Abstract

We investigate the probe-field transmission in a hybrid cavity quantum electrodynamic (CQED) system, where one optical cavity containing a quantum dot (QD) with high cavity dissipation is coupled to another auxiliary cavity with a high quality factor. We also investigate the hybrid system operating in the weak coupling regime of the light-matter interaction via comparing the QD photon interaction with the dipole decay rate and the cavity field decay rate. It is shown that the dipole induced transparency (DIT) regime similar to electromagnetically induced transparency (EIT) can be achieved due to the destructive interference of the cavity field in the weak coupling regime, which is extremely significant for the field of semiconductor CQED. The auxiliary cavity plays a key role in the hybrid system, which affords a quantum channel to affect the probe transmission leading to enhanced DIT. Further, DIT induced coherent optical propagation properties such as fast and slow light effects are also investigated based on the hybrid system for suitable parametric regimes. By controlling the coupling strength J and the decay rate ratio δ of the two cavities, tunable and controllable fast-to-slow light propagation can be achieved. This study provides a promising platform for understanding the dynamics of QD-CQED systems and may open up promising on chip applications in quantum information processing.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Due to the development of high quality (Q) factors optical microcavities, cavity quantum electrodynamics (CQED) [1,2] investigating light-matter interactions inside a resonator has witnessed rapid progress in the past few years due to it not only provides a test bed for quantum physics but also has potential applications in quantum-information processing [37]. Tremendous progress has been made by coupling single quantum dipole emitters to different cavities including a single quantum dot (QD)-semiconductor microcavity system [8], a single QD embedded in a microdisk microcavity system [9], and photonic crystal (PhC) micro/nanocavities with self-assembled QD system [10], which indicate a fascinating platform for studying solid-state CQED systems. Among the systems, PhC nanocavities coupled to QDs may be promising due to its highly confined ultra-small mode volume (V) in the order of the qubic wavelength and ultrahigh quality Q-factor [1116].

Usually, the approaches for light-dipole interaction in cavity QED are focused on the strong-coupling regime manifested by vacuum Rabi splitting (VRS) [8,10,1723], which is always referred to the high-Q regime with the emitter-photon coupling strength beyond both the decay rates of the emitter and the cavity field decay rate. In the past decades, the research of CQED focus on a single quantum emitter inside a cavity [14,16,2426] and how to improve the Q factor and reduce the mode V of the resonators to reach stronger interactions has made great efforts with different optical cavities [16,21,2730]. However, it remains difficult to achieve high Q factor and small mode V simultaneously for the same type resonator due to the diffraction limit. A smaller V corresponds to a larger radiative decay rate and more significant roughness scattering, leading to a lower Q. Different types of resonators possess their own unique properties, but the trade-off between high Q and small V still exists [3133].

In the high-Q regime (strong coupling regime), this result of VRS is clear, because the cavity mode is split into a lower and an upper polariton by more than a linewidth. In a bad cavity regime, where the cavity decay rate κ is much bigger than the dipole decay rate Γ (called the Purcell regime [34]), the similar result of analogous VRS (normal modes splitting) is surprising because the incident field can still drive both the cavity modes. Transparency in this regime is instead caused by destructive interference of the cavity field, which is analogous to the destructive interference of the excited state of a 3-level atomic system in electromagnetically induced transparency (EIT) [35]. For this reason, we refer to this effect as dipole induced transparency (DIT) [3641] which can be used for quantum information processing in the weak-coupling regime (low-Q regime), and DIT effect was also experimentally demonstrated in the photonic crystal cavity-waveguide system [42].

The demonstration of DIT effect in the low-Q regime is extremely important for the field of semiconductor CQED. Although the high-Q regime can be achieved with recent fabrication technique [43], it is extremely difficult to achieve using semiconductor technology, and strong coupling between solid state qubit and photon may be challenging with present experimental technology. Semiconductor implementations of CQED systems, such as photonic crystal cavities coupled to quantum dots, usually suffer from large out-of-plane losses, resulting in short cavity lifetimes. Furthermore, the spontaneous emission of solid state qubit and photon loss in the microcavity may be the main source of noise. The distinct advantage of DIT effect is that it can be achievable if large Purcell factor of CQED system is fulfilled without the requirement of strong coupling, thus it allows the system work in the bad cavity regime, which greatly relaxes the experimental requirement of CQED system.

In this work, we consider a hybrid CQED system, where the CQED system driven by two-tone fields consisting of a dipole quantum emitter and a cavity is coupled to an auxiliary cavity through a short-length single-mode waveguide. Here we take a Fabry-Perot CQED system as an example, and it allows generalization to other physical implementations, including solid-state circuit QED systems. We investigate the auxiliary cavity enhanced DIT effect and DIT induced a tunable and controllable fast-to-slow light propagation with manipulating the coupling strength J between the two cavities in a weak coupling regime where the QD-cavity field coupling strength g is less than κ. It has been theoretically predicted [44,45] that a weakly coupled QD can also control the photon transmission through a resonator, as long as the system is in the strong Purcell regime $({{{g^2}} \mathord{\left/ {\vphantom {{{g^2}} {\kappa \Gamma }}} \right.} {\kappa \Gamma }} > 1)$. Such a regime is much easier to achieve in the solid state systems, as $\Gamma < < g,\kappa$, as opposed to the atomic physics systems where κ is on the same order as Γ.

In addition, in the hybrid CQED system, the QED cavity a has high cavity dissipation $\kappa $ (without high Q) and the auxiliary cavity mode c with high Q but large V, then the requirement for high Q and small V for the same cavity can be removed. The protocol has been presented by Liu et al, [3133] for realizing effective strong coupling, here we introduce the protocol to investigate the enhanced DIT effect in weak coupling regime. We introduce a ratio parameter $\delta = {{{\kappa _a}} \mathord{\left/ {\vphantom {{{\kappa_a}} {{\kappa_c}}}} \right.} {{\kappa _c}}}$, where ${\kappa _a} = {{{\omega _a}} \mathord{\left/ {\vphantom {{{\omega_a}} {{Q_a}}}} \right.} {{Q_a}}}$ and ${\kappa _c} = {{{\omega _c}} \mathord{\left/ {\vphantom {{{\omega_c}} {{Q_c}}}} \right.} {{Q_c}}}$ are the decay rates of cavity modes a and c (${\omega _a}$ and ${\omega _c}$ are the frequencies of cavity a and c) to investigate DIT effect under different parameter regimes. The QED cavity a is coupled to auxiliary cavity c via evanescent field, and the coupling strength J between the two cavities can be controlled by varying the separation between them [44]. The results indicate that the fast-to-slow light can be controlled with adjusting the parameter $\delta$.

2. Model And theory

Our system, depicted in Fig. 1(a), consists of a QD embedded in an PhC nanocavity a with optical pump-probe technology [46] is coupled to an auxiliary cavity c with a single mode waveguide which is an ideal platform for the photon exchange between two optical cavities [47]. Here, we take a Fabry-Perot cavity QED system as an example, and it allows generalization to other physical implementations, such as PhC nanocavity QED [10] and solid-state circuit QED systems [48]. The cavity a and cavity c are coupled with the coupling strength J via exchanging energy and J depends on the distance between the two cavities, and the cavity-cavity coupling Hamiltonian [49] can be described as ${H_{a - c}} = \hbar J({a^{\dagger} }c + a{c^{\dagger} })$.

 

Fig. 1. (a) Schematic of the cavity QED system coupled to an auxiliary cavity, where a QED cavity a with high cavity dissipation driven by two-tone fields coupled to an auxiliary cavity c with high quality factor. The cavity a is coupled to cavity c via evanescent field, and the coupling strength $J$ between the two cavities can be controlled by varying the separation between them [44]. (b) The two energy levels of QD coupled to a single cavity mode and two optical fields. (c) and (d) are the energy level transitions with an entangled state $|{{n_{tot}}} \rangle $ (${n_{tot}} = {n_a} + {n_c}$ is the total photon number of the two cavities, where ${n_a}$ and ${n_c}$ represent the number state of the photon mode of cavity a and cavity c).

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In the QED cavity, we consider the QD as a two-level system including the ground state $|g \rangle $ and the single exciton state $|e \rangle $ [50] at low temperature, whose Hamiltonian is described as ${H_{QD}} = \hbar {\omega _e}{\sigma ^z}$, with the exciton frequency ${\omega _e}$, where ${\sigma ^z}$ and ${\sigma ^ \pm }$ are the Pauli operators. Considering a strong pump field with frequency ${\omega _p}$ and a weak probe field frequency ${\omega _s}$ simultaneously driving the cavity a, and their amplitudes can be defined as ${\varepsilon _p} = \sqrt {{{{P_c}} \mathord{\left/ {\vphantom {{{P_c}} {\hbar {\omega_p}}}} \right.} {\hbar {\omega _p}}}} $ and ${\varepsilon _s} = \sqrt {{{{P_s}} \mathord{\left/ {\vphantom {{{P_s}} {\hbar {\omega_s}}}} \right.} {\hbar {\omega _s}}}} $, respectively. The decay rate of cavities mode ${\kappa _c} = {\kappa _a} = {\kappa _{ex}} + {\kappa _0}$ with the intrinsic photon loss rate ${\kappa _0}$, and ${\kappa _{ex}}$ describes the rate at which energy leaves the optical cavity into propagating fields [51]. Here, we only consider the condition for simplicity:${\kappa _{ex}} = {\kappa _0} = {\kappa _{ae}} = {\kappa _{ce}}$, and we consider ${\omega _c} = {\omega _a}$. We use the rotating frame of the pump laser frequency ${\omega _p}$, and obtain the whole Hamiltonian of the system as

$$\begin{array}{l} H = \hbar {\Delta _p}{\sigma ^z} + \hbar {\Delta _a}{a^{\dagger} }a + \hbar {\Delta _c}{c^{\dagger} }c + \hbar J({a^{\dagger} }c + a{c^{\dagger} }) + \hbar g({\sigma ^ + }a + {\sigma ^ - }{a^{\dagger} })\\ + i\hbar \sqrt {{\kappa _{ae}}} {\varepsilon _p}({a^{\dagger} } - a) + i\hbar \sqrt {{\kappa _{ae}}} {\varepsilon _s}({a^{\dagger} }{e^{ - i\Omega t}} - a{e^{i\Omega t}}) \end{array}$$
where ${\Delta _p} = {\omega _e} - {\omega _p}$ is the exciton-pump field detuning, ${\Delta _a} = {\omega _a} - {\omega _p}$ is the PhC nanocavity pump field detuning, ${\Delta _c} = {\omega _c} - {\omega _p}$ is the auxiliary cavity-pump field detuning, and $\Omega = {\omega _s} - {\omega _p}$ is the probe-pump detuning. ${a^{\dagger} }({c^{\dagger} })$ and $a(c)$ are the creation and annihilation operators for cavity a and c, respectively. g denotes the coupling strength between the exciton in the QD and the photons in the PhC nanocavity.

In the present work, we are mainly interested in the mean response of the system to the probe field, so all the operators can be reduced to their expectation values. The quantum Langevin equations for the operators expectation values can be expressed as [52]:

$$\left\langle {\dot{a}} \right\rangle ={-} (i{\Delta _a} + {\kappa _a})\left\langle a \right\rangle - ig\left\langle {{\sigma^ - }} \right\rangle - iJ\left\langle c \right\rangle + \sqrt {{\kappa _{ae}}} ({\varepsilon _p} + {\varepsilon _s}{e^{ - i\Omega t}})$$
$$\left\langle {\dot{c}} \right\rangle ={-} (i{\Delta _c} + {\kappa _c})\left\langle c \right\rangle - iJ\left\langle a \right\rangle $$
$$\left\langle {{{\dot{\sigma }}^z}} \right\rangle ={-} {\Gamma _1}(\left\langle {{\sigma^z}} \right\rangle + 1) - ig(\left\langle {{\sigma^ + }} \right\rangle \left\langle a \right\rangle - \left\langle {{\sigma^ - }} \right\rangle \left\langle {{a^{\dagger} }} \right\rangle )$$
$$\left\langle {{{\dot{\sigma }}^ - }} \right\rangle ={-} (i{\Delta _p} + {\Gamma _2})\left\langle {{\sigma^ - }} \right\rangle + 2ig\left\langle a \right\rangle \left\langle {{\sigma^z}} \right\rangle $$
where ${\Gamma _1}({\Gamma _2})$ is the exciton relaxation rate (exciton dephasing rate). It should be noted that the factorization assumptions $\left\langle {{\sigma^ + }a} \right\rangle = \left\langle {{\sigma^ + }} \right\rangle \left\langle a \right\rangle $ and $\left\langle {{\sigma^z}a} \right\rangle = \left\langle {{\sigma^z}} \right\rangle \left\langle a \right\rangle $ are used in the above equations, so some minor quantum correlations are ignored for simplicity but without loss of generality.

As the probe laser is weaker than the pump laser, the Heisenberg operator O can be rewritten as the sum of its steady-state mean value ${O_s}$ and a small fluctuation $\delta O$, i.e. $O = {O_s} + \delta O$ ($O = {\sigma ^z},{\sigma ^ - },a,c$) with the standard methods of quantum optics. Then we can obtain two group equations as follows. The first group is the steady-state solutions of the Eqs. (2)–(5),

$$(i{\Delta _a} + {\kappa _a}){a_s} - ig{\sigma _s} - iJ{c_s} = \sqrt {{\kappa _{ae}}} {\varepsilon _p}$$
$$(i{\Delta _c} + {\kappa _c}){c_s} + iJ{a_s} = 0$$
$${\Gamma _1}(\sigma _s^z + 1) + ig(\sigma _s^\ast {a_s} - {\sigma _s}a_s^\ast ) = 0$$
$$(i{\Delta _p} + {\Gamma _2}){\sigma _s} - 2ig{a_s}\sigma _s^z = 0$$
Our analysis works in the weak excitation limit, where the QD is predominantly in the ground state. In this limit, $\sigma _s^z \approx 1$ for all time. For the second group equations, keeping only the linear terms of the fluctuation operators, we make the ansatz [53]$\left\langle {\delta O} \right\rangle = {O_ + }{e^{ - i\Omega t}} + {O_ - }{e^{i\Omega t}}$, and then obtain
$${a_ + } = \frac{{\sqrt {{\kappa _{ae}}} {\varepsilon _s}}}{{i({\Delta _a} - \Omega + g{\Lambda _4} + J{\eta _1}) + {\kappa _a}}}$$
where
$${\eta _1} = \frac{{ - iJ}}{{i({\Delta _c} - \Omega ) + {\kappa _c}}},{\eta _2} = \frac{{ - iJ}}{{i({\Delta _c} + \Omega ) + {\kappa _c}}},{\lambda _1} = \frac{{ - ig\sigma _s^\ast }}{{{\Gamma _1} - i\Omega }},{\lambda _2} = \frac{{ - ig{a_s}}}{{{\Gamma _1} - i\Omega }},$$
$${\lambda _3} = \frac{{ig{\sigma _s}}}{{{\Gamma _1} - i\Omega }},{\lambda _4} = \frac{{ - iga_s^\ast }}{{{\Gamma _1} - i\Omega }},{\Lambda _1} = \frac{{ig}}{{ - i({\Delta _a} + \Omega + J\eta _2^\ast ) + {\kappa _c}}}$$
$${\Lambda _2} = \frac{{ - 2iga_s^\ast {\lambda _1}}}{{ - i({\Delta _p} + \Omega ) + {\Gamma _2} + 2iga_s^\ast {\lambda _2} + 2ig(a_s^\ast {\lambda _3} + \sigma _s^z){\Lambda _1}}},$$
$${\Lambda _3} = \frac{{ - 2iga_s^\ast {\lambda _4}}}{{ - i({\Delta _p} + \Omega ) + {\Gamma _2} + 2iga_s^\ast {\lambda _2} + 2ig(a_s^\ast {\lambda _3} + \sigma _s^z){\Lambda _1}}},$$
$${\Lambda _4} = \frac{{2ig[({a_s}{\lambda _1} + \sigma _s^z) + {a_s}({\lambda _2} + {\lambda _3}{\Lambda _1}){\Lambda _2}]}}{{i({\Delta _p} - \Omega ) + {\Gamma _2} - 2ig{a_s}{\lambda _4} - 2ig{a_s}({\lambda _2} + {\lambda _3}{\Lambda _1}){\Lambda _3}}}$$
where ${\Re ^\ast }$ indicates the conjugate of $\Re$.

Using the standard input-output relation [54]${a_{out}}(t) = {a_{in}}(t) - \sqrt {2{\kappa _a}} a(t)$, where ${a_{out}}(t)$ is the output field operator. The transmission spectrum of the probe field defined by the ratio of the output and input field amplitudes at the probe frequency, which shows [51]

$$t({\omega _s}) = \frac{{{\varepsilon _s} - \sqrt {{\kappa _{ae}}} {a_ + }}}{{{\varepsilon _s}}}$$
The transmission group delay can be expressed as [51]
$${\tau _g} = \frac{{d{\phi _t}}}{{d{\omega _s}}}{|_{{\omega _s} = {\omega _p}}} = \frac{{d\{ \arg [t({\omega _s})]\} }}{{d{\omega _s}}}{|_{{\omega _s} = {\omega _p}}}$$
where ${\phi _t} = \arg [t({\omega _s})]$ is the phase dispersion relation, which can cause the transmission group delay. The positive group delay ${\tau _g} > 0$ corresponds to fast light and the negative delay group ${\tau _g} < 0$ corresponds to slow light, respectively. The phase change can play a significant role in slow and fast light propagation.

We choose the realistic coupled system of an InAs/GaAs QD embedded in a PhC nanocavity [10] in the simultaneous presence of a strong pump laser and a weak probe laser as shown in Fig. 1(a), and the realistic parameters [55] of the system are ${\kappa _a} = {\kappa _c} = 8\textrm{ MHz}$, ${\Gamma _1} = 2{\Gamma _2} = 5.2\textrm{ MHz}$, $g = 2\textrm{ MHz}$, $P = 1.0\textrm{ nW}$. $J$ is the coupling strength between the two cavities which strongly depends on the distance between the two cavities [56], and the coupling strength we expect $J/2\pi \sim MHz$.

3. Numerical results and discussions

Case A: Exciton-pump field detuning ${\Delta _p} = 0$

There are two kinds coupling in the hybrid CQED system, i.e. exciton-photon coupling g and cavity-cavity coupling J, which will affect the dynamics of the system. In order to demonstrate the auxiliary cavity enhanced DIT, here we keep the exciton-phonon coupling g unchanged (in weak coupling regime, $g = 2\textrm{ MHz}$) and vary the coupling strength $J$ of the two cavities. Firstly, in order to investigate the influence of coupling strength $J$ on the transmission spectrum at the resonant region (i.e., ${\Delta _p} = 0$, ${\Delta _a} = 0$, and ${\Delta _c} = 0$), we plot the probe transmission ${|{t({\omega_s})} |^2}$ as a function of the probe-cavity a detuning ${\Delta _s} = {\omega _s} - {\omega _a}$ with several coupling strengths J between the two cavities under the pump power of cavity a is $P = 1.0\textrm{ nW}$, as shown in Fig. 2(a). When there is no auxiliary cavity ($J = 0$) or the coupling strength $J$ is weak (such as $J = 1.0{\kappa _a}$) corresponding to the red curve in Fig. 2(a), a Lorentzian curve appears in the probe transmission spectrum, which shows the transmission of probe field. However, when increasing the coupling strength $J$ from $J = 1.0{\kappa _a}$ to $J = 4.0{\kappa _a}$, a transparency window starts appearing in the probe transmission spectrum, and the transparency peak increases with increasing the coupling strength $J$, which is analogous to the normal mode splitting [57]. Further, there is an interesting phenomenon in weak coupling regime, and we individually plot them as shown in Fig. 2(b) and Fig. 2(c) corresponding to the situation of $J = 0$ and $J = 1.0{\kappa _a}$. In the case of $J = 0$, except the transmission spectrum presents a Lorentzian curve, there are also two peaks locating symmetrically both sides of the Lorentzian peak marked by oval in Fig. 2(b), which are similar the bonding-(B) and antibonding-(AB) like modes [58]. If we consider the auxiliary cavity (such as $J = 1.0{\kappa _a}$,i.e., weak cavity-cavity coupling) as shown in Fig. 2(c), we find the B- and AB- like modes move to the center of peak (the Lorentzian peak). When we further increase the coupling strength $J$ from $J = 2.0{\kappa _a}$ to $J = 4.0{\kappa _a}$ (strong cavity-cavity coupling), the B- and AB- like modes disappear or wash away by the auxiliary cavity. According to measure the width of the splitting in the probe transmission spectrum with increasing the coupling strength $J$, we find that the width of the splitting and the coupling strength $J$ have a linear proportional relation. In Fig. 2(d), we plot the splitting distance in the probe transmission spectrum as a function of the coupling strength $J$, which indicate the splitting distance increases linearly with increasing the coupling strength J. The fact is that the coupling strength $J$ is directly proportional [actually, the coupling strength $J$ exponentially decreases with increasing the distance between the two cavities (Supplementary information in Ref. [44]). But, in a small region, the relationship between the distance of the two cavities and the coupling strength J is linear] to the distance of cavity a and cavity c. Once the distance between the two cavities is very large (i.e., the coupling strength $J$ tends to zero), there is no interaction between the two cavities and as a result no transparency window appears in the probe transmission spectrum (in weak coupling regime, $g = 2\textrm{ MHz}$).

 

Fig. 2. (a) The probe transmission ${|{t({\omega_s})} |^2}$ as a function of probe-cavity detuning ${\Delta _s}$ for several coupling strengths J at the condition of ${\Delta _p} = 0$. (b) The probe transmission ${|{t({\omega_s})} |^2}$ at $J = 0$. (c)The probe transmission ${|{t({\omega_s})} |^2}$ at $J = 1.0{\kappa _a}$. (d) The peak splitting distance as function of J.

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The physical process is almost similar to EIT in atoms system [35] and we call it as DIT [3641]. Recently, the analogous phenomenon of EIT in optomechanical systems was also demonstrated, and we termed them as optomechanically induced transparency (OMIT) and absorption (OMIA) [5961], which can be used as a quantum memory [62], quantum repeater [63]. The phenomenon of DIT can be interpreted with a dressed-state picture. When the QD coupled to the optical cavity, the excited state of the excition $|e \rangle $ are dressed by an entangled state $|{{n_{tot}}} \rangle $ satisfying the total photon number of the two cavities ${n_{tot}} = {n_a} + {n_c}$ (${n_a}$ and ${n_c}$ represent the number state of the photon mode of cavity a and cavity c) as shown in Fig. 1(b)-Fig. 1(d). Then the original eigenstates $|e \rangle $ are modified to form two dressed states, i.e. $|{e,{n_{tot}}} \rangle $ and $|{e,{n_{tot}} + 1} \rangle $. The left sharp peak indicates the transition from $|g \rangle $ to $|{e,{n_{tot}} + 1} \rangle $ and the right sharp peak is the transition from $|g \rangle $ to $|{e,{n_{tot}}} \rangle $. With increasing the coupling strength $J$ from $J = 1.0{\kappa _a}$ to $J = 4.0{\kappa _a}$, the splitting of two side peaks are more remarkable. In the excitation of a strong pump filed to cavity a, the steady-state entanglement state $|{{n_{tot}}} \rangle $ between cavity a and c, as a quantum channel, can be generated, which provides an indirect optical pathway to excite cavity c by means of the pump filed. Therefore, the coupling strength $J$ of the two cavities is an important factor of the quantum channel, which can influence the width of the mode splitting.

In fact, the exciton-photon coupling g can also affect the transmission characteristics of the probe field as investigated in previous works [64,65] in the strong coupling regime which manifested by vacuum Rabi splitting. For weak coupling strength g, a Lorentzian curve will appear in the probe transmission spectrum. It has been theoretically predicted [36,45] that a weakly coupled QD can also control the photon transmission through a resonator, as long as the system is in the strong Purcell regime (${{{g^2}} \mathord{\left/ {\vphantom {{{g^2}} {\kappa \Gamma }}} \right.} {\kappa \Gamma }} > 1$). Such a regime is much easier to achieve in the solid state systems, as $\Gamma < < g,\kappa $, as opposed to the atomic physics systems where $\kappa $ is on the same order as $\Gamma $. In the weak coupling regime, the Purcell effect [34] can either enhance or inhibit the decay rate of irreversible spontaneous emission. Therefore, the demonstration of DIT effect in weak coupling regime, in a way, is more significant than in the strong coupling regime manifested by VRS.

As we know, EIT in atoms system [66] will indicate a tendency to reach the slow- or fist light effect. In Eq. (12), the magnitude of group delay relies on the rapid phase dispersion in the transmitted probe field, and ${\tau _g} > 0$ and ${\tau _g} < 0$ corresponding to slow and fast light propagation, respectively. Here, in our system, we realize the tunable fast-to-slow light propagation or vice versa with controlling parameters in different regime. Using the relation of ${\phi _t} = \arg [t({\omega _s})]$, we study the phase of the transmitted probe field, and plot the phase as a function of ${\Delta _s}$ with several different coupling strength $J$ as shown in Fig. 3(a). Obviously, the transmitted probe field shows a negative slope at $J = 0$, which indicates a negative group delay or fast light propagation. This corresponds to the Lorentz line profile of the probe transmission spectrum in Fig. 2(a). When increasing the coupling strength $J$ from $J = 1.0{\kappa _a}$ to $J = 4.0{\kappa _a}$ at ${\kappa _a} = {\kappa _c}$, the phase of the transmitted probe field varies from negative to positive, which results in positive group delay or slow light propagation through the system. Therefore, the change of phase in the transmitted probe light from negative to positive slope with manipulating the coupling strength $J$ corresponds to the control of fast-to-slow light propagation. In Fig. 3(b) we plot the transmission group delay ${\tau _g}$ of the probe beam as a function of the coupling strength J. It is obvious that with increasing the coupling strength $J$, the group delay experiences the change of negative-positive, which corresponds to the fast light to slow light, respectively. This clearly demonstrates the control of positive group delay to negative (negative to positive) with changing the coupling strength $J$.

 

Fig. 3. (a) The phase ${\phi _t}$ of the probe transmission as a function of ${\Delta _s}$ for several coupling strengths $J$ at ${\Delta _p} = 0$. (b) The group delay ${\tau _g}$ as a function of the coupling strength $J$ at $g = 2\textrm{ MHz}$.

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Secondly, we consider a ratio parameter $\delta = {{{\kappa _a}} \mathord{\left/ {\vphantom {{{\kappa_a}} {{\kappa_c}}}} \right.} {{\kappa _c}}}$ (${\kappa _a} = {{{\omega _a}} \mathord{\left/ {\vphantom {{{\omega_a}} {{Q_a}}}} \right.} {{Q_a}}}$ and ${\kappa _c} = {{{\omega _c}} \mathord{\left/ {\vphantom {{{\omega_c}} {{Q_c}}}} \right.} {{Q_c}}}$) to investigate the parameters of the two cavities that influence DIT. $\kappa$ is the decay rate of the cavity mode, which is related to the frequency and Q factor of the cavity. As we know it is difficult to achieve high Q and small V simultaneously for a cavity mode due to the diffraction limit. For an optical cavity, a smaller V corresponding to a larger radiative decay rate results in a lower Q. Although different types of cavities possess their own unique properties, the weigh between high Q and small V still exists. However, when by coupling the originally QED cavity a with high cavity dissipation to an auxiliary cavity mode c with high Q but large V, the DIT can obtain easily. In Fig. 4(a), we plot probe transmission ${|{t({\omega_s})} |^2}$ as a function of the probe-cavity detuning ${\Delta _s}$ under several different $\delta = {{{\kappa _a}} \mathord{\left/ {\vphantom {{{\kappa_a}} {{\kappa_c}}}} \right.} {{\kappa _c}}}$ with unchanged coupling strength $J = 1.0{\kappa _a}$ and $g = 1.0\textrm{ MHz}$. With increasing the ratio $\delta$ from $\delta = 1.0$ to $\delta = 3.0$, a significant transparency window (i.e., the transparency window with a large width) appearing in the probe transmission spectrum. That is to say when ${Q_c} > {Q_a}$, the DIT effect can easily obtain. Therefore, when we investigate the enhanced DIT, we can design an optical cavity with high decay rate $\kappa$ without considering other parameters by coupling to an auxiliary optical cavity with high Q, which may provide a proposal to investigate the hybrid CQED system. In Fig. 4(b), we show the distance of peak-splitting (the red curve) and the peak height around ${\Delta _s} = 0$ (the blue curve) as a function of $\delta = {{{\kappa _a}} \mathord{\left/ {\vphantom {{{\kappa_a}} {{\kappa_c}}}} \right.} {{\kappa _c}}}$ under the conditions $J = 1.0{\kappa _a}$ and $g = 1.0\textrm{ MHz}$. For the distance of peak-splitting, when $\delta < 0.5$,the peak splitting is close to zero, i.e., no splitting. When $\delta > 0.5$, the splitting distance and $\delta$ display a prefect linearity. While the relation of peak height and $\delta$ show a smooth curve.

 

Fig. 4. (a) The probe transmission ${|{t({\omega_s})} |^2}$ as a function of ${\Delta _s}$ for several decay rate ratio $\delta = {{{\kappa _a}} \mathord{\left/ {\vphantom {{{\kappa_a}} {{\kappa_c}}}} \right.} {{\kappa _c}}}$ at the condition of ${\Delta _p} = 0$. (b) The peak-splitting and the peak height as a function of $\delta$ under the conditions $J = 1.0{\kappa _a}$ and $g = 1.0\textrm{ MHz}$. (c) The phase ${\phi _t}$ of the probe transmission as a function $\delta $. (d) The group delay ${\tau _g}$ as a function of $\delta $ for two coupling strength J.

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A transparency window, i.e., DIT in Fig. 4(a) induced by the ratio $\delta = {{{\kappa _a}} \mathord{\left/ {\vphantom {{{\kappa_a}} {{\kappa_c}}}} \right.} {{\kappa _c}}}$ will also leads to slow light or fast light. So with controlling the parameter $\delta$, the change from slow light to fast light can also obtain in the hybrid CQED system. We plot the phase as a function of ${\Delta _s}$ with several different parameter $\delta$ under $J = 1.0{\kappa _a}$ and $g = 1.0\textrm{ MHz}$ as shown in Fig. 4(c). Obviously, the slope in the phase varies significantly from negative to positive. Then in Fig. 4(d), we present the group delay ${\tau _g}$ vs. $\delta = {{{\kappa _a}} \mathord{\left/ {\vphantom {{{\kappa_a}} {{\kappa_c}}}} \right.} {{\kappa _c}}}$ under two different coupling strength $J$ such as $J = 1.0{\kappa _a}$ and $J = 3.0{\kappa _a}$ at $g = 1.0\textrm{ MHz}$. It is obvious that the group delay experiences the change of negative-positive, which corresponds to the fast light to slow light, respectively. Therefore, with controlling the cavity parameters, like the decay rate $\kappa$ or the Q factor of the cavities, the fast-to-slow light can achieve straightforward in the hybrid CQED system.

Case B: Exciton-pump field detuning ${\Delta _p} \ne 0$

On the other hand, we switch the exciton-pump field detuning ${\Delta _p}$ from resonant (${\Delta _p} = 0$)to detuning (${\Delta _p} \ne 0$), and further demonstrate the parameters of $J$ and $\delta$ that will affect the probe transmission and fast to slow slight. In Fig. 5(a), we plot the probe transmission ${|{t({\omega_s})} |^2}$ as a function of the detuning ${\Delta _s}$ for several different coupling strengths J with ${\Delta _p} = 100\textrm{ MHz}$ under weak coupling regime (g = 2.0 MHz). It is obvious that even no auxiliary cavity($J = 0$ corresponding to the black curve in Fig. 5(a)), DIT effect can also appear in the probe transmission spectrum when the hybrid CQDE system is in the detuning regime. With increasing the coupling strength $J$ from $J = 1.0{\kappa _a}$ to $J = 4.0{\kappa _a}$ at ${\kappa _a} = {\kappa _c}$, the probe transmission spectra present significant splitting. When we measure the splitting width of the two peaks, we find that the first four green dot deviate significantly from the black line in Fig. 5(b). However, the deviation becomes slighter with increases in the coupling strength. Therefore, the peak splitting width is dependent linearly on the coupling strength J. Thus, it is essential to enhance the coupling strength $J$ for a clear peak-splitting. In Fig. 5(c), we plot the phase as a function of ${\Delta _s}$ with several different coupling strength J. Obviously, the phase of the transmitted probe field shows a positive slope, which indicates a positive group delay or slow light propagation. In Fig. 5(d), we plot the transmission group delay ${\tau _g}$ of the probe beam as a function of the coupling strength J, and as it demonstrated in Fig. 5(c) the phenomenon of slow light propagation can appear in the system.

 

Fig. 5. (a) The probe transmission ${|{t({\omega_s})} |^2}$ as a function of ${\Delta _s}$ for five coupling strengths J at the condition of ${\Delta _p} = 100\textrm{ MHz}$. (b) The peak splitting distance as function of J. (c) The phase ${\phi _t}$ of the probe transmission for several different J. (d) The group delay ${\tau _g}$ as a function of J at g = 2.0 MHz.

Download Full Size | PPT Slide | PDF

As the same as in Fig. 4, we also investigate the parameter $\delta$ that will influence the probe transmission and coherent optical propagation but in the detuning regime (${\Delta _p} = 100\textrm{ MHz}$). In Fig. 6(a), we plot probe transmission ${|{t({\omega_s})} |^2}$ as a function of ${\Delta _s}$ for several different $\delta$ with unchanged coupling strength $J = 1.0{\kappa _a}$ and $g = 1.0\textrm{ MHz}$ under ${\Delta _p} = 100\textrm{ MHz}$. Obviously, the splitting of the two peaks in the probe transmission spectrum is enhanced with increasing the ratio $\delta$ from $\delta = 1.0$ to $\delta = 3.0$. The distance of peak-splitting (the black curve) and the peak height (the green curve) as a function of $\delta$ are shown in Fig. 6(b). For the peak-splitting, the black curve is divided into two parts, i.e., $\delta < 1({Q_c} < {Q_a})$ and $\delta > 1({Q_c} > {Q_a})$, which have different slope. As for the peak height, the black curve is also divided into two parts and there is a critical point, i.e., $\delta < 1.4$, which also have different slope. We also plot the phase as a function of ${\Delta _s}$ for several different parameter $\delta$ under ${\Delta _p} = 100\textrm{ MHz}$ rather than ${\Delta _p} = 0$ as shown in Fig. 6(c), where the slope in the phase varies from negative to positive. Thus the group delay ${\tau _g}$ vs. $\delta$ for two different coupling strength $J$ is shown in Fig. 6(d), where the group delay experiences the change of negative-positive, which corresponds to the fast light to slow light.

 

Fig. 6. (a) The probe transmission ${|{t({\omega_s})} |^2}$ for several decay rate ratio $\delta$ at the condition of ${\Delta _p} = 100\textrm{ MHz}$. (b) The peak-splitting and the peak height as a function of $\delta$ at the conditions $J = 1.0{\kappa _a}$ and $g = 1.0MHz$. (c) The phase ${\phi _t}$ of the probe transmission as a function $\delta$. (d) The group delay ${\tau _g}$ as a function of $\delta$ for two coupling strength J.

Download Full Size | PPT Slide | PDF

Compared Fig. 4 in the resonant (${\Delta _p} = 0$) with Fig. 6 in detuning (${\Delta _p} \ne 0$), we find that when we investigate the DIT effect and DIT induced coherent optical propagation (fast and slow light) as a function of the parameter $\delta$, the role of ${\Delta _p}$ is feeble. That is to say, when the QED cavity a coupled to an auxiliary cavity c and $({Q_c} > {Q_a})$, whether ${\Delta _p} = 0$ or ${\Delta _p} \ne 0$, the results of DIT effect and DIT induced coherent optical propagation are almost no difference. In addition, regardless of the value of the parameter $\delta$, DIT effect can still reach, in a sense, the DIT induced coherent optical propagation is not restricted by the parameter $\delta$. Therefore, with manipulating the parameters, the DIT effect is controllable.

4. Conclusion

We have demonstrated the DIT effect under the weak coupling regime in the hybrid CQED system, which consists of a QD implanted in a PhC cavity with high cavity dissipation driven by the optical pump-probe technology coupled to an auxiliary cavity with high quality factor. We investigate the transmission spectrum, the phase, and the group delay of the transmitted probe light with standard methods of quantum optics under different parameter regimes, such as the coupling strength $J$ between the two cavities and the decay rate ratio $\delta$ of the two cavities in the system. With controlling the parameters J, the switch of fast-to-slow light can be obtained. In addition, by employing the QED cavity to auxiliary cavity, the group delay is revealed to be capable of switching from fast light to slow light by tuning the parameter $\delta$. These results are beneficial for better understanding the light propagation and provide new tools for controlling light propagation using the hybrid CQED system.

Funding

National Natural Science Foundation of China (11647001, 11804004); Natural Science Foundation of Anhui Province (1708085QA11).

Acknowledgement

National Natural Science Foundation of China (Nos:11647001 and 11804004), Anhui Provincial Natural Science Foundation (No:1708085QA11).

Disclosures

The authors declare no conflicts of interest.

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2020 (1)

2019 (3)

M. H. Zheng, T. Wang, D. Y. Wang, C. H. Bai, S. Zhang, C. S. An, and H. F. Wang, ““Manipulation of multi-transparency windows and fast-slow light transitions in a hybrid cavity optomechanical system,” SCIENCE CHINA Physics,” Sci. China: Phys., Mech. Astron. 62(5), 950311 (2019).
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T. Wang, Y. Q. Hu, C. G. Du and G, and L. Long, “Multiple eit and eia in optical microresonators,” Opt. Express 27(5), 7344–7353 (2019).
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M. Wang, R. Wu, J. Lin, J. Zhang, and Y. Cheng, “Chemo-mechanical polish lithography: a pathway to low loss large scale photonic integration on lithium niobate on insulator (lnoi),” Quantum Engineering 1(1), e9 (2019).
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2018 (1)

M. Rossi, N. Kralj, S. Zippilli, R. Natali, A. Borrielli, G. Pandraud, E. Serra, G. D. Giuseppe, and D. Vitali, “Normal-mode splitting in a weakly coupled optomechanical system,” Phys. Rev. Lett. 120(7), 073601 (2018).
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2015 (1)

B.-C. Ren, G.-Y. Wang, and F.-G. Deng, “Universalhyperparallel hybrid photonic quantum gates with dipole-induced transparency in the weak-coupling regime,” Phys. Rev. A 91(3), 032328 (2015).
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2014 (5)

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
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R. Puthumpally-Joseph, M. Sukharev, O. Atabek, and E. Charron, “Dipole-Induced Electromagnetic Transparency,” Phys. Rev. Lett. 113(16), 163603 (2014).
[Crossref]

H. Jing, S. K. Ozdemir, X. Y. Lu, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref]

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity–time symmetry and variable optical isolation in active–passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Y. C. Liu, X. Luan, H. K. Li, Q. Gong, C. W. Wong, and Y. F. Xiao, “Coherent polariton dynamics in coupled highly dissipative cavities,” Phys. Rev. Lett. 112(21), 213602 (2014).
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2013 (3)

H. Kim, R. Bose, T. C. Shen, G. S. Solomon, and E. Waks, “A quantum logic gate between a solid-state quantum bit and a photon,” Nat. Photonics 7(5), 373–377 (2013).
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H. Toida, T. Nakajima, and S. Komiyama, “Vacuum Rabi Splitting in a Semiconductor Circuit QED System,” Phys. Rev. Lett. 110(6), 066802 (2013).
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Y. C. Yu, J. F. Liu, X. L. Zhuo, G. Chen, C. J. Jin, and X. H. Wang, “Vacuum Rabi splitting in a coupled system of single quantum dot and photonic crystal cavity: effect of local and propagation Green’s functions,” Opt. Express 21(20), 23486–23497 (2013).
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2012 (8)

T. J. Wang, S. Y. Song, and G. L. Long, “Quantum repeater based on spatial entanglement of photons and quantum-dot spins in optical microcavities,” Phys. Rev. A 85(6), 062311 (2012).
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Y. Li, L. Aolita, D. E. Chang, and L. C. Kwek, “Robust-fidelity atom-photon entangling gates in the weak-coupling regime,” Phys. Rev. Lett. 109(16), 160504 (2012).
[Crossref]

Y.-F. Xiao, Y.-C. Liu, B.-B. Li, Y.-L. Chen, Y. Li, and Q. Gong, “Strongly enhanced lightmatter interaction in a hybrid photonic-plasmonic resonator,” Phys. Rev. A 85(3), 031805 (2012).
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T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoglu, “Supplementary for Ultrafast all-optical switching by single photons,” Nat. Photonics 6(9), 605–609 (2012).
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R. Bose, D. Sridharan, H. Kim, G. S. Solomon, and E. Waks, “Low-photon-number optical switching with a single quantum dot coupled to a photonic crystal cavity,” Phys. Rev. Lett. 108(22), 227402 (2012).
[Crossref]

A. Reinhard, T. Volz, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoglu, “Strongly correlated photons on a chip,” Nat. Photonics 6(2), 93–96 (2012).
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A. Majumdar, A. Rundquist, M. Bajcsy, and J. Vučković, “Cavity quantum electrodynamics with a single quantum dot coupled to a photonic molecule,” Phys. Rev. B 86(4), 045315 (2012).
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A. Majumdar, M. Bajcsy, and J. Vuckovic, “Design and analysis of photonic crystal coupledcavity arrays for quantum simulation,” Phys. Rev. A 85(4), 041801 (2012).
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Y.-C. Liu, Y.-F. Xiao, B.-B. Li, X.-F. Jiang, Y. Li, and Q. Gong, “Coupling of a Single DiamondNanocrystal to a Whispering-Gallery Microcavity: Photon Transportation Benefittingfrom Rayleigh Scattering,” Phys. Rev. A 84(1), 011805 (2011).
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K. A. Atlasov, A. Rudra, B. Dwir, and E. Kapon, “Large mode splitting and lasing in optimally coupled photonic-crystal microcavities,” Opt. Express 19(3), 2619–2625 (2011).
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Y.-F. Xiao, M. Li, Y.-C. Liu, Y. Li, X. Sun, and Q. Gong, “Asymmetric Fano resonance analysis in indirectly coupled microresonators,” Phys. Rev. A 83(1), 019902 (2011).
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J. J. Li and K. D. Zhu, “A quantum optical transistor with a single quantum dot in aphotonic crystal nanocavity,” Nanotechnology 22(5), 055202 (2011).
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A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, “Dipole induced transparency in waveguide coupled photonic crystal cavities,” Opt. Express 16(16), 12154–12162 (2008).
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Quantum Engineering (1)

M. Wang, R. Wu, J. Lin, J. Zhang, and Y. Cheng, “Chemo-mechanical polish lithography: a pathway to low loss large scale photonic integration on lithium niobate on insulator (lnoi),” Quantum Engineering 1(1), e9 (2019).
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Figures (6)

Fig. 1.
Fig. 1. (a) Schematic of the cavity QED system coupled to an auxiliary cavity, where a QED cavity a with high cavity dissipation driven by two-tone fields coupled to an auxiliary cavity c with high quality factor. The cavity a is coupled to cavity c via evanescent field, and the coupling strength $J$ between the two cavities can be controlled by varying the separation between them [44]. (b) The two energy levels of QD coupled to a single cavity mode and two optical fields. (c) and (d) are the energy level transitions with an entangled state $|{{n_{tot}}} \rangle $ (${n_{tot}} = {n_a} + {n_c}$ is the total photon number of the two cavities, where ${n_a}$ and ${n_c}$ represent the number state of the photon mode of cavity a and cavity c).
Fig. 2.
Fig. 2. (a) The probe transmission ${|{t({\omega_s})} |^2}$ as a function of probe-cavity detuning ${\Delta _s}$ for several coupling strengths J at the condition of ${\Delta _p} = 0$. (b) The probe transmission ${|{t({\omega_s})} |^2}$ at $J = 0$. (c)The probe transmission ${|{t({\omega_s})} |^2}$ at $J = 1.0{\kappa _a}$. (d) The peak splitting distance as function of J.
Fig. 3.
Fig. 3. (a) The phase ${\phi _t}$ of the probe transmission as a function of ${\Delta _s}$ for several coupling strengths $J$ at ${\Delta _p} = 0$. (b) The group delay ${\tau _g}$ as a function of the coupling strength $J$ at $g = 2\textrm{ MHz}$.
Fig. 4.
Fig. 4. (a) The probe transmission ${|{t({\omega_s})} |^2}$ as a function of ${\Delta _s}$ for several decay rate ratio $\delta = {{{\kappa _a}} \mathord{\left/ {\vphantom {{{\kappa_a}} {{\kappa_c}}}} \right.} {{\kappa _c}}}$ at the condition of ${\Delta _p} = 0$. (b) The peak-splitting and the peak height as a function of $\delta$ under the conditions $J = 1.0{\kappa _a}$ and $g = 1.0\textrm{ MHz}$. (c) The phase ${\phi _t}$ of the probe transmission as a function $\delta $. (d) The group delay ${\tau _g}$ as a function of $\delta $ for two coupling strength J.
Fig. 5.
Fig. 5. (a) The probe transmission ${|{t({\omega_s})} |^2}$ as a function of ${\Delta _s}$ for five coupling strengths J at the condition of ${\Delta _p} = 100\textrm{ MHz}$. (b) The peak splitting distance as function of J. (c) The phase ${\phi _t}$ of the probe transmission for several different J. (d) The group delay ${\tau _g}$ as a function of J at g = 2.0 MHz.
Fig. 6.
Fig. 6. (a) The probe transmission ${|{t({\omega_s})} |^2}$ for several decay rate ratio $\delta$ at the condition of ${\Delta _p} = 100\textrm{ MHz}$. (b) The peak-splitting and the peak height as a function of $\delta$ at the conditions $J = 1.0{\kappa _a}$ and $g = 1.0MHz$. (c) The phase ${\phi _t}$ of the probe transmission as a function $\delta$. (d) The group delay ${\tau _g}$ as a function of $\delta$ for two coupling strength J.

Equations (17)

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H = Δ p σ z + Δ a a a + Δ c c c + J ( a c + a c ) + g ( σ + a + σ a ) + i κ a e ε p ( a a ) + i κ a e ε s ( a e i Ω t a e i Ω t )
a ˙ = ( i Δ a + κ a ) a i g σ i J c + κ a e ( ε p + ε s e i Ω t )
c ˙ = ( i Δ c + κ c ) c i J a
σ ˙ z = Γ 1 ( σ z + 1 ) i g ( σ + a σ a )
σ ˙ = ( i Δ p + Γ 2 ) σ + 2 i g a σ z
( i Δ a + κ a ) a s i g σ s i J c s = κ a e ε p
( i Δ c + κ c ) c s + i J a s = 0
Γ 1 ( σ s z + 1 ) + i g ( σ s a s σ s a s ) = 0
( i Δ p + Γ 2 ) σ s 2 i g a s σ s z = 0
a + = κ a e ε s i ( Δ a Ω + g Λ 4 + J η 1 ) + κ a
η 1 = i J i ( Δ c Ω ) + κ c , η 2 = i J i ( Δ c + Ω ) + κ c , λ 1 = i g σ s Γ 1 i Ω , λ 2 = i g a s Γ 1 i Ω ,
λ 3 = i g σ s Γ 1 i Ω , λ 4 = i g a s Γ 1 i Ω , Λ 1 = i g i ( Δ a + Ω + J η 2 ) + κ c
Λ 2 = 2 i g a s λ 1 i ( Δ p + Ω ) + Γ 2 + 2 i g a s λ 2 + 2 i g ( a s λ 3 + σ s z ) Λ 1 ,
Λ 3 = 2 i g a s λ 4 i ( Δ p + Ω ) + Γ 2 + 2 i g a s λ 2 + 2 i g ( a s λ 3 + σ s z ) Λ 1 ,
Λ 4 = 2 i g [ ( a s λ 1 + σ s z ) + a s ( λ 2 + λ 3 Λ 1 ) Λ 2 ] i ( Δ p Ω ) + Γ 2 2 i g a s λ 4 2 i g a s ( λ 2 + λ 3 Λ 1 ) Λ 3
t ( ω s ) = ε s κ a e a + ε s
τ g = d ϕ t d ω s | ω s = ω p = d { arg [ t ( ω s ) ] } d ω s | ω s = ω p

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