Coherent feedback induced transparency

Bixuan Fan; Jiahui Ning; Min Xie; Cunjin Liu; Shengguo Guan

doi:10.1364/OE.404053

1. Introduction

The effect of electromagnetically induced transparency (EIT) of light was first proposed by Harris et al. [1,2] in the context of a three-level atomic system. Subsequently, EIT has been investigated intensively leading to it being realized experimentally in a variety of quantum systems [3]. In an EIT effect, a narrow transparency window appears inside a wide absorption spectrum, which is usually accompanied by a rapid and positive change in light dispersion and results in a reduction in the group velocity of light. Today, the EIT effect is one of most mature and promising mechanisms of slowing and storing light [4–7], with such mechanisms playing an essential role in quantum information processing.

Conventionally, EIT is based on destructive interference between the transitions of atomic quantum states and can be achieved only through strict experimental techniques. However, various classical analogues of EIT have recently emerged, such as coupled-resonator-induced transparency (CRIT) [8–10] in optical systems and plasmon-induced transparency (PIT) in metamaterials [11,12]. These EIT-like phenomena present many advantages, such as operating at room temperature, avoiding precise manipulation of quantum systems, and being more suitable for integration.

Coherent feedback [13–15] is a type of feedback in which a system can be controlled by feeding its output into its input coherently through another controlling system without being disturbed by measurements. This concept was first introduced by Lloyd [13], before Wiseman and coworkers studied coherent feedback systematically in the context of optical systems using a more general model [14]. Coherent feedback has been applied extensively to manipulate quantum systems and enhance various quantum effects, such as coherent feedback enhanced squeezing [16,17], enhanced entanglement [18], quantum error correction [19], and quantum logic gate formation [20].

Motivated by these previous studies, in this work we propose a new mechanism for inducing the transparency of light that is based on linear coherent feedback control, in which the vanishing absorption of light occurs owing to the destructive interference between the original optical field in a system cavity and its feedback field passing through a linear coherent feedback loop. We refer to this effect as coherent feedback induced transparency (CFIT). To illustrate this CFIT effect, we use a system comprising two linear optical resonators in which one resonator represents the system to be controlled and the other represents the controller. The reason for choosing such simple linear elements is to ensure that the transparency effect results from the feedback control, rather than nonlinear factors within the system. The structure of the model is similar to that of CRIT in that both utilize two optical resonators. The key difference is that for CRIT the interaction between the two resonators is mutual, while for CFIT the interaction is unidirectional, i.e., the system resonator $\rightarrow$ the control resonator $\rightarrow$ the system resonator. Therefore, the forward and backward coupling processes can be tuned separately, which offers more flexible manipulation of the light transmission properties. By using the SLH formulism [21–23], we provide analytical expressions for the transmission properties of the system. We show that the transparency effect of light occurs for a large range of parameter regime. Interestingly, we find that, in our model, total transparency of light can be maintained while the group velocity is greatly reduced. This represents a distinct advantage over typical EIT or CRIT, in which the reduction of the group velocity is usually at the cost of decreased transmission rate. Moreover, we find that perfect transparency occurs when the linear coherent feedback loop contains a phase shifter as its sole element, whose structure is simpler than most EIT-like schemes.

2. Model and theory

As shown in Fig. 1(a), the system under consideration is composed of a linear optical cavity (the system to be controlled) and its coherent feedback control loop. The system cavity has two input and output channels, with two decay rates, $\kappa _{1}$ and $\kappa _{2}$. To ensure that the light transparency effect in our system is not due to nonlinearity, the controller here is chosen to be a linear cavity with decay rate $\kappa _c$. The signal field is injected into the system through channel 1 (with decay rate $\kappa _{1}$) and the feedback signal, via the controller, returns to the system cavity through channel 2 (with decay rate $\kappa _{2}$).

Fig. 1. (a) Schematic for coherent feedback induced transparency (CFIT). The coupling between the system cavity and controller in the feedback loop is unidirectional; therefore, the forward and backward coupling strengths can be manipulated separately. (b) Diagram for the input-output relation in our model. S denotes the system cavity, C denotes the controller, and BS denotes the beam splitter. In our analysis, the controller is a linear cavity or phase shifter. $E_{in}$ and $\hat {a}_{in,1}$ are the coherent and quantum fluctuation parts of the input to the system, respectively. $\hat {a} _{in,2}$ is the vacuum input to the dark port of the beam splitter.

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Owing to its effectiveness in treating classical and quantum networks with feedback and feedforward, we used the SLH quantum network formulism to establish our model. The SLH triples involved in our feedback network are

(1)$$\begin{aligned}G_{\mathrm{in}} =(1,E_{\mathrm{in}},0), \end{aligned}$$

(2)$$\begin{aligned}G_{a1} =(I,\sqrt{\kappa _{1}}\hat{a},\Delta _{a}\hat{a}^{\dagger }\hat{a}), \end{aligned}$$

(3)$$\begin{aligned}G_{a2} =(I,\sqrt{\kappa _{2}}\hat{a},0), \end{aligned}$$

(4)$$\begin{aligned}G_{\mathrm{BS}} &=\left( \left[ \begin{array}{cc} \sqrt{1-\eta ^{2}} & -\eta \\ \eta & \sqrt{1-\eta ^{2}} \end{array} \right] ,0,0\right), \end{aligned}$$

(5)$$\begin{aligned}G_{c} =(I,\sqrt{\kappa _{c}}\hat{c},\Delta _{c}\hat{c}^{\dagger }\hat{c}), \end{aligned}$$

(6)$$\begin{aligned}\mathbb{I}_{1} =(1,0,0), \end{aligned}$$

where the operators $\hat {a}$ $(\hat {a}^{\dagger })$ and $\hat {c}$ ($\hat {c} ^{\dagger })$ are the annihilation (creation) operators of the system cavity and control cavity, respectively. The cavity frequencies ($\omega _a$ and $\omega _c$) are rotated at the frequency of the input field ($\omega _{\mathrm { in}}$); therefore, the frequency detunings are $\Delta _{a}=\omega _{a}-\omega _{\mathrm {in}}$ and $\Delta _{c}=\omega _{c}-\omega _{\mathrm {in} }$. $G_{\mathrm {in}}$ represents the coherent light field applied to the system, $G_{a1}$ ($G_{a2}$) represents the mirror with decay rate $\kappa _{1}$ ($\kappa _{2}$) of the system cavity, $G_{\mathrm {BS}}$ represents the beam splitter with transmission coefficient $\eta$, $G_{c}$ represents the control cavity, and $\mathbb {I}_{1}$ is a padding component for describing the simple passthrough from an input to an output. Then, we can derive the $SLH$ triple for the entire network $G$:

(7)$$\begin{aligned}G &=(G_{a2}\boxplus \mathbb{I}_{1})\triangleleft (G_{c}\boxplus \mathbb{I} _{1})\triangleleft G_{\mathrm{BS}}\triangleleft (G_{a1}\boxplus \mathbb{I} _{1})\triangleleft (G_{\mathrm{in}}\boxplus \mathbb{I}_{1})\\ &=(\hat{S},\hat{L},\hat{H}), \end{aligned}$$

where

(8)$$\begin{aligned}\hat{S} &=\left[ \begin{array}{cc} \sqrt{1-\eta ^{2}} & -\eta \\ \eta & \sqrt{1-\eta ^{2}} \end{array} \right] , \end{aligned}$$

(9)$$\begin{aligned}\hat{L} &=\left[ \begin{array}{c} \sqrt{\kappa _{2}}\hat{a}+\sqrt{\kappa _{c}}\hat{c}+\sqrt{1-\eta ^{2}}(\sqrt{ \kappa _{1}}\hat{a}+E_{\mathrm{in}}) \\ \eta (\sqrt{\kappa _{1}}\hat{a}+E_{\mathrm{in}}) \end{array} \right] , \end{aligned}$$

(10)$$\begin{aligned}\hat{H} &=\Delta _{a}\hat{a}^{\dagger }\hat{a}+\Delta _{c}\hat{c}^{\dagger }\hat{c }+\frac{\sqrt{\kappa _{1}}E_{\mathrm{in}}}{2i}(\hat{a}^{\dagger }-\hat{a})+ \frac{1}{2i}(\sqrt{\kappa _{2}\kappa _{c}}-\sqrt{1-\eta ^{2}}\sqrt{\kappa _{1}\kappa _{c}})(\hat{a}^{\dagger }\hat{c}-\hat{c}^{\dagger }\hat{a})\\ &\quad +\frac{\sqrt{1-\eta ^{2}}E_{\mathrm{in}}}{2i}[\sqrt{\kappa _{2}}(\hat{a} ^{\dagger }-\hat{a})-\sqrt{\kappa _{c}}(\hat{c}^{\dagger }-\hat{c})]. \end{aligned}$$

In the calculation above, we have used the cascading rule $G_{2}\triangleleft G_{1}=\left ( \hat {S}_{2}\hat {S}_{1},\hat {L}_{2}+\hat {S} _{2}\hat {L}_{1},\hat {H}_{1}+\hat {H}_{2}+\frac {1}{2i}(\hat {L}_{2}^{\dagger }\hat { S}_{2}\hat {L}_{1}-\hat {L}_{1}^{\dagger }\hat {S}_{2}^{\dagger }\hat {L}_{2})\right )$ and the concatenation product $G_{1}\boxplus G_{2}=\left ( \left [ \begin {array}{cc} \hat {S}_{1} & 0 \\ 0 & \hat {S}_{2} \end {array} \right ] ,\left [ \begin {array}{c} \hat {L}_{1} \\ \hat {L}_{2} \end {array} \right ] ,\hat {H}_{1}+\hat {H}_{2}\right ).$

Following the theory outlined in [23], we can obtain the equations of motion for the operators corresponding to the $SLH$ triple:

(11)$$\begin{aligned}\frac{d\hat{a}}{dt} &=-i\Delta _{a}\hat{a}-\frac{\kappa _{1}+\kappa _{2}}{2} \hat{a}-\sqrt{\kappa _{1}\kappa _{2}}\sqrt{1-\eta ^{2}}\hat{a}-\sqrt{\kappa _{2}\kappa _{c}}\hat{c}-(\sqrt{\kappa _{1}}+\sqrt{\kappa _{2}}\sqrt{1-\eta ^{2}})E_{\mathrm{in}}\\ &-(\sqrt{\kappa _{1}}+\sqrt{\kappa _{2}}\sqrt{1-\eta ^{2}})\hat{a}_{\mathrm{ in},1}+\eta \sqrt{\kappa _{2}}\hat{a}_{\mathrm{in},2}, \end{aligned}$$

(12)$$\begin{aligned}\frac{d\hat{c}}{dt} &=-i\Delta _{c}\hat{c}-\frac{\kappa _{c}}{2}\hat{c}- \sqrt{\kappa _{1}\kappa _{c}}\sqrt{1-\eta ^{2}}\hat{a}-\sqrt{\kappa _{c}} \sqrt{1-\eta ^{2}}E_{\mathrm{in}}-\sqrt{\kappa _{c}}\sqrt{1-\eta ^{2}}\hat{a} _{\mathrm{in},1}\\&+\eta \sqrt{\kappa _{2}}\hat{a}_{\mathrm{in},2}. \end{aligned}$$

Here, $\hat {a}_{\mathrm {in},1}$ and $\hat {a}_{\mathrm {in},2}$ are assumed to be vacuum inputs. Then, we can obtain the steady-state intracavity amplitudes:

(13)$$\begin{aligned}\left\langle \hat{a}\right\rangle &=\frac{\sqrt{\kappa _{2}}\kappa _{c} \sqrt{1-\eta ^{2}}-(\sqrt{\kappa _{1}}+\sqrt{\kappa _{2}}\sqrt{1-\eta ^{2}} )\left( i\Delta _{c}+\frac{\kappa _{c}}{2}\right) }{\left( i\Delta _{a}+ \frac{\kappa _{1}+\kappa _{2}}{2}+\sqrt{\kappa _{1}\kappa _{2}}\sqrt{1-\eta ^{2}}\right) \left( i\Delta _{c}+\frac{\kappa _{c}}{2}\right) -\sqrt{\kappa _{1}\kappa _{2}}\kappa _{c}\sqrt{1-\eta ^{2}}}E_{\mathrm{in}}, \end{aligned}$$

(14)$$\begin{aligned}\left\langle \hat{c}\right\rangle &= -\frac{\sqrt{\kappa _{1}\kappa _{c}(1-\eta ^{2})}\left\langle \hat{a}\right\rangle -\sqrt{\kappa _{c}(1-\eta ^{2})}E_{\mathrm{in}}}{i\Delta _{c}+\kappa _{c}/2}. \end{aligned}$$

According to the input-output relation $\left [ \begin {array}{c} \hat {a}_{\mathrm {out},1} \\ \hat {a}_{\mathrm {out},2} \end {array} \right ] =\hat {S}\left [ \begin {array}{c} \hat {a}_{\mathrm {in},1} \\ \hat {a}_{\mathrm {in},2} \end {array} \right ] +\hat {L}$, we obtain the steady-state amplitude of the light field at the output of the beam splitter:

(15)$$\begin{aligned}\left\langle \hat{a}_{\mathrm{out},2}\right\rangle &= \eta (\sqrt{\kappa _{1}}\left\langle \hat{a}\right\rangle +E_{\mathrm{in}}) \end{aligned}$$

(16)$$\begin{aligned}&= \eta \frac{\left( i\Delta _{a}+\frac{\kappa _{2}-\kappa _{1}}{2}\right) \left( i\Delta _{c}+\frac{\kappa _{c}}{2}\right) }{\left( i\Delta _{a}+\frac{ \kappa _{1}+\kappa _{2}}{2}+\sqrt{\kappa _{1}\kappa _{2}}\sqrt{1-\eta ^{2}} \right) \left( i\Delta _{c}+\frac{\kappa _{c}}{2}\right) -\sqrt{\kappa _{1}\kappa _{2}}\kappa _{c}\sqrt{1-\eta ^{2}}}E_{\mathrm{in}}. \end{aligned}$$

Then, we can obtain the transmission coefficient $t_{f}$, transmission rate $T$, and phase $\phi$ of the output light [9]:

(17)$$\begin{aligned}t_{f} =\frac{\left\langle \hat{a}_{\mathrm{out,}2}\right\rangle }{E_{ \mathrm{in}}}=\frac{\eta (C+iD)}{A^{2}+B^{2}}, \end{aligned}$$

(18)$$\begin{aligned}T =\mathrm{abs}[t_{f}]^{2}=\frac{\eta ^{2}(C^{2}+D^{2})}{(A^{2}+B^{2})^{2}} , \end{aligned}$$

(19)$$\begin{aligned}\phi =\mathrm{arg}[t_{f}]=\tan ^{-1}\frac{D}{C}, \end{aligned}$$

where

(20)$$\begin{aligned}A =\kappa _{c}\left( \kappa _{2}+\kappa _{1}\right) /4-\Delta _{a}\Delta _{c}-\frac{\kappa _{c}}{2}\sqrt{\kappa _{1}\kappa _{2}\left( 1-\eta ^{2}\right) }, \end{aligned}$$

(21)$$\begin{aligned}B =\frac{ \Delta _{c}\left( \kappa _{2}+\kappa _{1}\right) +\Delta _{a}\kappa _{c}} {2}+\Delta _{c}\sqrt{\kappa _{1}\kappa _{2}\left( 1-\eta ^{2}\right) }, \end{aligned}$$

(22)$$\begin{aligned} C &=A\left( \frac{\kappa _{c}\left( \kappa _{2}-\kappa _{1}\right) }{4} -\Delta _{c}\Delta _{a}\right) +B\left( \frac{\kappa _{c}}{2}\Delta _{a}+ \frac{\Delta _{c}}{2}\left( \kappa _{2}-\kappa _{1}\right) \right) ,\\ D &=B\left(\Delta _{c}\Delta _{a}-\frac{\kappa _{c}\left( \kappa _{2}-\kappa _{1}\right) }{4}\right)+A\left( \frac{\kappa _{c}}{2}\Delta _{a}+\frac{\Delta _{c}}{2} \left( \kappa _{2}-\kappa _{1}\right) \right) . \end{aligned}$$

An additional important quantity in an EIT-like effect is the group delay [24]. It can be calculated by examining the differentiation of the phase shift as a function of the detuning frequency. Thus, we define the group delay as

(23)$$\tau _{g}=\frac{\partial \phi }{\partial \Delta _{a}}=\frac{D^{\prime }C-C^{\prime }D}{\left( C^{2}+D^{2}\right) ^{2}}C^{2}.$$

3. Results and discussions

The coherent feedback induced transparency (CFIT) effect of light can now be analyzed. For simplicity, we assume that the beam splitter has equal transmission and reflection coefficients ($\eta =1/\sqrt {2}$) and that the control cavity has the same resonance frequency as the system cavity, i.e., $\Delta _{a}=\Delta _{c}$. Under these assumptions, Fig. 2(a) shows the light transmission rate in the $\Delta _{a}-\kappa _{2}$ parameter plane. A transparency window with near-unity transmission rate at resonance ($\Delta _{a}=0$) is observed for most values of $\kappa _{2}$ except in the vicinity of $\kappa _{2}/\kappa _{1}=1$, where the light is completely absorbed. For a large range of the parameter $\kappa _2$, we can recognize the characteristic feature of EIT, i.e., a transparency window within a broad absorption profile. This transparency effect is caused by destructive interference between the original cavity field and feedback field. For the other values of $\kappa _2/\kappa _1$, the light is partially transmitted, with two examples ($\kappa _2/\kappa _1=0.5, 2$) shown in Figs. 2(c₂) and (c₄). Due to the presence of the $50:50$ beam splitter, when the incident light is significantly detuned from the cavity resonance frequency, the transmission rate approaches $\frac {1}{2}$, differing from that in typical EIT.

Fig. 2. (a) Light transmission rate in the $\Delta _{a}- \kappa _2$ parameter plane with the coherent feedback loop. (b) Light transmission rate in the $\Delta _{a}- \kappa _2$ parameter plane without the coherent feedback loop. ($c_1-c_5$) the transmission spectra corresponding to different values of $\kappa _2$ $(\kappa _2/\kappa _1=0.17,0.5, 1,2, 5.8)$. The solid curves are the results for the situation with feedback, while the dashed curves are the results for the situation without feedback. ($d_1-d_5$) and ($e_1-e_5$) are the corresponding phase $\phi$ (dispersion) and group delay $\tau _g$ of the output light, respectively. $\kappa _c$ is fixed at $0.3\kappa _1$ in this figure.

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To determine the conditions required to realize a perfect transparency effect qualitatively, we derived the transmission rate and group delay at the center of the transparency window ( $\Delta _{a}=\Delta _{c}=0$ ):

(24)$$T=\eta ^{2}\frac{\left( \kappa _{2}-\kappa _{1}\right) ^{2}}{\left( \kappa _{2}+\kappa _{1}-2\sqrt{\kappa _{1}\kappa _{2}\left( 1-\eta ^{2}\right) } \right) ^{2}}.$$

By solving $\frac {\partial T}{\partial (\kappa _{2}/\kappa _{1})}=0$, we know that the transmission rate $T$ is maximized at $\kappa _{2}/\kappa _{1}=3\pm \sqrt {8}\approx 0.17,5.8$, and the maximal value is $T=1$, indicating a perfect transparency effect. In addition, this expression shows that $T=0$ for $\kappa _{2}/\kappa _{1}=1.$ These conclusions are supported strongly by the transmission spectra in Figs. 2(c₁), (c₃), and (c₅).

For comparison, we also show the light transmission properties when there is no feedback loop. In such a situation, the equation of motion for the system cavity mode is $\frac {d\hat {a}}{dt}=-i\Delta _{a}\hat {a}-\frac {\kappa _{1}+\kappa _{2}}{2}\hat {a}-\sqrt {\kappa _{1}}E_{\mathrm {in}}$ and we can easily derive the transmission rate

(25)$$T=\eta ^{2}\frac{\Delta _{a}^{2}+\left( \frac{\kappa _{2}-\kappa _{1}}{2} \right) ^{2}}{\Delta _{a}^{2}+\left( \frac{\kappa _{1}+\kappa _{2}}{2} \right) ^{2}}.$$

By inspecting Eq. (25), we observe that the transmission rate in the case of no feedback can never reach 1. The corresponding transmission spectra are presented in Figs. 2(b) and 2(c₁)-(c₅) using dashed curves. Absorption windows are evident in all curves, which confirms that the transparency effect in our system is indeed induced by coherent feedback.

Next, we considered the dispersion (phase relation) and group delay of the output field of the system. As shown in Figs. 2(d₁)-(d₅), at $\kappa _{2}/\kappa _{1}=0.17,5.8$, the dispersion curves exhibit different shapes, although in both cases the perfect transparency of light occurs. When $\kappa _{2}/\kappa _{1}=5.8$, the dispersion curve demonstrates a similar profile as in a characteristic EIT, while the dispersion relation for $\kappa _{2}/\kappa _{1}=0.17$ reveals a profile that shows a similar slope at the center of transparency window but is different in other aspects. By comparing the group delay in Figs. 2(e₁)-(e₅), we can see that the group delays are positive for the cases of light transparency (both full and partial transparency) and negative for the case of light absorption. This is consistent with the Kramers–Kronig relation, which states that the dissipation and dispersion are related. Furthermore, we can see that the peak value of the group delay at $\kappa _{2}/\kappa _{1}=0.17$ is slightly greater than that at $\kappa _{2}/\kappa _{1}=5.8$.

Figure 3 shows the responses of the system to the variation of $\kappa _{c}$, i.e., the decay rate of the control cavity. The results demonstrate that the transparency window broadens as $\kappa _{c}$ increases. This indicates that the role $\kappa _{c}$ plays in CFIT is similar to that of the coupling strength in CRIT, in which a greater coupling strength results in a wider transparency window. This can be explained using the Hamiltonian of the system [Eq. (1)], in which $\sqrt { \kappa _{c}}$ is a factor in both the forward and backward coupling terms; therefore, its function is naturally similar to that of the coupling strength in CRIT. Notably, the transmission rate at the center of transparency window remains at unity while the group velocity is substantially suppressed. To explain this, we express the group velocity at the center of the transparency window:

(26)$$\tau _{g}=\frac{\kappa _{1}\kappa _{c}+\sqrt{\kappa _{1}\kappa _{2}(1-\eta ^{2})}\left( 2\kappa _{1}-2\kappa _{2}-\kappa _{c}\right) }{\kappa _{c}\left( \frac{ \kappa _{2}+\kappa _{1}} {4}-\frac{1}{2}\sqrt{\kappa _{1}\kappa _{2}(1-\eta ^{2})}\right) \left( \kappa _{2}-\kappa _{1}\right) }.$$

By comparing Eq. (24) and Eq. (26), it is clear that $\tau _{g}$ is dependent on $\kappa _{c}$, while $T$ is independent of $\kappa _{c}$, which means that one can control the group delay freely without reducing the transmission rate. Figure 3(c) shows the peak value of group velocity $\tau _{g}$ as a function of $\kappa _{c}$, revealing that the smaller the decay rate $\kappa _{c}$, the larger the group delay. This result indicates that, in principle, an arbitrarily large group delay can be achieved by choosing a sufficiently small $\kappa _{c}$. Of course, a smaller decay rate requires a control cavity of higher quality factor; therefore, using current technology, it is possible to choose a minimum $\kappa _{c}$ and achieve an impressive group delay for the incident light.

Fig. 3. (a) Transmission spectra for light at three different values of $\kappa _c$; (b) group delay of light corresponding to (a); (c) peak value of the group delay as a function of $\kappa _c$. $\kappa _2$ is fixed at $5.8\kappa _1$ in this figure.

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To generalize our proposal for CFIT, we also investigated the light transmission properties of our system in the case where a phase shifter represented the sole element of the feedback loop. This can be realized by inserting a phase shifter or adjusting the optical path length in the feedback loop (i.e., using a long fiber). In this case, the $SLH$ triple for the controller is $G_{c}=(e^{i\theta },0,0)$, with $\theta$ denoting the phase shift in the feedback loop. In this case, we derive the transmission rate of the incident light as

(27)$$T=\eta ^{2}\left\vert \frac{i\Delta _{a}+\frac{1}{2}\left( \kappa _{2}-\kappa _{1}\right) }{i\Delta _{a}+\frac{1}{2}\left( \kappa _{2}+\kappa _{1}\right) +e^{i\theta }\sqrt{\kappa _{1}\kappa_2}\sqrt{1-\eta ^{2}}}\right\vert ^{2}.$$

Figure 4 shows light transmission spectra with different phase shifts ($\theta$) and different cavity decay rate ratios ($\kappa _2/\kappa _1$). In the absence of phase shift ($\theta =0$), the incident light is always absorbed at resonance and no transparency effect occurs for all $\kappa _2/\kappa _1$. In contrast, for $\theta =\pi$, the perfect transparency effect is again observed for $\kappa _2/\kappa _1=0.17, 5.8$. This transparency results from the destructive interference between the original cavity field and feedback field with a $\pi$ phase shift. The profiles of the transmission spectra here differ from the previous case in that the transparency window here is not buried inside a wider absorption spectrum; instead, it arises directly from the line for which $T=1/2$. For the other phase shift angles, e.g., $\pi /2$, the transmission spectrum is deformed relative to the standard absorption dip or transparency peak; it contains a transparency part and an absorption part at different frequencies (or detunings) of the incident light. It should be noted that $\kappa _2/\kappa _1=1$ is a special case, in which the light is absorbed for all phase shift angles, consistent with Eq. (27).

Fig. 4. Transmission spectra, corresponding to the case in which a phase shifter acts as a feedback controller, for different values of $\kappa _2$ and different phase shift angles. (a) $\kappa _2/\kappa _1=0.17$; (b) $\kappa _2/\kappa _1=1$; (c) $\kappa _2/\kappa _1=5.8$.

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4. Conclusion

We have proposed a mechanism for inducing transparency that we describe as the coherent feedback induced transparency (CFIT). By using the SLH formulism, we derived analytical expressions for the transmission properties of the light propagating through the system. The results show novel transparency behaviors, for instance, the group velocity of light can be substantially reduced while preserving unity transmission rate, because the feedback control loop allows the forward and backward coupling strengths between the system and controller in the feedback loop to be manipulated separately.

Compared to EIT and OMIT, CFIT employs a simpler structure, consisting of linear optical components only. The light transparency effect in CFIT results from the destructive interference between the original cavity field and the feedback field, which can occur in both quantum and classical situations and therefore it does not require the precise control of quantum systems. Compared to CRIT, the manipulation of the transparency in CFIT is more flexible owing to the unidirectional coupling in the feedback loop, enable us to separately manipulate the transmission rate and group velocity of light. Our results offer a simple and effective way to preserve high transmission rate of light while the group velocity is substantially suppressed, which may help to build low-loss slow-light devices. Besides, our model has the potential to be extended to light transparency effect induced by a coherent feedback loop containing a nonlinear element, in which we expect more rich physics.

Funding

National Natural Science Foundation of China (11504145, 11664014, 11964014); Natural Science Foundation of Jiangxi Province (20161BAB201023, 20161BAB211013); Science and Technology Projects of Jiangxi Provincial Department of Education (GJJ191683).

Acknowledgment

The authors thank Dr Zhenglu Duan for his helpful discussions.

Disclosures

The authors declare no conflicts of interest.

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Coherent feedback induced transparency

Abstract

1. Introduction

2. Model and theory

3. Results and discussions

4. Conclusion

Funding

Acknowledgment

Disclosures

References

Cited By

Figures (4)

Equations (27)

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