Enhanced nonlinear characteristics with the assistance of a 𝒫𝒯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr{PT}$$\end{document}-symmetric trimer system

We study the parity-time (PT) symmetry characteristics and the applications to nonlinear optics in an optical trimer system consisting of two indirectly coupled standing-mode micro-cavities and a two-level quantum emitter (QE) placed at the intersection of two cavities. We find this trimer system can exhibit analogical phenomena as those in typical 𝒫𝒯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr{PT}$$\end{document}-symmetric dimer systems composed of a passive cavity directly coupled to an active cavity. This system, whose 𝒫𝒯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr{PT}$$\end{document} symmetry is demonstrated by our analytic results, can be transformed between the 𝒫𝒯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr{PT}$$\end{document}-symmetric phase and the 𝒫𝒯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr{PT}$$\end{document}-broken phase by adjusting relevant system parameters. Then, with this system, we observe both the linear and nonlinear parts of the transmission field become remarkably enhanced and can further reach peak values around the 𝒫𝒯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr{PT}$$\end{document} breaking point. In addition, we show the negative correlation between the gain degree of the linear (nonlinear) transmission part and decay rate of the QE. This trimer proposal is feasible for experiments and may provide a promising platform for 𝒫𝒯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr{PT}$$\end{document}-symmetric optics of low-light levels. Moreover, novel phenomena arising from the QE-cavity-coupling induced nonlinearity gain could be explored to fabricate photonic devices and controllable nonlinear optical media for quantum information process and communication of photons.

Recently, the enhanced nonlinearity based on PT symmetry of an optomechanical system or a cavity system coupled with a quantum emitter (QE) has a great deal of researches [38][39][40][41][42] . Especially, the QE-cavity-coupling system, where cavities are directly coupled with each other, is an ideal platform for investigating nonlinear optics due to its performance advantages and easiness to be implemented 27 . Most recently, the trimer system involving three cavities has been used to study the phase transition and shown its excellent controllability 43 . Here we expect to build a novel QE-cavity-coupling trimer system where two cavities are indirectly coupled via a QE to achieve PT symmetry and investigate optical nonlinearity based on it. In view of this, a flexible and valid trimer system with two orthogonal cavities and a two-level QE is proposed in this paper. With an enough small spontaneous decay of the QE, the trimer system exhibits PT-symmetric properties involving an approximate BP, around which both the linear and nonlinear parts of the transmission field can be significantly enhanced.

Model and Methods
We consider an optical trimer system made up of two standing-mode microresonators, i.e., Fabry-Pérot micro-cavities, which are not coupled directly due to their orthogonality with each other, and a two-level QE (or an atom) placed at the intersection of two cavities as shown in Fig. 1. The horizontal cavity of the resonance frequency ω a is described by the annihilation (creation) operator a (a † ); the vertical one of the resonance frequency ω b is described by the annihilation (creation) operator b (b † ), with assumption ω a = ω b = ω. The two-level QE has a transition between the ground state |g〉 and excited state |e〉 with frequency ω e , which can be driven by the intracavity fields. This transition process is described by the Pauli descending (ascending) operator σ − = |g〉〈e| (σ + = |e〉〈g|), obeying the commutation relation [σ + , σ − ] = 2σ z . Two uncoupled cavities couple to the QE with the coupling strength g a and g b , respectively. A monochromatic continuous-wave probe field of the amplitude ε p and the frequency ω p incident upon the horizontal cavity and coherently drive the cavity mode a. After the frame rotation with respect to the probe frequency, the Hamiltonian of this trimer system can be written as ae p with detunings Δ = ω − ω p and δ = ω e − ω as well as the QE-cavity coupling strength where μ eg is the transition dipole moment of the QE and V a,b is the mode volume of the cavity. We can assume g a = g b = g while assuming two cavities with appropriate parameters, such as the free length. The total loss of the horizontal cavity is described by κ a = κ ai + κ ae , where κ ai is the intrinsic decay rate and κ ae is the external loss rate due to the partial reflection of the driven mirror. Phenomenologically including losses in the cavities and the QE, the semiclassical Heisenberg-Langevin equations of motion for the complicated trimer system can be written as Figure 1. Schematic illustration of the PT-symmetric trimer system including a horizontal cavity, a vertical cavity and a two-level QE. The horizontal cavity is coherently driven by a weak probe field with the amplitude ε p and the frequency ω p . S out denotes the output field of the horizontal cavity. The vertical cavity can become active due to an additional external pumping field.
with the intrinsic decay rate κ b of the vertical cavity and the spontaneous decay rate γ of the QE. Here, we assume o = 〈o〉 (o = a, b, σ z , σ − ) in the semiclassical approximation within the weak-coupling regime (g < κ a /2) 40,44,45 . A perturbation expansion method can be applied to obtain the steady-state solution of Eq. (2) in the weak-excitation approximation which is always tenable here due to the weak probe field. In fact, by performing an analytic calculation, a probe field with amplitude  ε γ . 0 01 p , which is reasonable in experiments, is low enough to guarantee the weak-excitation approximation with the other parameters as shown in this paper. The expanded form of the steady-state solution is given by with a small continuous coefficient η ∈ [0, 1]. As the weak probe field results in the weak-excitation of the QE, the electrons mostly populate the ground state |g〉. Therefore, we could use σ = −1/2 for the zeroth-order steady-state solution 40,41 . With Eqs (2) and (3) and the expansions up to the third order of the given zeroth-order values, we can express the steady-state solution of a as  46,47 , the transmission field of the horizontal cavity can be written as with the first-order and the third-order transmission coefficient It is worth to note that = | | T t a a (3) can describe the linear transmission rate and the nonlinear intensity of the transmission field, respectively. Obviously, the nonlinear intensity strongly depends on the cavity-QE-coupling strength g. If setting g = 0, this trimer system is reduced to a single-cavity system, thereby K a = 0 means that nonlinear nature disappears in the absence of the QE.

Results and Discussion
PT symmetry characteristics. This double-cavity trimer system can be transformed between the passive-passive scheme (κ a > 0, κ b < 0) and passive-active scheme (κ a > 0, κ b < 0) by precisely adjusting the additional external pumping field on the vertical cavity 39,[48][49][50] . Now we consider that κ a = −κ b = κ and the decay rate of the QE satisfies γ κ  , we can safely neglect the QE loss (γ = 0) temporarily, then the matrix form of the Hamiltonian can be written as here, for simplicity, we assume that the transition of the two-level QE is resonant with the modes of two cavities, i.e. δ = 0. Then the eigenvalues of the Hamiltonian can be obtained as Obviously, the one of three eigenvalues λ 0 can maintain real and then satisfy the PT symmetry 43 the other two eigenvalues λ ± undergo PT phase transitions near the κ = g /2 2 , similar to the case in a typical dimer system, which comprises just one passive and one active cavities with balanced loss and gain coupled directly with each other 23,30,31 . That means our trimer system can provide an alternative platform for optical PT symmetry operation. In this system, the passive and active cavities with balanced loss and gain are coupled indirectly with each other, but both couple directly the QE. As the QE builds an interaction bridge for two cavities, the trimer system shows the similar PT-symmetric characteristics as the dimer one now. Moreover, this trimer proposal may be used to study PT-symmetric nonlinearity, and provide more controllable parameters due to introducing a QE. Relative characteristics can be directly observed in Fig. 2(a), where we plot the analytical results of the PT -symmetric structure according to Eq. (11). Then, we could consider the case of κ > g /2 2 as the PT -symmetric phase regime and the case of κ < g /2 2 as the PT-broken phase regime, respectively. In Fig. 2(b), we also plot the numerical results of the actual situation by taking into account a nonvanishing value of the spontaneous decay rate of the QE (γ = 0.025κ) 34 . Whether in panel (a) or (b), it represents the PT-symmetric characteristics that λ + and λ − are symmetrical and centered on λ 0 . In addition, Fig. 2(b) shows the transformation between the PT-symmetric phase and PT-broken phase around the BP, which is analogous to the one in Fig. 2(a). It can be found that the existence of γ just leads to a tiny shift of the BP position and a slight reduction of the change rate of the phase transition around BP, rather than destroying the PT symmetry. That means, even though taking into account a reasonable value of γ, we could also achieve PT symmetry involving a phase transition. Then, we further explore the applications of this PT-symmetric proposal for nonlinear optics in next section. Figure 3 shows the linear transmission rate T a and the third-order Kerr nonlinear intensity K a as functions of the detuning Δ with different values of κ b . In Fig. 3(a), when κ b = κ a = κ corresponding to a passive-passive system, two symmetric perfect absorption windows and an on-resonance dip appear; when κ b = 0.25κ corresponding to a reduction of the vertical cavity loss, it shows somewhat increasing linear output intensity, i.e., two symmetric peaks become obvious, but the system is still in an electromagnetically induced-absorption (EIA) regime; when κ b = −0.25κ corresponding to the presence of gain effect within the vertical cavity, forming a passive-active system, the amplification of the linear output intensity emerges, which is featured by two amplified sideband peaks. In addition, a controllable optical switch between the tunable perfect absorption and the flexible enhanced linear output intensity can be achieved. As shown in Fig. 3(b), when the trimer system enters into the passive-active scheme, the amplified sideband peaks of the nonlinear output intensity are far higher than those in the passive-passive scheme. Note, although both linear and nonlinear parts of the transmission field are largely enhanced, the nonlinear part exhibits much more remarkable enhancement than the linear part, from a originally poor value, meaning that the proportion of the nonlinear part gets gradually increased and the trimer system becomes more and more nonlinear. In view of this, in this trimer system, the nonlinearity of the transmission field is enhanced with the assistance of the gain effect of the vertical active cavity.

Enhanced nonlinear characteristics.
In order to further understand the role of the PT-symmetric nature to the transmission field within the trimer system, we plot in Fig. 4 the maximal logarithms of the linear transmission rate T a and the third-order nonlinear intensity K a versus the intrinsic decay rare κ b of the vertical cavity. As shown in Fig. 4(a) and (b), T a and K a begin to ascend quickly when the trimer system enters the passive-active regime of κ b < 0 and soon develop into a sharp peak close to the point of κ b = −κ, at which a PT-symmetric structure is realized with balanced gain and loss. Then, T a and K a descend quickly when κ b is decreased further beyond the PT-symmetric point of κ b = −κ but both maintain higher values than those in the passive-passive regime of κ b > 0. The underlying physics for this effect seems that the trimer system enters the weak coupling regime and thus tends to be decoupled, when the PT-symmetric point We set κ a = κ, γ = 0.025κ and g = 0.45κ. Other parameters are the same as those in Fig. 2.  40 , when the PT-symmetric point of κ b = −κ is far away from the boundary of |κ b | = 2g = 0.4κ between the weak and strong coupling regimes. The decrease of T a and K a for larger absolute values of κ b near the right peak can be attributed once again to the fact that the trimer system is entering the weak coupling regime. But the left peak should be more relevant to the PT-symmetric nature because it is always close to the PT -symmetric point, i.e. insensitive to the coupling strength g. In view of this, both the linear and nonlinear parts of the transmission field can be significantly enhanced when the trimer system exhibits the PT-symmetric nature no matter it works in the weak coupling regime or in the strong coupling regime.
As analytically predicted above, the PT-phase transition should occur near the BP where κ κ = .  g /2 2 0 354 43 . Then, we examine the behaviors of the linear part T a and the nonlinear part K a of the transmission field near the BP. In Fig. 5, we plot the maximal logarithms of T a and K a as functions of the QE-cavity-coupling strength g/κ. Here we change g/κ ∈ [0, 0.5] for satisfying the weak-coupling regime 40 . In Fig. 5, when g is very small, i.e. the QE-cavity coupling is extremely weak, which leads to a decoupled system located in the EIA regime. With g gradually increasing within the PT-broken phase κ < g ( /2 2) of the trimer system, the strong mutual coupling supports a compensation for the horizontal passive cavity loss and then T a increases exponentially until the BP. T a reaches the maximal value near the BP featured by a narrow sharp peak. Once g increases beyond the BP within the PT-symmetric phase, the enhanced T a may rapidly reduce. A similar behavior of the enhanced nonlinear parts can be observed in Fig. 5(b), and achieve a maximal value around the BP as well. Therefore, it is feasible to further enhance both the linear and the nonlinear parts of the transmission field by adjusting g approaching to the BP within the PT-broken phase. Physically, in the PT-broken phase, the field localization effect can leads to the dynamical accumulations of the intracavity energy, which will reach the acme near the BP 2,38,40 . Hence, when g is big enough, the originally weak linear part arising from the puny probe driving and the weak nonlinear part induced by the QE-cavity coupling are both gradually amplified and reach the maximal values near the BP. In view of this, the nonlinear characteristics of the system get enhanced. However, with the further increasing of g, a strong tunneling effect becomes dominant to suppress the localization effect, so that T a and K a will decrease rapidly and reduce to a stable level due to the QE saturation 41 .
Up to now, we have always used a reasonable and actual value γ/κ = 0.025, which is much smaller than the double-cavity loss, to maintain an satisfactory PT symmetry. As showed in Fig. 2, this trimer system may show more ideal PT symmetry with γ/κ = 0, and its PT-symmetric nature will gradually weaken with the increasing γ. Thus it is necessary to examine the enhanced T a and K a with varying γ in order to discuss the relation between the gain degree of the linear and nonlinear parts of the transmission field and the spontaneous decay rate of the QE. In Fig. 6, we plot the maximal logarithms of T a and K a as functions of γ/κ ∈ [0, 0.25]. Figure 6 shows a significant negative correlation between γ and T a (K a ) which reaches its maximum with γ/κ = 0. The results are in line with our expectation, which is the enhanced nonlinearity should be derived from the PT symmetry of this trimer system. The physical reason why the linear and nonlinear parts of the transmission field reduce with the increasing of γ is that the strong spontaneous decay of the QE will break the loss-gain balance, which is the precondition for PT symmetry. In addition, as shown in Fig. 6, both T a and K a could still stay at a satisfactory level (~30 and ~144, respectively) even though γ is considerable compared with the intrinsic decay rate κ ai .

Conclusions
In summary, we have proposed a trimer system composed of two uncoupled passive and active cavities and a two-level QE building an indirect interaction bridge for two cavities. Under the loss-gain balance of two cavities, the results show that, when the QE spontaneous decay rate γ = 0, the trimer system will show a strict PT symmetry; when γ has an actual value and is small enough compared with the total loss of the cavity, the PT -symmetric characteristics will weaken but still maintain the PT-broken point (BP). One of excellent applications of this PT-symmetric trimer system is the enhanced transmission field and nonlinear characteristics, i.e. both the linear and nonlinear parts of the transmission field can be significantly enhanced, and the proportion of the nonlinear part gets improved. Even though in the unbalanced loss-gain case, the linear and nonlinear transmission parts can also be enhanced to a satisfactory degree in this trimer system. One can further maximize the linear and nonlinear transmission parts by adjusting the QE-cavity coupling strength, which is in order to set the system around the BP. In addition, there is a significant negative correlation between the gain degree of the linear (nonlinear) transmission part and the QE spontaneous decay, because the increasing of γ will break the balanced loss-gain condition of PT symmetry. This flexible and novel trimer proposal of ideal PT symmetry with more tuned parameters is feasible for experiments and may open up a new path to investigating the PT-symmetric optics of the low-light level. And the enhanced nonlinearity induced by this novel QE-cavity coupling may provide a promising platform of controllable optical responses for fabricating photonic devices and controllable nonlinear optical media for quantum information and communication of photons.