Mechanically controllable nonreciprocal transmission and perfect absorption of photons

Nan-nan Zhou; Li-Qiang Zhang; Chang-Shui Yu

doi:10.1364/OE.460158

1. Introduction

Photons, as information carriers in quantum information and quantum communication, have been widely investigated in controlling various photon absorption [1–11], scattering [12,13], and localization [14,15] for a variety of fundamental studies and practical applications [16–19]. Photon perfect absorption originated from the interference between the transmitted field and the input field [16,17,20,21] has been addressed in many kinds of systems including the cavity quantum electrodynamics (CQED) and the cavity optomechanical system [4–10]. Photon perfect absorption, almost always accompanied with optical bistability or even multistability, provides the possibility of controlling light by light, has many wide potential applications in optical transistors, memory elements, all-optical switches, all-optical storage and logic circuits [22–31].

Nonreciprocity of photon transmission is another interesting effect that allows light to transmit from one side but to be prohibited from the other side. It has shown potential applications in signal processing and invisible sensing [32]. Recently, there is increasing interests in the connection of the nonreciprocity with other quantum effects such as nonreciprocal entanglement [33,34], nonreciprocal photon blockade [35,36], nonreciprocal high-order sidebands [37,38] and so on. Nonreciprocal optical devices have been realized in various systems including optomechanical systems [39,40], Kerr resonators [41,42], spinning optical resonator [33,35,36,43] etc. The experiment confirmed that optical isolation can achieve $99.6\%$ in spinning resonator [43], which provided strong support for spinning non-reciprocity. In Ref. [44], they utilized a microscopic optical resonator coupled with one atom to achieve photon blockade and photons controlled transport.

In this paper, we utilize a spinning optical resonator system to study both the nonreciprocal transmission and the perfect photon absorption. We introduce an atomic ensemble to interact with the system, which enables perfect photon absorption. We give the exact conditions of perfect photon absorption and exhibit the association with optical bistability behavior. It is especially important that our system can realize the amplified unidirectional photon transmission by controlling the spinning detuning or spinning velocity. That is, with the assistance of the spinning detuning, our system can realize not only the nonreciprocal photon propagation with amplifications but also the simultaneous block of light transmission from both sides. In this sense, our system provides new insight into the design of optical devices. The remaining of this paper is organized as follows: In Section 2, we briefly introduce the physical model. In Section 3, we show the perfect photon absorption. In Section 4, we show the mechanically controllable effects on nonreciprocal transmission, amplification, and absorption. We give the conclusions in section 6.

2. Physical model

The system we consider here consists of a spinning optical resonator coupled to an atomic ensemble. The input fields $a_{in}$ and $b_{in}$ with the same frequency $\omega _l$ are coupled to the resonator via a tapered fiber. The schematic diagram is sketched in Fig. 1. The optical resonator is rotating with an angular velocity $\Omega$, which can lead to the rotation-dependent frequency shift. Due to the rotation, the cavity mode undergoes a Fizeau shift $\Delta _{F}$ [43,45,46] with $\Delta _{F}=\pm \frac {nr\Omega \omega _{c}}{c}(1-\frac {1}{n^2}-\frac {\lambda }{n}\frac {dn}{d\lambda })$, where $n$ is the refractive index, $r$ is the cavity radius, $\omega _{c}$ is the resonance frequency of the non-spinning resonator, $c$ is the speed of light in vacuum, and $\lambda$ is the wavelength of the external classical light. The dispersion term $dn/d\lambda$ is relatively small up to $1\%$ which characterizes the relativistic origin of Sagnac effect [45]. $\Delta _{F} > 0$ ($\Delta _{F} < 0$) indicates the light propagating against (along) the direction of the rotating cavity as depicted in Fig. 1. The rotation is not only the root of the non-reciprocal transmission of light but also the main mechanically controlling mechanism. The Hamiltonian $H$ of the whole system reads ($\hbar =1$ hereafter)

(1)$$\begin{aligned} H = &(\Delta_{L} +\Delta_{F})\hat{a}^{\dagger}\hat{a}+(\Delta_{L} -\Delta_{F})\hat{b}^{\dagger}\hat{b}+J(\hat{a}^{\dagger}\hat{b}+\hat{b}^{\dagger}\hat{a})+\delta S^{z}+g((\hat{a}+\hat{b})S^{+}+(\hat{a}^{\dagger}+\hat{b}^{\dagger})S^{-})\\ &+i\sqrt{2\kappa_{1}}a_{in}\hat{a}^{\dagger}+i\sqrt{2\kappa_{2}}b_{in}\hat{b}^{\dagger}-i\sqrt{2\kappa_{1}}a_{in}^{\dagger}\hat{a}-i\sqrt{2\kappa_{2}}b_{in}^{\dagger}\hat{b},\end{aligned}$$

where $\hat {a} (\hat {a}^{\dagger })$ and $\hat {b} (\hat {b}^{\dagger })$ are the annihilation (creation) operators of the cavity modes, $\kappa _{i}$ denote the leakage rates of the corresponding mode, $\Delta _{L}=\omega _{c}-\omega _{l}$ is the laser detuning from the cavity mode, $\delta =\omega _{e}-\omega _{l}$ is laser detuning from the atoms, $J$ is the coupled strength of two counter-propagating modes, and $S^{z}=\frac {1}{2}\sum _{i=1}^{N}(\sigma _{i}^{+}\sigma _{i}^{-}-\sigma _{i}^{-}\sigma _{i}^{+})$ and $S^{\pm }=\sum _{i=1}^{N}\sigma _{i}^{\pm }$ stand for the collective atomic operators with $\sigma _{i}^{+}=\vert e\rangle _{i}\langle g \vert$ , $\sigma _{i}^{-}=\vert g\rangle _{i}\langle e\vert$.

Fig. 1. (a) The schematic diagram of a rotating resonator coupled with atomic ensemble. (b) The top view of the system. The resonator is spinning clockwise (CW) or counterclockwise (CCW) at a fixed angular velocity $\Omega$. $a$ and $b$ denote the two counter-propagating (CW and CCW) whispering-gallery modes in the resonator. The left and right input lights are marked as $a_{in}$ and $b_{in}$, respectively. $a_{out}$ and $b_{out}$ denote the output light field.

Download Full Size | PDF

The dynamics of the system is described by the Heisenberg-Langevin equation, which directly leads to the following equations for the expectations of the relevant observables as

(2)$$\langle \dot{a} \rangle ={-}i(\Delta_{+}-i\kappa_{1})\langle a \rangle-iJ \langle b \rangle-ig\langle S^{-} \rangle+\sqrt{2\kappa_{1}} a_{in},$$

(3)$$\langle \dot{b} \rangle={-}i(\Delta_{-}-i\kappa_{2})\langle b \rangle-iJ \langle a \rangle-ig\langle S^{-} \rangle+\sqrt{2\kappa_{2}} b_{in},$$

(4)$$ \langle \dot{S}^{z} \rangle={-}\gamma\langle S^{z} \rangle -\frac{N\gamma}{2}+[ig \langle a ^{\dagger}+ b^{\dagger}\rangle \langle S^{-}\rangle+h.c.],$$

(5)$$\langle \dot{S}^{-} \rangle={-}i\delta \langle S^{-} \rangle + 2ig \langle a+ b \rangle \langle S^{z}\rangle -\frac{\gamma}{2}\langle S^{-} \rangle,$$

where $\Delta _{\pm }=\Delta _{L}\pm \Delta _{F}$ and $\gamma$ is the atomic line width. Here we mainly consider the mean values of the related operators, which dominate the absorption and transmissions of lights [7]. The equations about the fluctuations are given in the Appendix, which guarantees all the used parameters in the stability regions. In particular, we will directly solve $\langle S^z\rangle$ instead of using the low-excitation limit. With the mean-field approximation $\left \langle o^\prime o\right \rangle =\left \langle o^\prime \right \rangle \left \langle o\right \rangle$ ($o^\prime$, $o$ stand for the operators), one will obtain the steady-state solutions of the intra-cavity light field as

(6)$$\langle a \rangle =\frac{\sqrt{2\kappa_{2}}b_{in}[A({-}iJ)-2 g^2N(\gamma-2i\delta)]+\sqrt{2\kappa_{1}}a_{in}[A(\kappa_{2}+i\Delta_{-})+2 g^2N(\gamma-2i\delta)]}{W_{2}+i W_{1}},$$

(7)$$\langle b \rangle =\frac{\sqrt{2\kappa_{1}}a_{in}[A({-}iJ)-2 g^2N(\gamma-2i\delta)]+\sqrt{2\kappa_{2}}b_{in}[A(\kappa_{1}+i\Delta_{+})+2 g^2N(\gamma-2i\delta)]}{W_{2}+i W_{1}},$$

where

(8)$$W_{1}=A(\Delta_{-}\kappa_{1}+\Delta_{+}\kappa_{2}) + 2 g^2 N [ \gamma (\Delta_{-}+\Delta_{+} -2J)- 2\delta (\kappa_{1}+\kappa_{2})],$$

(9)$$W_{2}=A(J^2-\Delta_+ \Delta_-{+} \kappa_{1}\kappa_{2})+ g^2N [4 \delta (\Delta_{+}+\Delta_{-} - 2J) + 2\gamma (\kappa_{1}+\kappa_{2})],$$

(10)$$A=8 g^2 \vert\langle a \rangle + \langle b \rangle\vert^2+\gamma^2+4\delta^2.$$

In addition, the atomic operators in the steady-state regime can be solved as

(11)$$\langle S^{-} \rangle =\frac{-i g N(\langle a \rangle + \langle b \rangle)}{(\frac{\gamma}{2}+i\delta)(1+\frac{2g^2 \vert\langle a \rangle + \langle b \rangle\vert^2}{\frac{\gamma^2}{4}+\delta^2})},$$

and $\langle S^z\rangle$ can be directly obtained via Eq. (4). When no ambiguity is possible, hereafter $\langle \cdot \rangle$ as given in Eqs. (6,7,11) is specialized to represent the steady-state expectation value. In addition, compared with $a_{in}$ we suppose that $b_{in}$ will get an additional phase $\theta$ during input process for simplicity, i.e., $b_{in}=a_{in}e^{i\theta }$. Thus, one can easily get the nonlinear relation between $\langle a + b \rangle$ and $\vert a_{in} \vert ^2$ as follows,

(12)$$\langle a + b \rangle = \frac{ A (\sqrt{2 \kappa_{1}} a_{in}(\kappa_{2} + i(\Delta_{-} - J))+\sqrt{2 \kappa_{2}} a_{in}(\kappa_{1}+ i (\Delta_{+}-J))e^{i \theta}))}{W_{2} + i W_{1}}.$$

Equation (12) can induce a cubic equation about $\vert a + b \vert ^2$. It shows the nonlinear relation between special intra-cavity intensity $\vert a+b \vert ^2$ and the input light intensity, which is the root of the photon bistability and further induces the intriguing behaviors of light.

3. Perfect photon absorption

To study the two output light fields, we employ the standard input-output relation [47] for the steady states as

(13)$$ a_{out}=\sqrt{2\kappa_{1}} \langle a\rangle-a_{in},$$

(14)$$ b_{out}=\sqrt{2\kappa_{2}} \langle b\rangle-b_{in}.$$

With $\kappa _{1}=\kappa _{2}=\kappa$, we can easily get the transmission rate based on the output intensity as

(15)$$T_a=\frac{\vert a_{out} \vert^2}{\vert a_{in} \vert^2}=\vert \frac{2\kappa e^{i\theta}[A({-}iJ)-2 g^2N(\gamma-2i\delta)]+2\kappa [A(\kappa+i\Delta_{-})+2 g^2N(\gamma-2i\delta)]}{W_{2}+i W_{1}} -1\vert^2,$$

(16)$$T_b=\frac{\vert b_{out} \vert^2}{\vert b_{in} \vert^2}=\vert \frac{2\kappa [A({-}iJ)-2 g^2N(\gamma-2i\delta)]+2\kappa e^{i\theta}[A(\kappa+i\Delta_{+})+2 g^2N(\gamma-2i\delta)]}{W_{2}+i W_{1}} - e^{i\theta}\vert^2 .$$

For simplicity, we first let $\theta =0$, which can eliminate the phase $\theta$ in Eq. (16) and

(17)$$\langle a + b \rangle = 2 \frac{\sqrt{2 \kappa} a_{in} A (\kappa + i(\Delta_{L} - J))}{W_{2} + i W_{1}}.$$

When perfect photon absorption occurs, the input fields are totally absorbed, which implies $a_{out}=b_{out}=0$. Thus one can easily get the perfect absorption condition

(18)$$\Delta_{F}=0, J + \Delta_{L} = 2\delta \frac{ \kappa}{\gamma}, A=4 g^2N\frac{ \gamma}{\kappa},$$

which indicates perfect photon absorption requires the system to be static. In particular, the perfect photon absorption condition reveals the importance of the atomic ensemble due to $A>0$. In addition, Eq. (18) determines a unique perfect absorption point with $\left \vert \langle a+b\rangle \right \vert ^2=\frac {4 g^2N \gamma -(\gamma ^2+4\delta ^2)\kappa }{16g^2\kappa }$, with which one can quite easily solve the explicit form of $\vert a_{in}\vert ^2$ from Eq. (17). An intuitive illustration of the perfect absorption is given in Fig. 2, where we plot the output light intensity with the input light intensity. With the current parameters, at the perfect absorption point, $\vert a_{in} \vert ^2=76840.41$ in Fig. 2. A roughly physical understanding of the perfect absorption is that the atomic ensemble adds interference channels to the system, which leads to destructive interference between the two input fields. In Fig. 2, besides the perfect photon absorption point, one can also find that the output light intensities of the CW and CCW modes are symmetric for $\Delta _{F}=0$, i.e., $\vert a_{in}\vert ^2=\vert b_{in} \vert ^2$, and there exists the bistability in the system. It is obvious that the bistability originates from the nonlinearity of the system again induced by the atomic ensemble.

Fig. 2. The output light intensity $\vert y_{out} \vert ^2$ as a function of $\vert a_{in} \vert ^2$. Here $y$ denotes $a$ or $b$, $\gamma / \kappa = 1.775$, $\Delta _{L}/ \kappa = 3.263$, $\Delta _{F} = 0$, $g^2N/\kappa = 126$, $\delta =10$, $J / \kappa = 8$, $g=0.02/\kappa$.

Download Full Size | PDF

To detailedly study the influence of atomic coupling, we demonstrate the transmission rates with different coupling parameters $g$ in Fig. 3. We are mainly interested in perfect photon absorption, so all the parameters but the free one $\Delta _L$ are subject to the perfect photon absorption. It is worth noting that the different marks with the same color mean the appearance of bistability. The perfect photon absorption occurs at $\Delta _L=3.26$ for $g=0.02$, which is consistent with Fig. 2, and the transmission rates will be suppressed when $g$ deviates from $0.02$, but $g \rightarrow 0.02$ leads to the bistability. $g=0$, i.e., no atomic coupling, corresponds to the full transmission. Figure 3 further approves the key role played by the coupling of the atomic ensemble in perfect photon absorption and bistability.

Fig. 3. The transmission $T$ as a function of $\Delta _{L} /\kappa$ under the different coupling parameter $g$. Here $\gamma / \kappa = 1.775$, $\Delta _{L}/ \kappa = 3.26$, $\Delta _{F} = 0$, $g^2N/\kappa = 126$, $\delta =10$, $J / \kappa = 8$, $\vert a_{in}\vert ^2=76840.41$. The different marks of the same color indicate the bistability.

Download Full Size | PDF

4. Mechanically controlled transmission

To show the mechanical controllability, i.e., the effects of the spinning on the transmission, we will study the transmission rates depending on $\Delta _{F}$ (or the velocity $\Omega$). The photon transmission characteristics is fully described by the transmission rates given in Eqs. (15) and (16). At first, let’s focus on the unidirectional transmission, i.e., the photon completely absorbed for CW or CCW spinning mode corresponding to one of $T_a$ and $T_b$ in Eqs. (15,16) vanishing but the other being a certain value. Such a condition directly leads to $W_2=2\kappa ^2 A$ and $W_1=2\kappa A(\Delta _\pm -J)$ which further give

(19)$$T_{a}=0,$$

(20)$$T_{b}=\frac{4\Delta_F^2}{\kappa^2+(\Delta_L-\Delta_F-J)^2}.$$

with

(21)$$\Delta_{F} = B \pm \sqrt{B^2 -(\frac{(J\gamma+\gamma\Delta_{L}-2\delta\kappa)((J-\Delta_{L})^2+\kappa^2)}{\gamma(J-\Delta_{L})+2\delta\kappa})}.$$

and $B=\kappa \frac {2 \delta (J-\Delta _L)-\gamma \kappa }{\gamma (J-\Delta _L)+2\delta \kappa }$; or

(22)$$T_{b}=0,$$

(23)$$T_{a}=\frac{4\tilde{\Delta}_F^2}{\kappa^2+(\Delta_L+\tilde{\Delta}_F-J)^2}.$$

with $\tilde {\Delta }_{F}= -\Delta _{F}$.

Thus one can find that the unidirectional transmission $T_{a,b}$ can be properly controlled based on different purposes. The maximum can reach

(24)$$T_{a,b}=\frac{4((\Delta_{L}-J)^2+\kappa^2)}{\kappa^2}.$$

Interestingly, if we let the system meet the perfect absorption condition, $\Delta _F$ given in Eq. (21) can be further simplified as

(25)$$\Delta_{F} =\{0,\frac{J^2-\Delta_L^2-\kappa^2}{J}\}.$$

It is obvious that $\Delta _{F} =0$ exactly corresponds to the perfect absorption point with $a_{out}=b_{out}=0$. However, $\Delta _{F} =\frac { J^2-\Delta _L^2-\kappa ^2}{J}$ means only $a_{out}=0$. In this sense, our system can not only realize the nonreciprocal transmission but also act as a light switch that forbids both directional lights to transmit.

As illustrations, we plot the transmission rates versus $\Delta _F$ under the perfect absorption condition in Fig. 4(a), where the blue dotted line and red solid line correspond to $T_{a}$ and $T_{b}$ separately. In this case, $a_{out}=b_{out}=0$, and the system becomes an absorber. In addition, from the figure, one can find that the transmissions of the CW and CCW modes are separated in a certain range of the nonzero spinning velocity, but the separation of the transmissions of the two modes is not so large. This can be intuitively understood in the following way. $T_a$ and $T_b$ actually represent two interference channels of the two input fields. The perfect absorption condition not only produces the perfect absorption point but also makes the two interference channels perfectly match together as shown in Fig. 2. Therefore, when the perfect absorption condition is gradually broken only by adjusting $\Delta _F\neq 0$, the two curves are only slightly separated.

Fig. 4. The transmission $T$ as a function of $\Delta _{F} /\kappa$. Here $J / \kappa =8$, $g=0.02$. In (a) $\gamma / \kappa =1.775$, $\Delta _{L}/ \kappa =8.895$, $g^2N/\kappa = 132$, $\delta =15$, $\vert a_{in}\vert ^2=5374.05$. In (b) $\delta =20$, $\Delta _{L}/ \kappa =3.89$, $\vert a_{in} \vert ^2=7720.111$, $g^2N/\kappa = 126$. In (c) $\delta =28$, $\Delta _{L}/ \kappa =-3.12775$, $\vert a_{in} \vert ^2=34655.357$, $g^2N/\kappa =100$. In (d) $\delta =10.79$, $\Delta _{L}/ \kappa =3.559$, $\vert a_{in} \vert ^2=39262.064$, $g^2N/\kappa = 126$.

Download Full Size | PDF

To get better controllability, we have to effectively separate the CW and CCW spinning modes, which can be easily realized by the deviation from the perfect absorption conditions. For explicit illustrations, we plot the nonreciprocal transmissions for $\Delta _{F} \neq 0$ with the changed input light intensity and the atomic detuning in Fig. 4(b) and (c).

One can observe in Fig. 4(b) that the two transmissions are slightly separated by the spinning velocity and the light can only transmit unidirectionally at a small rate. However, in Fig. 4(c) it is shown that one-directional light can transmit with $1$ probability and the other is completely absorbed, which indicates that the system can be a good optical isolator. Thus our system provides a good device to switch on/off and control the light to transmit nonreciprocically. In addition, one can note that the gain effect of the system cannot be ignored where the transmission of the system might be greater than unity. This is also analytically revealed in Eq. (24). Analogously, we also plot the gain effect in Fig. 4(d), which shows that our system contributes to the potential of amplified unidirectional transmission.

Figure 4(c-d) explicitly exhibits the key role of $\Delta _F$ played in the nonreciprocal transmissions. Since the perfect absorption condition is further broken, the two interference channels corresponding to $T_a$ and $T_b$ are well separated. The interference of each channel can be separately controlled by the system. In Fig. Figure 4(a), one can realize the simultaneously destructive interference of the two channels, which further produces the perfect absorption at $\Delta _F=0$, but in Fig. 4(c-d), we can skillfully adjust the system such that the destructive interference only happens in one channel which exhibits the vanishing transmission (absorption) through the channel at some given $\Delta _F$, and the constructive (enhanced) interference simultaneously happens in the other channel, which indicates the amplified transmission. It is worth noting that the amplification rate can also be simultaneously controlled.

Equation (25) implies that the CW and CCW modes can also be separated under perfect absorption conditions. From Eq. (25), one can find that when the spinning velocity is adjusted such that $\Delta _{F} =\frac { J^2-\Delta _L^2-\kappa ^2}{J}$, the system can realize the unidirectional transmission, and $\Delta _{F} =0$ will switch off the transmissions of both sides. In Fig. 5, we plot the transmission rate versus the detuning $\Delta _{F}$ under the perfect photon absorption different from Fig. 4 which is consistent with Fig. 2. The amplified unidirectional transmissions, as well as the switch functions, are apparently illustrated.

Fig. 5. Transmission rate $T$ as a function of $\Delta _{F} /\kappa$. $\gamma / \kappa = 1.775$, $g^2N/\kappa = 126$, $J / \kappa = 8$, $g/\kappa =0.02$, $\delta = 10$, $\Delta _{L} = 3.263$, $\vert a_{in} \vert ^2 = 76840.4$.

Download Full Size | PDF

In addition, we’d like to emphasize that all the mentioned interesting functions result from the system’s nonlinearity or bistability and the nonreciprocity caused by the Sagnac-Fizeau shift. Different phases $\theta$ have similar conclusions, even though some particular ones can directly destroy the bistability. To explicitly show the influence of the phase, we plot the output light intensity versus $|a_{in}|^2$ in Fig. 6. It is found that different phases induce the translation of the perfect absorption point along the $|a_{in}|^2$, and $\theta =\pi$ leads to the vanishing bistability and the vanishing perfect absorption.

Fig. 6. Output light intensity $\vert y_{out}\vert ^2$ as a function of $\vert a_{in}\vert ^2$ with different $\theta$. $\gamma / \kappa = 1.775$, $J / \kappa = 8$, $g/\kappa =0.02$. $\theta =0$: $g^2N/\kappa = 80$, $\delta = 10$, $\Delta _{L} = 3.26$, $\Delta _{F}=0$; $\theta =\frac {\pi }{3}$: $g^2N/\kappa = 126$, $\delta = 10.65$, $\Delta _{L} = 8$, $\Delta _{F}=-0.577$; $\theta =\frac {\pi }{2}$: $g^2N/\kappa = 126$, $\delta = 7.1$, $\Delta _{L} = 8$, $\Delta _{F}=-1$; $\theta =\frac {2\pi }{3}$: $g^2N/\kappa = 126$, $\delta = 3.55$, $\Delta _{L} = 8$, $\Delta _{F}=-1.73$; $\theta =\pi$: $g^2N/\kappa = 126$, $\delta = 10$, $\Delta _{L} = 8$, $\Delta _{F}=0$.

Download Full Size | PDF

Before the end, we would like to discuss the potential experimental parameters for our system. In our paper, the Sagnac-Fizeau shift $\Delta _F/\kappa \in [-40,40]$, which corresponds to the maximal estimated angular velocity $\Omega \sim 3KHz$ with the experimentally feasible parameter values [45] for the resonator $\lambda =1550nm$, $n=1.44$, $r=4.75mm$ and the quality factor $Q=1.5\times 10^8$ [42]. The mean photon number of the input field $|a_{in}|^{2}=|\epsilon _p|^2/2\kappa$ [44], where the input field amplitude $\epsilon _p=\sqrt { P/\hbar \omega }$ with $P$ denoting the laser power. With the above experimental parameter values, one can find that all the $|a_{in}|^{2}$ in the paper correspond to $P\in [1\mu W,10\mu W]$, which is obviously in the experimental region [39,42,48]. Considering the atomic ensemble, it is shown that the collective coupling can reach $g\sqrt {N}\sim 2\pi \times 11 MHZ$ [49] which entirely ensures the experimental feasibility of our used parameters. In addition, it is shown that the coupled strength of two counter-propagating modes and the line width can reach $J=2\pi \times 50MHz$ and $\gamma =2\pi \times 1MHz$ [44]. Here we utilize the atomic ensemble to realize the strong coupling, which could lead to additional phonon mode by the collective vibration. This could be avoided by other mediums such as quantum dots, and nanoparticles [50,51].

5. Conclusion

We have studied perfect photon absorption and nonreciprocal optical transmission in a spinning resonator system coupled to an atomic ensemble. With the atomic ensemble, we have given the condition of perfect photon absorption analytically and find that when $\Delta _{F} = 0$, the perfect photon absorption occurs accompanied by optical bistability. The phase difference of the input light fields generally results in the translated perfect absorption point, but the $\pi$ phase difference will destroy the perfect absorption and the bistability. We show that our system with the atomic ensemble can realize the amplified unidirectional transmission of light. In particular, the amplification rate can be designed based on the parameters of the system. It is crucial that, under the condition of perfect absorption, our system can manipulate both the amplified unidirectional transmission and the bidirectional light insulation by the spinning velocity $\Delta _F$. In this sense, by tuning input light intensity and spinning velocity, one can control the output intensity, which provides a possible design of an optical switch. The underlying physics in the model provides an important reference for observing the nonreciprocal and nonlinear transmission of light and gives a new sight into the potential of the nonlinear spinning resonator.

6. Appendix

In the main text, we have considered the dynamics of the expectation values of the related operators. Let $a=\alpha + \delta \hat {a}$, $b=\beta +\delta \hat {b}$, $S^{-}=S+\delta \hat {S^-}$, $S^{z}=S^z+\delta \hat {S^z}$. The accompanied equations on the fluctuations can be given as follows.

(26)$$\frac{d\overrightarrow{X}}{dt}=M \overrightarrow{X}+\overrightarrow{n},$$

where

(27)$$\overrightarrow{X}=(\delta\hat{a},\delta\hat{a}^{\dagger},\delta\hat{b},\delta\hat{b}^{\dagger},\delta\hat{S^-},\delta\hat{S^+},\delta\hat{S^z})^{T},$$

(28)$$\overrightarrow{n}=(\sqrt{2\kappa_{1}}a_{1in},\sqrt{2\kappa_{1}} a_{1in}^\dagger,\sqrt{2\kappa_{2}}b_{1in},\sqrt{2\kappa_{2}} b_{1in}^{\dagger},\sqrt{\gamma}S^{{-}in},\sqrt{\gamma}S^{{+}in},\sqrt{\gamma}S^{z in})^{T}.$$

(29)$$M=\begin{pmatrix} -\kappa-i\Delta_+ & 0 & -iJ & 0 & -ig & 0 & 0 \\ 0 & -\kappa+i\Delta_+ & 0 & iJ & 0 & ig & 0\\-iJ & 0 & -\kappa-i\Delta_- & 0 & -ig & 0 & 0\\0 & iJ & 0 & -\kappa+i\Delta_- & 0 & ig & 0\\ i2gS^z & 0 & i2gS^z & 0 & -i\delta-\gamma/2 & 0 & i2g(\alpha+\beta)\\0 & -i2gS^z & 0 & -i2gS^z & 0 & i\delta-\gamma/2 & -i2g(\alpha^*+\beta^*)\\-igS^+ & igS^- & -igS^+ & igS^- & ig(\alpha^*+\beta^*) & -ig(\alpha+\beta) & \gamma\\ \end{pmatrix}.$$

Thus one can use the Routh–Hurwitz-criterion [52] to test the validity of the system. The stability conditions are so tedious that we cannot list them. However, we have checked all the potential parameters in the main text to make sure the eigenvalues of $M$ have negative real parts, which means the stability of the system.

Funding

National Natural Science Foundation of China (No. 12011530014, No.11775040, No.12175029).

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No.12175029, No.11775040, and No. 12011530014.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

References

1. Q. Zhong, L. Simonson, T. Kottos, and R. El-Ganainy, “Coherent virtual absorption of light in microring resonators,” Phys. Rev. Res. 2(1), 013362 (2020). [CrossRef]

2. W. Xiong, J. Chen, B. Fang, C.-h. Lam, and J. Q. You, “Coherent perfect absorption in a weakly coupled atom-cavity system,” Phys. Rev. A 101(6), 063822 (2020). [CrossRef]

3. T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6(1), 7031 (2015). [CrossRef]

4. C. Jiang, H. Liu, Y. Cui, X. Li, G. Chen, and X. Shuai, “Controllable optical bistability based on photons and phonons in a two-mode optomechanical system,” Phys. Rev. A 88(5), 055801 (2013). [CrossRef]

5. G. S. Agarwal and S. Huang, “Nanomechanical inverse electromagnetically induced transparency and confinement of light in normal modes,” New J. Phys. 16(3), 033023 (2014). [CrossRef]

6. Y. mou Liu, X. dong Tian, J. Wang, C. hui Fan, F. Gao, and Q. qian Bao, “All-optical transistor based on Rydberg atom-assisted optomechanical system,” Opt. Express 26(9), 12330–12343 (2018). [CrossRef]

7. G. S. Agarwal, K. Di, L. Wang, and Y. Zhu, “Perfect photon absorption in the nonlinear regime of cavity quantum electrodynamics,” Phys. Rev. A 93(6), 063805 (2016). [CrossRef]

8. G. S. Agarwal and Y. Zhu, “Photon trapping in cavity quantum electrodynamics,” Phys. Rev. A 92(2), 023824 (2015). [CrossRef]

9. L. Wang, K. Di, Y. Zhu, and G. S. Agarwal, “Interference control of perfect photon absorption in cavity quantum electrodynamics,” Phys. Rev. A 95(1), 013841 (2017). [CrossRef]

10. Y. Zhang, A. Sohail, and C.-s. Yu, “Perfect photon absorption in hybrid atom-optomechanical system,” EPL 115(6), 64002 (2016). [CrossRef]

11. Y. Zhang, Y.-b. Ma, X.-p. Li, Y. Guo, and C.-s. Yu, “Perfect photon absorption based on the optical parametric process,” Chin. Phys. B 30(6), 064203 (2021). [CrossRef]

12. B. King, H. Hu, and B. Shen, “Three-pulse photon-photon scattering,” Phys. Rev. A 98(2), 023817 (2018). [CrossRef]

13. E. Lundström, G. Brodin, J. Lundin, M. Marklund, R. Bingham, J. Collier, J. T. Mendonça, and P. Norreys, “Using high-power lasers for detection of elastic photon-photon scattering,” Phys. Rev. Lett. 96(8), 083602 (2006). [CrossRef]

14. A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404(6780), 850–853 (2000). [CrossRef]

15. H. H. Jen, “Disorder-assisted excitation localization in chirally coupled quantum emitters,” Phys. Rev. A 102(4), 043525 (2020). [CrossRef]

16. Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: Time-reversed lasers,” Phys. Rev. Lett. 105(5), 053901 (2010). [CrossRef]

17. W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331(6019), 889–892 (2011). [CrossRef]

18. L. Zhou, Z. R. Gong, Y.-x. Liu, C. P. Sun, and F. Nori, “Controllable scattering of a single photon inside a one-dimensional resonator waveguide,” Phys. Rev. Lett. 101(10), 100501 (2008). [CrossRef]

19. M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96(6), 063904 (2006). [CrossRef]

20. K. N. Reddy, A. V. Gopal, and S. D. Gupta, “Nonlinearity induced critical coupling,” Opt. Lett. 38(14), 2517–2520 (2013). [CrossRef]

21. S. Longhi, “Coherent perfect absorption in a homogeneously broadened two-level medium,” Phys. Rev. A 83(5), 055804 (2011). [CrossRef]

22. A. Szöke, V. Daneu, J. Goldhar, and N. A. Kurnit, “Bistable optical element and its applications,” Appl. Phys. Lett. 15(11), 376–379 (1969). [CrossRef]

23. L. A. Lugiato, II Theory of Optical Bistability (Elsevier, 1984), vol. 21 of Progress in Optics, pp. 69–216.

24. H. Wang, D. Goorskey, and M. Xiao, “Controlling the cavity field with enhanced Kerr nonlinearity in three-level atoms,” Phys. Rev. A 65(5), 051802 (2002). [CrossRef]

25. H. Chang, H. Wu, C. Xie, and H. Wang, “Controlled shift of optical bistability hysteresis curve and storage of optical signals in a four-level atomic system,” Phys. Rev. Lett. 93(21), 213901 (2004). [CrossRef]

26. M. A. Antón, O. G. Calderón, S. Melle, I. Gonzalo, and F. Carre no, “All-optical switching and storage in a four-level tripod-type atomic system,” Opt. Commun. 268(1), 146–154 (2006). [CrossRef]

27. A. Brown, A. Joshi, and M. Xiao, “Controlled steady-state switching in optical bistability,” Appl. Phys. Lett. 83(7), 1301–1303 (2003). [CrossRef]

28. A. Joshi, W. Yang, and M. Xiao, “Effect of quantum interference on optical bistability in the three-level V-type atomic system,” Phys. Rev. A 68(1), 015806 (2003). [CrossRef]

29. V. R. Almeida, C. A. Barrios, R. R. Panepucci, M. Lipson, M. A. Foster, D. G. Ouzounov, and A. L. Gaeta, “All-optical switching on a silicon chip,” Opt. Lett. 29(24), 2867–2869 (2004). [CrossRef]

30. J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015). [CrossRef]

31. D. Ballarini, M. De Giorgi, E. Cancellieri, R. Houdré, E. Giacobino, R. Cingolani, A. Bramati, G. Gigli, and D. Sanvitto, “All-optical polariton transistor,” Nat. Commun. 4(1), 1778 (2013). [CrossRef]

32. D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11(12), 774–783 (2017). [CrossRef]

33. Y.-f. Jiao, S.-d. Zhang, Y.-l. Zhang, A. Miranowicz, L.-m. Kuang, and H. Jing, “Nonreciprocal optomechanical entanglement against backscattering losses,” Phys. Rev. Lett. 125(14), 143605 (2020). [CrossRef]

34. Z.-b. Yang, J.-s. Liu, A.-d. Zhu, H.-y. Liu, and R.-c. Yang, “Nonreciprocal transmission and nonreciprocal entanglement in a spinning microwave magnomechanical system,” Ann. Phys. 532(9), 2000196 (2020). [CrossRef]

35. R. Huang, A. Miranowicz, J.-q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018). [CrossRef]

36. H. Z. Shen, Q. Wang, J. Wang, and X. X. Yi, “Nonreciprocal unconventional photon blockade in a driven dissipative cavity with parametric amplification,” Phys. Rev. A 101(1), 013826 (2020). [CrossRef]

37. W.-a. Li, G.-y. Huang, J.-p. Chen, and Y. Chen, “Nonreciprocal enhancement of optomechanical second-order sidebands in a spinning resonator,” Phys. Rev. A 102(3), 033526 (2020). [CrossRef]

38. M. Wang, C. Kong, Z.-Y. Sun, D. Zhang, Y.-Y. Wu, and L.-L. Zheng, “Nonreciprocal high-order sidebands induced by magnon Kerr nonlinearity,” Phys. Rev. A 104(3), 033708 (2021). [CrossRef]

39. Z. Shen, Y.-l. Zhang, Y. Chen, C.-l. Zou, Y.-f. Xiao, X.-b. Zou, F.-w. Sun, G.-c. Guo, and C.-h. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016). [CrossRef]

40. N. R. Bernier, L. D. Tóth, A. Koottandavida, M. A. Ioannou, D. Malz, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” Nat. Commun. 8(1), 604 (2017). [CrossRef]

41. Q.-t. Cao, H. m. Wang, C.-h. Dong, H. Jing, R.-s. Liu, X. Chen, L. Ge, Q. h. Gong, and Y.-f. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118(3), 033901 (2017). [CrossRef]

42. L. D. Bino, J. M. Silver, M. T. M. Woodley, S. L. Stebbings, X. Zhao, and P. Del’Haye, “Microresonator isolators and circulators based on the intrinsic nonreciprocity of the Kerr effect,” Optica 5(3), 279–282 (2018). [CrossRef]

43. S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018). [CrossRef]

44. B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319(5866), 1062–1065 (2008). [CrossRef]

45. G. B. Malykin, “The sagnac effect: correct and incorrect explanations,” Phys.-Usp. 43(12), 1229–1252 (2000). [CrossRef]

46. H. Shi, Z. Xiong, W. Chen, J. Xu, S. Wang, and Y. Chen, “Gauge-field description of sagnac frequency shift and mode hybridization in a rotating cavity,” Opt. Express 27(20), 28114–28122 (2019). [CrossRef]

47. D. F. Walls and G. J. Milburn, Quantum optics (Springer Science & Business Media, 2007).

48. C. Wang, W. R. Sweeney, A. D. Stone, and L. Yang, “Coherent perfect absorption at an exception point,” Science 373(6560), 1261–1265 (2021). [CrossRef]

49. F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a bose–einstein condensate,” Nature 450(7167), 268–271 (2007). [CrossRef]

50. W.-X. Yang, A.-X. Chen, X.-T. Xie, and L. Ni, “Enhanced generation of higher-order sidebands in a single-quantum-dot–cavity system coupled to a $\mathcal {PT}$-symmetric double cavity,” Phys. Rev. A 96(1), 013802 (2017). [CrossRef]

51. L. Li, W.-X. Yang, T. Shui, and X. Wang, “Ultrasensitive sizing sensor for a single nanoparticle in a hybrid nonlinear microcavity,” IEEE Photonics J. 12(3), 1–8 (2020). [CrossRef]

52. E. X. DeJesus and C. Kaufman, “Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35(12), 5288–5290 (1987). [CrossRef]

Mechanically controllable nonreciprocal transmission and perfect absorption of photons

Abstract

1. Introduction

2. Physical model

3. Perfect photon absorption

4. Mechanically controlled transmission

5. Conclusion

6. Appendix

Funding

Acknowledgements

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (29)

Optics Express