1. Introduction

It is important to realize optical components that photons are only allowed to propagate in one direction, namely non-reciprocal photonic device, similar to diodes in electronic information science [1–4]. The magnetic-free optical diodes and chip isolators [5–7] controlling the flow of light are of principal significance for information processing and quantum networks, because it doesn’t need bulky magnets and is thus compatible with integrated circuit technology [8–10]. Versatile strategies have been Elaborated to achieve optical nonreciprocity, such as chiral quantum optical systems by asymmetric coupling of atomic internal states [11–15], optomechanically induced non-reciprocity on a few photon or even single-photon level has reported in theory and experiment [16–21], even at room temperature [22]. A similar effect can also be investigated in nonlinear regimes [23], giant-atom systems [24] or a chip by quantum squeezing one of two coupled resonator modes [20]. In particular, the perfect non-reciprocity reflection (unidirectional reflection) has been studied in a moving atomic lattice based on time reversal symmetry breaking [25–27], or in a stationary atomic lattice with applying standing-wave coupling fields based on parity-time (PT) symmetric or antisymmetric [28–30]. In addition, the unidirectional perfect reflection or reflectionless have been guaranteed to occur in PT-symmetric system of double-lattice photonic crystals or two-dimensional periodic structures [31,32].

The above physical systems have made significant progress in optical nonreciprocity photonic device. However, high performance optical amplification is an essential functionality in integrated photonic circuits. Therefore, harnessing non-reciprocial light amplification is more important, which can simultaneously amplify the weak signal output by the quantum system and isolate the sensitive quantum system from the backscattered external noise, and it is promising to explore more functions and applications [33–36]. Up to now, unidirectional amplification or asymmetric lasing mainly focuses on various physical systems, e.g., micro-ring resonators and silicon waveguides [37,38], optomechanical systems [39–41], an atomic ensemble in coupled cavities [42,43], and the PT-symmetry system [44,45]. These systems provide more possibilities for high-quality non-reciprocal photon devices, such as optical diodes, chip isolators and photonic filters.

Generally speaking, it is more applicable to achieve asymmetric light transport in homogeneous atomic medium. Recently, the perfectly asymmetric reflection can be achieved in homogeneous continuous medium, with refractive index of the plane electromagnetic wave obeying the spatial Kramers-Kronig (KK) relation, [46–51]. Based on the KK relation, many efficient schemes of controlled unidirectional reflection in uniform cold atomic medium have been proposed by suitable design of controlled Rydberg atom or the linear variation of coupling field intensities [52–54]. Homogeneous atomic system is more simple and controllable in experiment. However, the reflectivity is usually very low in a uniform atomic medium.

In this paper, we propose a theoretical scheme to realize amplified unidirectional reflections using cold rubidium (Rb) atoms homogeneously distributed, with ingenious designed of three-level $\Delta$-type atom model and linear variation of coupling field with position $x$. Specifically, the probe light can be amplified owing to the closed loop transitions of $\Delta$-type constructed by adding a microwave field, and the unidirectional reflection can be achieved by simple adjustment of linear changes in coupling field to destroy the spatial symmetry of susceptibility. It is worth emphasizing that the probe gain is achieved under the conservation of energy in our regime, unlike PT-symmetric metamaterial which requires a delicate balance of gain and loss, and the probe susceptibility does not require spatial KK relation. Our results provide a different perspective to implement unidirectional amplification in a uniform atomic medium, which is more applicable to design high-quality non-reciprocal devices. For example, optical diodes and isolators which can block light in one direction but allow light to pass in the opposite direction. This is important to protect a laser from back reflections, which can disturb the laser operation, or to mitigate multi-path interference in an optical communication system.

2. Theoretical model and equations

As shown in Fig. 1(a), the cold $^{87}$Rb atoms are driven into a closed three-level $\Delta$-type system, by a weak probe-field, a strong coupling field and a microwave field. The probe beam of Rabi frequency (detuning) $\Omega _{p}=\mathbf {E}_{p}\cdot \mathbf {d}_{31}/2\hbar$ ($\Delta _{p} = \omega _{p}-\omega _{31}$) and the coupling field of Rabi frequency (detuning) $\Omega _{c}=\mathbf {E}_{c}\cdot \mathbf {d}_{32}/2\hbar$ ($\Delta _{c} = \omega _{c}-\omega _{32}$) drives the dipole allowed transition $\left \vert 1\right \rangle \;{\leftrightarrow }\;\left \vert 3\right \rangle$ and $\left \vert 2\right \rangle \;{\leftrightarrow }\;\left \vert 3\right \rangle$ respectively, the resonant microwave field of Rabi frequency (detuning) $\Omega _{d}=\mathbf {E} _{d}\cdot \mathbf {d}_{21}/2\hbar$ ($\Delta _{d}=\omega _{d}-\omega _{21}$) couples the dipole forbidden transition $\left \vert 1\right \rangle \;{\leftrightarrow }\;\left \vert 2\right \rangle$. 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 of transition $\left \vert i\right \rangle$ to $\left \vert j\right \rangle$. Specifically, the energy levels $\left \vert 1\right \rangle$, $\left \vert 2\right \rangle$ and $\left \vert 3\right \rangle$ refer to states $\left \vert 5S_{1/2},F=1,m_{F}=-1\right \rangle$, $\left \vert 5S_{1/2},F=2,m_{F}=+1\right \rangle$, and $\left \vert 5P_{3/2},F=1,m_{F}=0\right \rangle$ 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), the probe light travels along the $x$-axis, the microwave and control fields travel vertical $x$-axis. It is worth to emphasize that the intensity of coupling field is linearly varied along $x$ direction, which can be easily modulated by a neutral density filter (NDF) [55]. Specifically, the Doppler broadening can be ignored safely in our cold atomic sample. Because the speed of atoms decreases after laser cooling, which can be cooled down to $\mu$K. Thus, the influence of atomic external state motion on the spectrum is greatly weakened, the frequency shift and broadening of atomic energy levels can be suppressed effectively.

 

Fig. 1. (a) Energy level diagram of a three-level $\Delta$-type atomic system driven by a weak probe field, a strong coupling field and a microwave field, respectively. (b) The Rabi-frequency of coupling field as a function of position $x$ for $G_{c}(x)=ax$, with $a=0.1$ MHz/${\mu }$m, the sample length $L=10\;{\mu }$m, $x\in (0$, $L)$, the average coupling field of Rabi-frequency $G_{c}^{0}=\overline {G_{c}(x)}=25$ MHz, and the constant microwave field with the Rabi-frequency $G_{d}=15$ MHz; (c) The diagram of 1D medium with atoms homogeneous distribution, illuminated by a coupling beam and a microwave field perpendicular to the $x$ direction, a probe field travels along the $x$ direction; (d) The probe susceptibility as a function of position $x$ with the same parameters as Fig. 3(c) at $\Delta _{p}=55$ MHz.

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With the electric-dipole and rotating-wave approximations, the interaction Hamiltonian of atom-field can be written as

Then, the equations of the density matrix are written as

According to the definition, the density matrix element$\ \rho _{ij}=\left \langle i\right \vert \rho \left \vert j\right \rangle =\left \langle i\right \vert \Psi \rangle \left \langle \Psi \right \vert j\rangle =C_{j}C_{i}^{\ast }$ 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. Additionally, $\gamma _{ij}=(\Gamma _{i}+\Gamma _{j})/2$ denotes the coherence dephasing rate on the transition $\left \vert i\right \rangle$ to $\left \vert j\right \rangle$, with population decay rates $\Gamma _{i}=\Sigma _{n} \Gamma _{in}$ and $\Gamma _{j}=\Sigma _{n}\Gamma _{jn}$; $n=1$, 2 and 3 describes the inevitable dissipation. The above equations constrained by $\rho _{11}+\rho _{22}+\rho _{33}=1$, conjugate conditions $\rho _{ij}=\rho _{ji}^{\ast }$ and the steady-state condition $\dot {\rho } _{ij}=0$.

We note that the Rabi frequencies of all fields cannot be treated as real numbers, because of the phase variation caused by microwave field. Thus, we make the standard transformation of the equations as follows: First, setting the initial phase of the probe, coupling and microwave field be $\phi _{p}$, $\phi _{c}$ and $\phi _{d}$ respectively; Then, the corresponding Rabi frequencies are redefined as $\Omega _{p}=G_{p}e^{i\phi _{p}}$, $\Omega _{c}=G_{c}^{0}e^{i\phi c}$ and $\Omega _{d}=G_{d}e^{i\phi _{d}}$, where $G_{p}$, $G_{c}^{0}$ and $G_{d}$ are all real numbers; Finally, the atomic variables are rewritten as $\sigma _{ii}=\rho _{ii}$ , $\sigma _{13}= \rho _{13}e^{i\phi _{p}}$, $\sigma _{32}=\rho _{32}e^{i\phi _{c}}$ and $\sigma _{12}=\rho _{12}e^{i(\phi _{p}-\phi _{c})}$. Therefore, it is not difficult to obtain the following density matrix equations:

Here, Re$[\chi _{p}(\Delta _{p},x)]$ and Im$[\chi _{p}(\Delta _{p},x)]$ represent the real and imaginary parts of susceptibility, corresponding to dispersion and absorption lines of probe beam, $\varepsilon _{0}$ is the dielectric constant in vacuum, and the atomic density $N_{0}$ is a constant.

In the following, we should check the light transport features, which can be examined by directly adopting standard transfer-matrix method [56]. First of all, we provide the $j$th $2\times 2$ unimodular transfer matrix $m_{j}(\Delta _{p},x_{j})$ by dividing the whole sample of length $L$ into $S$ thin layers, $j\in (1,S)$. With identical thickness $\delta =L/S$, but the susceptibilities exhibit slightly different which lead the various of transfer matrix

Note that $M^{l}(\Delta _{p},j\delta )$ is multiplied from left to right by layers, and $M^{^{r}}(\Delta _{p},j\delta )$ is multiplied from right to left. Thus, the probe reflectivities at $j$th layer that incidents from the left- and right-side are

It is worth emphasizing that $M_{(22)}^{l}(\Delta _{p},L) = M_{(22)}^{r}(\Delta _{p},L)$ and $M_{(21)}^{l}(\Delta _{p},L)=M_{(12)}^{r}( \Delta _{p},L)$ only at the ends of the sample. With the Eq. (10), we can check the nonreciprocal even unidirectional reflections. However, the quantification of nonreciprocal reflection can be expressed by the contrast factor:

3. Numerical results and discussions

In this section we explore, via full numerical calculations, how to realize and manipulate the unidirectional reflection amplification of probe field under the energy conservation $\Delta _{d}=\Delta _{p}-\Delta _{c}$. As shown in Fig. 1(a), we note that the probe beam should be amplified, because the microwave field $\mathbf {E}_{d}$ can transfer atoms from level $\left \vert 1\right \rangle$ to level $\left \vert 2\right \rangle$, and then to level $\left \vert 3\right \rangle$ with the help of coupling field $\mathbf {E}_{c}$. It is worth emphasizing that the various characteristics of this system become related to the relative phase because of the existence of microwave field. Thus, the Eq. (4) can be periodically modulated by the relative phase $\phi$. With the resonance of microwave field and two degenerate lower energy levels, we first check the probe gain Im$\chi _{p}$ versus the relative phase $\phi$ with the constant and resonance coupling field (we stress here the probe gain corresponds to Im$\chi _{p}<0$). It is found that, the probe gain can be achieved at different relative phase $\phi$ with different amplitude, for the chosen values of $\Delta _{p}$ in Fig. 2(a). Especially, when $\Delta _{p}=0$, the largest value of gain appears at $\phi =\pi /2+2k\pi$, where $k$ is an integer. And when $\Delta _{p}=30$ MHz ($\Delta _{p}=-30$ MHz), the maximum gain appears at $\phi <-\pi /2+2k\pi$ ($\phi >-\pi /2+2k\pi$). In other words, the probe field has two gain values in the frequency region at $\phi =-\pi /2+2k\pi$. It is easy to see that, the probe gain can be modulated periodically by the relative phase $\phi$, which is well displayed in Fig. 2(b). More specifically, one gain peak for $\phi =\pi /2+2k\pi$ split into double for $\phi =-\pi /2+2k\pi$ in the frequency region of probe field.

 

Fig. 2. Imaginary part of susceptibility Im$(\chi _{p})\;v.s.$ relative phase $\phi$ with different detunings $\Delta _{p}$ in (a), and $v.s.$ both relative phase $\phi$ and detuning $\Delta _{p}$ in (b). Other parameters are $N_{0}=2\times 10^{11}$ cm$^{-3}$, $G_{p}=0.3$ MHz, $G_{d}=15$ MHz, $G_{c}^{0}=25$ MHz, $\Delta _{c}= 0$, $\mathbf {d} _{13}=2.0\times 10^{-29\text { }}$C$\cdot$m, $\Gamma _{31}=\Gamma _{32}=6$ MHz.

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Next, we will investigate the behavior of unidirectional reflection in the gain region with chosen relative phase $\phi$. There are two gain peaks (the amplitude is almost $0.1$) around $\Delta _{p}=\pm 35$ MHz at $\phi =-\pi /2$, and only one gain peak (the amplitude is less than $0.05$) around $\Delta _{p}=0$ at $\phi =\pi /2$, as clearly shown in Figs. 3(a) and 3(b), respectively. Then, we consider that the coupling field varies linearly with position $x$ instead of being constant, by setting $G_{c}(x)=ax$ (here, what we need to emphasize is that the intensity of constant coupling field we used to check the probe gain is the average value of linearly varying coupling field, for $G_{c}^{0}=aL/2=25$ MHz in all figures). It is found that in Fig. 3(c), there are two unidirectional reflection bands with an amplified reflectivity of right-side $R^{r}>5.0$ accompanied by an almost zero reflectivity of left-side $R^{l}\simeq 0$, corresponding to two gain peaks at $\phi =-\pi /2$ around $\Delta _{p}=\pm 55$ MHz. Which can also be clarified by the high-contrast factor $C\simeq 1.0$, an important figure of merit on the asymmetric reflection, displayed in Fig. 3(e). It is also interesting to examine what would happen for reflectivities $R^{r}$ and $R^{l}$, when $\phi =\pi /2$ with a lower gain peak around $\Delta _{p}=0$. This is clearly shown in Fig. 3(d), the right- and left-side reflectivities are nonreciprocal in a large frequency region, with the high contrast $C\;\simeq \;1.0$ [see Fig. 3(f)]. However, the reflectivities are too small to achieve amplified unidirectional reflection with a lower imaginary part of susceptibility ($\left \vert \text {Im}\chi _{p}\right \vert <0.05$).

 

Fig. 3. Imaginary part of susceptibility Im$(\chi _{p})\;v.s.$ detuning $\Delta _{p}$ with $G_{c}^{0}=25$ MHz in (a) and (b); The reflectivities on the left-side $R^{l}$ and right-side $R^{r}$ in (c) and (d), and the contrast factor of the left- and right-side reflections $C$ in (e) and (f) $v.s.$ detuning $\Delta _{p}$, with $G_{c}(x)=ax$, $x\in (0$, $L)$. For $\phi =-\pi /2$ in (a), (c) and (e), and $\phi = \pi /2$ in (b), (d) and (f), respectively. Other parameters are the same as Fig. 2.

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Now, we need to explain why the reflectivity on the right-side is higher than left-side in the nonreciprocal frequency region. This is because the intensity of coupling field increases with position $x$, and reaches the maximum at the end of the sample (right-side), the probe beam incidents from right-side can be well reflected and amplified in the gain region. On the contrary, the intensity of coupling field is too weak to cause quantum destructive interference at the front of the medium (left-side), that the probe beam is difficult to form a high reflection band due to the large absorption. It can be well explained with Fig. 1(d), when $\Delta _{p}$ locates in the unidirectional amplified region, with position $x$ increasing, the Im$\chi _{p}$ becomes smaller and the corresponding probe gain increases, and the dispersion becomes steeper, which means that the probe light is well reflected. Thus, the light enters from the right-side of the sample can be well reflected and amplified corresponding to a reflection of almost zero on the left-side.

It is of special interest to study the unidirectional reflection at $\phi =0$ and $\phi =\pi$. We can see clearly that a gain peak around $\Delta _{p}=-23.5$ MHz ($\Delta _{p}=23.5$ MHz) appearing at $\phi =0$ ($\phi =\pi$) in Fig. 4(a) [Fig. 4(b)]. This is in good agreement with Fig. 2(b) that the probe gain region is $\Delta _{p}<0$ ($\Delta _{p}>0$) at $\phi =0+2k\pi$ ($\phi =\pi +2k\pi$). Note, the imaginary part of susceptibility amplitude is less than $0.05$ with the parameters in Fig. 2, but it can be improved over than $0.05$ with the increased atomic density. In the following, we plot the right- and left-side reflectivities at $\phi =0$ ($\phi =\pi$) with lower atomic density in Fig. 4(c) [Fig. 4(d)] and higher atomic density in Fig. 4(e) [Fig. 4(f)], respectively. It is easy to see that the right- and left-side reflectivities are obviously nonreciprocal in the gain region. In addition, the nonreciprocity of reflection becomes more perfect and the reflectivity is greatly improved, with the increasing atomic density. In fact, the probe beam can be amplified to varying degrees in the whole relative phase space with proper detuning $\Delta _{p}$ [see Fig. 2(b)]. This means that the probe gain is sensitive to the relative phase $\phi$. It is then justified that the unidirectional reflection can be well amplified and modulated by the relative phase $\phi$, with a sufficiently large atomic density (e.g. $N_{0}\geq 5\times 10^{11}$ cm$^{-3}$, it is easy to satisfy in the cold atomic sample).

 

Fig. 4. Imaginary part of susceptibility Im$(\chi _{p})\;v.s.$ detuning $\Delta _{p}$ with $G_{c}^{0}=25$ MHz in (a) and (b) for $N_{0}=2\times 10^{11}$ cm$^{-3}$ (blue line and circle) and $N_{0}=5\times 10^{11}$ cm$^{-3}$ (black line and triangle); the reflectivities of left-side $R^{l}$ and right-side $R^{r}$ in (c) and (d) with $N_{0}=2\times 10^{11}$ cm$^{-3}$, in (e) and (f) with $N_{0}=5\times 10^{11}$ cm$^{-3}\;v.s.$ detuning $\Delta _{p}$, with the linear variation of coupling field $G_{c}(x)$ same as Fig. 3. For $\phi =0$ in (a), (c) and (e) and $\phi =\pi$ in (b), (d) and (f), respectively. Other parameters are the same as Fig. 2.

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It is necessary to emphasize the origin of the phase sensitivity of the unidirectional reflection. The direct transition $\left \vert 3\right \rangle \;{\rightarrow }\;\left \vert 1\right \rangle$ ($\left \vert 1\right \rangle \;{\rightarrow }\;\left \vert 3\right \rangle$) only can take place in one way, and the emitted (absorbed) photons have identical initial phases of $\phi _{p}$, this leads the gain (absorption) coefficient Im$\chi _{p}$ does not depend on the laser beam phases $\phi _{p}$ and $\phi _{c}$ without microwave field. When the microwave field is included in this atomic system, there is an indirect transition $\left \vert 3\right \rangle \;{\rightarrow }\;\left \vert 2\right \rangle \;{\rightarrow }\;\left \vert 1\right \rangle$ ($\left \vert 1\right \rangle \;{\rightarrow }\;\left \vert 2\right \rangle \;{\rightarrow }\;\left \vert 3\right \rangle$) with the initial phases $\phi _{c}+\phi _{d}$, except the direct transition. The two gain or absorption channels interfere with one another and the total gain or absorption amplitude is the sum of two distinct ways. Therefore, the probe gain on phase difference $\phi = \phi _{c}+\phi _{d}-\phi _{p}$ is the consequence of interference between two competing transitions, which can directly modulate the unidirectional reflection amplification.

We then argue the modulation of unidirectional reflection amplification by coupling detuning $\Delta _{c}$. As can be seen from the results in Fig. 3(c), the second unidirectional reflection band is slightly lower than the first one at $\phi =-\pi /2$ in the resonance of coupling field. In further, we plot the reflectivities $v.s.\;{\Delta }_{p}$ with different $\Delta _{c}$ in Figs. 5(a) and  5(b). It is noteworthy that the second unidirectional reflection band decreases rapidly and moves to the large detuning with the increasing $\Delta _{c}$, which is clearly shown by compared with Fig. 3(c). Then, Fig. 5(c) depicts the imaginary part of susceptibility $v.s.\;{\Delta }_{c}$ and $\Delta _{p}$, it exhibits that there are two gain regions of probe field in a large coupling detuning $\Delta _{c}$. However, one of the gain peaks is significantly reduced or even disappeared with the increased $\left \vert {\Delta }_{c}\right \vert$. Judging from this, we can achieve manipulation of monochromatic unidirectional reflection amplification to two-color, by modulating $\Delta _{c}$ with proper relative phase $\phi$.

 

Fig. 5. The reflectivities on the left-side $R^{l}$ and right-side $R^{r}\;v.s$. detuning $\Delta _{p}$, corresponding to the coupling detuning $\Delta _{c}=10$ MHz and $\Delta _{c}=20$ MHz in (a) and (b) with the linear variation of coupling field $G_{c}(x)$ same as Fig. 3; The imaginary part of susceptibility Im($\chi _{p}$) $v.s.$ both detuning $\Delta _{p}$ and $\Delta _{c}$ in (c) with $G_{c}^{0}=25$ MHz; For $\phi =-\pi /2$. Other parameters are the same as Fig. 2.

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Last but not least, we wonder whether a single unidirectional reflection band ($e.g.$, corresponding to one gain peak at $\phi =\pi /2$ and $\Delta _{c}=0$) can split double ones, by modulating $\Delta _{c}$. In Figs. 6(a) and 6(b), the unidirectional reflection band of increased amplitude moves to large probe detuning with the increased $\Delta _{c}$. Specially, the nonreciprocity is more perfect with large coupling detuning $\Delta _{c}$. Unfortunately, there is always only one gain region of probe field in the whole coupling detuning $\Delta _{c}$, which is clearly exhibited in Fig. 6(c). However, Increasing $\left \vert \Delta _{c}\right \vert$ can lead to obvious broadening of the gain peak, this directly increases the width of unidirectional reflection band.

 

Fig. 6. The reflectivities on the left-side $R^{l}$ and right-side $R^{r}\;v.s$. detuning $\Delta _{p}$, corresponding to the coupling detuning $\Delta _{c}=20$ MHz and $\Delta _{c}=50$ MHz in (a) and (b) with the linear variation of coupling field $G_{c}(x)$ same as Fig.3; The imaginary part of susceptibility Im($\chi _{p}$) $v.s.$ both detuning $\Delta _{p}$ and $\Delta _{c}$ in (c) with $G_{c}^{0}=25$ MHz; For $\phi =\pi /2$. Other parameters are the same as Fig. 2.

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4. Conclusion

In this paper, we have presented a investigation of the amplified unidirectional reflection in a homogeneous cold atomic medium, the atoms are driven into a closed three-level $\Delta$-type. Note, in such a system, the linear increasing of coupling intensity not only breaks the spatial symmetry of probe susceptibility but also effectively suppresses the reflection of one side, this directly leads to the nonreciprocity even the unidirectionality of the left- and right-side reflections of probe beam. Another noteworthy feature of this system is that the amplified reflection band can be achieved in the probe gain region due to the presence of microwave field constructing a closed loop transitions, and the various characteristics of the system become related to the relative phase $\phi$, which can periodically control the imaginary part of susceptibility. The numerical results show that the amplified single- and two-color unidirectional reflection band can be switched freely by adjusting the relative phase $\phi$, while the originally lower unidirectional reflection band can be obviously improved by increasing atomic density in the resonance of coupling field. In addition, two-color unidirectional reflection band gradually degenerates into monochrome one by improving $\left \vert \Delta _{c}\right \vert$, with the chosen relative phase $\phi$ rather than arbitrary phase. Meanwhile, the width, amplitude and position of unidirectional reflection band can be easily tuned by $\Delta _{c}$. Therefore, the amplified and tunable unidirectional reflection bands are able to realized easily. Furthermore, there is no need to consider whether the electric dipole moment $\mathbf {d}_{31}$ and $\mathbf {d}_{32}$ are parallel or vertical, and no incoherent pumping is needed to amplify probe beam in this scheme. Thus, it is very convenient and feasible in the experimental realization of phase control of the unidirectional reflection amplification. This simple and effective scheme to explore the amplified unidirectional reflection has potential application in developing high-performance diodes and isolators. Such non-reciprocal photonic device generally improves the designability of an overall system as it suppresses spurious interferences, interactions between different devices and undesired light routing.