⇦Chinese Physics Letters, 2016, Vol. 33, No. 7, Article code 074202 Anderson Localization in the Induced Disorder System * Fei-Fei Lu(陆菲菲), Chun-Fang Wang(王春芳)** Affiliations Department of Physics, University of Shanghai for Science and Technology, Shanghai 200093 Received 23 April 2016 *Supported by the National Natural Science Foundation of China under Grant Nos 11104185, 11174084 and 10934011, and the National Basic Research Program of China under Grant No 2012CB921904.
**Corresponding author. Email: cfwang@usst.edu.cn Citation Text: Lu F F and Wang C F 2016 Chin. Phys. Lett. 33 074202 Abstract We propose a coherently prepared three-level atomic medium that can provide a flexible disordered scheme for realizing the Anderson localization. Different disorder levels can be attained by modulating the intensity ratio between the two control beams. Due to the real-time tunability, the localization of the signal beam is observable and controllable. The influences of the induced disorder level, atomic density and the initial waist radius of the signal beam on the Anderson localization in the medium are also discussed. DOI:10.1088/0256-307X/33/7/074202 PACS:42.50.-p, 42.65.-k, 42.25.Kb © 2016 Chinese Physics Society Article Text Anderson localization, that is, the complete suppression of wave diffusion caused by sufficiently strong disorder,[1] is one of the most important and intriguing phenomena studied in condensed-matter physics.[2,3] As an interference wave phenomenon, the concept has been extended to acoustic waves,[4] light,[5-7] Bose–Einstein condensed-mater waves,[8] and various quantum optical systems, such as propagating photons[9-12] and atomic lattices.[8] Moreover, the Anderson localization has also been studied in superlattices containing graphene layers[13] and PT-symmetric structures.[14] Therefore, Anderson localization has been a truly interdisciplinary topic, and the important contributions have emerged from different areas, including condensed-matter, acoustics, photonics, atomic physics, and seismology.[2] Elaborate studies on the Anderson localization are fundamental and essential. Moreover, it will also be beneficial to understanding a large number of physical phenomena occurring in systems ranging from light propagation in disordered materials,[15] disordered superconductors,[16] porous media, etc.[17] However, the Anderson localization in condensed-matter systems is difficult to observe directly, due to the existence of complicated multi-body interaction and other uncontrollable effects.[18] Instead, more recently, some engineered systems have been proposed in which transportation of particles or propagation of waves can be significantly manipulated and observed by controllable disorder. Among them, the successful examples include transverse Anderson localization, which has been observed in optically induced[15] and in fabricated lattices.[19-25] Impacts of refractive index gradients,[24] the inhomogeneity of disorder,[26] interfaces,[20,23,27,28] and nonlinearity[15,19,27,29,30] have been elucidated. In this Letter, we find that it is possible to use the electromagnetically induced transparency (EIT) to realize the disorder system and the observable Anderson localization, in which the disorder is induced by taking the hybrid of the speckled beam and plane wave[8] as control field. Due to the coherent manipulation, the properties of the induced system are flexible and controllable. The purpose of our work is to study the Anderson localization in the induced disorder system with real-time tunability. For the weak absorption,[31] observability, and controllable characteristics,[32] our work has the importantly theoretical and practical value, such as applications in four-wave mixing,[33,34] self-imaging[35] and waveguides.[36]
cpl-33-7-074202-fig1.png
Fig. 1. (Color online) Schematic diagram of the induced disorder EIT system. The control beam is combined with control beams 1 and 2. Control beam 1 is the plane wave and control beam 2 is the random speckled beam generated by the diffuser. The configuration of the ${\it \Lambda}$-type atomic system is shown in the upper right corner.
In our scheme, we consider a cold three-level atomic system with ${\it \Lambda}$-type configuration. As shown in the upper right corner of Fig. 1, the ground state $|0\rangle$ and the metastable state $|1\rangle$ are coupled individually with the excited state $|2\rangle$ via a weak signal field and an intense control laser field, respectively. Here ${\it \Omega}_{\rm s}$ and ${\it \Omega}_{\rm c}$ are the Rabi frequencies of the signal field and control field, which are corresponding to the transition of $|0\rangle\rightarrow|2\rangle$ and $|1\rangle\rightarrow|2\rangle$, respectively. The frequency $\omega_{\rm s}$ of the signal pulse is assumed to be close to the transition frequency of $|0\rangle\rightarrow|2\rangle$. Under these conditions, near $\omega_{20}$, the linear susceptibility of the signal field $\chi$ is[37] $$\begin{align} \chi=\,&\eta[{\it \Delta}_{\rm s}({\it \Omega}^{2}_{\rm c}-4{\it \Delta}^{2}_{\rm s}-4{\it \Gamma}^{2}_{1}) +2i{\it \Delta}^{2}_{\rm s}{\it \Gamma}_{2}\\ &+i{\it \Gamma}_{1}({\it \Omega}^{2}_{\rm c}+{\it \Gamma}_{1}{\it \Gamma}_{2})/2]/\{[{\it \Omega}^{2}_{\rm c} +{\it \Gamma}_{1}{\it \Gamma}_{2}\\ &-4{\it \Delta}^{2}_{\rm s}]+4{\it \Delta}^{2}_{\rm s}({\it \Gamma}_{1}+{\it \Gamma}_{2})^{2}\},~~ \tag {1} \end{align} $$ where $\eta=4\rho\mu^{2}/\varepsilon_{0}\hbar$, ${\it \Delta}_{\rm s}=\omega_{\rm s}-\omega_{20}$, $\mu$ is the electric dipole moment of transition $|0\rangle\rightarrow|2\rangle$, $\rho$ is the atomic density, $\varepsilon_{0}$ means the permittivity of vacuum, $\hbar$ is the Planck constant, and ${\it \Gamma}_{1}$ and ${\it \Gamma}_{2}$ are the decay rates of $|1\rangle$ and $|2\rangle$, respectively. To realize the Anderson localization, we use the hybrid beam as the control field (see Fig. 1). One is the plane wave (control beam 1), the other is the random speckled beam (control beam 2). The random speckled beam is generated by the automatically rotated diffuser. The disorder level of the control beam is determined by the intensity ratio of the two beams. As the combined control beam enters the atomic system, the medium can be coherently induced to disordered and inhomogeneous EIT media. Considering the propagation of a signal beam (the Gaussian beam) along the $z$-axis, perpendicular to the EIT medium, under the slowly varying envelope approximation, the wave equation of a paraxial propagating signal field in the EIT medium is[31] $$\begin{align} 2ik\phi_{z}+(\partial_{xx}+\partial_{yy})\phi+k^{2}\chi\phi=0,~~ \tag {2} \end{align} $$ where $\phi$ is the envelope of the signal field, ${\it \Omega}_{\rm s}(x,y,z)=\phi(x,y,z)\exp(-i\omega_{\rm s}t+ikz)$, $k=\omega_{\rm s}/c$, and the linear susceptibility $\chi$ has been given by Eq. (1). To study the Anderson localization effects, the disorder in the medium is induced by the random maximum amplitude ${\it \Omega}_{\rm cr}$, instead of the transversely homogeneous maximum amplitude ${\it \Omega}_{\rm c}$, which takes the values in the interval ${\it \Omega}_{\rm c}(1-Nr) < {\it \Omega}_{\rm cr} < {\it \Omega}_{\rm c}(1+Nr)$, where $r$ is the random number generator, and $N$ determines the degree of disorder. They both may take the values in the interval [0,1]. We quantify the degree of disorder by the ratio between the intensity of the random and the homogeneous EIT medium. In our simulation, we set $\eta=4\times10^{5}$ s$^{-1}$, ${\it \Gamma}_{1}=3\times10^{3}$ s$^{-1}$, ${\it \Gamma}_{2}=3\times10^{7}$ s$^{-1}$, ${\it \Omega}_{\rm c}=4\times10^{7}$ s$^{-1}$, ${\it \Delta}_{\rm s}=5\times10^{5}$ s$^{-1}$, $\lambda=800$ nm, and $w_{0}=0.04$ mm (the initial waist radius of signal beam). Since the signal field is absorbed partially during the propagation, the amplitude of the signal field is normalized at every plane $z$, that is, $E(x,y)=|\phi(x,y)|/\max\{|\phi(x,y)|\}$ is calculated. We simulate the propagations of the signal field in the EIT medium numerically by using Eq. (2) with different disorder levels $N$. The result of the normalized amplitude distribution of the signal field $E(x=0,y,z)$ is presented in Fig. 2. When $N=0$, there is no disorder, and the effective potential is purely transversely homogeneous. The signal field expands ballistically along the propagation direction (Fig. 2(a)). However, as $N$ increases, the signal field still expands while the expanding trend reduces. When $N$ increases to 75%, the signal field is localized along the $z$-axis direction. Notice that in the disordered EIT medium, with the influence of the disorder of the control field, the signal beam is scattered largely and then formed scattered waves with random amplitudes and phases, these scattered waves interfere with each other. Eventually, the signal beam transformed into a localized state, which is the Anderson localization. As shown in Fig. 2, when the disorder-level of the control field is large enough, the Anderson localization can be realized in the designed system.
cpl-33-7-074202-fig2.png
Fig. 2. (Color online) Normalized amplitude distribution of the signal field $E(x=0,y,z)$, with different disorder levels $N$. The parameters are $\eta=4\times10^{5}$ s$^{-1}$, ${\it \Gamma}_{1}=3\times10^{3}$ s$^{-1}$, ${\it \Gamma}_{2}=3\times10^{7}$ s$^{-1}$, ${\it \Omega}_{\rm c}=4\times10^{7}$ s$^{-1}$, ${\it \Delta}_{\rm s}=5\times10^{5}$ s$^{-1}$, $\lambda=800$ nm, and $w_{0}=0.04$ mm (the waist radius of the initial signal beam). (a) $N=0{\%}$, (b) $N=15{\%}$, (c) $N=30{\%}$, (d) $N=45{\%}$, (e) $N=60{\%}$ and (f) $N=75{\%}$.
For quantitative analysis, the effective beam width is defined as $w_{\rm eff}=P^{-1/2}$, where $$\begin{align} P=\frac{\int|E(x,y,z)|^{4}dxdy}{[\int|E(x,y,z)|^{2}dxdy]^{2}},~~ \tag {3} \end{align} $$ which is known as the inverse participation ratio.[15] To display the propagation properties of the signal field, on the output $z=100$ mm, the normalized amplitude distribution of the signal field $E(x,y)$ with the different control-field disorder levels (0%, 15%, 30%, 45%, 60% and 75%) are shown in Fig. 3. The corresponding effective beam width $w_{\rm eff}$ are 1.0563 mm, 0.8472 mm, 0.3992 mm, 0.2872 mm, 0.1183 mm, and 0.0208 mm, respectively. From Fig. 3, we can see that with the increase of the disorder level $N$, the effective beam width $w_{\rm eff}$ decreases, and the signal field gradually converges, and we can realize the Anderson localization.
cpl-33-7-074202-fig3.png
Fig. 3. (Color online) Normalized amplitude distribution of the signal field $E(x,y)$ on the output $z=100$ mm, with different disorder levels $N$. The other parameters are the same as those in Fig. 2. (a) $N=0{\%}$, (b) $N=15{\%}$, (c) $N=30{\%}$, (d) $N=45{\%}$, (e) $N=60{\%}$ and (f) $N=75{\%}$.
cpl-33-7-074202-fig4.png
Fig. 4. (Color online) The effective beam width $w_{\rm eff}$ versus the propagation distance $z$, for different disorder levels (0%, 15%, 30%, 45%, 60% and 75%). The other parameters are the same as those in Fig. 2.
The effective beam width $w_{\rm eff}$ as a function of the propagation distance is shown in Fig. 4. In order to investigate the influence of propagation distance on the Anderson localization process more clearly, we consider a longer propagation distance $z=100$ mm. For a transversely homogeneous medium (no disorder), the effective beam width $w_{\rm eff}$ linearly increases, as is expected. For weak disorder levels ($15{\%}$, $30{\%}$, $45{\%}$, $60{\%}$), the beam still expands diffusively, while the growth rate is slowed down. However, when the stronger disorder level ($75{\%}$) is introduced, the Anderson localization regime is reached after a short propagation distance.
cpl-33-7-074202-fig5.png
Fig. 5. (Color online) The effective beam width $w_{\rm eff}$ on the output $z=100$ mm versus the disorder level for $\rho_{1}=0.5\times10^{18}$ m$^{-3}$ (blue curve), $\rho_{2}=1.0\times10^{18}$ m$^{-3}$ (red curve), and $\rho_{3}=1.5\times10^{18}$ m$^{-3}$ (green curve), respectively. The other parameters are the same as those in Fig. 2.
cpl-33-7-074202-fig6.png
Fig. 6. (Color online) The effective beam width $w_{\rm eff}$ on the output $z=100$ mm versus the disorder level for the initial waist radius of the signal beam $w_{01}=0.04$ mm (blue curve), $w_{02}=0.056$ mm (red curve), and $w_{03}=0.072$ mm (green curve), respectively. The other parameters are the same as those in Fig. 2.
We study the effect of the atomic density $\rho$ on the Anderson localization, and measure the effective beam width $w_{\rm eff}$ versus the disorder level for $\rho_{1}=0.5\times10^{18}$ m$^{-3}$ ($\eta_{1}=2\times10^{5}$ s$^{-1}$, blue curve), $\rho_{2}=1.0\times10^{18}$ m$^{-3}$ ($\eta_{2}=4\times10^{5}$ s$^{-1}$, red curve), and $\rho_{3}=1.5\times10^{18}$ m$^{-3}$ ($\eta_{3}=6\times10^{5}$ s$^{-1}$, green curve), respectively (see Fig. 5). The general trend for all cases ($\rho_{1}=0.5\times10^{18}$ m$^{-3}$, $\rho_{2}=1.0\times10^{18}$ m$^{-3}$, and $\rho_{3}=1.5\times10^{18}$ m$^{-3}$) is similar, that is, the effective beam width $w_{\rm eff}$ decreases as the disorder level is increased, in other words, disorder always limited diffusion. It can also be seen that, generally speaking, under the same disorder level, the effective width decreases as the atomic density increases. However, here there is a high disorder level, the effective width is reduced to a rather small value, and the atomic density has no influence on the effective width. For example, when $N=75{\%}$, the effective beam width $w_{\rm eff}$ is 0.0208 mm for $\rho_{1}=0.5\times10^{18}$ m$^{-3}$, $\rho_{2}=1.0\times10^{18}$ m$^{-3}$ and $\rho_{3}=1.5\times10^{18}$ m$^{-3}$. We also study the influence of the initial waist radius $w_{0}$ of the signal beam on the Anderson localization. We consider three different cases: $w_{01}=0.04$ mm (blue curve), $w_{02}=0.056$ mm (red curve) and $w_{03}=0.072$ mm (green curve). We measure the effective beam width $w_{\rm eff}$ as a function of the disorder level (see Fig. 6). The general trends for all cases are similar: the effective beam width $w_{\rm eff}$ decreases as the level of disorder is increased, and finally reduces to the same value. It is interesting that, for the same disorder level $N$, the greater the initial waist radius of signal beam is, the smaller the effective beam width on the output is, which indicates that the induced disorder EIT medium has the stronger localization for the signal field with wider initial waist radius. In summary, we have investigated the Anderson localization of light in the induced disordered EIT medium. The results show that the disorder level, atomic density and initial waist radius of signal field have different influences on the Anderson localization. It is demonstrated that the stronger disorder level enhances light localization. For the same disorder level, increasing the atomic density and the initial waist radius can enhance light localization. It should be noticed that the proposal for realizing localization presented here is based on the coherent manipulation of the atomic system, in which one can easily modify the susceptibility of the induced medium as needed. The induced disorder provides another flexible and controllable method to realize the localization of the signal field. 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