&#x1D4AB;&#x1D4AF; symmetric phase transition and photonic transmission in an optical trimer system

L. F. Xue; Z. R. Gong; H. B. Zhu; Z. H. Wang

doi:10.1364/OE.25.017249

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].

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

\begin{matrix} 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}), \end{matrix}

where a_l (l = −1, 0, 1) is the annihilation operator for the lth cavity with resonant frequency ω_l. 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 = (\begin{matrix} a_{1}^{†} & a_{2}^{†} & a_{3}^{†} \end{matrix}) ℋ (\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \end{matrix}),

where

ℋ = (\begin{matrix} ω - i γ & J & 0 \\ J & ω & J \\ 0 & J & ω + i γ \end{matrix}) .

Solving the secular equation det(ℋ − EI) = 0, where I is a 3 × 3 identity matric, we will obtain the eigenvalues and the corresponding eigenstates of the Hamiltonian ℋ, yielding

E_{0} = ω, | E_{0} 〉 = \frac{1}{\sqrt{2 + {(γ / J)}^{2}}} (- 1, - i \frac{γ}{J}, 1),

E_{\pm} = ω \pm \sqrt{2 J^{2} - γ^{2}}, | E_{\pm} 〉 = \frac{1}{𝒩_{\pm}} (a_{\pm}, b_{\pm}, 1) .

where 𝒩_± are the normalized constants and we have defined

a_{\pm} : = \frac{J^{2} - γ^{2} \mp i γ \sqrt{2 J^{2} - γ^{2}}}{J^{2}},

b_{\pm} : = \frac{- i γ \pm \sqrt{2 J^{2} - γ^{2}}}{J} .

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^iπ |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., $J > γ / \sqrt{2}$ . It then satisfies 𝒩₊ = 𝒩₋ = 2 and the eigen-state |E_±〉 can be simplified as

| E_{\pm} 〉 = \frac{1}{2} (e^{\mp i θ_{1}}, 2 e^{\mp i θ_{2}}, 1),

where

θ_{1} : arctan [γ \sqrt{2 J^{2} - γ^{2}} / (J^{2} - γ^{2})]

and

θ_{2} : = arctan [\sqrt{2 J^{2} - γ^{2}} / γ]

. A simple calculation tells us

𝒫 𝒯 | E_{\pm} 〉 = e^{\pm i θ_{1}} | E_{\pm} 〉,

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 / \sqrt{2}

, the imaginary parts of E_± emerge and

E_{\pm} = ω \pm i \sqrt{γ^{2} - 2 J^{2}}

. Meanwhile, we can not find a global phase ϕ to satisfy 𝒫𝒯 |E_±〉 = e^±iϕ |E_±〉. In other words, when the system undergoes a spontaneous symmetry breaking as the inter-cavity coupling crosses the EP

J = γ / \sqrt{2}

. In this sense, we name the phase in the regime

J > γ / \sqrt{2}

as the 𝒫𝒯 symmetric phase and that for

J < γ / \sqrt{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 ( $J < γ / \sqrt{2}$ ), 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 ( $J > γ / \sqrt{2}$ ), 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.

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 $J = γ / \sqrt{2}$ 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 ⊗ |n〉₀ ⊗ |q〉₁, and |n〉_i (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

i \frac{d}{d t} α (t) = (ω - i γ) α (t) + J β (t),

i \frac{d}{d t} β (t) = ω β (t) + J [α (t) + ξ (t)],

i \frac{d}{d t} ξ (t) = (ω + i γ) ξ (t) + J β (t) .

3.1. Dynamics in 𝒫𝒯 symmetric phase

We now study the dynamical behavior of the system in its 𝒫𝒯 symmetric phase, that is $J > γ / \sqrt{2}$ . 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

α_{s} (t) = \frac{2 J^{2} e^{- i ω t}}{Δ^{2}} {cos}^{2} (\frac{Δ t + ϕ_{1}}{2}),

β_{s} (t) = \frac{2 i J^{2} e^{- i ω t}}{Δ^{2}} sin (\frac{Δ t}{2}) sin (\frac{Δ t - ϕ_{2}}{2}),

ξ_{s} (t) = - \frac{2 J^{2} e^{- i ω t}}{Δ^{2}} {sin}^{2} (\frac{Δ t}{2}),

where

Δ = \sqrt{2 J^{2} - γ^{2}}

, ϕ₁ = arctan[Δγ/(J² − γ²)], 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

α_{s}^{'} (t) = - \frac{2 J^{2} e^{- i ω t}}{Δ^{2}} {sin}^{2} (\frac{Δ t}{2}),

β_{s}^{'} (t) = \frac{2 i J^{2} e^{- i ω t}}{Δ^{2}} sin (\frac{Δ t}{2}) sin (\frac{Δ t + ϕ_{2}}{2}),

ξ_{s}^{'} (t) = \frac{2 J^{2} e^{- i ω t}}{Δ^{2}} {cos}^{2} (\frac{Δ t - ϕ_{1}}{2}) .

In Figs. 3(a) and 3(b), we plot the corresponding probabilities |h_s (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 |α_s(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/(2J² − γ²)]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 β_s (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 α_s (t, −γ) = ξ′_s (t, γ), β_s (t, −γ) = β′_s (t, γ) and ξ_s (t, −γ) = α′_s (t, γ). Meanwhile, it is obvious from Eq. (3) that ℋ (i ↔ −i) = ℋ (γ ↔ −γ), therefore, the asymmetric photonic transmission in the 𝒫𝒯 symmetric phase comes from the breaking of time reversal symmetry, that is [𝒯, ℋ] ≠ 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.

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 $J < γ / \sqrt{2}$ . On one hand, we consider that the photon is initially excited in the passive cavity, then the dynamics of system is obtained as

α_{b} (t) = - \frac{e^{- i ω t}}{δ^{2}} [J^{2} + (J^{2} - γ^{2}) cosh (δ t) + γ δ sinh (δ t)],

β_{b} (t) = - \frac{i J e^{- i ω t}}{δ^{2}} [γ cosh (δ t) - γ - δ sinh (δ t)],

ξ_{b} (t) = - \frac{2 J^{2} e^{- i ω t} {sinh}^{2} (\frac{δ t}{2})}{δ^{2}} .

where

δ = \sqrt{γ^{2} - 2 J^{2}}

. On the other hand, when the single photon is initially excited in the active cavity, the dynamics of the system is described by

α_{b}^{'} (t) = - \frac{2 J^{2} e^{- i ω t} {sinh}^{2} (\frac{δ t}{2})}{δ^{2}},

β_{b}^{'} (t) = - \frac{i J e^{- i ω t}}{δ^{2}} [γ cosh (δ t) - γ + δ sinh (δ t)],

ξ_{b}^{'} (t) = - \frac{e^{- i ω t}}{δ^{2}} [J^{2} + (J^{2} - γ^{2}) cosh (δ t) - γ δ sinh (δ t)] .

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/(γ² − J²)]/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.

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 α_s (t) and ξ′_s (t) does not hold any longer for α_b (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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𝒫𝒯 symmetric phase transition and photonic transmission in an optical trimer system

Abstract

1. Introduction

2. Model and 𝒫𝒯 phase transition

3. Dynamical behavior

3.1. Dynamics in 𝒫𝒯 symmetric phase

3.2. Dynamics in 𝒫𝒯 symmetry broken phase

4. Remark and conclusion

Funding

References and links

Cited By

Figures (4)

Equations (25)

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