## High-precision atom localization via controllable spontaneous emission in a cycle-configuration atomic system |

Optics Express, Vol. 20, Issue 7, pp. 7870-7885 (2012)

http://dx.doi.org/10.1364/OE.20.007870

Acrobat PDF (1237 KB)

### 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

## 1. Introduction

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

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

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

4. 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**, 1583–1586 (1998). [CrossRef] [PubMed]

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

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

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

11. J. Xu and 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]

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

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

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

*et al.*[17

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

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

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

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

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

22. J. E Thomas, “Uncertainty-limited position measurement of moving atoms using optical fields,” Opt. Lett. **14**, 1186–1188 (1989). [CrossRef] [PubMed]

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

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

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

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

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

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

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

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

35. Y. Wu, J. Saldana, and 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]

38. Y. P. Zhang, A. W. Brown, and 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]

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

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

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

46. 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**, 622–628 (2011). [CrossRef]

*et al.*[44

44. R. G. Wan, J. Kou, L. Jiang, Y. Jiang, and J. Y. Gao, “Two-dimensional atom localization via controlled spontaneous emission from a driven tripod system,” J. Opt. Soc. Am. B **28**, 10–17 (2011). [CrossRef]

## 2. Theoretical model and basic formula

*is used to resonantly couple the two hyperfine levels |1〉 and |2〉 through an allowed magnetic dipole transition. The excited level |3〉 is simultaneously coupled to the ground levels |1〉 and |2〉 by two coherent laser fields with Rabi frequencies*

_{m}*G*

_{1}(

*x, y*) and

*G*

_{2}(

*x, y*), respectively. Here, we consider two cases for the atom interacts with the standing-wave laser fields. The first case is that

*G*

_{1}(

*x, y*) and

*G*

_{2}(

*x, y*) correspond respectively to the two orthogonal standing-wave fields that couple the different atomic transitions, i.e.,

*G*

_{1}(

*x, y*) = Ω

_{1}sin(

*k*

_{1}

*x*) and

*G*

_{2}(

*x, y*) = Ω

_{2}sin(

*k*

_{2}

*y*) with

*k*

_{1}=

*ω*

_{1}/

*c*and

*k*

_{2}=

*ω*

_{2}/

*c*being the wave vectors of the two laser fields. The second case is that

*G*

_{1}(

*x, y*) corresponds to the combination of two orthogonal standing-wave fields with the same frequency that drive simultaneously the transition |1〉 ↔ |3〉, while

*G*

_{2}(

*x, y*) corresponds to a traveling-wave field, that is,

*G*

_{1}(

*x, y*) = Ω

_{1}[sin(

*k*

_{1}

*x*)+ sin(

*k*

_{1}

*y*)] and

*G*

_{2}(

*x, y*) = Ω

_{2}. An atom moves along the

*z*direction and passes through the intersectant region of the two orthogonal standing-wave fields in the

*x*–

*y*plane. As a result, the interaction between the atom and the standing-wave fields is spatial dependent on the

*x*–

*y*plane. Here we assume that the center-of-mass position of the atom along the directions of the standing-wave fields is nearly constant and we can neglect the kinetic part of the atom in the Hamiltonian by applying the Raman-Nath approximation [47]. Under these conditions, the resulting interaction Hamiltonian which describes the dynamics of this system in the rotating-wave approximation (RWA) and the electric dipole approximation (EDA) can be written in the following form (taking

*h̄*= 1) where the quantities Δ

_{1}=

*ω*

_{1}–

*E*

_{31}

*/h*̄ and Δ

_{2}=

*ω*

_{2}–

*E*

_{32}

*/h*̄ stand for the frequency detunings of the coherent laser fields from the corresponding atomic resonance frequencies. Here Ω

*is one-half Larmor frequency for the relevant driven transition, i.e., Ω*

_{m}*=*

_{m}*μ*

_{12}

*B*/(2

_{m}*h*̄), with

*B*being the amplitude of the microwave-driven field and

_{m}*μ*

_{12}=

*μ⃗*

_{12}

*·e⃗*(

_{L}*e⃗*is the unit polarization vector of the corresponding laser field) denoting the dipole matrix element for the transition |1〉 ↔ |2〉.

_{L}*b̂*and

_{k}*k*th vacuum mode with frequency

*ω*. The coefficient

_{k}*g*represents the coupling between the vacuum mode

_{k}*k*and the atomic transition |3〉 ↔ |0〉,

*δ*=

_{k}*ω*–

_{k}*ω*

_{30}is the corresponding frequency detuning. In the following calculations, we set Ω

_{1}and Ω

_{2}as real parameters, while Ω

*as a complex parameter, i.e., Ω*

_{m}*= |Ω*

_{m}*|*

_{m}*e*, here

^{iφ}*φ*is the phase of the microwave-driven field and can also be called the relative phase. It is remarkable that there exist two possible transition pathways from level |1〉 to level |3〉, i.e., the direct one

*φ*on the spontaneous emission spectra in such a four-level atomic system with a closed-loop structure can be explained from quantum interference caused by these two excitation decay channels. As a consequence, we can investigate the behavior of 2D atom localization by modulating the relative phase

*φ*, which can also be discussed in the following section.

*t*can be expressed as

*A*

_{j,0k}(

*x*,

*y*;

*t*) (

*j*= 1 – 3) and

*A*

_{0,1k}(

*x*,

*y*;

*t*) give the probability amplitude to find the atom at time

*t*. |

*j*,0

*〉 denotes the atom in the level |*

_{k}*j*〉 with no photons present and |0,1

*〉 represents the atom in its ground level |0〉 with a single photon in the*

_{k}*k*th vacuum mode. Besides,

*f*(

*x*,

*y*) is the center-of-mass wave function of the atom.

*x*–

*y*plane due to the spatial position-dependent interaction between atom and standing-wave fields. Therefore, the location of the atom can be determined by measuring the frequency of the spontaneously emitted photon. When we have detected a spontaneously emitted photon at time

*t*in the vacuum mode of wave vector

*k*, the atom is in its internal level |0〉 and the state vector of the system, by employing the following transformation over Ψ(

*t*), is transformed into where

*𝒩*is a normalization factor. Hence, the conditional position probability distribution, that is, the probability of finding the atom in the (

*x*,

*y*) position at time

*t*is which can be reduced to determine the probability amplitude

*A*

_{0,1k}(

*x*,

*y*;

*t*).

*A*

_{0,1k}by substituting the interaction Hamiltonian [Eq. (1)] and the atomic wave function of our system [Eq. (2)] into the time-dependent Schrödinger wave equation

*i∂*|Ψ(

*t*)〉/

*∂t*=

*H*|Ψ(

_{I}*t*)〉, and we can obtain the coupled equations of motion for the time evolution of the atomic probability amplitudes where Γ

_{0}= 2

*π*|

*g*|

_{k}^{2}

*D*(

*ω*) is the spontaneous decay rate from level |3〉 to level |0〉, with

_{k}*D*(

*ω*) being the density of states (DOS) at frequency

_{k}*ω*in the free space. Our calculations show that the decays of the excited level |3〉 to the ground levels |1〉 and |2〉 cannot affect the probability of finding the atom in the intersectant region of the standing-wave fields, but only slightly reduce the spatial resolution of the atom localization, and hence the two decay rates can be neglected here.

_{k}*A*

_{0,1k}in the long time limit can be obtained as here

*Ã*

_{3,0k}(

*s*) is the Laplace transform of

*A*

_{3,0k}(

*t*) with

*s*= −

*iδ*.

_{k}*Ã*

_{3,0k}(

*s*) as where

*ω*in the vacuum mode

_{k}*k*is then given by

*f*(

*x*,

*y*) is assumed to be nearly constant over many wavelengths of the standing-wave fields in the

*x*–

*y*plane, the conditional position probability distribution

*P*(

*x*,

*y*;

*t*→ ∞|0,1

*) is determined by the last term in Eq. (13). Therefore, we can define the filter function as*

_{k}## 3. Results and discussion

48. 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]

49. C. L. Wang, Z. H. Kang, S. C. Tian, and J. H. Wu, “Control of spontaneous emission from a micro-wave driven atomic system,” Opt. Express **20**, 3509–3518 (2012), http://www.opticsinfobase.org/abstract.cfm?URI=oe-20-4-3509. [CrossRef] [PubMed]

*k*

_{1}

*x*and

*k*

_{2}

*y*(

*k*

_{1}

*y*) when the filter function

*F*(

*x, y*) exhibits maxima. In the following discussion, we consider two different situations for the position measurement of the atom: (i) the two orthogonal standing-wave fields are respectively used to couple the different atomic transitions, i.e.,

*G*

_{1}(

*x, y*) = Ω

_{1}sin(

*k*

_{1}

*x*) and

*G*

_{2}(

*x, y*) = Ω

_{2}sin(

*k*

_{2}

*y*); (ii) the two standing-wave fields with the same frequency are applied to drive the same atomic transition, i.e.,

*G*

_{1}(

*x, y*) = Ω

_{1}[sin(

*k*

_{1}

*x*) + sin(

*k*

_{1}

*y*)] and

*G*

_{2}(

*x, y*) = Ω

_{2}. The spontaneous decay rate of the level |3〉 to level |0〉 is set as Γ

_{0}= 2

*γ*. All the parameters used in this paper are in units of

*γ*, which should be in the order of MHz for rubidium atoms.

### 3.1. Two standing-wave fields drive different atomic transitions

*φ*. In Fig. 2, we plot the filter function

*F*(

*x*,

*y*) versus the normalized positions (

*k*

_{1}

*x, k*

_{2}

*y*) by measuring the frequency of the spontaneously emitted photon under the condition of

*A*

_{3,0k}(0) = 1 when the two orthogonal standing-wave fields are both tuned to the resonant interaction with their respective atomic transition and the relative phase

*φ*= 0. Figure 3 shows the corresponding density plots of the filter function

*F*(

*x*,

*y*) in the

*x*–

*y*plane. It can be seen from Fig. 2(a) that the peak maxima of the filter function exhibit a latticelike pattern when we detect the detuning of the spontaneously emitted photon is

*δ*= 8

_{k}*γ*, and the atom is localized at the second and fourth quadrants in the

*x*–

*y*plane [see Fig. 3(a)]. As the detuning

*δ*increases, we observe that the localization peaks occur at

_{k}*k*

_{1}

*x*+

*k*

_{2}

*y*= 2

*mπ*or

*k*

_{1}

*x*–

*k*

_{2}

*y*= (2

*n*+ 1)

*π*(

*m,n*are integers), which indicates that the atom is distributed on the diagonal in the second and fourth quadrants [see Figs. 2(b) and 3(b)]. And these localization peaks become very sharp due to constructive interference of quantum pathways. When the detuning is detected at

*δ*= 16

_{k}*γ*, the conditional position probability distribution of the atom is contrary to that shown in Figs. 2(a) and 2(b), the maxima of the filter function in Fig. 2(c) are situated in the first and third quadrants with a craterlike pattern, and the atom is localized at the circular edges of the craters [see Fig. 3(c)]. Moreover, when the frequency detuning of the spontaneously emitted photon is measured at an appropriate value [e.g.,

*δ*= 19.3

_{k}*γ*in Fig. 2(d)], the resulting localization peaks display a spikelike pattern, which shows that the spatial resolution is greatly improved [see Fig. 3(d)]. As a result, we can achieve high-precision and high-resolution 2D atom localization by measuring the frequency of the spontaneously emitted photon under three-photon resonance conditions.

*x*–

*y*plane.

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

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

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

*P*(

*x*,

*y*;

*t*|0,1

*) carries the information about the atomic position. In Fig. 2, we plot the filter function*

_{k}*F*(

*x*,

*y*) for four cases of frequencies of the emitted photon

*ω*recorded during measuring time

_{k}*t*. It can be observed that if the detector records a larger frequency [e.g.,

*ω*=

_{k}*ω*

_{30}+ 19.3

*γ*in Fig. 2(d)], two sharp probability distributions centered at the antinodes of the standing-wave fields in the first and third quadrants are expected. The appearance of two steep peaks originates from the combined effects of the microwave coupling field and spontaneously generated coherence between dressed levels, which has been demonstrated in Ref. [49

49. C. L. Wang, Z. H. Kang, S. C. Tian, and J. H. Wu, “Control of spontaneous emission from a micro-wave driven atomic system,” Opt. Express **20**, 3509–3518 (2012), http://www.opticsinfobase.org/abstract.cfm?URI=oe-20-4-3509. [CrossRef] [PubMed]

*F*(

*x*,

*y*) versus the normalized positions (

*k*

_{1}

*x, k*

_{2}

*y*) by monitoring the spontaneously emitted photon under the condition of

*φ*=

*π*, as shown in Fig. 4. The corresponding density plots are illustrated in Fig 5. When the spontaneously emitted photon with detuning

*δ*= 8.5

_{k}*γ*is detected, the corresponding filter function in Figs. 4(a) and 5(a) is distributed in the first and third quadrants with a craterlike pattern. We find that only when the frequency of the spontaneously emitted photon is detected at an appropriate value [see Figs. 4(b) and 5(b)], that is, the quantum interference between |1〉 → |2〉 → |3〉 → |0〉 and |2〉 → |1〉 → |3〉 → |0〉 is so strong that we can observe two localization peaks with a spikelike pattern in the

*x*–

*y*plane. Under this situation, the high-spatial-resolution and high-precision localization of the atom can be achieved, and the probability of finding the atom within one period is 50%. However, when the detuning is increased to

*δ*= 17.2

_{k}*γ*, the peak maxima of the filter function in Figs. 4(c) and 5(c) are mostly distributed in the second and fourth quadrants with a lotus-like structure and little in the first and third quadrants. But, it is accompanied with a lower localization precision. With further increase of the detuning of the spontaneously emitted photon, it can be seen from Figs. 4(d) and 5(d) that the localization peaks in the first and third quadrants are completely vanished due to the destructive quantum interference in such a four-level atomic system with a closed-loop configuration. As can be seen from these figures, large detunings of the standing-wave fields do not alter qualitatively features revealed in our paper. That is to say, this kind of mismatch can not change the probability of finding the atom in the subwavelength regime and the precision of the atom localization. For simplicity, but without loss of generality, we focus our discussion on the resonant case in the following subsection.

### 3.2. Two standing-wave fields couple one atomic transition

*x*–

*y*plane, thus realizing the 2D atom localization, indeed. Although the values of the system parameters are the same as in Fig. 2, the behavior of the atom localization in Fig. 6 is different from those observed in the previous subsection. Similarly, the corresponding density plot of the filter function

*F*(

*x*,

*y*) is given in Fig. 7. When the detuning of the spontaneously emitted photon is detected at

*δ*= 7

_{k}*γ*, the peak maxima of the filter function are mostly distributed in the third quadrant with a bicycli-clike pattern, and little in the second and fourth quadrants, as can be seen from Figs. 6(a) and 7(a). For the case that

*δ*= 8.9

_{k}*γ*, the localization peaks with a bicycliclike pattern in the third quadrant evolve into a craterlike pattern, and the localization peaks distributed in the second and fourth quadrants become very sharp [see Figs. 6(b) and 7(b)]. This means that the spatial resolution of atomic position is greatly improved. When the detuning of the spontaneously emitted photon is measured at

*δ*= 13.5

_{k}*γ*, it can be seen from Figs. 6(c) and 7(c) that the peak maxima of

*F*(

*x*,

*y*) are situated on the diagonal in the second and fourth quadrants, and display a cross-shaped structure, while the localization peak in the third quadrant has a craterlike pattern. For a suitable detuning of the emitted photon, e.g.,

*δ*= 20

_{k}*γ*, the filter function in this case exhibits different patterns in one period, that is, the localization peaks distributed in the first quadrant has a craterlike pattern, while the localization peaks in the third quadrant display a spikelike pattern, as shown in Figs. 6(d) and 7(d). In particular, it can also be seen that, the localization peaks in the third quadrant have a higher precision and resolution than that shown in the first quadrant, which originated from the constructive quantum interference induced by the microwave, standing-wave and traveling-wave fields.

*x*–

*y*plane by modulating the system parameters. Under the conditions that the two orthogonal standing-wave fields couple the same atomic transition and the atom is initially prepared in a coherent superposition of two ground levels, i.e.,

*F*(

*x*,

*y*) and the corresponding density plots, respectively, as varying the combination values of the detuning

*δ*and the relative phase

_{k}*φ*. More precisely, in the case of

*φ*=

*π*/2 and

*δ*= 14.5

_{k}*γ*, the maxima of the filter function

*F*(

*x*,

*y*) are distributed in the four quadrants but with different probability, which shows that the spatial resolution is very poor [see Figs. 8(a) and 9(a)]. While, when the relative phase is tuned to

*φ*=

*π*/4 and the detuning of the spontaneously emitted photon is detected at

*δ*= 19.5

_{k}*γ*, the filter function exhibits a craterlike pattern as shown in Fig. 8(b) and its maxima are situated in the third quadrant [see Fig. 9(b)]. Furthermore, if the relative phase

*φ*is increased by a factor of

*π*/2 compared with Fig. 8(b) and the detuning of the emitted photon is increased to

*δ*= 26

_{k}*γ*, the atom is completely localized in the third quadrant, and the localization peak becomes very sharp, with a spikelike pattern [see Figs. 8(c) and 9(c)]. Therefore, high-precision and high-resolution of 2D atom localization is realized. In addition, the application of microwave field leads to an improvement of the probability of finding the atom at a particular position by a factor of up to 4 or 2 compared to the previous proposed schemes without the microwave-driven field [42

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

46. 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**, 622–628 (2011). [CrossRef]

*φ*=

*π*and the spontaneously emitted photon with detuning

*δ*= 6.75

_{k}*γ*is monitored, the peak maxima of the filter function in Fig. 8(d) are confined within the first quadrant with a spikelike pattern. In this case, the atom is localized at the position (

*k*

_{1}

*x, k*

_{1}

*y*) = (

*π*/2,

*π*/2) with very high spatial resolution during one period of the standing-wave fields [see Fig. 9(d)]. It is thereby possible to rigorously determine the position of the atom is localized when it passes through the standing-wave fields. It should be noted that a sharp single localization peak can be obtained in a subwavelength region [see, for instance, Figs. 8(c) and 8(d)]. However, because of the mechanical action of the standing-wave fields on the atom, it is inevitably accompanied by a wide momentum spread. Yet we find that the position-momentum uncertainty results from the mechanical action does not affect the precision position measurement of an atom such as that proposed by Storey

*et al.*[7

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

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

51. C. L. Ding, J. H. Li, X. 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**, 043840 (2011). [CrossRef]

52. J. H. Li, R. Yu, M. Liu, C. L. Ding, and 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]

53. R. G. Wan and T. Y. Zhang, “Two-dimensional sub-half-wavelength atom localization via controlled spontaneous emission,” Opt. Express **19**, 25823–25832 (2012), http://www.opticsinfobase.org/abstract.cfm?URI=oe-19-25-25823. [CrossRef] [PubMed]

*et al.*[54

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

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

^{87}Rb atoms [47,48

48. 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]

55. D. A. Steck, Rubidium 87 D line data, available online at http://steck.us/alkalidata.

*D*

_{2}line structure. The designated states can be chosen as follows: |0〉 = |5

*S*

_{1/2},

*F*= 2,

*m*= 2〉, |1〉 = |5

_{F}*S*

_{1/2},

*F*= 1,

*m*= 0〉, |2〉 = |5

_{F}*S*

_{1/2}

*, F*= 2,

*m*= 0〉, and |3〉 = |5

_{F}*P*

_{3/2},

*F*= 2,

*m*= 1〉, respectively. The spontaneous decay rate of the state |3〉 = |5

_{F}*P*

_{3/2},

*F*= 2,

*m*= 1〉 in this system is 6 MHz. In practical experiments, the transition between the states 5

_{F}*S*

_{1/2}and 5

*P*

_{3/2}is driven by standing-wave or traveling-wave laser fields at a wavelength of 780.2 nm. The hyperfine transition |5

*S*

_{1/2},

*F*= 1,

*m*= 0〉 ↔ |5

_{F}*S*

_{1/2},

*F*= 2,

*m*= 0〉 is resonantly coupled by a microwave field with frequency around 6.8 GHz, which can greatly enhance the localization precision and improve the spatial resolution. These fields can be obtained from the external cavity diode lasers [56

_{F}56. J. Vanier, A. Godone, and F. Levi, “Coherent population trapping in cesium: Dark lines and coherent microwave emission,” Phys. Rev. A **58**, 2345–2358 (1998), and references therein. [CrossRef]

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

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

46. 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**, 622–628 (2011). [CrossRef]

48. 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]

49. C. L. Wang, Z. H. Kang, S. C. Tian, and J. H. Wu, “Control of spontaneous emission from a micro-wave driven atomic system,” Opt. Express **20**, 3509–3518 (2012), http://www.opticsinfobase.org/abstract.cfm?URI=oe-20-4-3509. [CrossRef] [PubMed]

52. J. H. Li, R. Yu, M. Liu, C. L. Ding, and 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]

53. R. G. Wan and T. Y. Zhang, “Two-dimensional sub-half-wavelength atom localization via controlled spontaneous emission,” Opt. Express **19**, 25823–25832 (2012), http://www.opticsinfobase.org/abstract.cfm?URI=oe-19-25-25823. [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgments

## References and links

1. | W. D Phillips, “Nobel lecture: Laser cooling and trapping of neutral atoms,” Rev. Mod. Phys. |

2. | G. P Collins, “Experimenters produce new Bose-Einstein Condensate(s) and possible puzzles for theorists,” Phys. Today |

3. | Y. Wu, X. X. Yang, and C. P. Sun, “Systematic method to study the general structure of Bose-Einstein condensates with arbitrary spin,” Phys. Rev. A |

4. | 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 |

5. | 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. |

6. | K. T. Kapale, S. Qamar, and M. S. Zubairy, “Spectroscopic measurement of an atomic wave function,” Phys. Rev. A |

7. | P. Storey, M. Collett, and D. Walls, “Measurement-induced diffraction and interference of atoms,” Phys. Rev. Lett. |

8. | P. Storey, M. Collett, and D. Walls, “Atomic-position resolution by quadrature-field measurement,” Phys. Rev. A |

9. | R. Quadt, M. Collett, and D. F. Walls, “Measurement of atomic motion in a standing light field by homodyne detection,” Phys. Rev. Lett. |

10. | F. L. Kien, G. Rempe, W. P. Schleich, and M. S. Zubairy, “Atom localization via Ramsey interferometry: A coherent cavity field provides a better resolution,” Phys. Rev. A |

11. | J. Xu and X. M. Hu, “Sub-half-wavelength localization of an atom via trichromatic phase control,” J. Phys. B: At. Mol. Opt. Phys. |

12. | E. Paspalakis and P. L. Knight, “Localizing an atom via quantum interference,” Phys. Rev. A |

13. | S. Qamar, S. Y. Zhu, and M. S. Zubairy, “Atom localization via resonance fluorescence,” Phys. Rev. A |

14. | M. Macovei, J. Evers, C. H. Keitel, and M. S. Zubairy, “Localization of atomic ensembles via superfluorescence,” Phys. Rev. A |

15. | 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 |

16. | K. T. Kapale and M. S. Zubairy, “Subwavelength atom localization via amplitude and phase control of the absorption spectrum. II,” Phys. Rev. A |

17. | H. Nha, J. H. Lee, J. S. Chang, and K. An, “Atomic-position localization via dual measurement,” Phys. Rev. A |

18. | S. Qamar, A. Mehmood, and S. Qamar, “Subwavelength atom localization via coherent manipulation of the Raman gain process,” Phys. Rev. A |

19. | 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 |

20. | 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 |

21. | G. S. Agarwal and K. T. Kapale, “Subwavelength atom localization via coherent population trapping,” J. Phys. B: At. Mol. Opt. Phys. |

22. | J. E Thomas, “Uncertainty-limited position measurement of moving atoms using optical fields,” Opt. Lett. |

23. | K. D. Stokes, C. Schnurr, J. R. Gardner, M. Marable, G. R. Welch, and J. E. Thomas, “Precision position measurement of moving atoms using optical fields,” Phys. Rev. Lett. |

24. | J. R. Gardner, M. L. Marable, G. R. Welch, and J. E. Thomas, “Suboptical wavelength position measurement of moving atoms using optical fields,” Phys. Rev. Lett. |

25. | N. A. Proite, Z. J. Simmons, and D. D. Yavuz, “Observation of atomic localization using electromagnetically induced transparency,” Phys. Rev. A |

26. | F. Dell’Anno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. |

27. | A. Serafini, F. Illuminati, M. G. A. Paris, and S. De Siena, “Entanglement and purity of two-mode Gaussian states in noisy channels,” Phys. Rev. A |

28. | F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A |

29. | F. Dell’Anno, S. De Siena, G. Adesso, and F. Illuminati, “Teleportation of squeezing: Optimization using non-Gaussian resources,” Phys. Rev. A |

30. | Y. P. Niu, S. Q. Gong, R. X. Li, Z. Z. Xu, and X. Y. Liang, “Giant Kerr nonlinearity induced by interacting dark resonances,” Opt. Lett. |

31. | Y. Wu and X. X. Yang, “Giant Kerr nonlinearities and solitons in a crystal of molecular magnets,” Appl. Phys. Lett. |

32. | D. D. Yavuz and D. E. Sikes, “Giant Kerr nonlinearities using refractive-index enhancement,” Phys. Rev. A |

33. | Y. Wu, L. L. Wen, and Y. F. Zhu, “Efficient hyper-Raman scattering in resonant coherent media,” Opt. Lett. |

34. | G. Simon, B. Hehlen, E. Courtens, E. Longueteau, and R. Vacher, “Hyper-Raman scattering from vitreous boron oxide: Coherent enhancement of the boson peak,” Phys. Rev. Lett. |

35. | Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A |

36. | Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Controlling four-wave and six-wave mixing processes in multilevel atomic systems,” Appl. Phys. Lett. |

37. | Y. P. Zhang, B. Anderson, A. W. Brown, and M. Xiao, “Competition between two four-wave mixing channels via atomic coherence,” Appl. Phys. Lett. |

38. | Y. P. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. |

39. | E. Paspalakis and P. L. Knight, “Phase control of spontaneous emission,” Phys. Rev. Lett. |

40. | S. Evangelou, V. Yannopapas, and E. Paspalakis, “Modifying free-space spontaneous emission near a plasmonic nanostructure,” Phys. Rev. A |

41. | J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, and J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A |

42. | 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 |

43. | V. Ivanov and Y. Rozhdestvensky, “Two-dimensional atom localization in a four-level tripod system in laser fields,” Phys. Rev. A |

44. | 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 |

45. | 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. |

46. | 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 |

47. | P. Meystre and M. Sargent III, |

48. | J. H Li, “Control of spontaneous emission spectra via an external coherent magnetic field in a cycle-configuration atomic medium,” Eur. Phys. J. D |

49. | C. L. Wang, Z. H. Kang, S. C. Tian, and J. H. Wu, “Control of spontaneous emission from a micro-wave driven atomic system,” Opt. Express |

50. | J. T. Chang, J. Evers, M. O. Scully, and M. S. Zubairy, “Measurement of the separation between atoms beyond diffraction limit,” Phys. Rev. A |

51. | C. L. Ding, J. H. Li, X. 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 |

52. | J. H. Li, R. Yu, M. Liu, C. L. Ding, and 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 |

53. | R. G. Wan and T. Y. Zhang, “Two-dimensional sub-half-wavelength atom localization via controlled spontaneous emission,” Opt. Express |

54. | Z. P. Wang, B. L. Yu, J. Zhu, Z. G. Cao, S. L. Zhen, X. Q. Wu, and F. Xu, “Atom localization via controlled spontaneous emission in a five-level atomic system,” Ann. Phys. (New York) |

55. | D. A. Steck, Rubidium 87 D line data, available online at http://steck.us/alkalidata. |

56. | J. Vanier, A. Godone, and F. Levi, “Coherent population trapping in cesium: Dark lines and coherent microwave emission,” Phys. Rev. A |

**OCIS Codes**

(020.1670) Atomic and molecular physics : Coherent optical effects

(020.5580) Atomic and molecular physics : Quantum electrodynamics

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: December 1, 2011

Revised Manuscript: March 1, 2012

Manuscript Accepted: March 16, 2012

Published: March 21, 2012

**Citation**

Chunling Ding, Jiahua Li, Rong Yu, Xiangying Hao, and Ying Wu, "High-precision atom localization via controllable spontaneous emission in a cycle-configuration atomic system," Opt. Express **20**, 7870-7885 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7870

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### References

- W. D Phillips, “Nobel lecture: Laser cooling and trapping of neutral atoms,” Rev. Mod. Phys. 70, 721–741 (1998). [CrossRef]
- G. P Collins, “Experimenters produce new Bose-Einstein Condensate(s) and possible puzzles for theorists,” Phys. Today 49, 18–21 (1996). [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]
- 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]
- 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]
- K. T. Kapale, S. Qamar, M. S. Zubairy, “Spectroscopic measurement of an atomic wave function,” Phys. Rev. A 67, 023805 (2003). [CrossRef]
- P. Storey, M. Collett, D. Walls, “Measurement-induced diffraction and interference of atoms,” Phys. Rev. Lett. 68, 472–475 (1992). [CrossRef] [PubMed]
- P. Storey, M. Collett, D. Walls, “Atomic-position resolution by quadrature-field measurement,” Phys. Rev. A 47, 405–418 (1993). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- E. Paspalakis, P. L. Knight, “Localizing an atom via quantum interference,” Phys. Rev. A 63, 065802 (2001). [CrossRef]
- S. Qamar, S. Y. Zhu, M. S. Zubairy, “Atom localization via resonance fluorescence,” Phys. Rev. A 61, 063806 (2000). [CrossRef]
- M. Macovei, J. Evers, C. H. Keitel, M. S. Zubairy, “Localization of atomic ensembles via superfluorescence,” Phys. Rev. A 75, 033801 (2007). [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, M. S. Zubairy, “Subwavelength atom localization via amplitude and phase control of the absorption spectrum. II,” Phys. Rev. A 73, 023813 (2006). [CrossRef]
- H. Nha, J. H. Lee, J. S. Chang, K. An, “Atomic-position localization via dual measurement,” Phys. Rev. A 65, 033827 (2002). [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]
- 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]
- J. E Thomas, “Uncertainty-limited position measurement of moving atoms using optical fields,” Opt. Lett. 14, 1186–1188 (1989). [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]
- 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]
- 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]
- 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]
- 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, G. Adesso, F. Illuminati, “Teleportation of squeezing: Optimization using non-Gaussian resources,” Phys. Rev. A 82, 062329 (2010). [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]
- Y. Wu, X. X. Yang, “Giant Kerr nonlinearities and solitons in a crystal of molecular magnets,” Appl. Phys. Lett. 91, 094104 (2007). [CrossRef]
- D. D. Yavuz, D. E. Sikes, “Giant Kerr nonlinearities using refractive-index enhancement,” Phys. Rev. A 81, 035804 (2010). [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]
- 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]
- 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. 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]
- E. Paspalakis, P. L. Knight, “Phase control of spontaneous emission,” Phys. Rev. Lett. 81, 293–296 (1998). [CrossRef]
- S. Evangelou, V. Yannopapas, E. Paspalakis, “Modifying free-space spontaneous emission near a plasmonic nanostructure,” Phys. Rev. A 83, 023819 (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]
- 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]
- V. Ivanov, Y. Rozhdestvensky, “Two-dimensional atom localization in a four-level tripod system in laser fields,” Phys. Rev. A 81, 033809 (2010). [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]
- P. Meystre, M. Sargent, Elements of Quantum Optics (Springer-Verlag, Berlin, 1999).
- 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]
- C. L. Wang, Z. H. Kang, S. C. Tian, J. H. Wu, “Control of spontaneous emission from a micro-wave driven atomic system,” Opt. Express 20, 3509–3518 (2012), http://www.opticsinfobase.org/abstract.cfm?URI=oe-20-4-3509 . [CrossRef] [PubMed]
- 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]
- 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]
- R. G. Wan, T. Y. Zhang, “Two-dimensional sub-half-wavelength atom localization via controlled spontaneous emission,” Opt. Express 19, 25823–25832 (2012), http://www.opticsinfobase.org/abstract.cfm?URI=oe-19-25-25823 . [CrossRef] [PubMed]
- 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]
- D. A. Steck, Rubidium 87 D line data, available online at http://steck.us/alkalidata .
- 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]

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