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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 25 — Dec. 5, 2011
  • pp: 25823–25832
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Two-dimensional sub-half-wavelength atom localization via controlled spontaneous emission

Ren-Gang Wan and Tong-Yi Zhang  »View Author Affiliations


Optics Express, Vol. 19, Issue 25, pp. 25823-25832 (2011)
http://dx.doi.org/10.1364/OE.19.025823


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Abstract

We propose a scheme for two-dimensional (2D) atom localization based on the controlled spontaneous emission, in which the atom interacts with two orthogonal standing-wave fields. Due to the spatially dependent atom-field interaction, the position probability distribution of the atom can be directly determined by measuring the resulting spontaneously emission spectrum. The phase sensitive property of the atomic system leads to quenching of the spontaneous emission in some regions of the standing-waves, which significantly reduces the uncertainty in the position measurement of the atom. We find that the frequency measurement of the emitted light localizes the atom in half-wavelength domain. Especially the probability of finding the atom at a particular position can reach 100% when a photon with certain frequency is detected. By increasing the Rabi frequencies of the driving fields, such 2D sub-half-wavelength atom localization can acquire high spatial resolution.

© 2011 OSA

1. Introduction

Of great attention is the enhancement of precision in the localization of an atom. For 1D atom localization, in general, there are four localization peaks in a unit wavelength domain of the standing-wave [8

8. S. Qamar, S. Y. Zhu, and M. S. Zubairy, “Precision localization of single atom using Autler–Townes microscopy,” Opt. Commun. 176(4-6), 409–416 (2000). [CrossRef]

, 9

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

, 14

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

]. However, by using phase dependent absorption [12

12. 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]

, 13

13. 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]

] or spontaneous emission [10

10. F. Ghafoor, S. Qamar, and M. S. Zubairy, “Atom localization via phase and amplitude control of the driving field,” Phys. Rev. A 65(4), 043819 (2002). [CrossRef]

, 11

11. J. Xu and X. M. Hu, “Sub-half-wavelength atom localization via bichromatic phase control of spontaneous emission,” Phys. Lett. A 366(3), 276–281 (2007). [CrossRef]

] as well as the two standing-waves with different wavelength [19

19. 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]

,27

27. 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(8), 085508 (2008). [CrossRef]

], the atom localization peaks reduce to two in either of the two half-wavelength, and the detection probability at the particular position is then improved to 1/2. For 2D atom localization, the probability of finding the atom is 1/4 in each quadrant of the xy plane of the orthogonal standing-waves [20

20. 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]

22

22. 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]

, 26

26. C. L. Ding, J. H. Li, X. X. Yang, Z. M. Zhang, 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]

]. With the quenching in the controlled spontaneous emission [23

23. 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(1), 10–17 (2011). [CrossRef]

] and quantum interference [24

24. C. L. Ding, J. H. Li, Z. M. Zhang, and X. 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]

], the atom can be localized in two of the four quadrants of the xy plane, and then the uncertainty in position measurement is increased by a factor of 2. The similar results can be also obtained as the two standing-waves driving the same atomic transition where the spatial interference of the standing-waves plays an important role [25

25. 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]

].

2. Theoretical model

The schematic diagrams of the proposed scheme are shown in Fig. 1
Fig. 1 Schematic diagrams: (a) An atom moves along the z axis and interacting with two orthogonal standing-wave fields in the xy plane. (b) Four-level atomic system. Levels |0, |1 and |2 are driven by three fields Ω1, Ω2 and Ωc, and then form a closed loop. Δk=ωk(ω1j+ω2j)/2 is the detuning of the spontaneously emitted photon with frequency ωk from the average atomic transition frequency (ω10+ω20)/2.
. We consider an atom moves in the z direction and passes through the intersectant region of two orthogonal standing-wave fields, which are respectively aligned along the x and the y axises [see Fig. 1(a)]. The atomic system is shown in Fig. 1(b). The ground level |0 is coupled to the excited levels |1 and |2 by two laser fields E1 and E2 (carrier frequencies ω1 and ω2), while transition |1|2, usually dipole forbidden, is driven by a microwave Bc (carrier frequency ωc). The two upper levels decay to the lower level |j via interacting with the vacuum modes in free space. Those three fields form a closed loop subsystem, which results in phase dependent light-matter interaction and spontaneous emission. Coupling field E1 is the superposition of two orthogonal standing-waves with the same frequency and aligned along the x and the y directions, respectively. Therefore, the corresponding Rabi frequency, i.e., Ω1(x,y)=Ω10[sin(kx)+sin(ky)] (k=2π/λ1), is spatially dependent in the xy plane [see Fig. 1(a)]. Fields E2 and Bc are traveling-wave with position-independent Rabi frequencies Ω2 and Ωceiϕ, respectively, where actually ϕ is the collective phase of the closed loop.

We consider that the atom is moving with sufficiently high velocity in the z direction that the motion of atom can be treated classically. The interaction region and time are small enough so that the total kinetic energy acquired by the atom due to the recoil effect is small compared with its interaction energy with the optical fields, and therefore there is no significant variation of the x-(y-) velocity of the atoms as they interact with the coupling fields. Hence we can apply the Raman-Nath approximation and ignore the kinetic energy of the atom in the Hamiltonian. Under the electric-dipole and rotating-wave approximations, with the assumption of =1, the resulting interaction Hamiltonian for the system reads
Hint=Ω1(x,y)ei(Δ1+δ)t|01|+Ω2ei(Δ2δ)t|02|+Ωceiϕei(Δ2Δ12δ)t|12|+k[gk1ei(Δk+δ)tbk|j1|+gk2ei(Δkδ)tbk|j2|]+H.c.,
(1)
where δ=ω21/2 is half the energy of the transition |1|2.Δ1=ω1(ω10+ω20)/2 and Δ2=ω2(ω10+ω20)/2 represent the frequency detunings between the coupling lasers and the average atomic transition frequency (ω10+ω20)/2. Δk=ωk(ω1j+ω2j)/2 is the detuning of the spontaneously emitted photon with frequency ωk. bk is the annihilation operator for the k-th vacuum mode. gk1 and gk2 denote the coupling constants between the vacuum field and the corresponding atomic transitions.

The atom-field state vector of our considered system at time t, whose evolution obeys the Schrödinger equation, can be written as
|ψ(t)=dxdyf(x,y)|x,y{[a0(t)|0+a1(t)ei(Δ1+δ)t|1+a2(t)ei(Δ2δ)t|1]|{0}+kajk(t)|j|1k},
(2)
where the probability amplitude ai(t) (i=0,1,2) represents the state of atom at time t when there is no spontaneously emitted photon, ajk(t) is the probability amplitude that the atom is in level |j with one photon emitted spontaneously in the k-th vacuum mode, and f(x,y) is the center-of-mass wave function of the atom. In the following calculations, f(x,y) is assumed nearly constant over many wavelengths of the standing-wave fields, and it remains unchanged even after the interaction with the driving fields.

When we have detected at time t a spontaneously emitted photon in the vacuum mode of wave vector k, the atom is in the state |j and the state vector of the system, after making appropriate projection over |ψ(t), is reduced to
|ψj,1k=Nj,1k|ψ(t)=Ndxdyf(x,y)ajk(t)|x|y,
(3)
where N is a normalization factor. Hence the conditional position probability distribution, i.e. the probability of finding the atom in the (x,y) position is

W(x,y;t|j,1k)=|N|2|x|y|ψj,1k|2=|N|2|f(x,y)|2|ajk(t)|2.
(4)

As is well known, the spontaneous emission spectrum S(Δk) is proportional to |ajk(t)|2. Thus, the spontaneous emission from the atom can be used to characterize the conditional position probability distribution W(x,y;t|j,1k). In this scheme, the localization of the atom is conditioned on the detection of the spontaneously emitted photon. In fact, due to the standing-wave fields, the light-atom interaction is spatially dependent in the xy plane, and therefore the frequency of the spontaneously emission carries the position information of the atom.

The probability amplitude ajk(t) can be obtained by solving the Schrödinger wave equation with the interaction Hamiltonian [Eq. (1)] and the atom-field state vector [Eq. (2)]. With the Weisskopf-Wigner theory, the dynamical equations for the atomic probability amplitudes are given by
a˙0(t)=iΩ1(x,y)a1(t)iΩ2a2(t),
(5a)
a˙1(t)=[i(Δ1+δ)Γ12]a1(t)iΩ1(x,y)a0(t)iΩceiϕa2(t),
(5b)
a˙2(t)=[i(Δ2δ)Γ22]a2(t)iΩ2a0(t)iΩceiϕa1(t),
(5c)
a˙jk(t)=igk1ei(ΔkΔ1)ta1(t)igk2ei(ΔkΔ2)ta2(t),
(5d)
where Γi=2π|gik|2D(ωk) (i=1,2) represents the spontaneous decay rate from the excited level |i to the ground level |j with D(ωk) being the density of mode at frequency ωk in the vacuum.

By utilizing the Laplace transform method and the final-value theorem, we obtain ajk in the long-time limit as
ajk(t)=igk1a˜1[s=i(ΔkΔ1)]igk2a˜2[s=i(ΔkΔ2)],
(6)
where a˜i(s) (i=1,2) is the Laplace transform of ai(t). With resonant coupling fields, i.e. Δ1=δ and Δ2=δ, the position-dependent spontaneous emission spectrum is given by
S(Δk;x,y)Γ1|a˜1[s=i(Δk+δ)]|2+Γ2|a2[s=i(Δkδ)]|2,
(7a)
where
a˜1[s=i(Δk+δ)]=Ω1(x,y)(Δk+δ+iΓ22)+Ω2ΩceiϕA+(Δk),
(7b)
a˜2[s=i(Δkδ)]=Ω2(Δkδ+iΓ12)+Ω1(x,y)ΩceiϕA(Δk),
(7c)
and

A±(Δk)=(Δk±δ)(Δk±δ+iΓ12)(Δk±δ+iΓ22)Ω12(x,y)(Δk±δ+iΓ22)Ω22(Δk±δ+iΓ12)Ωc2(Δk±δ)2Ω1(x,y)Ω2Ωccosϕ.
(7d)

3. Numerical results and discussions

As can be seen from Eq. (7), since the spontaneous emission spectrum S(Δk) depends on sin(kx) and sin(ky) from Ω1(x,y), it is in principle possible to extract the 2D position information of the atom as it passes through the orthogonal standing-waves via frequency measurement of the spontaneously emitted photon. The probable positions of the atom are then given by those values of (x,y) where S(Δk) exhibits maxima. However, the form of S(Δk), which can show the position probability distribution clearly, is rather complicated. Therefore, we only present numerical results and discuss various conditions for the precise atom localization.

Due to the closed loop configuration of the coherently driven transitions |0Ω1|1Ωceiϕ|2Ω2|0, Autler-Townes splittings, i.e. dressed states of the atom, rely on the relative phase of the three fields. Consequently, the spontaneous emission is phase dependent as well. In the following discussions, we consider three cases according to different phase: ϕ=π/2, ϕ=0 and ϕ=π. In the calculations, we set Γ1=Γ2=γ and all the other parameters are scaled by γ.

According to the results, we can see that the conditional position probability distribution indicates a strong correlation between the detuning of the spontaneously emitted photon and the position of the atom. As a consequence, the measurement of a certain frequency leads to sub-wavelength or even sub-half-wavelength localization of the atom. To achieve sub-half-wavelength atom localization, the collective phase of the closed loop plays an important role. In order to explain the above results, we plot the spontaneous emission spectrum S(Δk) versus the Rabi frequency Ω1 in Fig. 4
Fig. 4 The spontaneous emission S(Δk) as a function of Ω1. In (a) ϕ=π/2 and (b) ϕ=0. (c) and (d) are the density plots of (a) and (b). Other parameters are the same as Fig. 2.
. In the case of ϕ=π/2, when we have detected one photon with detuning Δk (e.g. Δk=3.5), the corresponding Rabi frequency of the field E1 has two values ±Ω1 [see Fig. 4(a) and its density plot Fig. 4(b)]. That is to say, the potential locations of the atom have equal probability in quadrant I (II) and quadrant III (IV). However, if the phase is ϕ=0, one detuning Δk corresponds to only one Rabi frequency Ω1 [see Fig. 4(c) and its density plot Fig. 4(d)]. Moreover, due to the spatial interference of the two standing-waves, the Rabi frequencies Ω1(x,y) in the four quadrants have different values. Therefore, the uncertainty in a particular position measurement of the atom is reduced. Especially, for some conditions, the probability can be doubled. The case of ϕ=π is the same with ϕ=0. Consequently, the atom is localized in λ1/2×λ1/2 domain and sub-half-wavelength atom localization can be achieved.

Finally, we investigate the influence of Rabi frequencies of the coupling fields on the atom localization. Figures 5(a)
Fig. 5 The spontaneous emission S(Δk;x,y) (in arbitrary unit) which directly describes the conditional position probability distribution as a function of (kx,ky) in dependence on the detuning of the spontaneously emitted photon. Parameters are Δk=4, Ω10=3, Ω2=6, Ωc=6, and ϕ=0 (a); ϕ=π (b). Other parameters are the same as Fig. 2.
and 5(b) show the pattern of position probability distribution with Ω10=3, Ω2=6 and Ω3=6 in the cases of ϕ=0 and ϕ=π, respectively. Compared with Fig. 3(d), the localization peak structure is greatly narrowed as the Rabi frequencies increased. Therefore, we can obtain a better spatial resolution in position measurement of the atom, which is necessary in the application of atom lithography.

4. Conclusions

Acknowledgments

This work is supported by the National Basic Research Program of China (973 Program) (under Grant No. 2007CB310405) and by the National Natural Science Foundation of China (under Grant Nos. 61176084, 10834015, and 11174282).

References and links

1.

W. D. Phillips, “Nobel lecture: laser cooling and trapping of neutral atoms,” Rev. Mod. Phys. 70(3), 721–741 (1998). [CrossRef]

2.

K. S. Johnson, J. H. Thywissen, N. 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(5369), 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(2), 023805 (2003). [CrossRef]

4.

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

5.

R. Quadt, M. Collett, and D. F. Walls, “Measurement of atomic motion in a standing light field by homodyne detection,” Phys. Rev. Lett. 74(3), 351–354 (1995). [CrossRef] [PubMed]

6.

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

7.

S. Kunze, K. Dieckmann, and G. Rempe, “Diffraction of atoms from a measurement induced grating,” Phys. Rev. Lett. 78(11), 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(4-6), 409–416 (2000). [CrossRef]

9.

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

10.

F. Ghafoor, S. Qamar, and M. S. Zubairy, “Atom localization via phase and amplitude control of the driving field,” Phys. Rev. A 65(4), 043819 (2002). [CrossRef]

11.

J. Xu and X. M. Hu, “Sub-half-wavelength atom localization via bichromatic phase control of spontaneous emission,” Phys. Lett. A 366(3), 276–281 (2007). [CrossRef]

12.

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]

13.

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]

14.

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

15.

G. S. Agarwal and K. T. Kapale, “Subwavelength atom localization via coherent population trapping,” J. Phys. B 39(17), 3437–3446 (2006). [CrossRef]

16.

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(2), 025801 (2006). [CrossRef]

17.

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(10), 2180–2184 (2006). [CrossRef]

18.

J. Xu and X. M. Hu, “Sub-half-wavelength atom localization via phase control of a pair of bichromatic fields,” Phys. Rev. A 76(1), 013830 (2007). [CrossRef]

19.

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]

20.

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]

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(6), 805–810 (2009). [CrossRef]

22.

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]

23.

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(1), 10–17 (2011). [CrossRef]

24.

C. L. Ding, J. H. Li, Z. M. Zhang, and X. 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]

25.

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]

26.

C. L. Ding, J. H. Li, X. X. Yang, Z. M. Zhang, 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]

27.

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(8), 085508 (2008). [CrossRef]

28.

F. Ghafoor, S. Y. Zhu, and M. S. Zubairy, “Amplitude and phase control of spontaneous emission,” Phys. Rev. A 62(1), 013811 (2000). [CrossRef]

OCIS Codes
(020.1670) Atomic and molecular physics : Coherent optical effects
(270.0270) Quantum optics : Quantum optics

ToC Category:
Atomic and Molecular Physics

History
Original Manuscript: November 3, 2011
Revised Manuscript: November 17, 2011
Manuscript Accepted: November 17, 2011
Published: December 2, 2011

Citation
Ren-Gang Wan and Tong-Yi Zhang, "Two-dimensional sub-half-wavelength atom localization via controlled spontaneous emission," Opt. Express 19, 25823-25832 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-25823


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References

  1. W. D. Phillips, “Nobel lecture: laser cooling and trapping of neutral atoms,” Rev. Mod. Phys.70(3), 721–741 (1998). [CrossRef]
  2. K. S. Johnson, J. H. Thywissen, N. 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,” Science280(5369), 1583–1586 (1998). [CrossRef] [PubMed]
  3. K. T. Kapale, S. Qamar, and M. S. Zubairy, “Spectroscopic measurement of an atomic wave function,” Phys. Rev. A67(2), 023805 (2003). [CrossRef]
  4. J. Mompart, V. Ahufinger, and G. Birkl, “Coherent pattern of matter waves with subwavelength localization,” Phys. Rev. A79(5), 053638 (2009). [CrossRef]
  5. R. Quadt, M. Collett, and D. F. Walls, “Measurement of atomic motion in a standing light field by homodyne detection,” Phys. Rev. Lett.74(3), 351–354 (1995). [CrossRef] [PubMed]
  6. S. Kunze, G. Rempe, and M. Wilkens, “Atomic-position measurement via internal-state encoding,” Europhys. Lett.27(2), 115–121 (1994). [CrossRef]
  7. S. Kunze, K. Dieckmann, and G. Rempe, “Diffraction of atoms from a measurement induced grating,” Phys. Rev. Lett.78(11), 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(4-6), 409–416 (2000). [CrossRef]
  9. S. Qamar, S. Y. Zhu, and M. S. Zubairy, “Atom localization via resonance fluorescence,” Phys. Rev. A61(6), 063806 (2000). [CrossRef]
  10. F. Ghafoor, S. Qamar, and M. S. Zubairy, “Atom localization via phase and amplitude control of the driving field,” Phys. Rev. A65(4), 043819 (2002). [CrossRef]
  11. J. Xu and X. M. Hu, “Sub-half-wavelength atom localization via bichromatic phase control of spontaneous emission,” Phys. Lett. A366(3), 276–281 (2007). [CrossRef]
  12. 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. A72(1), 013820 (2005). [CrossRef]
  13. K. T. Kapale and M. S. Zubairy, “Subwavelength atom localization via amplitude and phase control of the absorption spectrum II,” Phys. Rev. A73(2), 023813 (2006). [CrossRef]
  14. E. Paspalakis and P. L. Knight, “Localizing an atom via quantum interference,” Phys. Rev. A63(6), 065802 (2001). [CrossRef]
  15. G. S. Agarwal and K. T. Kapale, “Subwavelength atom localization via coherent population trapping,” J. Phys. B39(17), 3437–3446 (2006). [CrossRef]
  16. 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. A73(2), 025801 (2006). [CrossRef]
  17. 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. B23(10), 2180–2184 (2006). [CrossRef]
  18. J. Xu and X. M. Hu, “Sub-half-wavelength atom localization via phase control of a pair of bichromatic fields,” Phys. Rev. A76(1), 013830 (2007). [CrossRef]
  19. S. Qamar, A. Mehmood, and S. Qamar, “Subwavelength atom localization via coherent manipulation of the Raman gain process,” Phys. Rev. A79(3), 033848 (2009). [CrossRef]
  20. V. Ivanov and Y. Rozhdestvensky, “Two-dimensional atom localization in a four-level tripod system in laser fields,” Phys. Rev. A81(3), 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(6), 805–810 (2009). [CrossRef]
  22. 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]
  23. 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. B28(1), 10–17 (2011). [CrossRef]
  24. C. L. Ding, J. H. Li, Z. M. Zhang, and X. X. Yang, “Two-dimensional atom localization via spontaneous emission in a coherently driven five-level M-type atomic system,” Phys. Rev. A83(6), 063834 (2011). [CrossRef]
  25. 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. B28(4), 622–628 (2011). [CrossRef]
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