Abstract

By slightly vibrating the mirrors in an interferometer at different frequencies, the photons’ trajectory information is stored in the light beam. To read out this information, we record the centroid location of the intensity distribution of output beam and Fourier analyze its time evolution. It is shown that every vibrating mirror contributes a peak in the Fourier spectrum. In other words, we can reveal the trajectory of the photons by figuring out the vibrating mirrors which ever interact with the light beam based on the Fourier spectrum. This techniques is not limited by the vibration amplitude of the mirrors.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Wave–particle duality of a photon in emission

Ming Lai and Jean-Claude Diels
J. Opt. Soc. Am. B 9(12) 2290-2294 (1992)

Nonlinear interferometers in quantum optics

M. V. Chekhova and Z. Y. Ou
Adv. Opt. Photon. 8(1) 104-155 (2016)

Coherent perfect absorption of path entangled single photons

Sumei Huang and G. S. Agarwal
Opt. Express 22(17) 20936-20947 (2014)

References

  • View by:
  • |
  • |
  • |

  1. R. P. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics: Vol. III (Addison Wesley, 1965).
  2. M. O. Scully and K. Drühl, “Quantum eraser: A proposed photon correlation experiment concerning observation and ‘delayed choice’ in quantum mechanics,” Phys. Rev. A 25, 2208 (1982).
    [Crossref]
  3. B. G. Englert, “Fringe visibility and which-way information: an inequality,” Phys. Rev. Lett. 77, 2154 (1996).
    [Crossref] [PubMed]
  4. V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Experimental realization of Wheeler’s delayed-choice Gedanken experiment,” Science 315, 966 (2007).
    [Crossref] [PubMed]
  5. V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-choice test of quantum complementarity with interfering single photons,” Phys. Rev. Lett. 100, 220402 (2008).
    [Crossref] [PubMed]
  6. A. Luis, “Complementarity and certainty relations for two-dimensional systems,” Phys. Rev. A 64, 012103 (2001).
    [Crossref]
  7. E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, “Dephasing in electron interference by a ‘which-path’ detector,” Nature (London) 391, 871–874 (1998).
    [Crossref]
  8. S. Dürr, T. Nonn, and G. Rempe, “Origin of quantum-mechanical complementarity probed by a which-way experiment in an atom interferometer,” Nature (London) 395, 33–37 (1998).
    [Crossref]
  9. S. Dürr, T. Nonn, and G. Rempe, “Fringe visibility and which-way information in an atom interferometer,” Phys. Rev. Lett. 81, 5705 (1998).
    [Crossref]
  10. A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman, “Asking Photons Where They Have Been,” Phys. Rev. Lett. 111, 240402 (2013).
    [Crossref]
  11. R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. ( McGraw-Hill, Boston2000).
  12. X. Z. Ou, G. A. Zheng, and C. H. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22(5), 4960–4972 (2014).
    [Crossref] [PubMed]
  13. E. B. Malm, N. C. Monserud, C. G. Brown, P. W. Wachulak, H. W. Xu, G. Balakrishnan, W. L. Chao, E. Anderson, and M. C. Marconi, “Tabletop single-shot extreme ultraviolet Fourier transform holography of an extended object,” Opt. Express 21(8), 9959–9966 (2013).
    [Crossref] [PubMed]
  14. B. E. Y. Svensson, “Comments to ‘Asking photons where they have been,’Phys. Rev. Lett.111, 240402 (2013)),” arXiv:1402.4315 [quant-ph] (2014).
    [Crossref]
  15. L. Greengard and J. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM Review 46(3), 443–454 (2004).
    [Crossref]
  16. D. G. Xu, Y. Huang, and J. U. Kang, “GPU-accelerated non-uniform fast Fourier transform-based compressive sensing spectral domain optical coherence tomography,” Opt. Express 22(12), 14871–14884 (2014).
    [Crossref] [PubMed]

2014 (2)

2013 (2)

2008 (1)

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-choice test of quantum complementarity with interfering single photons,” Phys. Rev. Lett. 100, 220402 (2008).
[Crossref] [PubMed]

2007 (1)

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Experimental realization of Wheeler’s delayed-choice Gedanken experiment,” Science 315, 966 (2007).
[Crossref] [PubMed]

2004 (1)

L. Greengard and J. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM Review 46(3), 443–454 (2004).
[Crossref]

2001 (1)

A. Luis, “Complementarity and certainty relations for two-dimensional systems,” Phys. Rev. A 64, 012103 (2001).
[Crossref]

1998 (3)

E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, “Dephasing in electron interference by a ‘which-path’ detector,” Nature (London) 391, 871–874 (1998).
[Crossref]

S. Dürr, T. Nonn, and G. Rempe, “Origin of quantum-mechanical complementarity probed by a which-way experiment in an atom interferometer,” Nature (London) 395, 33–37 (1998).
[Crossref]

S. Dürr, T. Nonn, and G. Rempe, “Fringe visibility and which-way information in an atom interferometer,” Phys. Rev. Lett. 81, 5705 (1998).
[Crossref]

1996 (1)

B. G. Englert, “Fringe visibility and which-way information: an inequality,” Phys. Rev. Lett. 77, 2154 (1996).
[Crossref] [PubMed]

1982 (1)

M. O. Scully and K. Drühl, “Quantum eraser: A proposed photon correlation experiment concerning observation and ‘delayed choice’ in quantum mechanics,” Phys. Rev. A 25, 2208 (1982).
[Crossref]

Anderson, E.

Aspect, A.

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-choice test of quantum complementarity with interfering single photons,” Phys. Rev. Lett. 100, 220402 (2008).
[Crossref] [PubMed]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Experimental realization of Wheeler’s delayed-choice Gedanken experiment,” Science 315, 966 (2007).
[Crossref] [PubMed]

Balakrishnan, G.

Bar-Ad, S.

A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman, “Asking Photons Where They Have Been,” Phys. Rev. Lett. 111, 240402 (2013).
[Crossref]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. ( McGraw-Hill, Boston2000).

Brown, C. G.

Buks, E.

E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, “Dephasing in electron interference by a ‘which-path’ detector,” Nature (London) 391, 871–874 (1998).
[Crossref]

Chao, W. L.

Danan, A.

A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman, “Asking Photons Where They Have Been,” Phys. Rev. Lett. 111, 240402 (2013).
[Crossref]

Drühl, K.

M. O. Scully and K. Drühl, “Quantum eraser: A proposed photon correlation experiment concerning observation and ‘delayed choice’ in quantum mechanics,” Phys. Rev. A 25, 2208 (1982).
[Crossref]

Dürr, S.

S. Dürr, T. Nonn, and G. Rempe, “Origin of quantum-mechanical complementarity probed by a which-way experiment in an atom interferometer,” Nature (London) 395, 33–37 (1998).
[Crossref]

S. Dürr, T. Nonn, and G. Rempe, “Fringe visibility and which-way information in an atom interferometer,” Phys. Rev. Lett. 81, 5705 (1998).
[Crossref]

Englert, B. G.

B. G. Englert, “Fringe visibility and which-way information: an inequality,” Phys. Rev. Lett. 77, 2154 (1996).
[Crossref] [PubMed]

Farfurnik, D.

A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman, “Asking Photons Where They Have Been,” Phys. Rev. Lett. 111, 240402 (2013).
[Crossref]

Feynman, R. P.

R. P. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics: Vol. III (Addison Wesley, 1965).

Grangier, P.

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-choice test of quantum complementarity with interfering single photons,” Phys. Rev. Lett. 100, 220402 (2008).
[Crossref] [PubMed]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Experimental realization of Wheeler’s delayed-choice Gedanken experiment,” Science 315, 966 (2007).
[Crossref] [PubMed]

Greengard, L.

L. Greengard and J. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM Review 46(3), 443–454 (2004).
[Crossref]

Grosshans, F.

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-choice test of quantum complementarity with interfering single photons,” Phys. Rev. Lett. 100, 220402 (2008).
[Crossref] [PubMed]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Experimental realization of Wheeler’s delayed-choice Gedanken experiment,” Science 315, 966 (2007).
[Crossref] [PubMed]

Heiblum, M.

E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, “Dephasing in electron interference by a ‘which-path’ detector,” Nature (London) 391, 871–874 (1998).
[Crossref]

Huang, Y.

Jacques, V.

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-choice test of quantum complementarity with interfering single photons,” Phys. Rev. Lett. 100, 220402 (2008).
[Crossref] [PubMed]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Experimental realization of Wheeler’s delayed-choice Gedanken experiment,” Science 315, 966 (2007).
[Crossref] [PubMed]

Kang, J. U.

Lee, J.

L. Greengard and J. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM Review 46(3), 443–454 (2004).
[Crossref]

Leighton, R.

R. P. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics: Vol. III (Addison Wesley, 1965).

Luis, A.

A. Luis, “Complementarity and certainty relations for two-dimensional systems,” Phys. Rev. A 64, 012103 (2001).
[Crossref]

Mahalu, D.

E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, “Dephasing in electron interference by a ‘which-path’ detector,” Nature (London) 391, 871–874 (1998).
[Crossref]

Malm, E. B.

Marconi, M. C.

Monserud, N. C.

Nonn, T.

S. Dürr, T. Nonn, and G. Rempe, “Fringe visibility and which-way information in an atom interferometer,” Phys. Rev. Lett. 81, 5705 (1998).
[Crossref]

S. Dürr, T. Nonn, and G. Rempe, “Origin of quantum-mechanical complementarity probed by a which-way experiment in an atom interferometer,” Nature (London) 395, 33–37 (1998).
[Crossref]

Ou, X. Z.

Rempe, G.

S. Dürr, T. Nonn, and G. Rempe, “Origin of quantum-mechanical complementarity probed by a which-way experiment in an atom interferometer,” Nature (London) 395, 33–37 (1998).
[Crossref]

S. Dürr, T. Nonn, and G. Rempe, “Fringe visibility and which-way information in an atom interferometer,” Phys. Rev. Lett. 81, 5705 (1998).
[Crossref]

Roch, J.-F.

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-choice test of quantum complementarity with interfering single photons,” Phys. Rev. Lett. 100, 220402 (2008).
[Crossref] [PubMed]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Experimental realization of Wheeler’s delayed-choice Gedanken experiment,” Science 315, 966 (2007).
[Crossref] [PubMed]

Sands, M.

R. P. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics: Vol. III (Addison Wesley, 1965).

Schuster, R.

E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, “Dephasing in electron interference by a ‘which-path’ detector,” Nature (London) 391, 871–874 (1998).
[Crossref]

Scully, M. O.

M. O. Scully and K. Drühl, “Quantum eraser: A proposed photon correlation experiment concerning observation and ‘delayed choice’ in quantum mechanics,” Phys. Rev. A 25, 2208 (1982).
[Crossref]

Svensson, B. E. Y.

B. E. Y. Svensson, “Comments to ‘Asking photons where they have been,’Phys. Rev. Lett.111, 240402 (2013)),” arXiv:1402.4315 [quant-ph] (2014).
[Crossref]

Treussart, F.

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-choice test of quantum complementarity with interfering single photons,” Phys. Rev. Lett. 100, 220402 (2008).
[Crossref] [PubMed]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Experimental realization of Wheeler’s delayed-choice Gedanken experiment,” Science 315, 966 (2007).
[Crossref] [PubMed]

Umansky, V.

E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, “Dephasing in electron interference by a ‘which-path’ detector,” Nature (London) 391, 871–874 (1998).
[Crossref]

Vaidman, L.

A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman, “Asking Photons Where They Have Been,” Phys. Rev. Lett. 111, 240402 (2013).
[Crossref]

Wachulak, P. W.

Wu, E.

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-choice test of quantum complementarity with interfering single photons,” Phys. Rev. Lett. 100, 220402 (2008).
[Crossref] [PubMed]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Experimental realization of Wheeler’s delayed-choice Gedanken experiment,” Science 315, 966 (2007).
[Crossref] [PubMed]

Xu, D. G.

Xu, H. W.

Yang, C. H.

Zheng, G. A.

Nature (London) (2)

E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, “Dephasing in electron interference by a ‘which-path’ detector,” Nature (London) 391, 871–874 (1998).
[Crossref]

S. Dürr, T. Nonn, and G. Rempe, “Origin of quantum-mechanical complementarity probed by a which-way experiment in an atom interferometer,” Nature (London) 395, 33–37 (1998).
[Crossref]

Opt. Express (3)

Phys. Rev. A (2)

A. Luis, “Complementarity and certainty relations for two-dimensional systems,” Phys. Rev. A 64, 012103 (2001).
[Crossref]

M. O. Scully and K. Drühl, “Quantum eraser: A proposed photon correlation experiment concerning observation and ‘delayed choice’ in quantum mechanics,” Phys. Rev. A 25, 2208 (1982).
[Crossref]

Phys. Rev. Lett. (4)

B. G. Englert, “Fringe visibility and which-way information: an inequality,” Phys. Rev. Lett. 77, 2154 (1996).
[Crossref] [PubMed]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-choice test of quantum complementarity with interfering single photons,” Phys. Rev. Lett. 100, 220402 (2008).
[Crossref] [PubMed]

S. Dürr, T. Nonn, and G. Rempe, “Fringe visibility and which-way information in an atom interferometer,” Phys. Rev. Lett. 81, 5705 (1998).
[Crossref]

A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman, “Asking Photons Where They Have Been,” Phys. Rev. Lett. 111, 240402 (2013).
[Crossref]

Science (1)

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Experimental realization of Wheeler’s delayed-choice Gedanken experiment,” Science 315, 966 (2007).
[Crossref] [PubMed]

SIAM Review (1)

L. Greengard and J. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM Review 46(3), 443–454 (2004).
[Crossref]

Other (3)

B. E. Y. Svensson, “Comments to ‘Asking photons where they have been,’Phys. Rev. Lett.111, 240402 (2013)),” arXiv:1402.4315 [quant-ph] (2014).
[Crossref]

R. P. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics: Vol. III (Addison Wesley, 1965).

R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. ( McGraw-Hill, Boston2000).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

A Mach-Zehnder interferometer. Two mirrors A and B vibrate at the frequency fA = 282Hz and fB = 296Hz, and the two beam splitters, BS1 and BS2, equally split the input light.

Fig. 2
Fig. 2

Fourier spectrum of the output beam based on the intensity difference (3). (a) In the case of constructive interference, two peaks fA and fB appear in the Fourier spectrum as the signature of the two vibrating mirrors A and B in the interferometer; (b) In the case of destructive interference, some “noisy” peaks appear in the Fourier spectrum, besides the two peaks fA and fB.

Fig. 3
Fig. 3

Fourier spectrum of the output beam based on CLSID. The peaks in the Fourier spectrum have a one-to-one correspondence with the vibrating mirrors, no matter which type of interference, constructive or destructive, is set for the output beam.

Fig. 4
Fig. 4

A modified Mach-Zehnder interferometer. A small Mach-Zehnder interferometer is nested in the upper arm of the larger one, and a destructive interference is set for the beam F initially. Five mirrors slightly vibrate in the experiment at the frequency fA = 282Hz, fB = 296Hz, fC = 309Hz, fE = 318Hz and fF = 337Hz, respectively.

Fig. 5
Fig. 5

Fourier spectrum of the output beam in the experiment with nested interferometer based on CLSID. Every mirror in the interferometer contributes a peak in the Fourier spectrum, see the five red peaks, denoted as fA, fB, fC, fE and fF. In addition, many “noisy” peaks appear in the spectrum as a result of the combined contribution of two or more vibrating mirrors, see the lower blue peaks, e.g. f1 and f2.

Equations (7)

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

Ψ ( x , y ) = 𝒜 2 ( e x 2 + ( y δ A ) 2 2 Δ 2 ± e x 2 + ( y δ B ) 2 2 Δ 2 ) ,
δ A = δ sin ( 2 π f A t ) , δ B = δ sin ( 2 π f B t ) .
I ( t ) = y > 0 | Ψ ( x , y ) | 2 d x d y y < 0 | Ψ ( x , y ) | 2 d x d y .
F ( ω ) = 1 N j = 1 N y j e i ω t j ,
y c ( t ) = 1 2 ( δ A + δ B ) = δ 2 [ sin ( 2 π f A t ) + sin ( 2 π f B t ) ] ,
Ψ ( x , y ) = 𝒜 3 ( e x 2 + ( y δ C ) 2 2 Δ 2 + e x 2 + ( y δ E δ A δ F ) 2 2 Δ 2 e x 2 + ( y δ E δ B δ F ) 2 2 Δ 2 ) ,
δ C = δ sin ( 2 π f C t ) , δ E = δ sin ( 2 π f E t ) , δ F = δ sin ( 2 π f F t ) .

Metrics