1. Introduction

In the past few decades, there has been a lot of attention paid to the high precision position measurement of an atom passing through a standing-wave field since it can find many applications in laser cooling and trapping of neutral atoms [1], atom lithography [2], measurement of the center of mass wave function of moving atoms [3], and coherent patterning of matter waves [4], etc. Due to the position dependent atom-field interaction, there have been several proposals for atom localization by using standing-wave fields. Earlier schemes include the measurement of the phase shift of the field inside the cavity [5], the atomic dipole [6], and the entanglement between atom’s position and its internal state [7]. More recently, atom localization based on atomic coherence and quantum interference effect has been attracted much interest. The spatially dependent spontaneous emission [8–11], absorption [12, 13], population [14–18], and gain [19] can directly provide the position information of the atom in sub-wavelength domain. These schemes have more advantages because the atom is localized when still inside the standing-wave. Compared with one-dimensional (1D) atom localization, the two-dimensional (2D) atom localization has a better prospect of application, and hence is extensively studied in recent years. By interacting with two orthogonal standing-waves, the population of a four-level atom [20–22], the controlled spontaneous emission from a driven tripod system [23] or M-type system with decay induced interference [24], and the absorption of N- [25] or N-tripod-type [26] atom are used to achieve 2D atom localization.

Of great attention is the enhancement of precision in the localization of an atom. For 1D atom localization, in general, there are four localization peaks in a unit wavelength domain of the standing-wave [8, 9, 14]. However, by using phase dependent absorption [12, 13] or spontaneous emission [10, 11] as well as the two standing-waves with different wavelength [19,27], the atom localization peaks reduce to two in either of the two half-wavelength, and the detection probability at the particular position is then improved to 1/2. For 2D atom localization, the probability of finding the atom is 1/4 in each quadrant of the $x-y$ plane of the orthogonal standing-waves [20–22, 26]. With the quenching in the controlled spontaneous emission [23] and quantum interference [24], the atom can be localized in two of the four quadrants of the $x-y$ plane, and then the uncertainty in position measurement is increased by a factor of 2. The similar results can be also obtained as the two standing-waves driving the same atomic transition where the spatial interference of the standing-waves plays an important role [25].

In order to further improve the precision, we propose a new scheme for 2D atom localization in a four-level atomic system with a closed loop. This phase sensitive system can exhibit strong line narrowing and selective cancellation of the spontaneous emission spectra [28], which is then exploited to enhance the efficiency of the 2D atom localization. By using two standing-waves to drive the same transition, the spatial interference together with the phase dependence can greatly reduce the periodicity in the position probability distribution. For the measurement of spontaneously emitted photon at particular frequency, we can localize the atom in sub-half-wavelength domain. The maximal probability of finding an atom at a certain position is 100%, which is increased by a factor of 2 or 4 compared with the previous schemes [20–26].

2. Theoretical model

The schematic diagrams of the proposed scheme are shown in Fig. 1 . We consider an atom moves in the $z$ direction and passes through the intersectant region of two orthogonal standing-wave fields, which are respectively aligned along the $x$ and the $y$ axises [see Fig. 1(a)]. The atomic system is shown in Fig. 1(b). The ground level $|0〉$ is coupled to the excited levels $|1〉$ and $|2〉$ by two laser fields $E_1$ and $E_2$ (carrier frequencies ${\omega }_1$ and ${\omega }_2$), while transition $|1〉\leftrightarrow |2〉$, usually dipole forbidden, is driven by a microwave $B_c$ (carrier frequency ${\omega }_c$). The two upper levels decay to the lower level $|j〉$ via interacting with the vacuum modes in free space. Those three fields form a closed loop subsystem, which results in phase dependent light-matter interaction and spontaneous emission. Coupling field $E_1$ is the superposition of two orthogonal standing-waves with the same frequency and aligned along the $x$ and the $y$ directions, respectively. Therefore, the corresponding Rabi frequency, i.e., ${\Omega }_1(x,y)={\Omega }_{10}[\sin (kx)+\sin (ky)]$ ($k=2\pi /{\lambda }_1$), is spatially dependent in the $x-y$ plane [see Fig. 1(a)]. Fields $E_2$ and $B_c$ are traveling-wave with position-independent Rabi frequencies ${\Omega }_2$ and ${\Omega }_c{e}^{i\varphi }$, respectively, where actually $\varphi$ is the collective phase of the closed loop.

 

Fig. 1 Schematic diagrams: (a) An atom moves along the $z$ axis and interacting with two orthogonal standing-wave fields in the $x-y$ plane. (b) Four-level atomic system. Levels $|0〉$, $|1〉$ and $|2〉$ are driven by three fields ${\Omega }_1$, ${\Omega }_2$ and ${\Omega }_c$, and then form a closed loop. ${\Delta }_k={\omega }_k-({\omega }_{1j}+{\omega }_{2j})/2$ is the detuning of the spontaneously emitted photon with frequency ${\omega }_k$ from the average atomic transition frequency $({\omega }_{10}+{\omega }_{20})/2$.

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We consider that the atom is moving with sufficiently high velocity in the $z$ direction that the motion of atom can be treated classically. The interaction region and time are small enough so that the total kinetic energy acquired by the atom due to the recoil effect is small compared with its interaction energy with the optical fields, and therefore there is no significant variation of the $x\text{-}$($y\text{-}$) velocity of the atoms as they interact with the coupling fields. Hence we can apply the Raman-Nath approximation and ignore the kinetic energy of the atom in the Hamiltonian. Under the electric-dipole and rotating-wave approximations, with the assumption of $\hslash =1$, the resulting interaction Hamiltonian for the system reads

The atom-field state vector of our considered system at time $t$, whose evolution obeys the Schrödinger equation, can be written as

When we have detected at time $t$ a spontaneously emitted photon in the vacuum mode of wave vector $k$, the atom is in the state $|j〉$ and the state vector of the system, after making appropriate projection over $|\psi (t)〉$, is reduced to

As is well known, the spontaneous emission spectrum $S({\Delta }_k)$ is proportional to ${\left|a_{jk}(t\to \infty )\right|}^2$. Thus, the spontaneous emission from the atom can be used to characterize the conditional position probability distribution $W(x,y;t|j,1_k)$. In this scheme, the localization of the atom is conditioned on the detection of the spontaneously emitted photon. In fact, due to the standing-wave fields, the light-atom interaction is spatially dependent in the $x-y$ plane, and therefore the frequency of the spontaneously emission carries the position information of the atom.

The probability amplitude $a_{jk}(t\to \infty )$ can be obtained by solving the Schrödinger wave equation with the interaction Hamiltonian [Eq. (1)] and the atom-field state vector [Eq. (2)]. With the Weisskopf-Wigner theory, the dynamical equations for the atomic probability amplitudes are given by

By utilizing the Laplace transform method and the final-value theorem, we obtain $a_{jk}$ in the long-time limit as

3. Numerical results and discussions

As can be seen from Eq. (7), since the spontaneous emission spectrum $S({\Delta }_k)$ depends on $\sin (kx)$ and $\sin (ky)$ from ${\Omega }_1(x,y)$, it is in principle possible to extract the 2D position information of the atom as it passes through the orthogonal standing-waves via frequency measurement of the spontaneously emitted photon. The probable positions of the atom are then given by those values of $(x,y)$ where $S({\Delta }_k)$ exhibits maxima. However, the form of $S({\Delta }_k)$, which can show the position probability distribution clearly, is rather complicated. Therefore, we only present numerical results and discuss various conditions for the precise atom localization.

Due to the closed loop configuration of the coherently driven transitions $|0〉\stackrel{{\Omega }_1}{\to }|1〉\stackrel{{\Omega }_c{e}^{i\varphi }}{\to }|2〉\stackrel{{\Omega }_2}{\to }|0〉$, Autler-Townes splittings, i.e. dressed states of the atom, rely on the relative phase of the three fields. Consequently, the spontaneous emission is phase dependent as well. In the following discussions, we consider three cases according to different phase: $\varphi =\pi /2$, $\varphi =0$ and $\varphi =\pi$. In the calculations, we set ${\Gamma }_1={\Gamma }_2=\gamma$ and all the other parameters are scaled by $\gamma$.

In the case of $\varphi =\pi /2$, we show the plots of the spontaneous emission $S({\Delta }_k)$ which can directly determine the conditional position probability distribution as a function of $(kx,ky)$ with four different detunings of the spontaneous emitted photon in Figs. 2(a) –2(d). It is clear that the conditional position probability distribution depends strongly on the detuning ${\Delta }_k$. Then the measurement of the frequency of emitted photon can lead to the localization of the atom in a sub-wavelength domain. When the spontaneous emission is resonant with the atomic transition $|1〉\leftrightarrow |j〉$, i.e., ${\Delta }_k=-10$, the peak maxima, which represent the most probable positions of the atom, are distributed in quadrants II and IV of the $x-y$ plane. In such case, the atom localization peak in a unit wavelength domain are determined by $kx-ky=\pm \pi$ or $kx+ky=0$ [see Fig. 2(a)]. As the detuning is tuned to ${\Delta }_k=-4.2$, the peak maxima are situated in all four quadrants but mainly in quadrants I and III [see Fig. 2(b)]. When ${\Delta }_k=-3.5$, the conditional position probability distribution displays crater-like pattern in quadrants I and III which leads to the localization of atom at these circles [see Fig. 2(c)]. As the detuning increases to ${\Delta }_k=-3$, the peaks maxima show spike-like pattern, and the atom is located at the positions $(kx,ky)=(\pi /2,\pi /2)$ or $(kx,ky)=(-\pi /2,-\pi /2)$ as shown in Fig. 2(d). Therefore, the probability of finding the atom at each position is $1/2$ when a photon with frequency ${\omega }_k=({\omega }_{1j}+{\omega }_{2j})/2-3$ is spontaneously emitted and then measured. This is the maximum probability obtained in our previous proposed schemes [23, 25], and is increased by a factor of 2 compared with other schemes [20–22, 26]. Moreover, it is notable that the position probability of the atom at other positions except for the peak maxima cannot be neglected in Figs. 2(a) and 2(b), and therefore the signal-to-background ratio of the atom localization is extremely low.

 

Fig. 2 The spontaneous emission $S({\Delta }_k;x,y)$ (in arbitrary unit) which directly describes the conditional position probability distribution as a function of $(kx,ky)$in dependence on the detuning of the spontaneously emitted photon when $\varphi =\pi /2$. (a) ${\Delta }_k=-10$; (b) ${\Delta }_k=-4.2$; (c) ${\Delta }_k=-3.5$; (d) ${\Delta }_k=-3$. Other parameters are ${\Gamma }_1=1$, ${\Gamma }_2=1$, ${\Omega }_{10}=2$, ${\Omega }_2=4$, ${\Omega }_c=4$, $\delta =10$, ${\Delta }_1=-\delta$ and ${\Delta }_2=\delta$.

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In the case of $\varphi =0$, the conditional position probability distributions versus the normalized position $(kx,ky)$ are shown in Fig. 3 with different detunings of the spontaneously emitted photon. On the condition of ${\Delta }_k=-10$, the peak maxima exhibit lattice-like pattern [see Fig. 3(a)], which is same as the case in Fig. 2(a). Nevertheless, the signal-to-noise ratio and the spatial resolution of the atom localization are significantly enhanced. When ${\Delta }_k=-9$, the peak maxima locate in quadrants II, III and IV of the $x-y$ plane, and the probability of finding the atom in quadrant I completely vanish [see Fig. 3(b)]. As the detuning is increased to a larger value [for example ${\Delta }_k=-7$ in Fig. 3(c)], we obtain crater-like probability distribution, which indicates that the atom passes through the standing-waves from the circular edges of the peak maxima only in quadrant III. On the condition that ${\Delta }_k=-6$, the position possibility distribution shows spike-like pattern which locates at position $(kx,ky)=(-\pi /2,-\pi /2)$ as shown in Fig. 3(d). That is to say, when we have detected a scattered photon with frequency ${\omega }_k=({\omega }_{1j}+{\omega }_{2j})/2-6$, the probability of finding the atom at such position is 100% in one period of standing-waves. Compared with the case of $\varphi =\pi /2$ and our previous schemes, this probability is increased by a factor of 2. From Figs. 3(c) and 3(d), it is clear that the atom is confined in ${\lambda }_1/2\times {\lambda }_1/2$ region. Hence we can achieve 2D sub-half-wavelength atom localization with high precision and high spatial resolution.

 

Fig. 3 The spontaneous emission $S({\Delta }_k;x,y)$ (in arbitrary unit) which directly describes the conditional position probability distribution as a function of $(kx,ky)$in dependence on the detuning of the spontaneously emitted photon when $\varphi =0$. (a) ${\Delta }_k=-10$; (b) ${\Delta }_k=-9$; (c) ${\Delta }_k=-7$; (d) ${\Delta }_k=-6$. Other parameters are the same as Fig. 2.

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From Eq. (10), we can obtain that the position dependent spontaneous emission remains unchanged under the transform $0\leftrightarrow \pi$ and $(x,y)\leftrightarrow (-x,-y)$, i.e. $S({\Delta }_k;x,y;\varphi =0)=S({\Delta }_k;-x,-y;\varphi =\pi )$. Therefore, in the case of $\varphi =\pi$, we expect a vice versa of the previous case ($\varphi =0$). The quenching of spontaneous emission takes place in quardrant III of the $x-y$ plane, and we can also achieve 2D sub-half-wavelength atom localization.

According to the results, we can see that the conditional position probability distribution indicates a strong correlation between the detuning of the spontaneously emitted photon and the position of the atom. As a consequence, the measurement of a certain frequency leads to sub-wavelength or even sub-half-wavelength localization of the atom. To achieve sub-half-wavelength atom localization, the collective phase of the closed loop plays an important role. In order to explain the above results, we plot the spontaneous emission spectrum $S({\Delta }_k)$ versus the Rabi frequency ${\Omega }_1$ in Fig. 4 . In the case of $\varphi =\pi /2$, when we have detected one photon with detuning ${\Delta }_k$ (e.g. ${\Delta }_k=-3.5$), the corresponding Rabi frequency of the field $E_1$ has two values $\pm {\Omega }_1$ [see Fig. 4(a) and its density plot Fig. 4(b)]. That is to say, the potential locations of the atom have equal probability in quadrant I (II) and quadrant III (IV). However, if the phase is $\varphi =0$, one detuning ${\Delta }_k$ corresponds to only one Rabi frequency ${\Omega }_1$ [see Fig. 4(c) and its density plot Fig. 4(d)]. Moreover, due to the spatial interference of the two standing-waves, the Rabi frequencies ${\Omega }_1(x,y)$ in the four quadrants have different values. Therefore, the uncertainty in a particular position measurement of the atom is reduced. Especially, for some conditions, the probability can be doubled. The case of $\varphi =\pi$ is the same with $\varphi =0$. Consequently, the atom is localized in ${\lambda }_1/2\times {\lambda }_1/2$ domain and sub-half-wavelength atom localization can be achieved.

 

Fig. 4 The spontaneous emission $S({\Delta }_k)$ as a function of ${\Omega }_1$. In (a) $\varphi =\pi /2$ and (b) $\varphi =0$. (c) and (d) are the density plots of (a) and (b). Other parameters are the same as Fig. 2.

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Finally, we investigate the influence of Rabi frequencies of the coupling fields on the atom localization. Figures 5(a) and 5(b) show the pattern of position probability distribution with ${\Omega }_{10}=3$, ${\Omega }_2=6$ and ${\Omega }_3=6$ in the cases of $\varphi =0$ and $\varphi =\pi$, respectively. Compared with Fig. 3(d), the localization peak structure is greatly narrowed as the Rabi frequencies increased. Therefore, we can obtain a better spatial resolution in position measurement of the atom, which is necessary in the application of atom lithography.

 

Fig. 5 The spontaneous emission $S({\Delta }_k;x,y)$ (in arbitrary unit) which directly describes the conditional position probability distribution as a function of $(kx,ky)$ in dependence on the detuning of the spontaneously emitted photon. Parameters are ${\Delta }_k=-4$, ${\Omega }_{10}=3$, ${\Omega }_2=6$, ${\Omega }_c=6$, and $\varphi =0$ (a); $\varphi =\pi$ (b). Other parameters are the same as Fig. 2.

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4. Conclusions

In summary, we have investigated the conditional position probability distribution of a coherently driven atom when passing through two orthogonal standing-wave fields. Owing to the spatially dependent interaction between the atom and the lasers, the frequency of the resulting spontaneously emitted photon can provide the position information of the atom in the $x-y$ plane of the standing-wave fields. Therefore, the measurement of the scattered light yields 2D atom localization. Due to the phase sensitive property of the atomic system, spontaneous emission quenching takes place in some regions of the $x-y$ plane, and then the uncertainty of the position probability distribution can be significantly reduced. When we have detected a photon with proper frequency, the atom can be localized in ${\lambda }_1/2\times {\lambda }_1/2$ domain, so sub-half-wavelength atom localization is obtained. The probability of the finding the atom at a particular position in one wavelength region of the standing-waves is 100% which is greatly improved as compared with the previous schemes. Moreover, large values of the Rabi frequency of the coupling fields can result in narrowed spontaneous emission, which provides a promising way to achieve 2D sub-half-wavelength atom localization with high spatial resolution.