Three methods for the single-photon transport in a chiral cavity quantum electrodynamics system

Jiang-Shan Tang; Lei Tang; Keyu Xia

doi:10.3788/COL202220.062701

Abstract

We investigate the single-photon transport problem in the system of a whispering-gallery mode microresonator chirally coupled with a two-level quantum emitter (QE). Conventionally, this chiral QE-microresonator coupling system can be studied by the master equation and the single-photon transport methods. Here, we provide a new approach, based on the transfer matrix, to assess the single-photon transmission of such a system. Furthermore, we prove that these three methods are equivalent. The corresponding relations of parameters among these approaches are precisely deduced. The transfer matrix can be extended to a multiple-resonator system interacting with two-level QEs in a chiral way. Therefore, our work may provide a convenient and intuitive form for exploring more complex chiral cavity quantum electrodynamics systems.

Keywords

chiral cavity QED system master equation method single-photon transport theory transfer matrix method

1. Introduction

The interaction of light and matter at the single-quantum level is the basis of essential physics of many phenomena and applications^[1], which has been extensively explored in various quantum systems, such as quantum emitters (QEs) coupling with single-mode waveguides^[2–9], cavities^[10–18], plasmons^[19–23], and whispering-gallery mode microresonators^[24–30]. In recent years, an emerging field of research, called “chiral quantum optics”^[31], exhibits chiral interactions of light and QEs^[32–38] and has received extensive attention in the field of optical nonreciprocity^{[32,35,39–41]}.

To realize the chiral light–matter interaction, an external magnetic field is usually required to induce the magneto-optical effect^[41] or initialize the states of QEs^[42]. It greatly limits the miniaturization and integration of single-photon devices. Recently, all-optical approaches, based on the optical Stark shift of quantum dots (QDs)^[35] and the valley-selective response in transition metal dichalcogenides^[43], have been proposed to release the requirement of magnetic biases. Towards on-chip chiral single-photon interfaces, non-magnetic schemes have been designed based on a whispering-gallery mode microresonator chirally coupled with a two-level QE^{[32,35,39,40]}.

Theoretically, the single-photon transport (SPT) problem in the system of a whispering-gallery mode microresonator coupled to a waveguide can be solved by methods such as the master equation (ME)^[44–49], the SPT^{[4,26,50–52]}, and the transfer matrix (TM)^[53]. The whispering-gallery mode microresonator system containing a two-level QE has also been discussed under the framework of the ME and SPT theory^[4,24,26,54], even extending to the chiral interactions^{[31,32,35,40]}. However, how to deal with the chiral interaction of a whispering-gallery mode microresonator with a two-level QE using the TM method is still not available. The inner link among these three methods also remains to be revealed.

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In this work, we study the SPT problem in a chiral QE-microresonator system using the TM method. By introducing a nonlinear coefficient related to the two-level QE into the transfer relation, we can use the TM method to solve the single-photon transmission. In this sense, the two-level QE can be regarded as a single-photon phase-amplitude modulator. Furthermore, we demonstrate that the ME, SPT, and TM methods are equivalent in dealing with such chiral cavity quantum electrodynamics (QED) systems. The correspondence between the parameters of the three methods is strictly deduced.

This paper is organized as follows. In Sec. 2, we review the ME and the SPT theory for the SPT problem in a chiral QE-microresonator system, respectively. Next, we discuss the TM approach and show that the three methods above are equivalent if we treat the two-level QE as a single-photon phase-amplitude modulator. In Sec. 3, we show the numerical results of these three methods. In the end, we present a conclusion in Sec. 4.

2. System and Model

The chiral QE-microresonator system, depicted in Fig. 1, consists of a whispering-gallery mode microresonator, a waveguide, and a two-level QE. The microresonator, which can be made with various material platforms, such as silicon oxynitride^[55,56], polymers^[57], or silicon on insulator^[58–60], supports two traveling wave modes, i.e., clockwise (CW) and counterclockwise (CCW) modes. Near the outer sidewall of the microresonator, the evanescent fields of the whispering-gallery modes are almost perfectly circularly polarized with its polarization locked to the propagation direction^[27,35]. We assume that the evanescent field of the CCW mode is $σ^{+}$ -polarized and that of the CW mode is $σ^{-}$ -polarized. As shown in Fig. 1, the QE is positioned near the outer sidewall of the microresonator. After initializing the QE in a specific spin ground state^[61–63] or shifting the transition energy with a polarization-selective optical Stark effect^[64–66], we can treat the QE as a two-level system with only $σ^{+}$ -polarization-driven transition. One can use a precisely positioned atom^[40,67], QD^{[34,42,68,69]}, or nanopillar covered by monolayers^[70–72] to construct that two-level QE. As a result, the QE-microresonator coupling strength is dependent on the propagating direction of light in our chiral system. In the forward case, the incident light from port 1 excites the CCW mode, and it strongly couples with the QE with the coupling strength $g$ . However, in the backward (port 2 incident) case, the CW mode is decoupled with the QE, and thus the coupling rate is negligible (i.e., $g \approx 0$ ). Practically, backscattering is usually present due to the surface roughness of the microresonator. In this paper, we treat the backscattering as one scatterer^[73], as depicted in Fig. 1.

Figure 1.Schematic of a chiral QE-microresonator system. A two-level QE is coupled to a whispering-gallery mode microresonator in a chiral way to form the QE-microresonator system. A waveguide is side coupled to the microresonator as input and output ports. A scatterer on the microresonator is considered to introduce backscattering. The arrows represent the propagating direction of a single photon for an input to port 1 (green) or port 2 (red).

Below, we first provide the ME, SPT, and TM methods to solve the response of the system. Then, we show that these three methods are equivalent if we treat the two-level QE as a single-photon phase-amplitude modulator. We only discuss the forward case ( $g \neq 0$ ) in detail, and the backward case corresponds to the system without the QE ( $g = 0$ ).

2.1. Master Equation Method

In this section, we discuss the ME method to solve our model. For a coupled atom-microresonator system, it has been analyzed^[24]. Here, we discuss the chiral coupling using the same approach. We consider that a two-level QE with transition frequency $ω_{qe}$ is coupled to the CCW mode and decoupled to the opposite mode. The two degenerate whispering-gallery modes, with same resonant frequency $Ω$ and dissipation $κ_{tol}$ , are assumed to be coupled with each other in a strength $h$ due to the scatterer. Here, we divide the dissipation $κ_{tol}$ into two parts, the intrinsic decay rate of $κ_{in}$ and the external loss of $κ_{ex}$ , satisfying $κ_{tol} = κ_{in} + κ_{ex}$ . A weak coherent field of frequency $ω$ with an amplitude $α_{in}$ drives the CCW mode a. In a good single-photon approximation, $α_{in} ≪ 1$ . In a frame rotating at the frequency $ω$ , the Hamiltonian of our system can be obtained^[74]: $H = - Δ_{1} a^{†} a - Δ_{2} σ^{+} σ^{-} - Δ_{1} b^{†} b + i \sqrt{2 κ_{ex}} α_{in} (a^{†} - a) + g (a^{†} σ^{-} + σ^{+} a) + h (a^{†} b + b^{†} a),$ (1)where $g$ represents the coupling strength between the CCW mode and the QE. $Δ_{1} = ω - Ω$ and $Δ_{2} = ω - ω_{qe}$ are the detunings. $b$ is the annihilation operator of the CW mode. $σ^{\pm}$ are the raising and lowering operators describing the two-level QE. It is worth noting that if we consider the coupling of two microresonators instead of the scatterer, the Hamiltonian has the same form as Eq. (1). In this case, $h$ describes the coupling strength between the two microresonators.

Introducing the dissipation of the QE, $γ$ , the evolution of the system can be found by solving the ME, $\dot{ρ} = - i [H, ρ] + κ_{tol} (2 a ρ a^{†} - a^{†} a ρ - ρ a^{†} a) + κ_{tol} (2 b ρ b^{†} - b^{†} b ρ - ρ b^{†} b) + γ (2 σ^{-} ρ σ^{+} - σ^{+} σ^{-} ρ - ρ σ^{+} σ^{-}),$ (2)where $ρ$ is the density operator. From Eq. (2), we can derive the equations of motion, $\dot{a} = i {\tilde{Δ}}_{1} a + α_{in} \sqrt{2 κ_{ex}} - i g σ^{-} - i h b,$ (3a) ${\dot{σ}}^{-} = i {\tilde{Δ}}_{2} σ^{-} + i g σ_{z} a,$ (3b) $\dot{b} = i {\tilde{Δ}}_{1} b - i h a,$ (3c)and obtain the steady-state solution, $〈 a 〉 = \frac{i α_{in} \sqrt{2 κ_{ex}} {\tilde{Δ}}_{1} {\tilde{Δ}}_{2}}{{\tilde{Δ}}_{1} ({\tilde{Δ}}_{1} {\tilde{Δ}}_{2} + 〈 σ_{z} 〉 g^{2}) - {\tilde{Δ}}_{2} h^{2}},$ (4)where $σ_{z} = σ^{+} σ^{-} - σ^{-} σ^{+}$ , ${\tilde{Δ}}_{1} = Δ_{1} + i κ_{tol}$ , and ${\tilde{Δ}}_{2} = Δ_{2} + i γ$ . According to the input–output relation, $〈 a_{out} 〉 = α_{in} - \sqrt{2 κ_{ex}} 〈 a 〉$ , the transmission amplitude is defined as $t_{ω} = 〈 a_{out} 〉 / α_{in}$ . Thus, we can get the transmission amplitude in port 2: $t_{ω} = \frac{{\tilde{Δ}}_{1} [(Δ_{1} + i κ_{in} - i κ_{ex}) {\tilde{Δ}}_{2} + σ_{z} g^{2}] - {\tilde{Δ}}_{2} h^{2}}{{\tilde{Δ}}_{1} ({\tilde{Δ}}_{1} {\tilde{Δ}}_{2} + σ_{z} g^{2}) - {\tilde{Δ}}_{2} h^{2}} .$ (5)

The transmission of port 2 can be obtained from $T = | t_{ω} |^{2}$ . Moreover, the full quantum dynamics of the system can be found by numerically solving Eq. (2) in a truncated space of photon number for the whispering-gallery modes.

2.2. Single-photon transport method

Hereafter, we consider only a single photon in our system. Based on the SPT theory^[4,26,50], our previous work^[35] has given a transmission amplitude for such a chiral system. The Hamiltonian for the single-excitation system takes the form^[35] $H = \int d x c_{F}^{†} (x) (ω_{0} - i v_{g} \frac{\partial}{\partial x}) c_{F} (x) + \int d x c_{B}^{†} (x) (ω_{0} + i v_{g} \frac{\partial}{\partial x}) c_{B} (x) + (Ω - i κ_{in}) a^{†} a + (Ω - i κ_{in}) b^{†} b + (Ω_{e} - i γ) a_{e}^{†} a_{e} + Ω_{g} a_{g}^{†} a_{g} + \int d x δ (x) [V_{a} c_{F}^{†} (x) a + V_{a}^{*} a^{†} c_{F} (x)] + \int d x δ (x) [V_{b} c_{B}^{†} (x) b + V_{b}^{*} b^{†} c_{B} (x)] + g a σ_{+} + g^{*} a^{†} σ_{-} + h b^{†} a + h^{*} a^{†} b,$ (6)where $c_{F / B}^{†} (x)$ is a Bosonic operator creating a forward- or back-moving photon with reference frequency $ω_{0}$ at $x$ in the waveguide, $σ^{+} = a_{e}^{†} a_{g}$ ( $σ^{-} = a_{g}^{†} a_{e}$ ) is the raising (lowering) operator for the QE with transition frequency $ω_{qe} = Ω_{e} - Ω_{g}$ , $g$ is the coupling strength for the interaction between the QE and the CCW mode $a$ , and the QE is decoupled to the CW mode $b$ in our system. $V_{a / b}$ is the waveguide-microresonator coupling strength of mode $a$ or $b$ . We set $V_{a} = V_{b} = V$ and thus have the external decay rate of the microresonator $κ_{ex} = V^{2} / 2 v_{g}$ , where $v_{g}$ is the group velocity of the photon in the waveguide.

A single-excitation state for the system is given by $| ψ 〉 = \int d x [{\tilde{φ}}_{F} (x, t) c_{F}^{†} (x) + {\tilde{φ}}_{B} (x, t) c_{B}^{†} (x)] | \emptyset 〉 + [{\tilde{e}}_{a} (t) a^{†} + {\tilde{e}}_{b} (t) b^{†} + {\tilde{e}}_{qe} (t) σ^{+}] | \emptyset 〉,$ (7)with the eigenfrequency $ω$ , where ${\tilde{φ}}_{F / B} (x, t)$ is the single-photon wave function of the forward- or backward-moving mode, ${\tilde{e}}_{a / b}$ is the excitation amplitude of mode $a$ or $b$ , and ${\tilde{e}}_{qe}$ is the excitation amplitude of the QE. Note that $ϒ = e^{- i ω t} ϒ$ with $ϒ \in {φ_{R}, φ_{L}, e_{a}, e_{b}, e_{q}}$ . $| \emptyset 〉$ is the vacuum state. To solve the single-photon transmission amplitude, we take $φ_{F} (x) = e^{i q x} [θ (- x) + t_{ω} θ (x)]$ and $φ_{B} (x) = r_{ω} e^{- i q x} θ (- x)$ with the Heaviside step function $θ (x)$ , where $t_{ω}$ ( $r_{ω}$ ) is the transmission (reflection) amplitude, and $q$ is the wave vector of the input field with the frequency around $ω$ . Based on the Schrödinger equation in real space, $H | ψ 〉 = i \partial | ψ 〉 / \partial t$ , we can derive the steady-state transmission amplitude in port 2: $t_{ω} = \frac{{\tilde{Δ}}_{1} [(Δ_{1} + i κ_{in} - i κ_{ex}) {\tilde{Δ}}_{2} - g^{2}] - {\tilde{Δ}}_{2} h^{2}}{{\tilde{Δ}}_{1} ({\tilde{Δ}}_{1} {\tilde{Δ}}_{2} - g^{2}) - {\tilde{Δ}}_{2} h^{2}},$ (8)where the detunings $Δ_{1}$ , ${\tilde{Δ}}_{1}$ , and ${\tilde{Δ}}_{2}$ are the same as the parameters in Sec. 2.1.

If we consider $σ_{z} = - 1$ in the ME method, that is, the weak probe field approximation^[74], we can find that Eq. (5) and Eq. (8) are equivalent. In Sec. 2.4, we will verify that the TM method is consistent with the SPT method.

2.3. Transfer matrix method

Next, we study the chiral QE-microresonator system using the TM method. Under the notation in Fig. 1, the coupling relation between the waveguide and the microresonator can be written as ${\begin{cases} a_{1} = t^{*} b_{1} - κ^{*} a_{0} \\ b_{0} = t a_{0} + κ b_{1} \end{cases}, {\begin{cases} c_{1} = t^{*} d_{1} - κ^{*} c_{0} \\ d_{0} = t c_{0} + κ d_{1} \end{cases},$ (9)where $t$ and $κ$ are the transmission and coupling coefficients, and $| t |^{2} + | κ |^{2} = 1$ for lossless coupling. We write Eq. (9) in a matrix form: $\begin{matrix} (\begin{array}{l} a_{0} \\ b_{0} \\ c_{0} \\ d_{0} \end{array}) \end{matrix} = \frac{1}{κ^{*}} (\begin{matrix} - 1 & t^{*} & 0 & 0 \\ - t & 1 & 0 & 0 \\ 0 & 0 & - 1 & t^{*} \\ 0 & 0 & - t & 1 \end{matrix}) \begin{matrix} (\begin{array}{l} a_{1} \\ b_{1} \\ c_{1} \\ d_{1} \end{array}) \end{matrix} \equiv M_{cpl} (\begin{array}{l} a_{1} \\ b_{1} \\ c_{1} \\ d_{1} \end{array}) .$ (10)

The size of the QE and the scatterer is much smaller than that of the structure of the microresonator, so theoretically they can be treated as particles. We assume the coupling point of the waveguide with the microresonator, QE, and scatterer divide the microresonator into three parts with lengths $L_{j}$ (j = 1, 2, 3), satisfying $L_{1} + L_{2} + L_{3} = 2 π R$ ; see Fig. 1. Here, $R$ is the radius of the microresonator. The field component notations are shown in Fig. 1. When a single photon propagates around the microresonator, it will accumulate propagation phases $θ_{j} = β L_{j}$ and may attenuate with loss $α_{j} (L_{j})$ ^[53]. We take $θ = θ_{1} + θ_{2} + θ_{3}$ and $α = α_{1} α_{2} α_{3}$ . The factor $β$ is the propagation constant in the microresonator as given by $β = n_{eff} ω / c$ , where $n_{eff}$ is the effective refractive index, and $ω$ is the frequency. Thus, we have the transfer relation $(\begin{array}{l} a_{1} \\ b_{1} \\ c_{1} \\ d_{1} \end{array}) = M_{pro} M_{x} (\begin{array}{l} a_{3} \\ b_{3} \\ c_{3} \\ d_{3} \end{array}),$ (11a) $b_{3} = α_{2} e^{i θ_{2}} a_{3},$ (11b) $d_{3} = α_{2} e^{i θ_{2}} c_{3},$ (11c)where $M_{pro} = (\begin{matrix} α_{1}^{- 1} e^{- i θ_{1}} & 0 & 0 & 0 \\ 0 & α_{3} e^{i θ_{3}} & 0 & 0 \\ 0 & 0 & α_{3}^{- 1} e^{- i θ_{3}} & 0 \\ 0 & 0 & 0 & α_{1} e^{i θ_{1}} \end{matrix}),$ (12) $M_{x}$ is derived from the contributions of the QE and the scatterer, and its exact form will be discussed below. We refer to $M_{cpl}$ and $M_{pro}$ as coupling and propagation matrices. Combining Eqs. (10) and (11), we obtain the TM as $(\begin{array}{l} a_{0} \\ b_{0} \\ c_{0} \\ d_{0} \end{array}) = M_{cpl} M_{pro} M_{x} (\begin{array}{l} a_{3} \\ b_{3} \\ c_{3} \\ d_{3} \end{array}) .$ (13)

We consider a single input of port 1 ( $c_{0} = 0$ ). It excites the CCW-direction whispering-gallery mode. In the following, we will discuss the single-photon transmission in four different cases.

2.3.1. No two-level QE and no scatterer

We first consider the case without two-level QEs and scatterers; the form of $M_{x}$ can be directly obtained: $M_{x} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) .$ (14)

In the absence of scatterers, the CCW and CW modes are decoupled. Substituting Eq. (14) and Eq. (11) into Eq. (13), we get the transmission amplitude in port 2^[53]: $t_{ω} = \frac{b_{0}}{a_{0}} = \frac{- t + α e^{i θ}}{- 1 + α t^{*} e^{i θ}} .$ (15)

2.3.2. No two-level QE and one scatterer

In this case, we consider the effect of the scatterer in the microresonator. The relation between the amplitudes can be written as $b_{2} = t_{s} b_{3} + r_{s} c_{2}$ , $c_{3} = t_{s} c_{2} + r_{s} b_{3}$ , $a_{3} = a_{2}$ , and $d_{2} = d_{3}$ . Thus, we have $M_{x} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 / t_{s} & r_{s} / t_{s} & 0 \\ 0 & - r_{s} / t_{s} & 1 / t_{s} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}),$ (16)where $t_{s}$ and $r_{s}$ are the transmission and reflection coefficients, respectively. They satisfy $| t_{s} |^{2} + | r_{s} |^{2} = 1$ when the dissipation of the scatterer is neglected. The two whispering-gallery modes are coupled to each other in this case. We assume the scatterer is weak, thus we can write $t_{s}$ and $r_{s}$ in the following forms^[73]: $t_{s} = \cos ε \approx 1 - \frac{ε^{2}}{2}, r_{s} = i \sin ε \approx i ε .$ (17)

Then, we have the transmission amplitude in port 2: $t_{ω} = \frac{b_{0}}{a_{0}} = \frac{- t + α e^{i θ} \frac{t_{s} - t^{*} α e^{i θ}}{1 - t_{s} t^{*} α e^{i θ}}}{- 1 + α t^{*} e^{i θ} \frac{t_{s} - t^{*} α e^{i θ}}{1 - t_{s} t^{*} α e^{i θ}}} .$ (18)

2.3.3. One two-level QE and no scatterer

Here, we study the effect of a two-level QE directionally coupled to a microresonator. Because the QE is in a specific spin ground state or the polarization-selective energy-level transition, the coupling of the QE and the evanescent field on the microresonator is direction-dependent. The reflection of single-photon propagation will vanish due to such chiral QE–light interaction^[32,35]. In this case, the single photon will not excite the CW mode, leading to decoupling between the CCW and CW modes. We assume the single photon through the two-level QE with a transmission coefficient $t_{qe}$ , i.e., $a_{3} = t_{qe} a_{2}$ and $d_{2} = d_{3}$ , such that $M_{x} = (\begin{matrix} t_{qe}^{- 1} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}),$ (19)and $t_{ω} = \frac{b_{0}}{a_{0}} = \frac{- t + α e^{i θ} t_{qe}}{- 1 + α {t^{*}}^{} e^{i θ} t_{qe}} .$ (20)

The specific form of $t_{qe}$ will be discussed below.

2.3.4. One two-level QE and one scatterer

Combining with the above discussions, we can obtain the form of $M_{x}$ , considering both a two-level QE directionally coupled to the microresonator and a scatterer: $M_{x} = (\begin{matrix} t_{qe}^{- 1} & 0 & 0 & 0 \\ 0 & 1 / t_{s} & r_{s} / t_{s} & 0 \\ 0 & - r_{s} / t_{s} & 1 / t_{s} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) .$ (21)

The transmission amplitude can be calculated as $t_{ω} = \frac{b_{0}}{a_{0}} = \frac{- t + α e^{i θ} t_{qe} \frac{t_{s} - t^{*} α e^{i θ}}{1 - t_{s} t^{*} α e^{i θ}}}{- 1 + α {t^{*}}^{} e^{i θ} t_{qe} \frac{t_{s} - t^{*} α e^{i θ}}{1 - t_{s} t^{*} α e^{i θ}}} .$ (22)

It can be found that the transmission amplitude $t_{ω}$ is independent of the relative distance $L_{2}$ between the QE and the scatterer on the microresonator from Eq. (22). This is because the chiral coupling of the QE with the microresonator only causes a modulation of the transmission $t_{qe}$ of the single photon propagating in the microresonator. Such modulation does not depend on the position of the QE on the microresonator; see Eq. (21). Therefore, the case of a single-emitter coupling can be generalized to multi-emitter cases by successively multiplying $t_{qe}$ in the transfer relation of the field amplitudes.

2.4. Single-photon phase-amplitude modulator

We define the round-trip time of the microresonator, $τ_{rt} = 2 π R n_{eff} / c$ , that a photon needs to make a round trip in the microresonator of length $2 π R$ . It is the inverse of the free spectral range $F$ , i.e., $τ_{rt} = 1 / F$ ^[75]. Since $F ≫ 1$ for a microresonator, $τ_{rt}$ is a small amount. We have $\exp (i θ) = \exp [i (ω - Ω) τ_{rt}] \approx 1 + i Δ_{1} τ_{rt}$ . On the one hand, for a single photon having travelled a round trip in the microresonator, we have $a_{1} (τ_{rt}) = α t^{*} a_{1} (0)$ from the transfer relation. The circulating power meets $| a_{1} (τ_{rt} {) |}^{2} = α^{2} t^{2} | a_{1} {(0) |}^{2}$ . On the other hand, we can obtain $| a_{1} (τ_{rt} {) |}^{2} = \exp (- 2 κ_{tol} τ_{rt}) | a_{1} {(0) |}^{2}$ from the dissipative properties of the microresonator. Hence, we have $α = e^{- κ_{in} τ_{rt}} \approx 1 - κ_{in} τ_{rt},$ (23a) $t = e^{- κ_{ex} τ_{rt}} \approx 1 - κ_{ex} τ_{rt} .$ (23b)

Because the size of the two-level QE is much smaller than that of the bend structure of the microresonator, the interaction between the evanescent field and the QE can be approximated as a waveguide coupling with a two-level QE directionally^[32], with a transmission coefficient $t_{qe} = \frac{ω - ω_{qe} + i (γ - Γ)}{ω - ω_{qe} + i (γ + Γ)},$ (24)where $Γ$ is the decay rate from the QE into the waveguide. Therefore, substituting Eqs. (17), (23), and (24) into Eq. (22) and ignoring the second-order small quantity, we have $t_{ω} = \frac{- t + α e^{i θ} t_{qe} \frac{t_{s} - t^{*} α e^{i θ}}{1 - t_{s} t^{*} α e^{i θ}}}{- 1 + α t^{*} e^{i θ} t_{qe} \frac{t_{s} - t^{*} α e^{i θ}}{1 - t_{s} t^{*} α e^{i θ}}} \approx \frac{κ_{ex} τ_{rt} - 1 + (1 - κ_{in} τ_{rt} + i Δ_{1} τ_{rt}) [(1 - \frac{2 i Γ}{{\tilde{Δ}}_{2} + i Γ}) (1 + \frac{ε^{2}}{i {\tilde{Δ}}_{1} τ_{rt} + ε^{2} / 2})]}{- 1 + (1 + i {\tilde{Δ}}_{1} τ_{rt}) [(1 - \frac{2 i Γ}{{\tilde{Δ}}_{2} + i Γ}) (1 + \frac{ε^{2}}{i {\tilde{Δ}}_{1} τ_{rt} + ε^{2} / 2})]} \approx \frac{{\tilde{Δ}}_{1} [(Δ_{1} + i κ_{in} - i κ_{ex}) {\tilde{Δ}}_{2} - Γ (2 / τ_{rt} - κ_{tol})] - {\tilde{Δ}}_{2} \frac{ε^{2}}{τ_{rt}^{2}}}{{\tilde{Δ}}_{1} [{\tilde{Δ}}_{1} {\tilde{Δ}}_{2} - Γ (2 / τ_{rt} - κ_{tol})] - {\tilde{Δ}}_{2} \frac{ε^{2}}{τ_{rt}^{2}}} = \frac{{\tilde{Δ}}_{1} [(Δ_{1} + i κ_{in} - i κ_{ex}) {\tilde{Δ}}_{2} - Γ (2 F - κ_{tol})] - {\tilde{Δ}}_{2} {(ε \times F)}^{2}}{{\tilde{Δ}}_{1} [{\tilde{Δ}}_{1} {\tilde{Δ}}_{2} - Γ (2 F - κ_{tol})] - {\tilde{Δ}}_{2} {(ε \times F)}^{2}} .$ (25)

Comparing Eq. (25) with Eq. (8), we can find that if we take $(2 F - κ_{tol}) Γ = g^{2}, ε \times F = h,$ (26)the TM method and the SPT method are consistent. This also proves that the assumption of Eq. (24) is well valid. For a system in which a two-level QE is chirally coupled to the waveguide, the light field interacts with the QE only once, and the decay rate from the QE into the waveguide is $Γ$ . However, when the QE is coupled to the microresonator, the photons in the microresonator interact with the QE many times. As a result, the microresonator has a feedback modulation to the decay rate $Γ$ . If we define $Γ_{eff} = \sqrt{(2 F - κ_{tol}) Γ}$ , then the physical meaning of $Γ_{eff}$ is the effective decay rate from the QE into the microresonator. It is equal to the coupling strength $g$ . For the second term in Eq. (26), $ε$ is a dimensionless parameter in the TM method. It is equal to the backscattering strength $h$ by multiplying the free spectral range $F$ , which has a dimension of frequency. Note that $ε$ only needs to vary from 0 to $π$ ; see Eq. (17). Thus, $ε$ reflects the normalized magnitude of backscattering strength in the microresonator.

Note that the chiral coupling of the two-level QE to the microresonator does not require additional auxiliary fields. This vacuum-induced interaction causes a phase shift and an amplitude modulation of a single photon passing through the QE. Therefore, the two-level QE can be treated as a single-photon phase-amplitude modulator. We divide Eq. (24) into two parts: $t_{qe} = \exp (i φ_{pha}) \exp (- φ_{dis}),$ (27)where $\exp (i φ_{pha}) = \arg (t_{qe})$ represents the change of the phase, and $\exp (- φ_{dis}) = | t_{qe} |$ describes the attenuation of the amplitude. The additional propagation phase introduced by the two-level QE can be equivalent to a shift of the effective resonance frequency of the microresonator. When a single photon travels around the microresonator, in the absence of the QE, we have $θ = 2 π R β = 2 π m ω / Ω$ , where $m = Ω n_{eff} R / c$ is the modal number. But, if we consider the chiral QE-microresonator interaction, the additional propagation phase $φ_{pha}$ leads to $θ + φ_{pha} = 2 π m ω / Ω_{eff}$ . The effective resonance frequency of the microresonator is $Ω_{eff} \approx Ω (1 - \frac{φ_{pha} Ω}{2 π m ω}),$ (28)and $Ω_{eff} \approx Ω (1 - φ_{pha} / 2 π m)$ for $Ω / ω \approx 1$ .

In general, by equating a two-level QE directionally coupled with a microresonator to a single-photon phase-amplitude modulator, we can use the TM method to solve the SPT problem in such chiral QE-microresonator systems. This only needs to be multiplied by a transmission coefficient $t_{qe}$ in the transfer relation. Furthermore, this approach can be extended to more complex systems such as a coupled-resonator optical waveguide interacting with an array of two-level QEs in a chiral way^[76].

3. Results

Below, we numerically study our system to prove the consistency of these three methods. For the TM and SPT methods, we solve Eqs. (22) and (8) directly, whereas, for the ME method, we perform a full quantum dynamics simulation using Eq. (2). We set a prepared QD as the two-level QE for coupling to a silicon-based microresonator in a chiral way. In Figs. 1 –4, the experimentally available parameters are chosen as^[60,68,77] $R = 10.5 μm$ , $n_{eff} = 1.5$ , $F / 2 π = 3 THz$ , $α_{in} / \sqrt{κ_{tol}} = 0.1$ , and $γ / 2 π = 6 MHz$ . The conversion relationships between the parameters of the three methods are given by Eqs. (23) and (26). We take $t = α = 0.99$ , thus satisfying the critical coupling condition, $κ_{ex} / 2 π = κ_{in} / 2 π = 30 GHz$ . The frequency of the QD is resonant with the microresonator, i.e., $ω_{qe} = Ω$ .

Figure 2.Transmission spectra of a waveguide coupled with a microresonator. The blue solid, red dashed, and green dotted curves are calculated by the TM, ME, and SPT methods, respectively. The settings in the following figures are the same: (a) in the absence of backscattering, (b) and (c) in presence of the backscattering with strengths h = κ_in and h = 10κ_in, respectively. See Sec. 3 for other parameters.

Figure 3.Transmission spectra for a chiral QE-microresonator system without considering the backscattering: (a)–(c) Γ = 0.1γ, Γ = γ, and Γ = 100γ, respectively.

Figure 4.Transmission spectra for a chiral QE-microresonator system with the QE and the scatterer: (a) Γ = 100γ, h = κ_in and (b) Γ = 100γ, h = 10κ_in.

We first consider the case without a two-level QD, corresponding to $Γ = 0$ ( $g = 0$ ), shown in Fig. 2. When the strength $h = 0$ , the deep transmission appears at the resonance point [see Fig. 2(a)]. As the strength $h$ increases, the transmission spectrum gradually splits [see Figs. 2(b) and 2(c)]. The calculation results of the three methods are exactly the same.

Then, we consider the chiral coupling of a two-level QD. By modeling the two-level QD chirally coupled to the microresonator as a single-photon phase-amplitude modulator, we can use the TM method to solve SPT problems. Figure 3 shows the transmission spectra without scatterers. The presence of the two-level QD causes the transmission spectrum to split^[35]. We can find that the transmission spectra calculated by the three methods are consistent regardless of whether it is under weak coupling, $Γ = 0.1 γ$ and $Γ = γ$ ( $g / κ_{tol} = 0.03$ and $g / κ_{tol} = 0.1$ ), or strong coupling, $Γ = 100 γ$ ( $g / κ_{tol} = 1$ ). The results taking into account the effect of backscattering are shown in Fig. 4. We consider the case of strong coupling, $Γ = 100 γ$ . Whether it is in the case of weak backscattering [see Fig. 4(a)] or strong backscattering [see Fig. 4(b)], the calculation results are consistent. Therefore, our numerical results further confirm the above theoretical analyses and prove the correctness of the parameter correspondence among these three methods.

We now discuss the effect of the pump power in the ME method. It is proportional to the driving amplitude $α_{in}$ . As we have analyzed above, the three methods are equivalent only if the probe field is a weak pump. It can be found that with the increase of the driving amplitude, the results of the TM method gradually have a difference with those of the ME method, especially at the resonant frequency $Δ / κ_{tol} = 0$ ; see Fig. 5(a). This is because the average photon number of the system reaches its maximum at the resonance, and it is no longer a single-photon case. Figure 5(b) shows the transmission spectra versus the driving amplitude $α_{in}$ at $Δ / κ_{tol} = 0$ . The result of the TM method is constant because it is independent of $α_{in}$ . In contrast, in the ME method, the transmission decreases as $α_{in}$ increases for both strong and weak coupling of the QE. As shown in Fig. 5(b), in the range of $α_{in} / \sqrt{κ_{tol}} \leq 0.23$ , the transmission calculated by the ME method is greater than 0.9 (see the black dotted line). At this time, the result calculated by the TM method, $T \approx 1$ , is almost consistent with it. For a larger pump power, the TM method is no longer accurate. It is worth noting that at off-resonance, the TM method is still valid, as shown in Fig. 5(a).

Figure 5.(a) Transmission spectra for different driving amplitudes α_in (corresponding to different pump powers), where the blue solid curve is calculated by the TM method, and the green dash-dotted curve (the red dashed curve, the purple dotted curve) is calculated by the ME method, with $α_{in} / \sqrt{κ_{tol}} = 0.1$ ( $α_{in} / \sqrt{κ_{tol}} = 0.5$ , $α_{in} / \sqrt{κ_{tol}} = 1$ ). (b) Transmission spectra as a function of α_in at different decay rates, where Γ = γ (the blue solid curve) and Γ = 100γ (the red dash-dotted curve) correspond to weak and strong coupling, respectively. Other parameters are κ_ex = κ_in = 0.5κ_tol, γ/κ_tol = 1 × 10⁻⁴, g/κ_tol = 1, and h = κ_in.

4. Conclusion

We demonstrate that a two-level QE can be treated as a single-photon phase-amplitude modulator in a chiral QE-microresonator system. Based on this, we can solve the SPT problem by the method of TM. Theoretical analyses confirm that the TM method is consistent with the ME and the SPT methods. Also, the results of numerical analysis prove the correctness of parameter relationships. Without loss of generality, the TM method can be extended to solve the single-photon transmission of any number of two-level QEs chirally coupled to multiple microresonators.

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