Abstract

The present study demonstrates the atom localization in the subwavelength domain by considering a $\Xi$-type atomic system with nearby upper levels. When an atom passes through standing-wave couple fields, it faces a position dependent probe absorption that is the key factor for the present study. The position corresponding to a large probe absorption refers the localization position in the subwavelength domain. Our numerical calculation based on the density matrix formalism suggests that nearby levels can change the localization pattern significantly in two dimensional (2D) as well as three dimensional (3D) subwavelength domains. In the 2D case, a narrower spike-like localization pattern can be obtained by optimizing the field parameters in the presence of nearby levels. Thus, precise information of the atom’s position can be achieved. Interestingly for the 3D case, the presence of nearby levels enhances the range of probe detuning; consequently, the probability of finding the atom becomes constant in the subwavelength domain. In addition, the maximal detection probability of finding the atom at a position, i.e., unity can be obtained in both the cases, 2D and 3D, by properly adjusting the fields’ parameter in the presence of nearby levels.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (5)

T. Shui, W.-X. Yang, A.-X. Chen, S. Liu, L. Li, and Z. Zhu, “High-precision two-dimensional atom localization from four-wave mixing in a double-$\lambda$λ four-level atomic system,” Laser Phys. 28(3), 035201 (2018).
[Crossref]

T. Shui, W.-X. Yang, S. Liu, L. Li, and Z. Zhu, “Asymmetric diffraction by atomic gratings with optical pt symmetry in the raman-nath regime,” Phys. Rev. A 97(3), 033819 (2018).
[Crossref]

D. Zhang, Y. Rong, Z. Sun, C. Ding, and M. S. Zubairy, “High-precision three-dimensional atom localization via phase-sensitive absorption spectra in a four-level atomic system,” J. Phys. B: At., Mol. Opt. Phys. 51(2), 025501 (2018).
[Crossref]

N. Singh and A. Wasan, “High-precision two-and three-dimensional atom localization via spatial dependent probe absorption in a closed-loop m-type atomic system,” J. Opt. Soc. Am. B 35(6), 1318–1327 (2018).
[Crossref]

A. N. Tuan, D. Le Van, and B. N. Huy, “Manipulating multi-frequency light in a five-level cascade-type atomic medium associated with giant self-kerr nonlinearity,” J. Opt. Soc. Am. B 35(6), 1233–1239 (2018).
[Crossref]

2017 (2)

2016 (4)

H. Hamedi and G. Juzeliūnas, “Phase-sensitive atom localization for closed-loop quantum systems,” Phys. Rev. A 94(1), 013842 (2016).
[Crossref]

H. Hamedi and M. Mehmannavaz, “Phase control of three-dimensional atom localization in a four-level atomic system in lambda configuration,” J. Opt. Soc. Am. B 33(1), 41–45 (2016).
[Crossref]

Z. Wang, D. Cao, and B. Yu, “Three-dimensional atom localization via electromagnetically induced transparency in a three-level atomic system,” Appl. Opt. 55(13), 3582–3588 (2016).
[Crossref]

Z. Zhu, W.-X. Yang, X.-T. Xie, S. Liu, S. Liu, and R.-K. Lee, “Three-dimensional atom localization from spatial interference in a double two-level atomic system,” Phys. Rev. A 94(1), 013826 (2016).
[Crossref]

2015 (3)

2014 (3)

J. Wu and B. Ai, “Two-dimensional sub-wavelength atom localization in an electromagnetically induced transparency atomic system,” Europhys. Lett. 107(1), 14002 (2014).
[Crossref]

V. S. Ivanov, Y. V. Rozhdestvensky, and K.-A. Suominen, “Three-dimensional atom localization by laser fields in a four-level tripod system,” Phys. Rev. A 90(6), 063802 (2014).
[Crossref]

Z. Wang, X. Wu, L. Lu, and B. Yu, “High-efficiency one-dimensional atom localization via two parallel standing-wave fields,” Laser Phys. 24(10), 105501 (2014).
[Crossref]

2013 (2)

S. Qamar, “Two-dimensional atom localization via probe-absorption spectrum,” Phys. Rev. A 88(1), 013846 (2013).
[Crossref]

B. K. Dutta, P. Panchadhyayee, and P. K. Mahapatra, “Coherent control of localization of a three-level atom by symmetric and asymmetric superpositions of two standing-wave fields,” Laser Phys. 23(4), 045201 (2013).
[Crossref]

2012 (2)

Y. Qi, F. Zhou, T. Huang, Y. Niu, and S. Gong, “Three-dimensional atom localization in a five-level m-type atomic system,” J. Mod. Opt. 59(12), 1092–1099 (2012).
[Crossref]

V. Bharti and A. Wasan, “Electromagnetic induced transparency in the doppler broadened cascade transition with multiple excited levels,” J. Phys. B: At., Mol. Opt. Phys. 45(18), 185501 (2012).
[Crossref]

2011 (5)

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Two-dimensional atom localization via interacting double-dark resonances,” J. Opt. Soc. Am. B 28(4), 622–628 (2011).
[Crossref]

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Two-dimensional atom localization via quantum interference in a coherently driven inverted-y system,” Opt. Commun. 284(4), 985–990 (2011).
[Crossref]

C. Ding, J. Li, X. Yang, Z. Zhan, and J.-B. Liu, “Two-dimensional atom localization via a coherence-controlled absorption spectrum in an n-tripod-type five-level atomic system,” J. Phys. B 44(14), 145501 (2011).
[Crossref]

C. Ding, J. Li, X. Yang, D. Zhang, and H. Xiong, “Proposal for efficient two-dimensional atom localization using probe absorption in a microwave-driven four-level atomic system,” Phys. Rev. A 84(4), 043840 (2011).
[Crossref]

C. Ding, J. Li, Z. Zhan, and X. Yang, “Two-dimensional atom localization via spontaneous emission in a coherently driven five-level m-type atomic system,” Phys. Rev. A 83(6), 063834 (2011).
[Crossref]

2010 (1)

V. Ivanov and Y. Rozhdestvensky, “Two-dimensional atom localization in a four-level tripod system in laser fields,” Phys. Rev. A 81(3), 033809 (2010).
[Crossref]

2009 (2)

S. Qamar, A. Mehmood, and S. Qamar, “Subwavelength atom localization via coherent manipulation of the raman gain process,” Phys. Rev. A 79(3), 033848 (2009).
[Crossref]

J. Mompart, V. Ahufinger, and G. Birkl, “Coherent patterning of matter waves with subwavelength localization,” Phys. Rev. A 79(5), 053638 (2009).
[Crossref]

2008 (1)

A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, “Coherent quantum optical control with subwavelength resolution,” Phys. Rev. Lett. 100(9), 093005 (2008).
[Crossref]

2007 (2)

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(5), 053809 (2007).
[Crossref]

J. Xu and X.-M. Hu, “Sub-half-wavelength localization of an atom via trichromatic phase control,” J. Phys. B 40(7), 1451–1459 (2007).
[Crossref]

2006 (1)

K. T. Kapale and M. S. Zubairy, “Subwavelength atom localization via amplitude and phase control of the absorption spectrum. ii,” Phys. Rev. A 73(2), 023813 (2006).
[Crossref]

2005 (2)

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(1), 013820 (2005).
[Crossref]

E. Paspalakis, A. Terzis, and P. Knight, “Quantum interference induced sub-wavelength atomic localization,” J. Mod. Opt. 52(12), 1685–1694 (2005).
[Crossref]

2003 (1)

K. T. Kapale, S. Qamar, and M. S. Zubairy, “Spectroscopic measurement of an atomic wave function,” Phys. Rev. A 67(2), 023805 (2003).
[Crossref]

2001 (1)

E. Paspalakis and P. Knight, “Localizing an atom via quantum interference,” Phys. Rev. A 63(6), 065802 (2001).
[Crossref]

2000 (2)

Y. Wu, X. Yang, and C. Sun, “Systematic method to study the general structure of bose-einstein condensates with arbitrary spin,” Phys. Rev. A 62(6), 063603 (2000).
[Crossref]

S. Qamar, S.-Y. Zhu, and M. S. Zubairy, “Atom localization via resonance fluorescence,” Phys. Rev. A 61(6), 063806 (2000).
[Crossref]

1998 (2)

W. D. Phillips, “Laser cooling and trapping of neutral atoms,” Rev. Mod. Phys. 70(3), 721–741 (1998).
[Crossref]

K. Johnson, J. Thywissen, N. Dekker, K. Berggren, A. Chu, R. Younkin, and M. Prentiss, “Localization of metastable atom beams with optical standing waves: nanolithography at the heisenberg limit,” Science 280(5369), 1583–1586 (1998).
[Crossref]

1997 (1)

A. Herkommer, W. Schleich, and M. Zubairy, “Autler-townes microscopy on a single atom,” J. Mod. Opt. 44(11–12), 2507–2513 (1997).
[Crossref]

1996 (1)

G. P. Collins, “Experimenters produce new bose einstein condensate (s) and possible puzzles for theorists,” Phys. Today 49(3), 18–21 (1996).
[Crossref]

1994 (1)

S. Kunze, G. Rempe, and M. Wilkens, “Atomic-position measurement via internal-state encoding,” Europhys. Lett. 27(2), 115–121 (1994).
[Crossref]

1993 (1)

P. Storey, M. Collett, and D. Walls, “Atomic-position resolution by quadrature-field measurement,” Phys. Rev. A 47(1), 405–418 (1993).
[Crossref]

1992 (1)

P. Storey, M. Collett, and D. Walls, “Measurement-induced diffraction and interference of atoms,” Phys. Rev. Lett. 68(4), 472–475 (1992).
[Crossref]

Ahufinger, V.

J. Mompart, V. Ahufinger, and G. Birkl, “Coherent patterning of matter waves with subwavelength localization,” Phys. Rev. A 79(5), 053638 (2009).
[Crossref]

Ai, B.

J. Wu and B. Ai, “Two-dimensional sub-wavelength atom localization in an electromagnetically induced transparency atomic system,” Europhys. Lett. 107(1), 14002 (2014).
[Crossref]

Berggren, K.

K. Johnson, J. Thywissen, N. Dekker, K. Berggren, A. Chu, R. Younkin, and M. Prentiss, “Localization of metastable atom beams with optical standing waves: nanolithography at the heisenberg limit,” Science 280(5369), 1583–1586 (1998).
[Crossref]

Bharti, V.

V. Bharti and A. Wasan, “Electromagnetic induced transparency in the doppler broadened cascade transition with multiple excited levels,” J. Phys. B: At., Mol. Opt. Phys. 45(18), 185501 (2012).
[Crossref]

Birkl, G.

J. Mompart, V. Ahufinger, and G. Birkl, “Coherent patterning of matter waves with subwavelength localization,” Phys. Rev. A 79(5), 053638 (2009).
[Crossref]

Cao, D.

Chen, A.-X.

T. Shui, W.-X. Yang, A.-X. Chen, S. Liu, L. Li, and Z. Zhu, “High-precision two-dimensional atom localization from four-wave mixing in a double-$\lambda$λ four-level atomic system,” Laser Phys. 28(3), 035201 (2018).
[Crossref]

Z. Zhu, W.-X. Yang, A.-X. Chen, S. Liu, and R.-K. Lee, “Two-dimensional atom localization via phase-sensitive absorption-gain spectra in five-level hyper inverted-y atomic systems,” J. Opt. Soc. Am. B 32(6), 1070–1077 (2015).
[Crossref]

Chu, A.

K. Johnson, J. Thywissen, N. Dekker, K. Berggren, A. Chu, R. Younkin, and M. Prentiss, “Localization of metastable atom beams with optical standing waves: nanolithography at the heisenberg limit,” Science 280(5369), 1583–1586 (1998).
[Crossref]

Collett, M.

P. Storey, M. Collett, and D. Walls, “Atomic-position resolution by quadrature-field measurement,” Phys. Rev. A 47(1), 405–418 (1993).
[Crossref]

P. Storey, M. Collett, and D. Walls, “Measurement-induced diffraction and interference of atoms,” Phys. Rev. Lett. 68(4), 472–475 (1992).
[Crossref]

Collins, G. P.

G. P. Collins, “Experimenters produce new bose einstein condensate (s) and possible puzzles for theorists,” Phys. Today 49(3), 18–21 (1996).
[Crossref]

Dekker, N.

K. Johnson, J. Thywissen, N. Dekker, K. Berggren, A. Chu, R. Younkin, and M. Prentiss, “Localization of metastable atom beams with optical standing waves: nanolithography at the heisenberg limit,” Science 280(5369), 1583–1586 (1998).
[Crossref]

Ding, C.

D. Zhang, Y. Rong, Z. Sun, C. Ding, and M. S. Zubairy, “High-precision three-dimensional atom localization via phase-sensitive absorption spectra in a four-level atomic system,” J. Phys. B: At., Mol. Opt. Phys. 51(2), 025501 (2018).
[Crossref]

C. Ding, J. Li, X. Yang, Z. Zhan, and J.-B. Liu, “Two-dimensional atom localization via a coherence-controlled absorption spectrum in an n-tripod-type five-level atomic system,” J. Phys. B 44(14), 145501 (2011).
[Crossref]

C. Ding, J. Li, Z. Zhan, and X. Yang, “Two-dimensional atom localization via spontaneous emission in a coherently driven five-level m-type atomic system,” Phys. Rev. A 83(6), 063834 (2011).
[Crossref]

C. Ding, J. Li, X. Yang, D. Zhang, and H. Xiong, “Proposal for efficient two-dimensional atom localization using probe absorption in a microwave-driven four-level atomic system,” Phys. Rev. A 84(4), 043840 (2011).
[Crossref]

Dutta, B. K.

B. K. Dutta, P. Panchadhyayee, and P. K. Mahapatra, “Coherent control of localization of a three-level atom by symmetric and asymmetric superpositions of two standing-wave fields,” Laser Phys. 23(4), 045201 (2013).
[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(5), 053809 (2007).
[Crossref]

Gao, J.-Y.

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Two-dimensional atom localization via quantum interference in a coherently driven inverted-y system,” Opt. Commun. 284(4), 985–990 (2011).
[Crossref]

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Two-dimensional atom localization via interacting double-dark resonances,” J. Opt. Soc. Am. B 28(4), 622–628 (2011).
[Crossref]

Gong, S.

Y. Qi, F. Zhou, T. Huang, Y. Niu, and S. Gong, “Three-dimensional atom localization in a five-level m-type atomic system,” J. Mod. Opt. 59(12), 1092–1099 (2012).
[Crossref]

Gorshkov, A. V.

A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, “Coherent quantum optical control with subwavelength resolution,” Phys. Rev. Lett. 100(9), 093005 (2008).
[Crossref]

Greiner, M.

A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, “Coherent quantum optical control with subwavelength resolution,” Phys. Rev. Lett. 100(9), 093005 (2008).
[Crossref]

Hamedi, H.

H. Hamedi and M. Mehmannavaz, “Phase control of three-dimensional atom localization in a four-level atomic system in lambda configuration,” J. Opt. Soc. Am. B 33(1), 41–45 (2016).
[Crossref]

H. Hamedi and G. Juzeliūnas, “Phase-sensitive atom localization for closed-loop quantum systems,” Phys. Rev. A 94(1), 013842 (2016).
[Crossref]

A. Raheli, H. Hamedi, and M. Sahrai, “Atom localization in 2d for five-level atomic schemes in x-configuration,” Laser Phys. 25(9), 095202 (2015).
[Crossref]

Herkommer, A.

A. Herkommer, W. Schleich, and M. Zubairy, “Autler-townes microscopy on a single atom,” J. Mod. Opt. 44(11–12), 2507–2513 (1997).
[Crossref]

Hu, X.-M.

J. Xu and X.-M. Hu, “Sub-half-wavelength localization of an atom via trichromatic phase control,” J. Phys. B 40(7), 1451–1459 (2007).
[Crossref]

Huang, T.

Y. Qi, F. Zhou, T. Huang, Y. Niu, and S. Gong, “Three-dimensional atom localization in a five-level m-type atomic system,” J. Mod. Opt. 59(12), 1092–1099 (2012).
[Crossref]

Huy, B. N.

Ivanov, V.

V. Ivanov and Y. Rozhdestvensky, “Two-dimensional atom localization in a four-level tripod system in laser fields,” Phys. Rev. A 81(3), 033809 (2010).
[Crossref]

Ivanov, V. S.

V. S. Ivanov, Y. V. Rozhdestvensky, and K.-A. Suominen, “Three-dimensional atom localization by laser fields in a four-level tripod system,” Phys. Rev. A 90(6), 063802 (2014).
[Crossref]

Jiang, L.

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Two-dimensional atom localization via quantum interference in a coherently driven inverted-y system,” Opt. Commun. 284(4), 985–990 (2011).
[Crossref]

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Two-dimensional atom localization via interacting double-dark resonances,” J. Opt. Soc. Am. B 28(4), 622–628 (2011).
[Crossref]

A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, “Coherent quantum optical control with subwavelength resolution,” Phys. Rev. Lett. 100(9), 093005 (2008).
[Crossref]

Jiang, X.

Jiang, Y.

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Two-dimensional atom localization via interacting double-dark resonances,” J. Opt. Soc. Am. B 28(4), 622–628 (2011).
[Crossref]

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Two-dimensional atom localization via quantum interference in a coherently driven inverted-y system,” Opt. Commun. 284(4), 985–990 (2011).
[Crossref]

Johnson, K.

K. Johnson, J. Thywissen, N. Dekker, K. Berggren, A. Chu, R. Younkin, and M. Prentiss, “Localization of metastable atom beams with optical standing waves: nanolithography at the heisenberg limit,” Science 280(5369), 1583–1586 (1998).
[Crossref]

Juzeliunas, G.

H. Hamedi and G. Juzeliūnas, “Phase-sensitive atom localization for closed-loop quantum systems,” Phys. Rev. A 94(1), 013842 (2016).
[Crossref]

Kapale, K. T.

K. T. Kapale and M. S. Zubairy, “Subwavelength atom localization via amplitude and phase control of the absorption spectrum. ii,” Phys. Rev. A 73(2), 023813 (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(1), 013820 (2005).
[Crossref]

K. T. Kapale, S. Qamar, and M. S. Zubairy, “Spectroscopic measurement of an atomic wave function,” Phys. Rev. A 67(2), 023805 (2003).
[Crossref]

Knight, P.

E. Paspalakis, A. Terzis, and P. Knight, “Quantum interference induced sub-wavelength atomic localization,” J. Mod. Opt. 52(12), 1685–1694 (2005).
[Crossref]

E. Paspalakis and P. Knight, “Localizing an atom via quantum interference,” Phys. Rev. A 63(6), 065802 (2001).
[Crossref]

Kou, J.

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Two-dimensional atom localization via quantum interference in a coherently driven inverted-y system,” Opt. Commun. 284(4), 985–990 (2011).
[Crossref]

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Two-dimensional atom localization via interacting double-dark resonances,” J. Opt. Soc. Am. B 28(4), 622–628 (2011).
[Crossref]

Kunze, S.

S. Kunze, G. Rempe, and M. Wilkens, “Atomic-position measurement via internal-state encoding,” Europhys. Lett. 27(2), 115–121 (1994).
[Crossref]

Le Van, D.

Lee, R.-K.

Z. Zhu, W.-X. Yang, X.-T. Xie, S. Liu, S. Liu, and R.-K. Lee, “Three-dimensional atom localization from spatial interference in a double two-level atomic system,” Phys. Rev. A 94(1), 013826 (2016).
[Crossref]

Z. Zhu, W.-X. Yang, A.-X. Chen, S. Liu, and R.-K. Lee, “Two-dimensional atom localization via phase-sensitive absorption-gain spectra in five-level hyper inverted-y atomic systems,” J. Opt. Soc. Am. B 32(6), 1070–1077 (2015).
[Crossref]

Li, J.

X. Jiang, J. Li, and X. Sun, “Two-dimensional atom localization based on coherent field controlling in a five-level m-type atomic system,” Opt. Express 25(25), 31678–31687 (2017).
[Crossref]

C. Ding, J. Li, Z. Zhan, and X. Yang, “Two-dimensional atom localization via spontaneous emission in a coherently driven five-level m-type atomic system,” Phys. Rev. A 83(6), 063834 (2011).
[Crossref]

C. Ding, J. Li, X. Yang, D. Zhang, and H. Xiong, “Proposal for efficient two-dimensional atom localization using probe absorption in a microwave-driven four-level atomic system,” Phys. Rev. A 84(4), 043840 (2011).
[Crossref]

C. Ding, J. Li, X. Yang, Z. Zhan, and J.-B. Liu, “Two-dimensional atom localization via a coherence-controlled absorption spectrum in an n-tripod-type five-level atomic system,” J. Phys. B 44(14), 145501 (2011).
[Crossref]

Li, L.

T. Shui, W.-X. Yang, S. Liu, L. Li, and Z. Zhu, “Asymmetric diffraction by atomic gratings with optical pt symmetry in the raman-nath regime,” Phys. Rev. A 97(3), 033819 (2018).
[Crossref]

T. Shui, W.-X. Yang, A.-X. Chen, S. Liu, L. Li, and Z. Zhu, “High-precision two-dimensional atom localization from four-wave mixing in a double-$\lambda$λ four-level atomic system,” Laser Phys. 28(3), 035201 (2018).
[Crossref]

Liu, J.-B.

C. Ding, J. Li, X. Yang, Z. Zhan, and J.-B. Liu, “Two-dimensional atom localization via a coherence-controlled absorption spectrum in an n-tripod-type five-level atomic system,” J. Phys. B 44(14), 145501 (2011).
[Crossref]

Liu, S.

T. Shui, W.-X. Yang, S. Liu, L. Li, and Z. Zhu, “Asymmetric diffraction by atomic gratings with optical pt symmetry in the raman-nath regime,” Phys. Rev. A 97(3), 033819 (2018).
[Crossref]

T. Shui, W.-X. Yang, A.-X. Chen, S. Liu, L. Li, and Z. Zhu, “High-precision two-dimensional atom localization from four-wave mixing in a double-$\lambda$λ four-level atomic system,” Laser Phys. 28(3), 035201 (2018).
[Crossref]

Z. Zhu, W.-X. Yang, X.-T. Xie, S. Liu, S. Liu, and R.-K. Lee, “Three-dimensional atom localization from spatial interference in a double two-level atomic system,” Phys. Rev. A 94(1), 013826 (2016).
[Crossref]

Z. Zhu, W.-X. Yang, X.-T. Xie, S. Liu, S. Liu, and R.-K. Lee, “Three-dimensional atom localization from spatial interference in a double two-level atomic system,” Phys. Rev. A 94(1), 013826 (2016).
[Crossref]

Z. Zhu, W.-X. Yang, A.-X. Chen, S. Liu, and R.-K. Lee, “Two-dimensional atom localization via phase-sensitive absorption-gain spectra in five-level hyper inverted-y atomic systems,” J. Opt. Soc. Am. B 32(6), 1070–1077 (2015).
[Crossref]

Lu, L.

Z. Wang, X. Wu, L. Lu, and B. Yu, “High-efficiency one-dimensional atom localization via two parallel standing-wave fields,” Laser Phys. 24(10), 105501 (2014).
[Crossref]

Lukin, M. D.

A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, “Coherent quantum optical control with subwavelength resolution,” Phys. Rev. Lett. 100(9), 093005 (2008).
[Crossref]

Mahapatra, P. K.

B. K. Dutta, P. Panchadhyayee, and P. K. Mahapatra, “Coherent control of localization of a three-level atom by symmetric and asymmetric superpositions of two standing-wave fields,” Laser Phys. 23(4), 045201 (2013).
[Crossref]

Mao, Y.

Mehmannavaz, M.

Mehmood, A.

S. Qamar, A. Mehmood, and S. Qamar, “Subwavelength atom localization via coherent manipulation of the raman gain process,” Phys. Rev. A 79(3), 033848 (2009).
[Crossref]

Metcalf, H. J.

H. J. Metcalf and P. Van der Straten, Laser Cooling and Trapping (Springer Science & Business Media, 2012).

Meystre, P.

P. Meystre and M. Sargent, Elements of Quantum Optics (Springer Science & Business Media, 2013).

Mompart, J.

J. Mompart, V. Ahufinger, and G. Birkl, “Coherent patterning of matter waves with subwavelength localization,” Phys. Rev. A 79(5), 053638 (2009).
[Crossref]

Niu, Y.

Y. Qi, F. Zhou, T. Huang, Y. Niu, and S. Gong, “Three-dimensional atom localization in a five-level m-type atomic system,” J. Mod. Opt. 59(12), 1092–1099 (2012).
[Crossref]

Panchadhyayee, P.

B. K. Dutta, P. Panchadhyayee, and P. K. Mahapatra, “Coherent control of localization of a three-level atom by symmetric and asymmetric superpositions of two standing-wave fields,” Laser Phys. 23(4), 045201 (2013).
[Crossref]

Paspalakis, E.

E. Paspalakis, A. Terzis, and P. Knight, “Quantum interference induced sub-wavelength atomic localization,” J. Mod. Opt. 52(12), 1685–1694 (2005).
[Crossref]

E. Paspalakis and P. Knight, “Localizing an atom via quantum interference,” Phys. Rev. A 63(6), 065802 (2001).
[Crossref]

Phillips, W. D.

W. D. Phillips, “Laser cooling and trapping of neutral atoms,” Rev. Mod. Phys. 70(3), 721–741 (1998).
[Crossref]

Prentiss, M.

K. Johnson, J. Thywissen, N. Dekker, K. Berggren, A. Chu, R. Younkin, and M. Prentiss, “Localization of metastable atom beams with optical standing waves: nanolithography at the heisenberg limit,” Science 280(5369), 1583–1586 (1998).
[Crossref]

Qamar, S.

S. Qamar, “Two-dimensional atom localization via probe-absorption spectrum,” Phys. Rev. A 88(1), 013846 (2013).
[Crossref]

S. Qamar, A. Mehmood, and S. Qamar, “Subwavelength atom localization via coherent manipulation of the raman gain process,” Phys. Rev. A 79(3), 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(3), 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(5), 053809 (2007).
[Crossref]

K. T. Kapale, S. Qamar, and M. S. Zubairy, “Spectroscopic measurement of an atomic wave function,” Phys. Rev. A 67(2), 023805 (2003).
[Crossref]

S. Qamar, S.-Y. Zhu, and M. S. Zubairy, “Atom localization via resonance fluorescence,” Phys. Rev. A 61(6), 063806 (2000).
[Crossref]

Qi, Y.

Y. Qi, F. Zhou, T. Huang, Y. Niu, and S. Gong, “Three-dimensional atom localization in a five-level m-type atomic system,” J. Mod. Opt. 59(12), 1092–1099 (2012).
[Crossref]

Raheli, A.

A. Raheli, H. Hamedi, and M. Sahrai, “Atom localization in 2d for five-level atomic schemes in x-configuration,” Laser Phys. 25(9), 095202 (2015).
[Crossref]

Rempe, G.

S. Kunze, G. Rempe, and M. Wilkens, “Atomic-position measurement via internal-state encoding,” Europhys. Lett. 27(2), 115–121 (1994).
[Crossref]

Rong, Y.

D. Zhang, Y. Rong, Z. Sun, C. Ding, and M. S. Zubairy, “High-precision three-dimensional atom localization via phase-sensitive absorption spectra in a four-level atomic system,” J. Phys. B: At., Mol. Opt. Phys. 51(2), 025501 (2018).
[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(3), 033809 (2010).
[Crossref]

Rozhdestvensky, Y. V.

V. S. Ivanov, Y. V. Rozhdestvensky, and K.-A. Suominen, “Three-dimensional atom localization by laser fields in a four-level tripod system,” Phys. Rev. A 90(6), 063802 (2014).
[Crossref]

Sahrai, M.

A. Raheli, H. Hamedi, and M. Sahrai, “Atom localization in 2d for five-level atomic schemes in x-configuration,” Laser Phys. 25(9), 095202 (2015).
[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(1), 013820 (2005).
[Crossref]

Sargent, M.

P. Meystre and M. Sargent, Elements of Quantum Optics (Springer Science & Business Media, 2013).

Schleich, W.

A. Herkommer, W. Schleich, and M. Zubairy, “Autler-townes microscopy on a single atom,” J. Mod. Opt. 44(11–12), 2507–2513 (1997).
[Crossref]

Shui, T.

T. Shui, W.-X. Yang, A.-X. Chen, S. Liu, L. Li, and Z. Zhu, “High-precision two-dimensional atom localization from four-wave mixing in a double-$\lambda$λ four-level atomic system,” Laser Phys. 28(3), 035201 (2018).
[Crossref]

T. Shui, W.-X. Yang, S. Liu, L. Li, and Z. Zhu, “Asymmetric diffraction by atomic gratings with optical pt symmetry in the raman-nath regime,” Phys. Rev. A 97(3), 033819 (2018).
[Crossref]

Singh, N.

Storey, P.

P. Storey, M. Collett, and D. Walls, “Atomic-position resolution by quadrature-field measurement,” Phys. Rev. A 47(1), 405–418 (1993).
[Crossref]

P. Storey, M. Collett, and D. Walls, “Measurement-induced diffraction and interference of atoms,” Phys. Rev. Lett. 68(4), 472–475 (1992).
[Crossref]

Sun, C.

Y. Wu, X. Yang, and C. Sun, “Systematic method to study the general structure of bose-einstein condensates with arbitrary spin,” Phys. Rev. A 62(6), 063603 (2000).
[Crossref]

Sun, X.

Sun, Z.

D. Zhang, Y. Rong, Z. Sun, C. Ding, and M. S. Zubairy, “High-precision three-dimensional atom localization via phase-sensitive absorption spectra in a four-level atomic system,” J. Phys. B: At., Mol. Opt. Phys. 51(2), 025501 (2018).
[Crossref]

Suominen, K.-A.

V. S. Ivanov, Y. V. Rozhdestvensky, and K.-A. Suominen, “Three-dimensional atom localization by laser fields in a four-level tripod system,” Phys. Rev. A 90(6), 063802 (2014).
[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(1), 013820 (2005).
[Crossref]

Terzis, A.

E. Paspalakis, A. Terzis, and P. Knight, “Quantum interference induced sub-wavelength atomic localization,” J. Mod. Opt. 52(12), 1685–1694 (2005).
[Crossref]

Thywissen, J.

K. Johnson, J. Thywissen, N. Dekker, K. Berggren, A. Chu, R. Younkin, and M. Prentiss, “Localization of metastable atom beams with optical standing waves: nanolithography at the heisenberg limit,” Science 280(5369), 1583–1586 (1998).
[Crossref]

Tuan, A. N.

Van der Straten, P.

H. J. Metcalf and P. Van der Straten, Laser Cooling and Trapping (Springer Science & Business Media, 2012).

Walls, D.

P. Storey, M. Collett, and D. Walls, “Atomic-position resolution by quadrature-field measurement,” Phys. Rev. A 47(1), 405–418 (1993).
[Crossref]

P. Storey, M. Collett, and D. Walls, “Measurement-induced diffraction and interference of atoms,” Phys. Rev. Lett. 68(4), 472–475 (1992).
[Crossref]

Wan, R.-G.

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Two-dimensional atom localization via quantum interference in a coherently driven inverted-y system,” Opt. Commun. 284(4), 985–990 (2011).
[Crossref]

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Two-dimensional atom localization via interacting double-dark resonances,” J. Opt. Soc. Am. B 28(4), 622–628 (2011).
[Crossref]

Wang, Z.

Wasan, A.

N. Singh and A. Wasan, “High-precision two-and three-dimensional atom localization via spatial dependent probe absorption in a closed-loop m-type atomic system,” J. Opt. Soc. Am. B 35(6), 1318–1327 (2018).
[Crossref]

V. Bharti and A. Wasan, “Electromagnetic induced transparency in the doppler broadened cascade transition with multiple excited levels,” J. Phys. B: At., Mol. Opt. Phys. 45(18), 185501 (2012).
[Crossref]

Wilkens, M.

S. Kunze, G. Rempe, and M. Wilkens, “Atomic-position measurement via internal-state encoding,” Europhys. Lett. 27(2), 115–121 (1994).
[Crossref]

Wu, J.

Y. Mao and J. Wu, “High-precision three-dimensional atom localization in a microwave-driven atomic system,” J. Opt. Soc. Am. B 34(6), 1070–1074 (2017).
[Crossref]

J. Wu and B. Ai, “Two-dimensional sub-wavelength atom localization in an electromagnetically induced transparency atomic system,” Europhys. Lett. 107(1), 14002 (2014).
[Crossref]

Wu, X.

Z. Wang, X. Wu, L. Lu, and B. Yu, “High-efficiency one-dimensional atom localization via two parallel standing-wave fields,” Laser Phys. 24(10), 105501 (2014).
[Crossref]

Wu, Y.

Y. Wu, X. Yang, and C. Sun, “Systematic method to study the general structure of bose-einstein condensates with arbitrary spin,” Phys. Rev. A 62(6), 063603 (2000).
[Crossref]

Xie, X.-T.

Z. Zhu, W.-X. Yang, X.-T. Xie, S. Liu, S. Liu, and R.-K. Lee, “Three-dimensional atom localization from spatial interference in a double two-level atomic system,” Phys. Rev. A 94(1), 013826 (2016).
[Crossref]

Xiong, H.

C. Ding, J. Li, X. Yang, D. Zhang, and H. Xiong, “Proposal for efficient two-dimensional atom localization using probe absorption in a microwave-driven four-level atomic system,” Phys. Rev. A 84(4), 043840 (2011).
[Crossref]

Xu, J.

J. Xu and X.-M. Hu, “Sub-half-wavelength localization of an atom via trichromatic phase control,” J. Phys. B 40(7), 1451–1459 (2007).
[Crossref]

Yang, W.-X.

T. Shui, W.-X. Yang, S. Liu, L. Li, and Z. Zhu, “Asymmetric diffraction by atomic gratings with optical pt symmetry in the raman-nath regime,” Phys. Rev. A 97(3), 033819 (2018).
[Crossref]

T. Shui, W.-X. Yang, A.-X. Chen, S. Liu, L. Li, and Z. Zhu, “High-precision two-dimensional atom localization from four-wave mixing in a double-$\lambda$λ four-level atomic system,” Laser Phys. 28(3), 035201 (2018).
[Crossref]

Z. Zhu, W.-X. Yang, X.-T. Xie, S. Liu, S. Liu, and R.-K. Lee, “Three-dimensional atom localization from spatial interference in a double two-level atomic system,” Phys. Rev. A 94(1), 013826 (2016).
[Crossref]

Z. Zhu, W.-X. Yang, A.-X. Chen, S. Liu, and R.-K. Lee, “Two-dimensional atom localization via phase-sensitive absorption-gain spectra in five-level hyper inverted-y atomic systems,” J. Opt. Soc. Am. B 32(6), 1070–1077 (2015).
[Crossref]

Yang, X.

C. Ding, J. Li, X. Yang, Z. Zhan, and J.-B. Liu, “Two-dimensional atom localization via a coherence-controlled absorption spectrum in an n-tripod-type five-level atomic system,” J. Phys. B 44(14), 145501 (2011).
[Crossref]

C. Ding, J. Li, Z. Zhan, and X. Yang, “Two-dimensional atom localization via spontaneous emission in a coherently driven five-level m-type atomic system,” Phys. Rev. A 83(6), 063834 (2011).
[Crossref]

C. Ding, J. Li, X. Yang, D. Zhang, and H. Xiong, “Proposal for efficient two-dimensional atom localization using probe absorption in a microwave-driven four-level atomic system,” Phys. Rev. A 84(4), 043840 (2011).
[Crossref]

Y. Wu, X. Yang, and C. Sun, “Systematic method to study the general structure of bose-einstein condensates with arbitrary spin,” Phys. Rev. A 62(6), 063603 (2000).
[Crossref]

Younkin, R.

K. Johnson, J. Thywissen, N. Dekker, K. Berggren, A. Chu, R. Younkin, and M. Prentiss, “Localization of metastable atom beams with optical standing waves: nanolithography at the heisenberg limit,” Science 280(5369), 1583–1586 (1998).
[Crossref]

Yu, B.

Zhan, Z.

C. Ding, J. Li, Z. Zhan, and X. Yang, “Two-dimensional atom localization via spontaneous emission in a coherently driven five-level m-type atomic system,” Phys. Rev. A 83(6), 063834 (2011).
[Crossref]

C. Ding, J. Li, X. Yang, Z. Zhan, and J.-B. Liu, “Two-dimensional atom localization via a coherence-controlled absorption spectrum in an n-tripod-type five-level atomic system,” J. Phys. B 44(14), 145501 (2011).
[Crossref]

Zhang, D.

D. Zhang, Y. Rong, Z. Sun, C. Ding, and M. S. Zubairy, “High-precision three-dimensional atom localization via phase-sensitive absorption spectra in a four-level atomic system,” J. Phys. B: At., Mol. Opt. Phys. 51(2), 025501 (2018).
[Crossref]

C. Ding, J. Li, X. Yang, D. Zhang, and H. Xiong, “Proposal for efficient two-dimensional atom localization using probe absorption in a microwave-driven four-level atomic system,” Phys. Rev. A 84(4), 043840 (2011).
[Crossref]

Zhou, F.

Y. Qi, F. Zhou, T. Huang, Y. Niu, and S. Gong, “Three-dimensional atom localization in a five-level m-type atomic system,” J. Mod. Opt. 59(12), 1092–1099 (2012).
[Crossref]

Zhu, S.-Y.

S. Qamar, S.-Y. Zhu, and M. S. Zubairy, “Atom localization via resonance fluorescence,” Phys. Rev. A 61(6), 063806 (2000).
[Crossref]

Zhu, Z.

T. Shui, W.-X. Yang, A.-X. Chen, S. Liu, L. Li, and Z. Zhu, “High-precision two-dimensional atom localization from four-wave mixing in a double-$\lambda$λ four-level atomic system,” Laser Phys. 28(3), 035201 (2018).
[Crossref]

T. Shui, W.-X. Yang, S. Liu, L. Li, and Z. Zhu, “Asymmetric diffraction by atomic gratings with optical pt symmetry in the raman-nath regime,” Phys. Rev. A 97(3), 033819 (2018).
[Crossref]

Z. Zhu, W.-X. Yang, X.-T. Xie, S. Liu, S. Liu, and R.-K. Lee, “Three-dimensional atom localization from spatial interference in a double two-level atomic system,” Phys. Rev. A 94(1), 013826 (2016).
[Crossref]

Z. Zhu, W.-X. Yang, A.-X. Chen, S. Liu, and R.-K. Lee, “Two-dimensional atom localization via phase-sensitive absorption-gain spectra in five-level hyper inverted-y atomic systems,” J. Opt. Soc. Am. B 32(6), 1070–1077 (2015).
[Crossref]

Zoller, P.

A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, “Coherent quantum optical control with subwavelength resolution,” Phys. Rev. Lett. 100(9), 093005 (2008).
[Crossref]

Zubairy, M.

A. Herkommer, W. Schleich, and M. Zubairy, “Autler-townes microscopy on a single atom,” J. Mod. Opt. 44(11–12), 2507–2513 (1997).
[Crossref]

Zubairy, M. S.

D. Zhang, Y. Rong, Z. Sun, C. Ding, and M. S. Zubairy, “High-precision three-dimensional atom localization via phase-sensitive absorption spectra in a four-level atomic system,” J. Phys. B: At., Mol. Opt. Phys. 51(2), 025501 (2018).
[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(5), 053809 (2007).
[Crossref]

K. T. Kapale and M. S. Zubairy, “Subwavelength atom localization via amplitude and phase control of the absorption spectrum. ii,” Phys. Rev. A 73(2), 023813 (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(1), 013820 (2005).
[Crossref]

K. T. Kapale, S. Qamar, and M. S. Zubairy, “Spectroscopic measurement of an atomic wave function,” Phys. Rev. A 67(2), 023805 (2003).
[Crossref]

S. Qamar, S.-Y. Zhu, and M. S. Zubairy, “Atom localization via resonance fluorescence,” Phys. Rev. A 61(6), 063806 (2000).
[Crossref]

Appl. Opt. (1)

Europhys. Lett. (2)

S. Kunze, G. Rempe, and M. Wilkens, “Atomic-position measurement via internal-state encoding,” Europhys. Lett. 27(2), 115–121 (1994).
[Crossref]

J. Wu and B. Ai, “Two-dimensional sub-wavelength atom localization in an electromagnetically induced transparency atomic system,” Europhys. Lett. 107(1), 14002 (2014).
[Crossref]

J. Mod. Opt. (3)

Y. Qi, F. Zhou, T. Huang, Y. Niu, and S. Gong, “Three-dimensional atom localization in a five-level m-type atomic system,” J. Mod. Opt. 59(12), 1092–1099 (2012).
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Figures (6)

Fig. 1.
Fig. 1. (a) The schematic diagram of $\Xi$ atomic scheme with nearby levels where realistic candidate is $^{85}$Rb. (b) 2D field configuration (c) 3D field configuration, Blue circle stands for the atom.
Fig. 2.
Fig. 2. Variation of $F(x,y)$ in 2D subwavelength domain in absence (left column) and presence (right column) of nearby levels for various $\Delta _p$. (a & e) $\Delta _p=0\times \Gamma _{21}$, (b & f) $\Delta _p=3.1\Gamma _{21}$, (c & g) $\Delta _p=5.0\Gamma _{21}$, (d & h) $\Delta _p=8.9\times \Gamma _{21}$. The other fields’parameter are: $\Delta _c=0\Gamma _{21}$, $\Omega =5.0\Gamma _{21}$, $G_p=0.001\Gamma _{21}$.
Fig. 3.
Fig. 3. Variation of $F(x,y)$ is 2D subwavelength domain in presence of nearby levels for varying $g_c$ and $\Delta _c$. (a) $g_c=0\Gamma _{21}$, $\Delta _c=0\Gamma _{21}$, (b) $g_c=0.5\Gamma _{21}$, $\Delta _c=0\Gamma _{21}$, (c) $g_c=1.0\Gamma _{21}$, $\Delta _c=0\Gamma _{21}$, (d) $g_c=1.0\Gamma _{21}$, $\Delta _c=2.0\Gamma _{21}$. The other fields’parameter are: $\Delta _p=8.9\Gamma _{21}$, $\Omega =5\Gamma _{21}$, $G_p=0.001\Gamma _{21}$.
Fig. 4.
Fig. 4. Isosurface for $F(x,y,z)=0.5$ in the absence (a$-$b) and presence (c$-$e) of nearby levels for varying $\Delta _p$. (a) $\Delta _p=5.5\Gamma _{21}$, (b) $\Delta _p=7.7\Gamma _{21}$, (c) $\Delta _p=7.7\Gamma _{21}$, (d) $\Delta _p=9.1\Gamma _{21}$, (e) $\Delta _p=13.5\Gamma _{21}$. The other fields’parameter are: $G_p=0.001\Gamma _{21}$, $\Omega =5\Gamma _{21}$, $\Delta _{c}=0\Gamma _{21}$.
Fig. 5.
Fig. 5. Isosurface for $F(x,y,z)=0.5$ in presence of nearby levels for varying $\Delta _c$. (a) $\Delta _c=0\Gamma _{21}$, (b) $\Delta _c=2.5\Gamma _{21}$, (c) $\Delta _c=5.0\Gamma _{21}$, (d) $\Delta _c=7.5\Gamma _{21}$. Others field parameters are: $g_c=1.0\Gamma _{21}$, $\Delta _p=11\Gamma _{21}$, $G_{p}=0.001\Gamma_{21}$
Fig. 6.
Fig. 6. Isosurface for various probe absorption. (a) $F(x,y,z)=0.1$, (b) $F(x,y,z)=0.4$, (c) $F(x,y,z)=0.7$. The other fields’parameter are: $\Omega =5.0\Gamma _{21}$, $\Delta _p=10\Gamma _{21}$, $\Delta _c=11\Gamma _{21}$, $g_c=1.0\Gamma _{21}$.

Equations (13)

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H = Δ p | 2 2 | ( Δ p + Δ c ) | 3 3 | ( Δ p + Δ c + δ 1 ) | 4 4 | ( Δ p + Δ c δ 2 ) | 5 5 | 1 2 [ G p | 1 2 | + a 32 G c | 2 3 | + a 42 G c | 2 4 | + a 52 G c | 2 5 | + H . c . ] ,
d ρ ( t ) d t = i [ H , ρ ] + L ρ ( t ) ,
L ρ = Γ 21 ( σ 12 ρ σ 21 1 2 ρ σ 22 1 2 σ 22 ρ ) + j = 3 5 Γ j 2 ( σ 2 j ρ σ j 2 1 2 ρ σ j j 1 2 σ j j ρ ) ,
d ρ 21 d t = ( i Δ p Γ 21 2 ) ρ 21 i 2 a 32 G c ρ 31 + i 2 G p ( ρ 22 ρ 11 ) i 2 a 42 G c ρ 41 i 2 a 52 G c ρ 51 ,
d ρ 31 d t = ( i ( Δ p + Δ c ) Γ 32 2 ) ρ 31 i 2 a 32 G c ρ 21 + i 2 G p ρ 32 ,
d ρ 41 d t = ( i ( Δ p + Δ c + δ 1 ) Γ 42 2 ) ρ 41 i 2 a 42 G c ρ 21 + i 2 G p ρ 42 ,
d ρ 51 d t = ( i ( Δ p + Δ c δ 2 ) Γ 52 2 ) ρ 51 i 2 a 52 G c ρ 21 + i 2 G p ρ 52 ,
ρ 21 ( 1 ) = i G p 2 ( F 21 + F 32 + F 42 + F 52 ) ,
F 21 = Γ 21 2 i Δ p , F 42 = a 42 2 ( G c / 2 ) 2 0.5 Γ 42 i ( Δ p + Δ c + δ 1 ) , F 32 = a 32 2 ( G c / 2 ) 2 0.5 Γ 32 i ( Δ p + Δ c ) , F 52 = a 52 2 ( G c / 2 ) 2 0.5 Γ 52 i ( Δ p + Δ c δ 2 ) ,
F ( x , y ) = I m [ ρ 21 ( 1 ) ( x , y ) ] Γ 21 G p .
F ( x , y , z ) = I m [ ρ 21 ( 1 ) ( x , y , z ) ] Γ 21 G p .
H d = Δ c | 3 3 | ( Δ c + δ 1 ) | 4 4 | ( Δ c δ 2 ) | 5 5 | 1 2 [ a 32 G c | 2 3 | + a 42 G c | 2 4 | + a 52 G c | 2 5 | + H . c . ] .
( λ + Δ c δ 2 ) ( λ 2 + λ Δ c ( a 32 G c 2 ) 2 ) ( λ + Δ c + δ 1 ) ( a 52 G c 2 ) 2 [ λ 2 + λ ( δ 1 + 2 Δ c ) + δ 1 Δ c + Δ c 2 ] ( a 42 G c 2 ) 2 × ( λ + Δ c ) ( λ + Δ c δ 2 ) = 0 ,

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