⇦Chinese Physics Letters, 2019, Vol. 36, No. 5, Article code 054201 Polarization Conversion of Single Photon via Scattering by a ${\Lambda}$ System in a Semi-Infinite Waveguide * Fu-Qiang Yu (于富强)1,2, Mu-Tian Cheng (程木田)1,2**, Shao-Ming Li (李绍铭)1,2, Xiao-San Ma (马小三)1,2, Zhi-Feng Zhu (朱志峰)1,2, Xian-Shan Huang (黄仙山)3 Affiliations 1Anhui Provincial Key Lab of Power Electronics and Motion Control, Anhui University of Technology, Maanshan 243002 2School of Electrical Engineering and Information, Anhui University of Technology, Maanshan 243002 3School of Mathematics and Physics, Anhui University of Technology, Maanshan 243002 Received 23 January 2019, online 17 April 2019 *Supported by the Anhui Provincial Natural Science Foundation under Grant Nos 1608085MA05 and 1608085MA09, and the National Natural Science Foundation of China under Grant Nos 11774262 and 11474003.
**Corresponding author. Email: mtcheng606@hotmail.com Citation Text: Yu F Q, Cheng M T, Li S M, Ma X S and Zhu Z F et al 2019 Chin. Phys. Lett. 36 054201    Abstract We theoretically investigate single-photon polarization conversion via scattering by an atom with ${\Lambda}$ configuration coupled to a semi-infinite waveguide and discuss the two cases in which the ${\Lambda}$ system is non-degenerated and degenerated. By applying the hard-wall boundary condition of the semi-infinite waveguide, it is found that single-photon polarization conversion can be realized with unit probability for both cases under the ideal condition. Together with the polarization conversion, the frequency conversion of a single photon can also be realized with unit probability in the ideal case if the ${\Lambda}$ system is not degenerated. DOI:10.1088/0256-307X/36/5/054201 PACS:42.50.Nn, 42.50.Ct, 32.70.Jz © 2019 Chinese Physics Society Article Text Quantum devices at single-photon level play important roles in quantum information processing and quantum computation. A waveguide can guide photons naturally. Furthermore, strong coupling between qubits and waveguides has been realized. Thus, waveguide electrodynamics can serve as a good platform for the realization of quantum computation.[1–6] Many advances, including quantum state preparation,[7] generation of entanglement between two qubits,[8–10] generation of frequency combs,[11–13] SWAP gates,[14] and quantum beats[15–17] based on waveguide electrodynamics, have been reported. Several kinds of quantum devices based on waveguide electrodynamics, such as single-photon switching,[18–28] transistors,[29,30] routers,[31–34] circulators,[35,36] isolators,[37–42] and frequency converters,[43,44] have been proposed and implemented. Polarization, which is an important property of light, has found wide applications in optical instruments. The manipulation of polarization has been investigated in various photonic structures, including topological insulators[45,46] and optical metamaterials.[47–50] Polarization of a single photon can carry quantum information. Thus polarization conversion of a single photon has attracted much attention recently based on the waveguide electrodynamics. Tsoi and Law reported the conversion of polarization of a single photon in a waveguide from an unknown state to a particular state via scattering by many atoms.[51] Later, Zhang et al. extended Tsoi and Law's results to a waveguide with nonlinear dispersion.[52] Recently, Gong's group investigated single-photon polarization conversion in a waveguide via scattering by a ${\Lambda}$ system assisted with a cavity.[53] We also reported on the manipulation of single-photon polarization via scattering a pair of atoms.[54] However, one needs to control many coupling strengths due to the many atoms interacting with the waveguide. In this Letter, we propose to manipulate the polarization of single photons via scattering by only one atom in a semi-infinite waveguide. The polarization conversion efficiency can reach 1 under the ideal condition. Compared to previous proposals, our method has many advantages. For example, the number of coupling strengths and transition frequencies of atoms needed to be manipulated is less than in previous proposals. The channels of dissipations are also less than those in the previous proposals. Thus our method may be easier to realize in the experiment.
cpl-36-5-054201-fig1.png
Fig. 1. The system considered in this study. A semi-infinite waveguide couples to an atom with a ${\Lambda}$-type three-level system. A single photon incidents from the left of the waveguide. The right-hand side of the waveguide reflects the incident photon with 100% probability.
The system under consideration is shown in Fig. 1. An atom or artificial atom with a ${\Lambda}$-type three-level system couples to a semi-infinite waveguide. The distance between the atom and the end of the waveguide is $a$. The transition between $|1\rangle$ and $|3\rangle$ couples to the photon with h-polarization while the transition between $|2\rangle$ and $|3\rangle$ interacts with the v-polarized photon. The coupling strengths are denoted by $g_{\rm h}$ and $g_{\rm v}$, respectively. They depend on the transition dipoles, transition frequencies and the polarizations of the photons.[55] Here $h$ and $v$ just denote two orthogonal polarizations. They can be horizontal and vertical. They can also represent left and right circularly polarized modes. The Hamiltonian in the real space is given by $H=H_{\rm wg}+H_{\rm a}+H_{\rm int}$, where $H_{\rm wg}$ describes the photon propagation in the waveguide. It is given by[18] $$\begin{align} H_{\rm wg}=\,&-iv_{\rm g}\sum_{p=h,v}\Big[\int dx C^†_{{\rm R},p}(x)\frac{\partial}{\partial x}C_{{\rm R},p}(x)\\ &-\int dx C^†_{{\rm L},p}(x)\frac{\partial}{\partial x}C_{{\rm L},p}(x)\Big],~~ \tag {1} \end{align} $$ where $C^†_{{\rm R},p}(x)(C^†_{{\rm L},p}(x))$ denotes creation of a right (left)-propagation photon at $x$ with p-polarization, and $v_{\rm g}$ is the photon propagation velocity in the waveguide. We have supposed that the group velocity for the two types of photons is the same.[56–58] The Hamiltonian of the atom $H_{\rm a}$ is given by $$ H_{\rm a}=\omega_{2}\sigma_{22}+(\omega_{3}-i\gamma/2)\sigma_{33},~~ \tag {2} $$ where $\omega_{2}$ ($\omega_{3}$) denotes the eigenfrequency of the state $|2\rangle$ $(|3\rangle)$. The energy of state $|1\rangle$ is the energy reference, and $\gamma$ represents the energy loss rate from state $|3\rangle$ to the non-guided waveguide modes. We have supposed that the state $|2\rangle$ has a long lifetime so that the energy loss of state $|2\rangle$ can be neglected. Here $\sigma_{ij}$ is the dipole transition operator between states $|i\rangle$ and $|j\rangle$. We have set $\hbar=1$. The interaction between the atom and the waveguide is given by $$\begin{align} H_{\rm int}=\,&g_{\rm h}\int dx \delta(x)[C^†_{{\rm R},h}(x)\sigma_{13}+C^†_{{\rm L},h}(x))\sigma_{13}+{\rm H.c.}] \\ &+g_{\rm v}\int dx \delta(x)[C^†_{{\rm R},v}(x)\sigma_{23}\\ &+C^†_{{\rm L},v}(x))\sigma_{23}+{\rm H.c.}].~~ \tag {3} \end{align} $$ Since we only consider single-photon polarization conversion, the scattering eigenstate can be written as $$\begin{align} |{\it \Psi}\rangle=\,&\int dx [\varphi_{{\rm R}h}(x)C^†_{{\rm R},h}(x)+\varphi_{{\rm L}h}(x)C^†_{{\rm L},h}(x)]|0,1\rangle \\ &+\int dx [\varphi_{{\rm R}v}(x)C^†_{{\rm R},v}(x)+\varphi_{{\rm L}v}(x)C^†_{{\rm L},v}(x)]|0,2\rangle\\ &+c_{3}|0,3\rangle,~~ \tag {4} \end{align} $$ where $|0,i\rangle$ denotes vacuum states of photons with the atom in state $|i\rangle$, $\varphi_{d,p}$ ($d$ = R, L) describes the wavefunction of the single photon, and $c_{3}$ is the probability amplitude of the state $|0,3\rangle$. From the Schrödinger equation $H|{\it \Psi}\rangle=\omega|{\it \Psi}\rangle$, one can obtain $$\begin{align} &-iv_{\rm g}\frac{\partial}{\partial x}\varphi_{{\rm R}h}(x)+g_{\rm h}\delta(x)c_{3}=\omega\varphi_{{\rm R}h}(x),~~ \tag {5a}\\ &iv_{\rm g}\frac{\partial}{\partial x}\varphi_{{\rm L}h}(x)+g_{\rm h}\delta(x)c_{3}=\omega\varphi_{{\rm L}h}(x),~~ \tag {5b}\\ &-iv_{\rm g}\frac{\partial}{\partial x}\varphi_{{\rm R}v}(x)+g_{\rm v}\delta(x)c_{3}=(\omega-\omega_{2})\varphi_{{\rm R}v}(x),~~ \tag {5c}\\ &iv_{\rm g}\frac{\partial}{\partial x}\varphi_{{\rm L}v}(x)+g_{\rm v}\delta(x)c_{3}=(\omega-\omega_{2})\omega\varphi_{{\rm L}v}(x),~~ \tag {5d}\\ &(\omega_{3}-i\frac{\gamma}{2})c_{3}+g_{\rm h}[\varphi_{{\rm R}h}(0)+\varphi_{{\rm L}h}(0)]\\ &+g_{\rm v}[\varphi_{{\rm R}v}(0)+\varphi_{{\rm L}v}(0)]=\omega c_{3},~~ \tag {5e} \end{align} $$ where $\omega$ is the eigenfrequency of the incident single photon. We suppose that the single photon incidents from the left of the waveguide with h-polarization and the atom in state $|1\rangle$. Then, the wavefunction of the single photon can be written as $$\begin{align} \varphi_{{\rm R}h}(x)=\,&e^{ik_{1}x}[s(-x)+a_{\rm 1r}s(x)],~~ \tag {6a}\\ \varphi_{{\rm L}h}(x)=\,&e^{-ik_{1}x}[r_{\rm h}s(-x)+a_{\rm 1l}s(x)],~~ \tag {6b}\\ \varphi_{{\rm R}v}(x)=\,&e^{ik_{2}x}b_{\rm 2r}s(x),~~ \tag {6c}\\ \varphi_{{\rm L}v}(x)=\,&e^{-ik_{2}x}[r_{\rm v}s(-x)+b_{\rm 2l}s(x)],~~ \tag {6d} \end{align} $$ where $k_{1}=\omega/v_{\rm g}$, $k_{2}=(\omega-\omega_{2})/v_{\rm g}$, $s(x)$ is the Heaviside step function with $s(0)=1/2$, $r_{\rm h}$ and $r_{\rm v}$ denoting the scattering amplitudes of the reflected photon with $h$ and $v$ polarizations, respectively, $a_{\rm 1r}$ ($a_{\rm 1l}$) represents the scattering amplitudes of the right (left) propagation photon with h-polarization in the region between $x=0$ and $x=a$, and $b_{\rm 2r}$ and $b_{\rm 2l}$ denote the scattering amplitudes of the right and left propagation photons with v-polarization, respectively. Substituting Eq. (6) into Eq. (5), one can obtain $$\begin{align} r_{\rm h}=\,&-e^{2i\theta_{1}}[g^{2}_{\rm h}(1-e^{-2i\theta_{1}}) +g^{2}_{\rm v}(e^{2i\theta_{2}}-1)\\ &+iv_{\rm g}({\it \Delta}+i\gamma/2)]/[g^{2}_{\rm h} (e^{2i\theta_{1}}-1)+g^{2}_{\rm v}(e^{2i\theta_{2}}-1)\\ &+iv_{\rm g}({\it \Delta}+i\gamma/2)],~~ \tag {7a}\\ r_{\rm v}=&[(e^{2i\theta_{1}}-1)(e^{2i\theta_{2}}-1)g_{\rm h}g_{\rm v}]/[g^{2}_{\rm h} (e^{2i\theta_{1}}-1)\\ &+g^{2}_{\rm v}(e^{2i\theta_{2}}-1)+iv_{\rm g}({\it \Delta}+i\gamma/2)],~~ \tag {7b} \end{align} $$ where $\theta_{1}=k_{1}a$, $\theta_{2}=k_{2}a$ and ${\it \Delta}=\omega-\omega_{3}$. The hard-wall boundary condition at the end of the waveguide $\varphi_{{\rm R}h}(a)+\varphi_{{\rm L}h}(a)=\varphi_{{\rm R}v}(a)+\varphi_{{\rm L}v}(a)=0$ is used to obtain the expressions of $r_{\rm h}$ and $r_{\rm v}$. To analyze the single-photon polarization conversion more clearly, we consider the case of ${\it \Delta}=0$, which means that the incident photon is resonant with the transition between states $|1\rangle$ and $|3\rangle$. The dissipation is neglected temporarily to catch the main physics. We also set $g_{\rm h}=g_{\rm v}=g$ for simplicity. Then, Eqs. (7a) and (7b) show $$\begin{align} &\theta_{1}+\theta_{2}=m\pi,~~ \tag {8a}\\ &\theta_{1},\theta_{2}\neq n\pi,~~ \tag {8b} \end{align} $$ $|r_{\rm h}|=0$ and $|r_{\rm v}|=1$ which means that the polarization conversion of the single photon is realized. Here $m$ and $n$ are integers. Figure 2 shows $R_{\rm p}=|r_{\rm p}|^{2}$ as functions of $\theta_{1}$ and $\theta_{2}$. It exhibits clearly that when $\theta_{1}+\theta_{2}=\pi$, $R_{\rm v}$ reaches the maximum value of 1. The single-photon polarization conversion from h-polarization to v-polarization is realized. The physical mechanism for the single-photon polarization conversion can be explained as follows: the single photon with h-polarization excites the ${\Lambda}$ system from the ground state $|1\rangle$ to the excited state $|3\rangle$. Then the atom decays through two channels. The first channel is that the final state is $|1\rangle$ and a photon with h-polarization is emitted. The second one is that the final state is $|2\rangle$ and emits a v-polarized photon. Then, the emitted photon propagates to the right and is then reflected back by the end of the waveguide. The interference between the emitted photon and the reflected photon induces the single-photon polarization rotation.
cpl-36-5-054201-fig2.png
Fig. 2. The values of $R_{\rm h}$ (a) and $R_{\rm v}$ (b) as functions of $\theta_{1}$ and $\theta_{2}$. The white dashed lines are for the equation $\theta_{1}+\theta_{2}=\pi$. In the calculations, ${\it \Delta}=\gamma=0$, and $g^{2}=10^{-5}v_{\rm g}\omega_{3}$.
In the previous discussions, we have set $\omega_{2}\neq\omega_{1}$, which means that the frequency of the reflected photon is different from the incident photon. Thus one can realize the frequency conversion of a single photon using this method. The frequency difference between the reflected and incident photon is $(\omega_{2}-\omega_{1})$. Now we consider the case of $\omega_{2}=\omega_{1}=0$. The frequency of the reflected photon is the same as that of the incident photon. Thus we obtain $k_{1}=k_{2}=k$. Equations (7a) and (7b) are simplified to $$\begin{align} r_{\rm h}=\,&-\frac{1+e^{2i\theta}}{2},~~ \tag {9a}\\ r_{\rm v}=\,&\frac{-1+e^{2i\theta}}{2},~~ \tag {9b} \end{align} $$ where $\theta=ka$. Equations (9a) and (9b) are the same as those reported in Refs.  [53,54]. One can obtain that when $\theta=(n+1/2)\pi$, $R_{\rm v}=1$ and $R_{\rm h}=0$. The polarization of the reflected photon becomes vertical. However, when $\theta=m\pi$, $R_{\rm h}=1$, and $R_{\rm v}=0$. The polarization of the reflected photon remains unchanged. Here we intend to emphasize that even though Eqs. (9a) and (9b) are the same as those reported in Refs.  [53,54], the model is different. In this work, the assistant cavity or the second atom is not involved. Thus the number of parameters needing to be controlled is less than that in Refs.  [53,54]. It may not be easy to control $g_{\rm h}=g_{\rm v}$ in practice. To show how $g_{\rm h}$ and $g_{\rm v}$ affect the probability of single-photon conversion, we plot Fig. 3 to exhibit $R_{\rm h}$ and $R_{\rm v}$ as functions of $g_{\rm h}/g_{\rm v}$ and $\theta_{1}$. One can find that the red region is large when $\theta_{1}$ is about 0.5$\pi$, which means that the polarization conversion fidelity is large even though the condition $g_{\rm h}=g_{\rm v}$ is not matched well.
cpl-36-5-054201-fig3.png
Fig. 3. The values of $R_{\rm h}$ (a) and $R_{\rm v}$ (b) as functions of $\theta_{1}$ and $g_{\rm h}/g_{\rm v}$. In the calculations, ${\it \Delta}=\gamma=0, g^{2}_{\rm v}=10^{-5}v_{\rm g}\omega_{3}$, and $\theta_{2}=\pi-\theta_{1}$.
cpl-36-5-054201-fig4.png
Fig. 4. The values of $R_{\rm h}$ (a) and $R_{\rm v}$ (b) as functions of ${\it \Delta}$ and $\gamma$ for the case of $\omega_{1}\neq\omega_{2}$. In the calculations, $g^{2}=10^{-5}v_{\rm g}\omega_{3}$, $\theta_{1}=\pi/4$, and $\theta_{2}=3\pi/4$.
cpl-36-5-054201-fig5.png
Fig. 5. The values of $R_{\rm h}$ (a) and $R_{\rm v}$ (b) as functions of ${\it \Delta}$ and $\gamma$ for the case of $\omega_{1}=\omega_{2}$. In the calculations, $g^{2}=10^{-5}v_{\rm g}\omega_{3}$, and $\theta=\pi/2$.
The influences of detuning and decay on the probability of single-photon polarization conversion are exhibited in Fig. 4. Here $\omega_{2}\neq\omega_{1}$ is considered. The frequency of the reflected photon is different from that of the incident photon. It shows that when ${\it \Delta}=0$, $R_{\rm h}$ can reach the maximum value. The maximum value decreases with increasing $\gamma$. However, $R_{\rm v}$ can still be around 0.9 even though $\gamma$ increases to about $10^{-6}\omega_{3}$. Figure 5 shows the influences of detuning and decay on $R_{\rm h}$ and $R_{\rm v}$ when $\omega_{2}=\omega_{1}$. The frequency of the reflected photon is the same as that of the incident one. One can find that when $\gamma$ reaches $10^{-6}\omega_{2}$, $R_{\rm v}$ for the resonant photon is about 0.94, which is larger than that for the case of $\omega_{2}\neq\omega_{1}$. In summary, we have investigated single-photon polarization conversion via scattering by a ${\Lambda}$ system coupled to a semi-infinite waveguide. We take the h-polarized photon incident as an example. For the non-degenerated ${\Lambda}$ system, the reflected photon becomes v-polarized when $\theta_{1}+\theta_{2}=m\pi$ and $\theta_{1},\theta_{2}\neq n\pi$ with $g_{\rm h}=g_{\rm v}$, and the frequency of the reflected v-polarized photon is different from that of the incident one. However, when $\theta_{1}=n\pi$ or $\theta_{2}=n\pi$, the polarization and frequency of the reflected photon are the same as the incident one. For the case of a degenerated ${\Lambda}$ system, the reflected photon becomes v-polarized when $\theta=(n+1/2)\pi$. However, it remains unchanged when $\theta=m\pi$. The frequency of the reflected photon is the same as that of the incident one. Our results may find applications in designing quantum information processing devices at single-photon level. 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