1. Introduction

Artificially engineered optical materials have attracted significant research interests in science and technology with the consideration that a flexible control of light propagation and interaction is essential for achieving various functionalities [1–5] required in optical information manipulation. Tunable processes of light transport underpin, in particular, a new generation of photonic circuits and devices such as optical switching or routing [6,7], unidirectional invisible devices [8–12], and all optical diodes [13–16]. These may be attained via dynamically controlled photonic bandgaps (PBGs) in one-dimensional (1D) lattices of cold atoms driven by traveling-wave coupling fields into the regime of electromagnetically induced transparency (EIT). The underlying physics lies in that controlled periodic structures of atomic lattices give rise to tunable Bragg scattering, depending on a flexible manipulation of coupling fields [17–24]. Recent researches further show that, in 1D driven atomic lattices, a probe field may experience parity-time (PT) symmetric or antisymmetric susceptibilities and thus exhibit unidirectional reflectionless behaviors as standing-wave coupling fields are applied with a phase shift relative to standing-wave lattice fields [25–28]. Such a nonreciprocal light manipulation can also be realized by breaking the time-reversal symmetry with moving atomic lattices [14,15,29]. It follows that 1D atomic lattices subject to the EIT condition displays a new paradigm on the control over light propagation, which can be easily extended to the two-deimensional or three-dimensional cases [22,30,31].

In reality, disorders characterized, e.g., by random fluctuations of atomic density in different cells may be caused by imperfect loading processes of cold atoms into optical lattices. In fact, ordered–disordered transitions take place in many physical systems [32–36], and obvious differences have been found between disordered and ordered lattices in investigations of Bose–Einstein condensate [37–40]. On the other hand, it has been shown that light reflection usually displays an enhanced robustness against various types of disorders in atomic lattices. To be more specific, unidirectional reflection attained with atomic lattices has been examined by considering random variations of both center and width of atomic spatial distributions in each lattice cell [41], and three-color reflection attained with atomic lattices has been examined by considering random fluctuations of atomic density in each lattice cell [42]. These considerations represent two types of most familiar disorders in PBGs materials made from atomic lattices. Note, however, that defective atomic lattices with random or continuous vacant cells have not been examined yet though may result in novel phenomena on controlled light transport properties. This then motivates us to extend one of our previous works [42] here by replacing atomic density disorders with vacant cell defects in atomic latices to reexamine the important issue of dynamical manipulation on light reflection in 1D atomic lattices.

In this work, we consider a specific model of 1D defective atomic lattices, in which most lattice cells contain tightly trapped $^{87}$Rb atoms with a Gaussian spatial distribution while others are vacant, i.e., contains no atoms, due to uncertain reasons. Numerical calculations will be done to examine various reflection behaviors of a probe field in two cases where vacant lattice cells are randomly or continuously distributed along the lattice direction. It is found that three reflection bands in two near-resonant EIT regions or a large detuning region exhibit very different robustness against random or continuous vacant cell defects. Detailed findings corresponding to appropriate parameters should be helpful in understanding and manipulating light transport properties in realistic atomic lattices. The defective archetypes of 1D atomic lattices and relevant equations are described in Sect. 2 while numerical results and qualitative discussions are shown in Sect. 3. The main conclusions are finally outlined in Sect. 4.

2. Theoretical model and equations

We start by considering cold $^{87}$Rb atoms driven into the four-level tripod configuration, as shown in Fig. 1(a), by three laser fields of frequencies (amplitudes) $\omega _{p}$ ($\mathbf {E}_{p}$), $\omega _{c1}$ ($\mathbf {E}_{c1}$) and $\omega _{c2}$ ($\mathbf {E}_{c2}$). The weak probe field $\omega _{p}$ and the strong coupling fields $\omega _{c1}$ and $\omega _{c2}$ interact, respectively, with the dipole-allowed transitions $\left \vert 0\right \rangle \leftrightarrow \left \vert 3\right \rangle$, $\left \vert 1\right \rangle \leftrightarrow \left \vert 3\right \rangle$, and $\left \vert 2\right \rangle \leftrightarrow \left \vert 3\right \rangle$ with Rabi frequencies (detunings) $\Omega _{p}=\mathbf {E}_{p}\cdot \mathbf {d} _{03}/2\hbar$ ($\Delta _{p}=\omega _{30}-\omega _{p}$), $\Omega _{c1}= \mathbf {E}_{c1}\cdot \mathbf {d}_{13}/2\hbar$ ($\Delta _{c1}=\omega _{31}-\omega _{c1}$) and $\Omega _{c2}=\mathbf {E}_{c2}\cdot \mathbf {d} _{23}/2\hbar$ ($\Delta _{c2}=\omega _{32}-\omega _{c2}$). Here we have defined $\mathbf {d}_{ij}$ and $\omega _{ij}$ as dipole moments and resonant frequencies, respectively, on relevant transitions. Typical spectra of probe absorption and dispersion, i.e., real and imaginary parts of probe susceptibility $\chi _{p}$ plotted against probe detuning $\Delta _{p}$ with the calculation method given later, are shown in Fig. 1(b) where three points $O_{1}$, $O_{2}$, and $O_{3}$ satisfy the Bragg condition. It is worth noting that $O_{1}$ and $O_{2}$ are located in two near-resonant EIT windows where absorption is suppressed due to destructive interference while $O_{3}$ is located in a large detuning region where absorption is negligible due to far off-resonance interaction. The probe field is expected to be be well reflected if its detuning is close to one of the three points as a result of both effective Bragg scattering and well suppressed absorption. Figure 1(c) shows a perfect atomic lattice of period $a_{0}$, where $^{87}$Rb atoms are filled in periodic optical traps formed by a red-detuned retroreflected laser beam of wavelength $\lambda _{Lat}=2a_{0}>\lambda _{30}$ along the $z$ axis. In this case, atomic density in each lattice cell exhibits a Gaussian distribution

 

Fig. 1. (a) Energy level diagram of a four-level tripod-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) $\Omega _{c1}$ ($\Delta _{c1}$) and $\Omega _{c2}$ ($\Delta _{c2}$ ), respectively. (b) Real (blue-solid) and imaginary (red-solid) parts of average probe susceptibility $ \chi _{p}$ against probe detuning $\Delta _{p}$ attained with relevant parameters used in Fig. 2. The black-dashed horizontal line is plotted for $-2\Delta \lambda / \lambda _{Lat}$ to identify three points $O_{1}$, $O_{2}$, and $O_{3}$ satisfying the Bragg condition. (c) A perfect 1D atomic lattice containing only filled cells exhibiting the same Gaussian distribution of atomic density. (d) A defective 1D atomic lattice with random vacant cells. (e) A defective 1D atomic lattice with continuous vacant cells. The incident and reflected probe fields may be misaligned with a small angle $2 \theta$.

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Under the electric dipole approximation and the rotating wave approximation, the Hamiltonian of our tripod system in Fig. 1(a) can be written as

We now introduce the transfer matrix method by starting with a 2$\times$2 unimodular transfer matrix $M_{p}^{f}$ describing the reflection and transmission properties of the $f$th lattice cell. To attain this transfer matrix, we divide the $f$th lattice cell into, e.g., 100 thin layers of identical thickness and refractive index such that each layer is described by the transfer matrix

With matrix $M_{p}^{K}$ on hand, it is easy to write down the probe reflectivity at the lattice end

3. Numerical results and discussions

In a previous work [42], we show that three bands of high probe reflection can be observed in a 1D disordered lattice of cold atoms driven into the tripod configuration. This is attained when the Bragg condition is satisfied around points $O_{1}$, $O_{2}$ in two near-resonant EIT regions and $O_{3}$ in a large detuning region as shown in Fig. 1(b). We now extend this work by considering a defective atomic lattice instead where atoms are absent ($N_{k}(z)=0$) in some vacant cells.

We first assess how probe reflection is perturbed in a defective atomic lattice where a variable number ($J_{r}$) of vacant cells are randomly distributed, as shown in Fig. 2. We can see that the wider reflection band in the large detuning region is obviously narrowed as $J_{r}$ increases, while a pair of narrower ones in two near-resonant EIT regions hardly change, even if the ratio of vacant cells to total cells has reached a quite large value $J_{r}/K\simeq 1/7$ with $J_{r}=1000$ and $K=L/a_{0}\simeq 7700$ [see Fig. 2(a) and Fig. 2(b)]. This indicates that the large-detuning wider reflection band is much more sensitive to the number of random vacant lattice cells, while the pair of near-resonant narrower ones are rather robust against this random defects of vacant lattice cells. It is worth noting that probe reflectivities in all three reflection bands are almost unchanged, which is also supported by our observations on DOS. At probe detuning $\Delta _{p}=-17.5$ MHz in a reflection band, DOS with different $J_{r}$ is reduced almost by the same value and remains small in most parts of the atomic lattice [see Fig. 2(c)]. At probe detuning $\Delta _{p}=-3.0$ MHz out of this reflection band, however, DOS will not decrease and maintains large values [see Fig. 2(d)].

 

Fig. 2. Probe reflection in (a) two near-resonant EIT regions or (b) the large detuning region. DOS for a probe field with $\Delta _{p}=-17.5$ MHz in an EIT reflection band (c) or $\Delta _{p}=-3$ MHz out of any reflection bands (d). Black lines with circles are attained for a perfect atomic lattice while red lines, olive lines, and blue lines with circles are attained for random defective atomic lattices with $J_{r}=100$, $500$, and $1000$, respectively. Other parameters are $\Gamma _{30}=\Gamma _{31}=\Gamma _{32}=6.0$ MHz, $\Omega _{p}=0.048$ MHz, $\Omega _{c1}=\Omega _{c2}=6.0$ MHz, $\Delta _{c1}=-\Delta _{c2}=-15.0$ MHz, $ \lambda _{Lat0}=781$ nm, $ \lambda _{p0}=780.24$ nm, $\Delta \lambda _{Lat}=0.9$ nm, $ \eta =5$, $L=3.0$ nm, $N_{0}=7\times 10^{11}$ cm$^{-3}$, and $ \mu _{03}=1.0357\times 10^{-29}$ Cm.

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We then consider how probe reflection is perturbed in another defective atomic lattice where a variable number ($J_{c}$) of vacant cells appear instead continuously. We can see from Fig. 3(a) and Fig. 3(b) that only the large-detuning reflection band becomes slightly wider, while the pair of near-resonant reflection bands remains indistinguishable. The corresponding DOS within ($\Delta _{p}=-17.5$ MHz) or outside ($\Delta _{p}=-3.0$ MHz) of relevant reflection bands exhibit no variations at different $J_{c}$ values as displayed in Fig. 3(c) and Fig. 3(d). It is worth noting that the overall positions of continuous vacant lattice cells are random. With above results, we may conclude that probe reflection is quite robust or roughly immune to continuous defects of vacant lattice cells even if the ratio of vacant lattice cells to total lattice cells approaches $1/7$ with $J_{c}=1000$.

 

Fig. 3. Probe reflection in (a) two near-resonant EIT regions or (b) the large detuning region. DOS for a probe field with $\Delta _{p}=-17.5$ MHz in an EIT reflection band (c) or $\Delta _{p}=-3$ MHz out of any reflection bands (d). Black lines with triangles are attained for a perfect atomic lattice while red lines, olive lines, and blue lines with triangles are attained for continuous defective atomic lattices with $J_{c}=100$, $500$, and $1000$, respectively. Other parameters are the same as in Fig. 2.

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It is also interesting to examine how the position of continuous vacant cells affects the reflection bands. We plot probe reflectivities with $J_{c}=100$ continuous vacant cells, which may be located in the front ($z=24\ \mu$m$\sim 64\ \mu$m), at the middle ($z=1480\ \mu$m$\sim 1520\ \mu$m), or near the tail ($z=2936\ \mu$m$\sim 2976\ \mu$m) of our atomic lattice in Fig. 4(a) and Fig. 4(b). It is found that all three reflection bands remains strong robustness against continuous vacant cells located at the middle or near the tail of our atomic lattice but are obviously widened with probe reflectivities in the widened parts approaching $50\% $. As continuous vacant cells appear in the front of our atomic lattice, however, the situation becomes different manifested as largely disrupted reflection bands, among which the pair of near-resonant ones change more than the large-detuning one. This promotes us to further assess probe reflection in both near-resonant EIT regions against $\Delta _{p}$ and $J_{c}$ in Fig. 4(c). It is of interest to note that the pair of near-resonant reflection bands exhibit periodic variations as $J_{c}$ is increased. We have shown here three specific points $A\ $($250$, $6.0$ MHz), $B\ $($550$, $11.0$ MHz), and $C\ $($1100$, $6.0$ MHz) around a near-resonant EIT reflection band corresponding to the detuning region $\Delta _{p}\sqsubseteq 10\sim 14.0$ MHz. With these points, we can see that this reflection band achieves the maximum width $8.0$ MHz with $J_{c}=250$, turns to the minimum width $3.0$ MHz with $J_{c}=550$, and recovers again the maximum width with $J_{c}=1100$.

 

Fig. 4. Probe reflection in two near-resonant EIT regions (a) or the large detuning region (b) for a perfect atomic lattice (black lines with stars) and continuous defective atomic lattices with vacant lattice cells in the front ($z=24\,\mu$m$\sim 64\,\mu$m, red lines with circles), at the middle ($z=1480\,\mu$m$\sim 1520\,\mu$m, olive lines with squares), and near the tail ($z=2936\,\mu$m$\sim 2976\,\mu$m, blue lines with diamonds), respectively. (c) Probe reflection in a near-resonant EIT region against probe detuning $\Delta _{p}$ and vacant cell number $J_{c}$ when continuous vacant lattice cells are in the lattice front. Other parameters are the same as in Fig. 2.

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To have a deeper insight into the interesting results in Fig. 4(c), we discuss the common effects on one EIT reflection band of the number and the position of continuous vacant lattice cells. We focus in particular on the one corresponding to $\Delta _{p}\sqsubseteq 10\sim 14$ MHz with probe reflection plotted against position $z$ with continuous vacant lattice cells with $J_{c}=250$ in Fig. 5(a), $550$ in Fig. 5(b), and $1100$ in Fig. 5(c), respectively. It is easy to see that when continuous vacant lattice cells start from position $z=0$, the reflection band remains the width of $4$ MHz, indicating that the increase of $J_{c}$ has almost no effects on probe reflection. As continuous vacant lattice cells appear at other positions ($z\ne 0$) in the front of our atomic lattice, the reflection band becomes clearly widened in a way depending also on $J_{c}$. That means, at a fixed position $z$, different $J_{c}$ will result in different width of the reflection band. Thus, points $A^{^{\prime }}$($24$ $\mu$m, $6.0$ MHz), $B^{^{\prime }}$($24$ $\mu$m, $11.0$ MHz), and $C^{^{\prime }}$($24$ $\mu$m, $6.0$ MHz) in Fig. 5 are in good agreement with $A\ $($250$, $6.0$ MHz), $B\ $($550$, $11.0$ MHz), and $C\ $($1100$, $6.0$ MHz) in Fig. 4(c), respectively.

 

Fig. 5. Probe reflection in two near-resonant EIT regions against probe detuning $\Delta _{p}$ and defect position $z$ of vacant lattice cells with $J_{c}=250$ (a), $550$ (b), and $1100$ (c), respectively. Other parameters are the same as in Fig. 2.

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In Fig. 6, we analyze the novel phenomena found in Fig. 4 and Fig. 5 through the Bragg condition. First, for a 1D atomic lattice with continuous vacant cells, we note that the refractive index $n_{0}$ in vacant lattice cells is smaller than the refractive index $n_{p}$ in filled lattice cells, yielding thus a reduced refractive index $n_{B}=n_{0}$ in part B. In this case, frequency $\omega _{p}$ will increase so as to satisfy the Bragg condition given by Eq. (14). Accordingly, relevant reflection bands move toward left and become widened, as shown by the blue solid line with squares in Fig. 6(a). For a fixed detuning, as the number $J_{c}$ of continuous vacant cells increases, the thickness $d_{B}$ in the unit cell of part $B$ will gradually increase so that the Bragg condition is established periodically. This well explains why the reflection band is widened in a periodic way in Fig. 4(c).

 

Fig. 6. Probe reflection in two near-resonant EIT regions against probe detuning $\Delta _{p}$ (a) in a perfect and a continuous defective atomic lattice, respectively; (b) in a perfect atomic lattice with $7700$ and $1000$ lattice cells, respectively. (c) Real and (d) imaginary parts of transfer matrix element $M(1,2)/M(2,2)$ against position $z$ in a perfect and a continuous defective atomic lattice ($J_{c}=100$), respectively, with probe detuning $\Delta _{p}=6$ MHz. Other parameters are the same as in Fig. 2.

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In Fig. 6(b), probe reflection in one EIT window is plotted for a perfect atomic lattice (with no defects) composed of $1000$ and $7700$ lattice cells, respectively. It is clear that $1000$ lattice cells are enough to form a good reflection band, which explains why in the case of $dA=0$ the reflection formed in part $C$ cannot be affected by part $B$. When position $z$ corresponding to the first vacant cell increases, the number of lattice cells ($d_{A}$) in part $A$ increases, thus resulting in a decrease of the number of lattice cells ($d_{C}$) in part $C$. As $d_{A}$ is increased, however, a reflection band is gradually formed in the frequency range of $\Delta _{p}\sqsubseteq 2\sim 10$ MHz. When the first vacant cell is at $z=24\mu$m, the width of this reflection band reaches the maximum. As $d_{A}$ continues to increase, the range of $\Delta _{p}\sqsubseteq 2\sim 10$ MHz will no longer satisfy the Bragg condition so that the width of this reflection band recovers. This stems from the fact that part $A$ is not a homogeneous medium, but a periodic structure composed of many filled lattice cells. When the number of lattice cells in part $A$ is large enough ($J_{c}>1000$, $d_{A}>400$ $\mu$m), a good reflection band can be formed which is hardly affected by the continuous vacant cells, otherwise the reflection band will be destroyed.

Finally, we analyze the widened reflection band by resorting to the normalized transmission matrix determining the probe reflectivity. This is done by plotting real and imaginary parts of $M(1,2)/M(2,2)$ against position $z$ for a perfect atomic lattice and a continuous defective atomic lattice with $\Delta _{p}=6$ MHz, respectively, in Fig. 6(c) and Fig. 6(d). Here part $B$ is assumed to contain $100$ vacant lattice cells and locates in the front ($z=24\ \mu$m$\sim 64\ \mu$m) of 1D optical lattice. A clear difference is found between the perfect atomic lattice and the continuous defect atomic lattice with respect to both real and imaginary parts of $M(1,2)/M(2,2)$. This may be attributed to different kinds of coherent interplay of matrix groups $M_{p}^{L_{-}}$ and $M_{p}^{L^{+}}$, which together lead to a widened reflection band in the continuous defective atomic lattice.

4. Conclusions

In summary, we have extended one previous work [42] on the dynamic generation and manipulation of three-color reflection in a 1D atomic lattice by replacing disorders characterized by varied atomic densities with defects characterized by vacant lattice cells. We consider in particular two types of defective atomic lattices where vacant lattice cells are randomly or continuously distributed. In the first case of random vacant cells, we find that a wider reflection band in the large detuning region is much more sensitive to, while the pair of narrower reflection bands in two near-resonant EIT regions are quite robust against, the number of random defects $J_{r}$ even if it is increased to $1000$. In the second case of continuous vacant cells, the pair of reflection bands in two near resonant EIT regions remain to be robust, i.e., change little as the number of continuous defects $J_{c}$ is increased. However, the robustness of all three reflection bands is destroyed when continuous vacant lattice cells appear near the lattice front. In this case, it is interesting to note that these reflection bands exhibit periodic variation at increased $J_{c}$ values. We also find that both position and number of continuous vacant lattice cells can destroy or move relevant reflection bands. This is due to the joint contributions of Bragg condition and coherent interplay (i.e., quantum interference) of matrix groups $M_{p}^{L_{-}}$ and $M_{p}^{L^{+}}$, which thus allows dynamic manipulation of probe reflection. Our results on defective atomic lattices should be instructive for developing novel optical devices inaccessible with other PBG materials.