1. Introduction

Waveguide quantum electrodynamics (QED) studies the interaction between the electromagnetic continuum confined to waveguides and quantum emitters (natural or artificial atoms). Owing to the advantage of scalability, waveguide QED is widely used in quantum information processing, such as quantum network [1], quantum communication [2–5] and quantum simulation [6]. In waveguide QED, an important research topic is how to control the photon transport in waveguides by quantum emitters. Over the past few years, single-photon or few-photon scattering in one-dimensional (1D) waveguides by two-level emitters have been studied intensively by several approaches [7–26], including Bethe-anatz approach [9–12], Lippmann-Schwinger formalism [13–18], input-output theory [19], path integral formalism [20], and dynamical theory [21–23]. A critical conclusion is that, in the single-transverse-mode region, the photon is perfectly reflected when its frequency resonates with the two-level emitter’s transition frequency. This can be explained as a result of Fano resonance, due to destructive interference between the directly transmitted photon and the photon re-emitted by the emitter [27]. Based on this result, several schemes of single-photon devices with different functions have been proposed, like optical switches [11], transistors [28], routings [2–5], and frequency converters [29].

Although many remarkable achievements have been made in the field of waveguide QED, most work is based on ideal 1D waveguide systems with linear dispersion relation and no cut-off frequencies. However, the cross section size of realistic waveguides is finite, and it leads to the existence of nonlinear dispersion relation and multiple transverse modes with corresponding cut-off frequencies. With the development of high-frequency coherent light sources such as X-ray lasers [30,31], when the wavelength of incident photon is smaller than the characteristic length of the waveguide’s cross-section, multiple transverse modes will be involved in the photon scattering process simultaneously. Due to the indirect interaction between waveguide’s transverse modes mediated by the emitter, the photon is inevitably scattered from one transverse mode to another [32–34]. At this point, the ideal 1D waveguide no longer holds. A few studies have also shown that the nonlinear dispersion relation of waveguides would lead to the frequency shift of the reflection resonance [11,16,35–37]. What is more, if two-level emitters are replaced by three-level emitters [38–45], the systems may exhibit richer behaviors such as electromagnetically induced transparency (EIT). To sum up, it is an urgent and meaningful scientific problem to study the photon transport property in the realistic waveguides with nonlinear dispersion relation and finite cross section.

In this paper, we study the single photon scattering by a V-type three-level emitter in an easy-to-compute infinite rectangular waveguide. Both the nonlinear dispersion relation and multi-transverse-mode effect are taken into account. According to the Maxwell equations and boundary conditions, rectangular waveguides can only support transverse magnet (TM) and transverse electric (TE) guiding modes. Specifically speaking, we consider the case that the emitter is locally embedded in the center of rectangular cross section and the electric dipole of the emitter is along the axis of waveguide. Due to the fact that $E_{z}=0$ for all TE modes, the electric-dipole interaction exists only between the emitter and the waveguide’s TM modes. This leads to the transition between the emitter’s ground state and two upper states. By using the scattering matrix method combined with resolvent operator techniques, we analytically derive the reflection and transmission coefficients for a single photon impinging onto the emitter. Meanwhile, the conditions for perfect transmission and reflection are analysed in detail. When the frequency of incident photon is between the lowest and the second lowest cutoff frequencies, the system is in the regime of single transverse mode. In the single-mode case, both the complete transmission and transmission can be achieved under EIT condition (destructive interference between two radiation pathways with different emitter’s upper states) and Fano resonance (destructive interference between the directly transmitted photon and re-emitted photon), respectively. Furthermore, with the frequency of incident photon increasing, the effect of multiple transverse modes becomes important and can not be ignored. The quantum interference between multiple scattering pathways with different transverse modes can be used to manipulate the single photon transport. We find in the regime of multiple transverse modes, the emitter becomes transparent when the superposition of waveguide modes at the position of the emitter is zero, and the perfect reflection is possible only for a specific coherent superposition state (CSS) [33].

The paper is organized as follows: In Sec. 2, we introduce the physical model and Hamiltonian. In Sec. 3, with the use of Green’s function formalism, the scattering matrix $S$ is derived. In Sec. 4, we aim to explore the regulation mechanism of the three-level emitter on the photon transport. The cases in both single-mode and multi-mode regions are taken into account. Finally, Sec. 5, is devoted to some conclusions.

2. Physical model

We consider that an infinite rectangular waveguide interacts with a V-type three-level emitter, as shown in Fig. 1. The axis of waveguide is along the $z$ direction. The $a$ and $b$ are the length and width of the cross section. Here the emitter is located in the center of the cross section with positions $\vec {r}_{0}=(a/2,b/2,z_{0})$. Both two upper states $\vert e_{1}\rangle$ and $\vert e_{2}\rangle$ of the emitter are coupled to the ground state $\vert g\rangle$ via the rectangular waveguide. The total Hamiltonian consists of the free part $H_{0}$ and interaction part $V$. $H_{w}$ and $H_{e}$ denote the free Hamiltonian of waveguide and emitter,

 

Fig. 1. Schematic of controlling single-photon scattering in an infinite rectangular waveguide by a V-type three-level emitter. The emitter is located in the center of cross section and dipole-coupled to waveguide modes. Both upper states $\vert e_{i}\rangle , i=1,2,$ couple to the waveguide with the strengths $\zeta _{1}$ and $\zeta _{2}$, respectively. The transition frequencies between two upper states $\vert e_{i}\rangle$ and ground state $\vert g\rangle$ are $\omega _{e_{i}}$.

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According to Maxwell equations and boundary conditions, two types of guiding modes exist in the rectangular waveguide, including TM modes ($H_{z}=0$) and TE modes ($E_{z}=0$). Due to the transverse spatial constraints, two positive integers $m,n$ are introduced to denote different TM$_{mn}$ modes and different TE$_{mn}$ modes. Based on the Coulomb gauge ($\nabla \cdot \vec {A} =0$ and $\varphi =0$), the vector potential $\vec {A}$ can be obtained by standard quantization [33], (the unit $\hbar =1$ is used)

The mode functions for the TE$_{mn}$ or TM$_{mn}$ modes are

Then the electric and magnetic fields can both be derived via $\vec {E}=-\frac {\partial \vec {A}}{\partial t},\vec {B}=\nabla \times \vec {A}$. After discarding the constant term, the Hamiltonian of rectangular waveguide reads

Furthermore, we adopt a three-level approximation for the emitter. The transition frequencies between two upper states $\vert e_{i}\rangle$ and ground state $\vert g\rangle$ are $\omega _{e_{i}} (i=1,2)$,

For the sake of simplicity, we consider the electric dipole of the emitter is oriented along the $z$ direction. Besides, when the wavelength of the photon in the waveguide is much larger than the size of the emitter (on the order of a few Bohr radii), the electric dipole approximation can be applied, with which the electric quadrupole and magnetic dipole interactions can be neglected [14]. Therefore, the emitter only couples to the TM modes because of $E_{z}=0$ for all TE modes. The interaction Hamiltonian can be obtained by the minimal coupling prescription, $V=-\frac {q}{m}\vec {A}\cdot \vec {p}=-\frac {q}{m}A_{z}p_{z}$. The matrix elements of emitter’s momentum can be derived as $\left \langle e_{i}\right \vert p_{z}\left \vert g\right \rangle =i\omega _{e_{i}}m\left \langle e_{i}\right \vert z\left \vert g\right \rangle$ based on the canonical commutation relation $\vec {p}=-im[\vec {r},H_{e}]$. Note that the $A_{z,{\textrm {TM}}}$ has sine modulations in the x and y directions from Eq. (4),

Equation (8) indicates that the emitter with position $\vec {r}_{0}=(a/2,b/2,{z_{0}})$ is exactly where the electric field maximizes and the magnetic field is zero. The $A_{z,{\textrm {TM}}}$ takes no-zero value only if both $m$ and $n$ take positive odd integers. Thus the TM$_{10}$ and TM$_{01}$ modes can be disregarded. What is more, as the first derivatives of $\sin (\frac {m\pi }{a}x) \sin ( \frac {n\pi }{b}y)$ with $x$ and $y$ are zero, it is legitimate to neglect the spatial variation of the electromagnetic field over the size of emitter. Even if the emitter is slightly away from the center $\delta {r}_{0}$, the influence on our following results is at most second-order in $\delta {r}_{0}$. All in all, the interaction Hamiltonian is

Hereafter the $k_{z}$ is replaced by $k$, and the subscript $(m,n)$ are replaced by a sequence number $j$ according to the ascending order of cut-off frequencies. Then $\omega _{j,k}\equiv \omega _{\vec {k}}$ and $\omega _{j}\equiv \omega _mn$. In this paper we set $a=1.5b$, thus $j=1,2,3\cdots$ denote $(m, n)=(1,1), (3,1), (1,3)\cdots$, respectively. In Eq. (9) the sum over $i=1,2$ includes two dipole-allowed transitions between $\vert e_{i}\rangle$ and $\vert g\rangle$, while the sum over $j$ denote the interaction between the emitter and different TM modes. It should be noted that the dependence of coupling strength $g^{(i)}_{j,k}$ on the emitter and waveguide is factored,

3. Scattering matrix $S$

We now derive the scattering matrix $S$ for the Hamiltonian in Eq. (1) using Green’s function formalism. We assume the input state is a single-photon state $\vert \varphi _{j,k}\rangle \equiv a^{\dagger }_{j,k}\vert 0\rangle \vert g\rangle$, a eigenstate of the free Hamiltonian $H_{0}$ with eigenenergy $E=\omega _{j,k}$. Then according to the time-independent scattering theory, the $S$ matrix is often presented by $T$ operator [13],

The first term on the right hand side of Eq. (11) represents the directly transmitted photon, and the second term reflects the single-photon scattering process by the emitter. The energy conservation is guaranteed by the delta function. Here the $T(\nu )$ operator can be defined in terms of Green’s function $G(\nu )$ as

As the state $\vert \varphi _{j,k}\rangle$ is only coupled to $\vert e_{i}\rangle$ $(i=1,2)$ through the interaction $V$, the $T$ matrix elements in Eq. (11) can be rewritten as $\langle \varphi _{j^{\prime },k^{\prime }}\vert T(\nu ) \vert \varphi _{j,k}\rangle =\langle \varphi _{j^{\prime },k^{\prime }}\vert V P G(\nu )P V \vert \varphi _{j,k}\rangle$ with projection operators $P=\vert e_{1}\rangle \langle e_{1}\vert +\vert e_{2}\rangle \langle e_{2}\vert$, $Q=1-P=\sum _{j}\int _{-\infty }^{+\infty }dk\vert \varphi _{j,k}\rangle \langle \varphi _{j,k}\vert$. According to $[PG(\nu )P]^{-1}=\nu -PH_{0}P-PW(\nu )P$, the projection of Green’s function $G(\nu )$ in the subspace $\mathcal {E}$ subtended by $\{\vert e_{1}\rangle , \vert e_{2}\rangle \}$ can be derived as [13,14]

Because the coupling strength $g_{j,k}^{(i)}$ in Eq. (10) is factored, it is obvious that $W_{p,q}\propto \zeta _{p}\zeta _{q}^{*}$ and the $\frac {W_{p,q}}{\zeta _{p}\zeta _{q}^{*}}$ is independent of $p$ and $q$. Then the $T$ matrix elements is obtained from Eqs. (12) and (13),

In the last line of Eq. (16), the $\vert \zeta _{1}\vert ^{2}W_{22}=\vert \zeta _{2}\vert ^{2}W_{11}=\zeta _{1}^{*}\zeta _{2}W_{12}=\zeta _{1}\zeta _{2}^{*}W_{21}$ is used due to $W_{p,q}\propto \zeta _{p}\zeta _{q}^{*}$, and the function $f(E)$ is introduced,

Equation (16) allows us to understand the scattering process. Reading the first line from right to left suggests that the system, starting from the input state $\vert \varphi _{j,k}\rangle$, evolves into two discrete states $\vert e_{i}\rangle$ by the interaction $V$, then evolves under the effect "Hamiltonian" $PH_{0}P+PW(z)P$ in the subspace $\mathcal {E}$, finally passes through the effect of the coupling $V$ to the output state $\vert \varphi _{j^{\prime },k^{\prime }}\rangle$. Note that the $PW(z)P$ can be considered as a non-Hermitian "Hamiltonian" in the subspace $\mathcal {E}$. Using the property $\lim _{\varepsilon \rightarrow 0^{+}}\frac {1}{x-x_{0}+i\varepsilon }=\frac {\mathcal {P}}{x-x_{0}}-i\pi \delta \left (x-x_{0}\right )$ with the principal value function $\mathcal {P}$, the $W_{p,q}(z)$ in Eq. (15) can be divided into the real and imaginary parts. The real part determines the perturbed levels shift and the imaginary part describes the dissipative phenomena,

Here in the region of $E<\omega _{j}$, the red Lamb shift is neglected and the decay rate $\Gamma _{pq}(E)$ vanishes due to the step function $\Theta (E-\omega _{j})$. Thus, the real and imaginary parts of $\mathcal {D}(E+i0^{+})$ in Eq. (14) can also be achieved through the self-energy $W_{11}(z)$, $W_{22}(z)$ and $W_{12}W_{21}=W_{11}W_{22}$,

Until now, the scattering matrix $S$ is obtained by inserting Eqs. (16)-(22) into Eq. (11).

4. Single-photon scattering by a V-type emitter with finite cross section

In this section, we explore the regulation mechanism of the emitter on the single-photon transport. The influence of input states and finite cross section on the photon scattering is also analyzed.

Assume the emitter is initially in the ground state $\vert g \rangle$ and the input photon in a superposition state with a given energy $E=\omega _{in}$,

Here the number of TM modes in the input and outgoing states are the same $j^{\prime }_{\max }=j_{\max }$, as a result of the conservation of energy. That is to say, the $j^{\prime }_{\max }$ is determined by $\omega _{j^{\prime }_{\max }}\leq \omega _{in}<\omega _{j^{\prime }_{\max }+1}$. And from Eq. (16) the reflection amplitude $r_{j^{\prime }}$ is

Equation (24) implies that, when a photon propagates in the waveguide, it may pass through directly, or it may be absorbed by the emitter and then re-radiated. Due to the symmetry in the coupling to left and right going modes, $\langle \varphi _{j^{\prime },k_{j^{\prime }}}\vert T \vert \varphi _{j,k_{j}}\rangle =\langle \varphi _{j^{\prime },-k_{j^{\prime }}}\vert T \vert \varphi _{j,k_{j}}\rangle$, thus the photon re-radiated by the emitter is either reflected back or transmitted forward with the same probability $r_{j^{\prime }}$. Noting that the $r_{j^{\prime }}$ in Eq. (25) not only depends on the photon state density $\rho _{j^{\prime }}$ in the $j^{\prime }$-th mode, but also results from multiple scattering channels: all the modes ($j=1,2,\ldots , j_{\max }$) in the input state evolving through the $T$ matrix to the $j^{\prime }$-th mode with corresponding probabilities $c_{j}$. Thus, the photon transport can be co-regulated by the input state and the emitter’s parameters through $T$ matrix.

The reflectivity and transmissivity in the $j^{\prime }$-th mode are defined with the group velocity of photon being $\frac {d\omega _{j^{\prime },k^{\prime }}}{dk^{\prime }}|_{k^{\prime }=k^{\prime }_{j^{\prime }}}=\rho _{j^{\prime }}^{-1}$,

The total reflectivity $R=\sum _{j^{\prime }=1}^{j^{\prime }_{\max }}R_{j}$ and total transmissivity $T=\sum _{j=1}^{j_{\max }}T_{j}$ are obtained from Eqs. (25) and (26) with $R+T=1$,

Noting that two special cases should be pointed out. In the cases of degenerate upper states $(\omega _{e_{1}}=\omega _{e_{2}}=\omega _{e})$ and two-level emitter $(\zeta _{2}=0)$, the $\mathcal {D}$ in Eq. (14) should be replaced as follows:

4.1 Perfect transmission and perfect reflection conditions

Now we begin to analyze the conditions for perfect transmission and perfect reflection from Eq. (27). It can be seen that $\left \vert \varphi _{out}\right \rangle =\left \vert \varphi _{in}\right \rangle$ occurs under either of two conditions,

The physical mechanism of complete transmission derives from two kinds of destructive interference. The first one corresponds to the sum of $j$ in Eq. (25): when the incident photon is in the superposition state, there are multiple scattering channels from $\vert \varphi _{j,k_{j}}\rangle$ to $\vert e_{i}\rangle$. Under the condition of $\sum _{j=1}^{j_{max}}c_{j}\chi _{j,k_{j}}=0$, the superposition of waveguide modes has zero amplitude at the position of the emitter. This leads to the decoupling between emitter and waveguide and the emitter becomes transparent with respect to this input photon. The second kind of interference corresponds to emitter’s excited states: the excited emitter after absorbing the photon has two transition pathways to the ground state. In the case of $f(\omega _{in})=\vert \zeta _{1}\vert ^{2}(\omega _{in}-\omega _{e_{2}})+\vert \zeta _{2}\vert ^{2}(\omega _{in}-\omega _{e_{1}})=0$, the destructive interference between these two radiation pathways leads to perfect transmission, so-called EIT mechanism. Here the $\zeta _{i}=\frac {q\omega _{e_{i}}\langle e_{i}\vert z \vert g\rangle }{\sqrt {\pi {\epsilon _{0}} ab}}$ affects the coupling strength between the TM modes of the waveguide and $\vert e_{i} \rangle$, and $\omega _{in}-\Omega _{i}$ is the detuning between the input photon and emitter’s transition frequency. For two-level and degenerate three-level emitters, the latter destructive interference vanishes. Two-level emitters have only one transition pathway, while in degenerate three-level emitters the photon from two radiation pathways cannot be distinguished. This exactly highlights the advantages of non-degenerate three-level emitters in photon transport regulation.

We next analyze the perfect reflection conditions by using Cauchy-Buniakowsky-Schwarz inequality. The case of incident photon resonating with cut-off frequencies is excluded.

There exist two kinds of input-state may be completely reflected. One is the energy in the single transverse mode region with $\omega _{1}< \omega _{in}<\omega _{2}$. The other one is the input-state in multi transverse modes with $c_{j}\propto \rho _{j}\chi _{j,k_{j}}$, we call coherent superposition states (CSS). In the CSS case, $r_{j^{\prime }}=-ic_{j^{\prime }}\textrm {Im}[{\mathcal {D}} ]/\mathcal {D}$ and $R=\textrm {Im}[{\mathcal {D}} ]^{2}/|\mathcal {D}|^{2}$, which means the effect of emitter on each mode is the same. Thus, the photon scattering in the CSS case is similar to the single transverse mode case. This is due to the coherent interference between different scattering channels from $\vert \varphi _{j,k_{j}}\rangle$ to $\vert e_{i}\rangle$. The other necessary condition of perfect reflection is requirement for the emitter’s parameters to meet $\textrm {Re}(\mathcal {D})\equiv (\omega _{in}-\omega _{e_{1}})(\omega _{in}-\omega _{e_{2}}) -(\omega _{in}-\omega _{e_{2}})\Delta _{11}-(\omega _{in}-\omega _{e_{1}})\Delta _{22}=0$. It forecasts that the energy of perfect reflected photon resonates with the emitter’s re-normalized transition frequency. The frequency shift of the resonant frequency with respect to the bare frequency $\Omega _{i}$ is determined by $\Delta _{11}$ and $\Delta _{22}$, which stems from the waveguide’s finite cross section and the nonlinear dispersion relation. From $\textrm {Re}(\mathcal {D})=0$, two different input energy for complete reflection could reasonably be expected to exist. But for the two-level or degenerate three-level emitters, there exists at most one from Eq. (28). The complete reflection can be explained by Fano interference between the directly transmitted photon and the re-emitted photon. Only if the resonance condition and either of two special input states are satisfied simultaneously, the photon directly transmitted and re-emitted only differ by a phase shift $\pi$, i.e., $r_{j^{\prime }}=-c_{j^{\prime }}$. Then the transmission amplitude vanishes and the perfect reflection occurs.

4.2 Single-photon transport in single-transverse-mode region

In order to explore how the photon transport is manipulated by the V-type three-level emitter, we first consider the single-transverse -mode case ($j_{\max }=1$) with the incident photon energy in the region of $\omega _{1}<\omega _{in}<\omega _{2}$. The $\omega _{1}$ and $\omega _{2}$ are the cut-off frequencies of dominant TM$_{11}$ mode and the second lowest TM$_{31}$ mode, respectively. Here we also consider the emitter’s transition frequency in the region of $\omega _{e_{i}}<\omega _{2}$. In the single-transverse-mode region, only the TM$_{11}$ mode is coupled to the emitter.

In Fig. 2 and Fig. 3, the dependence of reflectance spectrum on the emitter’s transition frequencies $\omega _{e_{i}}$ and the dimensionless coupling coefficients $\zeta _{i}/\omega _{e_{i}}$ are illustrated. These results not only help to choose appropriate emitters as function elements when constructing single-photon devices, but also further validate the conditions under which Fano resonance and EIT phenomena occur in the system. Note that the photon will be reflected completely when the photon energy resonates with the cutoff-frequency as a result of zero group velocity, i.e., $R(\omega _{in}=\omega _{1})=1$.

 

Fig. 2. Single-transverse-mode region: (a) Density plot of reflectivity $R$ as a function of input-photon energy $\omega _{1}<\omega _{in}<\omega _{2}$ and the transition frequency $\omega _{e_{1}}/2\pi$ with $\omega _{e_{2}}/2\pi =45$. The dotted (white) and dashed (black) curves correspond to complete transmission and complete reflection, respectively. (b) $R$ against $\omega _{in}/2\pi$ for different transition frequencies. (c) $R$ against $\omega _{e_{1}}/2\pi$ for a given input-photon energy $\omega _{in}/2\pi =40$ with different $\omega _{e_{2}}/2\pi =35,45$. Other parameters are chosen as $\zeta _{1}/\omega _{e_{1}}=\zeta _{2}/\omega _{e_{2}}=0.1$, $b=5$ um, $a=7.5$ um, with cutoff frequencies $\omega _{1}/2\pi =36$ and $\omega _{2}/2\pi =67$. All frequencies are in units of the THz.

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Fig. 3. Single-transverse-mode region: For different dimensionless coupling coefficients $\zeta _{i}/\omega _{e_{i}}$ caused by the change in electric dipole moments, (a) $R$ as a function of $\omega _{in}/2\pi$ with $\omega _{e_{1}}/2\pi =50$ and $\omega _{e_{2}}/2\pi =40$. (b) $R$ against $\omega _{e_{1}}/2\pi$ for a given input-photon energy $\omega _{in}/2\pi =40$ with $\omega _{e_{2}}/2\pi =45$. The size of cross section is $b=5$ um, $a=7.5$ um. All frequencies are in units of the THz.

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Firstly, the effect of $\omega _{e_{i}}$ on photon transport is illustrated in Fig. 2 with $\zeta _{1}/\omega _{e_{1}}=\zeta _{2}/\omega _{e_{2}}=0.1$. Here the size of cross section is set to be $b=5$um, $a=7.5$ um, with the cutoff frequencies $\omega _{1}/2\pi =36$ THz for TM$_{11}$ and $\omega _{2}/2\pi =67$ THz for TM$_{31}$ modes [46]. As shown in Fig. 2(a), the dotted white and dashed black lines hint that the conditions for perfect transition and reflection are exactly correspond to $f(\omega _{in})=0$ and $\textrm {Re}[\mathcal {D}(\omega _{in})]=0$, respectively. In the weak-coupling limit, the completely reflected photon resonates with the emitter’s transition frequency $\omega _{in}\approx \omega _{e_{i}}$. And in the case of $\zeta _{1}/\omega _{e_{1}}=\zeta _{2}/\omega _{e_{2}}$, the condition for perfect transition reduces to $\omega _{in}=\frac {\omega ^{2}_{e_{1}}\omega _{e_{2}}+\omega _{e_{1}}\omega ^{2}_{e_{2}}}{\omega ^{2}_{e_{1}}+\omega ^{2}_{e_{2}}}$. Looking more closely, we find that the transmission spectral width (purple region) increases with the frequency difference $\vert \omega _{e_{1}}-\omega _{e_{2}}\vert$. In the region of $\omega _{e_{1}}<\omega _{e_{2}}$, the reflection spectral width (red region) around $\omega _{e_{1}}/2\pi$ decreases with $\omega _{e_{1}}$, while in the region of $\omega _{e_{1}}>\omega _{e_{2}}$, the opposite is true. Figure 2(b) shows the detailed behavior of the reflectance spectrum primarily depends on the number of solutions to the two condition equations in Eqs. (29) and (30). The following conclusion may be helpful to realize narrow-band filter experimentally. (i) When $\omega _{e_{1}}, \omega _{e_{2}}< \omega _{1}$ (dotted green line), both two condition functions have no solutions. As the incident photon energy $\omega _{in}$ increases, the detuning between the photon and emitter increases, and the scattering effect of emitter on photon weakens, showing a monotonic decline of reflectivity. (ii) When $\omega _{e_{1}}<\omega _{1}<\omega _{e_{2}}<\omega _{2}$ (dot-dashed purple line), the perfect reflection occurs around $\omega _{in}\approx \omega _{e_{2}}$ as a result of Fano resonance, and the complete transmission due to EIT may also be achieved if $f(\omega _{in})=0$. Thus, $R$ first gets smaller and then goes up to perfect reflection and finally decreases again. (iii) When $\omega _{1}<\omega _{e_{i}}<\omega _{2}$ with$i=1,2$ (solid red line), there exist two $\omega _{in}$ satisfying the Fano resonance condition and one $\omega _{in}$ satisfying the EIT condition. Thus, the $R$ exhibits two complete reflection peaks and one complete transmission valley. (iv) In the special degenerate three-level case with $\omega _{e_{1}}/2\pi =\omega _{e_{2}}/2\pi$ (dashed blue line), the $R$ exhibits only one perfect reflection peak and the perfect transition is absent. This is due to the fact that the $\mathcal {D}(\omega _{in})$ in this degenerate case should be replaced as in Eq. (28). For a given input-photon energy $\omega _{in}/2\pi =40$ THz, the reflectivity $R$ as a function of the transition frequencies $\Omega _{i}$ is shown in Fig. 2(c). The photon transport can be co-regulated by two transition frequencies. When $\omega _{e_{2}}<\omega _{in}$, the $R$ increases to perfect reflection first, then decreases to perfect transmission and finally increases with $\omega _{e_{1}}$. But when $\omega _{e_{2}}>\omega _{in}$, the change of $R$ with respect to $\omega _{e_{1}}$ is just the opposite. Compared with the two-level emitter case, the regulation means is more flexible, which is beneficial to experimentally realize single-photon devices such as optical switches.

Secondly, we start to analyze the dependence of photon transport on the dimensionless coupling coefficient $\zeta _{i}/\omega _{e_{i}}=\frac {q\langle e_{i}\vert z \vert g\rangle }{\sqrt {\pi {\epsilon _{0}} ab}}$ in Fig. 3. The $\zeta _{i}/\omega _{e_{i}}$ is jointly determined by the emitter’s electric dipole moments and the size of cross section. Here we only discuss the change caused by the electric moments. In Fig. 3(a) with $\omega _{1}<\omega _{e_{i}}<\omega _{2}$ ($i=1,2$), the reflection $R$ exhibits two complete reflection peaks and one complete transition valley no matter what the nonzero coupling strengths are. This is consistent with the previous discussion on Fig. 2. And by comparing the dotted blue and the dashed red lines in Fig. 3(a), it is found that, for a given $\zeta _{1}/\omega _{e_{1}}$, the reflection frequency and the reflection spectral width around the $\omega _{e_{2}}/2\pi$ increases with $\zeta _{2}/\omega _{e_{2}}$, while the transmission frequency changes in the direction of $\omega _{e_{1}}/2\pi$ and the transmission spectral width decreases with increasing $\zeta _{2}/\omega _{e_{2}}$. This phenomenon can be explained as the blue shift of emitter’s transition frequency increases with the coupling strength, as shown in Eq. (19). What is more, in the special case with $\zeta _{2}/\omega _{e_{2}}=0$, the V-type emitter reduces to a two-level emitter. The solid green line in Fig. 3(a) shows that, the EIT-like phenomena induced by destructive interference between two radiation pathways disappears, and the single-photon can only be reflected completely under the Fano resonance condition.

Besides, for a given $\omega _{in}$ in the single-transverse-mode region, the regulatory capacity of two transition frequencies $\omega _{e_{i}}$ on photon transport is determined by the coupling strengths $\zeta _{i}/\omega _{e_{i}}$. As shown in Fig. 3(b), in the extreme case of $\zeta _{2}/\omega _{e_{2}}=0$, the photon transport characteristics are completely controlled by $\omega _{e_{1}}$. Under the weak coupling condition with $\zeta _{1}/\omega _{e_{1}}=0.05$, the two-level emitter is almost transparent to photons, unless the Fano resonance occurs. But as the $\zeta _{2}/\omega _{e_{2}}$ is increased to be comparable or even greater than $\zeta _{1}/\omega _{e_{1}}$, the effect of transition between $|e_{2}\rangle$ and $|g\rangle$ on the photon transport cannot be ignored. It not only induces the EIT-like phenomenon, but also increases the sensitivity of the reflectivity $R$ to $\omega _{e_{1}}$, as shown in the dotted blue and dashed red lines in Fig. 3(b). This means that with the help of the third-level $|e_{2}\rangle$, it is possible to realize the regulation of photon transport from perfect transmission to perfect reflection within a small range of $\omega _{e_{1}}$.

Until now, we can safely conclude that the V-type three-level emitter can be used to regulate the single-photon transport from complete transmission to complete reflection in the single-transverse-mode region. Compared with the two-level emitter case, in which the complete transmission can not be achieved, there are more tunable parameters in the V-type three-level emitter case.

4.3 Multi-transverse-mode effect induced by finite cross section

At last we study the influence of multi transverse modes on the single-photon scattering in the rectangular waveguide. Recall that the single-transverse-mode region requires the photon energy to be in the region $\omega _{1}<\omega _{in}<\omega _{2}$. As shown in Fig. 4, the single transverse mode bandwidth $\omega _{2}-\omega _{1}$ becomes narrow with the increase of cross size. For this reason, more than one waveguide mode are involved in photon scattering, so the effect of multi transverse modes should be considered. Besides, driven by the development of high-frequency laser, the research on multi-transverse-mode effect becomes more urgent.

 

Fig. 4. The cut-off frequencies $\omega _{j}$ against the size of cross section. For a given input-photon energy $\omega _{in}$, there exist cut-off region, single-transverse-mode region, and multi-transverse-mode region. The critical size $b_{j}$ is determined by $\omega _{in}=\omega _{j}$.

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For a given input-photon energy $\omega _{in}$, the number of TM modes involved in photon scattering is determined by the size of the waveguide’s cross section. As shown in the Fig. 4, the critical size $b_{j}=\frac {c\pi }{\omega _{in}}\sqrt {\frac {m^{2}}{(a/b)^{2}}+n^{2}}$ is determined by $\omega _{in}=\omega _{j}$. (i)When $b<b_{1}$ ($\omega _{in}<\omega _{1}$), the photon cannot propagate in the waveguide as the wavelength is larger than the cross section size. This is the so-called cut-off region. (ii) When $b_{1}<b<b_{2}$ ($\omega _{1}<\omega _{in}<\omega _{2}$), the principle of energy conservation leads to that only TM$_{11}$ mode can be propagated. This interval is the single transverse mode working region. (iii) When $b>b_{2}$ ($\omega _{in}>\omega _{2}$), multi transverse modes will appear simultaneously in the waveguide, which is called multi transverse modes working region. It is the finite rectangular cross section of the waveguide that leads to the appearance of multiple TM modes with different cut-off frequencies.

Next, we will carefully explore the multi-transverse-mode effect with different input states. When the input state is prepared in the CSS with $c_{j^{\prime }}\propto \rho _{j}\chi _{j,k_{j}}$, the reflection spectrum in the multi-transverse-mode region is similar to that in the single-transverse-mode region. As discussed behind Eq. (30), all the results in single-transverse-mode region almost apply to multi-transverse-mode case with input CSS, except that the frequency range changes. The dotted green line in Fig. 5(a) shows that, as long as the input state is prepared in the CSS, both the perfect reflection and perfect transmission can be achieved. But when the input photon is in one certain transverse mode, the photon-transport characteristics are quite different.

 

Fig. 5. Two-transverse-mode region: (a) The reflectivity $R$ against the input-photon energy $\omega _{2}<\omega _{in}<\omega _{3}$ with different input states. The dotted (green), solid (red) and dashed (blue) curves denote the input states are CSS, TM$_{11}$ and TM$_{31}$ sates, respectively. The reflection $R_{i}$ and transition $T_{i}$ in the TM$_{11}$ (i=1) and TM$_{31}$ (i=2) modes against $\omega _{in}$ when the input sate is in the (b) TM$_{11}$-mode and (c) TM$_{31}$-mode. The cross size is $b=5$ um, $a=7.5$ um with $\omega _{2}/2\pi =67$ PHz and $\omega _{3}/2\pi =92$ PHz. Other parameters are $\zeta _{1}/\omega _{e_{1}}=\zeta _{1}/\omega _{e_{1}}=0.05$, $\omega _{e_{1}}/2\pi =70$ PHz, $\omega _{e_{2}}/2\pi =75$ PHz.

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For simplicity but without loss of generality, take the two-transverse-mode case with $\omega _{2}<\omega _{in}<\omega _{3}$ $(j_{\max }=2)$ for example. The reflection and transmission components in the $j^{\prime }$th TM mode can be derived by insetting the input state parameter $c_{j}=\delta _{nj}$ and Eq. (25) into Eq. (26). Here we assume the input state in the $n$th TM mode.

Thus, the total reflectivity with the $n$th-mode input state can be obtained as

In Fig. 5(a), the solid red and dashed blue lines show that the perfect reflection is impossible when the input state is in either TM$_{11}$ or TM$_{31}$. But the perfect transmission can also be achieved under the condition of $f(\omega _{in})=0$, since the EIT mechanism still holds in the multi-transverse-mode region. All these results are consistent with the previous discussion on the perfect transmission and reflection conditions.

The physical picture of photon scattering in the multi-transverse-mode region can be described as follows: when a photon in the $n$th mode is incident, it may pass through directly, or it may be absorbed by the emitter with coupling strength $g_{n,k}^{(i)}$ and then re-radiated. It is worth noting that as the emitter is coupled to multiple TM modes, the incident $n$th-mode photon inevitably enters other TM modes when the emitter transitions from $|e_{i}\rangle$ to $|g\rangle$. And because of the symmetry in the coupling to left and right going modes, the transmittance and reflectivity in other TM modes is exactly the same, i.e., $T_{j}=R_{j}$ ($j\not =n$). But for the $n$th mode, the transmission probability consists of directly pass through component and re-radiated component, which leads to $T_{n}\not =R_{n}$. Also because of the coupling of emitter with multiple transverse modes, even under the Fano resonance condition, the total reflectivity can only reach a peak value of $R_{\max }=\rho _{n}\chi _{n,k_{n}}^{2}/\sum _{j^{\prime }}\rho _{j^{\prime }}\chi _{j^{\prime },k_{j^{\prime }}}^{2}<1$. As the photon states density is larger in the higher transverse mode for a given $\omega _{in}$, the reflectivity component in the high transverse mode is therefore larger than that in the low transverse modes ($R_{2}>R_{1}$). Last but not at least, the cut-off effect due to zero group velocity in the multi transverse modes region should be treated carefully. The photon will either be completely transmitted or completely reflected when the input-photon energy resonates with cut-off frequencies. It depends on whether the mode of incident photon and cut-off frequency is consistent, i.e., $R(\omega _{n})= 1$ and $R(\omega _{j\not =n})=0$. All above results are confirmed in the Fig. 5(b) and Fig. 5(c).

Based on the above analysis, the photon transport characteristics in the multi-transverse-mode region can be summarized as follows. Only when the incident state is CSS, the photon can be completely reflected under Fano resonance condition and be completely transmitted under EIT mechanism. But when the input state is in the single $n$th mode, due to the indirectly interaction between waveguide transverse modes mediated by the emitter, the complete reflection vanishes and the photon inevitably enters other TM modes with equal $R_{j}=T_{j}$ ($j\not =n$). We can make further predictions that, if the input $n$th mode is the lowest transverse mode, the photon will be transmitted with great probability in a large input-frequency range, except for the small probability of reflection near the Fano resonance point. If the input $n$th mode is the highest transverse mode that the energy allows, the reflection spectrum is similar to that in the single transverse mode region, except of the reduced reflectivity peak.

5. Conclusion

To realize all-optical devices at single-photon level, it is very crucial to control the coherent transport of a single photon in quantum waveguide systems. The most of investigations have focused on the 1D waveguide with linear dispersion, which gives the result of total reflection with Breit-Wigner or Fano line shapes when the frequency of the incident photon is near to the transition frequency of the coupled quantum emitter [27,47,48]. While the wavelength of the single photon decreases to be comparable to the size of the waveguide’s cross section, the effect of the cross section will play a part. A remarkable feature in this case is that the phenomenon of interference between multiple transverse modes will occur. In this paper, The single-photon scattering by a V-type three-level emitter in a rectangular waveguide has been studied for a large range of input-photon energy. In the single-transverse-mode regime, the perfect transmission and reflection can both be achieved as a result of EIT and Fano resonance, respectively. Nevertheless, as the photon energy is increased to be greater than the second lowest cut-off frequency, more transverse modes are involved in scattering process, and the multi-transverse-mode effect need to be emphasized. Although the perfect transmission still exists, the photon can not be complete reflected except that the input-state is prepared in the CSS. Noting that these conclusions cannot hold for the degeneracy three-level emitter or the two-level emitter cases, where only one complete reflection peak may appear and complete transmission will not occur. This exactly highlights the advantage of the non-degenerate three-level emitter in photon regulation. Besides, in all photon energy region, the frequency for perfect transmission and the width of this central transparency window can both be manipulated by adjusting the emitter’s transition frequencies and the emitter-waveguide coupling strengths. In a sense, the emitter is essentially a functional quantum node to control the photon transport through the absorption and emission of photons to- and fro- the waveguide. Last but not least, it is worth noting that our main results of Eqs. (27), (29), and (30) still hold for the emitter’s electric dipole along arbitrary directions. The only difference for different dipole directions is the value of $g_{j,k}^{(i)}$ in Eq. (10), and the contribution from all coupled TE and TM modes should be considered. To draw a conclusion, our results may not only promote the development of single-photon devices with wide applicable frequency region and narrow spectral width, like optical switches and optical filters, but also contribute to deeply understand the influence of the geometry constraint of electromagnetic field on the light-matter interaction.

At last, we discuss the physical implementation of such waveguide-QED system. Based on the selection rules, the magnetic sublevels can be taken as a possible experimental realization of the emitter. Then the employed magnetic field can be used to change the transition frequencies of emitter to control the photon transport. Besides, apart from the considered metal rectangular waveguide [32,33], the multi-mode scattering can also occur in other structures, such as the line defect in photonic crystal coupled with a quantum dot [49–51], an open superconduction transmission line with a superconduction transmon qubit [52], or an x-ray waveguide [30,31], as long as the wavelength of incident photon decreases to be comparable to the cross size.