Recent developments in chip-scale quantum optical technologies [1] have generated substantial interest in quantum dots (QDs), which act as “artificial atoms” in solid-state media. However, electron–phonon coupling in solid-state media has been shown to significantly modify the emission properties of a QD as compared to an isolated atom [2]. Studying phonon interactions in governing the emission properties of QDs has been an intense area of research, leading to a number of effects beyond a simple pure-dephasing model [3]. For driven QD excitons yielding Rabi oscillations, phonon coupling manifests in damping and frequency shifts [4 –6]. In QD-cavity systems, phonons cause intensity-dependent broadening of Mollow side-bands [7], off-resonant cavity feeding [8], and asymmetric vacuum Rabi doublets [9,10].

Semiconductor QDs coupled to structured photonic reservoirs provide a promising platform for tailoring light–matter interaction in a solid-state environment. One of the primary interests in coupling QDs to structured reservoirs is for modifying the spontaneous emission rate (SE), $\gamma$, via the Purcell effect [11,12]. Photonic crystals are a paradigm example of a structured photonic reservoir, and both photonic crystal cavities [Fig. 1(a)] and coupled-cavity optical waveguide (CROW) [Fig. 1(b)] structures have been investigated for modifying QD SE rates [12,13]. For an unstructured reservoir, $\gamma$ remains unchanged in the presence of phonons [14]. For structured reservoirs, previous theories have assumed phonon processes to be much faster than all relevant system dynamics [15,16], thus restricting them to structures with sharp variations of photon local density of states (LDOS) (e.g., high-$Q$ cavity or photonic band edge). A primary example of a structured reservoir is a microcavity, and existing theories [15,17,18] treat the cavity mode as a system operator and find that phonons modify the QD-cavity coupling rate, through $g\to 〈B〉g$ [17,19], where $〈B〉$ is the thermal average of the coherent phonon bath displacement operators $B_{\pm }$ [17]. Hence the Purcell factor is believed to scale as $g^2\to 〈B^2〉g^2$ [16]. However, such theories do not apply to large $\kappa$ cavities, where $\kappa$ is the cavity decay rate, and one would expect to recover the result that $\gamma$—and thus $g$—are not affected by phonons. Moreover, for an arbitrary photonic bath medium, it is not known how phonons affect the SE rates, yet clearly such an effect is of significant fundamental interest and is also important for understanding emerging QD experiments.

 

Fig. 1. Two photonic reservoir systems under consideration and an energy level picture of the various quantum states. Schematic of (a) semiconductor cavity and (b) waveguide using a photonic crystal platform, containing a single QD. (c) Energy level diagram of a neutral QD (electron–hole pair) interacting with a phonon bath and a photon bath. The operator ${\mathbf{f}}_{\mathbf{k}}^{†}$ creates a photon, and the $b_q^{†}$ operator creates a phonon.

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In this Letter we introduce a self-consistent master equation (ME) approach with both phonon and photon reservoirs included, and we explore in detail the influence of a photon reservoir on the phonon-modified SE rate. When the relaxation times of the photon and phonon baths compare, the frequency dependence of the LDOS is found to dictate how phonons modify the SE rates, causing a clear breakdown of Fermi’s golden rule. Such an effect arises due to dressing of QD excitations by phonons in a solid-state medium. Importantly, our theory can be applied to any general LDOS medium, and as specific examples we consider a semiconductor microcavity and a slow-light coupled-cavity waveguide.

We model the QD as a two-level system interacting with an inhomogeneous semiconductor-based photonic reservoir and an acoustic phonon bath [17] (see Fig. 1(c)). Assuming the QD of dipole moment $\mathbf{d}=d{\widehat{\mathbf{n}}}_d$ at spatial position ${\mathbf{r}}_d$, the total Hamiltonian of the system in a frame rotating at the QD exciton frequency, ${\omega }_x$, is [20]

To appreciate how phonons modify the SE rate in a structured photonic reservoir, we first consider a simple Lorentzian cavity [cf. Fig. 1(a)]. For a single cavity mode, in a dielectric with a dielectric constant $ϵ=n_b^2$,

To obtain a mean-field approximation in the high-$Q$ limit, i.e., in the limit $\kappa \ll 2\pi /{\tau }_{\mathrm{pn}}$ (where ${\tau }_{\mathrm{pn}}\approx 1\mathrm{ps}$ is the phonon relaxation time), then $\tilde{\gamma }\to {\tilde{\gamma }}^{\mathrm{mean}}={\mathrm{\Gamma }}^{a^{†}{\sigma }^{-}}+2g^2{〈B〉}^2\frac{\frac{\kappa }{2}}{{\mathrm{\Delta }}_{cx}^2+{\left(\frac{\kappa }{2}\right)}^2}$, where the phonon-mediated cavity scattering rate ${\mathrm{\Gamma }}^{a^{†}{\sigma }^{-}}=2{〈B〉}^2{g}^2\mathrm{Re}[{\int }_0^{\infty }\mathrm{d}\tau e^{-i{\mathrm{\Delta }}_{cx}\tau }(e^{\varphi (\tau )}-1)]$ [21]. This is exactly the expression for the SE rate [15,21], which is derived (see Supplement 1, Section 3) with a polaron ME [21] when treating the cavity mode as a system operator, with phenomenological damping $\kappa$ and using the bad cavity limit ($\kappa >g/2$). Our theory thus not only recovers previous (polaronic) cavity-QED results in the appropriate limit, but also reveals a fundamental limitation of these approaches for sufficiently large $\kappa$ (low-$Q$) cavities. Specifically, when the cavity relaxation time becomes comparable to ${\tau }_{\mathrm{pn}}$, or smaller, these formalisms break down.

To test this hypothesis, consider the example of three different cavity decay rates, $\kappa =0.06,0.6$, and 2.4 meV, corresponding to a $Q$ factor of around 23000, 2300, and 600, respectively (with ${\omega }_c/2\pi =1440\mathrm{meV}$). In Fig. 2 we plot the PF (left panels) and phonon-modified SE factor $\chi =\tilde{\gamma }/\gamma$ (right panels). For each cavity, we investigate two different bath temperatures (4 and 40 K), and the dashed lines on the left panels represent PFs without phonon modification. The clear asymmetry in $\chi$ about the LDOS peak arises since phonon emission is more probable than absorption [21] at low temperatures. The main results can be explained by writing $\tilde{\gamma }={〈B〉}^2\gamma +{\tilde{\gamma }}_{\mathrm{nl}}$, where $〈B{〉}^2\gamma$ is the coherently renormalized bare SE rate and arises due to local (${\omega }_x$) sampling of photonic LDOS, while ${\tilde{\gamma }}_{\mathrm{nl}}={〈B〉}^2\mathrm{Re}[2{\int }_0^{\infty }(e^{\varphi (\tau )}-1)J_{\mathrm{ph}}(\tau )]\mathrm{d}\tau$ accounts for the nonlocal contribution (i.e., frequencies that would not contribute to a Fermi’s golden rule expression). Note that when $\kappa$ is small, ${\tilde{\gamma }}_{\mathrm{nl}}\to {\mathrm{\Gamma }}^{a^{†}{\sigma }^{-}}$. Due to the nonlocal component, the reduction of the SE rate is always $\ge {〈B〉}^2$, at zero detuning. At large detunings, the nonlocal component dominates, leading to an overall enhancement of the SE rate. Figures 2(b), 2(d), and 2(f) show that $\chi$ varies significantly over several meV. The dashed lines on the right panels represent ${\chi }^{\mathrm{mean}}={\tilde{\gamma }}^{\mathrm{mean}}/\gamma$, which evidently differs from our full calculations in the limit $\kappa \approx {\tau }_{\mathrm{pn}}^{-1}$ [Figs. 2(d) and 2(f)]. This is because the reservoir structure of the photon bath is properly accounted in the present calculations and is not approximated as a high-$Q$ cavity (also see Supplement 1, Section 1). It is also important to note that low-$Q$ (several hundred) cavities are commonly employed for measuring the vertical emission from QDs in planar cavities [2] and for modifying the SE rates in simple photonic crystal cavities [12], while intermediate $Q$ ($\approx 3000$) cavities are used for all-optical switching [23] and enabling single photon sources [24].

 

Fig. 2. Phonon-modified spontaneous emission of a QD in a dielectric cavity. Near-resonant PFs (left panels) and phonon-modified SE factor $\chi$ (right panels) for cavities with (a),(b) $\kappa =0.06\mathrm{meV}$, (c),(d) 0.6 meV, and (e),(f) 2.4 meV. We use the continuous form of the phonon spectral function, $J_{\mathrm{pn}}(\omega )={\alpha }_p{\omega }^3\exp [-\frac{{\omega }^2}{2{\omega }_b^2}]$, for longitudinal acoustic phonon interaction, and adopt experimental numbers for InAs QDs [2]: cavity-QD coupling rate $g=0.08\mathrm{meV}$, phonon cutoff frequency ${\omega }_b=1\mathrm{meV}$, and exciton–phonon coupling strength ${\alpha }_p/{(2\pi )}^2=0.06{\mathrm{ps}}^2$ [21]. The dark (red) dashed lines on the left panels represent the PF without phonons. The dark and light solid lines correspond to the phonon-modified PF and $\chi$ at $T=4$ and 40 K, respectively. The dashed lines on the right panels plot ${\chi }^{\mathrm{mean}}$ (see text) at $T=4\mathrm{K}$ (light) and 40 K (dark).

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We now depart from the simple cavity, and consider the richer case of a photonic crystal CROW [cf. Fig. 1(b)]. Photonic crystal waveguides [Fig. 1(b)] can be used for realizing slow-light propagation [25] and for manipulating the emission properties of embedded QDs for on-chip single photon emission [26,27], with a number of recent experiments emerging. Current theories in this regime either ignore phonon coupling or assume a coherent renormalization factor (${〈B〉}^2\gamma$), consistent with Fermi’s golden rule (modified by a mean-field reduction factor). For the photon reservoir function, we adopt a model LDOS for a CROW [28], and use an analytical tight-binding technique to calculate the photonic band structure [29]; the photon reservoir spectral function is obtained analytically, as

 

Fig. 3. Phonon-modified spontaneous emission of a QD in a coupled-cavity waveguide. (a) Purcell factors and (b) phonon-modified SE factor $\chi$ in $a{\log }_{10}$ scale, for a PC CROW structure, where ${\omega }_0$ marks the band center. For the waveguide, $\mathrm{PF}=\gamma /{\gamma }_h$, where ${\gamma }_h=d^2{n}_b{\omega }^3/(6\pi \hslash {ϵ}_0{c}^3)$ is the SE rate within the homogenous slab material. The solid (dashed) line in (a) represents the PF with (without) phonons; the inset shows $J_{\mathrm{ph}}(t)$ at the upper mode edge (light line) and band center (dark line) in arbitrary units; the dashed line shows the simple exponential damping function $e^{-\lambda t}$ (see text). The phonon calculations are performed at $T=40\mathrm{K}$. Inset in (b) plots $\chi$ in a linear scale to show better display $\chi$ inside the waveguide band.

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At a sharp mode edge of the LDOS, the contribution from local (${\omega }_x$) photonic LDOS dominates and $\chi \approx {〈B〉}^2$ [16]. Away from the mode edge [Figs. 3(a) and 3(b)], the SE rate is enhanced due to nonlocal effects. This simple model discussion is, however, approximate as in reality the mode-edge LDOS is non-Lorentzian. For a symmetric Lorentzian with the same bandwidth, ${\lambda }^{-1}\approx 50\mathrm{ps}$, $J_{\mathrm{ph}}(t)$ damps much faster than ${\lambda }^{-1}$ initially and damps very slowly thereafter [Fig. 3(a), inset]. The long time decay rate is set by the linewidth of the sharper side of the mode-edge LDOS ($\approx 0.1\mathrm{ns}$). This non-Lorentzian mode edge in turn leads to a very strong enhancement of the PF ($\times 200$) outside the waveguide band [Fig. 3(b)], compared to a symmetric Lorentzian lineshape [see Fig. 2(b)].

In conclusion, we have demonstrated how the frequency dependence of the LDOS of a photonic reservoir determines the extent to which phonons modify the SE of a coupled QD. The relative dynamics between the phonon and the photon bath correlation functions is found to play a fundamentally important role; specifically, when the relaxation times are comparable, phonons strongly modify the emission spectra leading to non-Lorentzian cavity lineshapes and even enhanced SE. These effects are not obtained using the usual Fermi’s golden rule. Our formalism is important for understanding related experiments with QD-cavity systems, such as with photoluminescence intensity measurements with a coherent drive [33], and is broadly applicable to various photonic reservoir systems.