Phase-sensitive atom localization for closed-loop quantum systems

A scheme of high-precision two- and three-dimensional (3D) atom localization is proposed and analyzed by using a density matrix method for a five-level atom-light coupling scheme. In this system four strong laser components (which could be standing waves) couple a pair of atomic internal states to another pair of states in all possible ways to form a closed-loop diamond-shape configuration of the atom-light interaction. By systematically solving the density matrix equations of the motion, we show that the imaginary part of the susceptibility for the weak probe field is position dependent. As a result, one can obtain information about the position of the atom by measuring the resulting absorption spectra. Focusing on the signatures of the relative phase of the applied fields stemming from the closed- loop structure of the diamond- shape subsystem, we find out that there exists a significant phase dependence of the eigenvalues required to have a maximum in the probe absorption spectra. It is found that by properly selecting the controlling parameters of the system, a nearly perfect 2D atom localization can be obtained. Finally, we numerically explore the phase control of 3D atom localization for the present scheme and show the possibility to obtain 1/2 detecting probability of finding the atom at a particular volume in 3D space within one period of standing waves.

Figure 1

(a) Schematic diagram of the five-level quantum system. (b), (c), (d), and (e) Different situations considered in which the atom could interact with the position- dependent standing-wave fields.

Figure 2

Plots of probe absorption ${\chi }^{\prime \prime }\mathrm{versus} (kx,ky)$. The selected parameters are $({\mathrm{\Omega }}_{43},{\mathrm{\Omega }}_{32},{\mathrm{\Omega }}_{41},{\mathrm{\Omega }}_{21},\varphi )=$ (a) $(5\gamma ,5\gamma ,5\gamma ,5\gamma ,0)$, (b) $(4\gamma ,6\gamma ,4\gamma ,6\gamma ,0)$, (c) $(6\gamma ,4\gamma ,6\gamma ,4\gamma ,0)$, (d) $(5\gamma ,5\gamma ,5\gamma ,5\gamma ,\pi )$. The other parameters are ${\gamma }_{14}={\gamma }_{12}={\gamma }_{23}={\gamma }_{43}={\gamma }_{35}=\gamma , \mathrm{\Delta }={\mathrm{\Delta }}_{12}={\mathrm{\Delta }}_{14}={\mathrm{\Delta }}_{23}={\mathrm{\Delta }}_{43}=0$, as well as ${\mathrm{\Delta }}_p=8\gamma , {\mathrm{\Omega }}_p=0.01\gamma$.

Figure 3

Plots of probe absorption ${\chi }^{\prime \prime }\mathrm{versus} (kx,ky)$. The selected parameters are $({\mathrm{\Omega }}_{43},{\mathrm{\Omega }}_{32},{\mathrm{\Omega }}_{41},{\mathrm{\Omega }}_{21})=$ (a) $(10\gamma ,10\gamma ,10\gamma ,10\gamma )$, (b) $(8\gamma ,\gamma ,10\gamma ,10\gamma )$, (c) $(25\gamma ,25\gamma ,25\gamma ,25\gamma )$, (d) $(2.5\gamma ,2.5\gamma ,2.5\gamma ,2.5\gamma )$. Here, ${\mathrm{\Delta }}_p=10\gamma , \varphi =0$. The other parameters are the same as Fig. 2.

Figure 4

Plots of probe absorption ${\chi }^{\prime \prime }\mathrm{versus} (kx,ky)$. The selected parameters are (a) $\varphi =0$, (b) $\varphi =\pi /3$, (c) $\varphi =\pi /2$, (d) $\varphi =\pi$. Here, $({\mathrm{\Omega }}_{43},{\mathrm{\Omega }}_{32},{\mathrm{\Omega }}_{41},{\mathrm{\Omega }}_{21})=(5\gamma ,5\gamma ,5\gamma ,5\gamma ), {\mathrm{\Delta }}_p=5\gamma$. The other parameters are the same as Fig. 4.

Figure 5

Plots of probe absorption ${\chi }^{\prime \prime }$ versus $(kx,ky)$. The selected parameters are $({\mathrm{\Omega }}_{43},{\mathrm{\Omega }}_{32},{\mathrm{\Omega }}_{41},{\mathrm{\Omega }}_{21})=$ (a) $(10\gamma ,10\gamma ,10\gamma ,10\gamma )$, (b) $(10\gamma ,10\gamma ,20\gamma ,10\gamma )$, (c) $(10\gamma ,20\gamma ,10\gamma ,10\gamma )$, (d) $(18\gamma ,10\gamma ,10\gamma ,10\gamma )$, and (e) $(10\gamma ,10\gamma ,10\gamma ,14.8\gamma )$. Here, ${\mathrm{\Delta }}_{12}={\mathrm{\Delta }}_{14}={\mathrm{\Delta }}_{23}={\mathrm{\Delta }}_{43}=10\gamma , {\mathrm{\Delta }}_p=0$, and the other parameters are the same as Fig. 4.

Figure 6

Plots of probe absorption ${\chi }^{\prime \prime }$ versus $(kx,ky)$. The selected parameters are (a) $\varphi =0$, (b) $\varphi =\pi /4$, (c) $\varphi =\pi /2$, (d) $\varphi =3\pi /4$, (e) $\varphi =\pi$. Here, $({\mathrm{\Omega }}_{43},{\mathrm{\Omega }}_{32},{\mathrm{\Omega }}_{41},{\mathrm{\Omega }}_{21})=(10\gamma ,10\gamma ,10\gamma ,14.8\gamma )$. The other selected parameters are the same as Fig. 5.

Figure 7

Isosurface plots of probe absorption ${\chi }^{\prime \prime }$ versus $(kx,ky,kz)$. The selected parameters are (a) $\varphi =0$, (b) $\varphi =\pi /3$, (c) $\varphi =\pi /2$, and (d) $\varphi =\pi$. Here, $({\mathrm{\Omega }}_{43},{\mathrm{\Omega }}_{32},{\mathrm{\Omega }}_{41},{\mathrm{\Omega }}_{21})=(10\gamma ,10\gamma ,10\gamma ,10\gamma ),{\mathrm{\Delta }}_{12}={\mathrm{\Delta }}_{14}={\mathrm{\Delta }}_{23}={\mathrm{\Delta }}_{43}={\mathrm{\Delta }}_p=10\gamma$. The other selected parameters are the same as Fig. 3.