Efficient two-dimensional atom localization in a five-level conductive chiral atomic medium via bire
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Sajid Ali1, Muhammad Idrees,2,3,∗, Bakth Amin Bacha,2,∗, Arif Ullah2, Muhammad Haneef11Lab of Theoretical Physics, Department of Physics, Hazara University, Mansehra 21300, Pakistan 2Quantum Optics and Quantum Information Research Group, Department of Physics, University of Malakand, Chakdara Dir(L), Khyber Pakhtunkhwa, Pakistan 3Zhejiang Province Key Laboratory of Quantum Technology and Device, Department of Physics, Zhejiang University, Hangzhou 310027, China
First author contact:∗ Authors to whom any correspondence should be addressed. Received:2020-07-12Revised:2020-10-21Accepted:2020-10-26Online:2020-12-22
Abstract We have theoretically investigated two-dimensional atom localization using the absorption spectra of birefringence beams of light in a single wavelength domain. The atom localization is controlled and modified through tunneling effect in a conductive chiral atomic medium with absorption spectra of birefringent beams. The significant localization peaks are investigated in the left and right circularly polarized beam. Single and double localized peaks are observed in different quadrants with minimum uncertainty and significant probability. The localized probability is modified by controlling birefringence and tunneling conditions. These results may be useful for the capability of optical microscopy and atom imaging. Keywords：2D atom localization;birefringence beam absorption;chiral atomic medium
PDF (1184KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Sajid Ali, Muhammad Idrees, Bakth Amin Bacha, Arif Ullah, Muhammad Haneef. Efficient two-dimensional atom localization in a five-level conductive chiral atomic medium via birefringence beam absorption spectrum. Communications in Theoretical Physics, 2021, 73(1): 015102- doi:10.1088/1572-9494/abc46c
1. Introduction
During the past few decades, atom localization has attracted attention on both theoretical and experimental fronts, primarily due to its wide scope in atom nanolithography [1, 2], trapping and cooling of neutral atoms [3], moving atoms center-of-mass wave function measurement [4, 5], Bose–Einstein condensation [6, 7], matter waves coherent patterning [8], etc. Various kinds of methods have evolved over the course of its progress. While the initial techniques included phase measurement of standing-wave fields [9, 10], atomic dipole, atomic fluorescence, dark resonance interference, spontaneous emission spectrum, and Raman gain process measurement [11–17], they differ from modern ones. One such example is an atom, whose energy levels are placed under the influence of the standing wave fields in a subwavelength domain. This new technique has enabled us to witness some basic phenomena, which among many include electromagnetically induced absorption [18, 19], population trapping [20, 21], superluminal and subluminal propagation of light [22–26], Kerr nonlinearity [27, 28], optical bistability [29, 30], four-wave mixing [31, 32], electromagnetically induced transparency [33, 34] and optical multistability [35, 36]. Atom localization phenomena are also investigated by calculating the optical Bloch equations based on the formalism of the density matrix. It turns out that imaginary and real parts of the coherence term are directly related to the probe absorption and dispersion spectrum, respectively. Furthermore, highly efficient and high-resolution atomic localization in the subwavelength region is also observed via level population and probe absorption [37–47].
Recently, different schemes have been used for one-dimensional (1D) atom localization in subwavelength space through excited level population measurements [48–52], and probe field absorption spectra [53–57]. In such cases, the standing wave fields are applied along a single direction (i.e, x-axis) and the atom is localized in the 1D subwavelength space. Many new advanced (high-resolution and high-precision) proposals have emerged recently for two-dimensional (2D) and three-dimensional (3D) atom localization. 2D atom localization is experimentally realized by superimposing two standing wave fields. In 2010, Ivanov et al proposed a tripod four-level atomic scheme for 2D atom localization via excited-state population [41]. Ding and his coworkers, in 2011, introduced a four-level closed-loop system and observed 2D atom localization using the probe absorption measurement and the localization was done in a quadrant subwavelength space [38]. Hamedi et al proposed a five-level atomic scheme for the phase sensitive 2D atom localization via excited-state population [37]. A myriad of other proposals for the 2D atomic localization have appeared, as discussed in [58–62].
When linear polarized light passes through a chiral medium it splits into the left (LCP) and right circularly polarized (RCP) beams. The LCP and RCP beams have a slight difference in refractive indices (${n}_{r}^{(+)},{n}_{r}^{(-)}$). The difference in refractive indices $(\delta n={n}_{r}^{(+)}-{n}_{r}^{(-)})$ is known as circular birefringence [63–65]. The circular birefringence and chiral medium affects the properties of light–matter interaction [66–68]. On a thorough literature survey, it appears no analysis has been carried out for atom localization by the birefringence absorption spectrum using the tunneling effect and conductive chiral medium. Inspired from the above studies, we here introduce a new way to localize atoms in the 2D space using dynamic conductive chiral atomic medium under the effects of quantum tunneling. The information of localization can be extracted by the birefringence absorption spectrum in one wavelength domain. The significant localization peaks are investigated in the LCP and RCP beams’ absorption spectra. Single and double localized peaks in different quadrants with minimum uncertainty and significant probability are observed.
2. Model of the atomic system
A five-level atomic configuration is under consideration for the proposed aims and objectives, as shown in figure 1. Two electric probe fields Ep, that differ by phase φp, are driving between the states $\left|5,3,4\right\rangle $. The Rabi frequency of these probes is Ωp and common detuning Δp. Two magnetic probe fields Bm, differing by phase φm, are driving between the states $\left|1,2,5\right\rangle $. The Rabi frequency of these probes is Ωm and common detuning Δm. A control field E1 is driving between the states $\left|1,4\right\rangle $. The Rabi frequency of the field is Ω1 and its detuning Δ1. Another control field E2 is driving between the states $\left|2,3\right\rangle $. The Rabi frequency of the field is Ω2 and its detuning Δ2. The excited states’ spontaneous decay rates are γ32,γ35,γ41,γ45,γ51, and γ52. The system is closed between the states $\left|1,4,5\right\rangle $ and $\left|2,3,5\right\rangle $ by control and electric/magnetic fields. The levels $\left|3,4\right\rangle $ and $\left|1,2\right\rangle $ are so close to each other that atomic kinetic energy p2/2m obeys the uncertainty principle Δp.Δx≥ℏ and ΔE.Δt≥ℏ. The tunneling between two states $\left|3,4\right\rangle $ and $\left|1,2\right\rangle $ are possible when Δx is larger then the states’ separation ΔL. Under these circumstances, an atom tunnels between the levels $\left|3,4\right\rangle $ and $\left|1,2\right\rangle $ with frequencies ν1T and ν2T.
Figure 1.
New window|Download| PPT slide Figure 1.Schematic of five-level atomic configuration for the 2D atom localization through dynamic conductive chiral medium under the effects of quantum tunneling using birefringence absorption spectrum in one wavelength domain.
The position dependent tunneling frequencies are written as [58–61] ${\nu }_{1T}(x,y)=T[\sin ({\eta }_{1}{kx})+\sin ({\eta }_{2}{ky}+{\varphi }_{3})]$ and ${\nu }_{2T}(x,y)=T[\sin ({\eta }_{3}{kx})+\sin ({\eta }_{5}{ky}+{\varphi }_{4})]$. The parameter T represents the space independent parts of tunneling frequencies and φ3, φ4 its phases. Wave vectors associated with the standing-waves respectively are ki=kηi,(i=1,2,3,5). The interaction Hamiltonian for the system is written as:$\begin{eqnarray}\begin{array}{rcl}{H}_{I} & = & -\displaystyle \frac{\hslash }{2}[{{\rm{\Omega }}}_{m}{{\rm{e}}}^{-{\rm{i}}{{\rm{\Delta }}}_{m}t}\left|1\right. \rangle \left. \langle 5\right|+{{\rm{\Omega }}}_{m}{{\rm{e}}}^{-{\rm{i}}{\varphi }_{m}}{{\rm{e}}}^{-{\rm{i}}{{\rm{\Delta }}}_{m}t}\left|2\right. \rangle \left. \langle 5\right|\\ & & +{{\rm{\Omega }}}_{p}{{\rm{e}}}^{-{\rm{i}}{\varphi }_{p}}{{\rm{e}}}^{-{\rm{i}}{{\rm{\Delta }}}_{p}t}\left|5\right. \rangle \left. \langle 4\right|+{{\rm{\Omega }}}_{p}{{\rm{e}}}^{-{\rm{i}}{{\rm{\Delta }}}_{p}t}\left|5\right. \rangle \left. \langle 3\right|\\ & & +{{\rm{\Omega }}}_{1}{{\rm{e}}}^{-{\rm{i}}{{\rm{\Delta }}}_{1}t}\left|1\right. \rangle \left. \langle 4\right|+{{\rm{\Omega }}}_{2}{{\rm{e}}}^{-{\rm{i}}{{\rm{\Delta }}}_{2}t}\left|2\right. \rangle \left. \langle 3\right|\\ & & +\hslash [{\nu }_{1T}(x,y)\left|1\right. \rangle \left. \langle 2\right|+{\nu }_{2T}(x,y)\left|4\right. \rangle \left. \langle 3\right|]+{\rm{H}}.{\rm{C}}..\end{array}\end{eqnarray}$
The detunings of these fields are related to their corresponding angular frequencies and atomic states’ resonance frequencies as: Δ1=ω1−ω14, Δ2=ω2−ω23 Δp=ωp−ω45,35 and Δm=ωm−ω15,25. The master equation for the density matrix is given by [57, 69]$\begin{eqnarray}\dot{\rho }=-\displaystyle \frac{{\rm{i}}}{{\hslash }}[{H}_{I},\rho ]-\displaystyle \frac{1}{2}\sum {\gamma }_{ij}({\delta }^{\dagger }\delta \rho +\rho {\delta }^{\dagger }\delta -2\delta \rho {\delta }^{\dagger }),\end{eqnarray}$where HI represents the interaction Hamiltonian of the system, γij denotes the spontaneous decay rate from the excited state $\left|i\right\rangle $ to the ground state $\left|j\right\rangle $ (i.e., γ41,γ45,γ32,γ35,γ51,γ52) and δ† (δ) is the general raising (lowering) operator. Using ${\rho }_{{ij}}\,={\widetilde{\rho }}_{{ij}}\exp [-{\rm{i}}{{\rm{\Delta }}}_{j}t],j=1,2,p,m$ in the dynamical equations of motion and after simplification, the following coupled rates equations are obtained.$\begin{eqnarray}\begin{array}{rcl}{\mathop{\mathop{\rho }\limits^{\sim }}\limits^{\cdot }}_{35} & = & {A}_{1}{\widetilde{\rho }}_{35}+\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{p}({\widetilde{\rho }}_{55}-{\widetilde{\rho }}_{33})+\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{2}{\widetilde{\rho }}_{25}-\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{m}{\widetilde{\rho }}_{31}\\ & & -\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{m}{{\rm{e}}}^{{\rm{i}}{\varphi }_{m}}{\widetilde{\rho }}_{32}-\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{p}{{\rm{e}}}^{-{\rm{i}}{\varphi }_{p}}{\widetilde{\rho }}_{34}-{\rm{i}}{\nu }_{2T}{\widetilde{\rho }}_{45},\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{\mathop{\mathop{\rho }\limits^{\sim }}\limits^{\cdot }}_{45} & = & {A}_{1}{\widetilde{\rho }}_{45}+\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{p}{{\rm{e}}}^{-{\rm{i}}{\varphi }_{p}}({\widetilde{\rho }}_{55}-{\widetilde{\rho }}_{44})+\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{1}{\widetilde{\rho }}_{15}\\ & & -\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{m}{{\rm{e}}}^{{\rm{i}}{\varphi }_{m}}{\widetilde{\rho }}_{42}-\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{p}{\widetilde{\rho }}_{43}-{\rm{i}}{\nu }_{2T}{\widetilde{\rho }}_{35},\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{\mathop{\mathop{\rho }\limits^{\sim }}\limits^{\cdot }}_{25} & = & {A}_{2}{\widetilde{\rho }}_{25}+\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{m}{{\rm{e}}}^{{\rm{i}}{\varphi }_{m}}({\widetilde{\rho }}_{55}-{\widetilde{\rho }}_{22})+\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{2}{\widetilde{\rho }}_{35}\\ & & -\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{m}{\widetilde{\rho }}_{21}-\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{p}{\widetilde{\rho }}_{23}-\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{p}{{\rm{e}}}^{-{\rm{i}}{\varphi }_{p}}{\widetilde{\rho }}_{24}-{\rm{i}}{\nu }_{1T}{\widetilde{\rho }}_{15},\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{\mathop{\mathop{\rho }\limits^{\sim }}\limits^{\cdot }}_{15} & = & {A}_{3}{\widetilde{\rho }}_{15}+\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{m}{\widetilde{\rho }}_{55}-{\widetilde{\rho }}_{11}+\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{1}{\widetilde{\rho }}_{45}-\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{m}{{\rm{e}}}^{-{\rm{i}}{\varphi }_{m}}{\widetilde{\rho }}_{12}\\ & & -\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{p}{\widetilde{\rho }}_{13}-\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{p}{{\rm{e}}}^{-{\rm{i}}{\varphi }_{p}}{\widetilde{\rho }}_{14}-{\rm{i}}{\nu }_{1T}{\widetilde{\rho }}_{25}.\end{array}\end{eqnarray}$The terms A1−3 are written in the appendix. Applying the first order perturbation condition to the coupled rates equations, while taking Ωp,m in the first order and Ω1,2 in all orders the atoms are prepared in the metastable state $\left|5\right\rangle $. The population in the other states are assumed to be null. This implies that its density element ${\widetilde{\rho }}_{55}^{(0)}=1$. Then the population in other states are zero such as ${\widetilde{\rho }}_{22,33,44}^{(0)}=0$ and ${\widetilde{\rho }}_{42,43,21,23,12,13}^{(0)}=0$. After application of the first order perturbation condition, the coupled density matrix equations ${\dot{\widetilde{\rho }}}_{\mathrm{35,45},\mathrm{25,15}}$ are solved by considering the following integral [70]$\begin{eqnarray}L(t)={\int }_{-\infty }^{t}{{\rm{e}}}^{-G(t-t^{\prime} )}P{\rm{d}}t={G}^{-1}Y,\end{eqnarray}$where L(t) and Y are column matrices and G is a 4×4 matrix. For more details, we refer the reader to [55, 57]. The matrix form of this equation is written in the appendix. The effective coherence terms ${\widetilde{\rho }}_{35}^{(1)}+{\widetilde{\rho }}_{45}^{(1)}$ and ${\widetilde{\rho }}_{25}^{(1)}+{\widetilde{\rho }}_{15}^{(1)}$ are related to electric effective polarization and magnetization, such as $P\,={Nd}({\widetilde{\rho }}_{35}^{(1)}+{\widetilde{\rho }}_{45}^{(1)})$ and $M=N\mu ({\widetilde{\rho }}_{25}^{(1)}+{\widetilde{\rho }}_{15}^{(1)})$. The effective electric and magnetic dipole moments between the states are respectively $\mu =\sqrt{\tfrac{3{\hslash }({\gamma }_{51}+{\gamma }_{52}){\lambda }^{3}}{8{\pi }^{2}}}$ and $d=c\sqrt{\tfrac{3{\hslash }({\gamma }_{51}+{\gamma }_{52}){\lambda }^{3}}{8{\pi }^{2}}}$. Replacing Ωp=Epd/ℏ and Ωm=Bmμ/ℏ while Bm= μ0(H+M) in the polarization and magnetization equations $P={Nd}({\widetilde{\rho }}_{35}^{(1)}+{\widetilde{\rho }}_{45}^{(1)})$ and $M=N\mu ({\widetilde{\rho }}_{25}^{(1)}+{\widetilde{\rho }}_{15}^{(1)})$, we get$\begin{eqnarray}P={\alpha }_{1}{E}_{p}+{\alpha }_{2}H,M={\alpha }_{3}{E}_{p}+{\alpha }_{4}H.\end{eqnarray}$
The terms α1−4 are given in the appendix. The electric polarization and magnetization in terms of chiral coefficient and electric magnetic susceptibility are given in [28, 47]$\begin{eqnarray}P={\unicode{x0025B}}_{0}{\chi }_{e}{E}_{p}+\displaystyle \frac{{\xi }_{{E}_{p}H}}{c}H,M=\displaystyle \frac{{\xi }_{{{HE}}_{p}}}{{\mu }_{0}c}{E}_{p}+{\chi }_{m}H.\end{eqnarray}$
Comparing (8) and (9), we obtain the following chiral coefficients and electric and magnetic susceptibility$\begin{eqnarray}{\chi }_{e}=\displaystyle \frac{{{Nd}}^{2}}{{\varepsilon }_{0}{\hslash }}\left[\displaystyle \frac{{\alpha }_{1}({\hslash }-N{\mu }^{2}{\mu }_{0}{\alpha }_{4})+N{\mu }^{2}{\mu }_{0}{\alpha }_{2}{\alpha }_{3}}{{\hslash }-N{\mu }^{2}{\mu }_{0}{\alpha }_{4}}\right],\end{eqnarray}$$\begin{eqnarray}{\chi }_{m}=\displaystyle \frac{N{\mu }^{2}{\mu }_{0}{\alpha }_{4}}{{\hslash }-N{\mu }^{2}{\mu }_{0}{\alpha }_{4}},\end{eqnarray}$$\begin{eqnarray}{\xi }_{\mathrm{HE}}=\displaystyle \frac{{Nc}\mu {\mu }_{0}{\alpha }_{3}d}{{\hslash }-N{\mu }^{2}{\mu }_{0}{\alpha }_{4}},\end{eqnarray}$and$\begin{eqnarray}{\xi }_{\mathrm{EH}}=\displaystyle \frac{{Nc}\mu {\mu }_{0}{\alpha }_{2}d}{{\hslash }-N{\mu }^{2}{\mu }_{0}{\alpha }_{4}},\end{eqnarray}$where A1−4 and α1−4 are in the appendix. The refractive indices of the LCP and RCP beams are written as (for more details see [28, 47])$\begin{eqnarray}\begin{array}{l}{n}_{r}^{(\pm )}\\ =\,\sqrt{\left(1+{\chi }_{e}\right)\left(1+{\chi }_{m}\right)-\displaystyle \frac{{\left({\xi }_{\mathrm{EH}}+{\xi }_{\mathrm{HE}}\right)}^{2}}{4}\pm \displaystyle \frac{{\rm{i}}}{2}\left({\xi }_{\mathrm{EH}}-{\xi }_{\mathrm{HE}}\right)},\end{array}\end{eqnarray}$where ${n}_{r}^{(+)}$ stands for the RCP beam complex refractive index and ${n}_{r}^{(-)}$ stands for the LCP beam complex refractive index. If ξEH(HE)=0, then the medium is an ordinary refractive medium, when Re $({n}_{r}^{(\pm )})$ is positive. The medium is negative refractive, when Re $({n}_{r}^{(\pm )})$ is negative. However, if ξEH(HE)≠0, the medium is chiral. The chiral medium contributed additional terms of chiral coefficients to refractive indices along with relative permittivity ϵr=1+χe and permeability μr=1+χm terms. To find the dependence of the dielectric function dependent on conductivity, we use the following Maxwell equations for conductive medium$\begin{eqnarray}{\rm{\nabla }}.E=0,{\rm{\nabla }}.B=0,E=-\displaystyle \frac{\partial B}{\partial t},{\rm{\nabla }}\times B=\mu J+\mu \varepsilon \displaystyle \frac{\partial E}{\partial t},\end{eqnarray}$where E and B are the electric and magnetic fields, respectively. ϵ represents the permittivity and μ represents the permeability of the material. By applying curl to (14), we get$\begin{eqnarray}{{\rm{\nabla }}}^{2}E=\mu \varepsilon \displaystyle \frac{\partial E}{\partial t}+\mu \varepsilon \displaystyle \frac{{\partial }^{2}B}{\partial {t}^{2}},\end{eqnarray}$$\begin{eqnarray}{{\rm{\nabla }}}^{2}B=\mu \varepsilon \displaystyle \frac{\partial B}{\partial t}+\mu \varepsilon \displaystyle \frac{{\partial }^{2}B}{\partial {t}^{2}}.\end{eqnarray}$The solution for the above equations are$\begin{eqnarray}E(r,t)={E}_{0}\exp \,({\rm{i}}(k.r-\omega t)),\end{eqnarray}$$\begin{eqnarray}B(r,t)={B}_{0}\exp \,({\rm{i}}(k.r-\omega t)),\end{eqnarray}$where E0 and B0 are the complex amplitudes. For more details, see [57].
Equations (17) and (18) are the wave equations describing propagation of electromagnetic waves in the metal. Differentiating equation (17) twice and substituting in equation (15), we get$\begin{eqnarray}{k}_{m}^{2}=\mu \varepsilon {\omega }^{2}+{\rm{i}}\mu \sigma \omega ,\end{eqnarray}$$\begin{eqnarray}k={k}_{m1}+{\rm{i}}{k}_{m2},\varepsilon ={\varepsilon }_{r}+{\rm{i}}{\varepsilon }_{i},\sigma ={\sigma }_{r}+{\rm{i}}{\sigma }_{i},\end{eqnarray}$$\begin{eqnarray}{k}_{m}^{2}=(\mu {\varepsilon }_{r}{\omega }^{2}-\mu {\sigma }_{i}\omega )+{\rm{i}}(\mu {\varepsilon }_{i},{\omega }^{2}+\mu {\sigma }_{r}\omega ),\end{eqnarray}$$\begin{eqnarray}{k}_{m1}^{2}-{k}_{m2}^{2}=(\mu {\varepsilon }_{r}{\omega }^{2}-\mu {\sigma }_{i}\omega ),\end{eqnarray}$$\begin{eqnarray}2{k}_{m1}{k}_{m2}=(\mu {\varepsilon }_{i},{\omega }^{2}+\mu {\sigma }_{r}\omega ),\end{eqnarray}$$\begin{eqnarray}k=\pm \displaystyle \frac{{\beta }_{1}}{\sqrt{2}}\left[\sqrt{1\pm {\left(\displaystyle \frac{{\beta }_{2}}{{\beta }_{1}}\right)}^{4}}+{\rm{i}}\sqrt{-1\pm \sqrt{1+{\left(\displaystyle \frac{{\beta }_{2}}{{\beta }_{1}}\right)}^{4}}}\right],\end{eqnarray}$where$\begin{eqnarray}{\beta }_{1}={k}_{0}\sqrt{{\varepsilon }^{{\prime} }-\displaystyle \frac{{\sigma }_{i}}{{\varepsilon }_{0}\omega }},{\beta }_{2}={k}_{0}\sqrt{{\varepsilon }^{{\prime\prime} }+\displaystyle \frac{{\sigma }_{i}}{{\varepsilon }_{0}\omega }}.\end{eqnarray}$
Real and imaginary parts of the permittivity in terms of the corresponding dielectric constants, ${\varepsilon }_{r}={\varepsilon }_{0}\varepsilon ^{\prime} $ and ϵi=ϵ0ϵ″, we have$\begin{eqnarray}{k}_{m1}^{2}-{k}_{m2}^{2}=\mu {\varepsilon }_{0}{\omega }^{2}\left(\varepsilon ^{\prime} -\displaystyle \frac{{\sigma }_{i}}{{\varepsilon }_{0}\omega }\right)={k}_{0}^{2}\left(\varepsilon ^{\prime} -\displaystyle \frac{{\sigma }_{i}}{{\varepsilon }_{0}\omega }\right)={\beta }_{1}^{2},\end{eqnarray}$$\begin{eqnarray}2{k}_{m1}{k}_{m2}=\mu {\varepsilon }_{0}{\omega }^{2}\left(\varepsilon ^{\prime\prime} +\displaystyle \frac{{\sigma }_{r}}{{\varepsilon }_{0}\omega }\right)={k}_{0}^{2}\left(\varepsilon ^{\prime\prime} +\displaystyle \frac{{\sigma }_{r}}{{\varepsilon }_{0}\omega }\right)={\beta }_{2}^{2},\end{eqnarray}$where we used μ0ϵ0=1/c and k0=ω/c. Real and imaginary propagation parameters km1 and km2 are found by separating equations (23) and (24) we get$\begin{eqnarray}\begin{array}{rcl}{k}^{{}_{(R,L)}} & = & \pm {k}_{0}\displaystyle \frac{\sqrt{{\varepsilon }^{{\prime} }-\tfrac{{\sigma }_{i}}{{\varepsilon }_{0}\omega }}}{\sqrt{2}}\left[{\rm{i}}\sqrt{-1\pm \sqrt{1+\displaystyle \frac{{\left({\varepsilon }^{{\prime\prime} }+\tfrac{{\sigma }_{r}}{{\varepsilon }_{0}\omega }\right)}^{4}}{{\left({\varepsilon }^{{\prime} }-\tfrac{{\sigma }_{i}}{{\varepsilon }_{0}\omega }\right)}^{4}}}}\right.\\ & & \left.+\sqrt{1\pm \sqrt{1+\displaystyle \frac{{\left({\varepsilon }^{{\prime\prime} }+\tfrac{{\sigma }_{r}}{{\varepsilon }_{0}\omega }\right)}^{4}}{{\left({\varepsilon }^{{\prime} }-\tfrac{{\sigma }_{i}}{{\varepsilon }_{0}\omega }\right)}^{4}}}}\right].\end{array}\end{eqnarray}$
3. Results and discussion
The birefringence beams absorption coefficients through a dynamic conductive chiral medium are investigated under the effect of quantum tunneling for the atom localization. The decay rate of the atomic state is supposed to γ=36.1 MHz. Here $| {{\rm{\Omega }}}_{\mathrm{1,2}}| =| {{\rm{\Omega }}}_{\mathrm{1,2}}| \exp ({\rm{i}}{\varphi }_{\mathrm{1,2}})$, ω0=104γ and λ=2πc/ω. The other frequency parameters are scaled with γ. In the chiral atomic system, two probes of electric and magnetic fields are applied. The strong control fields are coupled in the system to modify the behavior of electric and magnetic probes. For the conductive chiral medium, we calculated 16 types of wave vectors for birefringence due to three±signs. This is related to 16 types of absorption coefficients and refractive indices. We have localized the atom by the birefringent beams for the cases of (+, +, +), (+, −, −), (−, +, +) and (−, −, −). The absorption coefficient is related to the imaginary part of the complex wave vector (Im $({k}^{{}_{(R,L)}})={A}^{{}_{(R,L)}}$), while the real part of the complex wave vector is related to phase velocity and group velocity.
In figure 2, the plots are traced for atom localization through the dynamic conductive chiral medium under the effects of quantum tunneling using the birefringence absorption spectrum in one wavelength domain. Under the specific parametric condition, we have noticed two localized peaks in both the LCP and RCP light beams in inverse space of kx and ky. For the case, (+, +, +) and (−, −, −) two localized peaks are investigated by LCP and RCP beams’ absorption spectra. At the condition (+, +, +) two localized peaks are noted at the LCP and RCP beams absorption spectrum in the first and fourth quadrants in the 2D inverse position space. Here, one localized peak is in the first and another one is in the fourth quadrant. It means that we have two possible localized positions for an atom localization as shown in figures 2(a) and (b). At the condition (−, −, −) two localized peaks are noted at the LCP and RCP beams amplification (negative absorption) spectrum in the first and fourth quadrants in 2D inverse position space. It means the amplification occurs in the birefringent beams at the conditions of (−, −, −). At the birefringence beams amplifications, two downward localized peaks are investigated in the 2D plane (shown as figures 2(c) and (d)). It has been pointed out that these localized peaks also show two possible positions for the atom localization using birefringence absorption spectrum in a single wavelength domain.
Figure 2.
New window|Download| PPT slide Figure 2.Birefringence absorption coefficients as functions of (kx, ky). The parameters supposed here are γ=36.1 GHz, γ32,35,41,45=2γ, γ51,52=0.2γ, Δp=0.5γ, Δ1=0γ, φ1=π/4, φ2=π/3, φp=π/2 φm=0, ∣Ω1,2∣=20γ, σi,r=1000S/m, η1,3=1, η2,4=−1, T=10, φ3=3π/2, φ4=π/2 (+, +, +) birefringence condition for (a) and (b) while (−, −, −) birefringence condition for (c) and (d).
In figure 3, the plots are traced for atom localization through dynamic conductive chiral medium under the effects of quantum tunneling using birefringence absorption/amplification spectra in one wavelength domain of 2D inverse space. The parametric condition for this figure is the same as given in figure 2, but we have changed only the birefringence condition from (+, +, +) and (−, −, −) to (+, −, −) and (−, +, +), respectively. Two large localized and two small localized peaks were reported by the absorption and amplification spectra of the RCP and LCP beams in the 2D inverse plane in one wavelength domain. At condition (+, −, −), the RCP beam shows two small localized peaks in the second and third quadrants and two large localized peaks in the first and fourth quadrants. The localized peaks at the absorption spectrum of the LCP beam are the mirror images of localized peaks, which occur by the absorption spectrum of RCP beams as shown in figures 3(a) and (b). Further, two large localized and two small localized peaks were reported by the amplification spectra of the RCP and LCP beams in the 2D inverse plane in one wavelength domain. At condition (−, +, +), the RCP beam shows two small localized peaks in the second and third quadrants and two large localized peaks in the first and fourth quadrants by its amplification spectra. The localized peaks at the amplification spectrum of the LCP beam are the mirror images of localized peaks that occur by the amplification spectrum of RCP beams as shown in figures 3(c) and (d).
Figure 3.
New window|Download| PPT slide Figure 3.Birefringence absorption coefficients as functions of (kx, ky). The parameters supposed here are γ=36.1 GHz, γ32,35,41,45=2γ, γ51,52=0.2γ, Δp=0.5γ, Δ1=0γ, φ1=π/4, φ2=π/3, φp=π/2 φm=0, ∣Ω1,2∣=20γ, σi, r=1000 S m−1, η1,3=1, η2,4=−1, T=10, φ3=3π/2, φ4=π/2 (+, −, −) birefringence condition for (a) and (b) while (−, +, +) birefringence condition for (c) and (d).
To explicitly show the high resolution and high precision atomic localization through the birefringence absorption probe spectra, we modify the 2D atom localization behaviors as shown in figure 4. In figure 4, the plots are traced for atom localization through the dynamic conductive chiral medium under the effects of quantum tunneling using birefringence absorption/amplification spectra in one wavelength domain. The condition of (+, +, +) the RCP and LCP beams’ absorption spectra show a single broad localized peak in the first quadrant. It means that we have a single possible localized position for an atom as shown in figures 4(a) and (b). Furthermore, when we changed the birefringent condition to (−, −, −), the downward localized peaks were investigated by RCP and LCP beam amplification spectra. In this figure, it is reported that the conditional position probability occurs at a specific point and is maximum there. The localized probability occurs at the LCP and RCP beams amplification spectra and are the mirror images of localized probability occurring at absorption spectra as shown in figures (c) and (d).
Figure 4.
New window|Download| PPT slide Figure 4.Birefringence absorption coefficients as functions of (kx, ky). The parameters supposed here are γ=36.1 GHz, γ41=0.27γ, γ32=0.21γ, γ51=0.97γ, γ52=0.7γ, γ45=2.7γ, γ35=0.3γ, Δp=0.5γ, Δ1=−1γ, φ1=π/2, φ2=π/3, φp=π/2 φm=π/4, ∣Ω1∣=15γ, ∣Ω2∣=10γ, σr=10000 S m−1, σi=1500 S m−1, η1=1.35, η2=0.5, η3=0.2, η4=0.8, T=10, φ3=π/2, φ4=π/3 (+, +, +) birefringence condition for (a) and (b) while (−, −, −) birefringence condition for (c) and (d).
To modify the 2D atom microscopy for advanced high resolution and high precision, the birefringence absorption/amplification spectra are checked at another parametric condition. In figure 5, the plots are traced for atom localization through the dynamic conductive chiral medium under the effect of quantum tunneling using birefringence absorption/amplification spectra in one wavelength domain. The shift in the localized peaks are reported in the absorption and amplification spectra. We have investigated more interesting results such that the shift occurs in the localized peaks for other parametric conditions under same birefringence condition. The localized peak was investigated in the joint region of the second and third quadrants of the 2D space by the absorption/amplification spectra of the RCP and LCP beams under the conditions (+, +, +) and (−, −, −). It means that we have a single possible localized position for an atom in the joint region of the second and third quadrants. The localized peaks that occur at birefringent absorption spectra are the mirror images of the localized peaks that occur at birefringent amplification spectra, as depicted by figures 5(a)–(d).
Figure 5.
New window|Download| PPT slide Figure 5.Birefringence absorption coefficients as functions of (kx, ky). The parameters supposed here are γ=36.1 GHz, γ41,32,51,52,45,35,=2.5γ, Δp=0.5, Δ1=0γ, φ1=π/4, φ2=π/3, φp=π/2 φm=0, ∣Ω1,2∣=21γ, σi, r=1000 S m−1, η1,3=1, η2,4=−1, T=10, φ3=3π/2, φ4=π/2 (+, +, +) birefringence condition for (a) and (b) while (−, −, −) birefringence condition for (c) and (d).
4. Conclusion
In conclusion, a five-level chiral atomic medium (driven by two electric and magnetic probes with the two extra driving control fields) are used to modify the birefringence under quantum tunneling conditions. Likewise, density matrix formalism and Maxwell’s equation are also used to modify optical birefringence influenced by complex conductivity and tunneling effect. Moreover, 16 types of absorption coefficients are reported for conductive chiral atomic medium under the complex conductivity conditions. Therefore, the two-dimensional atom localization is investigated using the absorption spectrum of birefringence beams of light in a single wavelength domain. The localized peaks are studied for the conditions of (+, +, +), (+, −, −), (−, +, +) and (−, −, −) of the left and right circularly polarized beams absorption spectra. Two and a single localized peaks are reported in different quadrants with minimum uncertainty and significant probability. The localized probability is modified with the control of birefringence, tunneling and complex conductivity amplitude. These results may be useful for the capability of optical microscopy and atom imaging.
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