⇦Chinese Physics Letters, 2019, Vol. 36, No. 3, Article code 034204 Superposed Transparency Effect and Entanglement Generation with Hybrid System of Photonic Molecule and Dipole Emitter * Ji-Bing Yuan (袁季兵)1,3, Zhao-Hui Peng (彭朝晖)2, Shi-Qing Tang (唐世清)1,3**, Deng-Yu Zhang (张登玉)1 Affiliations 1College of Physics and Electronic Engineering, and Hunan Provincial Key Laboratory of Intelligent Information Processing and Application, Hengyang Normal University, Hengyang 421008 2Institute of Modern Physics and Department of Physics, Hunan University of Science and Technology, Xiangtan 411201 3Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081 Received 1 September 2018, online 23 February 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 11547258, 11647129 and 11405052, the Hunan Provincial Natural Science Foundation of China under Grant Nos 2018JJ3006, 2017JJ3005 and 2016JJ3006, the Scientific Research Fund of Hunan Provincial Education Department under Grant Nos 16B036 and 15A028, the Science and Technology Plan Project of Hunan Province under Grant No 2016TP1020, the Open Fund Project of Hunan Provincial Key Laboratory of Intelligent Information Processing and Application for Hengyang Normal University under Grant No IIPA18K08, the Open Fund Project of the Hunan Provincial Applied Basic Research Base of Optoelectronic Information Technology under Grant No GD18K04, and the Open Fund Project of the Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of the Ministry of Education under Grant Nos QSQC1704 and QSQC1706.
**Corresponding author. Email: sqtang@hunnu.edu.cn Citation Text: Yuan J B, Peng C H, Tang S Q and Zhang D Y 2019 Chin. Phys. Lett. 36 034204    Abstract We theoretically investigate the transparency effect with a hybrid system composed of a photonic molecule and dipole emitter. It is shown that the transparency effect incorporates both the coupled resonator-induced transparency (CRIT) effect and the dipole-induced transparency (DIT) effect. It is found that the superposed transparency windows are consistently narrower than the CRIT and DIT transparency windows. Benefiting from the superposed transparency effect, the photonic Faraday rotation effect could be realized in the photonic molecule system, which is useful for entanglement generation and quantum information processing. DOI:10.1088/0256-307X/36/3/034204 PACS:42.50.Pq, 42.50.-p, 03.67.-a © 2019 Chinese Physics Society Article Text Coherent control of photon transport in the waveguide is of great importance in optical communication and quantum information processing (QIP). One of the potential ways is to employ two or more optical microcavities evanescently coupled to the waveguide. Generally, coupled optical microcavities are referred to photonic molecule (PM) systems,[1,2] because the formation of optical bonding and antibonding modes (or supermodes) is analogous to the electronic states in molecules formed by atoms. Compared with the single microcavity, the PM system provides an additional degree of freedom (DoF) to tailor the optical density of states and the spatial distribution of modes, and thus it may be especially attractive for optical switching[3] and other practical applications, e.g. single-mode lasing[4] and biosensors.[5] The transmission properties of the PM system are significantly modified due to coherent interference between the fields from two microcavities which is similar to the electromagnetically induced transparency (EIT) in the three-level atomic system,[6,7] and thus its transparency effect is often called coupled resonator-induced transparency (CRIT).[8] Moreover, many of the basic limitations on bandwidth and decoherence which result from the fragility of the electronic coherence may be fundamentally overcome, and thus it has broad implications for optical communications and QIP. Recently, parity-time symmetric photonics[9] has also been proposed in the PM system with active and passive microcavities, and novel phenomena, e.g. non-reciprocity of light transmission,[10] optical isolation[11] and loss-induced revival of lasing,[12] have been demonstrated. On the other hand, a single dipole emitter[13,14] or a dipole-microcavity system[15,16] may also be possible alternatives for the manipulation of photon transport in a waveguide, and they are usually referred to as a waveguide quantum electrodynamics (QED) system or cavity QED system. A single dipole emitter may function as a quantum switch dependent of its internal state, and thus the quantum transistor[13,14] and dipole-induced transparency (DIT) effect[15] have been proposed, respectively. In particular, the DIT effect only requires a large Purcell factor of a cavity QED system, and thus allows the system to work in the bad cavity regime, which greatly relaxes the experimental requirement of the cavity QED system. In the DIT effect, the dipole-microcavity system can readily modulate both the amplitude and phase of an incident photon by considering its polarization DoF. The amplitude modulation feature can be utilized to split a polarized light beam, and thus functions as a photonic Stern–Gerlach apparatus,[17] which may be useful for generating spatial entanglement of the photon and related QIP.[18] The photonic Faraday rotation effect[19] may be obtained by considering the phase shift of the incident photon, and it is also useful for QIP[20,21] and quantum computation.[22,23] Up to now, the CRIT and DIT effects have only been discussed independently, and thus it may be interesting to investigate the superposed transparency effect, which incorporates both the CRIT effect and DIT effect, with the PM system. In this Letter, we theoretically investigate the superposed transparency effect with a system composed of PMs and a dipole emitter. It is shown that the superposition between the CRIT and DIT effect plays a crucial role in the superposed transparency effect, and the full widths at half maximum (FWHMs) of the dips of the superposed transparency windows are narrower than those of the CRIT and DIT effect. Benefiting from the superposed transparency effect, the photonic Faraday rotation effect could be realizable in the PM system, which is useful for entanglement generation and QIP.
cpl-36-3-034204-fig1.png
Fig. 1. (a) The schematic of hybrid transparency effect in the system composed of a photonic molecule and dipole emitter. (b) The energy-level configuration of the dipole emitter. The transition $|g_{\rm L(R)}\rangle {\leftrightarrow} |e\rangle$ is driven by left (right) circularly polarized light.
The schematic diagram of the superposed transparency effect is illustrated in Fig. 1(a), where the PM system includes two microcavities with single field modes $a_{i} (i=1,2)$, respectively, where $a_{1}$ and $a_{2}$ are the counter-clockwise and clockwise traveling-wave modes, respectively. In fact, there are two degenerate whispering gallery modes in the resonator.[24] However, only one of the modes (clockwise or counter-clockwise mode) could be excited in our model if we only consider that the probe light transfers from the left port of the waveguide in this work. The microcavity with mode $a_{1}$ is evanescently coupled to the waveguide for inputting and outputting the light field. The dipole emitter $e_{i}$ with transition frequency $\omega_{\rm di}$ is evanescently coupled to the $i$th microcavity, and the Hamiltonian of the whole system is described as ($\hbar=1$) $$\begin{align} H=\,&\sum_{i=1}^{2}\Big[\omega_{i}a_{i}^†a_{i}+\frac{\omega_{\rm di} \sigma_{i}^{z}}{2}\\ &+(Ja_{i}a_{i+1}^†+g_{i}a_{i}\sigma_{i}^{+}+{\rm H.c.})\Big],~~ \tag {1} \end{align} $$ where $\omega_{i}$ is the central frequency of the $i$th microcavity; $\sigma_{i}^{z}$, $\sigma_{i}^{+}$ and $\sigma_{i}^-$ are the Pauli operator, raising and lower operators of the $i$th dipole emitter; and $J$ and $g_{i}$ are the coupling strength of two microcavities and the vacuum Rabi frequency of the $i$th dipole emitter. We take $a_{3}^†\equiv a_{1}^†$. In the rotating frame with respect to the frequency of the incident photon $\omega_{\rm p}$, the Heisenberg–Langevin equations of cavity modes are $$\begin{align} \frac{da_{1}(t)}{dt}=\,&-i(\omega_{1}-\omega_{\rm p})a_{1}(t)-\frac{\kappa_{01}+\kappa_{1}}{2}a_{1}(t)\\ &-iJa_{2}(t)-ig_{1}\sigma_{1}^-(t)\\ &-\sqrt{\kappa_{1}}a_{\rm in}(t)-\sqrt{\kappa_{01}}c_{\rm 1,in}(t),~~ \tag {2} \end{align} $$ $$\begin{align} \frac{da_{2}(t)}{dt}=\,&-i(\omega_{2}-\omega_{\rm p})a_{2}(t)-\frac{\kappa_{02}}{2}a_{2}(t)\\ &-iJa_{1}(t)-ig_{2}\sigma_{2}^-(t)\\ &-\sqrt{\kappa_{02}}c_{\rm 2,in}(t),~~ \tag {3} \end{align} $$ where $\kappa_{1}$ is the coupling strength between the PM system and the waveguide, $\kappa_{0i}$ is the intrinsic loss of the $i$th microcavity; $a_{\rm in}$ and $a_{\rm out}$ are the input and output fields of the first microcavity; and $c_{i,{\rm in}}(t)$ is the noise input operator of the $i$th microcavity which preserves the canonical commutation relations. The motion equation of the dipole lower operator is $$\begin{alignat}{1} \frac{d\sigma_{i}^-(t)}{dt}=\,&-i(\omega_{\rm di}-\omega_{\rm p}) \sigma_{i}^-(t)-\frac{\gamma_{\rm di}}{2}\sigma_{i}^-(t)\\ &-g_{i}a_{i}(t)\sigma_{i}^{z}(t)+\sqrt{\gamma_{\rm ai}}\sigma_{i}^{z}(t)e_{i,{\rm in}}(t),~~ \tag {4} \end{alignat} $$ where $\gamma_{\rm di}$ and $e_{i,{\rm in}}(t)$ are spontaneous emission rate of the $i$th dipole emitter and its noise input operator. In the following we consider the weak excitation limit,[25] and thus the dipole emitters are always in their ground states, which allows us to replace the dipole population operator with its expectation value $\langle\sigma_{i}^{z}\rangle{\approx}{-1}$. In this limit, it has $[\sigma_{i}^-,\sigma_{i}^{+}]\approx1$ and $\langle\sigma_{i}^{+}\sigma_{i}^-\rangle\approx0$. Therefore, we can take the dipole emitter as a boson almost in a vacuum state. Furthermore, we assume that the external fields of the dipole emitters are in the vacuum state and the contributions of the dipole noise operators are negligible, i.e. $\langle e_{\rm 1,in}(t)\rangle=\langle e_{\rm 2,in}(t)\rangle=0$. The output fields into the waveguides are related to the input fields by the input–output relation $a_{\rm out}(t)=a_{\rm in}(t)+\sqrt{\kappa_{1}}a(t)$.[26] We assume a weak monochromatic field with frequency $\omega_{\rm p}$ input from the left port of the waveguide as shown in Fig. 1(a), and neglect the contribution of input fields from the right port of the waveguide. Thus the transmission coefficient of the PM system can be expressed as $$\begin{alignat}{1} \!\!\!\!\!\!t_{\rm H}=\frac{a_{\rm out}(t)}{a_{\rm in}(t)}=\frac{(i{\it \Delta}_{1}'-\frac{\kappa}{2}+\kappa_{1})(i{\it \Delta}_{2}'-\frac{\kappa_{02}}{2})+J^{2}}{(i{\it \Delta}_{1}'-\frac{\kappa}{2})(i{\it \Delta}_{2}'-\frac{\kappa_{02}}{2})+J^{2}},~~ \tag {5} \end{alignat} $$ where $\kappa=\kappa_{01}+\kappa_{1}$, $i{\it \Delta}_{i}'=i{\it \Delta}_{i}+g_{i}^2/(i{\it \Delta}_{\rm di}-\gamma_{\rm di}/2)$, ${\it \Delta}_{i}=\omega_{\rm p}-\omega_{i}$ and ${\it \Delta}_{\rm di}=\omega_{\rm p}-\omega_{\rm di}$. To discuss the transmission properties of the PM system explicitly, we will compare the results with those in the CRIT and DIT effect. The transmission coefficients for the CRIT and DIT effect can be directly obtained from Eq. (5) just by setting $g_{1}=g_{2}=0$ and $J=0$ as $$\begin{align} t_{\rm CRIT}=\,&\frac{(i{\it \Delta}_{1}-\frac{\kappa}{2}+\kappa_{1}) (i{\it \Delta}_{2}-\frac{\kappa_{02}}{2})+J^{2}} {(i{\it \Delta}_{1}-\frac{\kappa}{2})(i{\it \Delta}_{2}-\frac{\kappa_{02}}{2})+J^{2}},~~ \tag {6} \end{align} $$ $$\begin{align} t_{\rm DIT}=\,&\frac{i{\it \Delta}_{1}'-\frac{\kappa}{2}+\kappa_{1}} {i{\it \Delta}_{1}'-\frac{\kappa}{2}}.~~ \tag {7} \end{align} $$ In what follows, we investigate the transmission spectrum of the PM system, and consider the case where the cavity loss to waveguides is far larger than its intrinsic loss. Therefore, it is reasonable to consider the overcoupling limit $\kappa_{1}{\gg}(\kappa_{01},\kappa_{02})$, e.g. $\kappa_{01}=\kappa_{02}{\approx}10^{-3}\kappa_{1}$. The crucial parameter of the cavity QED system is the cooperativity or the Purcell factor,[27] and it has been verified that the DIT effect can be achieved in a dipole-microcavity system with a large Purcell factor.[15] To facilitate the subsequent discussion without loss of generality, we assume that the Purcell factors of dipole-microcavity systems are identical and far larger than 1, e.g. $F_{\rm p}=2g^{2}/\kappa_{1}\gamma_{\rm di}{\gg}1$, when the incident photon is coupling with dipole emitters, whereas their Purcell factors are zero for the decoupling case. In the numerical calculation, we assume that the decay rates of dipole emitters satisfy $\gamma_{d1}=\gamma_{d2}=10^{-3}\kappa_{1}$.
cpl-36-3-034204-fig2.png
Fig. 2. The power transmission of the photonic molecule system for the Purcell factors (a) $F_{\rm p}=10$, (b) $F_{\rm p}=50$, (c) $F_{\rm p}=200$ and (d) $F_{\rm p}=1000$, respectively. The coupling strength between microcavities is $J=\kappa/2$.
In Fig. 2, we depict the power transmission spectrum $T=|t|^{2}$ for the superposed transparency effect (red solid), DIT effect (blue dashed) and CRIT effect (black dot-dashed) under different Purcell factors. For different Purcell factors, we consistently set $J=\kappa/2$, thus the CRIT transmission spectra are identical in Figs. 2(a)–2(d). We can see that the CRIT transmission amplitudes are close to 1 and the transmission spectrum displays two dips (transparency windows). Then we open the DIT effect and study the superposed transparency effect. The DIT effect becomes enhanced with the increase in the Purcell factor. When the Purcell factor $F_{\rm p}=10$, compared with the CRIT transmission spectrum, the superposed transmission amplitudes decrease, while the two dips become sharper and the FWHMs of the superposed transparency windows are evidently narrower than those of the CRIT and DIT effect, which is beneficial to realizing the photonic Faraday rotation effect. From Figs. 2(a)–2(d), we can see that the superposed transmission amplitudes increase and the FWHMs of the superposed transparency windows become wider. However, in any case, the FWHMs of the superposed transparency windows are smaller than the CRIT and DIT transparency windows. Therefore, the hybrid system may be more suitable for the realization of QIP using the photonic Faraday rotation effect.
cpl-36-3-034204-fig3.png
Fig. 3. The transmission amplitudes (a) and phases (b) for four cases in the PM system. The Purcell factors of each system are $F_{\rm p}=1000$ and the coupling strength satisfies $J=0.6\kappa_{1}$.
We now investigate the potential application of the superposed transparency effect in the PM system by considering the polarization DoF of photons. From Eqs. (5)-(7), we can see that the transmission coefficients for the superposed transparency effect, CRIT effect and DIT effect are complex, and they are closely related to both the frequency and polarization of the incident photon. Thus the PM system can readily modulate both the phase and amplitude of the incident photon by its frequency and polarization. To explain the mechanism of the photonic Faraday rotation effect in the PM system, we consider the dipole emitter with two degenerate ground states $|g_{\rm L}\rangle$ and $|g_{\rm R}\rangle$ as shown in Fig. 1(b). The transition of dipole emitter $|e\rangle {\leftrightarrow} |g_{\rm L(R)}\rangle$ is driven by the left (right) circularly polarized photon. The transmission coefficients for only one of the dipole emitters, e.g. the $i$th dipole emitter, coupling with the incident photon can be obtained from Eq. (5) as follows: $$\begin{align} t_{1}^{(1)}=\,&\frac{(i{\it \Delta}_{1}'-\frac{\kappa}{2}+\kappa_{1}) (i{\it \Delta}_{2}-\frac{\kappa_{0b}}{2})+J^{2}} {(i{\it \Delta}_{1}'-\frac{\kappa}{2})(i{\it \Delta}_{2}-\frac{\kappa_{0b}}{2})+J^{2}},~~ \tag {8} \end{align} $$ $$\begin{align} t_{1}^{(2)}=\,&\frac{(i{\it \Delta}_{1}-\frac{\kappa}{2}+\kappa_{1}) (i{\it \Delta}_{2}'-\frac{\kappa_{0b}}{2})+J^{2}} {(i{\it \Delta}_{1}-\frac{\kappa}{2})(i{\it \Delta}_{2}'- \frac{\kappa_{0b}}{2})+J^{2}}.~~ \tag {9} \end{align} $$ On the other hand, the transmission coefficients for $m=0,2$ dipole emitters coupling with the incident photon are just those of the CRIT effect and superposed transparency effect, which can be denoted as $t_{0}$ and $t_{2}$, respectively. In Figs. 3(a) and 3(b), we plot the transmission amplitudes and phases for the cases where the incident photon is coupling with $m(=0,1,2)$ dipole emitters, respectively. In the resonant regime, it is shown that all of the transmission amplitudes are close to 1, and thus only phase shifts of the incident photon occur, which are denoted by $\theta_{0}$, $\theta_{1}^{(1)}$, $\theta_{1}^{(2)}$ and $\theta_{2}$, respectively. To be more explicit, we assume that the incident photon is in the linearly polarized state $(|L\rangle_{\rm p}+|R\rangle_{\rm p})/\sqrt{2}$, and the quantum states of the two dipole emitters are $|g_{\rm L}\rangle_{1}|g_{\rm R}\rangle_{2}$. Therefore, the transmitted photon will be in the state $(e^{i\theta_{1}^{(1)}}|L\rangle_{\rm p}+e^{i\theta_{2}^{(2)}}|R\rangle_{\rm p})/\sqrt{2}$, and the polarized direction of the transmitted photon relative to the incident photon is subject to the Faraday rotation angle $\theta_{\rm F}=\theta_{1}^{(2)}-\theta_{1}^{(1)}$, which is a function of the parameters of the system. From Fig. 3(b), we can see that phase shifts of the transmitted photon are $\theta_{0}=\theta_{1}^{(1)}=\theta_{2}=0$ and $\theta_{1}^{(2)}=\pi$ in the resonant regime, and thus $\theta_{\rm F}=\pi$ can be obtained, which is large enough for realizing QIP. The evolution of the dipole emitters and photon can be expressed as $$\begin{align} &\otimes_{i=1}^{2}\frac{1}{\sqrt{2}}(|g_{\rm L}\rangle_{i}+|g_{\rm R}\rangle_{i})|L\rangle_{\rm p}\\ \rightarrow\,&\frac{1}{2}(|g_{\rm L}\rangle_{1}|g_{\rm L}\rangle_{2}+|g_{\rm L}\rangle_{1}|g_{\rm R}\rangle_{2}\\ &-|g_{\rm R}\rangle_{1}|g_{\rm L}\rangle_{2}+|g_{\rm R}\rangle_{1}|g_{\rm R}\rangle_{2})|L\rangle_{\rm p},~~ \tag {10} \end{align} $$ $$\begin{align} &\otimes_{i=1}^{2}\frac{1}{\sqrt{2}}(|g_{\rm L}\rangle_{i}+|g_{\rm R}\rangle_{i})|R\rangle_{\rm p}\\ \rightarrow\,&\frac{1}{2}(|g_{\rm L}\rangle_{1}|g_{\rm L}\rangle_{2}-|g_{\rm L}\rangle_{1}|g_{\rm R}\rangle_{2}\\ &+|g_{\rm R}\rangle_{1}|g_{\rm L}\rangle_{2}+|g_{\rm R}\rangle_{1}|g_{\rm R}\rangle_{2})|R\rangle_{\rm p}.~~ \tag {11} \end{align} $$ The evolutions just correspond to the conditional phase gate for dipole emitters, and thus quantum entanglement of dipole emitters can be generated. The distinct advantage of this entanglement generation scheme is that it can be realizable within only one step, so it may be more feasible than those based on elementary gate decomposition.[20,28] We briefly discuss the experimental feasibility of the superposed transparency effect in the PM system based on recent experimental accessible technology. One of the potential candidates for constructing the PM system may be the whispering-gallery microcavity because of its ultrahigh quality factor (${\sim} 10^{8}$) and small modal volume.[29,30] The tapered optical fibers can be employed to input and output the photon, and the nitrogen-vacancy (NV) center in a diamond, which is chosen as the dipole emitter, can be located on the surface of a microtoroid cavity to manipulate the transport of a photon. The parameters of the system $(g,\kappa,\kappa_{\rm s},\gamma_{\rm a})=(180,4.7,2.35,13){\times}2\pi$ MHz are reachable,[16] and thus the related Purcell factor $F_{\rm p}$ is of the order of $10^{3}$. Thus the experimental conditions for the superposed transparency effect and the photonic Faraday effect can be fulfilled with the present experimental technology.[10,11] In conclusion, we have theoretically investigated the superposed transparency effect in the PM system. It is shown that the superposed transparency windows are consistently narrower than the CRIT and DIT transparency windows. Benefiting from the superposed transparency effect, the photonic Faraday rotation effect can be found and quantum entanglement may be generated in the PM system. In the future, it may be interesting to investigate the superposed transparency effect in the PM system including more microcavities or with parity-time symmetry. 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