1. Introduction

Solid-state cavity quantum-electrodynamics [1–10] (QED) is the subject of keen interest due to its scalability and integrability, which has great potential for providing a platform for future quantum communications and informatics. In particular, the development of a coupled system combining a nanocavity (based on photonic crystals [1–5, 11, 12], microdisks [6–8], or micropillars [9, 10]) and a quantum dot [13–16] (QD) has made remarkable progress. In these systems, two important photon emission mechanisms, the Purcell effect [17] in the weak coupling regime and vacuum Rabi-splitting [18, 19] in the strong coupling regime are known to occur near the on-resonant condition. Here, we show that a third emission mechanism plays an important role in these systems when under off-resonant conditions. This is based on the quantum Anti-Zeno effect (AZE) [20–22] induced by the pure-dephasing effect in a QD due to the elastic interactions between the QD and the electrons in the surrounding bulk region [5]. We have found that this AZE emission based on the pure dephasing is significantly enhanced by the inherent characteristics of the nanocavity such as small modal volume, high-Q factor, and control of spontaneous emission. This emission mechanism successfully explains the origin of strong photon emission at a cavity mode largely detuned from a QD, which has so far been considered as a counterintuitive, prima facie non-energy-conserving, light emission phenomenon in these systems [1–10]. These findings not only provide a basis that could lead to methods for controlling the decay and emission characteristics for solid-state cavity QED, but also accelerate the development of various types of solid-state quantum devices.

According to quantum mechanics, the state of a microscopic system is modified when it is measured [20-22]. In this context, the quantum AZE and the quantum Zeno effect (QZE) are respectively known as the acceleration and deceleration of a quantum decay rate when subjected to frequent measurements [20, 21] (including non-referred ones) or dephasing [22]. This modified decay rate W is in general expressed as follows [20–22]:

ξ(ω) describes the coupling of the quantum system to a reservoir. D(ω) is the spectral density of states of the reservoir. F(ω) is the spectral function of the quantum system related to the acts of measuring. Equation (1) suggests that the quantum decay rate W accelerates or decelerates with respective increase or decrease of the spectral overlap between |ξ(ω)|² D(ω) and F(ω). In general, the spectral width of F(ω) is broadened by measurements (or dephasing). If this increases the spectral overlap, W also increases. Hence, the measurements (or dephasing) accelerate the decay process, which qualifies the AZE. Conversely, when the broadening of F(ω) caused by the measurements (or dephasing) decreases the spectral overlap, W decreases. In this case, because the measurements (or dephasing) decelerate the decay process, this qualifies as the QZE. In general terms, assuming that |ξ(ω)|² ρ(ω) and F(ω) take the form of peaks, the AZE occurs when the detuning between the spectral centres of |ξ(ω)|² D(ω) and F(ω) is sufficiently large, and the QZE occurs when this detuning is sufficiently small.

In a case of a radiative decay of a QD combined with a nanocavity, the general expression of AZE and QZE can be explained as follows. Importantly, an elastic interaction between the QD and electrons in the surrounding bulk region is considered to induce the pure-dephasing effect in the QD, which corresponds to (non-referred) measurements. First, consider the case in which a QD is placed in free space instead of a nanocavity. |ξ(ω)|² D(ω) in equation (1) is flat with respect to frequency ω, and although F(ω) is broadened by the puredephasing, the magnitude of the overlap integral in equation (1) is constant. Therefore, neither QZE nor AZE occurs. By contrast, when a QD is introduced into a nanocavity, it begins to couple to the nanocavity mode, and the additional radiative decay path through the nanocavity is generated. When we consider the case of a weak coupling regime, the reservoir spectrum seen from the QD looks like a strong peak at the cavity mode over the flat background. As described in the general expression, the spectrum overlap between |ξ(ω)|² D(ω) and F(ω) will be modified when F(ω) is broadened by the pure-dephasing effect; thus, AZE and QZE are expected to occur.

In reality, the situation is complicated because the coupling between the QD and the nanocavity can range from the weak to strong coupling regime, and various detuning between the cavity mode and the QD can occur. To treat the QZE near the on-resonant condition in the strong coupling regime, equation (1) is not appropriate and consideration in the time domain is required.

Therefore, we initially analyze the radiative decay rate of the actual cavity QED system shown in Fig. 1(a) by solving the quantum master equation [7, 23–25], and show that AZE (and QZE) occurs as a consequence of the pure-dephasing effect in the QD. Then, we analyze the emitted photon spectrum and individual emission rates for the QD transition energy and the cavity resonance energy. We find noticeable photon emission [5] from the cavity mode largely detuned from the QD transition energy by the AZE enhanced by the inherent characteristics of the nanocavity such as small modal volume, high-Q factor and control of spontaneous emission.

 

Fig. 1. Schematic illustration of the system investigated. (a) Example of a solid-state cavity QED system, where a single QD is embedded in a two-dimensional photonic crystal nanocavity. (b) Schematic illustration of our analysis model, a TLS interacting with a single-mode cavity. Under the initial condition of an excited TLS, photons can be emitted to free space by two pathways: a direct pathway to free space or an indirect pathway through the cavity mode. (c) Schematic illustration of elastic interactions between the TLS and electrons in the cladding layer.

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2. Theoretical analysis

2.1 Analysis model

Our analysis model is shown in Fig. 1(b). As QDs are known to possess discrete energy levels [13–16] due to their three-dimensional confinement of electrons, we can approximate a single QD as a two-level system (TLS) in a solid. This TLS (with transition energy ħω _TLS) can interact with a single-mode electromagnetic field formed in a cavity (with resonant energy ħω _cav) with a coupling constant g. The TLS and cavity mode can also interact with vacuum photon modes in free space. The excited state composed of the TLS and the cavity mode can spontaneously emit photons to these vacuum modes. The critical feature of this model is that the TLS can interact elastically with electrons in the surroundings (Fig. 1(c)).

We analyze the quantum dynamics in this model using the quantum master equation [7, 23–25], focusing on the system of interest consisting of the TLS and the cavity mode, and treating its surroundings (the vacuum photon modes in free space and the surrounding electrons) as heat baths (see also Appendix A and B). The quantum master equation is then expressed as

where ρ _S is a reduced density operator, H _S represents the interaction Hamiltonian between the TLS and the cavity mode (non-Markov process), and the L _m terms (Liouvillians) represent the irreversible damping that arises from the interactions between the system and its surroundings (Markov process). The vacuum fields in free space cause irreversible energy damping to the TLS and cavity mode, whose original damping rates (without taking into account any mutual interaction) are represented by 2Γ _spon and 2Γ _cav (≡ω _cav/Q), respectively. The elastic Coulomb interactions between the TLS and the surrounding electrons cause pure-dephasing of the TLS (with a rate denoted by γ _phase) but do not induce any energy damping [7, 23] (see also Appendix B). An analogy between the effect of elastic Coulomb interactions and superselection theories [26] can also be made. Unless otherwise stated, we used a standard set of parameters in the numerical calculations, 2ħΓ _spon=0.044 µeV (1/2Γ _spon=15 ns, corresponding to the decay time of a QD in the photonic band gap [4]) and 2ħΓ _cav=100 µeV (Q=1.1×10⁴) [1, 2]. For the coupling constant g, we used a value of ħg=76 µeV (2π/2g=27.0 ps), which is reasonable for a QD that is aligned to the centre of the cavity used in the experiments [1] mentioned above. For ħγ _phase, it is known to range from ~20 to ~70 µeV at cryogenic temperatures (< 50 K) [15, 16] and when the temperature or density of surrounding electrons is increased, ħγ _phase increases larger than 100 µeV [16]. In our analysis, in order to investigate the effect of pure-dephasing systematically, we used values from 0 to 350 µeV.

2.2 Decay rate

Initially, we analyze the decay rate W of the TLS using equation (2), setting the initial condition as an excited TLS. At the beginning of the time evolution for the parameters listed above (Appendix C), W can be expressed as

where

Here, Γ _total≡Γ _cav+Γ _spon+γ _phase (Γ _total is the total dephasing rate for the system composed of the TLS and the cavity mode), and δω _TLS,_cav≡ω _TLS-ω _cav (δω _TLS,_cav is the detuning between the resonant angular frequencies of the TLS and the cavity mode). The decay rate of the excited-state TLS is shown in Fig. 2(a) for ħγ _phase values of 0 µeV, 35 µeV, 70 µeV, and 350 µeV. Pure-dephasing significantly modifies the decay rate of the TLS. This is consistent with the analysis [22] by Kofman et al. showing that stochastic phase changes (or dephasing) induce changes in the decay rate. Under off-resonant conditions in Fig. 2(a), W increases as ħγ _phase increases from 0 µeV to 350 µeV. Hence, pure-dephasing induces the AZE under off-resonant conditions (Fig. 2(b)). Conversely, W decreases with ħγ _phase under on-resonant conditions (gray region in Figs. 2(a) and 2(b)), so pure-dephasing induces the QZE under on-resonant conditions. When we consider ħγ _phase is between ~20 and ~70 µeV at cryogenic temperatures, and increases at higher temperatures or when the density of surrounding electrons increases, AZE and QZE indeed occur in the solid-state nanocavity QED systems.

 

Fig. 2. Decay rate of the excited TLS. (a) Decay rate for pure-dephasing rates ħγ _phase of 0 µeV (blue line), 35 µeV (red line), 70 µeV (green line) and 350 µeV (black line). (b) Decay rate for ħγ _phase=35 µeV (red line), 70 µeV (green line) and 350 µeV (black line) normalized by the decay rate for ħγ _phase=0 µeV in Fig. 2(a). The inset shows the magnified image under the onresonant condition. The gray dashed line denotes that the value along the longitudinal axis equals to one.

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2.3 Emitted photon spectrum

The decay rate W does not provide information on the emitted photon spectrum, which is crucial for determining the optical performance of cavity QED systems. Therefore, we now analyze the emitted photon spectra from the correlation function of the electric fields in free space. The emitted photon spectrum S(ω) is the sum of the emission spectra for light escaping the TLS directly to free space and indirectly passing through the cavity (Fig. 1(b)). An analytical expression for S(ω) can be obtained using equation (2), which is applicable regardless of the weak or strong coupling regime between the TLS and the cavity:

Here, we define

Exact expressions for the coefficients f(γ±) and Λ± are given in Appendix C and D. The emission spectra S(ω) for various values of detuning are shown in Fig. 3, where the pure-dephasing rates are colour coded as in Fig. 2(a). The vertical axes represent the number of emitted photons per unit angular frequency.

 

Fig. 3. Emission spectra for various values of detuning. (a) No pure-dephasing. (b) With puredephasing rate ħγ _phase=35 µeV. (c) With pure-dephasing rate ħγ _phase=70 µeV. (d) With pure-dephasing rate ħγ _phase=350 µeV. The labels TLS and Cav denote the peaks at the transition energy of the TLS and at the cavity resonance energy, respectively.

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For the case with no pure-dephasing in Fig. 3(a), the emission spectra show two peaks when detuned < 0.4 meV, which is the typical Rabi-splitting spectra [19, 18] in the strong coupling regime. For larger detuning, only one emission peak at the transition energy of the TLS is seen. However, when a small degree of pure-dephasing (ħγ _phase=35 µeV) is introduced, two clear emission peaks can be seen even for large detuning as in Fig. 3(b). As ħγ _phase is further increased to 70 µeV, a large peak at the cavity resonance energy develops (Fig. 3(c)). Finally, when ħγ _phase is sufficiently large (350 µeV), a single emission peak at the cavity resonance energy is observed (Fig. 3(d)). The presence of a peak at the cavity resonance energy for large detuning has been experimentally observed, but the origin has been unresolved. Our analysis suggests that the pure-dephasing effect is the origin of the emission. In addition, the Rabisplitting under the on-resonant condition is observed when the pure-dephasing rate is relatively small (< 70 µeV), which indicates that the Rabi splitting and QZE are not exclusive when considering the result of Fig. 2 simultaneously.

2.4 Individual emission rate

Now we consider the phenomena under the off-resonant condition, where two peaks at the TLS transition energy and the cavity resonant energy are observed. To obtain the individual photon emission rates W _TLS and W _cav at the TLS transition energy and at the cavity resonance energy, we integrated the emission spectra of individual peaks with respect to the emission energy and multiplied them with the total decay rate W, which can then be written as W=W _TLS+W _cav. The obtained photon emission rates W _TLS and W _cav are shown as a function of the pure-dephasing rate in Fig. 4(a), where a large detuning of 4 meV is assumed. Notably, the value of detuning is much larger than the total spectral broadening of 2ħΓ _total=2ħ(γ _phase+Γ _cav+Γ _spon) (<~500 µeV). Nevertheless, Fig. 4(a) clearly indicates that the photon emission rate at the cavity resonance energy increases linearly with increasing ħγ _phase, while the photon emission rate at the TLS transition energy is independent of ħγ _phase. We conclude that the increase of the cavity mode emission rate is due to the AZE that is induced by the pure dephasing effect by simultaneously considering the results shown in Fig. 2. Figure 4(b) illustrates the ratio of the cavity mode emission rate to the total emitted photon rate, defined as a factor F,

where F increases with increasing ħγ _phase. Consequently, the photon emission at the cavity resonance dominates the emission spectra for higher ħγ _phase. These results explain the photon emission mechanism: despite large detuning, the AZE induced by the pure dephasing effect generates cavity-mode photons.

When the detuning is large enough, (|δω _TLS,_cav| ≫ g, Γ _total), the photon emission rates W _TLS and W _cav shown in Fig. 4(a) can be analytically expressed as:

where the terms of W _TLS indicate the direct emission rate to free space and an approximated expression for the Purcell effect [17], respectively. Note that the Purcell effect occurs at the energy of TLS instead of the cavity resonant energy under off-resonant conditions, where the cavity mode is forced to oscillate at the TLS transition energy. On the other hand, W _cav is proportional to the line width of the TLS expressed as 2(Γ _spon+γ _phase), which is multiplied by g ²/δω ² _TLS,_cav. Thus, F can be expressed as

for |δω _TLS,_cav| ≫ g, Γ _total. From this expression, a QD combined with a nanocavity gives rise to a highly advantageous situation for the enhancement of F. This is due to the low photonic damping rate of Γ _cav (owing to the high quality factor) and a large coupling constant g (owing to the small modal volume). Hence, the strong photon emission [1–3] at a nanocavity resonance energy largely detuned from a QD (TLS), previously unexplained, can be understood as a consequence of the AZE emission that is enhanced by the inherent nature of the nanocavity. If a nanocavity is constructed within photonic crystals [1-3, 11, 12], F increases further, because the Γ _spon is suppressed by the photonic bandgap effect [4, 27]. In the case of reference 1, F≈0.24 can be obtained for |δω _TLS,_cav| ~4 meV with ħγ _phase=35 µeV, which suggests that 24% of the total emitted photons are detected as cavity mode emission even though the detuning is large (Figs. 4(b) and 4(c)). This agrees with the experimental results (see also Appendix E).

 

Fig. 4. Pure-dephasing dependence of the photon emission rates for individual peaks in the emission spectrum. (a) Photon emission rates for detuning of 4 meV. The labels W _TLS and W _cav denote the photon emission rates from the peaks at the transition energy of the TLS and at the cavity resonance energy, respectively. (b) Factor of F is defined as W _cav/(W _TLS+W _cav), which represents the ratio of the cavity mode emission rate to the total emitted photon rate. (c) Corresponding emission spectrum calculated for ħγ _phase=35 µeV, where the percentage represents the ratio of each peak to the total integral value.

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3. Discussions

We have described a case in which the number of QD is single and a nanocavity is based on photonic crystals. Here, we will discuss more general cases. First, we consider a nanocavity containing tens to hundreds of QDs variously detuned by 10–20 meV [2–4]. Second, we discuss the case for various cavity geometries [6–10].

Most nanocavity systems studied to date [2–10] have many detuned QDs and show light emission at the cavity resonance energy. To discuss this effect, we assume that each QD independently contributes to the cavity mode emission for simplicity. If we assume the average coupling constant ħg ~30 µeV with (2ħΓ _spon, 2ħΓ _cav, γ _phase)=(0.044 µeV, 100 µeV, 35 µeV), the factor F from each QD is estimated to be ~1.4% for 10 meV detuning and ~0.4% for 20 meV detuning. The combined effect of tens to hundreds of QDs results in a contribution to the cavity mode comparable to, or even exceeding, the individual QD exciton peaks. This estimation is reasonable compared with experimental results. Next, we consider the case for various cavity geometries. While the nanocavity we discussed above was photonic-crystal based, the nanocavity-enhanced AZE emission is also important for the other cavity geometries. In the case of a microdisk [6], a cavity mode emission is dominant relevant to the QD emissions. In this case, while the factor F becomes smaller for individual QDs due to the increased modal volume (which corresponds to the reduction of the coupling coefficient g), the total number of QDs is expected to increase due to the larger modal volume. Therefore, the total contribution of all the QDs can result in the strong cavity mode emission reaching or exceeding the individual QD emissions. In the case of micropillars or microdisks for photon turnstile devices [8], the second-order correlation g ⁽²⁾(τ=0) does not become zero when the QD is on resonance with the cavity mode. This unexpected second-order correlation in the photon-turnstile device can also be explained as a consequence of the nanocavity-enhanced AZE emission from the off-resonant QDs. The reference points out that using higher pump powers increases the degree of unexpected correlation. This is interpreted as the AZE emission when we consider that higher pump powers cause many carriers and higher rates of pure-dephasing.

4. Conclusion

We have brought the concept of AZE (and QZE) to solid-state quantum systems, and found that the AZE enhanced by the inherent characteristics of the nanocavity produces a third emission mechanism in solid-state nanocavity QED systems. This emission mechanism based on the nanocavity-enhanced AZE explains the counterintuitive photon emission of a cavity mode largely detuned from QD peaks. We believe that our findings not only provide a basis that could lead to methods for controlling the decay and emission characteristics for solidstate cavity QED, but will also accelerate the development of various types of solid-state quantum devices.

Appendix A:

In this section we give a simple explanation for the AZE-induced emission from a quantitative point of view, where the emission channels from the excited two-level system (TLS) to free space are discussed in terms of the limited bases of the density operator. In general, a density operator ρ _T of a quantum system (a system of interest and a reservoir) follows the Liouvillevon Neumann form, which is directly derived from the Schrödinger equation

(11)$$\frac{d{\rho }_T}{dt}=-i{\hslash }^{-1}[H_T,{\rho }_T],$$

where H _T is the total Hamiltonian of a quantum system. In the model that we consider, H _T includes the Hamiltonians of the TLS, the cavity mode, the vacuum modes in free space, the surrounding electrons, and their interactions. Using this form together with the state vectors |E (or G), m, n _λ >, we can obtain equations of motion for each component of the density operator. Here, |E (or G), m, n _λ > represents the TLS in the excited (E) [or ground (G)] state, m photons in the cavity (m), and n photons in the λ-th photon mode in free space. The obtained equations of motion suggest that there are physical couplings between different states of the density operator, as shown in Fig. 5. The diagonal states of the density operator represent (A) an excited two-level state, (B) one photon in the cavity, and (C) one photon in the λ-th mode in free space. The off-diagonal states of the density operator represent the correlations (D) between the TLS and the cavity mode (electric polarization for the cavity mode), (E) between the cavity mode and the λ-th photon mode in free space (electric polarization for the λ-th mode in free space), and (F) between the TLS and the λ-th photon mode in free space. We note that (E) and (F) are functions of the energy of the free-space modes and have single spectral peaks at the cavity resonant energy and at the transition energy of the TLS, respectively. In contrast, (D) is not a function of the energy of the free-space modes, but rather a function of the detuning between the TLS and the cavity. In Fig. 5, the Hermitian conjugate states and their related couplings are omitted for reasons of simplicity; they do not affect our argument. We now discuss the main emission channels between (A) and (C). To clarify the underlying physical mechanism, we compromise on mathematical rigor to some extent and focus on the main emission channels, considering the lowest-order channels.

As can be seen in the figure, the direct emission from the TLS to free space is composed of all pathways that reach (C) via (F), whereas the indirect emission through the cavity mode is composed of all pathways that reach (C) via (E). Therefore, the lowest-order channel for direct emission is (A)-(F)-(C). This is intuitive because the channel (A)-(F)-(C) represents the original emission channel (without the cavity mode) from the excited TLS, where state (A) gives rise to an oscillating electric polarization (F) and the radiation of a photon to state (C) in free space. It is clear that the emission probability spectrum for the channel (A)-(F)-(C) has only one peak, located at the transition energy of the TLS. In contrast, there are two lowest-order channels for indirect emission through the cavity: channel 1 is along the pathway (A)-(D)-(B)-(E)-(C) and channel 2 is along the pathway (A)-(D)-(F)-(E)-(C). We note that channel 1 goes through only one correlation state (E), whereas channel 2 goes through two correlation states (F) and (E). As mentioned above, the states (E) and (F) give rise to spectral peaks when the energy of the free-space modes is at the cavity resonant energy and the transition energy of the TLS, respectively. Therefore, channel 1 gives rise to only one peak at the cavity resonant energy, whereas channel 2 results in peaks at both the cavity resonant energy and the transition energy of the TLS. Therefore, indirect emission can in principle give rise to peaks both at the transition energy of the TLS and at the cavity resonant energy. However, in the absence of pure dephasing, channels 1 and 2 can interfere at state (E), which leads to cancellation of the peak at the cavity resonant energy. The probability of emitting a photon to free space at the cavity resonant energy will then be zero; thus, there is only one peak at the transition energy of the TLS in the absence of pure dephasing. This process can be interpreted as the forced oscillation (or virtual excitation) of state (E) by the transition energy of the TLS. In contrast, in the presence of pure dephasing the coherence of the dipoles vanishes. Channels 1 and 2 are then unable to interfere completely. As a result, a peak at the cavity resonant energy can be obtained. These scenarios clearly explain the mechanism of emission from the spectrally detuned TLS at the cavity resonant energy.

 

Fig. 5. Emission channels through density operator states. (A)–(C) represent diagonal states and (D)–(F) represent off-diagonal (correlation) states. Gray arrows denote couplings between states within first-order perturbation. For simplicity, the Hermitian conjugate states and their related couplings are not shown.

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Appendix B:

In this section we discuss the Liouvillians in the quantum master equation. The total Hamiltonian of H _T can be decomposed as

(12)$$H_T=H_S+H_R+H_{\mathrm{int}},$$

where the constituent Hamiltonians represent the system (H _S), the environment of the heat bath (H _R), and their interaction (H _int). Under the Born and Markov approximation, an integro-differential equation for the quantum master equation [23, 24] in the interaction picture can be obtained from equations (11) and (12) under the assumption that ρ _T(t) ≈ ρ _S(t)⊗ρR(0):

(13)$$\frac{d{\rho }_{\text{S}}\left(t\right)}{dt}=-i{\hslash }^{-1}\left[H_{\text{S}}\left(t\right),{\rho }_{\text{S}}\left(t\right)\right]-i{\hslash }^{-1}{\tau }^{-1}\underset{t}{\overset{t+\tau }{\int }}d{t}_{1}T{r}_R\left[H_{\mathrm{int}}\left(t_1\right),{\rho }_{\text{S}}\left(t\right)\otimes {\rho }_R\left(0\right)\right]$$
$$-{\hslash }^{-2}{\tau }^{-1}\underset{t}{\overset{t+\tau }{\int }}d{t}_1\underset{t}{\overset{t1}{\int }}d{t}_{2}T{r}_R\left[H_{\mathrm{int}}\left(t_1\right),[H_{\mathrm{int}}\left(t_2\right),{\rho }_{\text{S}}\left(t\right)\otimes {\rho }_R\left(0\right)]\right],$$

where ρ _S(t)≡Tr _R[ρ _T(t)] represents the reduced density operator, and ρ _S and ρ _R represent individual density operators for a system of interest and its reservoirs. In addition, dρ _S(t)/dt≡{ρS(t+τ)-ρ _S(t)}/τ expresses a coarse temporal differentiation. The first term represents the coherent time evolution within the system, whereas the second and third terms represent irreversible damping. Therefore, each Liouvillian is formed by the second and third terms in equation (13).

We now derive the Liouvillian L _i,_phase for the Coulomb interactions in the second quantization framework. The Coulomb interaction Hamiltonian Ĥ _Coulomb in the solid can be expressed as

(14)$$H_{\mathrm{Coulomb}}=\frac{1}{2}\iint d^{3}r{d}^{3}r\prime {\hat{\psi }}^{†}\left(r\right){\hat{\psi }}^{†}\left(r\prime \right)\frac{e^2}{\mid r-r\prime \mid }\hat{\psi }\left(r\prime \right)\hat{\psi }\left(r\right),$$

where ψ̂(r) represents the electron field operator. ψ̂(r) can be decomposed as

(15)$$\hat{\psi }\left(r\right)={\hat{c}}_1{\phi }_1\left(r\right)+{\hat{c}}_2{\phi }_2\left(r\right)+{\Sigma }_m{\hat{c}}_m{\phi }_m\left(r\right),$$

where φ _j(r) (j=1, 2, m) represents the j-th electron state wave function, and ĉ ₁ and ĉ ₂ are the annihilation operators of the first (lower) and second (upper) levels in the TLS, respectively. In addition, ĉ _m is the annihilation operator of the m-th electron state in the solid. These are Fermi operators satisfying anti-commutation relations. By substituting equation (15) into (14) and ignoring the Auger processes [28] in the first approximation, we obtain

(16)$$H_{\mathrm{Coulomb}}\approx H_{12,\mathrm{El}}+H_{11,\mathrm{El}}+H_{22,\mathrm{El}},$$

where H ₁₂,_El represents the Coulomb interaction between the first and second levels in the TLS, and H ₁₁,_El (H ₂₂,_El) represents the interactions between the first (second) level in the TLS and the other electrons in the solid. The two Hamiltonians can be expressed as

(17)$$H_{\mathrm{ii},\mathrm{El}}={\hat{c}}_i^{†}{\hat{c}}_i{\Sigma }_m{\hat{c}}_m^{†}{\hat{c}}_{m}G(i,i,m,m)+{\hat{c}}_i^{†}{\hat{c}}_i{\Sigma }_{n,k}{\hat{c}}_n^{†}{\hat{c}}_{k}G(i,i,n,k),(n\ne k,i=1\mathrm{or}2)$$

(18)$$H_{12,\mathrm{El}}={\hat{c}}_1^{†}{\hat{c}}_1{\hat{c}}_2^{†}{\hat{c}}_{2}G(1,1,2,2),$$

where we define

(19)$$G(i,j,k,l)\equiv \iint d^{3}r{d}^{3}r\prime \frac{e^2}{\mid r-r\prime \mid }\left[{\phi }_i^{*}\left(r\right){\phi }_j\left(r\right){\phi }_k^{*}\left(r\prime \right){\phi }_l\left(r\prime \right)-{\phi }_i\left(r\prime \right){\phi }_j^{*}\left(r\right){\phi }_l\left(r\right){\phi }_k^{*}\left(r\prime \right)\right].$$

Here, the first term of equation (17) describes the Coulomb interactions between the TLS and the other fixed states in the solid, whereas the second term represents the elastic Coulomb scattering of the electrons in the solid. Because H _ii,_El(i=1 or 2) represents the interaction between the system (the TLS and the cavity mode) and its environment (the electron heat bath), it is treated as H _int in equation (13). The Hamiltonian H ₁₂,_El describes the interaction within the system. Therefore, H ₁₂,_El is treated as H _S. It should be noted that H ₁₁,_El and H ₂₂,_El satisfy the commutation relations [ĉ†_i ĉ _i,H ₁₁,_El]=[ĉ†_i ĉ _i,H ₂₂,_El]=0, which suggests that these Hamiltonians do not change the temporal differentiation of <ĉ†_d ĉ _i(t)> [26, 29]. As a result, the Liouvillians derived from these Hamiltonians do not contribute directly to the temporal change of the expectation value of < ĉ†_i ĉ _i(t)> either, leading to pure dephasing. After transforming equation (17) to the interaction picture and substituting into (13), standard procedures to obtain the Liouvillian [24] yield

(20)$$L_{i,\mathrm{phase}}X=-i{\Delta }_i[{\hat{c}}_i^{†}{\hat{c}}_i,X]-{\gamma }_{\mathrm{phase},1}\left({\hat{c}}_i^{†}{\hat{c}}_{i}X+X{\hat{c}}_i^{†}{\hat{c}}_i-2{\hat{c}}_i^{†}{\hat{c}}_{i}X{\hat{c}}_i^{†}{\hat{c}}_i\right),(i=1,\mathrm{or}2)$$

where Δ_i denotes the angular frequency shifts of the i-th level in the TLS due to the Coulomb interactions, and γ _phase, _i is the pure dephasing rate of the i-th level in the TLS. We have ignored the Δ_i in the analytical expressions and have defined the total pure dephasing rate as γ _phase≡γ _phase,₁+γ _phase,₂. In a similar manner, we can also obtain the Liouvillians L _cav and L _spon, representing the individual energy damping from the cavity mode and the TLS to the vacuum fields in free space. The Liouvillians are expressed as [24]

(21)$$L_{\mathrm{cav}}X=-{\Gamma }_{\mathrm{cav}}\left({\hat{a}}_{\mathrm{cav}}^{†}{\hat{a}}_{\mathrm{cav}}X+X{\hat{a}}_{\mathrm{cav}}^{†}{\hat{a}}_{\mathrm{cav}}-2{\hat{a}}_{\mathrm{cav}}X{\hat{a}}_{\mathrm{cav}}^{†}\right),$$

(22)$$L_{\mathrm{spon}}X=-{\Gamma }_{\mathrm{spon}}\left({\hat{c}}_2^{†}{\hat{c}}_1{\hat{c}}_1^{†}{\hat{c}}_{2}X+X{\hat{c}}_2^{†}{\hat{c}}_1{\hat{c}}_1^{†}{\hat{c}}_2-2{\hat{c}}_1^{†}{\hat{c}}_{2}X{\hat{c}}_2^{†}{\hat{c}}_1\right),$$

where â _cav (â ^† _cav) is the annihilation (creation) operator of the cavity mode. In addition, the Hamiltonian describing the interaction within the system can be expressed as

(23)$$H_{\text{S}}\left(t\right)=\hslash \left({\hat{c}}_2^{†}{\hat{c}}_1{\hat{a}}_{\mathrm{cav}}g\exp \left(i\delta {\omega }_{\mathrm{TLS},\mathrm{cav}}t\right)+H.c.\right)+H_{12,\mathrm{El}},$$

where the first terms represent the interaction between the cavity mode and the TLS, and δω _TLS,_cav≡ω _TLS-ω _cav is the detuning. Using equation (13), the dynamical time evolution of the system (the cavity mode and the TLS) can be expressed as

(24)$$\frac{d{\rho }_S\left(t\right)}{dt}=L{\rho }_S\left(t\right),$$

with the total Liouvillian

(25)$$L\equiv L_S+L_{\mathrm{cav}}+L_{\mathrm{spon}}+L_{1,\mathrm{phase}}+L_{2,\mathrm{phase}},$$

where we defined

(26)$$L_{S}X\equiv -i{\hslash }^{-1}[H_S,X].$$

Appendix C:

In this section we analyze the time evolution of the expectation values <ĉ†₂ ĉ ₂(t)> and <â ^† _cav â _cav(t)> in order to find an electron in the upper level and a photon in the cavity, respectively. The coupled equations for these expectation values are obtained from equation (24) as

(27)$$\frac{d<{\hat{c}}_2^{†}{\hat{c}}_2\left(t\right)>}{dt}=\mathrm{Re}\left[-ig<{\hat{c}}_2^{†}{\hat{c}}_1{\hat{a}}_{\mathrm{cav}}\left(t\right)>\right]-2{\Gamma }_{\mathrm{spon}}\left[<{\hat{c}}_2^{†}{\hat{c}}_2\left(t\right)>-<{\hat{c}}_2^{†}{\hat{c}}_1^{†}{\hat{c}}_1{\hat{c}}_2\left(t\right)>\right],$$

(28)$$\frac{d<{\hat{a}}_{\mathrm{cav}}^{†}{\hat{a}}_{\mathrm{cav}}\left(t\right)>}{dt}=\mathrm{Re}\left[ig<{\hat{c}}_2^{†}{\hat{c}}_1{\hat{a}}_{\mathrm{cav}}\left(t\right)>\right]-2{\Gamma }_{\mathrm{cav}}<{\hat{a}}_{\mathrm{cav}}^{†}{\hat{a}}_{\mathrm{cav}}\left(t\right)>,$$

(29)$$\frac{d<{\hat{c}}_2^{†}{\hat{c}}_1{\hat{a}}_{\mathrm{cav}}\left(t\right)>}{dt}=-i{g}^{*}<{\hat{c}}_2^{†}{\hat{c}}_2\left(t\right)>i{g}^{*}<{\hat{a}}_{\mathrm{cav}}^{†}{\hat{a}}_{\mathrm{cav}}\left(t\right)>+\left(i\delta {\omega }_{\mathrm{TLS},\mathrm{cav}}-{\Gamma }_{\mathrm{total}}\right)<{\hat{c}}_2^{†}{\hat{c}}_1{\hat{a}}_{\mathrm{cav}}\left(t\right)>,$$

(30)$$\frac{d<{\hat{c}}_2^{†}{\hat{c}}_1^{†}{\hat{c}}_1{\hat{c}}_2\left(t\right)>}{dt}=0,$$

where the total dephasing rate is Γ _total≡Γ _cav+Γ _spon+γ _phase. In these coupled equations, <ĉ ^† ₂ ĉ ₁ â _cav(t)> represents the expectation value of the electronic polarization for the cavity mode (state (D) in Fig. 5), and <ĉ ^† ₂ ĉ ^† ₁ ĉ ₁ ĉ ₂(t)> represents the probability that both levels of the TLS are occupied by electrons. Because we have ignored the Auger processes, <ĉ ^† ₂ ĉ ^† ₁ ĉ ₁ ĉ ₂(t)> remains the initial value, as described by equation (30). The analytical solution of these coupled equations is difficult. Therefore, we make the approximation that d/dt ≈ 0 in equation (29), which is similar to an adiabatic elimination of < ĉ ^† ₂ ĉ ₁ â _cav(t)>. It follows that this approximation is always valid for g << Γ _total or g << δω _TLS,_cav. When these conditions are not satisfied, the solutions obtained from the remaining coupled equations (27), (28), and (30) become inaccurate on the time-scale g ^-1, and only temporally averaged solutions are obtained. In this limit, we can obtain simplified coupled equations for (27)–(30) using the initial condition <ĉ ^† ₂ ĉ ^† ₁ ĉ ₁ ĉ ₂(0)>=0.

(31)$$\frac{d<{\hat{c}}_2^{†}{\hat{c}}_2\left(t\right)>}{dt}=R<{\hat{c}}_2^{†}{\hat{c}}_2\left(t\right)>+R<{\hat{a}}_{\mathrm{cav}}^{†}{\hat{a}}_{\mathrm{cav}}\left(t\right)>-2{\Gamma }_{\mathrm{spon}}<{\hat{c}}_2^{†}{\hat{c}}_2\left(t\right)>,$$

(32)$$\frac{d<{\hat{a}}_{\mathrm{cav}}^{†}{\hat{a}}_{\mathrm{cav}}\left(t\right)>}{dt}=R<{\hat{c}}_2^{†}{\hat{c}}_2\left(t\right)>-R<{\hat{a}}_{\mathrm{cav}}^{†}{\hat{a}}_{\mathrm{cav}}\left(t\right)>-2{\Gamma }_{\mathrm{cav}}<{\hat{a}}_{\mathrm{cav}}^{†}{\hat{a}}_{\mathrm{cav}}\left(t\right)>,$$

where we have defined the following:

(33)$$R+iK\equiv 2{\mid g\mid }^2\frac{1}{-i\delta {\omega }_{\mathrm{TLS},\mathrm{cav}}+{\Gamma }_{\mathrm{total}}},$$

Here, equations (31) and (32) are classical coupled-rate equations and can be solved analytically using two eigenfrequencies of Λ _± expressed as

(34)$${\Lambda }_{\pm }\equiv -\left({\Gamma }_{\mathrm{cav}}+{\Gamma }_{\mathrm{spon}}+R\right)\pm \sqrt{{\left({\Gamma }_{\mathrm{cav}}-{\Gamma }_{\mathrm{spon}}\right)}^2-R^2}.$$

These eigenfrequencies characterize the decay rate W of <ĉ ^† ₂ ĉ ₂(t)>. Particularly for Γ _cav > Γ _spon, we can make the approximation that W ≈ -Λ₊, which immediately leads to equation (3). We now discuss the validity of this exppression. As mentioned above, the approximation W ≈-Λ₊(for Γ _cav > Γ _spon) is evidently valid under the condition that g << Γ _total or g << δω _TLS,_cav. Furthermore, if we ignore the oscillation component (Rabi oscillation) that changes over the time-scale of g ^-1 and focus on the time evolution over a larger time-scale, the coupled equations (31) and (32) are also useful even when the conditions g << Γ _total or g << δω _TLS,_cav are not satisfied. In general, it should be possible to estimate the decay rate W on this larger time-scale, thus the approximation W ≈ -Λ₊ (for Γ _cav > Γ _spon) is also valid even under the conditions g >> Γ _total and g >> δω _TLS,_cav.

Appendix D:

In this section we describe the total emission spectrum S(ω). Following the Wiener- Khintchine theorem, the total emission spectrum S(ω) can be expressed as

(35)$$S\left(\omega \right)=2{\epsilon }_0{c}_0{\left(2\pi \right)}^{-1}\underset{0}{\overset{\infty }{\int }}d\tau \underset{0}{\overset{\infty }{\int }}dt<E^{-}\left(t+\tau \right)·E^{+}\left(t\right)>\exp \left(i\omega \tau \right),$$

where E ⁺(t) is the electric field operator in free space in the Heisenberg picture, ε ₀ is the dielectric constant, and c ₀ is the velocity of light in free space. Using the Heisenberg equations of motion, E ⁺(t) can be expressed in the form E ⁺(t)=G _cav aâ_cav(t)+G _spon ĉ ^† ₁ ĉ ₂(t), where we ignore the solution of the homogeneous or free-field wave equation. The factors G _cav and G _spon represent the coupling between the system (the cavity mode and the TLS) and the free-space photon modes. As a result, further steps yield the relation

(36)$$2{\epsilon }_0{c}_0<E^{-}\left(t+\tau \right)·E^{+}\left(t\right)>\propto 2{\Gamma }_{\mathrm{cav}}<{\hat{a}}_{\mathrm{cav}}^{†}\left(t+\tau \right){\hat{a}}_{\mathrm{cav}}\left(t\right)>+2{\Gamma }_{\mathrm{spon}}<{\hat{c}}_2^{†}{\hat{c}}_1\left(t+\tau \right){\hat{c}}_1^{†}{\hat{c}}_2\left(t\right)>,$$

in which we have ignored the cross-terms for simplicity; these are not essential for our main conclusion. In the case of τ=0, the first term 2Γ _cav< â ^† _cav â _cav(t)> represents the photon emission through the cavity (pathways (A)-(D)-(B)-(E)-(C) and (A)-(D)-(F)-(E)-(C) in Fig. 5). The second term 2Γ _spon< ĉ ^† ₂ ĉ ₁ ĉ ^† ₁ ĉ ₂(t)> represents the direct photon emission from the TLS (pathway (A)-(F)-(C) in Fig. 5). By substituting equation (36) into (35), we can express the total emission spectrum S(ω) as

(37)$$S\left(\omega \right)\equiv S_{\mathrm{spon}}\left(\omega \right)+S_{\mathrm{cav}}\left(\omega \right),$$

where

(38)$$S_{\mathrm{cav}}\left(\omega \right)=\frac{{\Gamma }_{\mathrm{cav}}}{\pi }\mathrm{Re}\left[\underset{0}{\overset{\infty }{\int }}d\tau \underset{0}{\overset{\infty }{\int }}dt<{\hat{a}}_{\mathrm{cav}}^{†}\left(t\right){\hat{a}}_{\mathrm{cav}}\left(t+\tau \right)>\exp \left(i\omega \tau \right)\right],$$

(39)$$S_{\mathrm{spon}}\left(\omega \right)=\frac{{\Gamma }_{\mathrm{spon}}}{\pi }\mathrm{Re}\left[\underset{0}{\overset{\infty }{\int }}d\tau \underset{0}{\overset{\infty }{\int }}dt<{\hat{c}}_2^{†}\left(t\right){\hat{c}}_1\left(t\right){\hat{c}}_1^{†}\left(t+\tau \right){\hat{c}}_2\left(t+\tau \right)>\exp \left(i\omega \tau \right)\right].$$

In order to calculate these emission spectra, the expectation values for the two-time correlation functions < â ^† _cav(t)aĉ_cav(t+τ)> and <ĉ ^† ₂(t)ĉ ₁(t) ĉ ^† ₁(t+τ)ĉ ₂(t+τ)> are required; they can be obtained using the quantum master equation derived above together with the quantum regression theorem [23–25]. In addition, the time integral values <ĉ ^† ₂ ĉ ₂(t)> and <â ^† _cav aĉ_cav(t)> discussed above are also required so that the analytical solutions of the simplified equations (31) and (32) can be used. As a result, each emission spectrum can be expressed analytically in a unified form:

(40)$$S_{\alpha }\left(\omega \right)\equiv \frac{2}{\pi {\Lambda }_{+}{\Lambda }_{-}}\mathrm{Re}\left[\frac{1}{{\gamma }_{+}-{\gamma }_{-}}\left\{\frac{f_{\alpha }\left({\gamma }_{+}\right)}{i\omega +{\gamma }_{+}}-\frac{f_{\alpha }\left({\gamma }_{-}\right)}{i\omega +{\gamma }_{-}}\right\}\right],\left(\alpha =\mathrm{spon},\mathrm{or}\mathrm{cav}\right)$$

where we define the coefficients γ _± and f _α(γ _±) (α=spon, or cav) as

(41)$$2{\gamma }_{\pm }\equiv -\left[{\Gamma }_{\mathrm{total}}+i\left({\omega }_{\mathrm{TLS}}+{\omega }_{\mathrm{cav}}\right)\right]\pm \sqrt{{\left({\Gamma }_{\mathrm{cav}}-{\Gamma }_{\mathrm{spon}}-{\gamma }_{\mathrm{phase}}-i\delta {\omega }_{\mathrm{TLS},\mathrm{cav}}\right)}^2-4{\mid g\mid }^2},$$

(42)$$f_{\mathrm{cav}}\left({\gamma }_{\pm }\right)\equiv {-\Gamma }_{\mathrm{cav}}\left[-R\left({\gamma }_{\pm }+i{\omega }_{\mathrm{TLS}}+{\Gamma }_{\mathrm{total}}\right)+iK{\Gamma }_{\mathrm{cav}}\right],$$

(43)$$f_{\mathrm{spon}}\left({\gamma }_{\pm }\right)\equiv -{\Gamma }_{\mathrm{spon}}\left[{\Gamma }_{\mathrm{cav}}\left(iK-2{\Gamma }_{\mathrm{cav}}\right)-\left({\gamma }_{\pm }+i{\omega }_{\mathrm{cav}}\right)\left(R+2{\Gamma }_{\mathrm{cav}}\right)\right].$$

Using equation (37), these expressions result in Eq. (5), where f(γ _±)≡f _cav(γ _±)+f _spon(γ _±).

Appendix E:

As described in the main text, we have demonstrated that the experimental results reported in reference [1] of main text, especially those concerning the intensity of the cavity mode emission under conditions where detuning from the exciton emission peak is large, can be explained very well by our theory. However, the reference [1] also reported very strange intensity correlations involving the cavity and exciton emission peaks. The cavity emission peak (when filtered) showed Poissonian-like intensity correlations, whereas the exciton emission peak (also filtered) showed clear anti-bunched intensity correlations. The cross-correlation of the cavity emission intensity and the exciton emission intensity, however, showed anti-bunched correlations. In the following discussion, we consider why such strange correlation features were observed.

We first point out that similar experiments to those in reference 1 were also performed in reference [9] of main text using a system composed of a micropillar-type nanocavity and a QD, where it was demonstrated that not only cross-correlation measurements between the bare-cavity mode and exciton emissions, but also auto-correlation measurements for the bare-cavity emission, showed anti-bunching. In this context, the experimental results reported in references [1] and [9] are quite different: all of the correlation measurements showed anti-bunching in reference [9], whereas the bare-cavity auto-correlation measurement showed Poissonian-like intensity correlations in reference [1].

Here, we also point out that the experimental conditions employed were different in references [1] and [9]. In reference [9] a resonant excitation method was utilized, where the excitation laser was tuned to resonantly pump the first excited state (p-shell) exciton in a selected QD. This exciton thermalizes to the QD ground state (s-shell), which implies that only one two-level system can interact with the cavity mode. These experimental conditions are identical to our theoretical model, and the experimental results in reference [9] agree well with our calculations.

In contrast, the study in reference [1] employed a higher energy pumping method (above-band pumping). Furthermore, in reference [1] the experimental conditions were different for the cross-correlation measurements between the bare-cavity and the exciton emission (CW excitation) and for the auto-correlation measurements of the cavity mode emission (pulsed excitation). It is likely that these different excitation conditions would affect the characteristics of the intensity correlation measurements. In particular, when the number of excited carriers increases (or under conditions of strong excitation), higher states in the QD will be simultaneously occupied. From this point of view, one possible explanation for the results reported in reference [1] could be as follows.

(1) For the cross-correlation measurements under CW excitation and off-resonant conditions, the number of carriers created might be relatively small, and the QD can be treated as a single two-level system. Consequently, the cavity emission can originate from the two-level system and shows anti-bunching in the cross-correlation measurements.

(2) For the auto-correlation measurements under pulsed excitation and off-resonant conditions, the number of instantaneously created carriers could be much greater, so that higher states in the QD could be simultaneously occupied by excess excited carriers, and these filled higher states may act as multiple two-level systems. For the case of large detuning, the cavity mode would simultaneously receive multiple photons from these multiple two-level systems. In this case, the auto-correlation measurements of the bare-cavity emission do not need to show anti-bunching.

Acknowledgment

This investigation was supported by Research Programs (Grant-in-Aid, Centre of Excellence and Special Coordination Fund) for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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