Abstract

We numerically calculate the reliability with which one can optically determine the presence or absence of an individual scatterer in a randomly occupied three-dimensional array of well-localized, coherently radiating scatterers. The reliability depends on the ratio of lattice spacing to wavelength and on the numerical aperture of the imaging system. The behavior can be qualitatively understood by considering the dependence of Bragg scattering modes on lattice spacing. These results are of interest for atomic implementations of quantum information processing.

© 2008 Optical Society of America

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References

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  1. G. K. Brennen, C. M. Caves, P. S. Jessen, and I. Deutsch, Phys. Rev. Lett. 82, 1060 (1999).
    [Crossref]
  2. D. Jaksch, H. J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 82, 1975 (1999).
    [Crossref]
  3. D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cote, and M. D. Lukin, Phys. Rev. Lett. 85, 2208 (2000).
    [Crossref] [PubMed]
  4. J. Vala, A. V. Thapliyal, S. Myrgren, U. Vazirani, D. S. Weiss, and K. B. Whaley, Phys. Rev. A 71, 032324 (2005).
    [Crossref]
  5. D. S. Weiss, J. Vala, A. V. Thapliyal, S. Myrgren, U. Vazirani, and K. B. Whaley, Phys. Rev. A 70, 040302(R) (2004).
    [Crossref]
  6. K. D. Nelson, X. Li, and D. S. Weiss, Nat. Phys. 3, 556 (2007).
    [Crossref]
  7. J. J. Bollinger, Phys. Plasmas 7, 7 (2000).
    [Crossref]
  8. H. Walther, Phys. Scr. T59, 360 (1995).
    [Crossref]
  9. Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, D. Meschede, D. Schrader, and A. Rauschenbeutel, Nature 442, 151 (2006).
    [Crossref] [PubMed]
  10. S. Bergamini, B. Darquie, M. Jones, L. Jacubowiez, A. Browaeys, and P. Grangier, J. Opt. Soc. Am. B 21, 1889 (2004).
    [Crossref]
  11. W. L. Bragg, Proc. Cambridge Philos. Soc. 17, 43 (1912).
  12. G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, Phys. Rev. Lett. 75, 2823 (1995).
    [Crossref] [PubMed]
  13. M. Weidemüller, A. Hemmerich, A. Gorlitz, T. Esslinger, and T. W. Hansch, Phys. Rev. Lett. 75, 4583 (1995).
    [Crossref] [PubMed]
  14. S. L. Winoto, M. T. DePue, N. E. Bramall, and D. S. Weiss, Phys. Rev. A 59, R19 (1999).
    [Crossref]
  15. M. T. DePue, C. McCormick, S. L. Winoto, S. Oliver, and D. S. Weiss, Phys. Rev. Lett. 82, 2262 (1999).
    [Crossref]
  16. M. Born and E. Wolf, Principles of Optics (Cambridge, 2002), pp. 484-487.
  17. When scatterers are delocalized over a wavelength, interferences wash out and intensities from different atoms can be simply added. Because the intensity from an out-of-focus atom scales as m−2 and the number of contributing atoms scales as m2, each out-of-focus plane on average contributes approximately the same amount of light to the background. The signal to background ratio in that case is ~2N−1πρ2[sin−1(η)]4(χ)−2.

2007 (1)

K. D. Nelson, X. Li, and D. S. Weiss, Nat. Phys. 3, 556 (2007).
[Crossref]

2006 (1)

Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, D. Meschede, D. Schrader, and A. Rauschenbeutel, Nature 442, 151 (2006).
[Crossref] [PubMed]

2005 (1)

J. Vala, A. V. Thapliyal, S. Myrgren, U. Vazirani, D. S. Weiss, and K. B. Whaley, Phys. Rev. A 71, 032324 (2005).
[Crossref]

2004 (2)

D. S. Weiss, J. Vala, A. V. Thapliyal, S. Myrgren, U. Vazirani, and K. B. Whaley, Phys. Rev. A 70, 040302(R) (2004).
[Crossref]

S. Bergamini, B. Darquie, M. Jones, L. Jacubowiez, A. Browaeys, and P. Grangier, J. Opt. Soc. Am. B 21, 1889 (2004).
[Crossref]

2000 (2)

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cote, and M. D. Lukin, Phys. Rev. Lett. 85, 2208 (2000).
[Crossref] [PubMed]

J. J. Bollinger, Phys. Plasmas 7, 7 (2000).
[Crossref]

1999 (4)

G. K. Brennen, C. M. Caves, P. S. Jessen, and I. Deutsch, Phys. Rev. Lett. 82, 1060 (1999).
[Crossref]

D. Jaksch, H. J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 82, 1975 (1999).
[Crossref]

S. L. Winoto, M. T. DePue, N. E. Bramall, and D. S. Weiss, Phys. Rev. A 59, R19 (1999).
[Crossref]

M. T. DePue, C. McCormick, S. L. Winoto, S. Oliver, and D. S. Weiss, Phys. Rev. Lett. 82, 2262 (1999).
[Crossref]

1995 (3)

G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, Phys. Rev. Lett. 75, 2823 (1995).
[Crossref] [PubMed]

M. Weidemüller, A. Hemmerich, A. Gorlitz, T. Esslinger, and T. W. Hansch, Phys. Rev. Lett. 75, 4583 (1995).
[Crossref] [PubMed]

H. Walther, Phys. Scr. T59, 360 (1995).
[Crossref]

1912 (1)

W. L. Bragg, Proc. Cambridge Philos. Soc. 17, 43 (1912).

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

Nat. Phys. (1)

K. D. Nelson, X. Li, and D. S. Weiss, Nat. Phys. 3, 556 (2007).
[Crossref]

Nature (1)

Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, D. Meschede, D. Schrader, and A. Rauschenbeutel, Nature 442, 151 (2006).
[Crossref] [PubMed]

Phys. Plasmas (1)

J. J. Bollinger, Phys. Plasmas 7, 7 (2000).
[Crossref]

Phys. Rev. A (3)

J. Vala, A. V. Thapliyal, S. Myrgren, U. Vazirani, D. S. Weiss, and K. B. Whaley, Phys. Rev. A 71, 032324 (2005).
[Crossref]

D. S. Weiss, J. Vala, A. V. Thapliyal, S. Myrgren, U. Vazirani, and K. B. Whaley, Phys. Rev. A 70, 040302(R) (2004).
[Crossref]

S. L. Winoto, M. T. DePue, N. E. Bramall, and D. S. Weiss, Phys. Rev. A 59, R19 (1999).
[Crossref]

Phys. Rev. Lett. (6)

M. T. DePue, C. McCormick, S. L. Winoto, S. Oliver, and D. S. Weiss, Phys. Rev. Lett. 82, 2262 (1999).
[Crossref]

G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, Phys. Rev. Lett. 75, 2823 (1995).
[Crossref] [PubMed]

M. Weidemüller, A. Hemmerich, A. Gorlitz, T. Esslinger, and T. W. Hansch, Phys. Rev. Lett. 75, 4583 (1995).
[Crossref] [PubMed]

G. K. Brennen, C. M. Caves, P. S. Jessen, and I. Deutsch, Phys. Rev. Lett. 82, 1060 (1999).
[Crossref]

D. Jaksch, H. J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 82, 1975 (1999).
[Crossref]

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cote, and M. D. Lukin, Phys. Rev. Lett. 85, 2208 (2000).
[Crossref] [PubMed]

Phys. Scr. (1)

H. Walther, Phys. Scr. T59, 360 (1995).
[Crossref]

Proc. Cambridge Philos. Soc. (1)

W. L. Bragg, Proc. Cambridge Philos. Soc. 17, 43 (1912).

Other (2)

M. Born and E. Wolf, Principles of Optics (Cambridge, 2002), pp. 484-487.

When scatterers are delocalized over a wavelength, interferences wash out and intensities from different atoms can be simply added. Because the intensity from an out-of-focus atom scales as m−2 and the number of contributing atoms scales as m2, each out-of-focus plane on average contributes approximately the same amount of light to the background. The signal to background ratio in that case is ~2N−1πρ2[sin−1(η)]4(χ)−2.

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Figures (4)

Fig. 1
Fig. 1

Schematic of the model for imaging the lattice.

Fig. 2
Fig. 2

Numerically calculated contour plots of four unit cells around the target site in the image plane. All images are for a half-filled 9 3 lattice imaged with an η = 0.514 lens. a, ρ = 6.1 , with no target atom; b, ρ = 6.1 , with a target atom; c, ρ = 6.2 with no target atom; d, ρ = 6.2 with a target atom. The same linear gray scale applies to all images. For ρ = 6.1 , the target intensity increases when a target atom is present. For ρ = 6.2 , the target intensity counterintuitively decreases when a target atom is present.

Fig. 3
Fig. 3

Histogram of binned intensities at the center of the 2D image plane for 2000 randomly half-occupied 15 3 lattices. a, ρ = 6.1 ; b, ρ = 6.2 . The lighter (darker) bars correspond to a target atom being present (absent). For ρ = 6.2 , the intensity tends to be higher in the absence of a target atom, which would make it impossible to resolve site occupancy.

Fig. 4
Fig. 4

a, Resolution error as a function of the rescaled lattice spacing, ρ, for η = 0.514 . The data are based on 2000 random distributions. b, Net power collected by the imaging lens as a function of ρ. The data are for η = 0.514 lens and a full 15 3 lattice. The minimum (maximum) error points in a are associated with power peaks (valleys) in b. The lines are to guide the eye.

Equations (3)

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E ( u n , ν n ) = i A 2 π λ ( a d n ) 2 exp [ i ( d n a ) 2 u n ] 0 1 J 0 ( ν n ξ ) exp [ i u n ξ 2 2 ] ξ d ξ .
E p q m ( x , y ) = exp [ 2 π i ( α p q m α 00 m ) λ ] E 00 m ( x + p , y + q ) .
I net ( x , y ) = m = N N p , q = N N β p q m E p q m ( x , y ) 2 .

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