Abstract

A scheme of two-dimensional (2D) atom localization based on the interference of double-dark resonances is proposed, in which the N-type atom interacts with two orthogonal standing-wave fields. Because of the spatial-dependent atom–field interaction, 2D atom localization can be realized via measuring the upper state population or the probe absorption. We obtain that the maximum probability of finding an atom at a particular position in a wavelength domain (λ1×λ2) is 1/2 when the atom is localized at the intersection of the antinodes of quadrants I and III of the standing-wave plane. This scheme shows more advantages than other schemes of 2D atom localization.

© 2011 Optical Society of America

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  1. W. D. Phillips, “Nobel lecture: laser cooling and trapping of neutral atoms,” Rev. Mod. Phys. 70, 721–741 (1998).
    [CrossRef]
  2. K. S. Johnson, J. H. Thywissen, W. H. Dekker, K. K. Berggren, A. P. Chu, R. Younkin, and M. Prentiss, “Localization of metastable atom beams with optical standing waves: nanolithography at the Heisenberg limit,” Science 280, 1583–1586 (1998).
    [CrossRef] [PubMed]
  3. K. T. Kapale, S. Qamar, and M. S. Zubairy, “Spectroscopic measurement of an atomic wave function,” Phys. Rev. A 67, 023805(2003).
    [CrossRef]
  4. A. M. Herkommer, H. J. Carmichael, and W. P. Schleich, “Localization of an atom by homodyne measurement,” Quantum Semiclass. Opt. 8, 189–203 (1996).
    [CrossRef]
  5. F. L. Kienm, G. Rempe, W. P. Schleich, and M. S. Zubairy, “Atom localization via Ramsey interferometry: a coherent cavity field provides a better resolution,” Phys. Rev. A 56, 2972–2977 (1997).
    [CrossRef]
  6. R. Quadt, M. Collett, and D. F. Walls, “Measurement of atomic motion in a standing light field by homodyne detection,” Phys. Rev. Lett. 74, 351–354 (1995).
    [CrossRef] [PubMed]
  7. S. Kunze, K. Dieckmann, and G. Rempe, “Diffraction of atoms from a measurement induced grating,” Phys. Rev. Lett. 78, 2038–2041 (1997).
    [CrossRef]
  8. S. Qamar, S. Y. Zhu, and M. S. Zubairy, “Precision localization of single atom using Autler—Townes microscopy,” Opt. Commun. 176, 409–416 (2000).
    [CrossRef]
  9. S. Qamar, S. Y. Zhu, and M. S. Zubairy, “Atom localization via resonance fluorescence,” Phys. Rev. A 61, 063806 (2000).
    [CrossRef]
  10. J. Xu and X. M. Hu, “Sub-half-wavelength atom localization via bichromatic phase control of spontaneous emission,” Phys. Lett. A 366, 276–281 (2007).
    [CrossRef]
  11. M. Sahrai, H. Tajalli, K. T. Kapale, and M. S. Zubairy, “Subwavelength atom localization via amplitude and phase control of the absorption spectrum,” Phys. Rev. A 72, 013820 (2005).
    [CrossRef]
  12. E. Paspalakis and P. L. Knight, “Localizing an atom via quantum interference,” Phys. Rev. A 63, 065802 (2001).
    [CrossRef]
  13. G. S. Agarwal and K. T. Kapale, “Subwavelength atom localization via coherent population trapping,” J. Phys. B 39, 3437–3446(2006).
    [CrossRef]
  14. C. P. Liu, S. Q. Gong, D. C. Cheng, X. J. Fan, and Z. Z. Xu, “Atom localization via interference of dark resonances,” Phys. Rev. A 73, 025801 (2006).
    [CrossRef]
  15. D. C. Cheng, Y. P. Niu, R. X. Li, and S. Q. Gong, “Controllable atom localization via double-dark resonances in a tripod system,” J. Opt. Soc. Am. B 23, 2180–2184 (2006).
    [CrossRef]
  16. J. Xu and X. M. Hu, “Sub-half-wavelength atom localization via phase control of a pair of bichromatic fields,” Phys. Rev. A 76, 013830 (2007).
    [CrossRef]
  17. S. Qamar, A. Mehmood, and S. Qamar, “Subwavelength atom localization via coherent manipulation of the Raman gain process,” Phys. Rev. A 79, 033848 (2009).
    [CrossRef]
  18. L. L. Jin, H. Sun, Y. P. Niu, and S. Q. Gong, “Sub-half-wavelength atom localization via two standing-wave fields,” J. Phys. B 41, 085508 (2008).
    [CrossRef]
  19. J. Evers, S. Qamar, and M. S. Zubairy, “Atom localization and center-of-mass wave-function determination via multiple simultaneous quadrature measurements,” Phys. Rev. A 75, 053809(2007).
    [CrossRef]
  20. V. Ivanov and Y. Rozhdestvensky, “Two-dimensional atom localization in a four-level tripod system in laser fields,” Phys. Rev. A 81, 033809 (2010).
    [CrossRef]
  21. L. L. Jin, H. Sun, Y. P. Niu, S. Q. Jin, and S. Q. Gong, “Two-dimension atom nano-lithograph via atom localization,” J. Mod. Opt. 56, 805–810 (2009).
    [CrossRef]
  22. R. G. Wan, J. Kou, L. Jiang, Y. Jiang, and J. Y. Gao, “Two-dimensional atom localization via controlled spontaneous emission from a driven tripod system,” J. Opt. Soc. Am. B 28, 10–17(2011).
    [CrossRef]
  23. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
    [CrossRef]
  24. Y. C. Chen, Y. A. Liao, H. Y. Chiu, J. J. Su, and I. A. Yu, “Observation of the quantum interference phenomenon induced by interacting dark resonances,” Phys. Rev. A 64, 053806 (2001).
    [CrossRef]
  25. S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68, 063801 (2003).
    [CrossRef]
  26. Y. P. Niu, S. Q. Gong, R. X. Li, Z. Z. Xu, and X. Y. Liang, “Giant Kerr nonlinearity induced by interacting dark resonances,” Opt. Lett. 30, 3371–3373 (2005).
    [CrossRef]

2011

2010

V. Ivanov and Y. Rozhdestvensky, “Two-dimensional atom localization in a four-level tripod system in laser fields,” Phys. Rev. A 81, 033809 (2010).
[CrossRef]

2009

L. L. Jin, H. Sun, Y. P. Niu, S. Q. Jin, and S. Q. Gong, “Two-dimension atom nano-lithograph via atom localization,” J. Mod. Opt. 56, 805–810 (2009).
[CrossRef]

S. Qamar, A. Mehmood, and S. Qamar, “Subwavelength atom localization via coherent manipulation of the Raman gain process,” Phys. Rev. A 79, 033848 (2009).
[CrossRef]

2008

L. L. Jin, H. Sun, Y. P. Niu, and S. Q. Gong, “Sub-half-wavelength atom localization via two standing-wave fields,” J. Phys. B 41, 085508 (2008).
[CrossRef]

2007

J. Evers, S. Qamar, and M. S. Zubairy, “Atom localization and center-of-mass wave-function determination via multiple simultaneous quadrature measurements,” Phys. Rev. A 75, 053809(2007).
[CrossRef]

J. Xu and X. M. Hu, “Sub-half-wavelength atom localization via phase control of a pair of bichromatic fields,” Phys. Rev. A 76, 013830 (2007).
[CrossRef]

J. Xu and X. M. Hu, “Sub-half-wavelength atom localization via bichromatic phase control of spontaneous emission,” Phys. Lett. A 366, 276–281 (2007).
[CrossRef]

2006

G. S. Agarwal and K. T. Kapale, “Subwavelength atom localization via coherent population trapping,” J. Phys. B 39, 3437–3446(2006).
[CrossRef]

C. P. Liu, S. Q. Gong, D. C. Cheng, X. J. Fan, and Z. Z. Xu, “Atom localization via interference of dark resonances,” Phys. Rev. A 73, 025801 (2006).
[CrossRef]

D. C. Cheng, Y. P. Niu, R. X. Li, and S. Q. Gong, “Controllable atom localization via double-dark resonances in a tripod system,” J. Opt. Soc. Am. B 23, 2180–2184 (2006).
[CrossRef]

2005

Y. P. Niu, S. Q. Gong, R. X. Li, Z. Z. Xu, and X. Y. Liang, “Giant Kerr nonlinearity induced by interacting dark resonances,” Opt. Lett. 30, 3371–3373 (2005).
[CrossRef]

M. Sahrai, H. Tajalli, K. T. Kapale, and M. S. Zubairy, “Subwavelength atom localization via amplitude and phase control of the absorption spectrum,” Phys. Rev. A 72, 013820 (2005).
[CrossRef]

2003

K. T. Kapale, S. Qamar, and M. S. Zubairy, “Spectroscopic measurement of an atomic wave function,” Phys. Rev. A 67, 023805(2003).
[CrossRef]

S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68, 063801 (2003).
[CrossRef]

2001

Y. C. Chen, Y. A. Liao, H. Y. Chiu, J. J. Su, and I. A. Yu, “Observation of the quantum interference phenomenon induced by interacting dark resonances,” Phys. Rev. A 64, 053806 (2001).
[CrossRef]

E. Paspalakis and P. L. Knight, “Localizing an atom via quantum interference,” Phys. Rev. A 63, 065802 (2001).
[CrossRef]

2000

S. Qamar, S. Y. Zhu, and M. S. Zubairy, “Precision localization of single atom using Autler—Townes microscopy,” Opt. Commun. 176, 409–416 (2000).
[CrossRef]

S. Qamar, S. Y. Zhu, and M. S. Zubairy, “Atom localization via resonance fluorescence,” Phys. Rev. A 61, 063806 (2000).
[CrossRef]

1999

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[CrossRef]

1998

W. D. Phillips, “Nobel lecture: laser cooling and trapping of neutral atoms,” Rev. Mod. Phys. 70, 721–741 (1998).
[CrossRef]

K. S. Johnson, J. H. Thywissen, W. H. Dekker, K. K. Berggren, A. P. Chu, R. Younkin, and M. Prentiss, “Localization of metastable atom beams with optical standing waves: nanolithography at the Heisenberg limit,” Science 280, 1583–1586 (1998).
[CrossRef] [PubMed]

1997

F. L. Kienm, G. Rempe, W. P. Schleich, and M. S. Zubairy, “Atom localization via Ramsey interferometry: a coherent cavity field provides a better resolution,” Phys. Rev. A 56, 2972–2977 (1997).
[CrossRef]

S. Kunze, K. Dieckmann, and G. Rempe, “Diffraction of atoms from a measurement induced grating,” Phys. Rev. Lett. 78, 2038–2041 (1997).
[CrossRef]

1996

A. M. Herkommer, H. J. Carmichael, and W. P. Schleich, “Localization of an atom by homodyne measurement,” Quantum Semiclass. Opt. 8, 189–203 (1996).
[CrossRef]

1995

R. Quadt, M. Collett, and D. F. Walls, “Measurement of atomic motion in a standing light field by homodyne detection,” Phys. Rev. Lett. 74, 351–354 (1995).
[CrossRef] [PubMed]

Agarwal, G. S.

G. S. Agarwal and K. T. Kapale, “Subwavelength atom localization via coherent population trapping,” J. Phys. B 39, 3437–3446(2006).
[CrossRef]

Berggren, K. K.

K. S. Johnson, J. H. Thywissen, W. H. Dekker, K. K. Berggren, A. P. Chu, R. Younkin, and M. Prentiss, “Localization of metastable atom beams with optical standing waves: nanolithography at the Heisenberg limit,” Science 280, 1583–1586 (1998).
[CrossRef] [PubMed]

Carmichael, H. J.

A. M. Herkommer, H. J. Carmichael, and W. P. Schleich, “Localization of an atom by homodyne measurement,” Quantum Semiclass. Opt. 8, 189–203 (1996).
[CrossRef]

Chen, Y. C.

Y. C. Chen, Y. A. Liao, H. Y. Chiu, J. J. Su, and I. A. Yu, “Observation of the quantum interference phenomenon induced by interacting dark resonances,” Phys. Rev. A 64, 053806 (2001).
[CrossRef]

Cheng, D. C.

C. P. Liu, S. Q. Gong, D. C. Cheng, X. J. Fan, and Z. Z. Xu, “Atom localization via interference of dark resonances,” Phys. Rev. A 73, 025801 (2006).
[CrossRef]

D. C. Cheng, Y. P. Niu, R. X. Li, and S. Q. Gong, “Controllable atom localization via double-dark resonances in a tripod system,” J. Opt. Soc. Am. B 23, 2180–2184 (2006).
[CrossRef]

Chiu, H. Y.

Y. C. Chen, Y. A. Liao, H. Y. Chiu, J. J. Su, and I. A. Yu, “Observation of the quantum interference phenomenon induced by interacting dark resonances,” Phys. Rev. A 64, 053806 (2001).
[CrossRef]

Chu, A. P.

K. S. Johnson, J. H. Thywissen, W. H. Dekker, K. K. Berggren, A. P. Chu, R. Younkin, and M. Prentiss, “Localization of metastable atom beams with optical standing waves: nanolithography at the Heisenberg limit,” Science 280, 1583–1586 (1998).
[CrossRef] [PubMed]

Collett, M.

R. Quadt, M. Collett, and D. F. Walls, “Measurement of atomic motion in a standing light field by homodyne detection,” Phys. Rev. Lett. 74, 351–354 (1995).
[CrossRef] [PubMed]

Dekker, W. H.

K. S. Johnson, J. H. Thywissen, W. H. Dekker, K. K. Berggren, A. P. Chu, R. Younkin, and M. Prentiss, “Localization of metastable atom beams with optical standing waves: nanolithography at the Heisenberg limit,” Science 280, 1583–1586 (1998).
[CrossRef] [PubMed]

Dieckmann, K.

S. Kunze, K. Dieckmann, and G. Rempe, “Diffraction of atoms from a measurement induced grating,” Phys. Rev. Lett. 78, 2038–2041 (1997).
[CrossRef]

Evers, J.

J. Evers, S. Qamar, and M. S. Zubairy, “Atom localization and center-of-mass wave-function determination via multiple simultaneous quadrature measurements,” Phys. Rev. A 75, 053809(2007).
[CrossRef]

Fan, X. J.

C. P. Liu, S. Q. Gong, D. C. Cheng, X. J. Fan, and Z. Z. Xu, “Atom localization via interference of dark resonances,” Phys. Rev. A 73, 025801 (2006).
[CrossRef]

Fleischhauer, M.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[CrossRef]

Gao, J. Y.

Gong, S. Q.

L. L. Jin, H. Sun, Y. P. Niu, S. Q. Jin, and S. Q. Gong, “Two-dimension atom nano-lithograph via atom localization,” J. Mod. Opt. 56, 805–810 (2009).
[CrossRef]

L. L. Jin, H. Sun, Y. P. Niu, and S. Q. Gong, “Sub-half-wavelength atom localization via two standing-wave fields,” J. Phys. B 41, 085508 (2008).
[CrossRef]

C. P. Liu, S. Q. Gong, D. C. Cheng, X. J. Fan, and Z. Z. Xu, “Atom localization via interference of dark resonances,” Phys. Rev. A 73, 025801 (2006).
[CrossRef]

D. C. Cheng, Y. P. Niu, R. X. Li, and S. Q. Gong, “Controllable atom localization via double-dark resonances in a tripod system,” J. Opt. Soc. Am. B 23, 2180–2184 (2006).
[CrossRef]

Y. P. Niu, S. Q. Gong, R. X. Li, Z. Z. Xu, and X. Y. Liang, “Giant Kerr nonlinearity induced by interacting dark resonances,” Opt. Lett. 30, 3371–3373 (2005).
[CrossRef]

Herkommer, A. M.

A. M. Herkommer, H. J. Carmichael, and W. P. Schleich, “Localization of an atom by homodyne measurement,” Quantum Semiclass. Opt. 8, 189–203 (1996).
[CrossRef]

Hu, X. M.

J. Xu and X. M. Hu, “Sub-half-wavelength atom localization via phase control of a pair of bichromatic fields,” Phys. Rev. A 76, 013830 (2007).
[CrossRef]

J. Xu and X. M. Hu, “Sub-half-wavelength atom localization via bichromatic phase control of spontaneous emission,” Phys. Lett. A 366, 276–281 (2007).
[CrossRef]

Ivanov, V.

V. Ivanov and Y. Rozhdestvensky, “Two-dimensional atom localization in a four-level tripod system in laser fields,” Phys. Rev. A 81, 033809 (2010).
[CrossRef]

Jiang, L.

Jiang, Y.

Jin, L. L.

L. L. Jin, H. Sun, Y. P. Niu, S. Q. Jin, and S. Q. Gong, “Two-dimension atom nano-lithograph via atom localization,” J. Mod. Opt. 56, 805–810 (2009).
[CrossRef]

L. L. Jin, H. Sun, Y. P. Niu, and S. Q. Gong, “Sub-half-wavelength atom localization via two standing-wave fields,” J. Phys. B 41, 085508 (2008).
[CrossRef]

Jin, S. Q.

L. L. Jin, H. Sun, Y. P. Niu, S. Q. Jin, and S. Q. Gong, “Two-dimension atom nano-lithograph via atom localization,” J. Mod. Opt. 56, 805–810 (2009).
[CrossRef]

Johnson, K. S.

K. S. Johnson, J. H. Thywissen, W. H. Dekker, K. K. Berggren, A. P. Chu, R. Younkin, and M. Prentiss, “Localization of metastable atom beams with optical standing waves: nanolithography at the Heisenberg limit,” Science 280, 1583–1586 (1998).
[CrossRef] [PubMed]

Kapale, K. T.

G. S. Agarwal and K. T. Kapale, “Subwavelength atom localization via coherent population trapping,” J. Phys. B 39, 3437–3446(2006).
[CrossRef]

M. Sahrai, H. Tajalli, K. T. Kapale, and M. S. Zubairy, “Subwavelength atom localization via amplitude and phase control of the absorption spectrum,” Phys. Rev. A 72, 013820 (2005).
[CrossRef]

K. T. Kapale, S. Qamar, and M. S. Zubairy, “Spectroscopic measurement of an atomic wave function,” Phys. Rev. A 67, 023805(2003).
[CrossRef]

Kash, M. M.

S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68, 063801 (2003).
[CrossRef]

Kienm, F. L.

F. L. Kienm, G. Rempe, W. P. Schleich, and M. S. Zubairy, “Atom localization via Ramsey interferometry: a coherent cavity field provides a better resolution,” Phys. Rev. A 56, 2972–2977 (1997).
[CrossRef]

Knight, P. L.

E. Paspalakis and P. L. Knight, “Localizing an atom via quantum interference,” Phys. Rev. A 63, 065802 (2001).
[CrossRef]

Kou, J.

Kunze, S.

S. Kunze, K. Dieckmann, and G. Rempe, “Diffraction of atoms from a measurement induced grating,” Phys. Rev. Lett. 78, 2038–2041 (1997).
[CrossRef]

Li, R. X.

Liang, X. Y.

Liao, Y. A.

Y. C. Chen, Y. A. Liao, H. Y. Chiu, J. J. Su, and I. A. Yu, “Observation of the quantum interference phenomenon induced by interacting dark resonances,” Phys. Rev. A 64, 053806 (2001).
[CrossRef]

Liu, C. P.

C. P. Liu, S. Q. Gong, D. C. Cheng, X. J. Fan, and Z. Z. Xu, “Atom localization via interference of dark resonances,” Phys. Rev. A 73, 025801 (2006).
[CrossRef]

Lukin, M. D.

S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68, 063801 (2003).
[CrossRef]

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[CrossRef]

Mehmood, A.

S. Qamar, A. Mehmood, and S. Qamar, “Subwavelength atom localization via coherent manipulation of the Raman gain process,” Phys. Rev. A 79, 033848 (2009).
[CrossRef]

Niu, Y. P.

L. L. Jin, H. Sun, Y. P. Niu, S. Q. Jin, and S. Q. Gong, “Two-dimension atom nano-lithograph via atom localization,” J. Mod. Opt. 56, 805–810 (2009).
[CrossRef]

L. L. Jin, H. Sun, Y. P. Niu, and S. Q. Gong, “Sub-half-wavelength atom localization via two standing-wave fields,” J. Phys. B 41, 085508 (2008).
[CrossRef]

D. C. Cheng, Y. P. Niu, R. X. Li, and S. Q. Gong, “Controllable atom localization via double-dark resonances in a tripod system,” J. Opt. Soc. Am. B 23, 2180–2184 (2006).
[CrossRef]

Y. P. Niu, S. Q. Gong, R. X. Li, Z. Z. Xu, and X. Y. Liang, “Giant Kerr nonlinearity induced by interacting dark resonances,” Opt. Lett. 30, 3371–3373 (2005).
[CrossRef]

Paspalakis, E.

E. Paspalakis and P. L. Knight, “Localizing an atom via quantum interference,” Phys. Rev. A 63, 065802 (2001).
[CrossRef]

Phillips, W. D.

W. D. Phillips, “Nobel lecture: laser cooling and trapping of neutral atoms,” Rev. Mod. Phys. 70, 721–741 (1998).
[CrossRef]

Prentiss, M.

K. S. Johnson, J. H. Thywissen, W. H. Dekker, K. K. Berggren, A. P. Chu, R. Younkin, and M. Prentiss, “Localization of metastable atom beams with optical standing waves: nanolithography at the Heisenberg limit,” Science 280, 1583–1586 (1998).
[CrossRef] [PubMed]

Qamar, S.

S. Qamar, A. Mehmood, and S. Qamar, “Subwavelength atom localization via coherent manipulation of the Raman gain process,” Phys. Rev. A 79, 033848 (2009).
[CrossRef]

S. Qamar, A. Mehmood, and S. Qamar, “Subwavelength atom localization via coherent manipulation of the Raman gain process,” Phys. Rev. A 79, 033848 (2009).
[CrossRef]

J. Evers, S. Qamar, and M. S. Zubairy, “Atom localization and center-of-mass wave-function determination via multiple simultaneous quadrature measurements,” Phys. Rev. A 75, 053809(2007).
[CrossRef]

K. T. Kapale, S. Qamar, and M. S. Zubairy, “Spectroscopic measurement of an atomic wave function,” Phys. Rev. A 67, 023805(2003).
[CrossRef]

S. Qamar, S. Y. Zhu, and M. S. Zubairy, “Precision localization of single atom using Autler—Townes microscopy,” Opt. Commun. 176, 409–416 (2000).
[CrossRef]

S. Qamar, S. Y. Zhu, and M. S. Zubairy, “Atom localization via resonance fluorescence,” Phys. Rev. A 61, 063806 (2000).
[CrossRef]

Quadt, R.

R. Quadt, M. Collett, and D. F. Walls, “Measurement of atomic motion in a standing light field by homodyne detection,” Phys. Rev. Lett. 74, 351–354 (1995).
[CrossRef] [PubMed]

Rempe, G.

S. Kunze, K. Dieckmann, and G. Rempe, “Diffraction of atoms from a measurement induced grating,” Phys. Rev. Lett. 78, 2038–2041 (1997).
[CrossRef]

F. L. Kienm, G. Rempe, W. P. Schleich, and M. S. Zubairy, “Atom localization via Ramsey interferometry: a coherent cavity field provides a better resolution,” Phys. Rev. A 56, 2972–2977 (1997).
[CrossRef]

Rozhdestvensky, Y.

V. Ivanov and Y. Rozhdestvensky, “Two-dimensional atom localization in a four-level tripod system in laser fields,” Phys. Rev. A 81, 033809 (2010).
[CrossRef]

Sahrai, M.

M. Sahrai, H. Tajalli, K. T. Kapale, and M. S. Zubairy, “Subwavelength atom localization via amplitude and phase control of the absorption spectrum,” Phys. Rev. A 72, 013820 (2005).
[CrossRef]

Sautenkov, V. A.

S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68, 063801 (2003).
[CrossRef]

Schleich, W. P.

F. L. Kienm, G. Rempe, W. P. Schleich, and M. S. Zubairy, “Atom localization via Ramsey interferometry: a coherent cavity field provides a better resolution,” Phys. Rev. A 56, 2972–2977 (1997).
[CrossRef]

A. M. Herkommer, H. J. Carmichael, and W. P. Schleich, “Localization of an atom by homodyne measurement,” Quantum Semiclass. Opt. 8, 189–203 (1996).
[CrossRef]

Scully, M. O.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[CrossRef]

Su, J. J.

Y. C. Chen, Y. A. Liao, H. Y. Chiu, J. J. Su, and I. A. Yu, “Observation of the quantum interference phenomenon induced by interacting dark resonances,” Phys. Rev. A 64, 053806 (2001).
[CrossRef]

Sun, H.

L. L. Jin, H. Sun, Y. P. Niu, S. Q. Jin, and S. Q. Gong, “Two-dimension atom nano-lithograph via atom localization,” J. Mod. Opt. 56, 805–810 (2009).
[CrossRef]

L. L. Jin, H. Sun, Y. P. Niu, and S. Q. Gong, “Sub-half-wavelength atom localization via two standing-wave fields,” J. Phys. B 41, 085508 (2008).
[CrossRef]

Tajalli, H.

M. Sahrai, H. Tajalli, K. T. Kapale, and M. S. Zubairy, “Subwavelength atom localization via amplitude and phase control of the absorption spectrum,” Phys. Rev. A 72, 013820 (2005).
[CrossRef]

Thywissen, J. H.

K. S. Johnson, J. H. Thywissen, W. H. Dekker, K. K. Berggren, A. P. Chu, R. Younkin, and M. Prentiss, “Localization of metastable atom beams with optical standing waves: nanolithography at the Heisenberg limit,” Science 280, 1583–1586 (1998).
[CrossRef] [PubMed]

Walls, D. F.

R. Quadt, M. Collett, and D. F. Walls, “Measurement of atomic motion in a standing light field by homodyne detection,” Phys. Rev. Lett. 74, 351–354 (1995).
[CrossRef] [PubMed]

Wan, R. G.

Welch, G. R.

S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68, 063801 (2003).
[CrossRef]

Xu, J.

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[CrossRef]

Opt. Lett.

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Phys. Rev. A

M. Sahrai, H. Tajalli, K. T. Kapale, and M. S. Zubairy, “Subwavelength atom localization via amplitude and phase control of the absorption spectrum,” Phys. Rev. A 72, 013820 (2005).
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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

K. T. Kapale, S. Qamar, and M. S. Zubairy, “Spectroscopic measurement of an atomic wave function,” Phys. Rev. A 67, 023805(2003).
[CrossRef]

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[CrossRef]

Y. C. Chen, Y. A. Liao, H. Y. Chiu, J. J. Su, and I. A. Yu, “Observation of the quantum interference phenomenon induced by interacting dark resonances,” Phys. Rev. A 64, 053806 (2001).
[CrossRef]

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[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Schematic diagrams. (a) An atom moves along the z axis and interacts with two orthogonal standing-wave fields. (b) Four-level N-type atomic system with two standing waves respectively driving different transitions. (c) Four-level N-type atomic system with two standing waves simultaneously driving one transition.

Fig. 2
Fig. 2

Filter function F ( x , y ) as a function of ( k 1 x , k 2 y ) in dependence on the probe detuning Δ p . Parameters are Γ 0 = 2 , Ω 1 = 5 , Ω 2 = 5 , Δ 1 = 0 , and Δ 2 = 0 . (a)  Δ p = 3 , (b)  Δ p = 5 , (c)  Δ p = 6 , (d)  Δ p = 5 2 .

Fig. 3
Fig. 3

Filter function F ( x , y ) as a function of ( k 1 x , k 2 y ) in dependence on the probe detuning Δ p . (a)  Δ p = 3 , (b)  Δ p = 4 , (c)  Δ p = 5 , (d)  Δ p = 34 . Other parameters are Γ 0 = 2 , Ω 1 = 5 , Ω 2 = 3 , Δ 1 = 0 , and Δ 2 = 0 .

Fig. 4
Fig. 4

Filter function F ( x , y ) as a function of ( k 1 x , k 2 y ) in dependence on the probe detuning Δ p . (a)  Δ p = 3 , (b)  Δ p = 4 , (c)  Δ p = 5 , (d)  Δ p = 34 . Other parameters are Γ 0 = 2 , Ω 1 = 3 , Ω 2 = 5 , Δ 1 = 0 , and Δ 2 = 0 .

Fig. 5
Fig. 5

Filter function F ( x , y ) as a function of ( k 1 x , k 1 y ) in dependence on the probe detuning Δ p . (a)  Δ p = 2 , (b)  Δ p = 4 , (c)  Δ p = 8 , (d)  Δ p = 101 . Other parameters are Γ 0 = 2 , Ω 1 = 5 , Ω 2 = 1 , Δ 1 = 0 , and Δ 2 = 0 .

Fig. 6
Fig. 6

Density plot of filter function F ( x , y ) shown in Fig. 5.

Equations (15)

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H I = ( Δ p Δ 1 ) | 2 2 | Δ p | 3 3 | ( Δ p Δ 1 + Δ 2 ) | 4 4 | + [ Ω p | 1 3 | + Ω 1 ( x , y ) | 2 3 | + Ω 2 ( x , y ) | 2 4 | + h.c. ] ,
| Ψ ( t ) = d x d y f ( x , y ) | x , y [ C 1 ( x , y ; t ) | 1 + C 2 ( x , y ; t ) | 2 + C 3 ( x , y ; t ) | 3 + C 4 ( x , y ; t ) | 4 ] ,
W ( x , y ; t | 3 ) = | N | 2 | f ( x , y ) | 2 | C 3 ( x , y ; t ) | 2 ,
i C ˙ 1 ( t ) = Ω p C 3 ( t ) ,
i C ˙ 2 ( t ) = ( Δ p Δ 1 ) C 2 ( t ) + Ω 1 ( x , y ) C 3 ( t ) + Ω 2 ( x , y ) C 4 ( t ) ,
i C ˙ 3 ( t ) = ( Δ p + Γ 3 2 ) C 3 ( t ) + Ω p C 1 ( t ) + Ω 1 ( x , y ) C 2 ( t ) ,
i C ˙ 4 ( t ) = [ ( Δ p Δ 1 + Δ 2 ) + Γ 4 2 ] C 4 ( t ) + Ω 2 ( x , y ) C 2 ( t ) ,
C 3 ( x , y ; t ) = Ω p ( Δ p + i Γ 3 2 ) Ω 1 2 ( x , y ) ( Δ p Δ 1 ) Ω 2 2 ( x , y ) Δ p Δ 1 + Δ 2 .
W ( x , y ; t | 3 ) = | N | 2 | f ( x , y ) | 2 · Ω p 2 [ Δ p Ω 1 2 ( x , y ) ( Δ p Δ 1 ) Ω 2 2 ( x , y ) Δ p Δ 1 + Δ 2 ] 2 + Γ 3 2 4 .
χ ( x , y ; Δ p ) = | f ( x , y ) | 2 Im [ 4 π N | d 13 | 2 Ω p · C 1 ( t ) C 3 * ( t ) ] = | f ( x , y ) | 2 · 2 π N | d 13 | 2 Γ 3 [ Δ p Ω 1 2 ( x , y ) ( Δ p Δ 1 ) Ω 2 2 ( x , y ) Δ p Δ 1 + Δ 2 ] 2 + Γ 3 2 4 ,
F ( x , y ) = 1 [ Δ p Ω 1 2 ( x , y ) ( Δ p Δ 1 ) Ω 2 2 ( x , y ) Δ p Δ 1 + Δ 2 ] 2 + Γ 3 2 4 .
Δ p Ω 1 2 ( x , y ) ( Δ p Δ 1 ) Ω 2 2 ( x , y ) Δ p Δ 1 + Δ 2 = 0.
Ω 1 2 ( x , y ) + Ω 2 2 ( x , y ) = Δ p 2 .
Ω 1 2 sin 2 ( k 1 x ) + Ω 2 2 sin 2 ( k 2 y ) = Δ p 2 .
Ω 1 2 [ sin ( k 1 x ) + sin ( k 1 y ) ] 2 + Ω 2 2 = Δ p 2 .

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