1. Introduction

In the past few decades, progress has been made in the study of nonreciprocal optical propagation in optical systems [1–5]. Especially in atomic systems, the non-reciprocal response have been investigate theoretically and experimentally [6,7]. The momentum locking of susceptibility can not only be generated by the random thermal motion of atoms to achieve a non-reciprocal optical system and an irreversible transmission-light amplification based on the Doppler effect [8,9]. However, the non-reciprocal light propagation can also be realized by using the parity-time (P-T) symmetric system combined with the one-dimensional (1D) atomic lattice and standing wave [10–13], so that the incident light signal from one side in the medium is suppressed while that on the other side is strengthened. Moreover, rapid progress has been made in designing optical non-reciprocity, and it may also be achieved in optomechanical and optical cavity systems [14–18]. In addition, an all-optical approach to achieve optical non-reciprocity on a chip has been attempted by the quantum squeezing of two coupled resonator modes. This protocol creates a new pathway to achieving integrable all-optical non-reciprocal devices, permitting chip-compatible optical isolation and non-reciprocal quantum-information processing [19]. lately, the nontrivial single-photon scattering properties of giant atoms has been investigated, which being an effective platform to realize nonreciprocal and chial quantum optics [20].

Recently, it has been found that non-reciprocal optical propagation can also be realized in an homogeneous medium, which requires that the refractive index, susceptibility or permittivity of plane electromagnetic waves are complex analytic functions, their real and imaginary parts satisfy the KK relation of spatial distribution [21–23]. The advantage of realizing unidirectional light propagation based on the spatial KK relation is that, there is no strict requirement for the incident-light angle, which is simpler and more controllable in experiments [24–28]. Furthermore, the susceptibility distribution under this relation does not require an extremely strict P-T symmetry, because only an isotropic and non-magnetic material is required [29–31]. Thus, unidirectional reflected electromagnetic waves based on the KK relation have wide applications, i.e., through the reasonable design of the spatial KK medium in a broadband absorber to achieve bidirectional and omnidirectional non-reflective properties [32–35].

What interests us is an efficient scheme for realizing the asymmetrical reflections by driving a homogeneous atomic sample. Here, we propose a mechanism to achieve two-color unidirectional reflections with a coherently driven four-level tripod atomic system, by reasonably setting the densities of the coupling fields accompanied by the corresponding variation of the position to break the spatial symmetry of probe susceptibility. In our system, there are two electromagnetically induced transparency (EIT) windows [36–38], through which two unidirectional reflection bands can be formed and modulated. In particular, our study can be easily extended to the study of polychromatic non-reciprocal reflections.

This paper is organized as follows: In Section 2, the mechanism we consider is illustrated through detailed computational designs, and the exact simulation with the transfer matrix method is also discussed. The main results of reflective non-reciprocity due to the coupling fields that vary with position are presented in Section 3. Lastly, we conclude with a summary of the study in Section 4.

2. Model and equations

We consider a cold $^{87}$Rb atomic sample driven by a weak probe-field and two strong coupling fields that form the four-level tripod configuration, as shown in Fig. 1(a). There are three ground energy states $\left \vert 1\right \rangle$, $\left \vert 2\right \rangle$ and $\left \vert 3\right \rangle$, and one excited energy state $\left \vert 4\right \rangle$, characterized by the Rabi frequency (detuning) $\Omega _{p}=\mathbf {E}_{p}\cdot \mathbf {d} _{41}/2\hbar$ ($\Delta _{p}$) for the transition $\left \vert 1\right \rangle$ to $\left \vert 4\right \rangle$, $\Omega _{c}=\mathbf {E}_{c}\cdot \mathbf {d}_{42}/2\hbar$ ($\Delta _{c}$) for the transition $\left \vert 2\right \rangle$ to $\left \vert 4\right \rangle$, and $\Omega _{d} =\mathbf {E}_{d}\cdot \mathbf {d}_{43}/2\hbar$ ($\Delta _{d}$) for the transition $\left \vert 3\right \rangle$ to $\left \vert 4\right \rangle$ respectively. The matrix element $\mathbf {d}_{ij}=\left \langle i\right \vert \mathbf {d} \left \vert j\right \rangle$ is used to denote the dipole moment for transitioning $\left \vert i\right \rangle$ to $\left \vert j\right \rangle$. Specifically, energy states $\left \vert 1\right \rangle$, $\left \vert 2\right \rangle$, $\left \vert 3\right \rangle$, and $\left \vert 4\right \rangle$ refer to states $\left \vert 5S_{1/2},F=1,m_{F}=-1\right \rangle$, $\left \vert 5S_{1/2},F=1,m_{F}=0\right \rangle$, $\left \vert 5S_{1/2},F=1,m_{F} =1\right \rangle$, and $\left \vert 5P_{3/2},F=0,m_{F}=0\right \rangle$ of cold $^{87}$Rb atoms’ D2 line broken by static magnetic fields, respectively. The homogeneous distribution of cold atoms is displayed in Fig. 1(c). It is worth to emphasize that the Doppler broadening can be ignored safely in our cold atomic sample.

 

Fig. 1. (a) Energy level diagram of a four-level tripod atomic system interacting with a weak probe-field of the Rabi frequency (detuning) $\Omega _{p}$ ($\Delta p$) and two strong coupling fields varying with position $x$ of the Rabi frequency (detuning) $\Omega _{c}(x)$ ($\Delta c$) and $\Omega _{d}(x)$ ($\Delta d$), respectively. (b) Rabi frequency as a function of the spatial position $\Omega _{c}^{2}(x)=\Omega _{c0}^{2}(k_{1}x+b_{1})$ and $\Omega _{d} ^{2}(x)=\Omega _{d0}^{2}(k_{2}x+b_{2})$, here is a simple consideration of $k_{1}=k_{2}=k=8.75\;\mu$m$^{-1}$, $b_{1}=b_{2}=b=5$, $\Omega _{c0}^{2} =\Omega _{d0}^{2}=\Omega _{0}^{2}=5$ MHz for convenience; (c) diagram of the 1D medium that shows tripod atoms homogeneous distribution, and illuminated by two control beams perpendicular to the $x$ direction, a probe field travels along the $x$ direction.

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With the electric-dipole and rotating-wave approximations, the atom-field Hamiltonian interaction in matrix form is

According to the definition, the density matrix element$\ \rho _{ij} =\left \langle i\right \vert \rho \left \vert j\right \rangle$ is the product of the probability amplitude of the population between states $\left \vert i\right \rangle$ and $\left \vert j\right \rangle$, and it indicates the state of coherence occurring in the system. Additionally, $\gamma _{ij}=(\Gamma _{i}+\Gamma _{j})/2$ denotes the complex coherence dephasing rate on the transition $\left \vert i\right \rangle$ to $\left \vert j\right \rangle$, with population decay rates $\Gamma _{i}=\Sigma _{k}\Gamma _{ik}$ and $\Gamma _{j}=\Sigma _{k}\Gamma _{jk}$; $k=1$, $2$, $3$ and $4$ describes the inevitable dissipation within the system. Working on the weak probe limit of $\Omega _{p}\ll \Omega _{c,d}(x)$, constrained by $\rho _{11}\approx 1$, $\rho _{24} =\rho _{34}=\rho _{44}\approx 0$ and the steady-state condition $\dot {\rho } _{ij}=0$, by solving Eq. (2) we can figure out $\rho _{41}$ as following

The real and imaginary parts of susceptibility in the frequency domain must satisfy the integral equation

$x^{\prime }$ is the spatial coordinate along the $x$ direction, the Cauchy principal value $P$ indicates the principal part of the integral after excluding the singular points $x^{\prime }=x$. To check the probe frequency region that satisfied the spatial KK relation in a finite atomic sample, we propose the following integral

3. Results and discussion

In this section, we first check the probe susceptibility by modulating the intensity of coupling fields by a neutral density filter of inhomogeneous transmissivity $\Omega _{c}^{2}(x)=\Omega _{c0}^{2}(k_{1}x+b_{1})$ and $\Omega _{d}^{2}(x)=\Omega _{d0}^{2}(k_{2}x+b_{2})$. For the simple case, which should be considered first, the two coupling fields have the same variation relationship, the settings $\Omega _{c0}=\Omega _{d0}=\Omega _{0}$, $k_{1} =k_{2}=k$ and $b_{1}=b_{2}=b$ are adjusted for convenience. In Figs. 2(a) and 2(b), we plot the real and imaginary parts of the complex susceptibility $\chi _{p}$ against position $x$ for $\Delta _{c}=\Delta _{d}=0$ MHz, with two different detuning values $\Delta _{p}=-30$ MHz and $\Delta _{p}=30$ MHz, respectively. Clearly, the dispersion (absorption) spectral lines Re$[\chi _{p}]$ (Im$[\chi _{p}]$) exhibits an odd (even) symmetry in space, the symmetric point centered at $x=L/2$ and practically fully contained by the atomic sample. This odd-even symmetry-variation trait is called the P-T symmetry of complex susceptibility, which is a critical characteristic for the non-reciprocity of probe reflection. This implies that, the complex susceptibility varies with spatial position due to the intensity variation of coupling fields. To reveal the frequency-dependent feature, we plot the real and imaginary parts of the probe susceptibility against both position $x$ and detuning $\Delta _{p}$ in Figs. 2(c) and 2(d). It can be realized that, the P-T symmetry points move simultaneously toward the right/left sample end with the increasing/decreasing $\left \vert \Delta _{p}\right \vert$.

 

Fig. 2. Real and imaginary parts of susceptibility Re$[\chi _{p}]$ and Im$[\chi _{p}]$ (green line with squares and violet line with triangles) against position $x$ with $\Delta _{p}=-30$ MHz in (a) and $\Delta _{p}=30$ MHz in (b); real and imaginary parts of susceptibility Re$[\chi _{p}]$ and Im$[\chi _{p}]$ against both position $x$ and detuning $\Delta _{p}$ in (c) and (d). The parameters are $N_{0}=7\times 10^{13}$ cm$^{-3}$, $\Omega _{p}=0.048$ MHz, $\Delta _{c}=\Delta _{d}=0$ MHz, $\mathbf {d}_{14}=2.0\times 10^{-29\text { }}$C$\cdot$m, $\Gamma _{41}=\Gamma _{42}=\Gamma _{43}=6$ MHz, medium length $L=20\;\mu$m. Other parameters are the same as Fig. 1(b).

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Next, we will examine the spatial KK relation in further by Eq. (7). In Figs. 3(a) and 3(b), we plot $K$ against position $x$ with a set of red and blue detuning probe light, respectively. It is then justified that, for a certain detuning $\Delta _{p}$ only one point in space satisfies $K(\Delta _{p},x)=0$ which is corresponding to P-T symmetry point marked by yellow circle, and in the frequency region of $K(\Delta _{p},x)\neq 0$ the P-T symmetry points have moved out of the atomic sample. Furthermore, we plot $K$ against both position $x$ and detuning $\Delta _{p}$ in Fig. 3(c). This clearly shows that two frequency regions guarantee $K(\Delta _{p},x)=0$, are also satisfy the P-T symmetry. All these can be clearly shown by comparing with Figs. 2(c) and 2(d). In our mechanism, the complex susceptibility can achieve PT symmetry, but it can’t satisfy the KK relation in the whole space.

 

Fig. 3. The value of $K$ v.s. position $x$ with red detunings $\Delta _{p}$ in (a) and blue detunings $\Delta _{p}$ in (b), the values of $K$ against position $x$ and detuning $\Delta _{p}$ in (c). Other parameters are the same as in Fig. 2.

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In the following, we focus on the unidirectional reflections based on the spatial modulation of probe susceptibility which is no longer satisfy the spatial KK relation. Then the reflection and transmission spectra are plotted in Fig. 4(a) for the parameters used in Fig. 2 based on Eq. (13). These spectra can be divided into two regions: (I) the EIT window with the average susceptibility for $\Delta _{p}\;\in$($-6$ MHz, $8$ MHz); (II) the PT symmetry are satisfied for $\Delta _{p}\;\in$($-45$ MHz, $-6$ MHz) and $\Delta _{p}\;\in$($8$ MHz, $45$ MHz). The generation of two different regions can be understood by examining in Fig. 4(b) the figure of contrast factor $C$ against probe detuning $\Delta _{p}$, which is clearly shown, as compared with Fig. 4(a), that $C$ governs the relation between $R^{r}$ and $R^{l}$. In Fig. 4(a), what excites us is that a remarkable nonreciprocal reflection can be clearly observed in the frequency region (II), and the unidirectional reflection (that is the left reflectivity is zero and the right reflectivity is rather high) can be realized in frequency region (I). These can also be clarified by the high-contrast factor $C$, which reveals the extent of non-reciprocal and unidirectional of probe reflections [see Fig. 4(b)]. Therefore, we may conclude that, if the complex susceptibility satisfies the P-T symmetry, the non-reciprocal reflection can be achieved. However, the realization of complete non-reciprocity, that is, one-way reflection, must be accompanied by a complete non-absorption. This means the unidirectional reflections should be located in the EIT window. In further, we need emphasize the reflections are reciprocal $R^{l}=R^{r}=0$ at the resonance point due to a high transmittivity as shown by the green solid line in Fig. 4(a), corresponding to the narrowest EIT window in the whole sample that locates the region of Im$\chi _{p}=$Re$\chi _{p}=0$ revealed in Figs. 2(c) and 2(d). This unidirectional reflection can be explained as following: for the intensity of coupling fields are increasing with the position $x$, a left (right) incident probe beam is most reflectionless (partially reflected) with a narrow (wide) band of tranmission in the EIT window.

 

Fig. 4. (a) Left and right reflectivities $R^{l}$ and $R^{r}$ (sapphire line with circles and orange line with stars), the transmittivity $T$ (green dash) and the imaginary part of average complex susceptibility $\overline {\chi }_{p}$ (purple dash dot line) against detuning $\Delta _{p}$, (b) the contrast factor $C$ against detuning $\Delta _{p}$. Other parameters are the same as in Fig. 2.

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It should be emphasized that our mechanism to achieve unidirectional reflections is to break the spatial symmetry of susceptibility by the spatial modulation coupling field intensity not depending on the spatial KK relation. In Fig. 5, we further investigate the unidirectional reflections in the case where one of the coupling fields is no longer resonant. In Fig. 5(a), there are two frequency regions of unidirectional reflections $\Delta _{p}\;\in$($-4$ MHz, $4$ MHz) and $\Delta _{p}\;\in$($45$ MHz , $54$ MHz) respectively, for $R_{p}^{r}>0.6$, $R_{p}^{l}=0$ in two EIT windows (containing two points of reciprocal reflection due to the strong transmission has been described in Fig. 4(a)). Fig. 5(c) further shows that the reflectivity contrast $C$, an important figure of merit on the asymmetric reflection, is almost $1.0$ in these two frequency regions of unidirectional reflection. For further check the unidirectional reflections, we set the intensity of coupling fields varies with position steadily by a decreased $k$ as shown in Fig. 5(b). The two frequency regions of unidirectional reflections have narrowed obviously, to $\Delta _{p}\;\in$($-2$ MHz, $2$ MHz) and $\Delta _{p}\;\in$($48$ MHz, $52$ MHz) respectively. It is clearly displayed by the contrast $C$ in Fig. 5(d). The main reason is that when decreasing $k$, the intensity of coupling fields can be reduced in the whole atom sample which narrowed the EIT windows. Thus, in this nontrivial case, a dynamically tunable two-color unidirectional reflections can be achieved.

 

Fig. 5. (a) and (b) Left and right reflectivities $R^{l}$ and $R^{r}$ (sapphire line with circles and orange line with stars) and the imaginary part of average complex susceptibility $\overline {\chi }_{p}$ (purple dash dot line) against detuning $\Delta _{p}$, (c) and (d) the contrast factor $C$ against detuning $\Delta _{p}$. with $\Delta _{c}=50$ MHz. Other parameters are the same as in Fig. 2, except for $k=5.0$ in (b) and (d).

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Then, we set the coupling strength of the control fields to experience a nonlinear increase, e.g. $\Omega _{c}^{2}(x)=\Omega _{d}^{2}(x)\propto x^{3}$ or $\Omega _{c}^{2}(x)=\Omega _{d}^{2}(x)\propto \frac {1}{x^{3}}$ as shown on the right side, and the corresponding reflectivity lines are shown on the left side in Figs. 6(a) and 6(b), respectively. It is found that the reflectivity lines have almost no change, which indicates that the reflectivity is not sensitive to the rate of the variation of the coupling strength with position $x$. It is of special interest that, if the coupling strength of one control field is linearly decreasing another one is increasing with $x$, the first unidirectional reflection region corresponds to $R^{r}>0.6$, $R^{l}\simeq 0$, and the second corresponds to $R^{r}\simeq 0$, $R^{l}>0.6$ [see Fig. 6(c)]. This means the manipulation of unidirectional reflection has more flexibility in our system.

 

Fig. 6. Left and right reflectivities $R^{l}$ and $R^{r}$ (sapphire line with circles and orange line with stars) against detuning $\Delta _{p}$ with $\Delta _{c}=50$ MHz, corresponding to different intensities variation of coupling fields in (a), (b) and (c). Other parameters are the same as Fig. 2.

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We analyse the variation of reflectivity with position $x$ as the probe light travels through the whole atomic sample. Figs. 7(a) and 7(b) show that when $\Delta _{p}=-25$ MHz, which is located in the non-reciprocal area, the reflectivity $R^{l}$ increases with position until to the end of the sample where it has reached $0.6$, and correspondingly, the reflectivity $R^{r}$ decreases with the position to $0.1$ at the end. When the reflectivity is in the unidirectional frequency region for $\Delta _{p}=52$ MHz, the reflectivity $R^{l}$ can be reduced to zero corresponding to $R^{r}$ over $0.6$. It is worth noting that in the unidirectional frequency region a left (right) incident beam is reflectionless (high reflected) because it first sees negative (positive) peak of Re$[\chi _{p}]$ [21], and the resonant absorption Im[$\chi _{p}$] is already strong enough to yield $T\rightarrow 0$ for forward photons while the dispersion profile Re$[\chi _{p}]$ is not too sharp to yield a high $R^{r}$. One good way to reduce $T$ and simultaneously increase $R^{r}$ is to produce enhanced absorption and sharper dispersion profiles in denser atomic sample. To further investigate the non-reciprocal and unidirectional reflections, we plot the left and right reflectivities against both position $x$ and detuning $\Delta _{p}$. Figs. 7(c) and 7(d) shows that, the reflectivity is different at different positions $x$ in non-reciprocal frequency regions due to the spatial variation of susceptibility. However, no matter which side of the atomic sample the light enters from, the reflectivity at each position can be determined. Thus, it is viable to convert the sample from left reflectionless to right reflectionless or vice versa. In addition, the non-reciprocal frequency regions are also determined by the length of atomic sample, e.g. the left and right reflections are completely reciprocal at the position $x=10$ $\mu$m, it means the non-reciprocal reflections can not be realized with the sample length $L=10\;\mu$m in our system. Then the non-reciprocal regions become wider and wider with the increasing $x$. Thus, we can also provide theoretical support for the experiment with accurate sample length.

 

Fig. 7. (a) Left reflectivity $R^{l}$ versus position $x$ with $\Delta _{p}=-25$ MHz (sapphire line with circles) and $\Delta _{p}=52$ MHz (blue line with triangles). (b) Right reflectivity $R^{r}$ v.s. position $x$ with $\Delta _{p}=-25$MHz (orange line with stars) and $\Delta _{p}=52$ MHz (red line with squares); (c) Left reflectivity $R^{l}$ and (d) right reflectivity $R^{r}$ against both position $x$ and detuning $\Delta _{p}$. with $\Delta _{c}=50$ MHz. Other parameters are the same as those in Fig. 2.

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4. Conclusion

In summary, we have explored the dynamically tunable two-color unidirectional reflections in a short and dense homogeneous cold atomic medium based on the spatial modulation of probe susceptibility, and the atoms were coherently driven into a four-level tripod system by a probe field and two strong coupling fields that vary with position $x$. Especially, there will appear two EIT windows by setting the detunings of two control fields. Thus, dynamically tunable two-color unidirectional reflections can be realized. We found that the reflectivity varies with $x$ due to the spatial variation of susceptibility, which can provide theoretical support for the experiment with accurate sample length. In fact, the essence of whether or not light propagation is reciprocal, hinges on the spatial variation of susceptibility, which can be modulated by the coupling fields and detunings. Thus, our mechanism can realize a more simple and easy to manipulate unidirectional reflection, which can provide a straightforward and feasible program for theoretically and experimentally exploring unidirectional light propagation, and it can be used to develop new photonic devices requiring an asymmetric light transport.