1. Introduction

Over the past few years, highly precision and resolution measurements of atomic position have been considerable interest of several research groups since it plays important role in several applications based on the atomic coherence and quantum interference effects. These applications mainly lie in various areas of atomic physics such as laser cooling and trapping of neutral atom [1], center-of-mass wave function’s measurement for moving atoms [2,3], Bose-Einstein condensation (BEC) [4,5], atom nano-lithography [6,7], coherent patterning of matter waves [8], etc. Werner Heisenberg had proposed that precise measurement of atom’s position is diffraction limited and it can not be determined more precisely than the half of field’s wavelength. In other words, atom can not be localized less than $\lambda /2$, where $\lambda$ is the field’s wavelength used in detection. In order to get precise information about atom’s position, the atomic coherence and quantum interference effect have been used via incorporating phase measurement of standing-wave field [9,10]/atomic dipole [11], atomic fluorescence [12], spontaneous emission spectrum [13–15] and coherent gain manipulation of Raman gain processes [16], four-wave mixing [17] etc. There are some other techniques also such as using the upper-level population [18] and probe field absorption [19]. This later technique is very easy to perform experimentally as compare to other such as spontaneous emission where emitted photon is hard to control due to its randomness.

Initially for precision measurement, one dimensional (1D) atom localization was proposed for various atomic systems [20–26] where 1D subwavelength domain is realized by preparing standing wave along one direction (say $x$-axis). Later on, it was extended for 2D as well as for 3D subwavelength domain due to its unique feature and extensive applications. Ivanova et al. utilized a four-level tripod-type atomic system and discussed the atom localization in both, 2D [18] and 3D subwavelength domain [27] using the upper level population. They reported the localization patterns such as crater-, spike-, wave-like in 2D and similarly spherical isosurface, hourglasses, bowls, donuts in 3D case. However, unity probability of finding atom at a position in subwavelength domain is not achieved in both the cases. Ding et al. analysed 2D atom localization in a closed-loop Y-type system [19], N-tripode-type atomic system [28] using the probe absorption. Hamedi et al. have discussed phase sensitivity of 2D atom localization in a closed-loop Kobrak Rice five-level (KR$5$) scheme [29]. In all these proposals, the maximum detection probability of finding atom at a position, i.e., unity have been reported that is caused by joint quantum inference between transitions. Similarly, atomic systems such as $\Lambda$-type [30], five-level inverted Y-type [31,32], tripod-type [33], five-level X-type [34], N-type [35] have also been studied theoretically for 2D atom localization measurement. 2D spatial modulation of control field is also being actively used for preparing periodic arrangement of atoms which is known as atomic grating [36].

In last few years, the atom localization has got attention to visualize it’s shape in 3D subwavelength domain which is prepared by three pairs of orthogonal standing-waves. The shapes are plotted in term of isosurface for various atomic scheme. For instance, Qi et al. demonstrated theoretically 3D atom localization in M-type atomic system by measuring the spatial dependent probe absorption [37]. Afterward, the different 3D atom localization were achieved by Hamedi et al. via probe field absorption [38]. In these two studies, the maximum probability of finding atom in the wavelength domain could not be achieved. Thereafter, Wang et al. showed the high-efficiency (100$\%$) 3D atom localization in a conventional three levels $\Lambda$ [39] and $\Xi$ [40] type system. This study has also been extended in several others atomic system such as double two-level system by Zhu et al. [41], closed-loop four-level scheme by Mao et al. [42], closed-loop four-level atomic system by Duo Zhang et al. [43] and closed-loop M-type atomic scheme by our group [44]. In all these studies, it has been shown that the maximum probability of finding the atom (i.e., unity) is achievable in 3D domain under specific parametric conditions.

In a real atomic system, the effect of closely spaced nearby levels can also change the probe absorption response [45]. Thus we can not neglect the presence of nearby levels while studying the atom localization. To show it, we have considered a conventional $\Xi$-type atomic system with two nearby upper levels [46] and this system is driven by two coherent e.m. fields. One of them is a probe field and another is a standing-wave couple field. As we are aware that in the presence of standing-wave field, there will be a position dependent atom-light interaction. This interaction results in a spatial dependent probe absorption that can be utilized for studying atom localization. The maxima of probe absorption provide precise information about the atom’s position in the subwavelength domain. This study can be performed for 2D and 3D subwavelength domain depending upon standing-wave field configuration. In 2D case, the nearby upper levels with proper probe detuning enhance the spike-like feature that infer about high precise atom’s position. While in 3D case, these nearby upper levels increase the range of fields’ parameter during which the probability of finding atom is constant. Besides this, the maximum probability of finding atom in subwavelength domain can also be obtained for specific parametric conditions. This paper is organized as follows: model and dynamical equations are discussed in section 2. Section 3 comprises numerical results regarding 2D and 3D atom localization followed by a conclusions in section 4.

2. Model and dynamical equations

We consider a conventional $\Xi$-type atomic system with two closely spaced nearby upper levels as shown in Fig. 1(a). This $\Xi$-type system is realized in $^{87}$Rb atom where $|1\rangle =|5S_{1/2}, F=3\rangle$, $|2\rangle =|5P_{3/2}, F=3\rangle$, $|3\rangle =|5D_{5/2}, F=3\rangle$, $|4\rangle =|5D_{5/2}, F=4\rangle$, and $|5\rangle =|5D_{5/2}, F=2\rangle$. $\delta _1$ ($=9.0$ MHz) and $\delta _2$ ($=7.6$ MHz) denote the frequency separations between the levels $|3\rangle -|4\rangle$ and $|3\rangle -|5\rangle$, respectively. The transition $|1\rangle \leftrightarrow |2\rangle$ of frequency $\omega _{21}$ is driven by a weak probe field ($\vec {E}_p$) with a Rabi frequency $G_{p}$ ($=\vec {d}_{12}\cdot \vec {E}_p/\hbar$). However, an intense couple field ($\vec {E}_c$) drives the transition $|2\rangle \leftrightarrow |3\rangle$ with a Rabi frequency $G_{c}$ ($=\vec {d}_{23}\cdot \vec {E}_c/\hbar$). Being $|4\rangle$ and $|5\rangle$ proximity to $|3\rangle$, this $\vec {E}_c$ field also couples the $|4\rangle$ and $|5\rangle$ levels having Rabi frequencies $a_{42}G_c$ and $a_{52}G_c$, respectively. Here, $a_{42}$($=d_{42}/d_{32}$) and $a_{52}$ ($=d_{52}/d_{32}$) are relative transition strengths. $d_{jk}$ is the dipole matrix element for a transition $|j\rangle \leftrightarrow |k\rangle$. We deal with only stationary atoms in numerical calculation. These stationary atom can be achieved experimentally in nano-cell or in ultracold BEC.

 

Fig. 1. (a) The schematic diagram of $\Xi$ atomic scheme with nearby levels where realistic candidate is $^{85}$Rb. (b) 2D field configuration (c) 3D field configuration, Blue circle stands for the atom.

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In our theoretical calculation, it is assumed that the center-of-mass position of atom is nearly constant along standing-wave fields. Therefore, the kinetic energy part of atom can be neglected in the total Hamiltonian under the Raman-Nath approximation [47]. The resultant Hamiltonian of atom-light system using rotating-wave approximation (RWA) [48] can be written as:

The Louville equation incorporating phenomenological relaxation contribution is used to obtain the evolution of atomic dynamics in form of density matrix.

In the limit of weak probe and intense couple field approximation ($G_p<<G_c$), We use perturbation approach to solve OBEs Eq. (3) by defining $\rho _{jk}=\rho _{jk}^{(0)}+\lambda \rho _{jk}^{(1)}+\lambda ^2\rho _{jk}^{(2)}\cdots$, where $\lambda$ represents perturbation parameter whose values lies between zero and one. Here, $\rho ^{(0)}_{jk}$, $\rho ^{(1)}_{jk}$, and $\rho ^{(2)}_{jk}$ are of the zeroth, first, and second order in $G_p$, respectively. In the weak probe field approximation, $\rho ^{(0)}_{11}=1$ and other zeroth order density matrix elements are zero. The analytic expression for $\rho ^{(1)}_{21}$ under the steady state consideration (i.e., $\dot {\rho }_{jk}=0$) is given as:

3. Numerical result and discussion

In this section, we demonstrate the atom localization with and without nearby levels cases for various parametric conditions such as $\Delta _c$, $\Delta _p$, additional running field ($g_c$). In our detailed calculation, the chosen atomic parameters are taken as: $\Gamma _{21}/2\pi =6.1$ MHz, $\Gamma _{32}/2\pi =\Gamma _{42}/2\pi =\Gamma _{52}/2\pi =0.97$ MHz, $\delta _1/2\pi =9$ MHz, $\delta _2/2\pi =7.6$ MHz and $a_{32}:a_{42}:a_{52}=1:1.4:0.6$. Equations (5) and (6) can not provide the exact analytic expression for positions ($kx$, $ky$, $kz$) of maxima filter function even though both fields are tuned resonantly to respective atomic transition. However, we can perform the numerical calculation for 2D and 3D filter functions which directly reflect the atom localization and conditional position probability distribution of atom in subwavelength domain.

3.1 2D localization structures

In order to see the effect of nearby levels on 2D atom localization in subwavelength domain, the plots ($F(x,y)$ vs $(kx, ky)$) are illustrated in Fig. 2 for various $\Delta _p$. The number of peaks in 2D subwavelength domain shows conditional position distribution. In the absence of nearby levels with $\Delta _p=0\times \Gamma _{21}$, the atom localizes along the diagonal in second ($-\pi <kx<0$, $0<ky<\pi$) and fourth ($0<kx<\pi$, $-\pi <ky<0$) quadrants of $x-y$ plane as shown in Fig. 2(a). It can be seen that $F(x,y)$ maxima is spatially uniform corresponding to $kx+ky=2m\pi$ [or $kx-ky=(2n+1)\pi$], where m and n are integer numbers. Making the probe field detuned by $3.1\Gamma _{21}$, atom localization structure become crater-like that is situated in the first ($0<kx<\pi$, $0<ky<\pi$) and third (-$\pi <kx<0$, $-\pi <ky<0$) quadrants of $x-y$ plane (Fig. 2(b)), and the Atom localizes along the circular edge of two crater. For $\Delta _p=5.0\Gamma _{21}$, atom localization appears in spike-like shape (Fig. 2(c)) in same quadrants. Thereafter, the degree of localization decreases rapidly with increasing $\Delta _p$ but the detection probability of finding atom remains constant as shown in Fig. 2(d) for $\Delta =8.9\Gamma _{21}$. Now considering the nearby levels, no change in atom localization is found for $\Delta _p=0\times \Gamma _{21}$ as can be seen in Fig. 2(e). For the same parametric case as in Fig. 2(b), atom localizes like very large crater shape situated in first, third quadrants as well as small localization in others two quadrants (Fig. 2(f)). Now taking $\Delta _p=5.0\Gamma _{21}$, we observe localization structure again in shape of crater in first and third quadrants as shown in Fig. 2(g). Further increasing the probe field detuning, i.e., at $\Delta _p=8.9\Gamma _{21}$, our numerical calculation exhibit spike-like atom localization for (Fig. 2(h)). If we compare the Fig. 2(c) and (h), the spike-like feature is more narrower in later case which is due to the combined effect of $\Delta _p$ and nearby levels. Thus, a better spatial resolution in conditional position probability distribution of atom can be achieved by considering the nearby levels for specific parametric condition. In such case, atom localization in two of four quadrants shows that the probability of finding the atom in subwavelength domain is limited, i.e., 1/2.

 

Fig. 2. Variation of $F(x,y)$ in 2D subwavelength domain in absence (left column) and presence (right column) of nearby levels for various $\Delta _p$. (a & e) $\Delta _p=0\times \Gamma _{21}$, (b & f) $\Delta _p=3.1\Gamma _{21}$, (c & g) $\Delta _p=5.0\Gamma _{21}$, (d & h) $\Delta _p=8.9\times \Gamma _{21}$. The other fields’parameter are: $\Delta _c=0\Gamma _{21}$, $\Omega =5.0\Gamma _{21}$, $G_p=0.001\Gamma _{21}$.

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So far, we could not acquire the atom localization in single quadrant of 2D subwavelength domain, i.e. unity probability of finding atom at a position. Therefore to achieve it, we modify the couple field configuration by introducing an additional running-wave couple field ($g_c$) that also drives the upper transitions. For this particular case, $G_c$ can be defined as $G_c(x,y)=\Omega (\sin (x)+\sin (y))+g_c$. We now discuss the atom localization for different values of $g_c$ and $\Delta _c$ in the presence of nearby levels. Figure 3(a) displays two spike-like atom localization with high resolution for the case $g_c=\Delta _c=0\times \Gamma _{21}$. This additional $g_c=0.5\Gamma _{21}$ makes change in the atom localization from spike-like to crater-like as shown in Fig. 3(b). As it can be seen that degree of localization enhances in one of those quadrants but still the localization probability is same like in previous case, i.e., 1/2. Now increase the $g_c$ value from $0.5\Gamma _{21}$ to $1.0\Gamma _{21}$, the crater-like structure starts broadening as shown in Fig. 3(c). This broadening reflects the reduction in precision measurement of atom’s position. One can also notice that the detection probability of atom at a position is enhanced to 1 because the crater localizes in one quadrant. Now set $\Delta _c=2.0\Gamma _{21}$ for same $g_c=\Gamma _{21}$, we observe very narrow spike-like atom localization situated at position ($kx=ky=\pi /2$) in one quadrant (Fig. 3(d)). Therefore, the probability of finding the atom at a position is increased by factor of 2 than in the case of Fig. 2. This can be understood as follows: When we substitute $G(x,y)$ in Eq. (4), it can be shown that there will be an additional quantum interference term $2\Omega (x,y)\times gc$. If gc = 0, there will be no quantum interference in upper transitions. Thus atom localizes at two position in 2D subwavelength domain as shown in Fig. 3(a) [because $F(x,y)$ does not alter under parity transformation $(x, y) \leftrightarrow (-x, -y)$]. On the other hand, when $g_{c}$ has some positive value, there will be a position dependent quantum interference and it will no longer hold parity transformation. This causes an asymmetry in the degree of atom localization. This asymmetry can be increased further such that atom localizes at a position under specific parametric condition (Fig. 3(d)). Thus for certain parameter values, enhanced precision with high resolution of atom position measurement is achievable by considering the nearby levels.

 

Fig. 3. Variation of $F(x,y)$ is 2D subwavelength domain in presence of nearby levels for varying $g_c$ and $\Delta _c$. (a) $g_c=0\Gamma _{21}$, $\Delta _c=0\Gamma _{21}$, (b) $g_c=0.5\Gamma _{21}$, $\Delta _c=0\Gamma _{21}$, (c) $g_c=1.0\Gamma _{21}$, $\Delta _c=0\Gamma _{21}$, (d) $g_c=1.0\Gamma _{21}$, $\Delta _c=2.0\Gamma _{21}$. The other fields’parameter are: $\Delta _p=8.9\Gamma _{21}$, $\Omega =5\Gamma _{21}$, $G_p=0.001\Gamma _{21}$.

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3.2 3D localization structures

In this section, we extend the numerical calculations to analyse the influence of nearby upper levels on 3D atom localization. For this purpose, isosurface is plotted as a function of positions ($kx, ky, kz$) for optimized value of $F(x,y,z)=0.5$ considering the standing-wave couple fields in all three directions, i.e., $G_c(x,y,z)=\Omega (\sin (kx)+\sin (ky)+\sin (kz))$. In Fig. 4, isosurface plots are given for different values of $\Delta _p$ in the absence and presence of nearby levels. It can be seen that the atom localization are quite sensitive to value of $\Delta _p$ due to strong correlation between $\Delta _p$ and $F(x,y,z)$. Figure 4(a) shows the isosurface plot for the case $\Delta _p=5.5\times \Gamma _{21}$ when the nearby levels are not considered. The plot shows that both isosurface are totally spherical of large diameter in two quadrants ($-\pi \leq kx,ky,kz\leq 0$ and $0\leq kx,ky,kz\leq \pi$) of 3D subwavelength domain. The presence of two symmetrical spherical isosurface can be explained using filter function ($F(x,y,z)$) that does not alter under parity transformation $(x,y,z)\leftrightarrow (-x,-y,-z)$. The diameter of both isosurfaces decreases with increasing the $\Delta _p$ (= $7.7\Gamma _{21}$) shown in Fig. 4(b). In this case, the probability of finding atom in 3D subwavelength domain is only 1/2 but with a high spatial resolution in the conditional position probability distribution. If keep the value of $\Delta _p$ same and incorporating the nearby levels then atom delocalizes in all the quadrants as shown in Fig. 4(c). Further an increase in $\Delta _p$ ($=9.1\Gamma _{21}$) makes the atom to localize only in two quadrants, shown in Fig. 4(d). On further increasing $\Delta _p$ ($=13.5\Gamma _{21}$), the size of both spherical isosurface reduces (Fig. 4(e)) which reflects the similar result as in Fig. 4(b), i.e., 1/2 atom localization probability with high resolution. A significance difference is found in these results depending whether the nearby levels are considered or not. In [Fig. 4(d-e)] plotted for the case of nearby levels, the localization probability remains same for a longer range of $\Delta _p$ than when no nearby levels are considered (Fig. 4(a-b)). In other words, the probability of finding atom at a position is less sensitive for a certain range of $\Delta _p$ when nearby levels are considered. This can be understood as follows: The proximity of the upper levels will reduce two photon detuning difference in three atomic sub-systems ($|1\rangle \leftrightarrow |2\rangle \leftrightarrow |3\rangle$ or $|4\rangle$ or $|5\rangle$). Thus, it will produce a longer range of probe detuning for a fixed probe absorption, i.e., $F(x, y, z)$.

 

Fig. 4. Isosurface for $F(x,y,z)=0.5$ in the absence (a$-$b) and presence (c$-$e) of nearby levels for varying $\Delta _p$. (a) $\Delta _p=5.5\Gamma _{21}$, (b) $\Delta _p=7.7\Gamma _{21}$, (c) $\Delta _p=7.7\Gamma _{21}$, (d) $\Delta _p=9.1\Gamma _{21}$, (e) $\Delta _p=13.5\Gamma _{21}$. The other fields’parameter are: $G_p=0.001\Gamma _{21}$, $\Omega =5\Gamma _{21}$, $\Delta _{c}=0\Gamma _{21}$.

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In measuring 3D atom localization, we prefer the maximum probability of finding atom, i.e., unity with a high spatial resolution. To achieve it, an additional running-wave couple field ($g_c$) is added in previously taken field configuration. Now, the expression for control field can be defined as $G_{c}=\Omega (\sin(kx)+\sin(ky)+\sin(kz))+g_{c}$. During the numerical calculation, $g_c$ value is taken to be a constant, i.e., $g_c=1.0\Gamma _{21}$ but vary $\Delta _c$ in presence of nearby levels. In Fig. 5(a), we show atom localization for a resonant control field ($\Delta _c=0\Gamma _{21}$) that displays two spherical isosurface but having different diameter situated in diagonal quadrants of 3D domain. The size of both spherical isosurface can be further reduced by making larger $\Delta _c$ as shown in Fig. 5(b). At higher $\Delta _c=5.0\Gamma _{21}$, atom completely localizes in single quadrant ($0\leq kx,ky,kz\leq \pi$) and it shows the unity probability of finding atom at position in 3D subwavelength domain, shown in Fig. 5(c). Further increase the $\Delta _c=7.5\Gamma _{21}$, the atom localization volume reduces more and it allows a high precision of atom’s position measurement. This is an unique feature of 3D atom localization. Our analysis conclude that precisioness of position measurement is inversely proportional to $\Delta _c$ because the size of spherical isosurface decreases with increasing $\Delta _c$.

 

Fig. 5. Isosurface for $F(x,y,z)=0.5$ in presence of nearby levels for varying $\Delta _c$. (a) $\Delta _c=0\Gamma _{21}$, (b) $\Delta _c=2.5\Gamma _{21}$, (c) $\Delta _c=5.0\Gamma _{21}$, (d) $\Delta _c=7.5\Gamma _{21}$. Others field parameters are: $g_c=1.0\Gamma _{21}$, $\Delta _p=11\Gamma _{21}$, $G_{p}=0.001\Gamma_{21}$

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Our numerical calculations also exhibit that efficient controlling of the probe absorption also helps in getting the precise information of atom’s position in the presence of nearby levels. For this particular purpose, the isosurfaces are plotted for various values of $F(x,y,z)$ while keeping the other parameters fixed as shown in Fig. 6. We first plot the isosurface for $F(x,y,z)=0.1$ (Fig. 6(a)) that show spherical isosurface localized in quadrant ($0\leq kx,ky,kz\leq \pi$) with maximum conditional probability of finding atom. Increasing $F(x,y,z)$ results in high spatial resolution in conditional probability position distribution as diameter of isosurface decreases, shown in Fig. 6(b,c). The reason for such localization control is that the absorption spectra line becomes narrower corresponding to a higher value of the probe absorption.

 

Fig. 6. Isosurface for various probe absorption. (a) $F(x,y,z)=0.1$, (b) $F(x,y,z)=0.4$, (c) $F(x,y,z)=0.7$. The other fields’parameter are: $\Omega =5.0\Gamma _{21}$, $\Delta _p=10\Gamma _{21}$, $\Delta _c=11\Gamma _{21}$, $g_c=1.0\Gamma _{21}$.

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Before summarizing, we give physical explanation for reported atom localization using dressed state approach. The dressed Hamiltonian ($H_d$) contains interaction term of atom-light system without probe field. Hence, it can be written as:

4 Conclusion

In conclusion, we have analysed the atom localization in a conventional $\Xi$ atomic system with nearby upper levels for 2D as well as for 3D cases. Our numerical calculations suggest that various kind of atom localization patterns such as diamond-, crater- and spike-like are observed for 2D case and different sizes spherical isosurface for 3D case. It is shown that nearby levels can change the atom localization patterns significantly. Most importantly in 2D case, proper adjustment of parametric conditions in the presence of nearby levels result in the narrowing of spike-like feature than in conventional three level $\Xi$-type atomic system. Thus, enhanced precise information of atom’s position can be achieved. Another main result in 3D case study, the probability of finding atom in 3D subwavelength domain remains constant over a long range of $\Delta _p$ in the presence of nearby levels. It means nearby levels reduce the sensitivity in certain range of $\Delta _p$ value, so in an experiment the atom localization can be found easily. Numerical calculations also reveal that the maximum probability of finding atom in subwavelength domain, i.e., unity is achievable in both 2D and 3D cases. Our study can be extended to any atomic system with nearby multiple levels for precise information of atom’s position.