Abstract

A scheme for realizing two-dimensional (2D) atom localization is proposed based on controllable spontaneous emission in a coherently driven cycle-configuration atomic system. As the spatial-position-dependent atom-field interaction, the frequency of the spontaneously emitted photon carries the information about the position of the atom. Therefore, by detecting the emitted photon one could obtain the position information available, and then we demonstrate high-precision and high-resolution 2D atom localization induced by the quantum interference between the multiple spontaneous decay channels. Moreover, we can achieve 100% probability of finding the atom at an expected position by choosing appropriate system parameters under certain conditions.

© 2012 OSA

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    [CrossRef]
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2012

2011

C. L. Ding, J. H. Li, X. X. Yang, D. Zhang, H. Xiong, “Proposal for efficient two-dimensional atom localization using probe absorption in a microwave-driven four-level atomic system,” Phys. Rev. A 84, 043840 (2011).
[CrossRef]

J. H. Li, R. Yu, M. Liu, C. L. Ding, X. X. Yang, “Efficient two-dimensional atom localization via phase-sensitive absorption spectrum in a radio-frequency-driven four-level atomic system,” Phys. Lett. A 375, 3978–3985 (2011).
[CrossRef]

S. Evangelou, V. Yannopapas, E. Paspalakis, “Modifying free-space spontaneous emission near a plasmonic nanostructure,” Phys. Rev. A 83, 023819 (2011).
[CrossRef]

R. G. Wan, J. Kou, L. Jiang, Y. Jiang, 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]

R. G. Wan, J. Kou, L. Jiang, Y. Jiang, J. Y. Gao, “Two-dimensional atom localization via quantum interference in a coherently driven inverted-Y system,” Opt. Commun. 284, 985–990 (2011).
[CrossRef]

R. G. Wan, J. Kou, L. Jiang, Y. Jiang, J. Y. Gao, “Two-dimensional atom localization via interacting double-dark resonances,” J. Opt. Soc. Am. B 28, 622–628 (2011).
[CrossRef]

N. A. Proite, Z. J. Simmons, D. D. Yavuz, “Observation of atomic localization using electromagnetically induced transparency,” Phys. Rev. A 83, 041803(R) (2011).
[CrossRef]

2010

F. Dell’Anno, S. De Siena, G. Adesso, F. Illuminati, “Teleportation of squeezing: Optimization using non-Gaussian resources,” Phys. Rev. A 82, 062329 (2010).
[CrossRef]

D. D. Yavuz, D. E. Sikes, “Giant Kerr nonlinearities using refractive-index enhancement,” Phys. Rev. A 81, 035804 (2010).
[CrossRef]

V. Ivanov, 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, S. Q. Gong, “Two-dimension atom nano-lithograph via atom localization,” J. Mod. Opt. 56, 805–810 (2009).
[CrossRef]

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

2007

J. Xu, X. M. Hu, “Sub-half-wavelength localization of an atom via trichromatic phase control,” J. Phys. B: At. Mol. Opt. Phys. 40, 1451–1459 (2007).
[CrossRef]

M. Macovei, J. Evers, C. H. Keitel, M. S. Zubairy, “Localization of atomic ensembles via superfluorescence,” Phys. Rev. A 75, 033801 (2007).
[CrossRef]

Y. P. Zhang, U. Khadka, B. Anderson, M. Xiao, “Controlling four-wave and six-wave mixing processes in multilevel atomic systems,” Appl. Phys. Lett. 91, 221108 (2007).
[CrossRef]

Y. P. Zhang, B. Anderson, A. W. Brown, M. Xiao, “Competition between two four-wave mixing channels via atomic coherence,” Appl. Phys. Lett. 91, 061113 (2007).
[CrossRef]

Y. P. Zhang, A. W. Brown, M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99, 123603 (2007).
[CrossRef] [PubMed]

F. Dell’Anno, S. De Siena, L. Albano, F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

J. H Li, “Control of spontaneous emission spectra via an external coherent magnetic field in a cycle-configuration atomic medium,” Eur. Phys. J. D 42, 467–473 (2007).
[CrossRef]

Y. Wu, X. X. Yang, “Giant Kerr nonlinearities and solitons in a crystal of molecular magnets,” Appl. Phys. Lett. 91, 094104 (2007).
[CrossRef]

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

2006

J. T. Chang, J. Evers, M. O. Scully, M. S. Zubairy, “Measurement of the separation between atoms beyond diffraction limit,” Phys. Rev. A 73, 031803(R) (2006).
[CrossRef]

G. Simon, B. Hehlen, E. Courtens, E. Longueteau, R. Vacher, “Hyper-Raman scattering from vitreous boron oxide: Coherent enhancement of the boson peak,” Phys. Rev. Lett. 96, 105502 (2006).
[CrossRef] [PubMed]

F. Dell’Anno, S. De Siena, F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006).
[CrossRef]

K. T. Kapale, M. S. Zubairy, “Subwavelength atom localization via amplitude and phase control of the absorption spectrum. II,” Phys. Rev. A 73, 023813 (2006).
[CrossRef]

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

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

2005

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

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

J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72, 023802 (2005).
[CrossRef]

2004

A. Serafini, F. Illuminati, M. G. A. Paris, S. De Siena, “Entanglement and purity of two-mode Gaussian states in noisy channels,” Phys. Rev. A 69, 022318 (2004).
[CrossRef]

2003

Y. Wu, L. L. Wen, Y. F. Zhu, “Efficient hyper-Raman scattering in resonant coherent media,” Opt. Lett. 28, 631–633 (2003).
[CrossRef] [PubMed]

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

Y. Wu, J. Saldana, Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67, 013811 (2003).
[CrossRef]

2002

H. Nha, J. H. Lee, J. S. Chang, K. An, “Atomic-position localization via dual measurement,” Phys. Rev. A 65, 033827 (2002).
[CrossRef]

2001

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

2000

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

Y. Wu, X. X. Yang, C. P. Sun, “Systematic method to study the general structure of Bose-Einstein condensates with arbitrary spin,” Phys. Rev. A 62, 063603 (2000).
[CrossRef]

1998

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

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

E. Paspalakis, P. L. Knight, “Phase control of spontaneous emission,” Phys. Rev. Lett. 81, 293–296 (1998).
[CrossRef]

J. Vanier, A. Godone, F. Levi, “Coherent population trapping in cesium: Dark lines and coherent microwave emission,” Phys. Rev. A 58, 2345–2358 (1998), and references therein.
[CrossRef]

1997

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

1996

G. P Collins, “Experimenters produce new Bose-Einstein Condensate(s) and possible puzzles for theorists,” Phys. Today 49, 18–21 (1996).
[CrossRef]

1995

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

1993

P. Storey, M. Collett, D. Walls, “Atomic-position resolution by quadrature-field measurement,” Phys. Rev. A 47, 405–418 (1993).
[CrossRef] [PubMed]

J. R. Gardner, M. L. Marable, G. R. Welch, J. E. Thomas, “Suboptical wavelength position measurement of moving atoms using optical fields,” Phys. Rev. Lett. 70, 3404–3407 (1993).
[CrossRef] [PubMed]

1992

P. Storey, M. Collett, D. Walls, “Measurement-induced diffraction and interference of atoms,” Phys. Rev. Lett. 68, 472–475 (1992).
[CrossRef] [PubMed]

1991

K. D. Stokes, C. Schnurr, J. R. Gardner, M. Marable, G. R. Welch, J. E. Thomas, “Precision position measurement of moving atoms using optical fields,” Phys. Rev. Lett. 67, 1997–2000 (1991).
[CrossRef] [PubMed]

1989

Adesso, G.

F. Dell’Anno, S. De Siena, G. Adesso, F. Illuminati, “Teleportation of squeezing: Optimization using non-Gaussian resources,” Phys. Rev. A 82, 062329 (2010).
[CrossRef]

Agarwal, G. S.

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

Albano, L.

F. Dell’Anno, S. De Siena, L. Albano, F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

An, K.

H. Nha, J. H. Lee, J. S. Chang, K. An, “Atomic-position localization via dual measurement,” Phys. Rev. A 65, 033827 (2002).
[CrossRef]

Anderson, B.

Y. P. Zhang, U. Khadka, B. Anderson, M. Xiao, “Controlling four-wave and six-wave mixing processes in multilevel atomic systems,” Appl. Phys. Lett. 91, 221108 (2007).
[CrossRef]

Y. P. Zhang, B. Anderson, A. W. Brown, M. Xiao, “Competition between two four-wave mixing channels via atomic coherence,” Appl. Phys. Lett. 91, 061113 (2007).
[CrossRef]

Berggren, K. K.

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

Brown, A. W.

Y. P. Zhang, B. Anderson, A. W. Brown, M. Xiao, “Competition between two four-wave mixing channels via atomic coherence,” Appl. Phys. Lett. 91, 061113 (2007).
[CrossRef]

Y. P. Zhang, A. W. Brown, M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99, 123603 (2007).
[CrossRef] [PubMed]

Cao, Z. G.

Z. P. Wang, B. L. Yu, J. Zhu, Z. G. Cao, S. L. Zhen, X. Q. Wu, F. Xu, “Atom localization via controlled spontaneous emission in a five-level atomic system,” Ann. Phys. (New York) 327, 1132–1145 (2012).
[CrossRef]

Chang, J. S.

H. Nha, J. H. Lee, J. S. Chang, K. An, “Atomic-position localization via dual measurement,” Phys. Rev. A 65, 033827 (2002).
[CrossRef]

Chang, J. T.

J. T. Chang, J. Evers, M. O. Scully, M. S. Zubairy, “Measurement of the separation between atoms beyond diffraction limit,” Phys. Rev. A 73, 031803(R) (2006).
[CrossRef]

Cheng, D. C.

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

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

Chu, A. P.

K. S. Johnson, J. H. Thywissen, N. H. Dekker, K. K. Berggren, A. P. Chu, R. Younkin, 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, D. F. Walls, “Measurement of atomic motion in a standing light field by homodyne detection,” Phys. Rev. Lett. 74, 351–354 (1995).
[CrossRef] [PubMed]

P. Storey, M. Collett, D. Walls, “Atomic-position resolution by quadrature-field measurement,” Phys. Rev. A 47, 405–418 (1993).
[CrossRef] [PubMed]

P. Storey, M. Collett, D. Walls, “Measurement-induced diffraction and interference of atoms,” Phys. Rev. Lett. 68, 472–475 (1992).
[CrossRef] [PubMed]

Collins, G. P

G. P Collins, “Experimenters produce new Bose-Einstein Condensate(s) and possible puzzles for theorists,” Phys. Today 49, 18–21 (1996).
[CrossRef]

Courtens, E.

G. Simon, B. Hehlen, E. Courtens, E. Longueteau, R. Vacher, “Hyper-Raman scattering from vitreous boron oxide: Coherent enhancement of the boson peak,” Phys. Rev. Lett. 96, 105502 (2006).
[CrossRef] [PubMed]

De Siena, S.

F. Dell’Anno, S. De Siena, G. Adesso, F. Illuminati, “Teleportation of squeezing: Optimization using non-Gaussian resources,” Phys. Rev. A 82, 062329 (2010).
[CrossRef]

F. Dell’Anno, S. De Siena, L. Albano, F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

F. Dell’Anno, S. De Siena, F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006).
[CrossRef]

A. Serafini, F. Illuminati, M. G. A. Paris, S. De Siena, “Entanglement and purity of two-mode Gaussian states in noisy channels,” Phys. Rev. A 69, 022318 (2004).
[CrossRef]

Dekker, N. H.

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

Dell’Anno, F.

F. Dell’Anno, S. De Siena, G. Adesso, F. Illuminati, “Teleportation of squeezing: Optimization using non-Gaussian resources,” Phys. Rev. A 82, 062329 (2010).
[CrossRef]

F. Dell’Anno, S. De Siena, L. Albano, F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

F. Dell’Anno, S. De Siena, F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006).
[CrossRef]

Ding, C. L.

C. L. Ding, J. H. Li, X. X. Yang, D. Zhang, H. Xiong, “Proposal for efficient two-dimensional atom localization using probe absorption in a microwave-driven four-level atomic system,” Phys. Rev. A 84, 043840 (2011).
[CrossRef]

J. H. Li, R. Yu, M. Liu, C. L. Ding, X. X. Yang, “Efficient two-dimensional atom localization via phase-sensitive absorption spectrum in a radio-frequency-driven four-level atomic system,” Phys. Lett. A 375, 3978–3985 (2011).
[CrossRef]

Ding, Y.

J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72, 023802 (2005).
[CrossRef]

Evangelou, S.

S. Evangelou, V. Yannopapas, E. Paspalakis, “Modifying free-space spontaneous emission near a plasmonic nanostructure,” Phys. Rev. A 83, 023819 (2011).
[CrossRef]

Evers, J.

M. Macovei, J. Evers, C. H. Keitel, M. S. Zubairy, “Localization of atomic ensembles via superfluorescence,” Phys. Rev. A 75, 033801 (2007).
[CrossRef]

J. Evers, S. Qamar, 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. T. Chang, J. Evers, M. O. Scully, M. S. Zubairy, “Measurement of the separation between atoms beyond diffraction limit,” Phys. Rev. A 73, 031803(R) (2006).
[CrossRef]

Fan, X. J.

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

Gao, J. Y.

R. G. Wan, J. Kou, L. Jiang, Y. Jiang, 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]

R. G. Wan, J. Kou, L. Jiang, Y. Jiang, J. Y. Gao, “Two-dimensional atom localization via interacting double-dark resonances,” J. Opt. Soc. Am. B 28, 622–628 (2011).
[CrossRef]

R. G. Wan, J. Kou, L. Jiang, Y. Jiang, J. Y. Gao, “Two-dimensional atom localization via quantum interference in a coherently driven inverted-Y system,” Opt. Commun. 284, 985–990 (2011).
[CrossRef]

J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72, 023802 (2005).
[CrossRef]

Gardner, J. R.

J. R. Gardner, M. L. Marable, G. R. Welch, J. E. Thomas, “Suboptical wavelength position measurement of moving atoms using optical fields,” Phys. Rev. Lett. 70, 3404–3407 (1993).
[CrossRef] [PubMed]

K. D. Stokes, C. Schnurr, J. R. Gardner, M. Marable, G. R. Welch, J. E. Thomas, “Precision position measurement of moving atoms using optical fields,” Phys. Rev. Lett. 67, 1997–2000 (1991).
[CrossRef] [PubMed]

Godone, A.

J. Vanier, A. Godone, F. Levi, “Coherent population trapping in cesium: Dark lines and coherent microwave emission,” Phys. Rev. A 58, 2345–2358 (1998), and references therein.
[CrossRef]

Gong, S. Q.

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

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

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

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

Hehlen, B.

G. Simon, B. Hehlen, E. Courtens, E. Longueteau, R. Vacher, “Hyper-Raman scattering from vitreous boron oxide: Coherent enhancement of the boson peak,” Phys. Rev. Lett. 96, 105502 (2006).
[CrossRef] [PubMed]

Hu, X. M.

J. Xu, X. M. Hu, “Sub-half-wavelength localization of an atom via trichromatic phase control,” J. Phys. B: At. Mol. Opt. Phys. 40, 1451–1459 (2007).
[CrossRef]

Illuminati, F.

F. Dell’Anno, S. De Siena, G. Adesso, F. Illuminati, “Teleportation of squeezing: Optimization using non-Gaussian resources,” Phys. Rev. A 82, 062329 (2010).
[CrossRef]

F. Dell’Anno, S. De Siena, L. Albano, F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

F. Dell’Anno, S. De Siena, F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006).
[CrossRef]

A. Serafini, F. Illuminati, M. G. A. Paris, S. De Siena, “Entanglement and purity of two-mode Gaussian states in noisy channels,” Phys. Rev. A 69, 022318 (2004).
[CrossRef]

Ivanov, V.

V. Ivanov, 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, S. Q. Gong, “Two-dimension atom nano-lithograph via atom localization,” J. Mod. Opt. 56, 805–810 (2009).
[CrossRef]

Jin, S. Q.

L. L. Jin, H. Sun, Y. P. Niu, S. Q. Jin, 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, N. H. Dekker, K. K. Berggren, A. P. Chu, R. Younkin, M. Prentiss, “Localization of metastable atom beams with optical standing waves: nanolithography at the Heisenberg limit,” Science 280, 1583–1586 (1998).
[CrossRef] [PubMed]

Kang, Z. H.

Kapale, K. T.

K. T. Kapale, M. S. Zubairy, “Subwavelength atom localization via amplitude and phase control of the absorption spectrum. II,” Phys. Rev. A 73, 023813 (2006).
[CrossRef]

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

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

Keitel, C. H.

M. Macovei, J. Evers, C. H. Keitel, M. S. Zubairy, “Localization of atomic ensembles via superfluorescence,” Phys. Rev. A 75, 033801 (2007).
[CrossRef]

Khadka, U.

Y. P. Zhang, U. Khadka, B. Anderson, M. Xiao, “Controlling four-wave and six-wave mixing processes in multilevel atomic systems,” Appl. Phys. Lett. 91, 221108 (2007).
[CrossRef]

Kien, F. L.

F. L. Kien, G. Rempe, W. P. Schleich, 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, P. L. Knight, “Localizing an atom via quantum interference,” Phys. Rev. A 63, 065802 (2001).
[CrossRef]

E. Paspalakis, P. L. Knight, “Phase control of spontaneous emission,” Phys. Rev. Lett. 81, 293–296 (1998).
[CrossRef]

Kou, J.

Lee, J. H.

H. Nha, J. H. Lee, J. S. Chang, K. An, “Atomic-position localization via dual measurement,” Phys. Rev. A 65, 033827 (2002).
[CrossRef]

Levi, F.

J. Vanier, A. Godone, F. Levi, “Coherent population trapping in cesium: Dark lines and coherent microwave emission,” Phys. Rev. A 58, 2345–2358 (1998), and references therein.
[CrossRef]

Li, A. J.

J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72, 023802 (2005).
[CrossRef]

Li, J. H

J. H Li, “Control of spontaneous emission spectra via an external coherent magnetic field in a cycle-configuration atomic medium,” Eur. Phys. J. D 42, 467–473 (2007).
[CrossRef]

Li, J. H.

J. H. Li, R. Yu, M. Liu, C. L. Ding, X. X. Yang, “Efficient two-dimensional atom localization via phase-sensitive absorption spectrum in a radio-frequency-driven four-level atomic system,” Phys. Lett. A 375, 3978–3985 (2011).
[CrossRef]

C. L. Ding, J. H. Li, X. X. Yang, D. Zhang, H. Xiong, “Proposal for efficient two-dimensional atom localization using probe absorption in a microwave-driven four-level atomic system,” Phys. Rev. A 84, 043840 (2011).
[CrossRef]

Li, R. X.

Liang, X. Y.

Liu, C. P.

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

Liu, M.

J. H. Li, R. Yu, M. Liu, C. L. Ding, X. X. Yang, “Efficient two-dimensional atom localization via phase-sensitive absorption spectrum in a radio-frequency-driven four-level atomic system,” Phys. Lett. A 375, 3978–3985 (2011).
[CrossRef]

Longueteau, E.

G. Simon, B. Hehlen, E. Courtens, E. Longueteau, R. Vacher, “Hyper-Raman scattering from vitreous boron oxide: Coherent enhancement of the boson peak,” Phys. Rev. Lett. 96, 105502 (2006).
[CrossRef] [PubMed]

Macovei, M.

M. Macovei, J. Evers, C. H. Keitel, M. S. Zubairy, “Localization of atomic ensembles via superfluorescence,” Phys. Rev. A 75, 033801 (2007).
[CrossRef]

Marable, M.

K. D. Stokes, C. Schnurr, J. R. Gardner, M. Marable, G. R. Welch, J. E. Thomas, “Precision position measurement of moving atoms using optical fields,” Phys. Rev. Lett. 67, 1997–2000 (1991).
[CrossRef] [PubMed]

Marable, M. L.

J. R. Gardner, M. L. Marable, G. R. Welch, J. E. Thomas, “Suboptical wavelength position measurement of moving atoms using optical fields,” Phys. Rev. Lett. 70, 3404–3407 (1993).
[CrossRef] [PubMed]

Mehmood, A.

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

Meystre, P.

P. Meystre, M. Sargent, Elements of Quantum Optics (Springer-Verlag, Berlin, 1999).

Nha, H.

H. Nha, J. H. Lee, J. S. Chang, K. An, “Atomic-position localization via dual measurement,” Phys. Rev. A 65, 033827 (2002).
[CrossRef]

Niu, Y. P.

Paris, M. G. A.

A. Serafini, F. Illuminati, M. G. A. Paris, S. De Siena, “Entanglement and purity of two-mode Gaussian states in noisy channels,” Phys. Rev. A 69, 022318 (2004).
[CrossRef]

Paspalakis, E.

S. Evangelou, V. Yannopapas, E. Paspalakis, “Modifying free-space spontaneous emission near a plasmonic nanostructure,” Phys. Rev. A 83, 023819 (2011).
[CrossRef]

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

E. Paspalakis, P. L. Knight, “Phase control of spontaneous emission,” Phys. Rev. Lett. 81, 293–296 (1998).
[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, N. H. Dekker, K. K. Berggren, A. P. Chu, R. Younkin, M. Prentiss, “Localization of metastable atom beams with optical standing waves: nanolithography at the Heisenberg limit,” Science 280, 1583–1586 (1998).
[CrossRef] [PubMed]

Proite, N. A.

N. A. Proite, Z. J. Simmons, D. D. Yavuz, “Observation of atomic localization using electromagnetically induced transparency,” Phys. Rev. A 83, 041803(R) (2011).
[CrossRef]

Qamar, S.

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

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

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

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

Quadt, R.

R. Quadt, M. Collett, 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.

F. L. Kien, G. Rempe, W. P. Schleich, 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, 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, M. S. Zubairy, “Subwavelength atom localization via amplitude and phase control of the absorption spectrum,” Phys. Rev. A 72, 013820 (2005).
[CrossRef]

Saldana, J.

Y. Wu, J. Saldana, Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67, 013811 (2003).
[CrossRef]

Sargent, M.

P. Meystre, M. Sargent, Elements of Quantum Optics (Springer-Verlag, Berlin, 1999).

Schleich, W. P.

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

Schnurr, C.

K. D. Stokes, C. Schnurr, J. R. Gardner, M. Marable, G. R. Welch, J. E. Thomas, “Precision position measurement of moving atoms using optical fields,” Phys. Rev. Lett. 67, 1997–2000 (1991).
[CrossRef] [PubMed]

Scully, M. O.

J. T. Chang, J. Evers, M. O. Scully, M. S. Zubairy, “Measurement of the separation between atoms beyond diffraction limit,” Phys. Rev. A 73, 031803(R) (2006).
[CrossRef]

Serafini, A.

A. Serafini, F. Illuminati, M. G. A. Paris, S. De Siena, “Entanglement and purity of two-mode Gaussian states in noisy channels,” Phys. Rev. A 69, 022318 (2004).
[CrossRef]

Sikes, D. E.

D. D. Yavuz, D. E. Sikes, “Giant Kerr nonlinearities using refractive-index enhancement,” Phys. Rev. A 81, 035804 (2010).
[CrossRef]

Simmons, Z. J.

N. A. Proite, Z. J. Simmons, D. D. Yavuz, “Observation of atomic localization using electromagnetically induced transparency,” Phys. Rev. A 83, 041803(R) (2011).
[CrossRef]

Simon, G.

G. Simon, B. Hehlen, E. Courtens, E. Longueteau, R. Vacher, “Hyper-Raman scattering from vitreous boron oxide: Coherent enhancement of the boson peak,” Phys. Rev. Lett. 96, 105502 (2006).
[CrossRef] [PubMed]

Stokes, K. D.

K. D. Stokes, C. Schnurr, J. R. Gardner, M. Marable, G. R. Welch, J. E. Thomas, “Precision position measurement of moving atoms using optical fields,” Phys. Rev. Lett. 67, 1997–2000 (1991).
[CrossRef] [PubMed]

Storey, P.

P. Storey, M. Collett, D. Walls, “Atomic-position resolution by quadrature-field measurement,” Phys. Rev. A 47, 405–418 (1993).
[CrossRef] [PubMed]

P. Storey, M. Collett, D. Walls, “Measurement-induced diffraction and interference of atoms,” Phys. Rev. Lett. 68, 472–475 (1992).
[CrossRef] [PubMed]

Sun, C. P.

Y. Wu, X. X. Yang, C. P. Sun, “Systematic method to study the general structure of Bose-Einstein condensates with arbitrary spin,” Phys. Rev. A 62, 063603 (2000).
[CrossRef]

Sun, H.

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

Tajalli, H.

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

Thomas, J. E

Thomas, J. E.

J. R. Gardner, M. L. Marable, G. R. Welch, J. E. Thomas, “Suboptical wavelength position measurement of moving atoms using optical fields,” Phys. Rev. Lett. 70, 3404–3407 (1993).
[CrossRef] [PubMed]

K. D. Stokes, C. Schnurr, J. R. Gardner, M. Marable, G. R. Welch, J. E. Thomas, “Precision position measurement of moving atoms using optical fields,” Phys. Rev. Lett. 67, 1997–2000 (1991).
[CrossRef] [PubMed]

Thywissen, J. H.

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

Tian, S. C.

Vacher, R.

G. Simon, B. Hehlen, E. Courtens, E. Longueteau, R. Vacher, “Hyper-Raman scattering from vitreous boron oxide: Coherent enhancement of the boson peak,” Phys. Rev. Lett. 96, 105502 (2006).
[CrossRef] [PubMed]

Vanier, J.

J. Vanier, A. Godone, F. Levi, “Coherent population trapping in cesium: Dark lines and coherent microwave emission,” Phys. Rev. A 58, 2345–2358 (1998), and references therein.
[CrossRef]

Walls, D.

P. Storey, M. Collett, D. Walls, “Atomic-position resolution by quadrature-field measurement,” Phys. Rev. A 47, 405–418 (1993).
[CrossRef] [PubMed]

P. Storey, M. Collett, D. Walls, “Measurement-induced diffraction and interference of atoms,” Phys. Rev. Lett. 68, 472–475 (1992).
[CrossRef] [PubMed]

Walls, D. F.

R. Quadt, M. Collett, 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.

Wang, C. L.

Wang, Z. P.

Z. P. Wang, B. L. Yu, J. Zhu, Z. G. Cao, S. L. Zhen, X. Q. Wu, F. Xu, “Atom localization via controlled spontaneous emission in a five-level atomic system,” Ann. Phys. (New York) 327, 1132–1145 (2012).
[CrossRef]

Welch, G. R.

J. R. Gardner, M. L. Marable, G. R. Welch, J. E. Thomas, “Suboptical wavelength position measurement of moving atoms using optical fields,” Phys. Rev. Lett. 70, 3404–3407 (1993).
[CrossRef] [PubMed]

K. D. Stokes, C. Schnurr, J. R. Gardner, M. Marable, G. R. Welch, J. E. Thomas, “Precision position measurement of moving atoms using optical fields,” Phys. Rev. Lett. 67, 1997–2000 (1991).
[CrossRef] [PubMed]

Wen, L. L.

Wu, J. H.

Wu, X. Q.

Z. P. Wang, B. L. Yu, J. Zhu, Z. G. Cao, S. L. Zhen, X. Q. Wu, F. Xu, “Atom localization via controlled spontaneous emission in a five-level atomic system,” Ann. Phys. (New York) 327, 1132–1145 (2012).
[CrossRef]

Wu, Y.

Y. Wu, X. X. Yang, “Giant Kerr nonlinearities and solitons in a crystal of molecular magnets,” Appl. Phys. Lett. 91, 094104 (2007).
[CrossRef]

Y. Wu, J. Saldana, Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67, 013811 (2003).
[CrossRef]

Y. Wu, L. L. Wen, Y. F. Zhu, “Efficient hyper-Raman scattering in resonant coherent media,” Opt. Lett. 28, 631–633 (2003).
[CrossRef] [PubMed]

Y. Wu, X. X. Yang, C. P. Sun, “Systematic method to study the general structure of Bose-Einstein condensates with arbitrary spin,” Phys. Rev. A 62, 063603 (2000).
[CrossRef]

Xiao, M.

Y. P. Zhang, B. Anderson, A. W. Brown, M. Xiao, “Competition between two four-wave mixing channels via atomic coherence,” Appl. Phys. Lett. 91, 061113 (2007).
[CrossRef]

Y. P. Zhang, A. W. Brown, M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99, 123603 (2007).
[CrossRef] [PubMed]

Y. P. Zhang, U. Khadka, B. Anderson, M. Xiao, “Controlling four-wave and six-wave mixing processes in multilevel atomic systems,” Appl. Phys. Lett. 91, 221108 (2007).
[CrossRef]

Xiong, H.

C. L. Ding, J. H. Li, X. X. Yang, D. Zhang, H. Xiong, “Proposal for efficient two-dimensional atom localization using probe absorption in a microwave-driven four-level atomic system,” Phys. Rev. A 84, 043840 (2011).
[CrossRef]

Xu, F.

Z. P. Wang, B. L. Yu, J. Zhu, Z. G. Cao, S. L. Zhen, X. Q. Wu, F. Xu, “Atom localization via controlled spontaneous emission in a five-level atomic system,” Ann. Phys. (New York) 327, 1132–1145 (2012).
[CrossRef]

Xu, J.

J. Xu, X. M. Hu, “Sub-half-wavelength localization of an atom via trichromatic phase control,” J. Phys. B: At. Mol. Opt. Phys. 40, 1451–1459 (2007).
[CrossRef]

Xu, Z. Z.

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

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

Yang, X. X.

C. L. Ding, J. H. Li, X. X. Yang, D. Zhang, H. Xiong, “Proposal for efficient two-dimensional atom localization using probe absorption in a microwave-driven four-level atomic system,” Phys. Rev. A 84, 043840 (2011).
[CrossRef]

J. H. Li, R. Yu, M. Liu, C. L. Ding, X. X. Yang, “Efficient two-dimensional atom localization via phase-sensitive absorption spectrum in a radio-frequency-driven four-level atomic system,” Phys. Lett. A 375, 3978–3985 (2011).
[CrossRef]

Y. Wu, X. X. Yang, “Giant Kerr nonlinearities and solitons in a crystal of molecular magnets,” Appl. Phys. Lett. 91, 094104 (2007).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of a four-level atomic system, which consists of one excited level |3〉, three ground levels |0〉, |1〉, and |2〉. The transitions | 1 G 1 ( x , y ) | 3 G 2 ( x , y ) | 2 Ω m | 1 form a cyclic configuration, in which G1(x, y) is a standing-wave field or a composition of two orthogonal standing waves, G2(x, y) is a standing-wave or traveling-wave field, and Ω m is one-half Larmor frequency for the relevant transition. Δ1 and Δ2 are the frequency detunings of the corresponding standing-wave or traveling-wave fields. And the transition |3〉 ↔ |0〉 is coupled to vacuum modes in the free space.

Fig. 2
Fig. 2

The filter function F(x,y), which directly reflects the conditional position probability distribution, as a function of (k1x, k2y) in dependence on the detuning of spontaneously emitted photon δk. (a) δk = 8γ; (b) δk = 9.05γ; (c) δk = 16γ; (d) δk = 19.3γ. The other parameters used are Ω1 = Ω2 = 10γ, |Ω m | = 9γ, Δ1 = Δ2 = 0, Γ0 = 2γ, and φ = 0. The atom is initially prepared in level |3〉, i.e., A3,0 k (0) = 1. All parameters are in units of γ.

Fig. 3
Fig. 3

Density plot of filter function F(x,y) in the xy plane shown in Fig. 2.

Fig. 4
Fig. 4

The filter function F(x,y) as a function of (k1x, k2y) in dependence on the detuning of spontaneously emitted photon. (a) δk = 8.5γ; (b) δk = 12.2γ; (c) δk = 17.2γ; (d) δk = 20γ. The system parameters used are the same as Fig. 2 except that Δ1 = Δ2 = 5γ, φ = π, and the atom is initially in Ψ ( 0 ) = ( | 1 + | 2 ) / 2 .

Fig. 5
Fig. 5

Density plot of filter function F(x,y) shown in Fig. 4.

Fig. 6
Fig. 6

The filter function F(x,y) as a function of (k1x, k1y) in dependence on the detuning of spontaneously emitted photon. (a) δk = 7γ; (b) δk = 8.9γ; (c) δk = 13.5γ; (d) δk = 20γ. The system parameters used are the same as Fig. 2.

Fig. 7
Fig. 7

Density plot of filter function F(x,y) in the xy plane shown in Fig. 6.

Fig. 8
Fig. 8

The filter function F(x,y) as a function of (k1x, k1y) for different combinations of the detuning δk and the phase φ. (a) δk = 14.5γ, φ = π/2; (b) δk = 19.5γ, φ = π/4; (c) δk = 26γ, φ = 3π/4; (d) δk = 6.75γ, φ = π. The system parameters used are the same as Fig. 2 except that the atom is initially in Ψ ( 0 ) = ( | 1 | 2 ) / 2 .

Fig. 9
Fig. 9

Density plot of filter function F(x,y) shown in Fig. 8.

Fig. 10
Fig. 10

The filter function F(x,y) as a function of the normalized positions for different system parameters. (a) and (b) denote the cases that the above Fig. 2(c) and Fig. 2(d) which are added a fluctuation 0.1γ, respectively, i.e., Ω1 = Ω2 = 10γ + 0.1γ. Other parameters are the same as that in Fig. 2(c) and Fig. 2(d). (c) and (d) correspond to Fig. 8(c) and Fig. 8(d) which are added a detuning fluctuation 0.05γ, i.e., Δ1 = Δ2 = 0.05γ. The other parameters used are the same as Fig. 8(c) and Fig. 8(d), respectively.

Equations (14)

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H I = G 1 ( x , y ) e Δ 1 t | 3 1 | + G 2 ( x , y ) e i Δ 2 t | 3 2 | + Ω m e i ( Δ 1 Δ 2 ) t | 2 1 | + k g k e i δ k t | 3 0 | b ^ k + H.c. ,
| Ψ ( t ) = d x d y f ( x , y ) | x | y [ A 1 , 0 k ( x , y ; t ) | 1 , 0 k + A 2 , 0 k ( x , y ; t ) | 2 , 0 k + A 3 , 0 k ( x , y ; t ) | 3 , 0 k + k A 0 , 1 k ( x , y ; t ) | 0 , 1 k ] ,
| ψ 0 , 1 k = 𝒩 0 , 1 k | Ψ ( t ) = 𝒩 d x d y f ( x , y ) | x | y A 0 , 1 k ( x , y ; t ) ,
P ( x , y ; t | 0 , 1 k ) = | 𝒩 | 2 | x | y | ψ 0 , 1 k | 2 = | 𝒩 | 2 | f ( x , y ) | 2 | A 0 , 1 k ( x , y ; t ) | 2 ,
i A 1 , 0 k ( t ) t = G 1 ( x , y ) e i Δ 1 t A 3 , 0 k ( t ) + Ω m * e i ( Δ 1 Δ 2 ) t A 2 , 0 k ( t ) ,
i A 2 , 0 k ( t ) t = G 2 ( x , y ) e i Δ 2 t A 3 , 0 k ( t ) + Ω m e i ( Δ 1 Δ 2 ) t A 1 , 0 k ( t ) ,
i A 3 , 0 k ( t ) t = G 1 ( x , y ) e i Δ 1 t A 1 , 0 k ( t ) + G 2 ( x , y ) e i Δ 2 t A 2 , 0 k ( t ) i Γ 0 2 A 3 , 0 k ( t ) ,
i A 0 , 1 k ( t ) t = g k * e i δ k t A 3 , 0 k ( t ) ,
A 0 , 1 k ( x , y ; t ) = i 0 g k * e i δ k t A 3 , 0 k ( t ) d t = i g k * A ˜ 3 , 0 k ( s = i δ k ) ,
A ˜ 3 , 0 k ( s = i δ k ) = C i D ,
C = [ | Ω m | 2 ( δ k Δ 1 ) ( δ k Δ 2 ) ] A 3 , 0 k ( 0 ) [ ( δ k Δ 2 ) G 1 ( x , y ) + Ω m G 2 ( x , y ) ] A 1 , 0 k ( 0 ) [ ( δ k Δ 1 ) G 2 ( x , y ) + Ω m * G 1 ( x , y ) ] A 2 , 0 k ( 0 ) ,
D = ( δ k + i Γ 0 2 ) [ | Ω m | 2 ( δ k Δ 1 ) ( δ k Δ 2 ) ] + ( δ k Δ 2 ) G 1 2 ( x , y ) + ( δ k Δ 1 ) G 2 2 ( x , y ) + ( Ω m + Ω m * ) G 1 ( x , y ) G 2 ( x , y ) .
P ( x , y ; t | 0 , 1 k ) = | 𝒩 | 2 | f ( x , y ) | 2 | A 0 , 1 k ( x , y ; t ) | 2 = | 𝒩 | 2 | f ( x , y ) | 2 | g k | 2 | C D | 2 .
F ( x , y ) = 1 [ δ k + δ k [ G 1 2 ( x , y ) + G 2 2 ( x , y ) ] + 2 | Ω m | G 1 ( x , y ) G 2 ( x , y ) | Ω m | 2 δ k 2 ] 2 + Γ 0 2 4 .

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