1. Introduction

Spontaneous generated coherence (SGC), one of the most intriguing effects in quantum optics, has attracted considerable interest in the last few decades. Many schemes [1–9] were proposed for control of spontaneous emission coherence. A rather typical work is that Paspalakis and Knight proposed a phase control scheme in a four-level atom driven by two lasers of the same frequencies [10] in 1998, where SGC exists in such atoms having two close-lying levels subject to the conditions that these levels are near degenerate and the corresponding dipole matrix elements are not orthogonal. As is well known, it is very difficult, if not impossible, to find a real atomic system with SGC to experimentally realize this scheme. Therefore, to overcome the above-mentioned difficulties several schemes have been proposed where the rigorous conditions are not required. For instance, Wu et al. studied the spontaneous-emission properties of a coherently driven four-level atom. They pointed out that the SGC could be observed in the experiment since the rigorous condition of nearly degenerate levels with nonorthogonal dipole moments was not required [11]. Later on, Li et al. investigated the features of spontaneous emission spectra in a coherently driven five-level atomic system by means of a radio frequency or microwave field driving a hyperne transition within the ground state [12]. Recently, Gao et al. [13] considered an atomic model can be found in real atoms, no stringent conditions have to be satisfied, so a corresponding experiment can be done to observe the expected phenomena related to SGC reported by Fountoulakis et al. [14]. More recently, two equivalent systems [15,16] have been experimentally performed to observe phenomena related to SGC without the rigorous requirement of close-lying levels.

In recent years, atomic localization has also been a topic of considerable interest due to its potential applications in laser cooling [17], atom nanolithograghy [18], Bose-Einstein condensation [19], and measurement of center-of-mass wave function [20,21]. From one-dimensional (1D) atomic localization to three-dimensional (3D) atomic localization, many researchers [22–48] have been put forward for localization of a single atom. In this paper, we present a new scheme to investigate the two-dimensional (2D) and 3D atomic localization behaviors based on SGC, in which we use a microwave field to drive the two highest lying levels of the four-level Y-type atomic system. Our scheme shows many advantages that the previous schemes [39–48] do not have. First, we show that the high-precision and high-resolution 2D and 3D atom localization behaviors can be realized by modulating the parameters of the system, which are well understood by qualitative explanations in the dressed-state picture. Second, we find that the localization precision and spatial resolution of the atom can be effectively modulated by the amplitude and phase of the microwave field. The effects of the initial populations on atom localization have also been studied. Particularly, by properly adjusting the system parameters, we can localize the atom at a particular position. Third, in this scheme, no stringent conditions have to be satisfied, so it is easy to carry out the corresponding experiment to observe the expected 2D and 3D atom localization phenomena related to SGC in atoms by the magneto-optical trap MOT technique. Otherwise, this system is also significant for control of 2D and 3D atom localization in atom-field systems since the amplitude and phase of the microwave field can be controlled conveniently.

2. The model and dynamic equations

We consider a four-level atom as depicted in Fig. 1(a). The level$|1〉$is coupled to the levels$|2〉$and$|3〉$by two coherent laser fields with carrier frequencies of${\omega }_1$, ${\omega }_3$and Rabi frequencies of${\Omega }_1$,${\Omega }_3$ respectively. A resonant microwave field with frequency${\omega }_m$and Rabi frequency${\Omega }_m$is used to couple the highest lying levels$|2〉$and$|3〉$through an allowed magnetic transition. We set${\Omega }_1$and${\Omega }_3$as real parameters, but microwave field ${\Omega }_m$as a complex parameter:${\Omega }_m={\Omega }_{m0}\exp \left(i\phi \right)$, and$\phi$is the phase of microwave field, and may also be called the relative phase. Here, we consider both 2D and 3D atom localization behaviors in which the atom interacts with the standing-wave fields. The 2D atom localization scheme is that${\Omega }_1$and${\Omega }_3$correspond, respectively, to the two orthogonal standing-wave fields that couple the different atomic transitions, i.e., ${\Omega }_1={\Omega }_{10}\sin \left({\text{k}}_{1}x\right)$ and${\Omega }_3={\Omega }_{30}\sin \left({\text{k}}_{2}y\right)$, with$k_1$and$k_2$being the wave vectors of the two laser fields. On the other hand, the 3D atom localization scheme is that${\Omega }_1$corresponds to the combination of two orthogonal standing-wave fields with the same frequency that drives simultaneously the transition$|1〉\leftrightarrow |2〉$,while${\Omega }_3$corresponds to a standing-wave field, that is, ${\Omega }_1={\Omega }_{xy}\left[\sin \left({\text{k}}_{1}x\right)+\sin \left(k_{1}y\right)\right]$and${\Omega }_3={\Omega }_z\sin \left({\text{k}}_{2}z\right)$.

 

Fig. 1 (a) The atomic energy levels in the bare state picture. (b) The atomic energy levels in the dressed-state picture.

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2.1 2D atomic localization scheme

For the 2D scheme, the transition$|1〉\leftrightarrow |2〉$is driven by a classical standing-wave field ${\Omega }_1={\Omega }_{10}\sin \left({\text{k}}_{1}x\right)$, while the transition$|1〉\leftrightarrow |3〉$is driven by another standing-wave field ${\Omega }_3={\Omega }_{30}\sin \left({\text{k}}_{2}y\right)$. We assume that the center-of-mass position of the atom is nearly constant along the directions of the laser waves and neglect the kinetic part of the atom from the Hamiltonian in the Raman-Nath approximation [49]. In the rotating-wave and the electric-dipole approximations, the interaction Hamiltonian for the system reads ($\hslash =1$)

At any time$t$, the atom-field state vector can be written as

The atom localization in our scheme is based on the fact that the spontaneously emitted photon carries information about the position of atom in the x-y plane as a result of the spatial position-dependent atom-field interaction. When we detected at time $t$a spontaneously emitted photon in the vacuum mode of wave vector$k$, the atom is in its internal state$|4〉$and the state vector of the system, after making appropriate projection over$|{\Psi }_{2D}〉$, is reduced to

By solving the Schrödinger equation$|{\dot{\Psi }}_{2D}\left(t\right)〉=-i{H}_{2D}|{\Psi }_{2D}\left(t\right)〉$, and then using the Weisskopf-Wigner theory [50], we can obtain the following dynamical equations for probability amplitudes

The spontaneous emission spectrum is the Fourier transform of${〈E^{-}\left(\text{t}+\tau \right)E^{+}\left(t\right)〉}_{t\to \infty }$, and can be written as $S\left(x,y\right)=\Gamma {\left|A_{4,k}\left(x,y;t\to \infty \right)\right|}^2/2\pi {\left|g_{k,14}\right|}^2$for the atom system. Here, we can see that the spontaneous emission spectrum$S\left(x,y\right)$is proportional to${\left|A_{4,k}\left(x,y;t\to \infty \right)\right|}^2$. Thus, the spontaneous emission from the atom can be used to characterize the conditional position probability distribution$W\left(x,y;t\to \infty |4,1_k\right)$.Using the Laplace transform method [51] and the position-dependent spontaneous emission spectrum is given by

2.2 3D atomic localization scheme

For the 3D scheme, ${\Omega }_1={\Omega }_{xy}\left[\sin \left(k_{1}x\right)+\sin \left(k_{1}y\right)\right]$corresponds to the combination of two orthogonal standing-wave fields with the same frequency that drives simultaneously the transition$|1〉\leftrightarrow |2〉$, while the transition$|1〉\leftrightarrow |3〉$is driven by a standing-wave field${\Omega }_3={\Omega }_z\sin \left(k_{2}z\right)$. In the Raman-Nath approximation, the rotating-wave and the electric-dipole approximations, the interaction Hamiltonian is ($\hslash =1$)

The dynamics of this system can be described by using the probability amplitude equations. Then the wave function of the system at time t can be expressed in terms of the state vectors as

As we detected at time$t$a spontaneously emitted photon in the vacuum mode of wave vector$k$, the atom is in its internal state$|4〉$and the state vector of the system, after making appropriate projection over${\Psi }_{3D}$, is reduced to

By solving the Schrödinger equation$|{\dot{\Psi }}_{3D}\left(t\right)〉=-i{H}_{3D}|{\Psi }_{3D}\left(t\right)〉$, and then using the Weisskopf-Wigner theory, we can obtain the following dynamical equations for probability amplitudes

The spontaneous emission spectrum is the Fourier transform of${〈E^{-}\left(\text{t}+\tau \right)E^{+}\left(t\right)〉}_{t\to \infty }$, and can be written as $S\left(x,y,z\right)=\Gamma {\left|A_{4,k}\left(x,y,z;t\to \infty \right)\right|}^2/2\pi {\left|g_{k,14}\right|}^2$for the atom system. Here, we can see that the spontaneous emission spectrum$S\left(x,y,z\right)$is proportional to${\left|A_{4,k}\left(x,y,z;t\to \infty \right)\right|}^2$. Thus, the spontaneous emission from the atom can be used to characterize the conditional position probability distribution$W\left(x,y,z;t\to \infty |4,1_k\right)$. And the position-dependent spontaneous emission spectrum is given by

3. Results and discussion

In this section, we analyze the position-dependent spontaneous emission spectra$S(x,y)$and$S\left(x,y,z\right)$, which directly reflect the conditional position probability distribution, and then demonstrate 2D and 3D atom localization schemes by measuring the frequency of spontaneously emitted photon. In this paper, all the parameters are in units of constant Γ, which should be in the order of MHz for rubidium atoms.

3.1 Case A: 2D atomic localization scheme

First of all, we consider the 2D atom localization where the transition$|1〉\leftrightarrow |2〉$is driven by a classical standing-wave field${\Omega }_1={\Omega }_{10}\sin \left(k_{1}x\right)$, while the transition$|1〉\leftrightarrow |3〉$is driven by another standing-wave field ${\Omega }_3={\Omega }_{30}\sin \left(k_{2}y\right)$.

In Fig. 2, we plot spontaneous emission spectrum$S\left(x,y\right)$versus positions$\left(k_{1}x/\pi ,k_{2}y/\pi \right)$for different detuning of spontaneously emitted photon. For${\delta }_k=0$, as shown in Fig. 2(a), the maxima of$S\left(x,y\right)$are located at$k_{1}x=n\pi$or$k_{2}y=m\pi$, where$n$and$m$are integers. The atom is localized around the nodes of the two standing-wave fields. When the detuning is tuned to${\delta }_k=5\Gamma$, the peak maxima are situated in four quadrants and shows a craterlike pattern at center of $x-y$plane [see Fig. 2(b)]. As we increase detuning${\delta }_k$to$10\Gamma$, the $S\left(x,y\right)$ in Fig. 2(c) displays a latticelike pattern, and the atom is localized at the edges of these lattices. Finally, when${\delta }_k=15\Gamma$, the peaks maxima show spikelike pattern, and the atom is located at the positions $\left(k_{1}x/\pi ,k_{2}y/\pi \right)=\left(-0.5,0.5\right)$ or $\left(k_{1}x/\pi ,k_{2}y/\pi \right)=\left(0.5,-0.5\right)$ as shown in Fig. 2(d). Therefore, the probability of finding the atom at each position is 50%. In fact, from Eq. (6), it is straightforward to show that the $S\left(x,y\right)$ depends strongly on the detuning${\delta }_k$. Consequently, we can observe different localization peaks by changing the detuning of spontaneously emitted photon.

 

Fig. 2 Spontaneous emission spectrum$S(x,y)$ versus positions $(k_{1}x/\pi ,k_{2}y/\pi )$ for (a) ${\delta }_k=0$; (b) ${\delta }_k=5\Gamma$; (c) ${\delta }_k=10\Gamma$; and (d) ${\delta }_k=15\Gamma$. The other parameters are ${\Delta }_1={\Delta }_3=0$, ${\Omega }_{10}=10\Gamma$, ${\Omega }_{\text{m0}}=1.5\Gamma$, ${\Omega }_{30}=10\Gamma$, and $\phi =0$. The initial populations are $A_1(x,y;0)=1$ and $A_2(x,y;0)=A_3(x,y;0)=0$.

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Keeping ${\delta }_k=15\Gamma$, we study the effect of the microwave field on the 2D atom localization numerically in Fig. 3. On the condition of${\Omega }_{m0}=0.5\Gamma$[see Fig. 3(a)], the peak maxima of spontaneous emission spectrum$S\left(x,y\right)$, which represent the most probable positions for the atom, distribute in all four quadrants but mainly in second and fourth quadrants of the$x-y$ plane. When the microwave field is ${\Omega }_{m0}=\Gamma$[see Fig. 3(b)], two localization peaks in the first and third quadrants in Fig. 3(a) are completely disappeared. In such a condition, the uncertainty of conditional position probability is reduced. In the case of ${\Omega }_{m0}=2\Gamma$ [see Fig. 3(c)], the spontaneous emission spectrum exhibits two craterlike patterns in second and fourth quadrants which lead to the localization of atom at these circles. Furthermore, when the microwave field is tuned to ${\Omega }_{m0}=4\Gamma$[see Fig. 3(d)], the localization peaks are also mainly distributed in the second and fourth quadrants but exhibit two large-caliber craterlike patterns compared with the case in Fig. 3(c). From the Fig. 3, we can conclude that only 50% detecting probability is obtained when we adjust the intensity of microwave field, but the localization precision can be controlled via this system parameter.

 

Fig. 3 Spontaneous emission spectrum$S(x,y)$ versus positions $(k_{1}x/\pi ,k_{2}y/\pi )$ for (a) ${\Omega }_{m0}=0.5\Gamma$; (b) ${\Omega }_{m0}=\Gamma$; (c) ${\Omega }_{m0}=2\Gamma$; and (d) ${\Omega }_{m0}=4\Gamma$. The other parameters are the same as Fig. 2(d).

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Next, we investigate the influence of the phase of the microwave field on the behavior of 2D atom localization in Fig. 4. When $\phi =0.5\pi$ or$\phi =1.5\pi$, as shown in Figs. 4(a) and 4(c), we can see that there are four spikelike patterns in the$x-y$plane. For the case that$\phi =\pi$, the peak maxima exhibit two spikelike patterns in the first and third quadrants. On the condition of$\phi =2\pi$, the peak maxima located at the positions $\left(k_{1}x/\pi ,k_{2}y/\pi \right)=\left(-0.5,0.5\right)$ or $\left(k_{1}x/\pi ,k_{2}y/\pi \right)=\left(0.5,\text{-}0.5\right)$ as shown in Fig. 4(d), which is the same as the case in Fig. 2(d). As can be seen, the spatial pattern of position probability distribution of atom localization is sensitive to the phase of the microwave field.

 

Fig. 4 Spontaneous emission spectrum$S(x,y)$ versus positions $(k_{1}x/\pi ,k_{2}y/\pi )$ for (a) $\phi =0.5\pi$; (b) $\phi =\pi$; (c) $\phi =1.5\pi$; and (d) $\phi =2\pi$. The other parameters are the same as Fig. 2(d).

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In order to understand the roles of the initial populations on the 2D atom localization clearly, we show the dependence of spontaneous emission spectrum$S\left(x,y\right)$on the initial populations in Fig. 5. For the case that $A_2\left(x,y;0\right)=1$ and $A_1\left(x,y;0\right)=A_3\left(x,y;0\right)=0$[see Fig. 5(a)], it can be seen that there are two localization patterns in$x-y$plane. For the case that$A_2\left(x,y;0\right)=0$and$A_1\left(x,y;0\right)=A_3\left(x,y;0\right)=1/\sqrt{2}$, the peak maxima of the spontaneous emission spectrum are only situated in the second quadrant with a spikelike pattern and thus the atom is localized at position, i.e., $\left(k_{1}x/\pi ,k_{2}y/\pi \right)=\left(-0.5,0.5\right)$as shown in Fig. 5(b), the localization peak in the fourth quadrant in Figs. 5(a) is completely disappeared. In this case, the probability of finding the atom in one period of the standing-wave fields is increased to 1, and the spatial resolution is about $\lambda \text{/10}$. For the case in Fig. 5(c), spontaneous emission spectrum shows only a spikelike pattern in the fourth quadrant, but the atom is localized at position $\left(k_{1}x/\pi ,k_{2}y/\pi \right)=\left(0.5,\text{-}0.5\right)$. When$A_1\left(x,y;0\right)=A_2\left(x,y;0\right)=A_3\left(x,y;0\right)=1/\sqrt{3}$ [see Fig. 4(d)], the result is more or less the same as Fig. 5(a), but the peak maxima of spontaneous emission spectrum become small and the detecting probability decreases from 1 to 1/2. By preparing the initial populations of the atoms in different levels, we can restrain certain localization peaks and confine the atom to half-wavelength regions, which is called sub-half-wavelength localization. This is a significant advantage that the previous scheme [43] does not have.

 

Fig. 5 Spontaneous emission spectrum$S(x,y)$ versus positions $(k_{1}x/\pi ,k_{2}y/\pi )$ for (a)$A_{\text{2}}(x,y;0)=1$,$A_{\text{1}}(x,y;0)=A_3(x,y;0)=0$;(b)$A_{\text{1}}(x,y;0)=A_3(x,y;0)=1/\sqrt{2}$, $A_{\text{2}}(x,y;0)=0$; (c)$A_{\text{3}}(x,y;0)=0$, $A_{\text{1}}(x,y;0)=A_{\text{2}}(x,y;0)=1/\sqrt{2}$; and (d)$A_{\text{1}}(x,y;0)=A_{\text{2}}(x,y;0)=A_{\text{3}}(x,y;0)=1/\sqrt{3}$. The other parameters are the same as Fig. 2(d).

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3.2 Case B: 3D atomic localization scheme

In the following, we analyze the conditional position probability distribution of the atom via a few numerical calculations based on the Eq. (12), and then address how the system parameters can be used to achieve 3D atom localization by controlled spontaneous emission.

Isosurfaces for spontaneous emission spectrum$S\left(x,y,z\right)=0.01$versus the positions$\left(k_{1}x,k_{1}y,k_{2}z\right)$ for different values of the detuning of spontaneously emitted photon is plotted in Fig. 6. As can be seen, the spatial pattern of position probability distribution and the resolution of 3D atom localization depend strongly on the detuning of spontaneously emitted photon. Figures 6(a)-6(c) show the isosurfaces for spontaneous emission spectrum distributed at positions$\left(k_{1}x/\pi =k_{1}y/\pi =0.5,-1\le k_{2}z/\pi \le 1\right)$ or$\left(k_{1}x/\pi =k_{1}y/\pi =-0.5,-1\le k_{2}z/\pi \le 1\right)$. However, when the detuning${\delta }_k=13\Gamma$, the localization structures display two small-size spheres and two large-size spheres with different localization precision in four different subspaces of the 3D space, and the atom localized at these spheres [see Fig. 6(d)]. From Eq. (12), one can see that there is a strong correlation between the frequency detuning of the spontaneously emitted photon and the position of the atom. The measurement of a particular frequency detuning corresponds to the localization of the atom within a sub-wavelength region of the standing-wave field in three dimensions.

 

Fig. 6 Isosurfaces for spontaneous emission spectrum$S(x,y,z)=0.01$ versus positions $(k_{1}x/\pi ,k_{1}y/\pi ,k_{2}z/\pi )$ for (a)${\delta }_k=10\Gamma$; (b) ${\delta }_k=11\Gamma$; (c) ${\delta }_k=12\Gamma$; and (d) ${\delta }_k=13\Gamma$. The other parameters are ${\Delta }_1={\Delta }_3=0$,${\Omega }_{xy}=5\Gamma$, ${\Omega }_{m0}=0.5\Gamma$, ${\Omega }_z=5\Gamma$, and $\phi =0$. The initial populations are $A_1(x,y,z;0)=1$and$A_2(x,y,z;0)=A_3(x,y,z;0)=0$.

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In Fig. 7, we study the effect of the intensity of the microwave field on the spatial distributions of the 3D atom localization. For the microwave field, i.e. ${\Omega }_{m0}=0.8\Gamma$or$\Gamma$, the isosurfaces for $S\left(x,y,z\right)$show two small-size spheres and two large-size spheres in 3D space [see Figs. 7(a) and 7(b)]. By comparing the Figs. 7(a) and 7(b), it is found that the spatial resolution of atom localization improves with adjusting the intensity of the microwave field, and the spatial resolution is about $\lambda \text{/20}$(see small-size spheres Fig. 7(b)). As a consequence, the atom can be localized with a high-precision within a certain range. When the microwave field is adjusted to${\Omega }_{m0}=1.5\Gamma$, two small size spheres in the two subspaces are completely disappeared, as shown in Fig. 7(c). In such a case, we can achieve much better probability of finding the atom at each position. Furthermore, with a further increase of${\Omega }_m$$\left({\Omega }_{m0}=2\Gamma \right)$ the size of the spheres becomes larger, which indicates that the resolution of atom localization is reduced.

 

Fig. 7 Isosurfaces for spontaneous emission spectrum$S(x,y,z)=0.01$ versus positions $(k_{1}x/\pi ,k_{1}y/\pi ,k_{2}z/\pi )$ for (a)${\Omega }_{m0}=0.8\Gamma$; (b) ${\Omega }_{m0}=\Gamma$; (c) ${\Omega }_{m0}=1.5\Gamma$; and (d) ${\Omega }_{m0}=2\Gamma$. The other parameters are the same as Fig. 6(d).

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To obtain a better understanding of how the phase of the microwave field $\phi$ modifies the 3D atom localization, we show isosurfaces for spontaneous emission spectrum$S\left(x,y,z\right)=0.01$ versus the positions$\left(k_{1}x,k_{1}y,k_{2}z\right)$ for different values of phase of the microwave field in Fig. 8. As shown in Fig. 8, one sees that the 3D localization structures in Fig. 8(a) are completely the same as the case in Fig. 8(c), while 3D localization structures in Fig. 8(b) are the mirror image of the case in Fig. 8(d) with respect to the plane$z=0$.

 

Fig. 8 Isosurfaces for spontaneous emission spectrum$S(x,y,z)=0.01$ versus positions $(k_{1}x/\pi ,k_{1}y/\pi ,k_{2}z/\pi )$ for (a)$\phi =0.5\pi$; (b) $\phi =\pi$; (c) $\phi =1.5\pi$; and (d) $\phi =2\pi$. The other parameters are the same as Fig. 6(d).

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Finally, we turn to illustrate the influences of the initial populations on the 3D atom localization are depicted in Fig. 9. Clearly, the 3D atom localization is very sensitive to the initial populations. The 100% probability of finding an atom within 3D space can be achieved for the case$A_2\left(x,y,z;0\right)=0$and $A_1\left(x,y,z;0\right)=A_3\left(x,y,z;0\right)=1/\sqrt{2}$[see Fig. 9(c)]. This is the maximum probability obtained in our previous proposed scheme [46], and is increased by a factor of 4 or 8 compared with the previous schemes [45,47].

 

Fig. 9 Isosurfaces for spontaneous emission spectrum$S(x,y,z)=0.01$ versus positions $(k_{1}x/\pi ,k_{1}y/\pi ,k_{2}z/\pi )$ for (a)$A_{\text{2}}(x,y,z;0)=1$, $A_{\text{1}}(x,y,z;0)=A_3(x,y,z;0)=0$; (b)$A_{\text{3}}(x,y,z;0)=0$,$A_{\text{1}}(x,y,z;0)=A_{\text{2}}(x,y,z;0)=1/\sqrt{2}$;(c)$A_{\text{1}}(x,y,z;0)=A_3(x,y,z;0)=1/\sqrt{2}$, $A_{\text{2}}(x,y,z;0)=\text{0}$; and (d) $A_{\text{1}}(x,y,z;0)=A_2(x,y,z;0)=A_3(x,y,z;0)=1/\sqrt{3}$. The other parameters are ${\Delta }_1={\Delta }_3=0$,${\Omega }_{xy}=5\Gamma$, ${\delta }_k=13\Gamma$, ${\Omega }_z=5\Gamma$, $\phi =0$, and ${\Omega }_{m0}=\Gamma$.

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Before ending this section, we will give some physical explanations of the above results. In fact, the interesting phenomena discussed above can be qualitatively explain in the dressed-state picture. By driving the system with three fields${\Omega }_1$,${\Omega }_m$and${\Omega }_3$, the levels$|1〉$,$|2〉$and$|3〉$in the bare state can be replaced by three new states$|+〉$, $|0〉$,and$|\text{-}〉$, as shown in Fig. 1(b). It is straightforward to show that there exists quantum interference between three different spontaneous emission pathways$|+〉\leftrightarrow |4〉$,$|0〉\leftrightarrow |4〉$, and$|\text{-}〉\leftrightarrow |4〉$. Multiple SGC arise among these competitive transition pathways and result in spectral-line narrowing, spectra-line enhancement, spectral-line suppression, and fluorescence quenching in the spontaneous emission spectra [11–13]. Consequently, we can observe different localization structures in two and three dimensions.

4. Conclusions

Before concluding, we discuss the possible experimental realization of our proposed scheme. The experimental system for this atomic scheme can be realized by the$Rb$atom with$5S_{1/2}$,$5P_{3/2}$,$5D_{3/2}$, and$5D_{5/2}$behaving as the$|4〉$, $|1〉$,$|2〉$,and$|3〉$state labels, respectively. The state$5D_{3/2}$is coupled to the state$5D_{5/2}$by a resonant microwave field with frequency around 120 GHz, and the microwave field can be obtained by means of a cavity as illustrated in paper [52]. The value of the spontaneous decay rate $\Gamma$in this system is 6 MHz. Moreover, in order to eliminate the Doppler broadening effect, atoms should be trapped and cooled by the technique of magneto-optical trap MOT [53].

To sum up, we have shown that by choosing appropriate parameters of the system, such as the phase, amplitude, and initial population distribution, we can obtain a very wide variety of interesting localization patterns, such as craterlike, spikelike, latticelike, spherelike patterns, and so on. The spontaneous emission spectrum is sensitive to variables of the phase and amplitude of microwave field, so we can control the two-dimensional and three-dimensional atom localization behaviors more conveniently. The corresponding experiment to observe the expected 2D and 3D atom localization phenomena related to SGC can be more conveniently realized in atoms, since no rigorous conditions are required.