Abstract

In a recent Letter, Brunner and Simon proposed an interferometric scheme using imaginary weak values with a frequency-domain analysis to outperform standard interferometry in longitudinal phase shifts [Phys. Rev. Lett 105, 010405 (2010)]. Here we demonstrate an interferometric scheme combined with a time-domain analysis to measure longitudinal velocities. The technique employs the near-destructive interference of non-Fourier limited pulses, one Doppler shifted due to a moving mirror in a Michelson interferometer. We achieve a velocity measurement of 400fm/s and show our estimator to be efficient by reaching its Cramér–Rao bound.

© 2013 Optical Society of America

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  1. C. R. Rao, Proc. Cambridge Philos. Soc. 43, 280283 (1946).
  2. T. Pfister, A. Fischer, and J. Czarske, Meas. Sci. Technol. 22, 055301 (2011).
    [CrossRef]
  3. D. J. Starling, P. B. Dixon, N. S. Williams, A. N. Jordan, and J. C. Howell, Phys. Rev. A 82, 011802 (2010).
    [CrossRef]
  4. C.-F. Li, X.-Y. Xu, J.-S. Tang, J.-S. Xu, and G.-C. Guo, Phys. Rev. A 83, 044102 (2011).
    [CrossRef]
  5. N. Brunner and C. Simon, Phys. Rev. Lett. 105, 010405 (2010).
    [CrossRef]
  6. J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, Phys. Rev. A 81, 033813 (2010).
    [CrossRef]
  7. P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett. 102, 173601 (2009).
    [CrossRef]
  8. V. Giovannetti, S. Lloyd, and L. Maccone, Nature 412, 417 (2001).
    [CrossRef]
  9. P. R. Kaczmarek, T. Rogowski, A. J. Antonczak, and K. M. Abramski, Opt. Appl. 34, 373 (2004).
  10. J. Guéna, R. Li, K. Gibble, S. Bize, and A. Clairon, Phys. Rev. Lett. 106, 130801 (2011).
    [CrossRef]
  11. A. H. Meier and T. Roesgen, Exp. Fluids 52, 1017 (2012).
    [CrossRef]
  12. T. O. H. Charrett, S. W. James, and R. P. Tatam, Meas. Sci. Technol. 23, 032001 (2012).
    [CrossRef]
  13. J. W. Czarske and H. Muller, Opt. Commun. 132, 421 (1996).
    [CrossRef]
  14. L. Scalise, Y. G. Yu, G. Giuliani, G. Plantier, and T. Bosch, IEEE Trans. Instrum. Meas. 53, 223 (2004).
    [CrossRef]
  15. S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
    [CrossRef]
  16. D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, Phys. Rev. A 82, 063822 (2010).
    [CrossRef]
  17. Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
    [CrossRef]
  18. A. Feizpour, X. Xing, and A. M. Steinberg, Phys. Rev. Lett. 107, 133603 (2011).
    [CrossRef]
  19. O. Hosten and P. Kwiat, Science 319, 787 (2008).
    [CrossRef]
  20. J. S. Laudeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Nature 474, 188 (2011).
    [CrossRef]
  21. S. Massar and S. Popescu, Phys. Rev. A 84, 052106 (2011).
    [CrossRef]
  22. H. F. Hofmann, M. E. Goggin, M. P. Almeida, and M. Barbieri, Phys. Rev. A 86, 040102 (2012).
    [CrossRef]
  23. J. Martínez-Rincón and J. C. Howell, “Cramer–Rao bound and weak values,” in preparation.
  24. G. Strübi and C. Bruder, Phys. Rev. Lett. 110, 083605 (2013).
    [CrossRef]
  25. Y. Kedem, Phys. Rev. A 85, 060102 (2012).
    [CrossRef]
  26. F. Zernike, Physica 9, 686 (1942).
    [CrossRef]
  27. The system is prepared in the state |Ψ〉∝exp[−t2/4τ2](|M〉+|P〉)⊗|α〉, where the states |M〉 and |P〉 correspond to which arm (mirror or piezo-driven mirror) the light goes though, and |α〉 is the coherent light state representation. The interaction can be expressed as U≈exp[−iωdt/2A^], where ωd=2πfd and A^=|M〉〈M|−|P〉〈P|. The final postselection is given by [|M〉−exp(i2ϕ)|P〉]/2. This gives a pure imaginary weak value protocol, where Aw=−i cot ϕ≈−i/ϕ.
  28. P. Réfrégier, Noise Theory and Application to Physics (Springer, 2004).
  29. S. M. Kay, Fundamentals of Statistical Signal Processing (Prentice-Hall, 1993).

2013 (1)

G. Strübi and C. Bruder, Phys. Rev. Lett. 110, 083605 (2013).
[CrossRef]

2012 (4)

Y. Kedem, Phys. Rev. A 85, 060102 (2012).
[CrossRef]

H. F. Hofmann, M. E. Goggin, M. P. Almeida, and M. Barbieri, Phys. Rev. A 86, 040102 (2012).
[CrossRef]

A. H. Meier and T. Roesgen, Exp. Fluids 52, 1017 (2012).
[CrossRef]

T. O. H. Charrett, S. W. James, and R. P. Tatam, Meas. Sci. Technol. 23, 032001 (2012).
[CrossRef]

2011 (7)

J. Guéna, R. Li, K. Gibble, S. Bize, and A. Clairon, Phys. Rev. Lett. 106, 130801 (2011).
[CrossRef]

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
[CrossRef]

A. Feizpour, X. Xing, and A. M. Steinberg, Phys. Rev. Lett. 107, 133603 (2011).
[CrossRef]

C.-F. Li, X.-Y. Xu, J.-S. Tang, J.-S. Xu, and G.-C. Guo, Phys. Rev. A 83, 044102 (2011).
[CrossRef]

T. Pfister, A. Fischer, and J. Czarske, Meas. Sci. Technol. 22, 055301 (2011).
[CrossRef]

J. S. Laudeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Nature 474, 188 (2011).
[CrossRef]

S. Massar and S. Popescu, Phys. Rev. A 84, 052106 (2011).
[CrossRef]

2010 (4)

D. J. Starling, P. B. Dixon, N. S. Williams, A. N. Jordan, and J. C. Howell, Phys. Rev. A 82, 011802 (2010).
[CrossRef]

N. Brunner and C. Simon, Phys. Rev. Lett. 105, 010405 (2010).
[CrossRef]

J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, Phys. Rev. A 81, 033813 (2010).
[CrossRef]

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, Phys. Rev. A 82, 063822 (2010).
[CrossRef]

2009 (1)

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett. 102, 173601 (2009).
[CrossRef]

2008 (1)

O. Hosten and P. Kwiat, Science 319, 787 (2008).
[CrossRef]

2004 (2)

L. Scalise, Y. G. Yu, G. Giuliani, G. Plantier, and T. Bosch, IEEE Trans. Instrum. Meas. 53, 223 (2004).
[CrossRef]

P. R. Kaczmarek, T. Rogowski, A. J. Antonczak, and K. M. Abramski, Opt. Appl. 34, 373 (2004).

2001 (1)

V. Giovannetti, S. Lloyd, and L. Maccone, Nature 412, 417 (2001).
[CrossRef]

1996 (1)

J. W. Czarske and H. Muller, Opt. Commun. 132, 421 (1996).
[CrossRef]

1988 (1)

Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
[CrossRef]

1946 (1)

C. R. Rao, Proc. Cambridge Philos. Soc. 43, 280283 (1946).

1942 (1)

F. Zernike, Physica 9, 686 (1942).
[CrossRef]

Abramski, K. M.

P. R. Kaczmarek, T. Rogowski, A. J. Antonczak, and K. M. Abramski, Opt. Appl. 34, 373 (2004).

Aharonov, Y.

Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
[CrossRef]

Albert, D. Z.

Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
[CrossRef]

Almeida, M. P.

H. F. Hofmann, M. E. Goggin, M. P. Almeida, and M. Barbieri, Phys. Rev. A 86, 040102 (2012).
[CrossRef]

Antonczak, A. J.

P. R. Kaczmarek, T. Rogowski, A. J. Antonczak, and K. M. Abramski, Opt. Appl. 34, 373 (2004).

Bamber, C.

J. S. Laudeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Nature 474, 188 (2011).
[CrossRef]

Barbieri, M.

H. F. Hofmann, M. E. Goggin, M. P. Almeida, and M. Barbieri, Phys. Rev. A 86, 040102 (2012).
[CrossRef]

Bize, S.

J. Guéna, R. Li, K. Gibble, S. Bize, and A. Clairon, Phys. Rev. Lett. 106, 130801 (2011).
[CrossRef]

Bosch, T.

L. Scalise, Y. G. Yu, G. Giuliani, G. Plantier, and T. Bosch, IEEE Trans. Instrum. Meas. 53, 223 (2004).
[CrossRef]

Braverman, B.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
[CrossRef]

Bruder, C.

G. Strübi and C. Bruder, Phys. Rev. Lett. 110, 083605 (2013).
[CrossRef]

Brunner, N.

N. Brunner and C. Simon, Phys. Rev. Lett. 105, 010405 (2010).
[CrossRef]

Charrett, T. O. H.

T. O. H. Charrett, S. W. James, and R. P. Tatam, Meas. Sci. Technol. 23, 032001 (2012).
[CrossRef]

Clairon, A.

J. Guéna, R. Li, K. Gibble, S. Bize, and A. Clairon, Phys. Rev. Lett. 106, 130801 (2011).
[CrossRef]

Czarske, J.

T. Pfister, A. Fischer, and J. Czarske, Meas. Sci. Technol. 22, 055301 (2011).
[CrossRef]

Czarske, J. W.

J. W. Czarske and H. Muller, Opt. Commun. 132, 421 (1996).
[CrossRef]

Dixon, P. B.

J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, Phys. Rev. A 81, 033813 (2010).
[CrossRef]

D. J. Starling, P. B. Dixon, N. S. Williams, A. N. Jordan, and J. C. Howell, Phys. Rev. A 82, 011802 (2010).
[CrossRef]

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, Phys. Rev. A 82, 063822 (2010).
[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett. 102, 173601 (2009).
[CrossRef]

Feizpour, A.

A. Feizpour, X. Xing, and A. M. Steinberg, Phys. Rev. Lett. 107, 133603 (2011).
[CrossRef]

Fischer, A.

T. Pfister, A. Fischer, and J. Czarske, Meas. Sci. Technol. 22, 055301 (2011).
[CrossRef]

Gibble, K.

J. Guéna, R. Li, K. Gibble, S. Bize, and A. Clairon, Phys. Rev. Lett. 106, 130801 (2011).
[CrossRef]

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, Nature 412, 417 (2001).
[CrossRef]

Giuliani, G.

L. Scalise, Y. G. Yu, G. Giuliani, G. Plantier, and T. Bosch, IEEE Trans. Instrum. Meas. 53, 223 (2004).
[CrossRef]

Goggin, M. E.

H. F. Hofmann, M. E. Goggin, M. P. Almeida, and M. Barbieri, Phys. Rev. A 86, 040102 (2012).
[CrossRef]

Guéna, J.

J. Guéna, R. Li, K. Gibble, S. Bize, and A. Clairon, Phys. Rev. Lett. 106, 130801 (2011).
[CrossRef]

Guo, G.-C.

C.-F. Li, X.-Y. Xu, J.-S. Tang, J.-S. Xu, and G.-C. Guo, Phys. Rev. A 83, 044102 (2011).
[CrossRef]

Hofmann, H. F.

H. F. Hofmann, M. E. Goggin, M. P. Almeida, and M. Barbieri, Phys. Rev. A 86, 040102 (2012).
[CrossRef]

Hosten, O.

O. Hosten and P. Kwiat, Science 319, 787 (2008).
[CrossRef]

Howell, J. C.

J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, Phys. Rev. A 81, 033813 (2010).
[CrossRef]

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, Phys. Rev. A 82, 063822 (2010).
[CrossRef]

D. J. Starling, P. B. Dixon, N. S. Williams, A. N. Jordan, and J. C. Howell, Phys. Rev. A 82, 011802 (2010).
[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett. 102, 173601 (2009).
[CrossRef]

J. Martínez-Rincón and J. C. Howell, “Cramer–Rao bound and weak values,” in preparation.

James, S. W.

T. O. H. Charrett, S. W. James, and R. P. Tatam, Meas. Sci. Technol. 23, 032001 (2012).
[CrossRef]

Jordan, A. N.

J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, Phys. Rev. A 81, 033813 (2010).
[CrossRef]

D. J. Starling, P. B. Dixon, N. S. Williams, A. N. Jordan, and J. C. Howell, Phys. Rev. A 82, 011802 (2010).
[CrossRef]

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, Phys. Rev. A 82, 063822 (2010).
[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett. 102, 173601 (2009).
[CrossRef]

Kaczmarek, P. R.

P. R. Kaczmarek, T. Rogowski, A. J. Antonczak, and K. M. Abramski, Opt. Appl. 34, 373 (2004).

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing (Prentice-Hall, 1993).

Kedem, Y.

Y. Kedem, Phys. Rev. A 85, 060102 (2012).
[CrossRef]

Kocsis, S.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
[CrossRef]

Kwiat, P.

O. Hosten and P. Kwiat, Science 319, 787 (2008).
[CrossRef]

Laudeen, J. S.

J. S. Laudeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Nature 474, 188 (2011).
[CrossRef]

Li, C.-F.

C.-F. Li, X.-Y. Xu, J.-S. Tang, J.-S. Xu, and G.-C. Guo, Phys. Rev. A 83, 044102 (2011).
[CrossRef]

Li, R.

J. Guéna, R. Li, K. Gibble, S. Bize, and A. Clairon, Phys. Rev. Lett. 106, 130801 (2011).
[CrossRef]

Lloyd, S.

V. Giovannetti, S. Lloyd, and L. Maccone, Nature 412, 417 (2001).
[CrossRef]

Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, Nature 412, 417 (2001).
[CrossRef]

Martínez-Rincón, J.

J. Martínez-Rincón and J. C. Howell, “Cramer–Rao bound and weak values,” in preparation.

Massar, S.

S. Massar and S. Popescu, Phys. Rev. A 84, 052106 (2011).
[CrossRef]

Meier, A. H.

A. H. Meier and T. Roesgen, Exp. Fluids 52, 1017 (2012).
[CrossRef]

Mirin, R. P.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
[CrossRef]

Muller, H.

J. W. Czarske and H. Muller, Opt. Commun. 132, 421 (1996).
[CrossRef]

Patel, A.

J. S. Laudeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Nature 474, 188 (2011).
[CrossRef]

Pfister, T.

T. Pfister, A. Fischer, and J. Czarske, Meas. Sci. Technol. 22, 055301 (2011).
[CrossRef]

Plantier, G.

L. Scalise, Y. G. Yu, G. Giuliani, G. Plantier, and T. Bosch, IEEE Trans. Instrum. Meas. 53, 223 (2004).
[CrossRef]

Popescu, S.

S. Massar and S. Popescu, Phys. Rev. A 84, 052106 (2011).
[CrossRef]

Rao, C. R.

C. R. Rao, Proc. Cambridge Philos. Soc. 43, 280283 (1946).

Ravets, S.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
[CrossRef]

Réfrégier, P.

P. Réfrégier, Noise Theory and Application to Physics (Springer, 2004).

Roesgen, T.

A. H. Meier and T. Roesgen, Exp. Fluids 52, 1017 (2012).
[CrossRef]

Rogowski, T.

P. R. Kaczmarek, T. Rogowski, A. J. Antonczak, and K. M. Abramski, Opt. Appl. 34, 373 (2004).

Scalise, L.

L. Scalise, Y. G. Yu, G. Giuliani, G. Plantier, and T. Bosch, IEEE Trans. Instrum. Meas. 53, 223 (2004).
[CrossRef]

Shalm, L. K.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
[CrossRef]

Simon, C.

N. Brunner and C. Simon, Phys. Rev. Lett. 105, 010405 (2010).
[CrossRef]

Starling, D. J.

J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, Phys. Rev. A 81, 033813 (2010).
[CrossRef]

D. J. Starling, P. B. Dixon, N. S. Williams, A. N. Jordan, and J. C. Howell, Phys. Rev. A 82, 011802 (2010).
[CrossRef]

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, Phys. Rev. A 82, 063822 (2010).
[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett. 102, 173601 (2009).
[CrossRef]

Steinberg, A. M.

A. Feizpour, X. Xing, and A. M. Steinberg, Phys. Rev. Lett. 107, 133603 (2011).
[CrossRef]

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
[CrossRef]

Stevens, M. J.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
[CrossRef]

Stewart, C.

J. S. Laudeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Nature 474, 188 (2011).
[CrossRef]

Strübi, G.

G. Strübi and C. Bruder, Phys. Rev. Lett. 110, 083605 (2013).
[CrossRef]

Sutherland, B.

J. S. Laudeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Nature 474, 188 (2011).
[CrossRef]

Tang, J.-S.

C.-F. Li, X.-Y. Xu, J.-S. Tang, J.-S. Xu, and G.-C. Guo, Phys. Rev. A 83, 044102 (2011).
[CrossRef]

Tatam, R. P.

T. O. H. Charrett, S. W. James, and R. P. Tatam, Meas. Sci. Technol. 23, 032001 (2012).
[CrossRef]

Vaidman, L.

Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
[CrossRef]

Vudyasetu, P. K.

J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, Phys. Rev. A 81, 033813 (2010).
[CrossRef]

Williams, N. S.

D. J. Starling, P. B. Dixon, N. S. Williams, A. N. Jordan, and J. C. Howell, Phys. Rev. A 82, 011802 (2010).
[CrossRef]

Xing, X.

A. Feizpour, X. Xing, and A. M. Steinberg, Phys. Rev. Lett. 107, 133603 (2011).
[CrossRef]

Xu, J.-S.

C.-F. Li, X.-Y. Xu, J.-S. Tang, J.-S. Xu, and G.-C. Guo, Phys. Rev. A 83, 044102 (2011).
[CrossRef]

Xu, X.-Y.

C.-F. Li, X.-Y. Xu, J.-S. Tang, J.-S. Xu, and G.-C. Guo, Phys. Rev. A 83, 044102 (2011).
[CrossRef]

Yu, Y. G.

L. Scalise, Y. G. Yu, G. Giuliani, G. Plantier, and T. Bosch, IEEE Trans. Instrum. Meas. 53, 223 (2004).
[CrossRef]

Zernike, F.

F. Zernike, Physica 9, 686 (1942).
[CrossRef]

Exp. Fluids (1)

A. H. Meier and T. Roesgen, Exp. Fluids 52, 1017 (2012).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

L. Scalise, Y. G. Yu, G. Giuliani, G. Plantier, and T. Bosch, IEEE Trans. Instrum. Meas. 53, 223 (2004).
[CrossRef]

Meas. Sci. Technol. (2)

T. O. H. Charrett, S. W. James, and R. P. Tatam, Meas. Sci. Technol. 23, 032001 (2012).
[CrossRef]

T. Pfister, A. Fischer, and J. Czarske, Meas. Sci. Technol. 22, 055301 (2011).
[CrossRef]

Nature (2)

V. Giovannetti, S. Lloyd, and L. Maccone, Nature 412, 417 (2001).
[CrossRef]

J. S. Laudeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Nature 474, 188 (2011).
[CrossRef]

Opt. Appl. (1)

P. R. Kaczmarek, T. Rogowski, A. J. Antonczak, and K. M. Abramski, Opt. Appl. 34, 373 (2004).

Opt. Commun. (1)

J. W. Czarske and H. Muller, Opt. Commun. 132, 421 (1996).
[CrossRef]

Phys. Rev. A (7)

D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, Phys. Rev. A 82, 063822 (2010).
[CrossRef]

J. C. Howell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, Phys. Rev. A 81, 033813 (2010).
[CrossRef]

D. J. Starling, P. B. Dixon, N. S. Williams, A. N. Jordan, and J. C. Howell, Phys. Rev. A 82, 011802 (2010).
[CrossRef]

C.-F. Li, X.-Y. Xu, J.-S. Tang, J.-S. Xu, and G.-C. Guo, Phys. Rev. A 83, 044102 (2011).
[CrossRef]

S. Massar and S. Popescu, Phys. Rev. A 84, 052106 (2011).
[CrossRef]

H. F. Hofmann, M. E. Goggin, M. P. Almeida, and M. Barbieri, Phys. Rev. A 86, 040102 (2012).
[CrossRef]

Y. Kedem, Phys. Rev. A 85, 060102 (2012).
[CrossRef]

Phys. Rev. Lett. (6)

G. Strübi and C. Bruder, Phys. Rev. Lett. 110, 083605 (2013).
[CrossRef]

N. Brunner and C. Simon, Phys. Rev. Lett. 105, 010405 (2010).
[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett. 102, 173601 (2009).
[CrossRef]

J. Guéna, R. Li, K. Gibble, S. Bize, and A. Clairon, Phys. Rev. Lett. 106, 130801 (2011).
[CrossRef]

Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
[CrossRef]

A. Feizpour, X. Xing, and A. M. Steinberg, Phys. Rev. Lett. 107, 133603 (2011).
[CrossRef]

Physica (1)

F. Zernike, Physica 9, 686 (1942).
[CrossRef]

Proc. Cambridge Philos. Soc. (1)

C. R. Rao, Proc. Cambridge Philos. Soc. 43, 280283 (1946).

Science (2)

O. Hosten and P. Kwiat, Science 319, 787 (2008).
[CrossRef]

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
[CrossRef]

Other (4)

The system is prepared in the state |Ψ〉∝exp[−t2/4τ2](|M〉+|P〉)⊗|α〉, where the states |M〉 and |P〉 correspond to which arm (mirror or piezo-driven mirror) the light goes though, and |α〉 is the coherent light state representation. The interaction can be expressed as U≈exp[−iωdt/2A^], where ωd=2πfd and A^=|M〉〈M|−|P〉〈P|. The final postselection is given by [|M〉−exp(i2ϕ)|P〉]/2. This gives a pure imaginary weak value protocol, where Aw=−i cot ϕ≈−i/ϕ.

P. Réfrégier, Noise Theory and Application to Physics (Springer, 2004).

S. M. Kay, Fundamentals of Statistical Signal Processing (Prentice-Hall, 1993).

J. Martínez-Rincón and J. C. Howell, “Cramer–Rao bound and weak values,” in preparation.

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Figures (3)

Fig. 1.
Fig. 1.

Optical modulator generates a non-Fourier limited Gaussian shaped pulse. We couple the pulse to a fiber and launch it to a Michelson interferometer where one mirror is moving with constant speed v. The interference is controlled by inducing a phase offset 2ϕ with the piezoelectric mirror. Photons exiting the interferometer are coupled into a fiber (not shown), and the arrival time of single photons is measured with an avalanche photodiode (APD) and a photon counting module.

Fig. 2.
Fig. 2.

Velocity of the mirror, v, is plotted as a function of τ={1.67,4.17,16.7,417,and833}ms. The phase offset angle is ϕ=0.31 radians. The points are the experimental results, and the lines are the theoretical predictions for different voltages. Signal-to-noise ratios are 54, 27.4, 14.7, and 5.7 for Vpp={105,52.5,26.25,10.5}mV, respectively.

Fig. 3.
Fig. 3.

Experimental error in Fig. 2 as a function of τN. The solid line is the CRB as in Eq. (4) for N54×106 photons. Note there are no error bars.

Tables (1)

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Table 1. Results of the Cut Gaussian Profile with τ=50s and N66×109a

Equations (5)

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Iout(t)Iin(t)|1exp(i2ϕ+i2kx(t))|2I0exp(t2/2τ2)sin2ϕ|sin(ϕ+kvt)sinϕ|2,
Iout(t)(I0sin2ϕ)exp[12τ2(t2kvτ2ϕ)2].
F(δt)=Nsin2ϕdtP(t;δt)[ddδtlnP(t;δt)]2Nϕ2τ2.
ΔvCRB=Δ(δt)ϕ2kτ2=12kτN.
SNR=δtτϕN=vΔv=fdΔfd.

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