidman, “Position measurements in the de Broglie-Bohm interpretation of quantum mechanics,” Ann. Phys.,  327, 2522 (2012).
[Crossref]

Nori, F.

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

Piani, M.

M. Piani and J. Watrous, “Necessary and sufficient quantum information characterization of Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 114, 060404 (2015).
[Crossref] [PubMed]

Poppe, A.

Pryde, G. J.

S. Wollmann, N. Walk, A. J. Bennet, H. M. Wiseman, and G. J. Pryde, “Observation of Genuine One-Way Einstein-Podolsky-Rosen Steering,” Phys. Rev. Lett. 116, 160403 (2016).
[Crossref] [PubMed]

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845 (2010).
[Crossref]

Quintino, M. T.

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein-Podolsky-Rosen Steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

M. T. Quintino, T. Vértesi, and N. Brunner, “Joint measurability, Einstein-Podolsky-Rosen steering, and Bell nonlocality,” Phys. Rev. Lett. 113, 160402 (2014).
[Crossref] [PubMed]

Ragy, S.

I. Kogias, A. R. Lee, S. Ragy, and G. Adesso, “Quantification of Gaussian quantum steering,” Phys. Rev. Lett. 114, 060403 (2015).
[Crossref] [PubMed]

Ravets, S.

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 photon in a two-slit Interferometer,” Science 332, 1170 (2011).
[Crossref] [PubMed]

Resch, K. J.

D. H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K. J. Resch, H. M. Wiseman, and A. Steinberg, “Experimental nonlocal and surreal Bohmian trajectories,” Sci. Adv. 2, e1501466 (2016).
[Crossref] [PubMed]

Rozema, L.

D. H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K. J. Resch, H. M. Wiseman, and A. Steinberg, “Experimental nonlocal and surreal Bohmian trajectories,” Sci. Adv. 2, e1501466 (2016).
[Crossref] [PubMed]

Saunders, D. J.

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845 (2010).
[Crossref]

Schrödinger, E.

E. Schrödinger, “Probability relations between separated systems,” Proc. Camb. Philos. Soc. 32, 446 (1936).
[Crossref]

E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Camb. Philos. Soc. 31, 555 (1935).
[Crossref]

Scully, M. O.

B.-G. Englert, M. O. Scully, G. Süssmann, and H. Walther, “Surrealistic Bohm trajectories,” Z. Naturforsch. 47a, 1175 (1992).

Shalm, L. K.

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 photon in a two-slit Interferometer,” Science 332, 1170 (2011).
[Crossref] [PubMed]

Simon, C.

B. Braverman and C. Simon, “Proposal to observe the nonlocality of Bohmian trajectories with entangled photons,” Phys. Rev. Lett. 110, 060406 (2013).
[Crossref] [PubMed]

Steinberg, A.

D. H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K. J. Resch, H. M. Wiseman, and A. Steinberg, “Experimental nonlocal and surreal Bohmian trajectories,” Sci. Adv. 2, e1501466 (2016).
[Crossref] [PubMed]

Steinberg, A. M.

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 photon in a two-slit Interferometer,” Science 332, 1170 (2011).
[Crossref] [PubMed]

Stevens, M. J.

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 photon in a two-slit Interferometer,” Science 332, 1170 (2011).
[Crossref] [PubMed]

Sun, K.

Y. Xiao, X.-J. Ye, K. Sun, J.-S. Xu, C.-F. Li, and G.-C. Guo, “Demonstration of multisetting one-way Einstein-Podolsky-Rosen steering in two-qubit systems,” Phys. Rev. Lett. 118, 140404 (2017).
[Crossref] [PubMed]

K. Sun, X.-J. Ye, J.-S. Xu, X.-Y. Xu, J.-S. Tang, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental quantification of asymmetric Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
[Crossref] [PubMed]

K. Sun, J.-S. Xu, X.-J. Ye, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental demonstration of the Einstein-Podolsky-Rosen steering game based on the all-versus-nothing proof,” Phys. Rev. Lett. 113, 140402 (2014).
[Crossref] [PubMed]

Süssmann, G.

B.-G. Englert, M. O. Scully, G. Süssmann, and H. Walther, “Surrealistic Bohm trajectories,” Z. Naturforsch. 47a, 1175 (1992).

Tamir, B.

B. Tamir and E. Cohen, “Introduction to weak measurements and weak values,” Quanta 2, 7 (2013).
[Crossref]

Tang, J.-S.

K. Sun, X.-J. Ye, J.-S. Xu, X.-Y. Xu, J.-S. Tang, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental quantification of asymmetric Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
[Crossref] [PubMed]

Vaidman, L.

G. Naaman-Marom, N. Erez, and L. Vaidman, “Position measurements in the de Broglie-Bohm interpretation of quantum mechanics,” Ann. Phys.,  327, 2522 (2012).
[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 (1988).
[Crossref] [PubMed]

Vermeyden, L.

D. H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K. J. Resch, H. M. Wiseman, and A. Steinberg, “Experimental nonlocal and surreal Bohmian trajectories,” Sci. Adv. 2, e1501466 (2016).
[Crossref] [PubMed]

Vértesi, T.

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein-Podolsky-Rosen Steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

M. T. Quintino, T. Vértesi, and N. Brunner, “Joint measurability, Einstein-Podolsky-Rosen steering, and Bell nonlocality,” Phys. Rev. Lett. 113, 160402 (2014).
[Crossref] [PubMed]

Walk, N.

S. Wollmann, N. Walk, A. J. Bennet, H. M. Wiseman, and G. J. Pryde, “Observation of Genuine One-Way Einstein-Podolsky-Rosen Steering,” Phys. Rev. Lett. 116, 160403 (2016).
[Crossref] [PubMed]

Walther, H.

B.-G. Englert, M. O. Scully, G. Süssmann, and H. Walther, “Surrealistic Bohm trajectories,” Z. Naturforsch. 47a, 1175 (1992).

Watrous, J.

M. Piani and J. Watrous, “Necessary and sufficient quantum information characterization of Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 114, 060404 (2015).
[Crossref] [PubMed]

Wiseman, H. M.

D. H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K. J. Resch, H. M. Wiseman, and A. Steinberg, “Experimental nonlocal and surreal Bohmian trajectories,” Sci. Adv. 2, e1501466 (2016).
[Crossref] [PubMed]

S. Wollmann, N. Walk, A. J. Bennet, H. M. Wiseman, and G. J. Pryde, “Observation of Genuine One-Way Einstein-Podolsky-Rosen Steering,” Phys. Rev. Lett. 116, 160403 (2016).
[Crossref] [PubMed]

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845 (2010).
[Crossref]

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[Crossref] [PubMed]

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

H. M. Wiseman, “Bohmian analysis of momentum transfer in Welcher Weg measurements,” Phys. Rev. A 58, 1740 (1998).
[Crossref]

Wollmann, S.

S. Wollmann, N. Walk, A. J. Bennet, H. M. Wiseman, and G. J. Pryde, “Observation of Genuine One-Way Einstein-Podolsky-Rosen Steering,” Phys. Rev. Lett. 116, 160403 (2016).
[Crossref] [PubMed]

Wootters, W. K.

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 78, 5022 (1997).

Wu, Y.-C.

K. Sun, X.-J. Ye, J.-S. Xu, X.-Y. Xu, J.-S. Tang, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental quantification of asymmetric Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
[Crossref] [PubMed]

K. Sun, J.-S. Xu, X.-J. Ye, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental demonstration of the Einstein-Podolsky-Rosen steering game based on the all-versus-nothing proof,” Phys. Rev. Lett. 113, 140402 (2014).
[Crossref] [PubMed]

Xiao, Y.

Y. Xiao, X.-J. Ye, K. Sun, J.-S. Xu, C.-F. Li, and G.-C. Guo, “Demonstration of multisetting one-way Einstein-Podolsky-Rosen steering in two-qubit systems,” Phys. Rev. Lett. 118, 140404 (2017).
[Crossref] [PubMed]

Xu, J.-S.

Y. Xiao, X.-J. Ye, K. Sun, J.-S. Xu, C.-F. Li, and G.-C. Guo, “Demonstration of multisetting one-way Einstein-Podolsky-Rosen steering in two-qubit systems,” Phys. Rev. Lett. 118, 140404 (2017).
[Crossref] [PubMed]

K. Sun, X.-J. Ye, J.-S. Xu, X.-Y. Xu, J.-S. Tang, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental quantification of asymmetric Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
[Crossref] [PubMed]

K. Sun, J.-S. Xu, X.-J. Ye, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental demonstration of the Einstein-Podolsky-Rosen steering game based on the all-versus-nothing proof,” Phys. Rev. Lett. 113, 140402 (2014).
[Crossref] [PubMed]

Xu, X.-Y.

K. Sun, X.-J. Ye, J.-S. Xu, X.-Y. Xu, J.-S. Tang, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental quantification of asymmetric Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
[Crossref] [PubMed]

Ye, X.-J.

Y. Xiao, X.-J. Ye, K. Sun, J.-S. Xu, C.-F. Li, and G.-C. Guo, “Demonstration of multisetting one-way Einstein-Podolsky-Rosen steering in two-qubit systems,” Phys. Rev. Lett. 118, 140404 (2017).
[Crossref] [PubMed]

K. Sun, X.-J. Ye, J.-S. Xu, X.-Y. Xu, J.-S. Tang, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental quantification of asymmetric Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
[Crossref] [PubMed]

K. Sun, J.-S. Xu, X.-J. Ye, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental demonstration of the Einstein-Podolsky-Rosen steering game based on the all-versus-nothing proof,” Phys. Rev. Lett. 113, 140402 (2014).
[Crossref] [PubMed]

Zanghi, N.

D. Durr, S. Goldstein, and N. Zanghi, “On the weak measurement of velocity in Bohmian mechanics,” J. Stat. Phys. 134, 1023 (2009).
[Crossref]

Zeilinger, A.

Ann. Phys. (1)

G. Naaman-Marom, N. Erez, and L. Vaidman, “Position measurements in the de Broglie-Bohm interpretation of quantum mechanics,” Ann. Phys.,  327, 2522 (2012).
[Crossref]

J. Stat. Phys. (1)

D. Durr, S. Goldstein, and N. Zanghi, “On the weak measurement of velocity in Bohmian mechanics,” J. Stat. Phys. 134, 1023 (2009).
[Crossref]

Nat. Phys. (1)

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845 (2010).
[Crossref]

New J. Phys. (1)

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

Opt. Express (1)

Phys. Rep. (1)

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

Phys. Rev. (2)

D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables. I,” Phys. Rev. 85, 166 (1952).
[Crossref]

D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables. II,” Phys. Rev. 85, 180 (1952).
[Crossref]

Phys. Rev. A (2)

Y. Aharonov, E. Cohen, and A. C. Elitzur, “Foundations and applications of weak quantum measurements,” Phys. Rev. A 89, 052105 (2014).
[Crossref]

H. M. Wiseman, “Bohmian analysis of momentum transfer in Welcher Weg measurements,” Phys. Rev. A 58, 1740 (1998).
[Crossref]

Phys. Rev. Lett. (12)

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 78, 5022 (1997).

B. Braverman and C. Simon, “Proposal to observe the nonlocality of Bohmian trajectories with entangled photons,” Phys. Rev. Lett. 110, 060406 (2013).
[Crossref] [PubMed]

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 (1988).
[Crossref] [PubMed]

K. Sun, J.-S. Xu, X.-J. Ye, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental demonstration of the Einstein-Podolsky-Rosen steering game based on the all-versus-nothing proof,” Phys. Rev. Lett. 113, 140402 (2014).
[Crossref] [PubMed]

K. Sun, X.-J. Ye, J.-S. Xu, X.-Y. Xu, J.-S. Tang, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental quantification of asymmetric Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
[Crossref] [PubMed]

S. Wollmann, N. Walk, A. J. Bennet, H. M. Wiseman, and G. J. Pryde, “Observation of Genuine One-Way Einstein-Podolsky-Rosen Steering,” Phys. Rev. Lett. 116, 160403 (2016).
[Crossref] [PubMed]

Y. Xiao, X.-J. Ye, K. Sun, J.-S. Xu, C.-F. Li, and G.-C. Guo, “Demonstration of multisetting one-way Einstein-Podolsky-Rosen steering in two-qubit systems,” Phys. Rev. Lett. 118, 140404 (2017).
[Crossref] [PubMed]

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[Crossref] [PubMed]

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein-Podolsky-Rosen Steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

M. T. Quintino, T. Vértesi, and N. Brunner, “Joint measurability, Einstein-Podolsky-Rosen steering, and Bell nonlocality,” Phys. Rev. Lett. 113, 160402 (2014).
[Crossref] [PubMed]

M. Piani and J. Watrous, “Necessary and sufficient quantum information characterization of Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 114, 060404 (2015).
[Crossref] [PubMed]

I. Kogias, A. R. Lee, S. Ragy, and G. Adesso, “Quantification of Gaussian quantum steering,” Phys. Rev. Lett. 114, 060403 (2015).
[Crossref] [PubMed]

Phys. Scr. (1)

B. J. Hiley and R. E. Callaghan, “Delayed-choice experiments and the Bohm approach,” Phys. Scr. 74, 336 (2006).
[Crossref]

Proc. Camb. Philos. Soc. (2)

E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Camb. Philos. Soc. 31, 555 (1935).
[Crossref]

E. Schrödinger, “Probability relations between separated systems,” Proc. Camb. Philos. Soc. 32, 446 (1936).
[Crossref]

Quanta (1)

B. Tamir and E. Cohen, “Introduction to weak measurements and weak values,” Quanta 2, 7 (2013).
[Crossref]

Rev. Mod. Phys. (1)

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 (2014).
[Crossref]

Sci. Adv. (1)

D. H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K. J. Resch, H. M. Wiseman, and A. Steinberg, “Experimental nonlocal and surreal Bohmian trajectories,” Sci. Adv. 2, e1501466 (2016).
[Crossref] [PubMed]

Science (1)

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 photon in a two-slit Interferometer,” Science 332, 1170 (2011).
[Crossref] [PubMed]

Z. Naturforsch. (1)

B.-G. Englert, M. O. Scully, G. Süssmann, and H. Walther, “Surrealistic Bohm trajectories,” Z. Naturforsch. 47a, 1175 (1992).

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Figures (3)

Fig. 1
Fig. 1 Experimental setup. Preparation of polarization-entangled states: An ultraviolet laser is reflected by a dichroic mirror (DM) to pump the PPKTP crystal located in a Sagnac interferometer. Dual-wavelength half-wave plates (HWP1 and HWP2) are used to rotate the polarizations of the pump laser and the down-converted photons which are separated by the dual-wavelength polarizing beam splitter (PBS1). Interference filters (IF) are used in both paths. Projective measurement: The polarization of photon A (Alice’s side) is projected using HWP3 and PBS2. It is then coupled by a fiber coupler (FC) and detected by the single-photon avalanche detector (SPAD) with the electric signals using as a trigger. Trajectories measurement: Photon B (Bob’s side) is coupled to an 85 m long single-mode fiber and then separated into two paths by a beam displacer (BD1). HWP4 is used to make the polarization of the two beams identical and birefrigent crystals (PC) are used to compensate the difference in the optical length. The polarization of the two beams is rotated by HWP5 and the photon goes through a thin calcite crystal to perform weak measurement. A quarter wave plate (QWP) and BD2 are used to detect the polarization of the photon. Lens L1 (plano-convex), L2 (aspherical, movelable) and L3 (plano-convex cylindrical) are used to image different planes in the ICCD camera which is triggered by the SPAD.
Fig. 2
Fig. 2 Steered Bohmian trajectories. The lines are reconstructed trajectories of photon B, using the measured Bohmian velocities. The trajectories start at one point and follow the non-projected velocity until at some point, a measurement on photon A takes place, and this modifies the velocity of photon B. The state in which photon A is found, determines the new direction for photon B, and from that point we plot the trajectories according to a number of states cos θ|H〉 − sin θ|V〉, shown in different colors for different θ (the non-projected trajectory is represented by the black dashed line after the point). The different panels represent different times for the measurement on photon A, which correspond to the time photon B has passed a longitudinal distance of (a) z=1.492 m, (b) z=2.245 m, (c) z=3.038 mm, and (d) z=3.749 m. The black dot line indicates the center between the two slits.
Fig. 3
Fig. 3 The change in velocity of photon B due to projection of photon A. Particle B is in a position (x, z), traveling with transverse velocity vx (x, z) (without projective-measurement on particle A) and then a remote measurement finds particle A in a state cos θ|H〉 − sin θ|V〉. The velocity of particle B is changed and this change is shown for (a) θ = 18.5°, (b) θ = 31.4°, (c) θ = 45.1°, (d) θ = 62.9°.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

v j = Re ψ * P j ψ m j | ψ | 2 ,
ψ = 1 2 f ( q B ) g ( q A ) ( | B | A + | B | A ) ,
ψ = 1 2 g ( q A ) ( f ( q B + d ) | B | A + f ( q B ) | B | A ) .
| θ A = cos θ | A sin θ | A ,
ψ = g ( q A + L ) 2 [ f ( q B + d ) cos ( θ ) | B f ( q B ) sin ( θ ) | B ] | θ A + g ( q A ) 2 [ f ( q B + d ) sin ( θ ) | B + f ( q B ) cos ( θ ) | B ] | θ ¯ A ,
v x ( x , z ) = c k x w k ,
ϕ ( k x ) = ζ k k x w = arcsin ( I R I L I R + I L ) .
ϕ ( k x ) = ζ k x w k + ϕ 0 .
θ in sin ( θ in ) = k x w k .
ζ θ in + ϕ 0 = arcsin ( I R I L I R + I L ) ,
k x w i | k | = 1 ζ arcsin [ N R i / S R N L i / S L N R i / S R + N L i / S L ] .
x rel z = ( R c [ R c ] + i ) * 0.013 β ,

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