1. Introduction

Asymmetric reflection control of the flow of a light beam is a key technique to perform photonic and quantum communication manipulations. Unidirectional reflectionless, in particular, has attracted much research efforts because of its perspective applications in developing novel photonic devices [1–9]. Control over reflection of a light beam incident from opposite sides of a device is usually reciprocal and static, i.e., takes place with identical reflectivity values and cannot be changed. This can be achieved, e.g., using photonic band-gap materials exhibiting a given periodic structure of the real refractive index [10,11]. Tunable photonic band gap structures have also been proven to be viable, where control of the periodic structure of the complex susceptibility can be done, e.g., through electromagnetically induced transparency (EIT) [12–14], with homogeneous atomic clouds dressed by standing-wave coupling fields [15–22] or periodic atomic lattices dressed by traveling-wave coupling fields [23–33]. Generally speaking, it is hard to achieve asymmetric light transport through standard linear optical processes [34–36], though significant progresses have been made recently in coherently driven moving atomic lattices [1,2,32] and in materials exhibiting parity-time (PT) symmetry or asymmetry [3,4,37–41]. Realistic implementations of these schemes are challenging owing to complex atom-light coupling configurations, precise light field arrangement in space and peculiar balanced gain and loss over a single period.

Alternatively, asymmetric and unidirectional reflection can be realized in an inhomogeneous continuous medium as the usual Kramers-Kronig (KK) relation, satisfied by its complex optical response function, is mapped from the frequency domain to the space domain [5–8,42–49]. In the pioneering work [5], Horsley et al. found that a non-Hermitian medium would not reflect light from one side for all incident angles if its complex permittivity satisfies the spatial KK relation, which promises also the realization of omnidirectional perfect absorber [7] and media with no transmission [50]. When extended into discrete lattices, complex potentials exhibiting the spatial KK relation may further become invisible to support a bidirectional reflectionless behavior [48,49]. Notice, in addition, that the original idea has been already experimentally demonstrated via a suitable design of different inhomogeneous media [7,43,51,52]. All above findings not only deepen our understanding of light propagation, but also provide new platforms toward multi-functional optical elements, especially those requiring perfect antireflection properties. Yet, schemes so far studied rely on fixed complex refractive index, susceptibility, or permittivity structures and hence lack of dynamic tunability.

Here we propose an efficient scheme for mapping the KK relation of a probe susceptibility from the frequency domain into the space domain in a cold atomic sample. The scheme hinges on devising a position-dependent ground level shift with a far detuned external control field whose intensity can be linearly varied in space (see Sect. 2). Other spatially dependent EIT systems have also been used to manipulate structured light beams, albeit via phase-dependent dark-states [53–55]. It is worth noting that in our scheme, depending on the probe frequency, spatial KK relations may be completely or partially fulfilled or even destroyed; in the following these are termed respectively as "unbroken", "transitional", and "broken" regimes. The unbroken regime is of particular interest here because it enables us to attain reflectionless manipulation of a probe beam incident on one sample end over a tunable range of frequencies. The transitional regime may instead be exploited to attain reflectionless manipulation from both sample side ends, albeit at different frequencies. Furthermore, changing the "sign" of the ground level shift enables us to directly swap the direction of the vanishing reflectivity whereas increasing the level shift "magnitude" leads to an enhancement of the nonzero reflectivity while maintaining (on the other side) a vanishing reflectivity. Bragg scattering may also be incorporated into the spatial KK relation to further increase the forward-backward reflectivity contrast.

2. Model and equations

We consider in Fig. 1(a) a cold atomic sample extending from $x=0$ to $x=L$, driven by a weak probe field of amplitude (frequency) $\mathbf {E}_{p}$ ($\omega _{p}$) and a strong control field of amplitude (frequency) $\mathbf {E}_{c}$ ($\omega _c$). The control field is assumed to illuminate the sample along the $-y$ direction while the probe field can travel through the sample along either $x$ or $-x$ direction. All atoms are driven into the three-level $V$ configuration, as shown in Fig. 1(b), characterized by Rabi frequencies (detunings) $\Omega _{p}=\mathbf {E}_{p}\cdot \mathbf {d}_{ge}/2\hbar$ on transition $\vert g\rangle \leftrightarrow \vert e\rangle$ ($\Delta _{p}=\omega _{p}-\omega _{eg}$) and $\Omega _{c}=\mathbf {E}_{c}\cdot \mathbf {d}_{ga}/2\hbar$ on transition $\vert g\rangle \leftrightarrow \vert a\rangle$ ($\Delta _{c}=\omega _{c}-\omega _{ag}$), being $\mathbf {d}_{\mu \nu }$ and $\omega _{\mu \nu }$ dipole moments and resonant frequencies of relevant transitions. We have also used $\Gamma _{ag}$ and $\Gamma _{eg}$ to describe the population decay rates from levels $|a\rangle$ and $|e\rangle$ to level $|g\rangle$, respectively. To be more concrete, levels $|a\rangle$, $|e\rangle$, and $|g\rangle$ may refer to states $\vert 5P_{3/2},F=3,m_{F}=3\rangle$, $\vert 5P_{1/2},F=1,m_{F}=1\rangle$, and $\vert 5S_{1/2},F=2,m_{F}=2\rangle$ of $^{87}$Rb atoms, respectively. This choice ensures that ($i$) dipole moments $|\mathbf {d}_{ge}|$ and $|\mathbf {d}_{ga}|$ take the largest possible values, which could relax the requirement of a very dense atomic sample for achieving a high reflectivity contrast; ($ii$) the control field doesn’t couple level $|g\rangle$ to a fourth level even in the case of a large $|\Delta _{c}|$, because no others except level $|a\rangle$ has $m_{F}=3$ on the $D_{2}$ line of $^{87}$Rb atoms. Most importantly, we will assume that the control field is linearly varied in intensity along the $x$ direction, e.g., by a neutral density filter (NDF). In this case, $\vert \Omega _{c}\vert ^{2}$ should be replaced by $\vert \Omega _{c}(x)\vert ^{2}=x\vert \Omega _{c0}\vert ^{2}/L$ with $\Omega _{c0}$ denoting the maximal Rabi frequency at $x=L$.

 

Fig. 1. (a) A cold atomic sample illuminated by a control beam $\mathbf {E}_{c}(x)$ along the $-y$ direction exhibits a strongly asymmetric reflection for a probe beam $\mathbf {E}_{p}$ incident either along the $+x$ or the $-x$ direction. (b) A three-level atomic system driven by a probe field of Rabi frequency $\Omega _{p}$ (detuning $\Delta _{p}$) and a control field of Rabi frequency $\Omega _c(x)$ (detuning $\Delta _c$) into the $V$ configuration. (c) A two-level atomic system with a dynamic shift $\delta (x)$ of level $\vert g\rangle$ upon the adiabatic elimination of level $\vert a\rangle$.

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With the electric-dipole and rotating-wave approximations, working in the weak probe limit, we can solve density matrix equations for the three-level $V$ configuration to attain the steady-state probe susceptibility

The real and imaginary parts of $\chi _{2}$ are well known to be related via the KK relation in the frequency domain based on the causality principle and Cauchy’s theorem if we set $\delta (x)\equiv 0$ [56]. Thus, for appropriate probe detunings and sufficiently long samples such that the spatial variation of the effective probe detuning induced by $\delta (x)$ is fully developed, as shown by the red solid lines in Figs. 2(a) and 2(b), the KK relation holds in the space domain and can be defined by the following Cauchy’s principal value integral

 

Fig. 2. (a) Real and (b) imaginary parts of susceptibility $\chi _{2}$ against position $x$ with $\delta _{0}/\gamma _{ge}$=100 and $\Delta _p/\gamma _{ge}=-50$ (red-solid); $\delta _{0}/\gamma _{ge}$=20 and $\Delta _p/\gamma _{ge}=-10$ (blue-dashed). (c) Real and (d) imaginary parts of susceptibility $\chi _{2}$ against position $x$ and detuning $\Delta _p$ with $\delta _{0}/\gamma _{ge}$=100. Other parameters used in calculations are $\gamma _{ge}=2 \pi \times 2.87$ MHz, $d_{ge}=1.79\times 10^{-29}$ C$\cdot$m, $N_{0}=2.0 \times 10^{13}$ cm$^{-3}$, $L=5.0$ $\mu$m, and $\lambda _{p}=795$ nm.

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A medium described by $\chi _{2}$ in Eq. (2) is expected to exhibit asymmetric light transport features, which can be examined via the standard transfer-matrix method [57–59] as sketched below. First, the atomic sample is partitioned along the $x$ direction into a large number ($J$) of slices labeled by $j\in [1,J]$, which exhibit slightly different susceptibilities $\bar {\chi }_{j}(\Delta _{p})=\chi _{2}(\Delta _{p},jl)$ but identical length $l=L/J=10$ nm. Second, a $2\times 2$ unimodular transfer matrix $M_{j}(\Delta _{p},l)$ characterized by $l$ and $\bar {\chi }_{j}(\Delta _{p})$ can be established to describe the propagation of an incident probe field of wavelength $\lambda _{p}$ through the $jth$ slice by

3. Results and discussion

In this section, we show via numerical calculations how to implement the spatial KK relation in a narrow spectral range by tailoring the complex probe susceptibility, and how to implement the unidirectional reflection of a high reflectivity contrast by utilizing the spatial KK relation. All numerical calculations will be done with realistic parameters corresponding to the three states of $^{87}$Rb atoms chosen above, though our driving configuration can also be realized, e.g., in other alkali metal atoms.

First, we plot in Figs. 2(a) and 2(b), respectively, the real and imaginary parts of $\chi _{2}$ against position $x$ for two sets of parameters making the choice $\Delta _{p}=-\delta _{0}/2$, which allows the effective probe detuning $\Delta _{p}^{eff}(x)$ to be in the range of $\{-\delta _{0}/2,\delta _{0}/2\}$. It is clear that $\chi _{2}^{\prime }$ and $\chi _{2}^{\prime \prime }$ show an odd profile and an even profile, respectively, centered at $z=L/2$ and practically fully contained by the atomic sample, so they should satisfy the spatial KK relation described by Eq. (3). We also can see that smaller (larger) values of $\delta _{0}/\gamma _{12}$, a key dimensionless parameter in our reduced two-level system, will result in broader (narrower) spatial profiles of $\chi _{2}^{\prime }$ and $\chi _{2}^{\prime \prime }$ yet without changing their peak amplitudes. To reveal the frequency-dependent feature, we plot in Figs. 2(c) and 2(d), respectively, the real and imaginary parts of $\chi _{2}$ against both position $x$ and detuning $\Delta _{p}$ instead. The profiles of $\chi _{2}^{\prime }$ and $\chi _{2}^{\prime \prime }$ are found to move simultaneously toward the left (right) sample end with the increasing (decreasing) of $\Delta _{p}$. Accordingly, the spatial KK relation will be gradually destroyed because Eq. (3) becomes less and less satisfied.

In order to assess the extent to which the spatial KK relation is satisfied for different probe detunings in a finite atomic sample, here we propose the following integral

Then we plot in Fig. 3(a) the reflection and transmission spectra for the parameters used in Figs. 2(a) and 2(b) based on Eq. (5). It is easy to see that these spectra can be divided into three regions: ($I$) where we have $R_{l}=R_{r}\to 0$ and $T\to 1$; ($II$) where $R_{l}\ne R_{r}$ and $T$ are sensitive to $\Delta _{p}$; ($III$) where $T\simeq 0.05$, $R_{l}\to 0$, but $R_{r}$ oscillates around $0.2$. The generation of three different regions can be understood by examining in Fig. 3(b) the figure of merit $D_{kk}$ against probe detuning $\Delta _{p}$, which clearly shows, as compared to Fig. 3(a), that $D_{kk}$ governs the relation between $R_{l}$, $R_{r}$, and $T$. The symmetric ($I$), asymmetric ($II$), and unidirectional ($III$) reflection regions correspond, respectively, to the broken ($D_{kk}=\pm 1$), transitional ($0<|D_{kk}|<1$), and unbroken ($D_{kk}=0$) regimes, and to the cases when the absorption ($\chi _{2}^{\prime \prime }$) and dispersion ($\chi _{2}^{\prime }$) profiles move out of the sample, lie at the sample boundaries, and are well contained by the sample. It is worth noting that in region ($III$) a left (right) incident probe beam is reflectionless (partially reflected) because it first sees the negative (positive) peak of $\chi _{2}^{\prime }$ [5], and the resonant absorption ($\chi _{2}^{\prime \prime }$) is already strong enough to yield $T \to 0.0$ for forward photons while the dispersion profile ($\chi _{2}^{\prime }$) is not too sharp to yield $R_{r}\to 1.0$ for backward photons. One way for further reducing $T$ and simultaneously increasing $R_{r}$ is to produce enhanced absorption profiles and sharper dispersion profiles in denser atomic samples. The restricted range of densities of cold atoms available in experiment, however, places a constraint on this approach.

 

Fig. 3. (a) Reflectivity $R_{l}$, reflectivity $R_{r}$, and half transmissivity $T$; (b) figure of merit $D_{kk}$ against detuning $\Delta _{p}$. Other parameters used in calculations are the same as in Fig. 2(a,b).

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Figure 4(a) further shows the different regimes on a diagram with $D_{kk}$ plotted against $\delta _{0}$ and $\Delta _{p}$, in which the green region ($D_{kk}=-1$) and the red region ($D_{kk}=1$) refer to the broken regime ($I$); the four narrow blue regions ($0<|D_{kk}|<1$) refer to the transitional regime ($II$); the two triangular yellow regions ($D_{kk}=0$) refer to the unbroken regime ($III$). It is also clear that the widths of two yellow regions depend critically on the magnitude of $\delta _{0}$; a broken regime may be converted into an unbroken regime and vice versa for a given $\Delta _{p}$ by changing the sign of $\delta _{0}$. Accordingly, it is viable to enlarge or reduce the reflectionless frequency band by varying the magnitude of $\delta _{0}$ and convert the sample from left reflectionless to right reflectionless or vice versa by changing the sign of $\delta _{0}$. This potentially dynamic controllability, a chief feature of our proposal, is well demonstrated in Figs. 4(b) and 4(c) in terms of reflectivities $R_{l}$ and $R_{r}$.

 

Fig. 4. (a) Figure of merit $D_{kk}$, (b) reflectivity $R_{l}$, and (c) reflectivity $R_{r}$ (c) against shift $\delta _{0}$ and detuning $\Delta _{p}$. Other parameters used in calculations are the same as in Figs. 2(a) and (b).

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It is also interesting to examine what could happen for reflectivities $R_{l}$ and $R_{r}$ when sample length $L$ is multiplied while atomic density $N_{0}$ remains invariant. In this case, we can see from $\delta (x)=x\delta _{0}/L$ that the linear variation occurs in a much larger range while its magnitude $\delta _{0}$ is unchanged. Then, as shown in Fig. 5, the spatially wider/smoother dispersion ($\chi _{2}^{\prime }$) and absorption ($\chi _{2}^{\prime \prime }$), of $L$-independent maxima and minima, together result in a notable reduction of $R_{r}$ while $R_{l}$ remains vanishing in the unbroken regime. The peak of $R_{l}$ ($R_{r}$) accompanied by $R_{r}\to 0$ ($R_{l}\to 0$) in the transitional regime has a $L$-independent position because it only appears as the main profiles of $\chi _{2}^{\prime }(x)$ and $\chi _{2}^{\prime \prime }(x)$ approach and even partially leave the left (right) sample end. It is clearly not a result of the spatial KK relation and allows two probe beams of different frequencies to be simultaneously reflected or not when they are incident upon the opposite sample ends. The damped oscillations of $R_{r}$ against $\Delta _{p}$ in the unbroken regime can be understood as a multiple interference effect due to the discontinuities of the probe susceptibility at the right sample end and at the resonant position inside the sample. It is clear that stronger (weaker) oscillations occur at larger (smaller) values of $|\Delta _{p}|$ because the resonant position of $\chi _{2}^{\prime }(x)$ and $\chi _{2}^{\prime \prime }(x)$ is close to (far from) the right sample end, yielding thus stronger (weaker) discontinuities. The oscillation period can be roughly estimated as $d\Delta _{p}\simeq \delta _{0}/L\cdot \lambda _{p}/2$ by considering that the interval $d\Delta _{p}$ of two adjacent maxima corresponds to a $2\pi$ phase shift ($\lambda _{p}/2$ spatial shift) gained by the reflected photons ($\chi _{2}^{\prime }(x)$ and $\chi _{2}^{\prime \prime }(x)$).

 

Fig. 5. Reflectivities (a) $R_{l}$ and (b) $R_{r}$ against detuning $\Delta _{p}$ for $L=10$ $\mu$m (black-squares), $L=15$ $\mu$m (red-circles), and $L=20$ $\mu$m (blue-triangles). Other parameters used in calculations are the same as in Figs. 2(a) and (b). Black (red) curves are shown with a vertical offset $0.1$ ($0.05$) in both insets.

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Finally, we note that an experiment may typically have a lower atomic density and a larger sample length than in simulations presented so far. To overcome this difficulty, we need to find an alternative way to enhance the nonzero reflectivity in the unbroken regime. This can be done by loading cold atoms into a 1D optical lattice to create a spatially periodic density $N_{j}(x)$ as described in the caption of Fig. 6, yielding thus Bragg scattering incorporated into the spatial KK relation. As shown in Fig. 6(a) and Fig. 6(b), both dispersion $\chi _{2}^{\prime }(x)$ and absorption $\chi _{2}^{\prime \prime }(x)$ of one-order lower values now exhibit the comb-like spatial profiles while satisfying to a less extent the spatial KK relation. In this case, we can find from Fig. 6(c) that $R_{l}$ and $R_{r}$ are strongly asymmetric in a much smaller frequency range, e.g., with $R_{r}$ exhibiting a maximal value up to $0.54$ while $R_{l}\lesssim 0.01$ for $-25.5\lesssim \Delta _{p}/\gamma _{ge}\lesssim -11.0$. Figure 6(d) further shows that the reflectivity contrast $C=(R_{r}-R_{l})/(R_{r}+R_{l})$, an important figure of merit on the asymmetric reflection, could be up to $0.97$ and is over $0.90$ for $-27.0\lesssim \Delta _{p}/\gamma _{ge}\lesssim -4.5$. It is noticeable that the incorporation of Bragg scattering, typically yielding symmetric reflectivities, has negligible effects on the vanishing reflectivity but largely enhances the nonzero reflectivity and the reflectivity contrast. That means, replacing a constant density $N_{0}$ with a periodic density $N_{j}(x)$ does not hamper the implementation of spatial KK relation, which is essential for developing nonreciprocal optical devices requiring a high reflectivity contrast.

 

Fig. 6. Spatial profiles of (a) $\chi _{2}^{\prime }$ and (b) $\chi _{2}^{\prime \prime }$ as well as corresponding spectra of (c) reflectivities $R_{l,r}$ and (d) contrast $C$ for an atomic lattice of density $N_{j}(x)=N_{0}e^{-(x-x_{j})^{2}/\delta x^{2}}$ in the $jth$ trap of center $x_{j}=(j-1/2)\Lambda$, width $\delta x$, and period $\Lambda$. Parameters are the same as in Fig. 2(b) except $N_{0}=2.0 \times 10^{12}$ cm$^{-3}$, $L=60$ $\mu$m, $\delta _{0}/\gamma _{ge}=30$, $\Lambda =400$ nm, and $\delta x=\Lambda /6$.

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It has been shown that atoms could be trapped and guided using nanofabricated wires and surfaces to form atom chips [60]. These chips provide a versatile experimental platform with cold atoms and constitute the basis for wide and robust applications ranging from atom optics to quantum optics. They have been used, for instance, in diverse experiments involving quantum simulation, metrology, and information processing [61–63]. We then believe that our proposal is well poised to atom-chip implementations in integrated optical devices.

4. Conclusions

In summary, we have investigated the spatial KK relation and relevant reflection features in a short and dense sample of cold $^{87}$Rb atoms. This nontrivial relation in regard of the probe susceptibility is enabled by generating a position-dependent ground level shift $\delta (x)$ with a far detuned control field of intensity linearly varied along the $x$ direction. We find, in particular, that the figure of merit $D_{kk}$ characterizing the spatial KK relation may switch from the unbroken regime of unidirectional reflection, via the transitional regime of asymmetric reflection, to the broken regime of symmetric reflection, or vice versa. This is attained by increasing the maximal level shift $\delta _{0}$ from a negative value to a positive value or considering an inverse process, depending on the sign of probe detuning $\Delta _{p}$. A swapping between the nonzero reflectivity and the vanishing reflectivity at opposite sample ends is also viable by changing the sign of maximal level shift $\delta _{0}$. It is of more interest that the nonzero reflectivity can be well enhanced to result in a high reflectivity contrast for lower densities and larger lengths in a cold atomic lattice, indicating that Bragg scattering does not hamper or spoil the main effects of spatial KK relation. Such a feasible scheme may inspire further applications in nonreciprocal optical devices and provide a promising platform for atom-chip implementations.