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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 7 — Mar. 26, 2012
  • pp: 7870–7885
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High-precision atom localization via controllable spontaneous emission in a cycle-configuration atomic system

Chunling Ding, Jiahua Li, Rong Yu, Xiangying Hao, and Ying Wu  »View Author Affiliations


Optics Express, Vol. 20, Issue 7, pp. 7870-7885 (2012)
http://dx.doi.org/10.1364/OE.20.007870


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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

In the quantum optics and quantum mechanics, studies of the precision position measurement of an atom are mainly motivated by the idea that it has many applications in laser cooling and neutral atoms trapping [1

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

], Bose-Einstein condensation [2

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

, 3

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]

], atom lithography [4

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

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]

], the measurement of center-of-mass wave function of moving atoms [6

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

], and so on. Considerable progress has been made in establishing precision position of an atom [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]

11

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]

] both from theoretical and experimental points of view. For example, Paspalakis and Knight [12

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

] proposed a scheme for the localization of an atom using the measurement of the population in the upper level. Zubairy and colleagues [13

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

16

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]

] have discussed atom localization via resonance fluorescence or phase and amplitude control of the absorption spectrum. Nha 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]

] have studied the localization of atomic position via dual measurement when a three-level atom interacts with a quantized standing-wave field in the Ramsey interferometer setup. Qamar and coworkers [18

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]

] have presented a scheme of atom localization in a subwavelength domain via manipulation of Raman gain process. Also, controllable atom localization can be obtained via dark resonances [19

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

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]

] or via coherent population trapping (CPT) [21

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]

]. Moreover, Thomas and coworkers have proposed and experimentally demonstrated subwavelength position localization of atoms using spatially varying energy shifts [22

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

24

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]

] in the early years. Recently, atomic localization using the technique of electromagnetically induced transparency (EIT) has also been experimentally observed in [25

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]

]. These schemes for realizing one-dimensional (1D) atom localization are mainly based on atomic coherence and quantum interference effects. Besides, quantum coherence and interference have led to the observation of many useful effects and techniques in atomic physics and quantum optics, including quantum multiphoton and quantum information processes [26

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

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]

], giant Kerr nonlinearities [30

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

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

], hyper-Raman scattering [33

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

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]

], four-wave mixing and EIT [35

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

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]

], control of spontaneous emission [39

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

41

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]

], as well as two-dimensional (2D) atom localization [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

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]

].

In order to address this question, we put forward a scheme for realizing 2D atom localization in a microwave-driven four-level atomic system with a closed-loop configuration. The results show that we can increase the probability of finding the atom at a particular position by employing the microwave-driven field when the two standing-wave fields drive simultaneously the same atomic transition. The new aspect of the present paper is the introduction of an external microwave field with respect to the model proposed by Wan 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]

], which drives a hyperfine transition between the two ground-state hyperfine levels. Of particular interest is the application of a microwave-driven field, because the microwave source is easier to obtain and manipulate than other extra laser fields, the microwave field plays a crucial role in determining the position of the atom localization, and this is the situation considered in the context. Our motivation is to explore whether or not new localization phenomena arise when the microwave field is applied to drive the ground-state hyperfine transition. Two important results are found: first, when the two standing-wave fields are respectively used to drive the different atomic transitions, the maximum probability of finding the atom in one period is 50%; second, for our considered model, we find that the atom can be localized at a certain position with a probability of 100% when the two standing-wave fields couple the same atomic transition. This is the problem extensively explored. Additionally, our work has the following features: the interaction model considered here are most fundamental in the theoretical studies of dynamic behavior of the atom, and we do not need to assume any specific conditions for the structured environment. Thus the results of our study possess good adaptability. Moreover, our deductions are completely analytical and hence the physical explanation of the results is more transparent. These investigations have potential applications in laser cooling, Bose-Einstein condensation, and trapping of neutral atoms, etc.

The article is organized as follows. In Section 2, we present the physical model and its theoretical description, and then we derive an analytical expression of the conditional position probability distribution for the system in the process of atom-field interaction. In Section 3, we give a detailed analysis and explanation for the behavior of 2D atom localization. Finally, the main conclusions are presented in Section 4.

2. Theoretical model and basic formula

Let us start by considering a microwave-driven four-level atomic system, which consists of one excited level |3〉, and three ground levels |0〉, |1〉, and |2〉 as depicted in Fig. 1. The transition from the excited level |3〉 to the ground level |0〉 is coupled by the vacuum modes in the free space. An external microwave-driven field with a Larmor frequency 2Ωm 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 G1(x, y) and G2(x, y), respectively. Here, we consider two cases for the atom interacts with the standing-wave laser fields. The first case is that G1(x, y) and G2(x, y) correspond respectively to the two orthogonal standing-wave fields that couple the different atomic transitions, i.e., G1(x, y) = Ω1 sin(k1x) and G2(x, y) = Ω2 sin(k2y) with k1 = ω1/c and k2 = ω2/c being the wave vectors of the two laser fields. The second case is that G1(x, y) corresponds to the combination of two orthogonal standing-wave fields with the same frequency that drive simultaneously the transition |1〉 ↔ |3〉, while G2(x, y) corresponds to a traveling-wave field, that is, G1(x, y) = Ω1[sin(k1x)+ sin(k1y)] and G2(x, y) = Ω2. An atom moves along the z direction and passes through the intersectant region of the two orthogonal standing-wave fields in the xy plane. As a result, the interaction between the atom and the standing-wave fields is spatial dependent on the xy 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

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

]. 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 = 1)
HI=G1(x,y)eΔ1t|31|+G2(x,y)eiΔ2t|32|+Ωmei(Δ1Δ2)t|21|+kgkeiδkt|30|b^k+H.c.,
(1)
where the quantities Δ1 = ω1E31/h̄ and Δ2 = ω2E32/h̄ stand for the frequency detunings of the coherent laser fields from the corresponding atomic resonance frequencies. Here Ωm is one-half Larmor frequency for the relevant driven transition, i.e., Ωm = μ12Bm/(2h̄), with Bm being the amplitude of the microwave-driven field and μ12 = μ⃗12 ·e⃗L (e⃗L is the unit polarization vector of the corresponding laser field) denoting the dipole matrix element for the transition |1〉 ↔ |2〉. k and b^k are interpreted as the annihilation and creation operators, respectively, corresponding to the kth vacuum mode with frequency ωk. The coefficient gk represents the coupling between the vacuum mode k and the atomic transition |3〉 ↔ |0〉, δk = ωkω30 is the corresponding frequency detuning. In the following calculations, we set Ω1 and Ω2 as real parameters, while Ωm as a complex parameter, i.e., Ωm = |Ωm|e, here 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 |1Ω1|3 and the indirect one |1Ωm|2Ω2|3, as can be seen from the atomic energy-level structure in Fig. 1. The influence of the relative phase 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.

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

The dynamics of this system can be described by using the probability amplitude equations. Then the wave function of our considered system at time t can be expressed as
|Ψ(t)=dxdyf(x,y)|x|y[A1,0k(x,y;t)|1,0k+A2,0k(x,y;t)|2,0k+A3,0k(x,y;t)|3,0k+kA0,1k(x,y;t)|0,1k],
(2)
where Aj,0k (x,y;t) ( j = 1 – 3) and A0,1k (x,y;t) give the probability amplitude to find the atom at time t. |j,0k〉 denotes the atom in the level |j〉 with no photons present and |0,1k〉 represents the atom in its ground level |0〉 with a single photon in the kth vacuum mode. Besides, f(x,y) is the center-of-mass wave function of the atom.

It should be explicitly pointed out that the 2D atom localization scheme in our system relies on the fact that the spontaneously emitted photon carries information about the position of atom in the xy 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
|ψ0,1k=𝒩0,1k|Ψ(t)=𝒩dxdyf(x,y)|x|yA0,1k(x,y;t),
(3)
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
P(x,y;t|0,1k)=|𝒩|2|x|y|ψ0,1k|2=|𝒩|2|f(x,y)|2|A0,1k(x,y;t)|2,
(4)
which can be reduced to determine the probability amplitude A0,1k (x,y;t).

We now deduce an analytical expression for the probability amplitude A0,1k 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 = HI|Ψ(t)〉, and we can obtain the coupled equations of motion for the time evolution of the atomic probability amplitudes
iA1,0k(t)t=G1(x,y)eiΔ1tA3,0k(t)+Ωm*ei(Δ1Δ2)tA2,0k(t),
(5)
iA2,0k(t)t=G2(x,y)eiΔ2tA3,0k(t)+Ωmei(Δ1Δ2)tA1,0k(t),
(6)
iA3,0k(t)t=G1(x,y)eiΔ1tA1,0k(t)+G2(x,y)eiΔ2tA2,0k(t)iΓ02A3,0k(t),
(7)
iA0,1k(t)t=gk*eiδktA3,0k(t),
(8)
where Γ0 = 2π|gk|2D(ωk) is the spontaneous decay rate from level |3〉 to level |0〉, with D(ωk) being the density of states (DOS) at frequency ωk 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.

Making use of the Laplace transform method and the final value theorem, the probability amplitude A0,1k in the long time limit can be obtained as
A0,1k(x,y;t)=i0gk*eiδktA3,0k(t)dt=igk*A˜3,0k(s=iδk),
(9)
here Ã3,0k (s) is the Laplace transform of A3,0k (t) with s = −k.

Next, after carrying out the Laplace transformations for Eqs. (5)(7), we get the solution to the probability amplitude Ã3,0k (s) as
A˜3,0k(s=iδk)=CiD,
(10)
where
C=[|Ωm|2(δkΔ1)(δkΔ2)]A3,0k(0)[(δkΔ2)G1(x,y)+ΩmG2(x,y)]A1,0k(0)[(δkΔ1)G2(x,y)+Ωm*G1(x,y)]A2,0k(0),
(11)
D=(δk+iΓ02)[|Ωm|2(δkΔ1)(δkΔ2)]+(δkΔ2)G12(x,y)+(δkΔ1)G22(x,y)+(Ωm+Ωm*)G1(x,y)G2(x,y).
(12)

Finally, the conditional probability of finding the atom in its internal level |0〉 with a spontaneously emitted photon of frequency ωk in the vacuum mode k is then given by
P(x,y;t|0,1k)=|𝒩|2|f(x,y)|2|A0,1k(x,y;t)|2=|𝒩|2|f(x,y)|2|gk|2|CD|2.
(13)

Due to the center-of-mass wave function of the atom f(x,y) is assumed to be nearly constant over many wavelengths of the standing-wave fields in the xy plane, the conditional position probability distribution P(x,y;t → ∞|0,1k) is determined by the last term in Eq. (13). Therefore, we can define the filter function as F(x,y)=|CD|2, which shows that the conditional position probability distribution depends upon the frequency detunings of the standing-wave driving fields and the population in the upper or lower levels, as well as the detuning of the spontaneously emitted photon. As a result, we can obtain the position information of the atom by measuring the frequency of spontaneously emitted photon under proper conditions.

Under the conditions A3,0k (0) = 1, A1,0k (0) = A2,0k (0) = 0, Δ1 = Δ2 = 0, and φ = 0, the filter function F(x,y) can be explicitly expressed in the following form
F(x,y)=1[δk+δk[G12(x,y)+G22(x,y)]+2|Ωm|G1(x,y)G2(x,y)|Ωm|2δk2]2+Γ024.
(14)

3. Results and discussion

It has been reported that the spontaneous emission can be coherently manipulated by the microwave field in a cycle-configuration four-level atomic system [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]

, 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]

]. Some of the interesting phenomena involving spectral-line narrowing, spectral-line enhancement, and spectral-line suppression can be observed by adjusting the system parameters. Here we are interested in the precise position measurement of the atom when it passes through the standing-wave fields using the measurement of the frequency of the spontaneously emitted photon. As we mentioned earlier, the Rabi frequencies of the standing-wave fields are position dependent for our proposed scheme. Consequently, the spontaneous emission spectra become position dependent and thus the position of the atom as it passes through the standing-wave fields can be determined as soon as we monitor the frequency of the spontaneously emitted photon. The precise location of the atom can be given by those values of k1x and k2y (k1y) 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., G1(x, y) = Ω1 sin(k1x) and G2(x, y) = Ω2 sin(k2y); (ii) the two standing-wave fields with the same frequency are applied to drive the same atomic transition, i.e., G1(x, y) = Ω1[sin(k1x) + sin(k1y)] and G2(x, y) = Ω2. The spontaneous decay rate of the level |3〉 to level |0〉 is set as Γ0 = 2γ. 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

Initially, we consider the situation that the two orthogonal standing-wave fields drive the different atomic transitions, respectively. In such a case, we will analyze the behavior of 2D atom localization by adjusting the initial state preparation, the frequency detuning of two standing-wave fields, and the relative phase φ. In Fig. 2, we plot the filter function F(x,y) versus the normalized positions (k1x, k2y) by measuring the frequency of the spontaneously emitted photon under the condition of A3,0k (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 xy 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 δk = 8γ, and the atom is localized at the second and fourth quadrants in the xy plane [see Fig. 3(a)]. As the detuning δk increases, we observe that the localization peaks occur at k1x + k2y = 2 or k1xk2y = (2n + 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 δk = 16γ, 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., δk = 19.3γ 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.

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

These interesting localization phenomena can be explained using quantum interference effect either in the bare-state picture or in the dressed-state picture. Under the condition that the atom is initially prepared in the excited level |3〉, there exist three spontaneous decay channels in the bare-state picture: |3〉 → |0〉, |3〉 → |1〉 → |2〉 → |3〉 → |0〉, and |3〉 → |2〉 → |1〉 → |3〉 → |0〉. Quantum interference among these three pathways results in the spectral-line narrowing and the quenching of spontaneous emission. Consequently, we can observe two sharp localization peaks in the xy plane.

It is desirable to obtain the position of the atom when the atom passes through the standing-wave fields. However, how to extract the localization information, i.e., relatively to what point in space is the localization? We now discuss some possible solutions, which are similar to those reported in Refs. [13

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

, 23

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

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]

]. Our scheme is based on the fact that the conditional position probability P(x,y;t|0,1k) carries the information about the atomic position. In Fig. 2, we plot the filter function F(x,y) for four cases of frequencies of the emitted photon ωk recorded during measuring time t. It can be observed that if the detector records a larger frequency [e.g., ωk = ω30 + 19.3γ 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]

]. We thus get a much precise position information due to a strong spectral-line narrowing effect when the frequency of the emitted photon is large enough. As a result, we can extract the immediate position information by making a measurement on the spontaneously emitted photon when the atom goes through the intersectant region of two standing-wave fields.

In order to further show the influence of the system parameters on the behavior of 2D atom localization, we give the filter function F(x,y) versus the normalized positions (k1x, k2y) by monitoring the spontaneously emitted photon under the condition of Ψ(0)=(|1+|2)/2 when the two standing-wave fields are tuned to nonresonant with the corresponding atomic transitions and the relative phase φ = π, as shown in Fig. 4. The corresponding density plots are illustrated in Fig 5. When the spontaneously emitted photon with detuning δk = 8.5γ 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 xy 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 δk = 17.2γ, 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.

Fig. 4 The filter function F(x,y) as a function of (k1x, k2y) in dependence on the detuning of spontaneously emitted photon. (a) δk = 8.5γ; (b) δk = 12.2γ; (c) δk = 17.2γ; (d) δk = 20γ. The system parameters used are the same as Fig. 2 except that Δ1 = Δ2 = 5γ, φ = π, and the atom is initially in Ψ(0)=(|1+|2)/2.
Fig. 5 Density plot of filter function F(x,y) shown in Fig. 4.

3.2. Two standing-wave fields couple one atomic transition

Fig. 6 The filter function F(x,y) as a function of (k1x, k1y) in dependence on the detuning of spontaneously emitted photon. (a) δk = 7γ; (b) δk = 8.9γ; (c) δk = 13.5γ; (d) δk = 20γ. The system parameters used are the same as Fig. 2.
Fig. 7 Density plot of filter function F(x,y) in the xy plane shown in Fig. 6.

Fig. 8 The filter function F(x,y) as a function of (k1x, k1y) for different combinations of the detuning δk and the phase φ. (a) δk = 14.5γ, φ = π/2; (b) δk = 19.5γ, φ = π/4; (c) δk = 26γ, φ = 3π/4; (d) δk = 6.75γ, φ = π. The system parameters used are the same as Fig. 2 except that the atom is initially in Ψ(0)=(|1|2)/2.
Fig. 9 Density plot of filter function F(x,y) shown in Fig. 8.

Before concluding this section, we now turn our attention to estimate the influence of the perturbations from the perfect microwave standing configurations of the two driving fields on the atom localization behavior. This is mainly because the perturbations of intensity and detuning are unavoidable in the process of experimental realization. The resulting localization patterns by considering an intensity fluctuation (for instance 0.1γ) are plotted in Figs. 10(a) and 10(b), which correspond to the localization profiles without the intensity fluctuation as shown in Figs. 2(c) and 2(d), respectively. By contrast, we found that the intensity perturbation does not affect the most probable positions of finding the atom in the subwavelength regime, but the resolution of atom localization is slightly reduced. Similarly, the perturbations of detuning may cause a small change of the atomic resolution, but cannot affect the precise location of the atom, these can be clearly seen by comparing the localization peaks with a small fluctuation of Figs. 10(c) and 10(d) to that shown in Figs. 8(c) and 8(d) without any perturbation. It can be concluded that the behavior of the atom localization does not vary with the fluctuations of laser intensity and detuning, therefore, we use the exact values of the intensity and detuning in the above-mentioned calculation and analysis is reasonable.

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

4. Conclusion

In summary, we have demonstrated the scheme for 2D atom localization of a four-level atomic system with a cyclic configuration based on the controllable spontaneous emission. In particular, the behavior of 2D atom localization in this kind of system in the presence of microwave field has been discussed in detail, giving rise to some interesting localization phenomena such as latticelike, craterlike, spikelike, bicycliclike patterns and lotus-like structure. These results show that the conditional position probability distribution of the atom is very sensitive to the tunable system parameters. Furthermore, the maximum probability of finding the atom in one period of the standing-wave fields is reached to 100%, that is, we can localize the atom at a particular position. Also, according to our predictions, the atom can be localized at an expected position for different sets of parameters due to the quantum interference effects. Finally, our proposal may have potential applications in improving the performance of laser cooling and neutral atoms trapping, Bose-Einstein condensation, atom lithography, etc. With the development of optical techniques, we hope that this 2D atom localization scheme proposed here will be implemented by actual experiments in the near future.

Our scheme may be extended to the subwavelength localization of an ensemble of atoms. In other words, the localization properties of a region containing multiple atoms can be treated using our single-atom localization approach if the atoms evolve independently. The assumption of independent atoms is reasonable, if it satisfies the following conditions: (i) The atoms do not interact strongly with one another in the process of atoms traveling through the standing-wave fields. (ii) These atoms must be separated far enough from each other that we can neglect the quantum interference from the two atoms interacting with the same mode of the reservoir. (iii) The applied fields are classical.

Acknowledgments

We would like to thank Professor Xiaoxue Yang for her encouragement and helpful discussion. This research was supported in part by the National Natural Science Foundation of China under Grants No. 11004069, No. 11104210, and No. 91021011, by the Doctoral Foundation of the Ministry of Education of China under Grant No. 20100142120081, by the National Basic Research Program of China under Contract No. 2012CB922103, and by the Fundamental Research Funds from Huazhong University of Science and Technology (HUST) under Grant No. 2010MS074.

References and links

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

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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]

8.

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

9.

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

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 56, 2972–2977 (1997). [CrossRef]

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]

14.

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

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 72, 013820 (2005). [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]

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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]

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]

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]

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 69, 022318 (2004). [CrossRef]

28.

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

31.

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

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. 91, 221108 (2007). [CrossRef]

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. 91, 061113 (2007). [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]

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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]

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]

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]

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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]

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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]

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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

  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, 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, 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, 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, M. S. Zubairy, “Spectroscopic measurement of an atomic wave function,” Phys. Rev. A 67, 023805 (2003). [CrossRef]
  7. P. Storey, M. Collett, D. Walls, “Measurement-induced diffraction and interference of atoms,” Phys. Rev. Lett. 68, 472–475 (1992). [CrossRef] [PubMed]
  8. P. Storey, M. Collett, D. Walls, “Atomic-position resolution by quadrature-field measurement,” Phys. Rev. A 47, 405–418 (1993). [CrossRef] [PubMed]
  9. 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]
  10. 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]
  11. 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]
  12. E. Paspalakis, P. L. Knight, “Localizing an atom via quantum interference,” Phys. Rev. A 63, 065802 (2001). [CrossRef]
  13. S. Qamar, S. Y. Zhu, M. S. Zubairy, “Atom localization via resonance fluorescence,” Phys. Rev. A 61, 063806 (2000). [CrossRef]
  14. M. Macovei, J. Evers, C. H. Keitel, M. S. Zubairy, “Localization of atomic ensembles via superfluorescence,” Phys. Rev. A 75, 033801 (2007). [CrossRef]
  15. 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]
  16. 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]
  17. H. Nha, J. H. Lee, J. S. Chang, K. An, “Atomic-position localization via dual measurement,” Phys. Rev. A 65, 033827 (2002). [CrossRef]
  18. S. Qamar, A. Mehmood, 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, 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, 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, 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]
  23. 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]
  24. 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]
  25. 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]
  26. F. Dell’Anno, S. De Siena, F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006). [CrossRef]
  27. 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]
  28. 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]
  29. 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]
  30. 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]
  31. Y. Wu, X. X. Yang, “Giant Kerr nonlinearities and solitons in a crystal of molecular magnets,” Appl. Phys. Lett. 91, 094104 (2007). [CrossRef]
  32. D. D. Yavuz, D. E. Sikes, “Giant Kerr nonlinearities using refractive-index enhancement,” Phys. Rev. A 81, 035804 (2010). [CrossRef]
  33. Y. Wu, L. L. Wen, 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, 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, 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]
  36. 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]
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