1. Introduction

High precision and high spatial resolution position measurement of moving atom has potential applications in neutral atom trapping [1], atom nanolithography [2], and quantum information processing [3]. Several schemes based on spontaneous emission [4–8], absorption [9–13] and gain [14] have been proposed due to the dependence of the position on the atom-field interaction. For the scheme based on spontaneous emission, when an atom passing through a standing-wave field the spontaneous emitted photon carries information about the atom’s position which is closely associated with the Rabi frequency of the standing-wave field. Therefore, information about the atom’s position can be obtained by detecting the spontaneously emitted photon.

For the multi-level atomic system driven by multi-fields, the atom localization has been studied by controlling the phase and amplitude of driving fields. To further improve the precision, atomic systems with external fields in a closed loop were proposed. Ding et al [6] explored the high-precision atom localization with a cyclic configuration via controllable spontaneous emission, and achieved sub-wavelength atom localization. Wan et al [7] proposed a new scheme for two-dimensional(2D) atom localization in a four-level atomic system with a closed loop three external fields, and obtained the sub-half-wavelength domain.

The quantum interference is a basic phenomenon for controlling spontaneous emission [15–19], the atom localization based on quantum interference effect has attracted much interest. Ghafoor and associates [18] demonstrated subwavelength atom localization via quantum interference in a three-level atomic system. For the coherently driven five-level M-type atomic system, Ding et al [20] explored the 2D atom localization. However, the localization still limited in subwavelength domain and the narrow localization peak cannot be obtained. In our paper, in order to further improve precision and spatial resolution of atom localization we try to introduce microwave coupling field between two upper levels in the M-type atomic system, and investigate the influence of coherent fields and SGC effect on atom localization. The 2D sub-half-wavelength atom localization with high-precision and high spatial resolution is obtained through adjusting the detuning and the Rabi frequency, and the atom can be localized in a region smaller than$\lambda /10\times \lambda /10$.

2. Theoretical model and equations

As shown in Fig. 1, we consider a system of a five-level atom interacting with three coherent fields, which resonantly couple the corresponding transitions. The upper levels$|3〉$and$|4〉$are coupled by a microwave field${\Omega }_C$, and the two upper levels decay to the lower level$|0〉$via interactions with the vacuum field modes. ${\Gamma }_1$and${\Gamma }_2$are the spontaneous decay rates. The lower level$|1〉$is coupled to the excited levels$|3〉$by a 2D standing-wave field${\Omega }_1(x,y)$, and lower level$|2〉$is coupled to the excited levels$|4〉$by a coherent driving field${\Omega }_2$.

 

Fig. 1 Schematic diagrams (a) An atom moving along$z$axis and passing through the two orthogonal standing-wave fields in$x-y$plane, (b) Coherently driven M-type five-level atomic system.

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The 2D standing-wave field${\Omega }_1(x,y)={\Omega }_{10}(\sin kx+\sin ky)$($k=2\pi /\lambda$) is the superposition of two orthogonal standing-wave with the same frequency. When an atom moves along the z direction and passes through the 2D standing-wave field with a high enough velocity, the kinetic energy of the atom along the direction of the field(z direction) is very large, and can be treated as a classical case. However, the velocity of the atom along the direction of wave-vector of the standing-wave is considered very small, and the enter-of-mass position of the atom through the standing-wave does not change during the interaction time. Thus, the kinetic energy term along wave-vector direction is far less than the kinetic energy term along z direction in the Hamiltonian, and can be neglected under the Ramman-Nath approximation [21]. Applying the rotating-wave approximation the Hamiltonian for the system in the interaction picture reads($\hslash =1$)

At any time$t$, the wave function of our system is

The CPP distribution$P\left(x,y;t|0,1_k〉\right)$is defined as the probability of finding the atom at position$(x,y)$in the 2D standing-wave fields, where a spontaneously emitted photon is detected at time$t$in the reservoir mode of wave vector$k$, which can be determined by taking the appropriate projection over the atom-field state$|\psi (t)〉$ [4,10]

Under the Raman-Nath approximation the center-of-mass wave function of the atom$f(x,y)$is assumed nearly constant over many wavelengths of the standing-wave fields in the$x-y$plane, thus the CPP distribution$P\left(x,y\right)$is determined by$a_k(t\to \infty )$. It is noted that the spontaneous emission spectrum$S({\omega }_k)$is proportional to${\left|a_k(t\to \infty )\right|}^2$. Thus, the $P\left(x,y\right)$ can be characterized by spontaneous emission spectrum. The probability amplitude$a_k(t\to \infty )$ can be found by solving the Schrodinger wave equation Eq. (1) and the wave function Eq. (2)

From Eq. (7) we can’t get analytical expression of$a_3(s)$and$a_4(s)$, here, setting$a_{30}(s)=B(s)/A(s)$and$a_{40}(s)=F(s)/E(s)$, where the$a_{30}(s)$and$a_{40}(s)$represent the zero-order approximation of$a_3(s)$and$a_4(s)$. Using the iterative method [22,23] once in Eq. (7) we obtain

3. Results and discussion

Since the frequency of spontaneously emitted photon depends on the position-dependent Rabi frequency${\Omega }_1(x,y)$, in principle the 2D position information of the atom passing through the orthogonal standing-wave can be extracted via frequency measurement of the emitted photons. The probable positions of the atom are then given by those values of$(x,y)$, where the CPP$P(x,y)$exhibits its maxima. From Eq. (9) we find that the CPP depends not only on the frequency of spontaneously emitted photon and the parameters of the laser fields, but also on the quantum interference effects. Firstly, we discuss the CPP distribution without the consideration of quantum interference($p=0$). All the parameters are scaled by$\Gamma$, and$\Gamma$is an arbitrary constant.

Considering the system is initially in the superposition of state$|1〉$and$|2〉$that$a_1(0)=a_2(0)=\sqrt{2}/2$. The CPP distribution is plot in Fig. 2 for${\delta }_k=3.2\Gamma$and${\delta }_k=3.53\Gamma$, respectively. In the case of${\delta }_k=3.2\Gamma$, the CPP displays two crater-like patterns in quadrant I and III as shown in Fig. 2(a). When${\delta }_k$is increased from${\delta }_k=3.2\Gamma$to${\delta }_k=3.53\Gamma$, the atom is localized mainly in quadrant III and the CPP distribution shows a broad localization peak in Fig. 2(b). In this case we just get sub-wavelength atom localization with a low spatial resolution. In order to describe clearly the localization properties we give the three-dimensional(3D) plot of CPP as shown in Figs. 2(c) and 2(d), which correspond the Figs. 2(a) and 2(b), respectively.

 

Fig. 2 The conditional position probability distribution versus the position$(x,y)$for different values of the detuning of spontaneously emitted photon when$p=0$and$a_1(0)=a_2(0)=\sqrt{2}/2$. (a) ${\delta }_k=3.2\Gamma$, (b) ${\delta }_k=3.53\Gamma$, (c) three-dimensional plot associated with (a), (d) three-dimensional plot associated with (b), other parameters are${\omega }_{34}=2\Gamma$,${\Gamma }_1={\Gamma }_2=\Gamma$,${\Omega }_{10}=2\Gamma$,${\Omega }_2=4\Gamma$,${\Omega }_c=\Gamma$.

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As shown in Fig. 2, we can’t get the narrow localization peak when$p=0$. Therefore, we will discuss the CPP considering the quantum interference between the two decay channels. When$p=1$, the CPP distribution as a function of$x$and$y$with four different detuning of the spontaneous emitted photon is shown in Fig. 3. Under the condition of${\delta }_k=-2.0\Gamma$, the CPP distributes in quadrants I and III but mainly in quadrants I, where two broad spark-like patterns can be found. When the detuning is tuned to${\delta }_k=-1.5\Gamma$and${\delta }_k=1.9\Gamma$, the maxima of the CPP distribution in Figs. 3(b) and 3(c) are situated in the quadrants I and III with a crater-like pattern, respectively. As the detuning is increased to an appropriate value (${\delta }_k=3.96\Gamma$), we obtain a narrow localization peak in quadrants III, which implies the spatial resolution is improved significantly. However, there is a very weak localization peak in quadrant I, therefore, we still can’t get the sub-half-wavelength atom localization.

 

Fig. 3 The conditional position probability distribution versus the position$(x,y)$for different values of the detuning of spontaneously emitted photon when$p=1$and$a_1(0)=a_2(0)=\sqrt{2}/2$. (a)${\delta }_k=-2.0\Gamma$, (b)${\delta }_k=-1.5\Gamma$, (c)${\delta }_k=1.9\Gamma$, (d)${\delta }_k=3.96\Gamma$, other parameters are${\omega }_{34}=2\Gamma$,${\Gamma }_1={\Gamma }_2=\Gamma$,${\Omega }_{10}=\Gamma$,${\Omega }_2=4\Gamma$,${\Omega }_c=\Gamma$.

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To further improve the precision and spatial resolution of atom localization, we will explore the dependence of the CPP distribution on the Rabi frequency${\Omega }_c$of microwave field. In order to compare the results under different microwave fields, the CPP for${\Omega }_c=\Gamma$ is given once again in Fig. 4(a). When the coupling between the two upper levels is ${\Omega }_c=\Gamma$, as plotted in Fig. 4(a), the CPP distributes in quadrant I and III but mainly in quadrant III. As the intensity of coupling field is decreased, for instance, ${\Omega }_c$is varied from${\Omega }_c=\Gamma$to${\Omega }_c=0.5\Gamma$, the atom is localized in quadrant III and a ultra-narrow localization peak was found as shown in Fig. 4(b), which means that the precision and the spatial resolution is improved significantly. The Figs. 4(c) and 4(d) are the 3D plot of CPP associated with Figs. 4(a) and 4(b), respectively. In Fig. 4(d), we can see a single ultra-narrow localization peak. From Fig. 4, we find that through introducing coupling field between two upper levels and considering the SGC effect in the atomic system, the spatial resolution of the atom localization is improved significantly compared with the results in Ref [20]. We can concluded that the ultra-high precision and spatial resolution of atom localization can be achieved by choosing the appropriate detecting frequency and intensity of microwave coupling field. In addition, when the system is initially in the superposition of state$|\text{3}〉$and$|\text{4}〉$($a_{\text{3}}(0)=a_4(0)=\sqrt{2}/2$), only the sub-wavelength atom localization with broad localization peak can be achieved at specific detuning. Therefore, we only give the results when the system is initially in the superposition of state$|1〉$and$|2〉$.

 

Fig. 4 The influence of microwave coupling field on the conditional position probability distribution versus the position$(x,y)$ $p=1$and$a_1(0)=a_2(0)=\sqrt{2}/2$. (a) ${\Omega }_c=\Gamma$,${\delta }_k=3.96\Gamma$, (b) ${\Omega }_c=0.5\Gamma$,${\delta }_k=1.99\Gamma$, (c) three-dimension plot associated with (a), (d) three-dimension plot associated with (b), other parameters are${\omega }_{34}=2\Gamma$,${\Gamma }_1={\Gamma }_2=\Gamma$,${\Omega }_{10}=\Gamma$,${\Omega }_2=4\Gamma$.

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Finally, we discuss the real atomic system that could be used for possible experiment realization. The cold atom ⁸⁷Rb (D₂ line) can be as a possible candidate, so that levels$|\text{0}〉$,$|\text{1}〉$,$|\text{2}〉$,$|\text{3}〉$and$|\text{4}〉$corresponds to states$5S_{1/2}|\text{F=1,}\text{m=0}〉$,$5S_{1/2}|\text{F=2,}\text{m=+2}〉$,$5S_{1/2}|\text{F=2,}\text{m=-2}〉$,$5P_{3/2}|\text{F=2,}\text{m=+1}〉$, and$5P_{3/2}|\text{F=2,}\text{m=-1}〉$, respectively. The two upper states decay with rates${\Gamma }_1={\Gamma }_2=\Gamma =6\text{MHz}$.

Last but not the least, for 2D atom localization based on spontaneous emission controlling, when the SGC effect and multi-coherent fields are introduced, we can get the high-resolution and precision atom localization by adjusting detecting frequency and intensity of external fields. As we know that the CPP is proportional to the spontaneous emission spectrum, therefore, the spontaneous emission properties of atomic system determine the localization properties. Based on the coherent control of spontaneous emission, the spectral line elimination and ultra-narrow spectral line can be achieved. The ultra-narrow spectral line results in the ultra-narrow localization peak in CPP distribution. That is why we can get the high-resolution and precision atom localization through controlling atomic spontaneous emission.

4. Summary

In summary, we have studied 2D sub-half-wavelength atom localization based on coherent fields controlling and SGC effect. The analytical expression of CPP was obtained by using iterative method when$p=1$. We found that the microwave coupling field between two upper levels and the SGC affect the atom localization significantly. When$p=1$, compared with the case without coupling field, the spatial resolution of the 2D atom localization is improved remarkably. By properly adjusting the detuning and the Rabi frequency, the atom can be localized in a region smaller than$\lambda /10\times \lambda /10$and the probability of the finding the atom at a particular position can reach 100%. Our work greatly improved the previous results that the atom is confined in$\lambda /2\times \lambda /2$region.

As a concluding remark, we mention that the experimental realization of our proposed scheme. The main challenge for realizing SGC effect is to have both near-degenerate atomic levels and non-orthogonal dipole moments, a condition rather difficult to find in real systems. To the best of our knowledge, there are two ways of observing the SGC effects. One is embedding the atom in anisotropy vacuum or coupling the atom with cavity fields [24,25]. The other way is simulating the SGC in the dressed-state picture [17,26–28]. In several coherently driven atomic systems, the rigorous condition for SGC can be achieved in the dressed-state picture. In these schemes, SGC effect with maximal quantum interference exists between the dressed states and the lower state, but does not need the rigorous atomic conditions. For the detection of spontaneously emitted photon we can achieve the measurement using the 4$\text{π}$ detectors [29]. However, it is not necessary to measure the position of each atom, it is sufficient to select those atoms whose spontaneous emitted radiation has indeed been detected by the detector.