An atomic system with four cascade energy levels is studied and its concrete atomic structures are shown in Figure 1. State |1〉 is a ground state, |2〉 is an intermediate state, and the top two states |3〉 and |4〉 are two-folded levels. Where we use three standing-wave fields those are orthogonal to each other and whose Rabi frequency 2*Ω*(*j*)sin (*kj*) (*j* = *x*, *y*, *z*; *k* = *ω*/*c*) depends on position to couple with the transition between levels |2〉 and |3〉, and the total Rabi frequency 2*Ω*_{s}(*x*,*y*,*z*) may be expressed as 2*Ω*_{s}(*x*,*y*,*z*) = 2*Ω*(*x*)sin(*kx*)+2*Ω*(*y*)sin(*ky*)+2*Ω*(*z*)sin(*kz*). A weak probe field with angular frequency *ω*_{p} and Rabi frequency 2*Ω*_{p} interacts with the atomic system and excites the transition between levels |2〉 and |1〉. A radio-frequency field with angular frequency *ω*_{rf} and Larmor frequency 2*Ω*_{rf} is applied to drive transition between the two-folded levels |3〉 and |4〉.

Figure 1 Schematic diagram of a four-level atomic system

Here we neglect the kinetic part of the atom from the Hamiltonian according to the Raman-Nath approximation, and suppose that the center-of-mass coordinate of the atom almost does not change with time along with the directions of the laser when the intensity of Rabi frequency of the laser field is big enough. Under the condition of rotation wave approximation, the interaction Hamiltonian of this system is given by

$$\begin{array}{}{H}_{I}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& =\left({{\mathit{\Delta}}_{xyz}+\mathit{\Delta}}_{p}\right)\left|3\u3009\u30083\right|+{\mathit{\Delta}}_{p}\left|2\u3009\u30082\right|\\ & +\left({{\mathit{\Delta}}_{rf}+{\mathit{\Delta}}_{xyz}+\mathit{\Delta}}_{p}\right)\left|4\u3009\u30084\right|\\ & -({{\mathit{\Omega}}_{s}\left(x,y,z\right)\left|3\u3009\u30082\right|+\mathit{\Omega}}_{p}\left|2\u3009\u30081\right|+{\mathit{\Omega}}_{rf}\left|4\u3009\u30083\right|\\ & +H.c.),\end{array}$$(1)

where we let *ħ*=0, *Δ*_{rf}=*ω*_{43}−*ω*_{rf}, *Δ*_{xyz}=*ω*_{32}−*ω*_{xyz}, and *Δ*_{p}=*ω*_{21}−*ω*_{p} are corresponding to detuning of the radio-frequency field, the standing-wave field and the probe field, respectively, and *ω*_{43}, *ω*_{32}, and *ω*_{21} are corresponding to transition frequency between levels |4〉 and |3〉, between levels |3〉 and |2〉, and between levels |2〉 and |1〉, respectively.

According to the equation satisfied by the evolution of density matrix elements

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{\rho}_{mn}}{\mathrm{\partial}t}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}}& ={\displaystyle \frac{1}{i\hslash}\sum _{k}\left({H}_{mk}{\rho}_{kn}-{\rho}_{mk}{H}_{kn}\right)}\\ & -{\displaystyle \frac{1}{2}\sum _{k}\left({\mathit{\Gamma}}_{mk}{\rho}_{kn}+{\rho}_{mk}{\mathit{\Gamma}}_{kn}\right),\phantom{\rule{1em}{0ex}}(k=1,2,3,4)}\end{array}$$(2)

where *Γ*_{mn} stands for relaxation matrix elements, and it can be expressed as 〈*n*|*Γ*|*m*〉=*γ*_{n}*δ*_{nm}, and *γ*_{n} is the decay rate of level |*n*〉. Based on Eqs. (1) and (2), we can obtain the first derivative of density matrix elements with time:

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{\rho}_{11}}{\mathrm{\partial}t}={\gamma}_{2}{\rho}_{22}-i{{\mathit{\Omega}}_{p}\rho}_{12}+i{{\mathit{\Omega}}_{p}^{\ast}\rho}_{21},}\end{array}$$(3a)

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{\rho}_{22}}{\mathrm{\partial}t}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}}& =-{\gamma}_{2}{\rho}_{22}+{\gamma}_{3}{\rho}_{33}+{\gamma}_{4}{\rho}_{44}+i{{\mathit{\Omega}}_{p}\rho}_{12}\\ & -i{{\mathit{\Omega}}_{p}^{\ast}\rho}_{21}-i{{\mathit{\Omega}}_{s}\left(x,y,z\right)\rho}_{23}+i{{\mathit{\Omega}}_{s}^{\ast}(x,y,z)\rho}_{32},\end{array}$$(3b)

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{\rho}_{33}}{\mathrm{\partial}t}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}}& =-{\gamma}_{3}{\rho}_{33}+i{{\mathit{\Omega}}_{rf}^{\ast}\rho}_{43}-i{{\mathit{\Omega}}_{rf}\rho}_{34}\\ & +i{{\mathit{\Omega}}_{s}\left(x,y,z\right)\rho}_{23}-i{{\mathit{\Omega}}_{s}^{\ast}(x,y,z)\rho}_{32},\end{array}$$(3c)

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{\rho}_{44}}{\mathrm{\partial}t}=-{\gamma}_{4}{\rho}_{44}+i{{\mathit{\Omega}}_{rf}\rho}_{34}-i{{\mathit{\Omega}}_{rf}^{\ast}\rho}_{43},}\end{array}$$(3d)

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{\rho}_{21}}{\mathrm{\partial}t}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}}& {\displaystyle =-\left(\frac{{\gamma}_{2}}{2}+i{\mathit{\Delta}}_{p}\right){\rho}_{21}}\\ & +i{{\mathit{\Omega}}_{s}^{\ast}\left(x,y,z\right)\rho}_{31}-i{{\mathit{\Omega}}_{p}(\rho}_{22}-{\rho}_{11}),\end{array}$$(3e)

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{\rho}_{31}}{\mathrm{\partial}t}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}}& {\displaystyle =-\left[\frac{{\gamma}_{3}}{2}+i\left({\mathit{\Delta}}_{xyz}+{\mathit{\Delta}}_{p}\right)\right]{\rho}_{31}+i{{\mathit{\Omega}}_{s}\left(x,y,z\right)\rho}_{21}}\\ & +i{{\mathit{\Omega}}_{rf}^{\ast}\rho}_{41}-i{\mathit{\Omega}}_{p}{\rho}_{32},\end{array}$$(3f)

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{\rho}_{41}}{\mathrm{\partial}t}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}}& {\displaystyle =-\left[\frac{{\gamma}_{4}}{2}+i\left({{\mathit{\Delta}}_{xyz}+\mathit{\Delta}}_{p}+{\mathit{\Delta}}_{rf}\right)\right]{\rho}_{41}}\\ & +i{{\mathit{\Omega}}_{rf}\rho}_{31}-i{\mathit{\Omega}}_{p}{\rho}_{42},\end{array}$$(3g)

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{\rho}_{32}}{\mathrm{\partial}t}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}}& {\displaystyle =-\left(\frac{{\gamma}_{2}+{\gamma}_{3}}{2}+i{\mathit{\Delta}}_{xyz}\right){\rho}_{32}}\\ & +i{{\mathit{\Omega}}_{rf}^{\ast}\rho}_{42}-i{{\mathit{\Omega}}_{p}^{\ast}\rho}_{31}-i{{\mathit{\Omega}}_{s}(x,y,z)(\rho}_{33}-{\rho}_{22}),\end{array}$$(3h)

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{\rho}_{42}}{\mathrm{\partial}t}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}}& {\displaystyle =-\left[\frac{{\gamma}_{2}+{\gamma}_{4}}{2}+i\left({\mathit{\Delta}}_{xyz}+{\mathit{\Delta}}_{rf}\right)\right]{\rho}_{42}+i{{\mathit{\Omega}}_{rf}\rho}_{32}}\\ & -i{{\mathit{\Omega}}_{p}^{\ast}\rho}_{41}-i{\mathit{\Omega}}_{s}(x,y,z){\rho}_{43},\end{array}$$(3i)

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{\rho}_{43}}{\mathrm{\partial}t}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}}& {\displaystyle =-\left[\frac{{\gamma}_{3}+{\gamma}_{4}}{2}+i{\mathit{\Delta}}_{rf}\right]{\rho}_{43}+i{{\mathit{\Omega}}_{rf}(\rho}_{33}-{\rho}_{44})}\\ & -i{{\mathit{\Omega}}_{s}^{\ast}(x,y,z)\rho}_{42},\end{array}$$(3j)

where *ρ*_{11}+*ρ*_{22}+*ρ*_{33}+*ρ*_{44}=1, *γ*_{2}, *γ*_{3} and *γ*_{4} are decay rates for |2〉→|1〉, |3〉→|2〉and |4〉→|2〉, respectively. Based on the theory of light propagating in atomic medium, we get the expression of the complex susceptibility for the probe field

$$\begin{array}{}{\displaystyle {\chi}_{p}=\frac{N{\left|{\mu}_{21}\right|}^{2}}{{\u03f5}_{0}\hslash {\mathit{\Omega}}_{p}}{\rho}_{21}}\end{array}$$(4)

where physical parameter *ε*_{0} is the free space permittivity and *N* is the atom number density of four-level atomic system.

According to the theory about light propagating in medium, we know the imaginary part *Im*(*χ*_{p}) of the susceptibility can be applied to describe absorption of probe field. To simplify the formula, we let *δ*_{2}=
$\begin{array}{}\frac{{\gamma}_{2}}{2}\end{array}$ +*iΔ*_{p}, *δ*_{3}=
$\begin{array}{}\frac{{\gamma}_{3}}{2}\end{array}$ +*i*(*Δ*_{xyz}+*Δ*_{p}), and *δ*_{4}=
$\begin{array}{}\frac{{\gamma}_{4}}{2}\end{array}$
+*i*(*Δ*_{xyz}+*Δ*_{p}+*Δ*_{rf}). So from Eqs. (3) and (4), the expression about *Im*(*χ*_{p}) is given by

$$\begin{array}{}{\displaystyle Im\left({\chi}_{p}\right)=Im\frac{{\delta}_{3}{\delta}_{4}+{\mathit{\Omega}}_{rf}^{2}}{{\delta}_{2}{\mathit{\Omega}}_{rf}^{2}+{\delta}_{4}{\mathit{\Omega}}_{s}^{2}\left(x,y,z\right)+{\delta}_{4}{\delta}_{3}{\delta}_{2}}i}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}=\frac{A\left(\frac{{\gamma}_{4}{\gamma}_{3}}{4}-{\mathit{\Delta}}_{4}{\mathit{\Delta}}_{3}+{\mathit{\Omega}}_{rf}^{2}\right)-B(\frac{{\gamma}_{4}{\mathit{\Delta}}_{3}}{2}+\frac{{\gamma}_{3}{\mathit{\Delta}}_{4}}{2})}{{A}^{2}+{B}^{2}}}\end{array}$$(5)

where

*Δ*_{3}=*Δ*_{xyz}+*Δ*_{p},

*Δ*_{4}=*Δ*_{p}+*Δ*_{rf}+*Δ*_{xyz},

$\begin{array}{}{\mathit{\Delta}}_{3}={\mathit{\Delta}}_{xyz}+{\mathit{\Delta}}_{p},\\ {\mathit{\Delta}}_{4}={\mathit{\Delta}}_{p}+{\mathit{\Delta}}_{rf}+{\mathit{\Delta}}_{xyz},\\ A=({\gamma}_{4}{\mathit{\Omega}}_{s}^{2}\left(x,y,z\right)+{\gamma}_{2}{\mathit{\Omega}}_{rf}^{2}-{\gamma}_{4}{\mathit{\Delta}}_{p}{\mathit{\Delta}}_{3}-{\mathit{\Delta}}_{3}{\mathit{\Delta}}_{4}{\gamma}_{2}-{\mathit{\Delta}}_{4}{\mathit{\Delta}}_{p}{\gamma}_{3}+\\ \frac{{\gamma}_{4}{\gamma}_{3}{\gamma}_{2}}{4})/2,\\ B={\mathit{\Delta}}_{4}{\mathit{\Omega}}_{s}^{2}\left(x,y,z\right)+{\mathit{\Delta}}_{p}{\mathit{\Omega}}_{rf}^{2}-{\mathit{\Delta}}_{4}{\mathit{\Delta}}_{3}{\mathit{\Delta}}_{p}+({\mathit{\Delta}}_{3}{\gamma}_{2}{\gamma}_{4}+{\mathit{\Delta}}_{p}{\gamma}_{3}{\gamma}_{4}+\\ {\mathit{\Delta}}_{4}{\gamma}_{3}{\gamma}_{2})/4.\end{array}$

Eq. (5) tells us that the imaginary part *Im*(*χ*_{p}) depends on not only space coordinates (*x*,*y*,*z*) and the detunings *Δ*_{p}, *Δ*_{xyz}, *Δ*_{rf} between natural frequency of atom and light fields, but also the Rabi frequencies *Ω*_{rf} and *Ω*_{s}(*x*,*y*,*z*). Accordingly, we can use absorption of the probe field to detect distribution of atoms in space. In the process of numerical calculations, we will determine probe absorption *Im*(*ρ*_{21}/*Ω*_{p}) and measure the probe absorption depending on space position. Then informations about the atomic space distribution are obtained, and finally realize atom localization in three dimensional. Through choosing appropriate parameters of system, we can control the localization behavior of the atom in 3D space and improve the precision of detection of atomic position.

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