Dressed state analysis of two- and three-dimensional atom localization in the $\Lambda -\Xi$ configuration

Abstract

The $\Lambda -\Xi$ atomic system is employed to examine the localization behavior of atom in two-dimensional (2D) and three-dimensional (3D) cases. The variation in atom-field interaction across space results in a position-dependent probe susceptibility. This enables the determination of an atom’s position probability distribution through the analysis of probe spectra. We have elucidated the system behavior by scrutinizing the dressed states approach, which provide the fundamental framework for its physical interpretation. High-resolution and precise sub-wavelength atom localization can be attained by properly tuning the system parameters. The implementation of running wave field to create destructive quantum interference phenomenon is pivotal in enhancing the precision and efficiency of locating atomic positions with $100\%$ probability in 2D and 3D subspaces. We also observe how the Doppler broadening affects atom positioning in both two-dimensional and three-dimensional space. Our findings demonstrate that the Doppler broadening significantly diminishes the accuracy of spatial localization.

Appendix

1.1 Impact of coupling field intensities on 2D and 3D atom localization

Fig. 10

The 2D probe absorption plot of filter function versus (kx,ky) for different values of $|\Omega _{c_1}|$, $\Omega _{c_2}$ and $\Omega _R$, a $|\Omega _{c_1}|=\Omega _{c_2}=\Omega _R=6\Gamma _3$, b $|\Omega _{c_1}|=\Omega _{c_2}=\Omega _R=5\Gamma _3$, c $|\Omega _{c_1}|=\Omega _{c_2}=\Omega _R=4\Gamma _3$. The values of other parameters are $\Omega _p=0.0016\Gamma _3$, $\Omega _0=2\Gamma _3$, $\Delta _p=5\Gamma _3$ and $\Delta _{c_1}=\Delta _{c_2}=\Delta _{R}=0\Gamma _3$

Fig. 11

The 3D probe absorption isosurface plot of filter function versus (kx, ky, kz) for different values of $|\Omega _{c_1}|$, $\Omega _{c_2}$ and $\Omega _R$, a $|\Omega _{c_1}|=\Omega _{c_2}=\Omega _R=5\Gamma _3$, b $|\Omega _{c_1}|=\Omega _{c_2}=\Omega _R=4\Gamma _3$, c $|\Omega _{c_1}|=\Omega _{c_2}=\Omega _R=3\Gamma _3$. The values of other parameters are $\Omega _p=0.0016\Gamma _3$, $\Omega _0=1\Gamma _3$, $\Delta _p=5\Gamma _3$ and $\Delta _{c_1}=\Delta _{c_2}=\Delta _{R}=0\Gamma _3$

Let’s now discuss the impact of the coupling field intensities on the behavior of 2D atom localization, which stands out as one of the most intriguing attributes of the current atomic scheme. In our numerical calculation, we fix the parameters as $\Omega _p=0.0016\Gamma _3$, $\Omega _0=2\Gamma _3$, $\Delta _p=5\Gamma _3$ and $\Delta _{c_1}=\Delta _{c_2}=\Delta _{R}=0\Gamma _3$. When $|\Omega _{c_1}|=\Omega _{c_2}=\Omega _R=6\Gamma _3$, the spatial distribution of F(x,y) form crater-like pattern in quadrant I $(0<kx,ky<\pi )$ and spike-like in III $(-\pi<kx,ky<0)$ quadrant [Fig. 10(a)]. Furthermore, when $|\Omega _{c_1}|=\Omega _{c_2}=\Omega _R=5\Gamma _3$, we observed a crater-like localization pattern in quadrant I [Fig. 10(b)]. In this case atom is localized along the circular edges of the crater. For $|\Omega _{c_1}|=\Omega _{c_2}=\Omega _R=4\Gamma _3$, the probability of locating an atom within a single quadrant becomes maximum. Here, the filter function displays a spike-like pattern in quadrant I $(0<kx,ky<\pi )$ [Fig. 10(c)], indicating a significantly higher level of localization precision compared to that depicted in Fig. 10(a). The preceding discussion makes it clear that the precise localization of atoms is predominantly related to the coupling field intensities.

The isosurface plots depicted in Fig. 11 illustrate how probe absorption varies throughout three-dimensional space (kx, ky, kz) across different magnitudes of $|\Omega _{c_1}|$, $\Omega _{c_2}$, and $\Omega _R$. Based on the observation from Fig. 11(a), when $|\Omega _{c_1}|=\Omega _{c_2}=\Omega _R=5\Gamma _3$, we observe that the absorption of the probe is distributed across various subspaces, exhibiting spherical structures with wide dimensions in 3D space. Here, the measurement of the atom’s position lacks accuracy due to the dispersion of the probe absorption pattern across multiple subspaces. This leads to difficulty in precisely pinpointing the atom’s location. With a subsequent decrease in intensities, specifically when $|\Omega _{c_1}|=\Omega _{c_2}=\Omega _R=4\Gamma _3$, the isosurface now extend in two subspaces and the probability of finding the atom within a single octant becomes 50% [Fig. 11(b)]. Moreover, it becomes evident that when $|\Omega _{c_1}|=\Omega _{c_2}=\Omega _R=3\Gamma _3$, the isosurface appears within one subspace, resulting in the efficient attainment of high-precision atom positioning in three-dimensional space [Fig. 11(c)]. Therefore, it becomes apparent that the ability to localize the atom is directly influenced by the intensity of the fields.

1.2 Impact of thermal velocities on 2D atom and 3D localization

In this section, we explore the impact of thermal velocities on localizing atoms in both 2D and 3D spaces. When an atom is subjected to laser fields, there remains a possibility that the atom may not be entirely stationary [39]. Because of atomic motion, it becomes essential to account for the impact of Doppler broadening on the positional measurement of a single atom within sub-wavelength space in both 2D and 3D dimensions. This yields a velocity-dependent filter function $F^{(v)}$.

Fig. 12

The 2D probe absorption plot of Doppler broadened filter function versus (kx, ky) at different temperatures, a T = 0.05K, b T = 0.1K, c T = 0.5 K. The values of other parameters are $\Omega _p=0.0016\Gamma _3$, $|\Omega _{c_1}|= \Omega _{c_2}=\Omega _R=4\Gamma _3$, $\Omega _0=2\Gamma _3$, $\Delta _p=5\Gamma _3$ and $\Delta _{c_1}=\Delta _{c_2}=\Delta _{R}=0\Gamma _3$

Fig. 13

The 3D probe absorption plot of Doppler broadened filter function versus (kx, ky, kz) at different temperature, a T = 0.05 K, b T = 0.1 K, c T = 0.5 K. $\Omega _{c_2}=\Omega _R=20\Gamma _3$. The values of other parameters are $\Omega _p=0.0016\Gamma _3$, $|\Omega _{c_1}|=\Omega _{c_2}=\Omega _R=3\Gamma _3$, $\Omega _0=1\Gamma _3$, $\Delta _p=5\Gamma _3$ and $\Delta _{c_1}=\Delta _{c_2}=\Delta _{R}=0\Gamma _3$

Since the standing-wave field ($\Omega _{c_1}$) is produced by two counter-propagating beams, any frequency shifts in the $\Omega _{c_1}$ due to the resonant interaction between the atom and the $\Omega _{c_1}$ are absent. This cancellation occurs because the counter-propagating fields neglect each other’s effects. Here, we examine $\Omega _p$, $\Omega _{c_2}$ and $\Omega _R$ to explore the impact of Doppler broadening. Therefore, the impact of Doppler broadening is only taken into account for running wave fields. Incorporating the Doppler effect corresponding to the probe and control fields involves replacing the detunings $\Delta _p$, $\Delta _{c_2}$ and $\Delta _R$ with $\Delta _p$ + $k_pv$, $\Delta _{c_2} - k_{c_2}v$ and $\Delta _R$ + $k_Rv$ respectively, in Eq. (1). The Doppler-averaged filter function is derived by integrating Eq. (5) over one-dimensional Maxwell–Boltzmann distribution. The resulting modified filter function is as follows:

$$\begin{aligned} F^{(v)}=\sqrt{\frac{m}{2\pi k_BT}}\int \limits _{-\infty }^\infty F(\Delta _p + k_pv,\Delta _p + k_pv,\Delta _{c_2} - k_{c_2}v)e^{-\frac{mv^2}{2k_BT}}dv, \end{aligned}$$

(18)

where m is the atomic mass, $k_B$ is Boltzmann constant and T is the absolute temperature.

In Fig. 12, we illustrate the Doppler-broadened 2D filter function of Eq. (18) as a function of the normalized positions kx and ky at different temperatures. All other variables remain the same as in Fig. 10(c). At T = 0.05 K, we observe the emergence of crater-like pattern in quadrant I, which was absent in the absence of Doppler broadening, as depicted in Fig. 10(c). As the temperature escalates to two distinct values, T = 0.1 K and 0.5 K as depicted in Fig. 12(b and c), we observe that the craters become significantly broader. Thus, it is evident from our observations that the inclusion of the Doppler effect diminishes the precision of atom position in 2D. Fig. 13(a) depicts the isosurface plots of the velocity dependent filter function with same parameters as in Fig. 11(a). It is evident that the single precise sphere transforms into two spheres of unequal sizes. This indicates a degradation in the resolution of atom localization resulting from the introduction of the Doppler effect. Increasing the temperature results in the emergence of a dispersed filter function $F^{(v)}(x,y,z)$ within distinct subspaces in the 3D volume space. This demonstrates that the atom’s position is significantly imprecise across all directions within three-dimensional space. As the temperature value is raised further, the isosurface plots show greater ambiguity. Hence, the precision of the atom’s position in 3D space is diminished. Accurate information regarding the atom’s position cannot be reliably obtained when the Doppler effect is introduced in the calculation.