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𝒫𝒯 symmetric phase transition and photonic transmission in an optical trimer system

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Abstract

Parity-time (𝒫𝒯) symmetric structures have exhibited potential applications in developing various robust quantum devices. In an optical trimer with balanced loss and gain, we analytically study the 𝒫𝒯 symmetric phase transition by investigating the spontaneous symmetric breaking. We also illustrate the asymmetric photonic transmission behaviors in both of the 𝒫𝒯 symmetric and 𝒫𝒯 symmetry broken phases. We find (i) the non-periodical dynamics of photonic transmission in the 𝒫𝒯 symmetry broken phase instead of 𝒫𝒯 symmetric phase can be regarded as a signature of phase transition; and (ii) it shows asymmetric photonic transmission behavior in both of the phases but comes from different underlying physical mechanisms. The obtained results may be useful to implement the photonic devices based on coupled-cavity system.

© 2017 Optical Society of America

1. Introduction

Since Bender and Boettcher proposed the concept of parity-time (𝒫𝒯) symmetry [1, 2], it has attracted a lot of attentions due to its potential applications. Remarkably, the system with 𝒫𝒯 symmetry can undergo a phase transition when the parameter that controls the non-Hermiticity surpasses a critical value which is usually called exceptional point (EP) [2,3]. Below the EP, all of the eigen-values of the non-Hermit Hamiltonian are real, and some or all of the eigen values become complex beyond the EP.

The broken of 𝒫𝒯 symmetry will lead to a lot of interesting phenomena. For example, people have observed the non-reciprocal photonic transmission in the 𝒫𝒯 symmetric structure [4–12], and predicted the enhancement of nonlinear interaction due to field localization [13, 14]. Moreover, the unique property of the system with 𝒫𝒯 symmetry has potential applications in various fields, such as loss-induced or gain-induced transparency [15, 16], efficient photon or phonon lasing [17–21], ultralow-threshold optical chaos [22,23] as well as quantum metrology [24].

On the other hand, the coupled-cavity system is widely used to coherently control the photon transfer. In the coupled-cavity-array with infinite length, the defect can be introduced to construct a singe-photon switcher [25, 26], router [27] and frequency converter [28]. Moreover, within the capacity of current experiments, a lot of attentions have been paid on the optical dimer (also called optical molecule [29, 30], which is composed of two coupled cavities), such as the coherent polariton [31] and state transfer [32]. Furthermore, motivated by the simulation of photosynthesis harvest system in recent years, there are also studies about the photon [33] and thermal transport [34] in the trimer structure.

In this paper, we focus on an optical trimer with 𝒫𝒯 symmetry. Here, our scheme is composed of an active gain cavity and a passive loss cavity, which simultaneously couple with a third cavity without loss and gain, to form a coupled-cavity-array as shown in Fig. 1. With balanced gain and loss, which are described phenomenally in this paper, we show a 𝒫𝒯 symmetric phase transition in the non-Hermitian system. Different from previous work about the 𝒫𝒯 symmetry in optical trimer structure [35–38], we here focus on the property of the eigen-state of the non-Hermite Hamiltonian, and find that the field localization and spontaneous symmetry broken occurs along with the 𝒫𝒯 symmetry phase transition, in which a smaller critical coupling strength is needed than that in the dimer system [4,5,13,14].

 figure: Fig. 1

Fig. 1 Schematic illustration of 𝒫𝒯 symmetric optical trimer. In this setup, the passive cavity −1 and active cavity 1 simultaneously couple to the central cavity 0, which is without loss or/and gain.

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To wit the 𝒫𝒯 phase transition, we study the dynamical evolution of the system in both of the 𝒫𝒯 symmetric phase (where all the eigen values of the Hamiltonian are real) and 𝒫𝒯 symmetry broken phase (where the complex eigen values emerge). Under the semi-classical approximation, we neglect the effect of quantum noise which contributes to the photonic correlation [39], and study the dynamical behavior of the system by solving the Schödinger equation analytically. As a result, we show the asymmetric photonic transmission in both of the phases. Actually, the authors have studied the asymmetric photonic transmission in an optical dimer [35–38], where some nonlinearities are evolved. On contrary, we here consider only the linear interaction, and give an intuitive explanation based on the symmetry of the system.

The rest of the paper is organized as follows. In Sec. 2, we present a 𝒫𝒯 symmetry model by an optical trimer with balanced loss and gain and discuss the 𝒫𝒯 symmetric phase transition. In Sec. 3, we illustrate the dynamical evolution of the system and show the asymmetric photonic transmission in both of the 𝒫𝒯 symmetric phase and 𝒫𝒯 symmetry broken phase. At last, we give a brief remark and conclusion in Sec. 4.

2. Model and 𝒫𝒯 phase transition

As shown in Fig. 1, our model consists of an array of three single-mode cavities [33], where a passive and an active cavity (labelled by “−1” and “1” respectively) simultaneously couple to the third cavity (labelled by “0”) without loss and gain. By describing the gain and loss in our scheme phenomenologically, the Hamiltonian is written as

H=(ω1iγ1)a1a1+ω0a0a0+(ω1+iγ1)a1a1+J(a1a0+a0a1+a0a1+a1a0),
where al (l = −1, 0, 1) is the annihilation operator for the lth cavity with resonant frequency ωl. J is the photon-tunneling strength between the two nearest cavities, which can be adjusted by changing the distance of them. In addition, we use γ−1 (> 0) to denote the decay of the passive cavity and γ1(> 0) to denote the gain of active cavity. Hereafter, we consider the case that ω−1 = ω0 = ω1 = ω and γ−1 = γ1 = γ, therefore the Hamiltonian satisfies a 𝒫𝒯 symmetry, that is [H, 𝒫𝒯] = 0. Here 𝒫 represents the mirror reflection 1 ↔ −1 and 𝒯 denotes the time reversal i ↔ −i [2].

To deeply investigate the 𝒫𝒯 phase transition in our system, we write the Hamiltonian in the form of

H=(a1a2a3)(a1a2a3),
where
=(ωiγJ0JωJ0Jω+iγ).
Solving the secular equation det(EI) = 0, where I is a 3 × 3 identity matric, we will obtain the eigenvalues and the corresponding eigenstates of the Hamiltonian , yielding
E0=ω,|E0=12+(γ/J)2(1,iγJ,1),
E±=ω±2J2γ2,|E±=1𝒩±(a±,b±,1).
where 𝒩± are the normalized constants and we have defined
a±:=J2γ2iγ2J2γ2J2,
b±:=iγ±2J2γ2J.

We note that the eigenvalue E0 is always real and is independent of J and γ, and the eigen-state satisfies the 𝒫𝒯 symmetry, that is 𝒫𝒯 |E0〉 = e |E0〉. However, the other paired eigenvalues E± are dependent not only on ω, but also on γ and J.

To obtain a purely real spectrum, we need a strong inter-cavity coupling strength, i.e., J>γ/2. It then satisfies 𝒩+ = 𝒩 = 2 and the eigen-state |E±〉 can be simplified as

|E±=12(eiθ1,2eiθ2,1),
where θ1:arctan[γ2J2γ2/(J2γ2)] and θ2:=arctan[2J2γ2/γ]. A simple calculation tells us
𝒫𝒯|E±=e±iθ1|E±,
which implies that the wave functions |E±〉 are transformation invariant under the 𝒫𝒯 operation (except for a global phase). On the other hand, for the situation of J<γ/2, the imaginary parts of E± emerge and E±=ω±iγ22J2. Meanwhile, we can not find a global phase ϕ to satisfy 𝒫𝒯 |E±〉 = e± |E±〉. In other words, when the system undergoes a spontaneous symmetry breaking as the inter-cavity coupling crosses the EP J=γ/2. In this sense, we name the phase in the regime J>γ/2 as the 𝒫𝒯 symmetric phase and that for J<γ/2 as the 𝒫𝒯 symmetry broken phase.

In Figs. 2(a) and 2(b), we give the real and imaginary parts of E± respectively. In the 𝒫𝒯 symmetry broken phase (J<γ/2), the small coupling strength protects the gained energy flowing from the active cavity to the passive one, and the long lifetime supermode E+ is localized at the active cavity as shown in Fig. 2(c), where we plot |a+| as a function of the coupling strength J. On contrary, in the 𝒫𝒯 symmetric phase (J>γ/2), the gained energy is transferred to the passive cavity quickly, and the photon yields an equal weight distribution in the passive and active cavities, that is |a±| ≡ 1.

 figure: Fig. 2

Fig. 2 The real parts of E± (a), imaginary parts of E± (b), and |a+| as a function of the coupling strength J. We have chosen ω = 5γ, and all of the other parameters are in units of γ = 1.

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We point out that here, the similar 𝒫𝒯 symmetric phase transition also occurs in optical dimer which consists of two cavities with balanced loss and gain (see Refs. [4, 5] and the references therein). The differences stem in the following two aspects: On one hand, in our optical trimer system, there exist a single real energy level E0, and the corresponding wave function is always invariant under the 𝒫𝒯 operation, independent of whether the phase transition occurs. On the other hand, the EP of optical trimer system is J=γ/2 instead of J = γ in optical dimer [4,5,13,14], that is, a smaller coupling strength is needed.

3. Dynamical behavior

As a witness of the 𝒫𝒯 symmetry phase transition, we here consider the dynamical evolution of the system in both of the 𝒫𝒯 symmetric phase and 𝒫𝒯 symmetry broken phase. For the sake of simplicity and to grasp the major physics, we restrict ourselves in the subspace spanned by the basis {|1; 0; 0〉, |0; 1; 0〉, |0; 0; 1〉}. Here, |m; n; q〉 := |m−1 ⊗ |n0 ⊗ |q1, and |ni (i = −1, 0, 1) represents that the ith cavity is in the Fock state |n〉. Then, the wave function at arbitrary time t can be assumed as

|ψ(t)=α(t)|1;0;0+β(t)|0;1;0+ξ(t)|0;0;1,

Under the semi-classical approximation [40], we will focus on the photonic distribution in different cavities (that is |α(t)|2, |β(t)|2 and |ξ(t)|2). Therefore, we will neglect the quantum noise from the environment, which makes significant contributions to the photonic correlation, and therefore beyond our consideration in this paper. In this sense, the dynamics of the system is governed by the Schödinger equation i∂t |ψ〉 = H|ψ〉, where the Hamiltonian H is given in Eq. (1), and it yields

iddtα(t)=(ωiγ)α(t)+Jβ(t),
iddtβ(t)=ωβ(t)+J[α(t)+ξ(t)],
iddtξ(t)=(ω+iγ)ξ(t)+Jβ(t).

3.1. Dynamics in 𝒫𝒯 symmetric phase

We now study the dynamical behavior of the system in its 𝒫𝒯 symmetric phase, that is J>γ/2. Firstly, we consider the situation that the photon is excited in the passive cavity initially, that is α(0) = 1, β(0) = ξ(0) = 0, the probability amplitudes for finding the photon in the three cavities can be obtained explicitly as

αs(t)=2J2eiωtΔ2cos2(Δt+ϕ12),
βs(t)=2iJ2eiωtΔ2sin(Δt2)sin(Δtϕ22),
ξs(t)=2J2eiωtΔ2sin2(Δt2),
where Δ=2J2γ2, ϕ1 = arctan[Δγ/(J2γ2)], ϕ2 = 2 arctan(Δ/γ). Secondly, when the photon is initially excited in the active cavity, that is α(0) = β(0) = 0, ξ(0) = 1, the solution of Eqs. (3) are obtained as
αs(t)=2J2eiωtΔ2sin2(Δt2),
βs(t)=2iJ2eiωtΔ2sin(Δt2)sin(Δt+ϕ22),
ξs(t)=2J2eiωtΔ2cos2(Δtϕ12).

In Figs. 3(a) and 3(b), we plot the corresponding probabilities |hs (t)|2 and |h′s (t)|2 (h = α, β, ξ) as functions of evolution time t. As shown in the figure, when the system is in the 𝒫𝒯 symmetric phase, the dynamics shows regular periodical oscillations. If the photon is initially excited in the passive cavity, i.e., α(0) = 1, the decay makes |αs(t)|2 directly decrease to zero and then the revival occurs. However, if it is excited in the active cavity, i.e., ξ(0) = 1, with the assistance of the gain effect, |ξ′s (t)|2 firstly reaches its maximal value [2J2/(2J2γ2)]2, which is obviously larger than 1 and then oscillates between the maximal value and zero. As for the central cavity, we can also observe that βs (t) ≠ β′s (t). In this sense, our system exhibits an asymmetric photon transmission behavior even in the 𝒫𝒯 symmetric phase. Furthermore, we find that the initial phases ϕ1 and ϕ2 are γ dependent and ϕ1(−γ) = −ϕ1(γ), ϕ2(−γ) = −ϕ2(γ). As a result, by regarding the amplitudes as functions of t and γ, we will reach the relationship αs (t, −γ) = ξ′s (t, γ), βs (t, −γ) = β′s (t, γ) and ξs (t, −γ) = α′s (t, γ). Meanwhile, it is obvious from Eq. (3) that (i ↔ −i) = (γ ↔ −γ), therefore, the asymmetric photonic transmission in the 𝒫𝒯 symmetric phase comes from the breaking of time reversal symmetry, that is [𝒯, ] ≠ 0. The breaking of time reversal symmetry can also be observed from the dynamical behavior of the system. For example, it is shown that |g(t)|2 ≠ |g(−t)|2 and |g′(t)|2 ≠ |g′(−t)|2 for g = α, β.

 figure: Fig. 3

Fig. 3 The dynamics of the system in its 𝒫𝒯 symmetric phase. The parameters are set as ω = 5γ, J = 5γ, and all parameters are in units of γ = 1. The initial condition is set as (a) α(0) = 1, β(0) = ξ(0) = 0. (b) α(0) = β(0) = 0, ξ(0) = 1.

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3.2. Dynamics in 𝒫𝒯 symmetry broken phase

In this subsection, we will continue to study the dynamics of the system in the 𝒫𝒯 symmetry broken phase, where J<γ/2. On one hand, we consider that the photon is initially excited in the passive cavity, then the dynamics of system is obtained as

αb(t)=eiωtδ2[J2+(J2γ2)cosh(δt)+γδsinh(δt)],
βb(t)=iJeiωtδ2[γcosh(δt)γδsinh(δt)],
ξb(t)=2J2eiωtsinh2(δt2)δ2.
where δ=γ22J2. On the other hand, when the single photon is initially excited in the active cavity, the dynamics of the system is described by
αb(t)=2J2eiωtsinh2(δt2)δ2,
βb(t)=iJeiωtδ2[γcosh(δt)γ+δsinh(δt)],
ξb(t)=eiωtδ2[J2+(J2γ2)cosh(δt)γδsinh(δt)].

In Fig. 4, we depict the dynamics of the system when it is in the 𝒫𝒯 symmetry broken phase. Obviously, it shows a completely different behavior compared with the case in the 𝒫𝒯 symmetric phase. As shown in Fig. 4(a), when the photon is initially excited in the passive cavity, it will experience a loss firstly and then the gain in the active cavity will compensate the loss. As a result, the probability for finding the photon in the cavities will increase as the time elapse. As for the central cavity, the incident photon will hop to it, but the obtained photons will jump to the other two cavities due to the coherent coupling until the gained photon from the active cavity jumped back to it. Furthermore, at the time t = arctanh[γδ/(γ2J2)]/δ, the probability for finding the photon in the passive or the central cavities achieve their smallest values simultaneously. On the other hand, the probability for finding a photon in the active cavity will increase monotonously due to the combinational effect of the photonic hopping from the central cavity and the gain from the surrounding environment.

 figure: Fig. 4

Fig. 4 The dynamics of the system in its 𝒫𝒯 symmetric phase. The parameters are set as ω = 5γ, J = 0.5γ, and all parameters are in units of γ = 1. The initial condition is set as (a) α(0) = 1, β(0) = ξ(0) = 0. (b) α(0) = β(0) = 0, ξ(0) = 1.

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In Fig. 4(b), we show the results when the photon is initially excited in the active cavity. In such a situation, the gain effect will take action from the very beginning and the probabilities for finding photons in all of the cavities will undoubtedly increase as the time evolution.

Comparing the results in Figs. 4(a) and 4(b), we also observe the asymmetric photonic transmission in the 𝒫𝒯 symmetry broken phase. It can be explained from the following two aspects. Firstly, similar to that in the 𝒫𝒯 symmetric phase, the time reversal symmetry breaking of the Hamiltonian naturally results in the different transmission behaviors for the photons initially excited in the left and right sides of the system. However, the periodical triangle functions which describe the dynamics of the system in the 𝒫𝒯 symmetric phase is replaced by the monotonous hyperbola function in the broken phase. Therefore, the fixed phase difference between αs (t) and ξ′s (t) does not hold any longer for αb (t) and ξ′b (t). Secondly, the asymmetric transmission also comes from the field localization in the 𝒫𝒯 symmetry broken phase. As shown in Sec. 2, the long lifetime eigen-state |E+〉 has a lager distribution weight in the 1th cavity, that is, the photon is localized in the active cavity when the system is in the 𝒫𝒯 symmetry broken phase. As a result, for the photon excited in the active cavity, the overlap between the initial state and |E+〉 is much larger than that excited in the passive cavity, and leading to a different transmission behavior.

In dealing with the dynamics of the system in this section, we have applied the semi-classical approximation, where the quantum noise is neglected. The effect of noise to the photonic correlation is deserved to be explored more deeply, especially in the 𝒫𝒯 symmetry broken phase, where the amplification induced by the active cavity becomes significant. However, it is beyond the discussions in this paper.

4. Remark and conclusion

In this paper, we have constructed an optical trimer system by coupling a passive and active cavities to a third one without gain and loss. Experimentally speaking, the active cavity can be realized by doping the ions inside the material of the cavity [4, 5]. On the other hand, the lossless cavity can be realized by coupling a decay cavity to an auxiliary one with a much larger decay rate, which is similar to the scheme proposed by Liu et.al. [31]. Following the similar approach to eliminate the mode in the auxiliary cavity, we can obtain the effective decay rate of the considered cavity, which can achieve zero by choosing the relative parameters properly. In other words, we are able to obtain an effectively lossless cavity.

In conclusion, we have studied the 𝒫𝒯 symmetric phase transition by demonstrating the spontaneous symmetry breaking in an optical trimer with balanced loss and gain. In the 𝒫𝒯 symmetric phase, all of the eigen values are real and the corresponding eigen states show a balanced distribution in the passive and active cavities. In the 𝒫𝒯 symmetry broken phase, the imaginary parts of the eigen values appear and the supermode with long lifetime is characterized by a strong field localization in the active cavity. Comparing with the optical molecule/dimer system, which was broadly studied recently, the critical coupling strength of the phase transition is much smaller in our trimer structure. As a signature of the 𝒫𝒯 symmetric phase transition, we subsequently find the dramatically different dynamical behaviors when the system is in the two phases. Our results show that the regular periodical oscillation is replaced by the non-periodical behavior as the system transfers from the 𝒫𝒯 symmetric phase to 𝒫𝒯 symmetry broken phase. Moreover, we find an asymmetric transmission phenomenon in both phases. We hope our study about the 𝒫𝒯 symmetry in optical trimer will be helpful for the designing of photonic device based on coupled-cavity system.

Note added. Recently, we have learned of recent related work about the 𝒫𝒯 symmetry trimer systems, which cares about the dynamical behavior just at the EP [41]

Funding

National Natural Science Foundation of China (NSFC) (11404021 and 11504241); Jilin Province Science and Technology Development Plan (20170520132JH); Fundamental Research Funds for the Central Universities (2412016KJ015 and 2412016KJ004).

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35. M. Duanmu, K. Li, R. L. Horne, P. G. Kevrekidis, and N. Whitaker, “Linear and nonlinear parity-time-symmetric oligomers: a dynamical systems analysis,” Phil. Trans. R. Soc. A 371, 20120171 (2013). [CrossRef]   [PubMed]  

36. S. V Suchkov, A. A Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photonics Rev. 10, 177(2016). [CrossRef]  

37. K. Li, P. G. Kevrekidis, D. J. Frantzeskakis, C. E. Ruter, and D. Kip, “Revisiting the PT -symmetric trimer: bifurcations, ghost states and associated dynamics,” J. Phys. A: Math. Theor. 46, 375304 (2013). [CrossRef]  

38. S. V Suchkov, F. F.- Ngaffo, A. K.- Jiotsa, A. D Tikeng, T. C Kofane, Y. S Kivshar, and A. A Sukhorukov, “Non-Hermitian trimers: PT-symmetry versus pseudo-Hermiticity,” New J. Phys. 18, 065005 (2016). [CrossRef]  

39. G. S. Agarwal and K. Qu, “Spontaneous generation of photons in transmission of quantum fields in PT-symmetric optical systems,” Phys. Rev. A 85, 031802(R) (2012). [CrossRef]  

40. Z. R. Gong, H. Ian, Lan Zhou, and C. P. Sun, “Controlling quasibound states in a one-dimensional continuum through an electromagnetically-induced-transparency mechanism,” Phys. Rev. A 78, 053806 (2008). [CrossRef]  

41. L. Jin, “𝒫𝒯-symmetric trimer systems,” arXiv: 1701.00316v1 (2017).

References

  • View by:

  1. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
    [Crossref]
  2. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947 (2007).
    [Crossref]
  3. A. Mostafazadeh, “Pseudo-Hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 205 (2002).
    [Crossref]
  4. B. Peng, S. K. Ödemir, 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, 394 (2014).
    [Crossref]
  5. L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524 (2014).
    [Crossref]
  6. J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
    [Crossref] [PubMed]
  7. C. Li, L. Jin, and Z. Song, “Non-Hermitian interferometer: Unidirectional amplification without distortion,” Phys. Rev. A 95, 022125 (2017).
    [Crossref]
  8. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear 𝒫𝒯-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
    [Crossref]
  9. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
    [Crossref]
  10. L. Feng, Y. Xu, W. S. Fegadolli, M. Lu, J. E. B. Oliveira, V. R. Almeida, Y. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108 (2012).
    [Crossref] [PubMed]
  11. N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
    [Crossref] [PubMed]
  12. J. D’Ambroise, P. G. Kevrekidis, and S. Lepri, “Asymmetric wave propagation through nonlinear PT-symmetric oligomers,” J. Phys. A: Math. Theor. 45, 444012 (2012).
    [Crossref]
  13. J. Li, R. Yu, and Y. Wu, “Proposal for enhanced photon blockade in parity-time-symmetric coupled microcavities,” Phys. Rev. A 92, 053837 (2015).
    [Crossref]
  14. J. Li, X. Zhan, C. Ding, D. Zhang, and Y. Wu, “Enhanced nonlinear optics in coupled optical microcavities with an unbroken and broken parity-time symmetry,” Phys. Rev. A 92, 043830 (2015).
    [Crossref]
  15. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier- Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
    [Crossref]
  16. H. Jing, S. K. Ödemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in partiy-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015).
    [Crossref]
  17. B. Peng, S. K. Ödemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
    [Crossref] [PubMed]
  18. L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972 (2014).
    [Crossref] [PubMed]
  19. H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975 (2014).
    [Crossref] [PubMed]
  20. H. Jing, S. K. Ödemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “𝒫𝒯-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
    [Crossref]
  21. B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A 94, 031802 (2016).
    [Crossref]
  22. C. T. West, T. Kottos, and T. Prosen, “𝒫𝒯-symmetric wave chaos,” Phys. Rev. Lett. 104, 054102 (2010).
    [Crossref]
  23. X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
    [Crossref]
  24. Z.-P. Liu, J. Zhang, S. K. Ödemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
    [Crossref] [PubMed]
  25. 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, 100501 (2008).
    [Crossref] [PubMed]
  26. Z. H. Wang, Y. Li, D. L. Zhou, C. P. Sun, and P. Zhang, “Single-photon scattering on a strongly dressed atom,” Phys. Rev. A 86, 023824 (2012).
    [Crossref]
  27. L. Zhou, L.-P. Yang, Y. Li, and C. P. Sun, “Quantum routing of single photons with a cyclic three-level system,” Phys. Rev. Lett. 111, 103604 (2013).
    [Crossref]
  28. Z. H. Wang, L. Zhou, Y. Li, and C. P. Sun, “Controllable single-photon frequency converter via a one-dimensional waveguide,” Phys. Rev. A 89, 053813 (2014).
    [Crossref]
  29. M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
    [Crossref]
  30. Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photon. Rev. 4, 179 (2010).
    [Crossref]
  31. Y.-C. Liu, X. Luan, H.-K. Li, Q. Gong, C. W. Wong, and Y.-F. Xiao, “Coherent polariton dynamics in coupled highly dissipative cavities,” Phys. Rev. Lett. 112, 213602 (2014).
    [Crossref]
  32. C. D. Ogden, E. K. Irish, and M. S. Kim, “Dynamics in a coupled-cavity array,” Phys. Rev. A 78, 063805 (2008).
    [Crossref]
  33. S. Felicetti, G. Romero, D. Rossini, R. Fazio, and E. Solano, “Photon transfer in ultrastrongly coupled three-cavity arrays,” Phys. Rev. A 89, 013853 (2014).
    [Crossref]
  34. K. Joulain, J. Drevillon, Y. Ezzahri, and J. O.- Miranda, “Quantum thermal transistor,” Phys. Rev. Lett. 116, 200601 (2016).
    [Crossref] [PubMed]
  35. M. Duanmu, K. Li, R. L. Horne, P. G. Kevrekidis, and N. Whitaker, “Linear and nonlinear parity-time-symmetric oligomers: a dynamical systems analysis,” Phil. Trans. R. Soc. A 371, 20120171 (2013).
    [Crossref] [PubMed]
  36. S. V Suchkov, A. A Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photonics Rev. 10, 177(2016).
    [Crossref]
  37. K. Li, P. G. Kevrekidis, D. J. Frantzeskakis, C. E. Ruter, and D. Kip, “Revisiting the PT -symmetric trimer: bifurcations, ghost states and associated dynamics,” J. Phys. A: Math. Theor. 46, 375304 (2013).
    [Crossref]
  38. S. V Suchkov, F. F.- Ngaffo, A. K.- Jiotsa, A. D Tikeng, T. C Kofane, Y. S Kivshar, and A. A Sukhorukov, “Non-Hermitian trimers: PT-symmetry versus pseudo-Hermiticity,” New J. Phys. 18, 065005 (2016).
    [Crossref]
  39. G. S. Agarwal and K. Qu, “Spontaneous generation of photons in transmission of quantum fields in PT-symmetric optical systems,” Phys. Rev. A 85, 031802(R) (2012).
    [Crossref]
  40. Z. R. Gong, H. Ian, Lan Zhou, and C. P. Sun, “Controlling quasibound states in a one-dimensional continuum through an electromagnetically-induced-transparency mechanism,” Phys. Rev. A 78, 053806 (2008).
    [Crossref]
  41. L. Jin, “𝒫𝒯-symmetric trimer systems,” arXiv: 1701.00316v1 (2017).

2017 (1)

C. Li, L. Jin, and Z. Song, “Non-Hermitian interferometer: Unidirectional amplification without distortion,” Phys. Rev. A 95, 022125 (2017).
[Crossref]

2016 (5)

B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A 94, 031802 (2016).
[Crossref]

Z.-P. Liu, J. Zhang, S. K. Ödemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref] [PubMed]

K. Joulain, J. Drevillon, Y. Ezzahri, and J. O.- Miranda, “Quantum thermal transistor,” Phys. Rev. Lett. 116, 200601 (2016).
[Crossref] [PubMed]

S. V Suchkov, A. A Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photonics Rev. 10, 177(2016).
[Crossref]

S. V Suchkov, F. F.- Ngaffo, A. K.- Jiotsa, A. D Tikeng, T. C Kofane, Y. S Kivshar, and A. A Sukhorukov, “Non-Hermitian trimers: PT-symmetry versus pseudo-Hermiticity,” New J. Phys. 18, 065005 (2016).
[Crossref]

2015 (4)

X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

J. Li, R. Yu, and Y. Wu, “Proposal for enhanced photon blockade in parity-time-symmetric coupled microcavities,” Phys. Rev. A 92, 053837 (2015).
[Crossref]

J. Li, X. Zhan, C. Ding, D. Zhang, and Y. Wu, “Enhanced nonlinear optics in coupled optical microcavities with an unbroken and broken parity-time symmetry,” Phys. Rev. A 92, 043830 (2015).
[Crossref]

H. Jing, S. K. Ödemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in partiy-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015).
[Crossref]

2014 (10)

B. Peng, S. K. Ödemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
[Crossref] [PubMed]

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972 (2014).
[Crossref] [PubMed]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975 (2014).
[Crossref] [PubMed]

H. Jing, S. K. Ödemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “𝒫𝒯-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref]

B. Peng, S. K. Ödemir, 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, 394 (2014).
[Crossref]

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524 (2014).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

S. Felicetti, G. Romero, D. Rossini, R. Fazio, and E. Solano, “Photon transfer in ultrastrongly coupled three-cavity arrays,” Phys. Rev. A 89, 013853 (2014).
[Crossref]

Z. H. Wang, L. Zhou, Y. Li, and C. P. Sun, “Controllable single-photon frequency converter via a one-dimensional waveguide,” Phys. Rev. A 89, 053813 (2014).
[Crossref]

Y.-C. Liu, X. Luan, H.-K. Li, Q. Gong, C. W. Wong, and Y.-F. Xiao, “Coherent polariton dynamics in coupled highly dissipative cavities,” Phys. Rev. Lett. 112, 213602 (2014).
[Crossref]

2013 (4)

M. Duanmu, K. Li, R. L. Horne, P. G. Kevrekidis, and N. Whitaker, “Linear and nonlinear parity-time-symmetric oligomers: a dynamical systems analysis,” Phil. Trans. R. Soc. A 371, 20120171 (2013).
[Crossref] [PubMed]

L. Zhou, L.-P. Yang, Y. Li, and C. P. Sun, “Quantum routing of single photons with a cyclic three-level system,” Phys. Rev. Lett. 111, 103604 (2013).
[Crossref]

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref] [PubMed]

K. Li, P. G. Kevrekidis, D. J. Frantzeskakis, C. E. Ruter, and D. Kip, “Revisiting the PT -symmetric trimer: bifurcations, ghost states and associated dynamics,” J. Phys. A: Math. Theor. 46, 375304 (2013).
[Crossref]

2012 (4)

G. S. Agarwal and K. Qu, “Spontaneous generation of photons in transmission of quantum fields in PT-symmetric optical systems,” Phys. Rev. A 85, 031802(R) (2012).
[Crossref]

J. D’Ambroise, P. G. Kevrekidis, and S. Lepri, “Asymmetric wave propagation through nonlinear PT-symmetric oligomers,” J. Phys. A: Math. Theor. 45, 444012 (2012).
[Crossref]

Z. H. Wang, Y. Li, D. L. Zhou, C. P. Sun, and P. Zhang, “Single-photon scattering on a strongly dressed atom,” Phys. Rev. A 86, 023824 (2012).
[Crossref]

L. Feng, Y. Xu, W. S. Fegadolli, M. Lu, J. E. B. Oliveira, V. R. Almeida, Y. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108 (2012).
[Crossref] [PubMed]

2011 (1)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

2010 (3)

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear 𝒫𝒯-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[Crossref]

C. T. West, T. Kottos, and T. Prosen, “𝒫𝒯-symmetric wave chaos,” Phys. Rev. Lett. 104, 054102 (2010).
[Crossref]

Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photon. Rev. 4, 179 (2010).
[Crossref]

2009 (1)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier- Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

2008 (3)

C. D. Ogden, E. K. Irish, and M. S. Kim, “Dynamics in a coupled-cavity array,” Phys. Rev. A 78, 063805 (2008).
[Crossref]

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, 100501 (2008).
[Crossref] [PubMed]

Z. R. Gong, H. Ian, Lan Zhou, and C. P. Sun, “Controlling quasibound states in a one-dimensional continuum through an electromagnetically-induced-transparency mechanism,” Phys. Rev. A 78, 053806 (2008).
[Crossref]

2007 (1)

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947 (2007).
[Crossref]

2002 (1)

A. Mostafazadeh, “Pseudo-Hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 205 (2002).
[Crossref]

1998 (2)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
[Crossref]

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
[Crossref]

Agarwal, G. S.

G. S. Agarwal and K. Qu, “Spontaneous generation of photons in transmission of quantum fields in PT-symmetric optical systems,” Phys. Rev. A 85, 031802(R) (2012).
[Crossref]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier- Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Almeida, V. R.

L. Feng, Y. Xu, W. S. Fegadolli, M. Lu, J. E. B. Oliveira, V. R. Almeida, Y. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108 (2012).
[Crossref] [PubMed]

Artoni, M.

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

Bayer, M.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
[Crossref]

Bender, C. M.

B. Peng, S. K. Ödemir, 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, 394 (2014).
[Crossref]

B. Peng, S. K. Ödemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
[Crossref] [PubMed]

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947 (2007).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
[Crossref]

Bender, N.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref] [PubMed]

Bodyfelt, J. D.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref] [PubMed]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
[Crossref]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Chang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524 (2014).
[Crossref]

Chen, Y.

L. Feng, Y. Xu, W. S. Fegadolli, M. Lu, J. E. B. Oliveira, V. R. Almeida, Y. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108 (2012).
[Crossref] [PubMed]

Christodoulides, D. N.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975 (2014).
[Crossref] [PubMed]

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear 𝒫𝒯-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier- Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

D’Ambroise, J.

J. D’Ambroise, P. G. Kevrekidis, and S. Lepri, “Asymmetric wave propagation through nonlinear PT-symmetric oligomers,” J. Phys. A: Math. Theor. 45, 444012 (2012).
[Crossref]

Ding, C.

J. Li, X. Zhan, C. Ding, D. Zhang, and Y. Wu, “Enhanced nonlinear optics in coupled optical microcavities with an unbroken and broken parity-time symmetry,” Phys. Rev. A 92, 043830 (2015).
[Crossref]

Dmitriev, S. V.

S. V Suchkov, A. A Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photonics Rev. 10, 177(2016).
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H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear 𝒫𝒯-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
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B. Peng, S. K. Ödemir, 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, 394 (2014).
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M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
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H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975 (2014).
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M. Duanmu, K. Li, R. L. Horne, P. G. Kevrekidis, and N. Whitaker, “Linear and nonlinear parity-time-symmetric oligomers: a dynamical systems analysis,” Phil. Trans. R. Soc. A 371, 20120171 (2013).
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S. V Suchkov, A. A Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photonics Rev. 10, 177(2016).
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Z. R. Gong, H. Ian, Lan Zhou, and C. P. Sun, “Controlling quasibound states in a one-dimensional continuum through an electromagnetically-induced-transparency mechanism,” Phys. Rev. A 78, 053806 (2008).
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C. D. Ogden, E. K. Irish, and M. S. Kim, “Dynamics in a coupled-cavity array,” Phys. Rev. A 78, 063805 (2008).
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L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524 (2014).
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L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524 (2014).
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C. Li, L. Jin, and Z. Song, “Non-Hermitian interferometer: Unidirectional amplification without distortion,” Phys. Rev. A 95, 022125 (2017).
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Z.-P. Liu, J. Zhang, S. K. Ödemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
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X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
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H. Jing, S. K. Ödemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in partiy-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015).
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H. Jing, S. K. Ödemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “𝒫𝒯-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
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S. V Suchkov, F. F.- Ngaffo, A. K.- Jiotsa, A. D Tikeng, T. C Kofane, Y. S Kivshar, and A. A Sukhorukov, “Non-Hermitian trimers: PT-symmetry versus pseudo-Hermiticity,” New J. Phys. 18, 065005 (2016).
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M. Duanmu, K. Li, R. L. Horne, P. G. Kevrekidis, and N. Whitaker, “Linear and nonlinear parity-time-symmetric oligomers: a dynamical systems analysis,” Phil. Trans. R. Soc. A 371, 20120171 (2013).
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J. D’Ambroise, P. G. Kevrekidis, and S. Lepri, “Asymmetric wave propagation through nonlinear PT-symmetric oligomers,” J. Phys. A: Math. Theor. 45, 444012 (2012).
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H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975 (2014).
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C. D. Ogden, E. K. Irish, and M. S. Kim, “Dynamics in a coupled-cavity array,” Phys. Rev. A 78, 063805 (2008).
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Kip, D.

K. Li, P. G. Kevrekidis, D. J. Frantzeskakis, C. E. Ruter, and D. Kip, “Revisiting the PT -symmetric trimer: bifurcations, ghost states and associated dynamics,” J. Phys. A: Math. Theor. 46, 375304 (2013).
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S. V Suchkov, F. F.- Ngaffo, A. K.- Jiotsa, A. D Tikeng, T. C Kofane, Y. S Kivshar, and A. A Sukhorukov, “Non-Hermitian trimers: PT-symmetry versus pseudo-Hermiticity,” New J. Phys. 18, 065005 (2016).
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S. V Suchkov, A. A Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photonics Rev. 10, 177(2016).
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M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
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S. V Suchkov, F. F.- Ngaffo, A. K.- Jiotsa, A. D Tikeng, T. C Kofane, Y. S Kivshar, and A. A Sukhorukov, “Non-Hermitian trimers: PT-symmetry versus pseudo-Hermiticity,” New J. Phys. 18, 065005 (2016).
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N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
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Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
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H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear 𝒫𝒯-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
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M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
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B. Peng, S. K. Ödemir, 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, 394 (2014).
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J. D’Ambroise, P. G. Kevrekidis, and S. Lepri, “Asymmetric wave propagation through nonlinear PT-symmetric oligomers,” J. Phys. A: Math. Theor. 45, 444012 (2012).
[Crossref]

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C. Li, L. Jin, and Z. Song, “Non-Hermitian interferometer: Unidirectional amplification without distortion,” Phys. Rev. A 95, 022125 (2017).
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Li, C.-W.

Z.-P. Liu, J. Zhang, S. K. Ödemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
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L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524 (2014).
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Y.-C. Liu, X. Luan, H.-K. Li, Q. Gong, C. W. Wong, and Y.-F. Xiao, “Coherent polariton dynamics in coupled highly dissipative cavities,” Phys. Rev. Lett. 112, 213602 (2014).
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J. Li, R. Yu, and Y. Wu, “Proposal for enhanced photon blockade in parity-time-symmetric coupled microcavities,” Phys. Rev. A 92, 053837 (2015).
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J. Li, X. Zhan, C. Ding, D. Zhang, and Y. Wu, “Enhanced nonlinear optics in coupled optical microcavities with an unbroken and broken parity-time symmetry,” Phys. Rev. A 92, 043830 (2015).
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K. Li, P. G. Kevrekidis, D. J. Frantzeskakis, C. E. Ruter, and D. Kip, “Revisiting the PT -symmetric trimer: bifurcations, ghost states and associated dynamics,” J. Phys. A: Math. Theor. 46, 375304 (2013).
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Z. H. Wang, L. Zhou, Y. Li, and C. P. Sun, “Controllable single-photon frequency converter via a one-dimensional waveguide,” Phys. Rev. A 89, 053813 (2014).
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B. Peng, S. K. Ödemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
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Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
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Y.-C. Liu, X. Luan, H.-K. Li, Q. Gong, C. W. Wong, and Y.-F. Xiao, “Coherent polariton dynamics in coupled highly dissipative cavities,” Phys. Rev. Lett. 112, 213602 (2014).
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Z.-P. Liu, J. Zhang, S. K. Ödemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
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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, 100501 (2008).
[Crossref] [PubMed]

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Z.-P. Liu, J. Zhang, S. K. Ödemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
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B. Peng, S. K. Ödemir, 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, 394 (2014).
[Crossref]

Lu, M.

L. Feng, Y. Xu, W. S. Fegadolli, M. Lu, J. E. B. Oliveira, V. R. Almeida, Y. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108 (2012).
[Crossref] [PubMed]

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Z.-P. Liu, J. Zhang, S. K. Ödemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref] [PubMed]

X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
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H. Jing, S. K. Ödemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in partiy-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015).
[Crossref]

H. Jing, S. K. Ödemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “𝒫𝒯-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref]

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Y.-C. Liu, X. Luan, H.-K. Li, Q. Gong, C. W. Wong, and Y.-F. Xiao, “Coherent polariton dynamics in coupled highly dissipative cavities,” Phys. Rev. Lett. 112, 213602 (2014).
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X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
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Ma, R.-M.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972 (2014).
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K. Joulain, J. Drevillon, Y. Ezzahri, and J. O.- Miranda, “Quantum thermal transistor,” Phys. Rev. Lett. 116, 200601 (2016).
[Crossref] [PubMed]

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H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975 (2014).
[Crossref] [PubMed]

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B. Peng, S. K. Ödemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
[Crossref] [PubMed]

B. Peng, S. K. Ödemir, 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, 394 (2014).
[Crossref]

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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier- Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
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Z.-P. Liu, J. Zhang, S. K. Ödemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref] [PubMed]

H. Jing, S. K. Ödemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in partiy-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015).
[Crossref]

B. Peng, S. K. Ödemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
[Crossref] [PubMed]

H. Jing, S. K. Ödemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “𝒫𝒯-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref]

B. Peng, S. K. Ödemir, 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, 394 (2014).
[Crossref]

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, 100501 (2008).
[Crossref] [PubMed]

Ödemir, S. K.

Z.-P. Liu, J. Zhang, S. K. Ödemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref] [PubMed]

H. Jing, S. K. Ödemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in partiy-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015).
[Crossref]

B. Peng, S. K. Ödemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
[Crossref] [PubMed]

H. Jing, S. K. Ödemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “𝒫𝒯-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref]

B. Peng, S. K. Ödemir, 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, 394 (2014).
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C. D. Ogden, E. K. Irish, and M. S. Kim, “Dynamics in a coupled-cavity array,” Phys. Rev. A 78, 063805 (2008).
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L. Feng, Y. Xu, W. S. Fegadolli, M. Lu, J. E. B. Oliveira, V. R. Almeida, Y. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108 (2012).
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Z.-P. Liu, J. Zhang, S. K. Ödemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
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H. Jing, S. K. Ödemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in partiy-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015).
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B. Peng, S. K. Ödemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
[Crossref] [PubMed]

B. Peng, S. K. Ödemir, 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, 394 (2014).
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C. T. West, T. Kottos, and T. Prosen, “𝒫𝒯-symmetric wave chaos,” Phys. Rev. Lett. 104, 054102 (2010).
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G. S. Agarwal and K. Qu, “Spontaneous generation of photons in transmission of quantum fields in PT-symmetric optical systems,” Phys. Rev. A 85, 031802(R) (2012).
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Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photon. Rev. 4, 179 (2010).
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N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
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H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear 𝒫𝒯-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
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M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
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S. Felicetti, G. Romero, D. Rossini, R. Fazio, and E. Solano, “Photon transfer in ultrastrongly coupled three-cavity arrays,” Phys. Rev. A 89, 013853 (2014).
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S. Felicetti, G. Romero, D. Rossini, R. Fazio, and E. Solano, “Photon transfer in ultrastrongly coupled three-cavity arrays,” Phys. Rev. A 89, 013853 (2014).
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Rotter, S.

B. Peng, S. K. Ödemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
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Ruter, C. E.

K. Li, P. G. Kevrekidis, D. J. Frantzeskakis, C. E. Ruter, and D. Kip, “Revisiting the PT -symmetric trimer: bifurcations, ghost states and associated dynamics,” J. Phys. A: Math. Theor. 46, 375304 (2013).
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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier- Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
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L. Feng, Y. Xu, W. S. Fegadolli, M. Lu, J. E. B. Oliveira, V. R. Almeida, Y. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108 (2012).
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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier- Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
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S. Felicetti, G. Romero, D. Rossini, R. Fazio, and E. Solano, “Photon transfer in ultrastrongly coupled three-cavity arrays,” Phys. Rev. A 89, 013853 (2014).
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C. Li, L. Jin, and Z. Song, “Non-Hermitian interferometer: Unidirectional amplification without distortion,” Phys. Rev. A 95, 022125 (2017).
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S. V Suchkov, A. A Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photonics Rev. 10, 177(2016).
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S. V Suchkov, F. F.- Ngaffo, A. K.- Jiotsa, A. D Tikeng, T. C Kofane, Y. S Kivshar, and A. A Sukhorukov, “Non-Hermitian trimers: PT-symmetry versus pseudo-Hermiticity,” New J. Phys. 18, 065005 (2016).
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Sukhorukov, A. A

S. V Suchkov, F. F.- Ngaffo, A. K.- Jiotsa, A. D Tikeng, T. C Kofane, Y. S Kivshar, and A. A Sukhorukov, “Non-Hermitian trimers: PT-symmetry versus pseudo-Hermiticity,” New J. Phys. 18, 065005 (2016).
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S. V Suchkov, A. A Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photonics Rev. 10, 177(2016).
[Crossref]

Sun, C. P.

Z. H. Wang, L. Zhou, Y. Li, and C. P. Sun, “Controllable single-photon frequency converter via a one-dimensional waveguide,” Phys. Rev. A 89, 053813 (2014).
[Crossref]

L. Zhou, L.-P. Yang, Y. Li, and C. P. Sun, “Quantum routing of single photons with a cyclic three-level system,” Phys. Rev. Lett. 111, 103604 (2013).
[Crossref]

Z. H. Wang, Y. Li, D. L. Zhou, C. P. Sun, and P. Zhang, “Single-photon scattering on a strongly dressed atom,” Phys. Rev. A 86, 023824 (2012).
[Crossref]

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, 100501 (2008).
[Crossref] [PubMed]

Z. R. Gong, H. Ian, Lan Zhou, and C. P. Sun, “Controlling quasibound states in a one-dimensional continuum through an electromagnetically-induced-transparency mechanism,” Phys. Rev. A 78, 053806 (2008).
[Crossref]

Tikeng, A. D

S. V Suchkov, F. F.- Ngaffo, A. K.- Jiotsa, A. D Tikeng, T. C Kofane, Y. S Kivshar, and A. A Sukhorukov, “Non-Hermitian trimers: PT-symmetry versus pseudo-Hermiticity,” New J. Phys. 18, 065005 (2016).
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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier- Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
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Wang, G.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524 (2014).
[Crossref]

Wang, Y.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972 (2014).
[Crossref] [PubMed]

Wang, Z. H.

Z. H. Wang, L. Zhou, Y. Li, and C. P. Sun, “Controllable single-photon frequency converter via a one-dimensional waveguide,” Phys. Rev. A 89, 053813 (2014).
[Crossref]

Z. H. Wang, Y. Li, D. L. Zhou, C. P. Sun, and P. Zhang, “Single-photon scattering on a strongly dressed atom,” Phys. Rev. A 86, 023824 (2012).
[Crossref]

Wen, J.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524 (2014).
[Crossref]

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C. T. West, T. Kottos, and T. Prosen, “𝒫𝒯-symmetric wave chaos,” Phys. Rev. Lett. 104, 054102 (2010).
[Crossref]

Whitaker, N.

M. Duanmu, K. Li, R. L. Horne, P. G. Kevrekidis, and N. Whitaker, “Linear and nonlinear parity-time-symmetric oligomers: a dynamical systems analysis,” Phil. Trans. R. Soc. A 371, 20120171 (2013).
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Y.-C. Liu, X. Luan, H.-K. Li, Q. Gong, C. W. Wong, and Y.-F. Xiao, “Coherent polariton dynamics in coupled highly dissipative cavities,” Phys. Rev. Lett. 112, 213602 (2014).
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L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972 (2014).
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J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
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J. Li, R. Yu, and Y. Wu, “Proposal for enhanced photon blockade in parity-time-symmetric coupled microcavities,” Phys. Rev. A 92, 053837 (2015).
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J. Li, X. Zhan, C. Ding, D. Zhang, and Y. Wu, “Enhanced nonlinear optics in coupled optical microcavities with an unbroken and broken parity-time symmetry,” Phys. Rev. A 92, 043830 (2015).
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B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A 94, 031802 (2016).
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L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524 (2014).
[Crossref]

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Y.-C. Liu, X. Luan, H.-K. Li, Q. Gong, C. W. Wong, and Y.-F. Xiao, “Coherent polariton dynamics in coupled highly dissipative cavities,” Phys. Rev. Lett. 112, 213602 (2014).
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L. Feng, Y. Xu, W. S. Fegadolli, M. Lu, J. E. B. Oliveira, V. R. Almeida, Y. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108 (2012).
[Crossref] [PubMed]

Yang, C.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524 (2014).
[Crossref]

Yang, L.

B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A 94, 031802 (2016).
[Crossref]

Z.-P. Liu, J. Zhang, S. K. Ödemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref] [PubMed]

H. Jing, S. K. Ödemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in partiy-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015).
[Crossref]

B. Peng, S. K. Ödemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
[Crossref] [PubMed]

H. Jing, S. K. Ödemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “𝒫𝒯-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref]

B. Peng, S. K. Ödemir, 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, 394 (2014).
[Crossref]

Yang, L.-P.

L. Zhou, L.-P. Yang, Y. Li, and C. P. Sun, “Quantum routing of single photons with a cyclic three-level system,” Phys. Rev. Lett. 111, 103604 (2013).
[Crossref]

Yilmaz, H.

B. Peng, S. K. Ödemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
[Crossref] [PubMed]

Yu, R.

J. Li, R. Yu, and Y. Wu, “Proposal for enhanced photon blockade in parity-time-symmetric coupled microcavities,” Phys. Rev. A 92, 053837 (2015).
[Crossref]

Zhan, X.

J. Li, X. Zhan, C. Ding, D. Zhang, and Y. Wu, “Enhanced nonlinear optics in coupled optical microcavities with an unbroken and broken parity-time symmetry,” Phys. Rev. A 92, 043830 (2015).
[Crossref]

Zhang, D.

J. Li, X. Zhan, C. Ding, D. Zhang, and Y. Wu, “Enhanced nonlinear optics in coupled optical microcavities with an unbroken and broken parity-time symmetry,” Phys. Rev. A 92, 043830 (2015).
[Crossref]

Zhang, J.

Z.-P. Liu, J. Zhang, S. K. Ödemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref] [PubMed]

H. Jing, S. K. Ödemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in partiy-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015).
[Crossref]

H. Jing, S. K. Ödemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “𝒫𝒯-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref]

Zhang, P.

Z. H. Wang, Y. Li, D. L. Zhou, C. P. Sun, and P. Zhang, “Single-photon scattering on a strongly dressed atom,” Phys. Rev. A 86, 023824 (2012).
[Crossref]

Zhang, X.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972 (2014).
[Crossref] [PubMed]

Zhou, D. L.

Z. H. Wang, Y. Li, D. L. Zhou, C. P. Sun, and P. Zhang, “Single-photon scattering on a strongly dressed atom,” Phys. Rev. A 86, 023824 (2012).
[Crossref]

Zhou, L.

Z. H. Wang, L. Zhou, Y. Li, and C. P. Sun, “Controllable single-photon frequency converter via a one-dimensional waveguide,” Phys. Rev. A 89, 053813 (2014).
[Crossref]

L. Zhou, L.-P. Yang, Y. Li, and C. P. Sun, “Quantum routing of single photons with a cyclic three-level system,” Phys. Rev. Lett. 111, 103604 (2013).
[Crossref]

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, 100501 (2008).
[Crossref] [PubMed]

Zhou, Lan

Z. R. Gong, H. Ian, Lan Zhou, and C. P. Sun, “Controlling quasibound states in a one-dimensional continuum through an electromagnetically-induced-transparency mechanism,” Phys. Rev. A 78, 053806 (2008).
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J. D’Ambroise, P. G. Kevrekidis, and S. Lepri, “Asymmetric wave propagation through nonlinear PT-symmetric oligomers,” J. Phys. A: Math. Theor. 45, 444012 (2012).
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Laser Photon. Rev. (1)

Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photon. Rev. 4, 179 (2010).
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S. V Suchkov, A. A Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photonics Rev. 10, 177(2016).
[Crossref]

Nat. Mater. (1)

L. Feng, Y. Xu, W. S. Fegadolli, M. Lu, J. E. B. Oliveira, V. R. Almeida, Y. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108 (2012).
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Nat. Photon. (1)

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524 (2014).
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Nat. Phys. (1)

B. Peng, S. K. Ödemir, 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, 394 (2014).
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S. V Suchkov, F. F.- Ngaffo, A. K.- Jiotsa, A. D Tikeng, T. C Kofane, Y. S Kivshar, and A. A Sukhorukov, “Non-Hermitian trimers: PT-symmetry versus pseudo-Hermiticity,” New J. Phys. 18, 065005 (2016).
[Crossref]

Phil. Trans. R. Soc. A (1)

M. Duanmu, K. Li, R. L. Horne, P. G. Kevrekidis, and N. Whitaker, “Linear and nonlinear parity-time-symmetric oligomers: a dynamical systems analysis,” Phil. Trans. R. Soc. A 371, 20120171 (2013).
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Phys. Rev. A (11)

C. D. Ogden, E. K. Irish, and M. S. Kim, “Dynamics in a coupled-cavity array,” Phys. Rev. A 78, 063805 (2008).
[Crossref]

S. Felicetti, G. Romero, D. Rossini, R. Fazio, and E. Solano, “Photon transfer in ultrastrongly coupled three-cavity arrays,” Phys. Rev. A 89, 013853 (2014).
[Crossref]

Z. H. Wang, L. Zhou, Y. Li, and C. P. Sun, “Controllable single-photon frequency converter via a one-dimensional waveguide,” Phys. Rev. A 89, 053813 (2014).
[Crossref]

Z. H. Wang, Y. Li, D. L. Zhou, C. P. Sun, and P. Zhang, “Single-photon scattering on a strongly dressed atom,” Phys. Rev. A 86, 023824 (2012).
[Crossref]

B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A 94, 031802 (2016).
[Crossref]

C. Li, L. Jin, and Z. Song, “Non-Hermitian interferometer: Unidirectional amplification without distortion,” Phys. Rev. A 95, 022125 (2017).
[Crossref]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear 𝒫𝒯-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[Crossref]

J. Li, R. Yu, and Y. Wu, “Proposal for enhanced photon blockade in parity-time-symmetric coupled microcavities,” Phys. Rev. A 92, 053837 (2015).
[Crossref]

J. Li, X. Zhan, C. Ding, D. Zhang, and Y. Wu, “Enhanced nonlinear optics in coupled optical microcavities with an unbroken and broken parity-time symmetry,” Phys. Rev. A 92, 043830 (2015).
[Crossref]

G. S. Agarwal and K. Qu, “Spontaneous generation of photons in transmission of quantum fields in PT-symmetric optical systems,” Phys. Rev. A 85, 031802(R) (2012).
[Crossref]

Z. R. Gong, H. Ian, Lan Zhou, and C. P. Sun, “Controlling quasibound states in a one-dimensional continuum through an electromagnetically-induced-transparency mechanism,” Phys. Rev. A 78, 053806 (2008).
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Phys. Rev. Lett. (14)

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H. Jing, S. K. Ödemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in partiy-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015).
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Other (1)

L. Jin, “𝒫𝒯-symmetric trimer systems,” arXiv: 1701.00316v1 (2017).

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Figures (4)

Fig. 1
Fig. 1 Schematic illustration of 𝒫𝒯 symmetric optical trimer. In this setup, the passive cavity −1 and active cavity 1 simultaneously couple to the central cavity 0, which is without loss or/and gain.
Fig. 2
Fig. 2 The real parts of E± (a), imaginary parts of E± (b), and |a+| as a function of the coupling strength J. We have chosen ω = 5γ, and all of the other parameters are in units of γ = 1.
Fig. 3
Fig. 3 The dynamics of the system in its 𝒫𝒯 symmetric phase. The parameters are set as ω = 5γ, J = 5γ, and all parameters are in units of γ = 1. The initial condition is set as (a) α(0) = 1, β(0) = ξ(0) = 0. (b) α(0) = β(0) = 0, ξ(0) = 1.
Fig. 4
Fig. 4 The dynamics of the system in its 𝒫𝒯 symmetric phase. The parameters are set as ω = 5γ, J = 0.5γ, and all parameters are in units of γ = 1. The initial condition is set as (a) α(0) = 1, β(0) = ξ(0) = 0. (b) α(0) = β(0) = 0, ξ(0) = 1.

Equations (25)

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H = ( ω 1 i γ 1 ) a 1 a 1 + ω 0 a 0 a 0 + ( ω 1 + i γ 1 ) a 1 a 1 + J ( a 1 a 0 + a 0 a 1 + a 0 a 1 + a 1 a 0 ) ,
H = ( a 1 a 2 a 3 ) ( a 1 a 2 a 3 ) ,
= ( ω i γ J 0 J ω J 0 J ω + i γ ) .
E 0 = ω , | E 0 = 1 2 + ( γ / J ) 2 ( 1 , i γ J , 1 ) ,
E ± = ω ± 2 J 2 γ 2 , | E ± = 1 𝒩 ± ( a ± , b ± , 1 ) .
a ± : = J 2 γ 2 i γ 2 J 2 γ 2 J 2 ,
b ± : = i γ ± 2 J 2 γ 2 J .
| E ± = 1 2 ( e i θ 1 , 2 e i θ 2 , 1 ) ,
𝒫 𝒯 | E ± = e ± i θ 1 | E ± ,
| ψ ( t ) = α ( t ) | 1 ; 0 ; 0 + β ( t ) | 0 ; 1 ; 0 + ξ ( t ) | 0 ; 0 ; 1 ,
i d d t α ( t ) = ( ω i γ ) α ( t ) + J β ( t ) ,
i d d t β ( t ) = ω β ( t ) + J [ α ( t ) + ξ ( t ) ] ,
i d d t ξ ( t ) = ( ω + i γ ) ξ ( t ) + J β ( t ) .
α s ( t ) = 2 J 2 e i ω t Δ 2 cos 2 ( Δ t + ϕ 1 2 ) ,
β s ( t ) = 2 i J 2 e i ω t Δ 2 sin ( Δ t 2 ) sin ( Δ t ϕ 2 2 ) ,
ξ s ( t ) = 2 J 2 e i ω t Δ 2 sin 2 ( Δ t 2 ) ,
α s ( t ) = 2 J 2 e i ω t Δ 2 sin 2 ( Δ t 2 ) ,
β s ( t ) = 2 i J 2 e i ω t Δ 2 sin ( Δ t 2 ) sin ( Δ t + ϕ 2 2 ) ,
ξ s ( t ) = 2 J 2 e i ω t Δ 2 cos 2 ( Δ t ϕ 1 2 ) .
α b ( t ) = e i ω t δ 2 [ J 2 + ( J 2 γ 2 ) cosh ( δ t ) + γ δ sinh ( δ t ) ] ,
β b ( t ) = i J e i ω t δ 2 [ γ cosh ( δ t ) γ δ sinh ( δ t ) ] ,
ξ b ( t ) = 2 J 2 e i ω t sinh 2 ( δ t 2 ) δ 2 .
α b ( t ) = 2 J 2 e i ω t sinh 2 ( δ t 2 ) δ 2 ,
β b ( t ) = i J e i ω t δ 2 [ γ cosh ( δ t ) γ + δ sinh ( δ t ) ] ,
ξ b ( t ) = e i ω t δ 2 [ J 2 + ( J 2 γ 2 ) cosh ( δ t ) γ δ sinh ( δ t ) ] .

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