Three-color reflections in one-dimensional ordered and disordered atomic lattices with trapped N-type cold atoms

Maohua Wang; Maohua Wang; Tianming Li; Yan Zhang; Xiaoshan Pei; Hong Yang

doi:10.1364/OE.463767

1. Introduction

In the last decades, many quantum effects have been proposed to create various types of coherent quantum systems, which play a fundamental role in quantum optics and information processing [1–6]. Electromagnetically induced transparency (EIT) is a significant quantum interference effect where the absorption of a weak probe field can be suppressed in a small frequency range [7–12]. Based on EIT, applying a standing-wave (SW) field in the uniformly distributed atomic system makes absorption and dispersion properties spatially periodic for viability in attaining tunable photonic bandgaps (PBGs) [13–16] and light storage and retrieval [17–20]. The main advantage of this regime is that the light propagation can be controlled on demand. However, a one-dimensional (1D) optical lattice trapping cold atoms of Gaussian density distribution in each period provides another regime of tunable light manipulation; for example, achieving dynamically tunable PBGs driven by traveling-wave coupling fields into the EIT regime [21–28]. Further, if the traveling-wave coupling fields are replaced by the standing-wave fields, the nonreciprocity Bragg reflections [21,29–32] and unidirectional reflectionless can be realized [33–36] in a 1D atomic lattice. This method has also been used to realize structures of electromagnetically induced gratings due to the spatially periodic absorption and dispersion in the 1D or 2D optical lattice [33,37]. In particular, non-reciprocal reflections can be achieved in a moving optical lattice system [38,39].

The above photon probability phenomena have been mainly investigated in the ordered atomic lattice. The disorder arising from imperfect manufacturing or random distributions of atoms should be considered, which may destroy the periodicity of probe susceptibility [40–45]. Thus, the robustness of light propagation in the presence of variant types of disorder should be assessed [46–48]. In particular, the probe reflection exhibits a blue shift in a defective atomic lattice that can be regarded as a special disordered atomic lattice with a large atomic number fluctuation [49]. This motivates the examination of a different disordered atomic lattice with a fluctuation of the average density of atoms in each lattice cell, which may result in novel phenomena for manipulating light transport properties.

This study proposes a disordered atomic lattice of tightly trapped $^{87}$Rb atoms with a Gaussian spatial distribution that is driven by traveling-wave fields into the four-level $N$ configuration. Here, the disorder caused by the fluctuation of the average atomic density in each lattice cell has been considered to determine the robustness of probe reflections by (i) a fixed and (ii) random fluctuation. The theoretical description of the study is given in Sec. II, the numerical computation is given in Sec. III, and the main conclusions of our work are summarized in Sec. IV.

2. Model and methods

The atomic configuration is shown in Fig. 1(a). All $^{87}$Rb atoms are driven into the four-level $N$ system, where a weak probe field of frequency $\omega _{p}$ (amplitude $\mathbf {E}_{p}$) interacts with the transition $\left \vert 1\right \rangle \leftrightarrow \left \vert 3\right \rangle$ of wavelength $\lambda _{31}$ while two strong control fields of frequencies $\omega _{c1,c2}$ (amplitude $\mathbf {E}_{c1,c2}$) drive the transitions $\left \vert 2\right \rangle \leftrightarrow \left \vert 3\right \rangle$ and $\left \vert 2\right \rangle \leftrightarrow \left \vert 4\right \rangle$. The relevant Rabi frequencies (detunings) are $\Omega _{p}=\mathbf {E}_{p} \cdot \mathbf {d}_{13}/2\hbar$($\Delta _{p}=\omega _{31}-\omega _{p}$ ), $\Omega _{c1}=\mathbf {E}_{c1}\cdot \mathbf {d}_{23}/2\hbar$ ($\Delta _{c1}=\omega _{32}-\omega _{c1}$) and $\Omega _{c2}=\mathbf {E}_{c2} \cdot \mathbf {d}_{24}/2\hbar$ ($\Delta _{c2}=\omega _{42}-\omega _{c2}$), with $\mathbf {d}_{ij}$ ($\omega _{ij}$) being the dipole moment (resonant frequency) on relevant transitions. More concretely, $\left \vert 4\right \rangle$, $\left \vert 3\right \rangle$, $\left \vert 2\right \rangle$ and $\left \vert 1\right \rangle$ may refer to $\left \vert 5P_{3/2}\text {, }F=1\text {, } m_{F}=+1\right \rangle$, $\left \vert 5P_{3/2}\text {, }F=1\text {, } m_{F}=0\right \rangle$, $\left \vert 5S_{1/2}\text {, }F=1\text {, } m_{F}=+1\right \rangle$ and $\left \vert 5S_{1/2}\text {,}F=1\text {, } m_{F}=-1\right \rangle$ of $^{87}$Rb atoms broken by static magnetic fields in order. All atoms are trapped in a 1D optical lattice of the periodicity $a_{0}$ with the Gaussian density distribution. Here, we consider a special disorder atomic lattice where the atomic number density along $z$ in each period is described by

(1)$$N_{k}(z)=\frac{N_{0}(1+\Delta_{k})a_{0}}{\sigma_{z}\sqrt{2\pi}}\cdot e^{\left[ -(z-z_{k})^{2}/2\sigma_{z}^{2}\right] },$$

as shown in Fig. 1(b). $N_{0}(1+\Delta _{k})$ is the average atomic density with $\Delta _{k}$ modulated by period $k$, that is, $N_{k}(z)$ can describe a disorder with an average atomic density fluctuation. Here, we define $K^{+}$ ($K^{-}$) as the number of the lattice cells corresponding to the atomic density $N_{k}^{+}(z)$ ($N_{k}^{-}(z)$) for $\Delta _{k}>0$ ($\Delta _{k}<0$) and $K$ ($\simeq 7700$ for $L=3.0$mm) as the total number of cells. $N_{0}$ is regarded to be small enough to guarantee that the average distance between two atoms is much longer than the size of each atom so that the interaction of atoms can be neglected. In addition, $a_{0}=\lambda _{L}/2$ with a red-detuned retroreflected laser beam with a wavelength of $\lambda _{L}$, which forms the optical lattice, and a half width $\sigma _{z}=\lambda _{Lat}/(2\pi \sqrt {\eta } )$, where $\eta =2U_{0}/(\kappa _{B}T)$ is connected to the trapping depth $U_{0}$ of the optical potential, and a temperature $T$ of the atomic sample. When the half width can satisfy both $\sigma _{z}\rightarrow 0$ ($T\rightarrow 0$) and $\sigma _{z}\ll a_{0}$, the width $\sigma _{z}$ should be limited by the size of the Wannier function in a given lattice minimum, and the center of the $k$th lattice cell becomes $z_{k}=(k-1/2)a_{0}$.

Fig. 1. (color online) (a) Energy level diagram of a four level $N$-type atomic system interacting with a weak probe field of Rabi-frequency (detuning) $\Omega _{p}$ ($\Delta _{p}$) and two strong coupling fields of Rabi-frequencies (detunings) $\Omega _{c1,c2}$ ($\Delta _{c1,c2}$). (b) 1D atomic lattice of periodicity $a_{0}$ trapped these driven $N$-type atoms with a Gaussian distribution via the fluctuation of atomic density in each lattice cell .

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Under the rotating-wave and electric-dipole approximations, the atom-field interaction Hamiltonian is described by

(2)$$\begin{aligned} H_{I}& =\hbar \lbrack (\Delta _{p}-\Delta _{c1})\left\vert 2\right\rangle \left\langle 2\right\vert +\Delta _{p}\left\vert 3\right\rangle \left\langle 3\right\vert +(\Delta _{p}-\Delta _{c1}+\Delta _{c2})\\ \left\vert 4\right\rangle \left\langle 4\right\vert ]& -\hbar \lbrack \Omega _{p}\left\vert 3\right\rangle \left\langle 1\right\vert +\Omega _{c1}\left\vert 3\right\rangle \left\langle 2\right\vert +\Omega _{c2}\left\vert 4\right\rangle \left\langle 2\right\vert ]+\text{H.c.} \end{aligned}$$

Then, the equations of motion of the density matrix are written as

(3)$$\begin{aligned} \partial_{t}\rho_{14} & ={-}(i\Delta_{p}+\gamma_{14}-\Delta_{c1}+\Delta _{c2})\rho_{14}-i\Omega_{c2}^{{\ast}}\rho_{12},\\ \partial_{t}\rho_{13} & =\left[{-}i\Delta_{p}-\gamma_{13}\right] \rho _{13}-i\Omega_{p}^{{\ast}}-i\Omega_{c2}^{{\ast}}\rho_{12},\\ \partial_{t}\rho_{12} & =\left[ i(\Delta_{c1}-\Delta_{p})-\gamma _{12}\right] \rho_{12}-i\Omega_{c2}\rho_{14}-i\Omega_{c1}\rho_{13}, \end{aligned}$$

where $\gamma _{ij}=\sum _{k}(\Gamma _{ik}+\Gamma _{jk})$ denotes the coherent decay rate with $\Gamma _{ik}$ representing the spontaneous decay rate, and $\Gamma _{31}=\Gamma _{32}=\Gamma _{42}=\Gamma$. The above equations satisfy the footprint condition $\rho _{11}+\rho _{22}+\rho _{33}+\rho _{44}=1$ and conjugate conditions $\rho _{ij}=\rho _{ji}^{\ast }$. In the steady state and weak field approximation, $\rho _{11}\simeq 1$, $\rho _{22}\simeq$ $\rho _{33}\simeq$ $\rho _{44}$ $\simeq 0$ and $\partial t\rho _{ij}=0$. Then, we obtain

(4)$$\rho_{31}=\frac{i\Omega_{p}^{{\ast}}}{\frac{\left\vert \Omega_{c1}\right\vert ^{2}(i\Delta_{p}-i\Delta_{c1}+i\Delta_{c2}+\gamma_{41})}{(i\Delta_{p} -i\Delta_{c1}+\gamma_{21})(i\Delta_{p}-i\Delta_{c1}+i\Delta_{c2}+\gamma _{41})+\left\vert \Omega_{c2}\right\vert ^{2}}+\gamma_{31}+i\Delta_{p}}.$$

The complex probe susceptibility is obtained with Eq. (4) as

(5)$$\chi_{p}(z)=\chi_{p}^{\prime}(z)+i\chi_{p}^{\prime\prime}(z)=\frac {N_{k}(z)\left\vert \mathbf{d_{31}}\right\vert ^{2}}{\varepsilon_{0}\hbar }\cdot\frac{\rho_{31}}{\Omega_{p}}\text{,}$$

where the complex refractive index is $n_{p}(z)=\sqrt {1+\chi _{p}(z)}$. It is obvious that $\chi _{p}(z)$ and $n_{p}(z)$ are periodic in space depending on the modulation of the atom polarizability. The essential Bragg condition $n_{p}(z)=\lambda _{p0}/\lambda _{p}=\lambda _{Lat0}/\lambda _{Lat}$ is satisfied, with the shift $\Delta \lambda _{Lat}=\lambda _{Lat}-\lambda _{Lat0}$ and a nonzero propagation angle cos$\theta =\lambda _{p0}/\lambda _{Lat0}$ subject to the following condition:

(6)$$\chi_{p}(z)\simeq\frac{-2\Delta\lambda_{Lat}}{\lambda_{Lat}}$$

which can predict the points where $\Delta _{p}$ reflective bands appear.

To explore the detection of light reflection and transmission, we divided each period into $S$ layers of uniform thickness, and the atomic density of each thin layer is $N_{k}(z_{j}),j\in (1$, $S)$. Then, the transfer matrix [50,51] of a layer is

(7)$$m_{p}(z_{j})=\frac{_{1}}{t_{p}(z_{j})}\left[ \begin{array}{cc} (t_{p}(z_{j})^{2}-r_{p}(z_{j})^{2}) & r_{p}(z_{j})\\ -r_{p}(z_{j}) & 1 \end{array} \right] ,$$

where $r_{p}$ denotes the reflection coefficient and $t_{p}$ the transmission coefficient. Thus, the total transfer matrix for each period can be obtained as

(8)$$M_{p}^{k}=m_{p}(z_{1})\times{\cdot}\cdot{\cdot}\times m_{p}(z_{j})\times{\cdot} \cdot{\cdot}\times m_{p}(z_{S}).$$

We can then solve the Bloch equation. In the equation, the Bloch vector is $\kappa _{p}=\kappa _{p}^{\prime}+i\kappa _{p}^{\prime \prime}$. The photonic band structure can be obtained from solutions of the equation

(9)$$e^{2i\kappa_{p}a_{0}}-Tr(M_{p}^{k})e^{i\kappa_{p}a_{0}}+1=0.$$

Here $\kappa _{p}^{\prime}$ and $\kappa _{p}^{\prime \prime}$ denote, respectivity, the dispersion and extinction rates of probe photons $\omega _{p}$ incident upon the 1D atomic lattice of infinite length $L\rightarrow \infty$. As is known, the probe reflection band opens up for $\kappa _{p}^{\prime}=\pi /a$ and $\kappa _{p}^{\prime \prime}\neq 0$ at the Brillouin zone boundary. Such restriction can be used to estimate the position and size of the bandgap. Then multiplying the single transfer matrix of each atomic period, the total transfer matrix can be expressed as $M_{p}^{L}=M_{p}^{1}\cdot M_{p}^{2} \cdot \cdot \cdot M_{p}^{k}\cdot \cdot \cdot M_{p}^{K}$. Finally, the reflectivities and transmissivities on both atomic lattice ends can be expressed as

(10)$$\begin{aligned} R_{p}^{l}(L) &=\left\vert \frac{M_{p}^{L}(12)}{M_{p}^{L}(22)}\right\vert ^{2},R_{p}^{r}(L)=\left\vert \frac{M_{p}^{L}(21)}{M_{p}^{L}(22)}\right\vert ^{2}, \\ T_{p}^{l}(L) &=T_{p}^{r}(L)=\left\vert \frac{1}{M_{p}^{L}(22)}\right\vert ^{2}.\ \end{aligned}$$

The spectrum property can be expressed by a sufficiently reduced density of state (DOS) for probe photons, i.e., the reflection appears the reduction of DOS [52], which is given by

(11)$$\text{DOS}(z)=\operatorname{Re}\left\vert \frac{2+r_{p}^{+}(z)+r_{p}^{-} (z)}{1-r_{p}^{+}(z)r_{p}^{-}(z)}-1\right\vert$$

by using the complex reflection coefficients $r_{p}^{+}(z)$ and $r_{p}^{-}(z)$ of probe fields emitting along +$z$-axis and -$z$-axis, respectively, and considering a position in the middle of the lattice.

3. Results and discussions

We first examine the three-color reflections in an ordered atomic lattice with the constant average atomic density $N_{0}$ (for $\Delta _{k}=0$) in each lattice cell, as shown in Fig. 2. There are three high reflection bands with high reflectivities of over 90% [ see Figs. 2(c) and 2(d)] at the edge of the first Brillouin zone $\kappa _{p}^{\prime}a/{\pi} =1$ and $\kappa _{p}^{\prime\prime}\neq 0$ in the Bragg condition [see Figs. 2(a) and 2(b)], indicating that the positions of probe reflections arise around points $O_{1}$, $O_{2}$ (in the EIT windows) and $O_{3}$ (in the large detuning region) for $\chi _{p}^{\prime}=-2\Delta \lambda _{Lat}/\lambda _{Lat}$ and $\chi _{p}^{\prime\prime}=0$ [see Figs. 2(e) and 2(f)]. The purpose of our study is checking the robustness of two EIT reflections in a disordered atomic lattice with a fluctuation of average atomic density, the more specific discussions are displayed in the following.

Fig. 2. (Color online) (a) and (b) Real parts (black dash dot) and imaginary parts (red solid) of the Bloch wave vector $\kappa _{p}$; (c) and (d) reflectivities (red solid); (e) and (f) real parts (black dash dot) and imaginary parts (red solid) of the average probe susceptibilities versus probe detuning $\Delta _{p}$, where the olive-dash-dotted line is given by $-2\Delta \lambda _{Lat}$ /$\lambda _{Lat}$, with the parameters $\Omega _{p}=0.3$ MHz, $\Omega _{c1}=\Omega _{c2}=18$ MHz, $\Gamma =6$ MHz, $\Delta _{c1}=10$ MHz, $\Delta _{c2}=0$ MHz, $\lambda _{Lat0}=781$ nm, $\lambda _{p0}=780.24$ nm, $\Delta \lambda _{Lat}=0.9$ nm, $\sigma _{z}=$ $\lambda _{Lat}/(4\pi \sqrt {2\eta })$, $\eta =5$, $L=3.0$ mm, and $N_{0}=7\times 10^{11}$ cm$^{-3}$, $d_{14}=1.0357\times 10^{-29}$ C$\cdot$m, $\Delta _{k}=0$.

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Then, we consider a fixed average atomic density fluctuation $N_{k}(z)$ (for $\Delta _{k}=0$) that is replaced by $N_{k}^{-}(z)$ or $N_{k}^{+}(z)$ corresponding to $\Delta _{k}=-0.7$ or $\Delta _{k}=2.0$ in some lattice cells, which decreases the average atomic density to $N_{0}/3$ or increases it to $3N_{0}$ for an exaggerated average atomic density fluctuation. Here, we assess how EIT reflections are perturbed in the disordered atomic lattice where the lattice cells with $N_{k}^{-}(z)$ or $N_{k}^{+}(z)$ are randomly distributed. Figures 3(a) and 3(b) show that the width and height of two EIT reflections are reduced with increased $K^{-}$, mainly due to the reduced refractive index $n_{p}(z)$ accompanied by the corresponding decrease of the average atomic density. To further check the variation of probe reflections by the number of lattice cells with $N_{k}^{-}(z)$, we plot the maximum reflectivities and the width of reflections via $K^{-}/K$ in Fig. 3(c). The width of two reflections decreases significantly as the ratio $K^{-}/K$ increases, and gradually tend to be equal when the ratio reaches a large value $K^{-}/K=1/5$. However, the height of two reflections decreases slowly and maintains consistent reflectivities with increasing $K^{-}/K$, indicating that the height of reflections only depends on the density of atoms. Thus, the reduction of the atom number density in some lattice cells can weaken the probe reflections.

Fig. 3. (Color online) (a) and (b) Two probe reflections in the EIT windows of the ordered atomic lattice (black-triangle); the other two lines denote those of the disordered atomic lattice corresponding to the number ${K}^{-}$ ($\Delta _{k}=-0.7$) when it is $500$ (red-circle) and $1500$ (green-square). (c) The width of the EIT reflections $R_{1}$ and $R_{2}$ (orange-pentacle curves and blue-pentacle curves) and the maximum of the EIT reflections $R_{1}$ and $R_{2}$ (red-ring curves, green-circle curves) versus ${K}^{-}/K$. Other parameters are as in Fig. 2.

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We also investigated the influence of the increasing atomic density ($N_{k}^{+}(z)$) in some of the randomly distributed lattice cells, as shown in Fig. 4. The widths of the two EIT reflections have been widened remarkably via the increased $K^{+}/K$, but the heights remain robust against $K^{+}/K$ when the atomic density is sufficiently high [see Figs. 4(a) and 4(b)]. Figure 4(c) further verifies that the height of two EIT reflections Does not change, only the widths increase gradually. From Figs. 3 and 4, we can conclude that the reflectivities are generally robust against the disordered atomic lattice when the atomic number density is sufficiently large to produce high reflections. However, the width of the reflection is particularly sensitive to the increasing number of lattice cells with a large or small atomic density.

Fig. 4. (Color online) (a) and (b) Probe reflections in two EIT windows of an ideal atomic lattice (black-triangle); the two lines denote those of the disordered atomic lattice corresponding to the ${K}^{+}$ ($\Delta _{k}=2.0$) of 500 (red-circle) and 1500 (green-square). (c) The width of EIT reflections $R_{1}$ and $R_{2}$ (orange-pentacle curves and blue-pentacle curves) and the maximum of EIT reflections $R_{1}$ and $R_{2}$ (red-ring curves, green-circle curves) versus ${K}^{+}/K$. Other parameters are as in Fig. 2.

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Examining the effect of the position of a cluster disordered lattice cells (consisting of $K^{+}$ lattice cells with $N_{k}^{+}(z)$ and $K^{-}$ lattice cells with $N_{k}^{-}(z)$) on the EIT reflection bands is of special interest, because the cluster disorder can be regard as a special case of the random distribution. We plotted probe reflectivities with the cluster disordered lattice cells located in the front (corresponding to the $1^{\sim}2000$ cells), at the middle (corresponding to the $3001^{\sim}5000$ cells), or near the tail (corresponding to the $5701^{\sim}7700$ cells) of the atomic lattice for $K^{+}+K^{-}=2000$. The two EIT reflection bands remain robust against the cluster disordered lattice cells located at the middle or near the tail of our atomic lattice [see Figs. 5(a) and 5(b)]. However, the reflectivities dramatically decrease to 50% for $K^{-}>K^{+}$, when the cluster disordered lattice cells located at the front of the atomic lattice. The reflectivities can be raised by the increased $K^{+}$, and it is robust against this disorder when the ratio is $K^{+}/K>3/4$ as shown in Fig. 5(c). This is because a lower average atomic density ($N_{k}^{-}(z)$) reduces the reflectivity, but when this cluster disordered lattice cells appears at the middle or near the tail of the atomic lattice, the number of ordered lattice cells ($N_{k}(z)$) in the front is enough to form a high reflection band. In fact, the spatial symmetry of the refractive index will be broken. If this disordered lattice cell cluster appears in the atomic lattice with asymmetry at the position. Therefore, we plot the reflectivities on the right side in Fig. 5(d). A comparison of Figs. 5(c) and 5(d) shows that the probe light incident on the right side forms a wide and high reflection band, corresponding to extremely low and narrow band gaps on the left side. Similarly, when this disordered lattice cell cluster appears in the tail, the reflectivities on the right and left have the opposite behavior, which is not discussed here. In this condition, the reciprocity of probe light reflectivity is broken and is no longer robust. We are also interested in the transmissivities on the left and right sides, they are ploted in Figs. 5(e) and 5(f). It is clearly shown that the transmissivities are reciprocial, which can be easy abtained by equation (10).

Fig. 5. (Color online) In (a) and (b) the reflectivities on the left side in the EIT windows versus probe field detuning $\Delta _{p}$ and $K^{+}$, the starting position of the lattice cell with $N_{k}^{+}{(z)}$ is $5701$ and $3001$, respectively; The refletivities on the left and right sides in (c) and (d), the transmissivities on the left and right sides in (e) and (f) versus probe field detuning $\Delta _{p}$ and $K^{+}$, for the starting position of the lattice cell with $N_{k}^{+}{(z)}$ is $1$. Other parameters are as in Fig. 2.

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Finally, the fluctuation of average atomic density in each cell is random and can be regarded as a common disorder, with the random number $\Delta _{k}$ varying with the period $k$. As displayed in Fig. 6, we plot the photon probabilities considering a slight (moderate and drastic) vertical fluctuation for $\Delta _{k}\in (-0.1$, $+0.1)$ [$\Delta _{k}\in (-0.5$, $+0.5)$ and $\Delta _{k}\in (-1.0$, $+1.0)$]. Figures 6(a) and 6(b) show that the two EIT reflections have almost no change compared with that in the ordered atomic lattice ($\Delta _{k}=0$) under a slight fluctuation, even with the large random number $\Delta _{k}\in (-0.5$, $+0.5)$ or $\Delta _{k}\in (-1.0$, $+1.0)$. More specifically, the reflectivity remains robust against the disordered atomic lattice with the random fluctuation of average atomic density. Furthermore, the robustness can be investigated using the DOSs in ($\Delta _{p}=-3.0$ MHz) or out ($\Delta _{p}=+25.0$ MHz) of the reflection band. Figures 6(c) and 6(d) show that all the DOSs in the frequency regions where the photons are prevented or not have a slight decrease with the increasing random fluctuation range, mainly due to the average effect of fluctuations.

Fig. 6. (Color online) (a) and (b) Reflections R1 and R2 versus probe detuning $\Delta _{p}$, (c) the DOS of probe photons $\omega _{p}$ at the center of R1 for $\Delta _{p}=-3.0$ MHz, (d) the DOS of probe photons $\omega _{p}$ outside R1 and R2 for $\Delta _{p}=25.0$ MHz in the 1D ordered atomic lattice $\Delta _{k}=0$ (black solid line with triangle) and in the 1D disordered atomic lattice for $\Delta _{k}\in (-0.1$, $+0.1)$ (red solid line with circle), $\Delta _{k}\in (-0.5$, $+0.5)$ (green solid line with square) and $\Delta _{k}\in (-1.0$, $+1.0)$ (magenta solid line with pentacle). Other parameters are as in Fig. 2.

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In fact, an extreme case of the average atomic density fluctuations is $N_{k}(z)=0$ (for $\Delta _{k}=-1$) in one period which can be regard as a vacant lattice cell. A continous vacant lattice cells can break the spatial symmetry of 1D atomic lattice that the probe reflections exhibit non-reciprocity, unless its position located in the whole periodic structure is symmetry (e.g. a continous vacant lattice cells locates in the middle of the 1D atomic lattice). However, the atomic lattice is still symmetry in space, even if the fluctuations of average atomic density are random, the probe reflections have strong robustness. It is worth noting that the special design of atomic lattice to realize non-reciprocal reflections requires more elaborate manipulations in experiment. As far as we know, the waist of the laser beam is smaller than the length of our 1D atomic lattice, the atomic density distribution can be controlled by pump laser heating or cooling. Thus, it is feasible to manipulate the atomic density of a continuous lattice cells, e.g. improving the atomic density at middle or tail of atomic lattice.

4. Conclusions

In summary, we proposed a regime for investigating probe reflections in a 1D ordered atomic lattice characterized by a Gaussian distribution in each period and 1D disordered atomic lattice characterized by the fluctuation of average atomic density, where the atoms are driven into a four-level $N$ configuration by three traveling-wave coherent fields. Our regime had three reflection regions of high reflectivity. One of the reflections in the large-detuning region is much wider, and the other two thinner reflections located in the EIT windows are of particular interest owing to their flexible adjustability. Then, we considered two types of disorder where the fluctuation of the average atomic density is fixed ($\Delta _{k}=-0.7$ or $+2.0$ corresponding to $K^{-}$ or $K^{+}$ lattice cells with the Gaussian density distribution $N_{k}^{-}(z)$ or $N_{k}^{+}(z)$) or random ($\Delta _{k}$ varies by period $k$) in the atomic lattice. In the first case, the lattice cells with $N_{k}^{-}(z)$ or $N_{k}^{+}(z)$ are randomly distributed in the whole periodic structure. The numerical results indicate that the width and height of the reflection bands can be reduced by the increased $K^{-}$, and an increase in $K^{+}$ can only widen the reflection, but the reflections from the right or left side are symmetrical. However, a cluster of disordered lattice cells with $N_{k} ^{-}(z)$ and $N_{k}^{+}(z)$ may break this symmetry when they appear in the front or tail of the atomic lattice for $K^{-}>K^{+}$ due to spatial symmetry breaking for the refractive index. Symmetry can be preserved if they are located in the middle of the atomic lattice. In the second case, the photon probability can maintain the symmetry and is robust against fluctuation, even with a large $\Delta _{k}\in (-1.0$, $+1.0)$. For this disordered atomic lattice system, We theoretically showed how the robustness of probe reflection can be maintained for this system and provide a new idea for non-reciprocal reflections.

Funding

the specific research fund of The Innovation Platform for Academics of Hainan Province (YSPTZX202215); National Natural Science Foundation of China (11861031, 12126314, 12126351); Natural Science Foundation of Hainan Province (121RC539, 2019RC190).

Acknowledgments

This work is supported by the Hainan Provincial Natural Science Foundation of China (Grant No.121RC539, 2019RC190) and the National Natural Science Foundation of China (Grant Nos. 12126314, 12126351, 11861031). This project is also supported by the specific research fund of The Innovation Platform for Academics of Hainan Province (Grant No.YSPTZX202215). Key Laboratory of Laser Technology and Optoelectronic Functional Materials of Hainan Province.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

References

1. J. P. Marangos, “Electromagnetically induced transparency,” J. Mod. Opt. 45(3), 471–503 (1998). [CrossRef]

2. M. D. Lukin and A. Imamoǧlu, “Controlling photons using electromagnetically induced transparency,” Nature 413(6853), 273–276 (2001). [CrossRef]

3. Z. Ficekand and S. Swain, “Quantum interference in optical fields and atomic radiation,” J. Mod. Opt. 49(1-2), 3–42 (2002). [CrossRef]

4. K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82(2), 1041–1093 (2010). [CrossRef]

5. J. J Garcia-Ripoll, P. Zoller, and J. I. Cirac, “Quantum information processing with cold atoms and trapped ions,” J. Phys. B 38(9), S567–S578 (2005). [CrossRef]

6. S. C. Tian, R. G. Wan, L. J. Wang, S. L. Shu, H. Y. Lu, X. Zhang, C. Z. Tong, J. L. Feng, M. Xiao, and L. J. Wang, “Asymmetric light diffraction of two dimensional electromagnetically induced grating with PT symmetry in asymmetric double quantum wells,” Opt. Express 26(25), 32918–32930 (2018). [CrossRef]

7. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997). [CrossRef]

8. H. Yang, D. Yan, M. Zhang, B. Fang, Y. Zhang, and J.-H. Wu, “Absorption and dispersion control in a five-level M-type atomic system,” Chin. Phys. B 21(11), 114207 (2012). [CrossRef]

9. D. Yan, J. W. Gao, Q.-Q. Bao, H. Yang, H. Wang, and J.-H. Wu, “Electromagnetically induced transparency in a five-level Λ system dominated by two-photon resonant transitions,” Phys. Rev. A 83(3), 033830 (2011). [CrossRef]

10. Y. Zhang, J.-W. Gao, C.-L. Cui, Y. Jiang, and J.-H. Wu, “Comparison of steady and transient optical responses between a four-level Tripod system and a three-level Lambda system,” Phys. Lett. A 374(8), 1088–1092 (2010). [CrossRef]

11. Y.-W. Guo, S.-L. Xu, J.-R. He, P. Deng, M. R. Belić, and Y. Zhao, “Transient optical response of cold Rydberg atoms with electromagnetically induced transparency,” Phys. Rev. A 101(2), 023806 (2020). [CrossRef]

12. G.-H. Wang, X.-M. Hu, Q.-P. Hu, and L. Hu, “Giant second-order cross nonlinearities via direct perturbation to the dark state in coherent population trapping,” Phys. Rev. A 101(2), 023816 (2020). [CrossRef]

13. X.-M. Su and B. S. Ham, “Dynamic control of the photonic band gap using quantum coherence,” Phys. Rev. A 71(1), 013821 (2005). [CrossRef]

14. M. Artoni and G. C. La Rocca, “Optically Tunable Photonic Stop Bands in Homogeneous Absorbing Media,” Phys. Rev. Lett. 96(7), 073905 (2006). [CrossRef]

15. J.-H. Wu, M. Artoni, and G. C. La Rocca, “Controlling the photonic band structure of optically driven cold atoms,” J. Opt. Soc. Am. B 25(11), 1840–1849 (2008). [CrossRef]

16. J.-W. Gao, J.-H. Wu, N. Ba, C.-L. Cui, and X.-X. Tian, “Efficient all-optical routing using dynamically induced transparency windows and photonic band gaps,” Phys. Rev. A 81(1), 013804 (2010). [CrossRef]

17. J.-H. Wu, M. Artoni, and G. C. La Rocca, “All-Optical Light Confinement in Dynamic Cavities in Cold Atoms,” Phys. Rev. Lett. 103(13), 133601 (2009). [CrossRef]

18. Y. W. Cho and Y. H. Kim, “Storage and retrieval of thermal light in warm atomic vapor,” Phys. Rev. A 82(3), 033830 (2010). [CrossRef]

19. D. Devashish, O. S. Ojambati, S. B. Hasan, J. J. W. vander Vegt, and W. L. Vos, “Three-dimensional photonic band gap cavity with finite support: Enhanced energy density and optical absorption,” Phys. Rev. B 99(7), 075112 (2019). [CrossRef]

20. W. Li, P. Islam, and P. Windpassinger, “Controlled Transport of Stored Light,” Phys. Rev. Lett. 125(15), 150501 (2020). [CrossRef]

21. J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014). [CrossRef]

22. H. Yang, T.-G. Zhang, and Y. Zhang, “Probe gain via four-wave mixing based on spontaneously generated coherence,” Chin. Phys. B 26(2), 024204 (2017). [CrossRef]

23. J. Lang, D. Chang, and F. Piazza, “Interaction-Induced Transparency for Strong-Coupling Polaritons,” Phys. Rev. Lett. 125(13), 133604 (2020). [CrossRef]

24. A. F. Tzortzakakis, K. G. Makris, S. Rotter, and E. N. Economou, “Shape-preserving beam transmission through non-Hermitian disordered lattices,” Phys. Rev. A 102(3), 033504 (2020). [CrossRef]

25. A. Schilke, C. Zimmermann, P. W. Courteille, and W. Guerin, “Photonic Band Gaps in One-Dimensionally Ordered Cold Atomic Vapors,” Phys. Rev. Lett. 106(22), 223903 (2011). [CrossRef]

26. A. Schilke, C. Zimmermann, and W. Guerin, “Photonic properties of one-dimensionally-ordered cold atomic vapors under conditions of electromagnetically induced transparency,” Phys. Rev. A 86(2), 023809 (2012). [CrossRef]

27. H. Yang, L. Yang, X.-C. Wang, C.-L. Cui, Y. Zhang, and J.-H. Wu, “Dynamically controlled two-color photonic band gaps via balanced four-wave mixing in one-dimensional cold atomic lattices,” Phys. Rev. A 88(6), 063832 (2013). [CrossRef]

28. Y. Zhang, Y.-M. Liu, X.-D. Tian, T.-Y. Zheng, and J.-H. Wu, “Tunable high-order photonic band gaps of ultraviolet light in cold atoms,” Phys. Rev. A 91(1), 013826 (2015). [CrossRef]

29. L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature 12(2), 108–113 (2013). [CrossRef]

30. C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic Nonreciprocity,” Phys. Rev. Appl. 10(4), 047001 (2018). [CrossRef]

31. P.-F. Yang, X.-W. Xia, H. Hai, S.-K. Li, H. Xing, P. Zhang, G. Li, P.-F. Zhang, J.-P. Xu, Y.-P. Yang, and T.-C. Zhang, “Realization of Nonlinear Optical Nonreciprocity on a Few-Photon Level Based on Atoms Strongly Coupled to an Asymmetric Cavity,” Phys. Rev. Lett. 123(23), 233604 (2019). [CrossRef]

32. G.-W. Li, S.-C. Zhang, Y.-Q. Hu, Y.-P. Niu, J.-B. Gong, and S.-Q. Gong, “Nonreciprocal Amplification with Four-Level Hot Atoms,” Phys. Rev. Lett. 123(3), 033902 (2019). [CrossRef]

33. Y.-M. Liu, F. Gao, C.-H. Fan, and J.-H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42(21), 4283–4286 (2017). [CrossRef]

34. X. Wang and J.-H. Wu, “Optical PT-symmetry and PT-antisymmetry in coherently driven atomic lattices,” Opt. Express 24(4), 4289–4298 (2016). [CrossRef]

35. S. A. R. Horsley, M. Artoni, and G. C. La Rocca, “Spatial Kramers-Kronig relations and the reflection of waves,” Nat. Photonics 9(7), 436–439 (2015). [CrossRef]

36. Y. Zhang, J.-H. Wu, M. Artoni, and G. C. La Rocca, “Controlled unidirectional reflection in cold atoms via the spatial Kramers-Kronig relation,” Opt. Express 29(4), 5890–5900 (2021). [CrossRef]

37. Y.-M Liu, F. Gao, J.-H. Wu, M. Artoni, and G. C. La Rocca, “Lopsided diffractions of distinct symmetries in two-dimensional non-Hermitian optical gratings,” Phys. Rev. A 100(4), 043801 (2019). [CrossRef]

38. L. Yang, Y. Zhang, X.-B. Yan, Y. Sheng, C.-L. Cui, and J.-H. Wu, “Dynamically induced two-color nonreciprocity in a tripod system of a moving atomic lattice,” Phys. Rev. A 92(5), 053859 (2015). [CrossRef]

39. D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical Diode Made from a Moving Photonic Crystal,” Phys. Rev. Lett. 110(9), 093901 (2013). [CrossRef]

40. L. Martin, G. Di Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, “Anderson localization in optical waveguide arrays with off-diagonal coupling disorder,” Opt. Express 19(14), 13636–13646 (2011). [CrossRef]

41. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007). [CrossRef]

42. S. Karbasi, C. R. Mirr, P. G. Yarandi, R. J. Frazier, K. W. Koch, and A. Mafi, “Detailed investigation of the impact of the fiber design parameters on the transverse Anderson localization of light in disordered optical fibers,” Opt. Express 20(17), 18692–18706 (2012). [CrossRef]

43. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices,” Phys. Rev. Lett. 100(1), 013906 (2008). [CrossRef]

44. H. Yang, T.-G. Zhang, Y. Zhang, and J.-H. Wu, “Dynamically tunable three-color reflections immune to disorder in optical lattices with trapped cold ⁸⁷Rb atoms,” Phys. Rev. A 101(5), 053856 (2020). [CrossRef]

45. J.-H. Wu, M. Artoni, and G. C. La Rocca, “Perfect absorption and no reflection in disordered photonic crystals,” Phys. Rev. A 95(5), 053862 (2017). [CrossRef]

46. H. P. Lüschen, S. Scherg, T. Kohlert, M. Schreiber, P. Bordia, X. Li, S. D. Sarma, and I. Bloch, “Single-Particle Mobility Edge in a One-Dimensional Quasiperiodic Optical Lattice,” Phys. Rev. Lett. 120(16), 160404 (2018). [CrossRef]

47. A. F. Tzortzakakis, K. G. Makris, and E. N. Economou, “Non-Hermitian disorder in two-dimensional optical lattices,” Phys. Rev. B 101(1), 014202 (2020). [CrossRef]

48. A. Maksymov, P. Sierant, and J. Zakrzewski, “Many-body localization in a one-dimensional optical lattice with speckle disorder,” Phys. Rev. B 102(13), 134205 (2020). [CrossRef]

49. T.-M. Li, M.-H. Wang, C.-P. Yin, J.-H. Wu, and H. Yang, “Dynamic manipulation of three-color light reflection in a defective atomic lattice,” Opt. Express 29(20), 31767–31777 (2021). [CrossRef]

50. M. Artoni, G. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72(4), 046604 (2005). [CrossRef]

51. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312(5781), 1780–1782 (2006). [CrossRef]

52. G. Boedecker and C. Henkel, “All-frequency effective medium theory of a photonic crystal,” Opt. Express 11(13), 1590 (2003). [CrossRef]

Three-color reflections in one-dimensional ordered and disordered atomic lattices with trapped N-type cold atoms

Abstract

1. Introduction

2. Model and methods

3. Results and discussions

4. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (11)

Optics Express