Open Access
Add to BibSonomy
Abstract
In this paper, we propose a 1-dimensional optomechanical lattice which possesses non-Hermitian property due to its nonreciprocal couplings. We calculated the energy spectrum under periodical boundary condition and open boundary condition, respectively. To investigate the transmission property of the system, we calculate the Green function of the system using non-Bloch band theory. By analyzing the Green function and the periodical boundary condition results, we studied the directional amplification of the system and found the frequency that supports the amplification. By adding probe laser on one site and detect the output of the same site, we found that optomechanically induced transparency (OMIT) can be achieved in our system. Different from the traditional OMIT spectrum, quantum interference due to a large number of modes can be observed in our system. When varying the nonreciprocal and other parameters of the system, the OMIT peak can be effectively modulated or even turned into optomechanically induced amplification. Our system is very promising to act as a one-way signal filter. Our model can also be extended to other non-Hermitian optical systems which may possess topological features and bipolar non-Hermitian skin effect.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Hermiticity of the Hamiltonian is the basic assumption in quantum mechanics. Based on this assumption the eigenvalue of the system is real and the particle number is conserved. However, there will inevitably be particle and energy exchange between the real physics system we are intereted and the environment. The non-Hermiticity is ubiquitous in nature, including the gain and loss of optical system, the friction of the mechanical system, the finite lifetime of the quasiparticle in condensed matter physics, the dissipation of open quantum system, the measurement backaction in quantum measurement, etc [1]. In theory, the way to investigate the non-Hermitian system is to consider it as an open quantum system whose behavior is described by the Lindblad quantum master equation [2]. With the development of the fabrication and experimental skills, there are also plenty of experiments showing novel non-Hermitian physics, including skin effect [3–5], parity-time symmtry [6–9] and topology [7,10,11].
In a Hermitian system with translation invariance, the Bloch theorem shows its power since it describes many physical properties of the system, such as the topological invariant, the existence of edge states and the bulk-boundary correspondence (BBC) [12–15]. However, the traditional BBC in Hermitian system has failed in a non-Hermitian system due to a higher sensitivity to the boundary effect [16–18]. Specifically, in a Hermitian system in open boundary condition (OBC) the eigenstates are Bloch waves while in a non-Hermitian system in OBC the eigenstates are localized at the boundaries of the system with an exponential decay into the bulk, namely the non-Hermitian skin effect (NHSE) [16,19–25]. So the non-Bloch band theory to describe the topological feature of a non-Hermitian system was established [17,26–31]. In this theory, the Brillouin zone (BZ) in Bloch theorem was replaced by the generalized Brillouin zone (GBZ). Accordingly, the topological invariant, e.g. the winding number, was redefined as changing the integration zone from the BZ to the GBZ. Another important example of this generalization is the Green function formulas given by the non-Bloch band theory [32].
Optical system is naturally a non-Hermitian system since its onsite loss and gain and nonreciprocal coupling between different optical modes [33,34]. Since the Maxwell equation can be written in the form of the Schrödinger equation, optical systems with translation invariance can also form the energy band, breeding the research of non-Hermtian effect in topological photonics [35–39]. Inspired by this, we decided to investigate a non-Hermitian optomechanical lattice.
Light will interact with the cavity that confines it via radiation pressure. This interaction is the well-known optomechanical interaction and has attracted wide attention among researchers over the past decade [40–47]. Inspired by the interaction between the optical field and the atoms, people have also been finding novel phenomena in optomechanical system. Electromagnetically induced transparency (EIT) in atom systems have found its analogy in optomechanical system, i.e. optomechanically induced transparency (OMIT) [48–51]. A traditional OMIT system consists of one optical mode and one mechanical mode. A peak in the transmission spectrum of a weak probe laser can be observed when the optical mode are coupled by a strong control field. In recent years, OMIT has been shown in various experimental setups [49,52]. Many theoretical proposals based on OMIT in whispering gallery modes system have also been discussed [53–64]. OMIT is also suggested for applications such as the manipulation of light propagation [65], precision measurement [66] and ground state cooling of mechanical motion [67–69].
Realizing the directional amplification is an important issue in quantum information processing since it allows signals to propagate in a single way and strong enough to cover the noise [32,70–74]. Devices such as circulators and isolators have played important roles in optical and microwave systems [75]. In our work we utilize these elementary devices to form a long array which supports amplification for some frequency. Note that not all frequency can be amplified so that our system can also be considered as a filter.
2. Model and method
2.1 Non-Hermitian optomechanical lattice Hamiltonian
We begin with a finite 1D non-Hermitian optomechanical lattice shown in Fig. 1 below. The lattice consists of N sublattices and each sublattice is composed of one optical mode $\hat {a}_{n}$ and one mechanical mode $\hat {b}_{n}$. The adjacent optical mode and mechanical mode are coupled to each other via standard optomechanical interaction with coupling constant $g$. At the same time the optical mode in adjacent sublattices are coupled to each other in a nonreciprocal pattern, i.e. the left optical mode hops to the right one with strength $t-\frac {\gamma }{2}$ while the right optical mode hops to the left one with strength $t+\frac {\gamma }{2}$. This can be easily done in experiments because theoretical proposal for nonreciprocal transport in optical system has been widely discussed in recent years [75]. For example, we can use the Faraday magneto-optical effect to achieve this. We note that this nonreciprocity will introduce dissipation to the system and is contributed to the non-Hermitian behavior of the system. Each optical mode is driven by a control laser with frequency $\omega _{L}$ and probe waveguides can added to the system, e.g. in Fig. 1 one probe waveguide is attached to the $n$th optical mode and probe laser photon with frequency $\omega _{p}$ is injected into this site. The optical mode and the mechanical mode has decay rate $\kappa$ and $\Gamma$, respectively.
Fig. 1. A finite 1D non-Hermitian optomechanical lattice. Each sublattice consists of one optical mode (red circle) and one mechanical mode (blue circle). The adjacent optical mode and mechanical mode are coupled to each other through standard optomechanical interaction. The adjacent optical modes are coupled to each other in a nonreciprocal way. Each optical mode is driven by a control laser and the probe waveguide is added to the site which we are interested in. The probe waveguide is used to send or extract probe laser photon.
Our system can be described by the Hamiltonian below,
The optical mode and the mechanical mode has onsite energy $\omega _{c}$ and $\omega _{m}$, respectively. Note that we didn’t include the driving term $\hat {H}_{\rm drive}=\sum _{n=1}^{n=N} (i \sqrt {\kappa _{ex}}\alpha _{in} \hat {a}^{\dagger }_{n} e^{-i \omega _{L}t}+{\rm H.c.})$ into the Hamiltonian here because it only provides a steady optical field here. What we care aboute here is the small fluctuation over this steady optical field. Using the rotation frame transformation $H_{\rm rot}=UHU^{\dagger }-i\hbar U \frac {\partial U^{\dagger }}{\partial t}$ where $U=\exp [\sum _{n}^{n=N}i\omega _{L} t \hat {a}^{\dagger }_{n}\hat {a}_{n}]$ and following the standard linearization procedure $\hat {a}_{n}=\bar {\alpha }_{n}+\delta \hat {a}_{n}$ , $\hat {b}_{n}=\bar {\beta }_{n}+\delta \hat {b}_{n}$ one would have the linearized Hamiltonian
2.2 Generalized Brillouin zone and energy spectrum of the system
The eigenstates and eigenvalues can be solved numerically for the finite non-Hermitian optomechanical lattice. While in order to understand the features (e.g. the topology , the non-Hermitian skin effect) capturing the system more accurately, it’s still worth investigating the system analytically. Since the analytical method for non-Hermitian system, i.e. non-Bloch band theory [17,26–31], has been well established , we will follow its standard way and give the generalized Brillouin zone (GBZ) which influences many physical properties of the system.
The eigenstate of the system in real space can be written as $| \psi \rangle =(\psi _{1,A},\psi _{1,B},\psi _{2,A},\psi _{2,B},\ldots )^{T}$ and the eigenfunction $H| \psi \rangle =E| \psi \rangle$ can be divided into a set of cascaded equations
Take the ansatz $\psi _{n,\mu }=\sum _{j} \phi ^{(j)}_{n,\mu },(\mu =A,B)$, where $(\phi ^{(j)}_{n,A},\phi ^{(j)}_{n,B})=(\beta _{j})^{n}(\phi ^{(j)}_{A},\phi ^{(j)}_{B})$ , one have $H(\beta )| \psi \rangle =E| \psi \rangle$ where the generalized Bloch Hamiltonian
To have a continuous energy band one must have $\vert \beta _{1}\vert =\vert \beta _{2}\vert$ for Eq. (6) and all $\beta$ satisfy this condition will form the GBZ we want [18,26]. That is to say, if we have $\beta$ satisfies $f(\beta,E)=0$ , there will be another $\tilde {\beta }=\beta e^{i\theta }$ satisfying $f(\beta e^{i\theta },E)=0$. Using the method of resultant one can eliminate $E$ by getting $R_{E}(\beta,e^{i\theta })=0$ [29]. For any $\theta \in [0,2\pi ]$ the resultant equation will give solution $\beta$ that satisfies $f(\beta,E)=f(\beta e^{i\theta },E)=0$. By varying $\theta$ one can get a set of $\beta$ which constitutes the auxiliary generalized Brillouin zone (aGBZ). For equation $f(\beta,E)=0$ with higher order $2M$ ($M>1$) for $\beta$ will have roots ${\beta _{1},\beta _{2},\ldots,\beta _{2M}}$ listed by the modulus ($\vert \beta _{1}\vert \leq \vert \beta _{2}| \leq \cdots \leq \vert \beta _{2M}\vert$), one have to verifying whether $\beta _{0}$ and $\beta _{0}e^{i\theta _{0}}$ given by resultant equation $R_{E}(\beta,e^{i\theta })=0$ is the middle two root $\beta _{M}$ and $\beta _{M+1}$. That’s so so-called picking up the GBZ from the aGBZ [29], while we don’t have to do this because we have only two roots here and they are apparently the middle two roots. So the aGBZ we get is actually the GBZ we want.
Typical parameters for optomechanical system can be found in previous experimental work [40]. Usually we have optical and mechanical damping rates in the range $10^{5}\sim 10^{9} {\rm Hz}$ and $1\sim 10^{4} {\rm Hz}$. And the optomechanical interaction constant ranges from $1 {\rm Hz}$ to $10^{5} {\rm Hz}$. The optical mode coupling constant $t$ and nonreciprocity $\gamma$ can be tuned on a large scale by adjusting the waveguide that connects them. In this paper, we set optical mode coupling constant $t$ as the unit of the energy, and all other parameters have their value relative to it.
We have shown two examples of how GBZ affects the energy spectrum of the system in Fig. 2. In Fig. 2(a) and (c), we have shown the BZ and the GBZ for different optomechanical interaction, i.e., $G=0.0008$ for Fig. 2(a) and $G=0.08$ for Fig. 2(c). Fig. 2(b) and (d) are the energy spectrum for Fig. 2(a) and (c), respectively. We can see two red intersecting circles constitute the GBZ in Fig. 2(a). Each red circle is the sub-GBZ which forms the energy band of its own. Specifically, the left bigger red circle is the sub-GBZ that forms the mechanical-like energy band focused around the origin of coordinates. This is easy to understand because the mechanical mode has an energy scale much smaller than the optical mode. The right smaller circle in Fig. 2(a) is the sub-GBZ responsible for forming the flat optical-like energy band with the imaginary part around $-0.1$ and the real part ranging from $-1.76$ to $1.76$. We can see two different ways to calculate the energy spectrum, i.e. solving the real space Hamiltonian eigenequation or using the GBZ to give the energy spectrum, leading to the same result. In Fig. 2(b) and (c), the red points represent the brute force calculation of a 60-site real space Hamiltonian and the blue dots represent the GBZ way. By denoting our system 60-site we mean our system are composed of 60 optical modes and 60 mechanical modes. Although both methods can give the energy spectrum, the GBZ way can give calculational points more accurately when system grows larger. In Fig. 2(c) and (d) the optomechanical interaction $G$ is stronger, so the mechanical mode and the optical mode interacts with each other strongly and they form the new polariton modes. This can be shown both in the GBZ and the energy spectrum. In Fig. 2(c), the GBZ becomes two red circle lines with the smaller one surrounded by the bigger one. One can check that the left part of the bigger circle and the right part of the smaller circle is the sub-GBZ for the mechanical-like polariton mode. However, the right part of the bigger circle and the left part of the smaller circle is the sub-GBZ for the optical-like polariton mode. So the sub-GBZ for different energy band mingles with each other because of a stronger optomechanical interaction $G$. We can also see this feature in Fig. 2(d), where the optical-like polariton mode transforms from the flat band to a parabola-like band. By comparing Fig. 2(b) and (d), we can also see the imaginary gap for two bands shrinks from $0.1$ to $0$. Since the imaginary part of the eigen energy corresponds to the decay of the eigenstate, we can conclude that with the increase of the optomechanical interaction strength $G$ the eigenstates become less dissipative. Besides the open boundary condition (OBC) energy spectrum we discussed above, we have also calculated the periodical boundary condition (PBC) energy spectrum in Fig. 2(b) and (d), one can see that PBC spectrum is a orange closed loop that surrounds the blue OBC spectrum.
Fig. 2. Brillouin zone, generalized Brillouin zone and energy spectrum with different parameters. (a)(b)$t=1,\kappa =0.1,G=0.0008,\Gamma =10^{-6},\gamma =0.94$ (c)(d) $t=1,\kappa =0.1,G=0.08,\Gamma =10^{-6},\gamma =0.94$. In (a) and (c), the blue unit circle represents the BZ while red circles represent the GBZ. In (b) and (d), the red dots are eigenvalues calculated by solving eigen equation of a 60-site lattice system, while the blue dots are OBC eigenvalues given by the GBZ in (a) and (c), respectively. The orange loop represents the PBC energy spectrum.
Fig. 3. Skin effect of our 60-site lattice system. From (a) to (d), all parameters are the same except for nonreciprocity $\gamma$. They are $t=1,\kappa =0.1,G=0.0008,\Gamma =10^{-6}$. In (a) and (c), $\gamma =0.94$. In (b) $\gamma =0.14$ and in (d) $\gamma =-0.94$. In (a),(b) and (d), different color lines represent different eigenstate. In (c), the black line represents the optical mode while the red line represents the mechanical mode.
2.3 Non-Hermitian skin effect of the system
By looking at Fig. 2(a) and (c), one would find the GBZ could be constituted by $\beta$ with modulus less than $1$ under the choice of system parameters, so our system have non-Hermtian skin effect [16,17,21–25]. For a Hermitian system under open boundary condition, the wavefunctions are extended plain wave. On the contrary, the non-Hermitian skin effect is the phenomenon that all wavefunctions are localized at the edge of the system. This is shown in Fig. 3. By comparing Fig. 3(a) and (d), one can see that the nonreciprocity $\gamma$ will enhance the skin effect since (a) has a shorter decay length than (d). By comparing Fig. 3(a) and (d), one can see that the symbol of $\gamma$ will affect the side that skin effect appears. In Fig. 3(c) we show two specific eigenstate, where the black line represents the optical skin mode while the red line represents the mechanical skin mode. Comparing Fig. 3(a) and (d), it seems like that the sign of $\gamma$ determines the hopping preference, which in result determines the side that the skin effect appears. In fact, this is not true. For some certain models with a given hopping preference, e.g. $\gamma >0$, there may be skin effect at both ends of the system. This phenomenon is called as bipolar non-Hermitian skin effect [76].
3. Green function of the system
Our main goal is to discuss the transmission property of our non-Hermitian optomechanical lattice, so it’s worth calculating the Green function of the system. Similarly, there are two ways to calculate the Green function of the system, one is solving the real space matrix and the other one is using the formulas of Green function in non-Bloch band theory given by Xue [32].
3.1 Solving the real space Green function matrix
Our system is an open quantum system described by the quantum master equation
By setting $\left \{ L_{\mu }\right \}=\left \{ \sqrt {2\Gamma }b_{i},\sqrt {\gamma }(a_{i}-ia_{i+1}),\sqrt {\kappa _{ex}}a_{i}\right \}$ ,one can show the expectations of the operator $a_{i}$ and $b_{i}$ , i.e. $<a_{i}(t)>=Tr[a_{i}{\rho }(t)]$ and $<b_{i}(t)>=Tr[b_{i}{\rho }(t)]$ , are governed by the Langevin equations
By denoting $(\psi _{1},\psi _{2},\psi _{3},\ldots,\psi _{2n-1},\psi _{2n},\ldots )=(<a_{1}>,<b_{1}>,<a_{2}>, \ldots,<a_{n}>,<b_{n}>\cdots )$, one would have the matrix equation:
where $H$ is exactly the Hamiltonian in Eq. (21) with $\frac {\kappa _{ex}+2\gamma }{2}=\kappa$ and $\psi _{in}=(\bar {\alpha }_{in},0,\bar {\alpha }_{in},0,\ldots,\bar {\alpha }_{in},0,\ldots )$. If we are interested in the transmission property of the frequency $\omega$ component , by performing the Fourier transform for both ends, we will get where $\vec {\psi }(\omega )=(\psi _{1}(\omega ),\psi _{2}(\omega ),\ldots,\psi _{2n-1}(\omega ),\psi _{2n}(\omega ),\ldots )$ and similar for the definition of $\vec {\psi }_{in}(\omega )$. $G(\omega )=\frac {1}{\omega -H}$ is the Green function of our optomechanical lattice with uniform driving field for each optical mode. Using the input-output relation $\psi _{j,out}=\psi _{j,in}+\sqrt {\kappa _{ex}}\psi _{j}$ we can immediately get $\psi _{j,out}=(\mathbb {I} -i\kappa _{ex}G)_{jk}\psi _{k,in}$. And the scattering matrix is naturally defined as $S_{jk}(\omega )=(\mathbb {I} -i\kappa _{ex}G(\omega ))_{jk}$. We note that we didn’t consider the effect of the probe waveguide here. In fact, the introduction of the probe waveguide can be viewed as an effective imaginary potential [72] where $\kappa _{p,j}$ is the loss introduced by probe waveguide at $j$ site. Using the Dyson’s equation one can get the Green equation with the probe waveguide’s contribution $\tilde {G}(\omega )$ And similarly the scattering matrix element in probe waveguide’s frequency domain is3.2 Green function given by non-Bloch band theory
In this subsection, we will show how to use the non-Bloch band theory to calculate the Green function of our system and compare it with the brute force calculation of the real space calculation. According to Xue [32], the formulas for Green function of a non-Hermitian system is given by
Since $h(\beta )$ is $2\times 2$ matrix, $G_{ij}$ is also a $2 \times 2$ matrix with4. Transmission property of the system
4.1 Amplification of the system
We have calculated the Green function of the system in two different ways. As shown in Fig. 4, one can see that two different calculational ways give almost the same result for a 100-site system. Each subfigure in Fig. 4 has two lines, one represents weak coupling region $G=0.0008$ and the other represents strong coupling region $G=0.08$. In Fig. 4(a) one can see that the optical mode favors transporting from right to left rather than the opposite direction, which serves as an evidence for non-Hermitian skin effect. From Fig. 4(d) we can see there is a peak at $G_{50_{B},50_{B}}$. A closer inspection tells us that this peak becomes smaller with the increase of the optomechanical interaction $G$. This means under the parameter $G=0.0008$ the phonon is more likely to be localized rather than hopping to the neighboring optical mode, which comes as a natural result of weak optomechanical interaction. By increasing the optomechanical interaction $G$ , e.g. $G=0.08$, one can find this peak will become less pronounced. Except for the amplification between optical modes, we can also also achieve the amplification from the optical mode on the right side to the mechanical mode on the left side. For example, one can have $\vert G_{1_{B},50_{A}} \vert = 58.58$ which is not shown in the blue line in Fig. 4(c). Similarly, we can have amplification from mechanical mode to optical mode and amplification from mechanical mode to mechanical mode by choosing proper $\omega$ with other parameters unchanged. The direction amplification is always from right to left due to non-Hermitian skin effect, so that our system can be considered as a one-way filter.
Fig. 4. Green function calculated by two ways for a 100-site system. From (a) to (d), the parameters are $t=1,\kappa =0.1,\Gamma =10^{-6},\gamma =0.94,\omega =1.8$. The orange and green dots are Green function calculated by solving the real space matrix with 100 sites. The blue and red line are calculated by the formulas for Green function in non-Bloch band theory given by Eq. (16). The blue line and the orange dots ($G=0.0008$) fit very well, so do the red line and the green dots ($G=0.08$). One can see that two calculational ways have almost the same result .
Now we discuss the amplification behavior of the system with help of Eq. (15). In Fig. 5, we have shown the scattering property of a 40-site system where two probe waveguides are added on the 1st and the 23rd optical mode. From Fig. 5(b) and (c) one can see evident amplification from site 23 to site 1 while no amplification from site 1 to site 23 when $\gamma =0.94$. From Fig. 5(d) we can see no amplification even from site 23 to site 1 with $\gamma =0.04$. This can be explained by Eq. (19). In Fig. 5(a) we plot the roots $\beta _{1}, \beta _{2}$ for $\omega \mathbb {I} -h(\beta )=0$. From Fig. 5(a) we can see under parameter $\gamma =0.94, \omega =1.47$ even the bigger root $\vert \beta _{2} \vert <1$, so that we have $G_{i_{A}j_{A}}\gg 1$ for $i\ll j$. While this behavior will vanish when $\gamma =0.04,\omega =1.47$ because we have $\vert \beta _{2} \vert >1$. This will lead to $G_{i_{A}j_{A}}\ll 1$ for $i\ll j$ even though from an intuitive sight we might think the amplification still exists since we still have $t+\frac {\gamma }{2}>t-\frac {\gamma }{2}$. Another important feature is that the scattering spectrum is symmetric about $\omega =0$ and there is a sharp dip at $\omega =0$ no matter there is amplification or not. We note that $\omega =0$ corresponds to condition $\omega _{p}=\omega _{a}$ here and we will show the origin of the dip at $\omega =0$ in our next subsection.
Fig. 5. Scattering spectrum of a 40-site lattice optomechanical system. We have added two probe waveguides at site 1 and site 23 and investigate the transmission property between them. In (b),(c) and (d), all parameters are the same except for nonreciprocity $\gamma$ : $t=1,\kappa =0.1,G=0.0008,\Gamma =10^{-6},\kappa _{p,1}=\kappa _{p,23}=0.1$. In (b) and (c) $\gamma =0.94$ while in (d) $\gamma =0.04$. Similarly, we have valued $\omega$ as its strength relative to $t$. Since we have performing rotation frame transformation before, $\omega =0$ here means $\omega _{p}=\omega _{a}$. In (a) we show the roots for $\omega \mathbb {I} -h(\beta )=0$ with $\omega =1.47$ and their position with respect to the BZ. Other parameters in (a) are: $t=1,\kappa =0.1,G=0.0008,\Gamma =10^{-6}$.
Another important question is to determine the region of $\omega$ that supports amplification. From Fig. 5(b) we can see this region is roughly $-2<\omega <2$. This can be seen from our PBC spectrum in Fig. 2(b) and (c). Suppose we have a vertical line which intersects with the real axis at $(\omega,0)$ in Fig. 2(b) and (c). When $\omega \rightarrow \pm \infty$ there are no interaction point between this vertical line and the PBC spectrum. One can prove that the smaller root for $\omega \mathbb {I} -h(\beta )=0$, i.e. $\beta _{1}$ stays inside the PBC spectrum while the bigger root $\beta _{2}$ stays out of the PBC spectrum as long as this vertical line has no intersection point with the PBC spectrum [18]. By moving this vertical line from $\pm \infty$ to the center , when the vertical line has intersection points with the PBC spectrum, the outer $\beta _{2}$ will move into the BZ since the BZ generates the PBC spectrum. And that’s the moment when amplification starts existing. Since the first intersection point between the vertical line and the PBC spectrum has real part about $\pm 2$, the region $(-2,2)$ is region that supports amplification.
4.2 Optomechanically induced transparency in our system
By adding just one probe waveguide to some site of our system, e.g. the first left optical mode of our system, we can investigate the onsite transmission property of the system. From Eq. (15) we know the onsite transmission for the first optical mode is
Fig. 6. Optomechanically induced transparency of the first optical site. System parameters: (a) OMIT of a 30-site system. $t=1,\kappa =0.1,G=0.0008,\Gamma =10^{-6},\kappa _{p,1}=0.1$. By increasing the nonreciprocity $\gamma$, one can see that the OMIT peak is increasing. (b) OMIT effect with different system size: $\gamma =0.94,t=1,\kappa =0.1,G=0.0008,\Gamma =10^{-6},\kappa _{p,1}=0.1$. One can see the realization of OMIA is possible by varing the size of the sytem. (c) The relation ship between the eigenvalues of a 20-site system and the onsite transmission spectrum, prarameters are $\gamma =0.94,t=1,\kappa =0.1,G=0.0008,\Gamma =10^{-6},\kappa _{p,1}=0.1$. The red points represents the real part of the eigenvalues. (d) The bulk onsite transmission property of a 20-site system with parameters $\gamma =0.94,t=1,\kappa =0.1,G=0.0008,\Gamma =10^{-6}$. We added probe waveguide at site 2 (site 10) with $\kappa _{p,2}=0.1(\kappa _{p,10}=0.1)$ for red dashed line (black solid line). (e) A zoom-in picture of the OMIT peak of a 20-site system. Other parameters are: $t=1,\kappa =0.1,\Gamma =10^{-6},\kappa _{p,1}=0.1,\gamma =0.94$ (f) OMIT with disorder. Other parameters are: $t=1,\kappa =0.1,G=0.0008,\Gamma =10^{-6},\kappa _{p,1}=0.1,\gamma =0.94$. From (a) to (f), we have valued $\omega$ as its strength relative to $t$.
Besides, we also investigate the effect of the system’s size on the first site transmission property. From Fig. 6(b) we can see that one can even achieve optomechanically induced amplification (OMIA) in a 10-site system. From Eq. (20) we know that this is because $\tilde {G}_{1_{A}1_{A}}(\omega )$ is a complex number, so a different ${\rm arg}(\tilde {G}_{1_{A}1_{A}}(\omega ))$ may turn OMIT into OMIA. This explanation is similar to the famous quantum pathways interference theory in the literature before [49]. We can use this quantum pathways interference theory to explain why an increasing $\gamma$ will lead to an increasing OMIT peak under the parameters in Fig. 6(a). For a N-site system, there will be N optical eigenstates and N mechanical eigenstates, namely $\left \{\psi _{1,A},\psi _{2,A},\ldots,\psi _{N,A}\right \}$ and $\left \{\psi _{1,B},\psi _{2,B},\ldots,\psi _{N,B}\right \}$. These eigenstates are listed by the order of the modulus of their eigenvalues, i.e. $\left \{\vert E_{1,A}\vert > \vert E_{2,A}\vert > \cdots > \vert E_{N,A}\vert \right \}$ and $\left \{ \vert E_{1,B}\vert >\vert E_{2,B}\vert >\cdots >\vert E_{N,B}\vert \right \}$ where the subscript A represents the optical mode and B represents the mechanical mode. Among all optical eigenstates, there will be an eigenstate $\psi _{N,A}$ which has the smallest eigenvalues $E_{N,A}$. For example, the black line in Fig. 3(c) is $\psi _{60,A}$ with ${\rm \vert} Re(E_{60,A})\vert =0.0454$. By increasing the nonreciprocity $\gamma$, one can show that the absolute value of the real part of the smallest optical eigenvalue, i.e. ${\rm \vert} Re(E_{N,A}) \vert$ , approaches to the zero more closely. So the optical mode $\psi _{N,A}$ has a larger probability to accept the probe laser photon injected into the system with relative frequency $\omega =0$, which finally results in a larger OMIT peak due to a stronger quantum pathways interference. From Fig. 6(a) and (b) we also find that the transmission spectrum is almost symmetric around $\omega =0$. A further study on Fig. 6(b) tells us that the number of the absorption peaks on the left (right) of $\omega =0$ is increasing when the system becomes larger. In fact the absorption peaks on the left (right) of $\omega =0$ are actually the real part of the eigenvalues of the optical-like modes while the OMIT peak (OMIA dip) centering around $\omega =0$ is connected with the real part of the eigenvalues of the mechanical-like modes. This argument can be testified in Fig. 6(c), (d) and (e). In Fig. 6(c), we plot the transmission spectrum of the first optical mode of a 20 site system. At the same time, we calculate the eigenvalues of this 20 site system and mark the real part of the eigenvalues in Fig. 6(c), i.e. the red dots in Fig. 6(c). One can see that they fit quite well with the absorption peaks. To show that the OMIT (OMIA) peak around the center is actually the mechanical absorption peak, we investigate the detail of the peak with a variable optomechanical interaction $G$. In Fig. 6(e), one can see that the OMIT peak in weak optomechanical region ($G=0.0008$) splits into various peaks in a stronger optomechanical interaction ($G=0.0058$ or $G=0.008$). Comparing black dashed line and red dashed line in Fig. 6(e), one would find that the region of split peaks becomes wider with the increase of $G$. This phenomenon is similar to the broadening of the energy band in a tight-banding model. When $G$ is small, the mechanical modes almost decouple from each other. So the mechanical-like eigenvalues form energy levels centering around $\omega =0$, i.e. the OMIT peak. When $G$ is larger, the mechanical modes couple with each other through optomechanical interaction. So the energy levels interact with each other and the mechanical energy band forms consequently. When $G$ is larger, the width of the split region is wider due to a stronger interference between different energy levels.
From Fig. 6(b) we also conclude that the optical absorption peaks on the right (left) side of $\omega =0$ become shallower and less pronounced with system size increases. This is due to the interference of different bulk optical modes. With the increase of the system size, this many body physics phenomenon becomes more apparent. One can also see this phenomenon by investigating the onsite transmission property of a bulk site, e.g. the 10th optical mode of a 20-site system as shown as the black dashed line in Fig. 6(d). One can find that the optical absorption peaks at the bulk site become more chaotic than the first site because the bulk has a stronger interference effect than the edge. Moreover, one can also conclude that changing the site that the probe laser is added to can turn OMIT into OMIA by comparing the red dashed line and black solid line in Fig. 6(d).
It’s worth discussing the difference between the traditional OMIT and the OMIT in our lattice system. For a traditional OMIT system, there is only one absorption peak in the center $\omega =0$. While in our system, there are symmetric absorption valleys around $\omega =0$ which correspond to the real part of eigenvalues of the optical-like modes. So our system provides a new perspective to explain the OMIT other than the quantum pathways interference, i.e. it’s the absorption peak of the mechanical-like modes. Besides, in Fig. 6(e), we investigated the detail of the OMIT peak by increasing the optomechanical interaction strength $G$. We found that the OMIT peak splits into small peaks with $G$ increasing, which comes as a result of a stronger interference of different mechanical-like modes. Another important difference is that OMIT and OMIA can both be observed in the same system in our model such as the example shown in Fig. 6(d). This phenomenon doesn’t exist in traditional OMIT system.
We finally discuss possibility of the experimental realization of our model. To see this, we consider the effect of inevitable disorders introduced by imperfect fabrication. We define a positive parameter $\epsilon$ which describes the strength of the disorder. With this parameter, the Hamiltonian under the effect of disorders is:
This means that any parameter $\xi$ in our model can be varied in the region $[\xi -\epsilon _{\xi,n} \xi, \xi +\epsilon _{\xi,n} \xi ]$ where $0<\epsilon _{\xi,n}<\epsilon$. As shown in Fig. 6(f), numerical result shows that the OMIT peak can survive even under a disorder strength as strong as $\epsilon =0.2$, which shows the robustness of this phenomenon in our model.
5. Conclusion
We have investigated the energy spectrum and the transmission property of a long non-Hermitian optomechanical lattice. The non-Hermitian property of our model is introduced by not only the onsite decay but also the nonreciprocal coupling of different sites. To find out the open boundary condition (OBC) spectrum of the system we calculated the generalized Brillouin Zone (GBZ) of the system based on the concept of the resultant. To investigate the transmission property of the system, the Green function is calculated in two ways, i.e. solving the inverse of the real space matrix and using the non-Bloch theory formula. Two different methods give the same result.
With the help of the Green function we investigated the scattering of different sites. The directional amplification from right to left is observed. So our model can serve as a single pathway filter. The frequency that supports amplification is discussed by considering the periodical boundary condition energy spectrum of our model. The amplification direction has a close relationship between the non-Hermitian skin effect. Since our system only supports skin modes on the left side of the lattice, the amplification is always from right to left. As an extension, one can investigate other non-Hermitian bosonic systems that support bipolar non-Hermitian skin effect.
Using the Green function, we have also investigated the onsite transmission property of the system. The optomechanically induced transparency (OMIT) is observed in our model. We find that the OMIT peak can be modulated by changing the nonreciprocity $\gamma$ and system size in our model. The absorption peaks are actually corresponding to the real part of the eigenvalues of the system. The mechanical-like mode eigenvalues are related to the OMIT peak in the center and the optical-like mode eigenvalues are related to the absorption peak on both sides. Our model can also be extended to other non-Hermitian bosonic systems which have topological features. The possible relationship between the OMIT and the topology is worth discussing.
Funding
National Natural Science Foundation of China (61727801, 62131002); National Key Research and Development Program of China (2017YFA0303700); Special Project for Research and Development in Key areas of Guangdong Province (2018B030325002); Beijing Innovation Center for Future Chip; Tsinghua Initiative Scientific Research Program.
Acknowledgments
We thank Yu-Min Hu and Dr.Jinghui Pi for helpful discussion.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. Y. Ashida, Z. Gong, and M. Ueda, “Non-hermitian physics,” Adv. Phys. 69(3), 249–435 (2020). [CrossRef]
2. M. O. Scully and M. S. Zubairy, “Quantum optics,” (1999).
3. L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, “Non-hermitian bulk–boundary correspondence in quantum dynamics,” Nat. Phys. 16(7), 761–766 (2020). [CrossRef]
4. X. Zhang, Y. Tian, J.-H. Jiang, M.-H. Lu, and Y.-F. Chen, “Observation of higher-order non-hermitian skin effect,” Nat. Commun. 12(1), 1–8 (2021). [CrossRef]
5. D. Zou, T. Chen, W. He, J. Bao, C. H. Lee, H. Sun, and X. Zhang, “Observation of hybrid higher-order skin-topological effect in non-hermitian topolectrical circuits,” Nat. Commun. 12(1), 7201 (2021). [CrossRef]
6. C. M. Bender and S. Boettcher, “Real spectra in non-hermitian hamiltonians having p t symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998). [CrossRef]
7. S. Xia, D. Kaltsas, D. Song, I. Komis, J. Xu, A. Szameit, H. Buljan, K. G. Makris, and Z. Chen, “Nonlinear tuning of pt symmetry and non-hermitian topological states,” Science 372(6537), 72–76 (2021). [CrossRef]
8. L. Feng, R. El-Ganainy, and L. Ge, “Non-hermitian photonics based on parity–time symmetry,” Nat. Photonics 11(12), 752–762 (2017). [CrossRef]
9. O. Sigwarth and C. Miniatura, “Time reversal and reciprocity,” AAPPS Bull. 32(1), 23 (2022). [CrossRef]
10. L. Xiao, X. Zhan, Z. Bian, K. Wang, X. Zhang, X. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. Sanders, and P. Xue, “Observation of topological edge states in parity–time-symmetric quantum walks,” Nat. Phys. 13(11), 1117–1123 (2017). [CrossRef]
11. Y. Meng, X. Wu, Y. Shen, D. Liu, Z. Liang, X. Zhang, and J. Li, “Non-hermitian topological coupler for elastic waves,” Sci. China Phys. Mech. Astron. 65(2), 224611 (2022). [CrossRef]
12. X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011). [CrossRef]
13. M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010). [CrossRef]
14. J. K. Asbóth, L. Oroszlány, and A. Pályi, “A short course on topological insulators,” Lect. Notes Phys. 919, 166 (2016). [CrossRef]
15. S.-Q. Shen, Topological insulators, vol. 174 (Springer, 2012).
16. N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, “Topological origin of non-hermitian skin effects,” Phys. Rev. Lett. 124(8), 086801 (2020). [CrossRef]
17. S. Yao and Z. Wang, “Edge states and topological invariants of non-hermitian systems,” Phys. Rev. Lett. 121(8), 086803 (2018). [CrossRef]
18. Y.-M. Hu, F. Song, and Z. Wang, “Generalized brillouin zone and non-hermitian band theory,” Acta Phys. Sin. 70(23), 230307 (2021). [CrossRef]
19. S. Longhi, “Probing non-hermitian skin effect and non-bloch phase transitions,” Phys. Rev. Res. 1(2), 023013 (2019). [CrossRef]
20. L. Li, C. H. Lee, S. Mu, and J. Gong, “Critical non-hermitian skin effect,” Nat. Commun. 11(1), 1–8 (2020). [CrossRef]
21. S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, “Topological funneling of light,” Science 368(6488), 311–314 (2020). [CrossRef]
22. K. Wang, A. Dutt, K. Y. Yang, C. C. Wojcik, J. Vučković, and S. Fan, “Generating arbitrary topological windings of a non-hermitian band,” Science 371(6535), 1240–1245 (2021). [CrossRef]
23. K. Wang, A. Dutt, C. C. Wojcik, and S. Fan, “Topological complex-energy braiding of non-hermitian bands,” Nature 598(7879), 59–64 (2021). [CrossRef]
24. Y. Song, Y. Chen, W. Xiong, and M. Wang, “Flexible light manipulation in non-hermitian frequency su–schrieffer–heeger lattice,” Opt. Lett. 47(7), 1646–1649 (2022). [CrossRef]
25. X. Zhu, H. Wang, S. K. Gupta, H. Zhang, B. Xie, M. Lu, and Y. Chen, “Photonic non-hermitian skin effect and non-bloch bulk-boundary correspondence,” Phys. Rev. Res. 2(1), 013280 (2020). [CrossRef]
26. K. Yokomizo and S. Murakami, “Non-bloch band theory of non-hermitian systems,” Phys. Rev. Lett. 123(6), 066404 (2019). [CrossRef]
27. K. Kawabata, N. Okuma, and M. Sato, “Non-bloch band theory of non-hermitian hamiltonians in the symplectic class,” Phys. Rev. B 101(19), 195147 (2020). [CrossRef]
28. S. Yao, F. Song, and Z. Wang, “Non-hermitian chern bands,” Phys. Rev. Lett. 121(13), 136802 (2018). [CrossRef]
29. Z. Yang, K. Zhang, C. Fang, and J. Hu, “Non-hermitian bulk-boundary correspondence and auxiliary generalized brillouin zone theory,” Phys. Rev. Lett. 125(22), 226402 (2020). [CrossRef]
30. K. Zhang, Z. Yang, and C. Fang, “Correspondence between winding numbers and skin modes in non-hermitian systems,” Phys. Rev. Lett. 125(12), 126402 (2020). [CrossRef]
31. S. Longhi, “Non-bloch-band collapse and chiral zener tunneling,” Phys. Rev. Lett. 124(6), 066602 (2020). [CrossRef]
32. W.-T. Xue, M.-R. Li, Y.-M. Hu, F. Song, and Z. Wang, “Simple formulas of directional amplification from non-bloch band theory,” Phys. Rev. B 103(24), L241408 (2021). [CrossRef]
33. L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011). [CrossRef]
34. L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011). [CrossRef]
35. T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019). [CrossRef]
36. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014). [CrossRef]
37. D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, “Nonlinear topological photonics,” Appl. Phys. Rev. 7(2), 021306 (2020). [CrossRef]
38. A. B. Khanikaev and G. Shvets, “Two-dimensional topological photonics,” Nat. Photonics 11(12), 763–773 (2017). [CrossRef]
39. M. Kim, Z. Jacob, and J. Rho, “Recent advances in 2d, 3d and higher-order topological photonics,” Light: Sci. Appl. 9(1), 130 (2020). [CrossRef]
40. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014). [CrossRef]
41. F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009). [CrossRef]
42. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321(5893), 1172–1176 (2008). [CrossRef]
43. M. Metcalfe, “Applications of cavity optomechanics,” Appl. Phys. Rev. 1(3), 031105 (2014). [CrossRef]
44. T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007). [CrossRef]
45. C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012). [CrossRef]
46. Y.-P. Gao, Z.-X. Wang, T.-J. Wang, and C. Wang, “Optomechanically engineered phononic mode resonance,” Opt. Express 25(22), 26638–26650 (2017). [CrossRef]
47. K. Wang, Y.-P. Gao, R. Jiao, and C. Wang, “Recent progress on optomagnetic coupling and optical manipulation based on cavity-optomagnonics,” Front. Phys. 17(4), 42201 (2022). [CrossRef]
48. J. P. Marangos, “Electromagnetically induced transparency,” J. Mod. Opt. 45(3), 471–503 (1998). [CrossRef]
49. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010). [CrossRef]
50. H. Xiong and Y. Wu, “Fundamentals and applications of optomechanically induced transparency,” Appl. Phys. Rev. 5(3), 031305 (2018). [CrossRef]
51. A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the nonlinear quantum regime,” Phys. Rev. Lett. 111(13), 133601 (2013). [CrossRef]
52. C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Transient optomechanically induced transparency in a silica microsphere,” Phys. Rev. A 87(5), 055802 (2013). [CrossRef]
53. H. Lü, C. Wang, L. Yang, and H. Jing, “Optomechanically induced transparency at exceptional points,” Phys. Rev. Appl. 10(1), 014006 (2018). [CrossRef]
54. D.-G. Lai, X. Wang, W. Qin, B.-P. Hou, F. Nori, and J.-Q. Liao, “Tunable optomechanically induced transparency by controlling the dark-mode effect,” Phys. Rev. A 102(2), 023707 (2020). [CrossRef]
55. H. Lü, Y. Jiang, Y.-Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photonics Res. 5(4), 367–371 (2017). [CrossRef]
56. F.-C. Lei, M. Gao, C. Du, Q.-L. Jing, and G.-L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Opt. Express 23(9), 11508–11517 (2015). [CrossRef]
57. G.-Q. Qin, H. Yang, X. Mao, J.-W. Wen, M. Wang, D. Ruan, and G.-L. Long, “Manipulation of optomechanically induced transparency and absorption by indirectly coupling to an auxiliary cavity mode,” Opt. Express 28(1), 580–592 (2020). [CrossRef]
58. X. Mao, G.-Q. Qin, H. Yang, Z. Wang, M. Wang, G.-Q. Li, P. Xue, and G.-L. Long, “Tunable partial polarization beam splitter and optomechanically induced faraday effect,” Phys. Rev. A 105(3), 033526 (2022). [CrossRef]
59. G.-Q. Qin, R.-R. Xie, H. Zhang, Y.-Q. Hu, M. Wang, G.-Q. Li, H. Xu, F. Lei, D. Ruan, and G.-L. Long, “Experimental realization of sensitivity enhancement and suppression with exceptional surfaces,” Laser Photonics Rev. 15(5), 2000569 (2021). [CrossRef]
60. B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity–time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014). [CrossRef]
61. R.-R. Xie, G.-Q. Qin, H. Zhang, M. Wang, G.-Q. Li, D. Ruan, and G.-L. Long, “Phase-controlled dual-wavelength resonance in a self-coupling whispering-gallery-mode microcavity,” Opt. Lett. 46(4), 773–776 (2021). [CrossRef]
62. X. Jiang, M. Wang, M. C. Kuzyk, T. Oo, G.-L. Long, and H. Wang, “Chip-based silica microspheres for cavity optomechanics,” Opt. Express 23(21), 27260–27265 (2015). [CrossRef]
63. X. Mao, G.-Q. Qin, H. Yang, H. Zhang, M. Wang, and G.-L. Long, “Enhanced sensitivity of optical gyroscope in a mechanical parity-time-symmetric system based on exceptional point,” New J. Phys. 22(9), 093009 (2020). [CrossRef]
64. W. Chen and A. A. Clerk, “Photon propagation in a one-dimensional optomechanical lattice,” Phys. Rev. A 89(3), 033854 (2014). [CrossRef]
65. H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015). [CrossRef]
66. J.-Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012). [CrossRef]
67. Y. Guo, K. Li, W. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90(5), 053841 (2014). [CrossRef]
68. T. Ojanen and K. Børkje, “Ground-state cooling of mechanical motion in the unresolved sideband regime by use of optomechanically induced transparency,” Phys. Rev. A 90(1), 013824 (2014). [CrossRef]
69. Y.-C. Liu, Y.-F. Xiao, X. Luan, and C. W. Wong, “Optomechanically-induced-transparency cooling of massive mechanical resonators to the quantum ground state,” Sci. China Phys. Mech. Astron. 58(5), 1–6 (2015). [CrossRef]
70. C. C. Wanjura, M. Brunelli, and A. Nunnenkamp, “Topological framework for directional amplification in driven-dissipative cavity arrays,” Nat. Commun. 11(1), 3149 (2020). [CrossRef]
71. B. Abdo, K. Sliwa, L. Frunzio, and M. Devoret, “Directional amplification with a josephson circuit,” Phys. Rev. X 3(3), 031001 (2013). [CrossRef]
72. A. McDonald, T. Pereg-Barnea, and A. Clerk, “Phase-dependent chiral transport and effective non-hermitian dynamics in a bosonic kitaev-majorana chain,” Phys. Rev. X 8(4), 041031 (2018). [CrossRef]
73. S. Mikki and M. Herde, “Analysis and design of secure quantum communication systems utilizing electromagnetic schrodinger coherent states,” Quantum Eng. 3(3), e72 (2021). [CrossRef]
74. X.-P. Zhang, L.-T. Shen, Y. Zhang, L. Sun, H. Wu, Z.-B. Yang, and Z.-Q. Yin, “External control of qubit-photon interaction and multi-qubit reset in a dissipative quantum network,” Sci. China Phys. Mech. Astron. 64(5), 250311 (2021). [CrossRef]
75. E. Turner and R. Stolen, “Fiber faraday circulator or isolator,” Opt. Lett. 6(7), 322–323 (1981). [CrossRef]
76. F. Song, S. Yao, and Z. Wang, “Non-hermitian topological invariants in real space,” Phys. Rev. Lett. 123(24), 246801 (2019). [CrossRef]
