Couplings between Photons and Ensemble of Emitters in a High-$Q$ Cavity at Mesoscopic Excitation Levels

We consider a single cavity mode with energy $\omega_{C}$ and linewidth $\gamma_{C}$, coupled to an ensemble of $N_{X}$ individual two-level emitters. Each emitter has the same energy $\omega_{X}$, linewidth $\gamma_{X}$, and coupling strength $g_{i}$ to the cavity mode. The state of cavity QED system consists of the state of emitters and the number of photons in the cavity. As depicted in Fig. 1(a)-(c), the emitter state is collectively described by the number of excited emitters, e.g., $G$ means all emitters are at the ground level, $X$ means one emitter is excited, and $2X$ means two emitters are excited. Following is the number of photons in the cavity mode. $|G,0\rangle$ is thereby the ground state of the whole system. In the framework of the rotating wave approximation, the coupling between photons and emitters is described by the Jaynes-Cummings or Tavis-Cummings model [37, 15, 16]. The system Hamiltonian is given by

where $\sigma_{Ai,Bi}=|Ai\rangle\langle Bi|$ ($A,B\in{X,G}$) is the Dirac operator for emitter $i$ and $a^{+}$/$a$ is the ladder operators for photons. This model is valid when the cavity mode volume is small i.e., the free spectral range is much larger than the coupling strength [38, 39].

The simple case is the first order coupling in the weak excitation limit, as depicted in Fig. 1(a). This is the focus of usual investigations. In this case, the energy symmetrically and coherently exchanges between the $|G,1\rangle$ and $|X,0\rangle$ state. Since the $X\rightarrow G$ transition is a collective of $N_{X}$ transitions, the coupling strength between these two states is $g_{X}=\sqrt{N_{X}}g_{i}$ [15, 16]. This coupling results in the two first-order polariton eigenstates denoted by $P_{1a}$ and $P_{1b}$. The corresponding two eigenfrequencies can be calculated by solving the Hamiltonian matrix

and results are

where $\Omega_{X}=\omega_{X}-i\gamma_{X}/2$ and $\Omega_{C}=\omega_{C}-i\gamma_{C}/2$. The well-known Rabi splitting at resonance $\omega_{X}=\omega_{C}$ occurs when the coupling strength is larger than the decay rate as $g_{X}>\left(\gamma_{X}-\gamma_{C}\right)/4$ [40, 41].

In addition, the cavity QED system also supports high energy states e.g., the $|G,2\rangle$, $|X,1\rangle$ and $|2X,0\rangle$ states depicted in Fig. 1(b). The coupling strength between $|G,2\rangle$ and $|X,1\rangle$ is $\sqrt{2}g_{X}$, where the $\sqrt{2}$ arises from the ladder operator for two photons. The $2X\rightarrow X$ transition is a collective of $N_{X}\left(N_{X}-1\right)$ transitions, thus the coupling strength between $|X,1\rangle$ and $|2X,0\rangle$ is $g_{2X}=\sqrt{N_{X}\left(N_{X}-1\right)}g_{i}$ [15, 16]. The couplings between these three states result in three second-order polariton eigenstates denoted by $P_{2a}$, $P_{2b}$, and $P_{2c}$. The eigenfrequency of polaritons can be calculated by solving the Hamiltonian matrix

As such, when the excitation is enough to pump the system to these three states, 6 emission peaks representing the transition from second-order to first-order polaritons would be observed as depicted by the downward arrows in Fig. 1(b). Similar couplings can be expected between higher energy states involving more excited emitters and photons.

However, the Hamiltonian matrix in Eqs. 4 and 8 are only valid with a relative weak excitation, which means the pump has little impact on the energy exchange between emitters and photons. As the excitation level increases, the pump-induced dephasing will suppress the coherent energy exchange [27, 28, 29, 30]. In the limit of strong excitation, emitters are approximately forced at the excited state, with only the unidirectional energy transfer from emitters to photons as depicted in Fig. 1(c). In this case, the spectral emission degenerates to the bare cavity mode, which includes the transitions $|G,1\rangle\rightarrow|G,0\rangle$, $|G,2\rangle\rightarrow|G,1\rangle$, etc., no matter the number of photons and emitters in the system.

As discussed above, the increasing excitation level has two effects: pump the system to high energy states and suppress the coherent energy exchange. To further investigate the dominance of the two effects, we use Quantum Optics Toolbox [31] to solve the maser equation of the coupling system. The master equation is given by

where $\rho$ is the density matrix including all allowed states with maximum $N_{C}$ photons, $H$ is the Hamiltonian defined in Eq. 1, and the Liouvillian superoperator

describes Markovian processes corresponding to the decay of photons, the decay of emitters, and the pump of emitters with rate $P$, respectively. The pump in Eq. 10 is incoherent thereby introduces the pump-induced dephasing [27, 28, 29, 30]. For brevity, we use energy unit for all the parameters $\omega_{X\left(C\right)}$, $\gamma_{X\left(C\right)}$, and $P$ (e.g. $\omega_{X}$ is actually $\hbar\omega_{X}$) and omit the unit meV. The emission spectrum is calculated from the steady state of Eq. 9.

In Fig. 1(d) we present typical emission spectra calculated for narrow-linewidth emitters with various $N_{C(X)}$. The parameters are set as $\omega_{X}=0$, $\omega_{C}=10$, $\gamma_{X}=0.25$, and $\gamma_{C}=0.2$. The coupling strength of the cavity mode to a single emitter is set to $g_{i}=g_{X}/\sqrt{N_{X}}$, to keep a same value of the first-order coupling strength $g_{X}=10$ between the cases with different $N_{X}$. We emphasize that the collective coupling strength e.g., $g_{X}$, $\sqrt{2}g_{X}$, and $g_{2X}$ in Eq. 4 and 8, is not a precondition of master equation Eq. 9 (only $g_{i}$ is), but is a result which can be achieved by solving the master equation.

$N_{C(X)}=1$ is the simplest case having four states of the system as $|G,0\rangle$, $|G,1\rangle$, $|X,0\rangle$, and $|X,1\rangle$. With the relative small pump $P=0-1.5$, the coherent energy exchange between $|G,1\rangle$ and $|X,0\rangle$ results in two polariton eigenstates with the eigenfrequencies as

Due to the degeneration of transitions $P_{1a}\rightarrow|G,0\rangle$ and $|X,1\rangle\rightarrow P_{1b}$ (energy of $-6.2$), as well as $P_{1b}\rightarrow|G,0\rangle$ and $|X,1\rangle\rightarrow P_{1a}$ (energy of $16.2$), only two emission peaks are observed in the spectra. In contrast, with the maximum photon number $N_{C}=2$, the system has two additional states $|G,2\rangle$ and $|X,2\rangle$. The coherent energy exchange between $|G,2\rangle$ and $|X,1\rangle$ results in two second-order polariton eigenstates with the eigenfrequencies

As shown in Fig. 1(d), with increasing pump $P$, emission peaks of the first-order transitions $P_{1a}\rightarrow|G,0\rangle$ at $-6.2$ and $P_{1b}\rightarrow|G,0\rangle$ at $16.2$ quickly suppress when $P>0.4$. Emission peaks from the second-order transitions $P_{2a}\rightarrow P_{1a}$ at $6.2$, $P_{2a}\rightarrow P_{1b}$ at $-16.2$, and $P_{2b}\rightarrow P_{1b}$ at $13.8$ are firstly enhanced when $P<1$ and then suppressed. The other second-order transition $P_{2b}\rightarrow P_{1a}$ at $36.2$ (out of plot range) behaviors similar to these three. The emission peaks from highest order transitions $|X,2\rangle\rightarrow P_{2a}$ at $20$ and $|X,2\rangle\rightarrow P_{2b}$ at $-10$ are continuously enhanced in the calculation range of $P=0-1.5$. As the pump increases, the converting of dominated emission peak is consistent with the coupling pumped from low to high orders.

When more photons and emitters are involved in the system, the coupling becomes complex, and more transitions appear in the emission spectra [21, 42] e.g., the results with $N_{C(X)}=2$ and $N_{C(X)}=3$ presented in Fig. 1(d). Despite the complex details, we clearly observe the central result i.e., the increasing $P$ pumps the system from low to high order couplings, as denoted by the black arrows in Fig. 1(d). In addition, although the first-order coupling strength $g_{X}$ is set to same, the high order coupling strength increases with $N_{C(X)}$ [15, 16]. Thereby, larger energy shift between low and high order transitions are observed, e.g., by comparing the spectra with $N_{C(X)}=3$ and $N_{C(X)}=2$. In contrast to the relative small $P$ in Fig. 1(d), we present the case with $N_{C(X)}=3$ but with a much larger $P$ in Fig. 1(e). As $P$ approaches the strong limit $>100$, all emission peaks degenerate to the bare cavity energy at $\omega_{C}=10$, in accordance to the pump-induced dephasing depicted in Fig. 1(c).

For the system with narrow-linewidth emitters ($\gamma_{X}=0.25$) presented in Fig. 1, the mesoscopic excitation level $P\in\left(0.4,\ 100\right)$ is clearly observed in the emission peaks. Next, we investigate the dependence of this phenomenon on the emitter decay rate (linewidth) $\gamma_{X}$. In Fig. 2, we present the emission spectra calculated with various $\gamma_{X}$. Other parameters $\omega_{X}=0$, $\omega_{C}=10$, $\gamma_{C}=0.2$, $g_{X}=10$ are same to those used in Fig. 1. The number of photons (emitters) is fixed at $N_{C(X)}=2$. We focus on the spectral emission around the bare cavity energy $\omega_{C}=10$. As shown in Fig. 2, for $\gamma_{X}=2$, two distinguishable peaks from the low (bottom) and high (top) order coupling are clearly observed, along with the converting of dominated emission peak at $P=5$. For $\gamma_{X}=5$, although only one emission peak is distinguishable, the nonmonotonic energy shift clearly reveals that the coupling is pumped to high orders before suppressed by the pump-induced dephasing. In contrast, for $\gamma_{X}=15$, only one emission peak with monotonic energy shift is observed, indicating the dephasing effect dominates throughout. These results reveal that the nonmonotonic peak energy shift as the feature of high order couplings is only observed for $\gamma_{X}$ below a threshold (lifetime above a threshold).

The threshold value of $\gamma_{X}$ is further calculated as 10.0 for the case in Fig. 2. This is much smaller compared to the threshold of Rabi splitting in the first-order coupling i.e., $4g_{X}+\gamma_{C}$ which has the value 40.2 here [40, 41]. The calculation of threshold value in more cases are presented in Appendix A.

To bridge our theoretical calculation with the experiment, we start by discussing a cavity mode that couples to totally $N_{total}$ number of emitters, with the same coupling strength $g_{j}$ to each emitter. As an example, Fig. 3 shows a typical L3 photonic crystal cavity that couples to totally 12 emitters at the antinodes [40, 43]. Usually, not all emitters can be at the excited state simultaneously [44], since the emitter-emitter interaction will be significant when two excited emitters are close to each other in spatial. The emitter-emitter interaction will result in complex effects [45, 46, 47] thus dephase the coupling to the cavity [48, 49]. Therefore, the cavity equally couples to $N_{X}$ subcollective emitters. Each subcollective emitter has $N_{total}/N_{X}$ emitters which are close to each other in spatial, and among the $N_{total}/N_{X}$ at most one excitation is allowed to exclude the emitter-emitter interaction. E.g., in Fig. 3 the 12 emitters are subgrouped into three subcollective emitters. Thus, each subcollective emitter has a coupling strength to the cavity mode $g_{i}=\sqrt{N_{total}/N_{X}}g_{j}$ [39, 50]. The first-order coupling strength between the cavity mode and $N_{X}$ subcollective emitters is $g_{X}=\sqrt{N_{X}}g_{i}=\sqrt{N_{total}}g_{j}$. Meanwhile, the system also supports high energy states since the $N_{X}$ subcollective emitters can be simultaneously excited. As shown, this case is exactly described by the calculation with $N_{X}$ emitters in Sec. II.2. Meanwhile, the limited number of photons in the calculation is allowed to exclude the dephasing effects such as the photon-phonon interaction [51]. Therefore, our model is a good approximation of the cavity QED system involving multiple excitations in experiment.