## Abstract

Semiconductor quantum dots are inevitably coupled to the vibrational modes of their host lattice. This interaction reduces the efficiency and the indistinguishability of single-photons emitted from semiconductor quantum dots. While the adverse effects of phonons can be significantly reduced by embedding the quantum dot in a photonic cavity, phonon-induced signatures in the emitted photons cannot be completely suppressed and constitute a fundamental limit to the ultimate performance of single-photon sources based on quantum dots. In this paper, we present a self-consistent theoretical description of phonon effects in such sources and describe their influence on the figures of merit.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## Corrections

3 January 2020: A typographical correction was made to the body text.

## 1. Introduction

Sources of single indistinguishable photons are key components in optical quantum information processing [1], and deterministic single-photon sources based on self-assembled semiconductor quantum dots (QDs) have developed significantly over the past few years. QDs can be grown with excellent optical properties, and advances in fabrication of photonic nanostructures have led to demonstrations of bright and coherent single-photon sources [2–5]. However, the inevitable coupling of the QD to the vibrational phonon modes of the host lattice constitutes a significant challenge [6], which will only be increasingly important as the performance of the sources is pushed from the current state-of-the-art towards the high level necessary for realising large-scale optical quantum computation.

There are several figures of merit that characterise the performance of a single photon source. Naturally, the photon number statistics of the emitted light is important. The Hanbury Brown and Twiss second-order correlation function, $g^{(2)}(\tau )$, quantifies the probability of simultaneously detecting two photons when evaluated at $\tau =0$ [8]. Multiphoton components in the emitted light leads to a non-zero value of $g^{(2)}(0)$, which can stem from the excitation process [9–11]. Similarly, the efficiency of the source is the probability of sending a single photon into the detection channel when the source is triggered [7]. Another important feature, which characterises the coherence properties of the source, is the indistinguishability of the emitted photons, which is a measure of the degree to which two photons can interfere. The indistinguishability is measured by sending the two photons into the two input ports of a beamsplitter. If the two photons are completely indistinguishable, the Hong-Ou-Mandel effect results in the photons leaving the beamsplitter in the same output port [12]. This effect is a fundamental necessity for linear quantum computing schemes, forming the basis for two-photon gates [13]. Alternatively, the quantum dot can be operated by exciting a biexciton, thereby emitting a polarisation-entangled photon pair [4,14,15]. In this scenario, the entanglement fidelity of the emitted photon pair constitutes an additional figure of merit of the source. Suppressing phonon-assisted photon emission in these pair-sources requires individual, polarisation-degenerate Purcell enhancement of the exciton and biexciton lines, which relies on highly challenging photonic engineering [16].

The focus of the present paper is the impact of phonons on the efficiency and indistinguishability in the context of single-photon generation. Phonons influence these figures of merit by 1) generating detuned, distinguishable photons in a sideband by exchanging energy with the QD [17–20] and 2) dephasing the QD exciton through scattering of thermal phonons [21–23]. When the QD relaxes from its excited state to its ground state, a phonon wavepacket can be simultaneously created, thereby leading to the emission of a photon with energy lower than that of the QD exciton (see Fig. 1(a)). The precise energy of the photon is not known, since it depends on the energy of the phonon wavepacket, which is lost into the environment. As a result, photons produced through this phonon-mediated transition are inherently distinguishable. Similarly, if the temperature is high enough to allow for thermal population of the phonon modes, a phonon wavepacket can be absorbed, leading to emission of a blue-detuned photon. In the emission spectrum of the source, these processes are observed as a broad sideband in the vicinity of a narrow zero-phonon line, corresponding to a phononless $|{e}\rangle \rightarrow |{g}\rangle$ transition (see Fig. 1(b)). Furthermore, if the thermal occupation of phonon modes is appreciable, thermal phonons can interact with the QD and drive virtual transitions to higher excited states. Although such scattering processes do not induce population transfer, they dephase the dipole moment of the QD during the photon emission process [21], which also leads to a decrease in the photon indistinguishability [23]. This effect can be suppressed simply by operating the source at temperatures far below the characteristic phonon energies.

Naturally, the incoherent photons can be removed by applying a narrow filter around the coherent zero-phonon line in the spectrum, which is done in most indistinguishability measurements on single-photon sources, e.g. Refs. [2,5]. Although restoring the indistinguishability, such filtering will also decrease the efficiency of the source. The fraction of photons emitted into the phonon sideband can also be reduced by placing the QD in a photonic cavity with a narrow resonance line that selectively enhances zero-phonon emission processes through the Purcell effect (see Fig. 1(c)). However, this funneling strategy only works up to the limit where the Purcell regime breaks down and strong light–matter coupling sets in. In the strong-coupling regime, a relaxation from the upper to the lower polariton state can occur by emitting a phonon wavepacket, thereby reducing the coherence of the system and the indistinguishability of the emitted photons [24]. Furthermore, in the entangled photon pair application, a single broadband Purcell enhancement [4] covering both the exciton and biexciton lines cannot be used to suppress the phonon sideband. Here, the funneling strategy would require a twin-cavity approach [16].

The vibrational environment is not the only important noise mechanism that degrades the single-photon coherence. Another dominant mechanism is the slow charge fluctuations in nearby crystal impurities [23,25] and fluctuations in the local nuclear spins [26–28]. Since the typical time scale of these fluctuations can be as slow as a few tens of $\mathrm {kHz}$ they do not necessarily degrade the indistinguishability of photons emitted subsequently from the same source [3]. However, when multiple single-photon sources are used in combination, the charge fluctuations in two distinct sources are completely uncorrelated, and thus the photons become distinguishable [29].

An additional source of decoherence is the timing jitter introduced by non-resonant or quasi-resonant excitation schemes [30,31]. In these excitation schemes, the stochastic relaxation process from a higher excited state to the lowest-energy exciton via nonradiative processes leads to an uncertainty in the emission time, which degrades the Hong-Ou-Mandel interference visibility. However, by using strictly resonant excitation, this problem can be largely overcome [32–34].

In this paper, we focus on the influence of phonon interactions on the emission properties of QD single photon sources. Further, the effect of a structured electromagnetic environment is included as a central feature of our analysis. Thus, we will consider a total Hamiltonian of the form

where $H_{\mathrm {E}}=\hbar \omega _{eg}|{e}\rangle \langle {e}|$ is the free evolution of the emitter (with $|{g}\rangle$ and $|{e}\rangle$ denoting the ground state and lowest exciton state, respectively, separated by an energy $\hbar \omega _{eg}$), $H_{\mathrm {F}}$ and $H_{\mathrm {P}}$ describe the free evolution of the electromagnetic field and phonon environment, respectively and $H_{\mathrm {EF}}$ and $H_{\mathrm {EP}}$ describe the interaction of these environments with the emitter. In the following sections we will describe all environmental terms in the Hamiltonian in detail and derive the fundamental properties of light emitted from the system.## 2. Properties and numerical model for the photonic structure

To control the light emission and ensure a good coupling to the collection optics, the QD is typically placed in a photonic structure. Such structures often fall in two categories: A waveguide or a cavity. A waveguide offers highly directional emission and broadband screening of the optical radiation modes, which increases the overall efficiency of the source, but not the indistinguishability due to lack of spectral selectivity. Cavities with linewidths below the frequency range of the sideband selectively funnels emission into the zero-phonon line, thereby increasing the indistinguishability. Simultaneously, the efficiency is increased due to Purcell enhancement.

Many nanophotonic cavities are essentially realized by adding a pair of mirrors to an underlying waveguide. The waveguide as well as the mirrors can be implemented in various ways and have different characteristics, but we will here treat the cavity using a unified framework (see Fig. 2(a)) allowing analysis of many different geometries. One example is the micropillar configuration, where the central cavity section is an unstructured, waveguide, while distributed Bragg reflectors (DBRs) constitute the mirrors (see Fig. 2(b)). Another example is a line-defect photonic crystal cavity, where a missing row of air holes constitutes a line-defect waveguide and the re-instated holes form the mirrors. The waveguide structure is characterised by a spontaneous emission rate into the fundamental mode, $\Gamma _0$, and the effective index of this mode, $\bar {n}$, while the cavity is formed by adding two mirrors with amplitude reflectivities $r_1$ and $r_2$, separated by a distance $L$.

#### 2.1 Optical local density of states of a nanocavity

The Hamiltonians describing the free evolution of the electromagnetic field and its interaction with the emitter are given by $H_{\mathrm {F}}=\hbar \sum _\mu \omega _\mu a_\mu ^{\dagger } a_\mu$, $H_{\mathrm {EF}}=\hbar \sum _\mu (h_\mu a_\mu \sigma ^{\dagger } + h_\mu ^*a_\mu ^{\dagger } \sigma )$. Here, $\sigma = |{g}\rangle \!\langle {e}|$ is the dipole operator of the emitter, $a_\mu$ and $a_\mu ^{\dagger }$ are the bosonic annihilation and creation operators for the $\mu ^{\textrm {th}}$ electric field mode with frequency $\omega _\mu$ and coupling strength to the emitter $h_\mu$. These coupling strengths depend both on the structure of the electromagnetic field and the position and orientation of the emitter, and are thus neither an intrinsic property of the field nor the emitter alone. A central object that characterises the emitter–field dynamics is the projected local density of states (LDOS) of the electromagnetic field in the optical structure, which is related to the optical field normal modes $\textbf {u}_\mu (\textbf {r},\omega )$ as [35,36]

**r**as well as the orientation $\textbf {n}_{\textrm {p}}$ of the emitter dipole. The projected LDOS is related to the spectral density of the electric field in the light–matter interaction [35,36],

**r**and $\textbf {n}_{\textrm {p}}$ for simplicity, and furthermore we generally refer to $J$ as defined in Eq. (3) as the LDOS or the spectral density of the field. In the weak light–matter coupling limit, $J(\omega )$ gives the spontaneous emission rate, when evaluated at the emitter frequency, $\omega _{eg}$. In the strong coupling regime, one needs to identify a discrete cavity mode from the LDOS, in order to resolve coherent, reversible exchange of energy between the emitter and the field in the form of vacuum Rabi oscillations.

Exact calculations of the LDOS are numerically demanding, and the wide range of modes $\mu$ generally contributing to $J(\omega )$ complicates the analysis of the physics. However, if the QD is placed in a single-mode waveguide supporting only a single fundamental transverse mode in the frequency regime of interest, the LDOS can be well described using a single-mode model as the sum

where $J_0(\omega )$ and $\Gamma _{\mathrm {R}}(\omega )$ are the contributions from the fundamental mode and radiation modes, respectively. The contribution $\Gamma _{\mathrm {R}}(\omega )$ varies slowly and can be approximated to its value $\Gamma _{\mathrm {R}}(\omega _0)$ at the cavity resonance frequency $\omega _0$, while $J_0(\omega )$ is given in a single-mode picture [37] asAn example of a single-photon source geometry well described by Eq. (5) is the multi-mode micropillar geometry illustrated in Fig. 2(b). Spectra for $J_0(\omega )$ and $\Gamma _{\mathrm {R}}(\omega )$ for the micropillar computed using the single-mode model Eq. (5) as well as from full numerical simulations are presented in Fig. 2(c). Excellent agreement between the single-mode model and the full simulation is observed, and the physics of the LDOS can thus be fully understood from the simple Eq. (5). Our model, Eqs. (4)–(5), can be applied to etalon-type structures, including photonic crystal waveguides [39], where modal reflection coefficients can be defined. However, for other classes of resonant modes, e.g. plasmonic resonances [40], the modal reflectivity may not be well-defined, and in this case, an alternative model based e.g. on a quasi-normal mode formulation [40] is required.

#### 2.2 Waveguide and cavity contributions to the local density of states

If the reflectivity of the mirrors is sufficiently high, the LDOS $J_0(\omega )$ in the vicinity of a resonant frequency can be described by a Lorentzian function corresponding to a single damped cavity mode. However, as the system reduces to a photonic waveguide for vanishing reflectivity, there exists an intermediate reflectivity regime where the LDOS around a cavity resonance cannot be described by a single Lorentzian cavity mode, but also contains a flat background contribution, $\Gamma _{\mathrm {WG}}$ [38], as illustrated in Figs. 2(d)–2(g). When calculating the quantum dynamics and emission properties of the system, one is typically interested in the cavity quantum electrodynamical (cQED) parameters that characterise the optical structure: The cavity decay rate, $\kappa$, the cavity–emitter coupling strength, $g$, and the spontaneous emission rate into the underlying waveguide, $\Gamma _{\mathrm {WG}}$. All of these properties can be derived from the LDOS, which is described by its corresponding microscopic parameters: The spontaneous emission rate into the bare waveguide, $\Gamma _0$, the mirror reflectivities, $r_i$, the effective index of the guided mode, $\bar {n}$, and the cavity length, $L$. The spontaneous emission rate into the bare waveguide, $\Gamma _0$, depends strongly on the structure of the waveguide. For example, a photonic crystal waveguide uses slow-light effects to enhance $\Gamma _0$ relative to the spontaneous emission into radiation modes, $\Gamma _{\mathrm {R}}$ [42–44]. The mapping between the microscopic and QED parameters can be performed by requiring that the effective spectral density generated by the QED system is equivalent to the optical LDOS [38,45]. In Fig. 3, we show how the QED parameters depend on the mirror reflectivity for three different cavity lengths and all other parameters kept constant. For vanishing mirror reflectivity, the emitter–cavity coupling strength approaches zero and the waveguide contribution to the LDOS, $\Gamma _{\mathrm {WG}}$, approaches $\Gamma _0$, consistent with the fact that the optical structure simply becomes identical to the underlying waveguide. For increasing reflectivities, the waveguide component vanishes, and the coupling strength saturates at a value, which is proportional to $L^{-1/2}$. This scaling is essentially linked to the underlying proportionality $g\propto V^{-1/2}$. The cavity decay rate, $\kappa$, monotonically decreases with mirror reflectivity, as expected. Importantly, $\kappa$ does not diverge as $r\rightarrow 0$, but reaches a finite value. This feature is attributed to causality, meaning that light cannot escape a finite spatial region infinitely fast.

The approximation that the spontaneous emission rates into the fundamental waveguide mode and the radiation modes are frequency independent might not be fully justified for all parameter regimes of a given nanocavity structure. However, as discussed above, our analysis in the present paper qualitatively captures the behaviour of the important optical parameters as the properties of the underlying structure are varied.

## 3. Electron–phonon coupling and its signatures in the optical emission

State-of-the-art experiments have demonstrated that electron–phonon processes play a crucial role in determining the coherence properties of photons emitted from solid-state quantum emitters [23,46,47]. However, developing a theoretical model to accurately account for such processes remains a significant challenge. Phenomenological treatments typically assume a pure dephasing form for phonon dissipation, that is, an exponential suppression of the off-diagonal elements in the QD density operator [48]. This is only valid in very particular regimes [49] and misses key physics necessary to understand phonon processes, while also failing to capture key signatures of electron–phonon interactions, e.g. the presence of a phonon sideband.

In the following section we will outline how one may go beyond a simple pure dephasing treatment, introducing a microscopic model for electron–phonon processes in QDs. We will go on to discuss the polaron formalism, which allows one to capture complex non-perturbative effects—such as the phonon sideband—while maintaining a simple and intuitive dynamical description of the QD degrees of freedom.

#### 3.1 Electron–phonon interactions in QDs

The dominant phonon coupling in self-assembled semiconductor QDs is to longitudinal acoustic (LA) phonons through a deformation potential coupling [50,51]. These phonons are natural oscillations of the host lattice, and can be approximated as a collection of harmonic oscillators. In the absence of electron-phonon coupling, the lattice dynamics are described by the free Hamiltonian $H_{\mathrm {P}} = \hbar \sum _{\textbf {k}}\nu _{\textbf {k}}\hat {b}^{\dagger }_{\textbf {k}}\hat {b}_{\textbf {k}}$, where $\nu _{\textbf {k}}$ is the frequency of a phonon mode with wavevector $\textbf {k}$, and $\hat {b}_{\textbf {k}}$ ($\hat {b}^{\dagger }_{\textbf {k}}$) is the corresponding annihilation (creation) operator.

When the charge configuration of the QD changes, for example, during the creation of an exciton, the lattice of the host material will reorganise itself in response. If the resulting displacement of the ions is small, the interaction between the emitter and the phonon environment is given by $H_{\mathrm {EP}} = |{e}\rangle \langle {e}|(V_{\mathrm {L}} + V_{\mathrm {Q}})$. This interaction consists of two terms corresponding to quite different phonon mechanisms: $V_{\mathrm {L}}$ gives rise to emission/absorption of a phonon wavepacket in the photon emission process, while $V_{\mathrm {Q}}$ describes virtual transitions to higher excitonic levels generated by thermal phonons (see Fig. 4(a)).

The first term is linear in phonon operators [52,53]:

where $g_{\textbf {k}}$ is the linear electron-phonon interaction strength for the $\textbf {k}^{\mathrm {th}}$-phonon mode, and may be defined in terms of the phonon matrix elements $\hbar g_{\textbf {k}} = \sum _{a = e,h} M^{11}_{a,\textbf {k}}$, where for deformation potential coupling, the matrix element takes the form:The second term is quadratic in phonon operators, and takes the form [21–23]:

*s*-shell) and the next state (

*p*-shell) $\sim 40~\mathrm {meV}$, is thus orders of magnitude greater than phonon energies at cryogenic temperatures [23]. With this in mind, we can intuitively understand the quadratic interaction term as virtual scattering of a phonon from mode $\textbf {k}$ into $\textbf {k}^\prime$ mediated by the excitonic state, as shown schematically in Fig. 4(a). This scattering process imparts a random phase kick to the exciton, the cumulative effect of which is a temperature dependent broadening of the zero phonon line [54] (see Fig. 4(c)), which is absent when there is only a linear electron-phonon coupling term. The effective coupling strength for the quadratic coupling may be written in terms of the matrix elements $\hbar f_{\textbf {k},\textbf {k}^\prime } = \sum _{a=e,h}\sum _{j\;>\;1}M^{1j}_{a,\textbf {k}}M^{j1}_{a,\textbf {k}^\prime }[\hbar \omega _j^a - \hbar \omega _1^a]^{-1}$ and $\hbar \omega ^{e/h}_{j}$ is the energy of the $j^{\mathrm {th}}$ electronic state.

#### 3.2 Optical properties of a QD in a photonic environment

Calculating the dynamical and optical properties of a QD including phonon effects constitutes a challenging theoretical problem. Strong coupling between electronic and vibrational degrees of freedom means that standard perturbative treatments are insufficient, a fact that is compounded when the QD is interfaced with a complex photonic environment (e.g. a optical micropillar). In this section, we will outline the polaron formalism, which allows one to account for strong electron-phonon coupling in addition to strong coupling to complex photonic structures [55–57]. Numerically exact approaches based on real-time path integrals [58–61], non-equilibrium Green’s functions [62] and diagonalisation of the Liouvillian [20,24,63,64] have previously been employed to solve similar problems. In contrast to these computationally demanding approaches, using perturbation theory in a polaron representation leads to numerically inexpensive implementations, and even allow analytical results in certain cases, while maintaining a high accuracy in the regimes relevant for single-photon sources.

We now consider the full Hamiltonian, Eq. (1), where the electromagnetic environment is described by a cavity mode with annihilation operator $a$, coupling strength $g$ to the emitter, and decay rate $\kappa$, as described in Sec. 2.2.

First, a canonical transformation to the *polaron frame* generated by the unitary $\mathcal {U} = |{g}\rangle \!\langle {g}| + |{e}\rangle \!\langle {e}| e^{-S}$, with $S = \sum _{\mathbf {k}}\nu _{\mathbf {k}}^{-1}g_{\mathbf {k}}(b_{\mathbf {k}}^{\dagger }-b_{\mathbf {k}})$, is applied to total Hamiltonian. This transformation dresses the system with vibrational modes of the host material, removing the linear electron-phonon interaction term in the process, and provides an optimised basis for perturbation theory.

In the polaron frame we may derive a $2^{\mathrm {nd}}$-order master equation for the emitter and cavity mode which is non-perturbative in electron-phonon coupling strength, taking the form:

In the more general case, the background electromagnetic modes that give rise to $\Gamma _{\mathrm {R}}$ and $\Gamma _{\mathrm {WG}}$ can potentially have a spectral density that varies appreciably over the frequency range of the phonon sideband. In such cases, phonon interactions renormalise the resulting spontaneous emission rate [65].

The pure dephasing rate, $\gamma$, entering the master equation, contains contributions from charge/spin noise, $\gamma _0$, and thermal phonon scattering induced by virtual $p$-shell transitions, $\gamma _{\mathrm {P}}$ [22,23], such that $\gamma =\gamma _0 + \gamma _{\mathrm {P}}$. The form of the phonon-induced pure-dephasing rate will be introduced in the next section.

The master equation Eq. (9) can be written in the compact form $\dot {\rho }(t)=\mathcal {L}\rho (t)$, where $\mathcal {L}$ is the Liouvillian superoperator. The two-time correlation function of the emitter dipole can be numerically calculated using the Liouvillian as

To understand the spectral features of the emitted light, we may write the dipole power spectrum as $\bar {S}_0(\omega )=\bar {S}_0^{\mathrm {zpl}}(\omega )+ \bar {S}_0^{\mathrm {psb}}(\omega )$, with two distinct spectral components that correspond to the zero-phonon line and the phonon sideband, respectively,

The second contribution, $\bar {S}_0^{\mathrm {psb}}$, describes photons emitted while a phonon wavepacket is absorbed or emitted. This spectral component emerges through the linear electron–phonon coupling. Since the temporal features of the phonon correlation function, $G_{\mathrm {P}}(\tau )$, are very fast, on the order of a few ps, the spectral features of $\bar {S}_0^{\mathrm {psb}}$ are very broad – on the order of a few meV.

#### 3.3 Electron-phonon coupling parameters

In order to calculate the influence of phonons on the optical properties of a QD, it is necessary to specify the linear and quadratic electron-phonon coupling strengths. To do so, it is clear from Eq. (7) that we must first specify the wave function of electrons and holes confined within the QD. However, calculating exciton wave functions for realistic QDs is a highly non-trivial task, requiring sophisticated and expensive computational methods [68–70]. Instead, we can accurately reproduce both the qualitative and quantitative effects of phonons by considering an idealised model for a QD. Here we assume the QD confinement to be parabolic, spherically symmetric, and the electrons and holes have the same confinement length [53,64]. Relaxing such approximations leads to the emergence of complex phonon behaviour, including the quenching of phonon modes for specific asymmetries between the electron-hole confinement potentials [71].

Using the above approximations we may obtain analytic forms for the polaron frame propagator [55,57] $\varphi (\tau ) = \alpha \int _0^\infty \nu e^{-\nu ^2/\nu _c^2}(\cos (\nu \tau )\coth (\hbar \nu /2k_{\mathrm {B}}T) - i\sin (\nu \tau ))$, where we have taken a continuum limit over the phonon modes assuming a linear dispersion relation. We have also introduced the electron-phonon coupling strength, which may be written in terms of material parameters $\alpha = [4\pi ^2\hbar \varrho c_s^5]^{-1}(D_e - D_h)^2$. Using typical material parameters for GaAs QDs (see Table 1), we find $\alpha = 0.025~\mathrm {ps}^{2}$. The electron-phonon interaction strength is also determined by the cut-off frequency $\nu _c = \sqrt {2}c_s/d$, where $d$ is the confinement length of the QD.

With the electron-phonon coupling fully characterised, it is worth briefly considering the regime of validity of the polaron master equation described in Sec. 3.2. The polaron transformation results in a master equation that incorporates electron-phonon coupling to all orders, with the light-matter coupling strength becoming the relevant perturbative parameter in the transformed frame. Following McCutcheon and Nazir [57], the resulting master equation can be found to be valid in regimes where $(2 g/\nu _c)^2(1-B^4)\;<<\;1$.

If we assume that the dominant contribution to the quadratic phonon-interaction arises from virtual transition between the first and second excited states of the QD (i.e. the *s*- and *p*-shells), then we may also find an analytic form for the pure dephasing rate [22,23]:

*s-*and

*p*-states for the electron/hole, and naturally depends on the confinement potential of the QD. For a spherically symmetric, harmonic potential, it is simply given by $\Delta _{e/h}=\hbar ^2/(d^2m_{e/h})$, where $m_{e/h}$ is the effective electron/hole mass.

Both the FC factor and the pure dephasing rate are dependent on temperature. In the case of the FC factor, as temperature increases, $B$ decreases due to an increasing fraction of light being emitted by the phonon sideband rather than by the ZPL. The pure dephasing rate, on the other hand, increases, as more phonons are available to drive virtual transitions between the electronic states. Crucially this dephasing rate freezes out at low temperatures owing to the quadratic dependence of the rate on the phonon occupation, physically this corresponds to an absence of phonons in the environment available to drive transitions.

Table 1 shows realistic parameters for GaAs QDs, which are generally used for calculations in this paper unless otherwise stated [72,73].

## 4. Phonon-induced limitations to single-photon sources

Due to the presence of the phonon sideband in the emission spectrum, a quantum dot–based single photon source cannot simultaneously have unity efficiency and indistinguishability [67]: If the emission spectrum is left unfiltered, the indistinguishability is reduced below unity due to the photons in the sideband, which are uncorrelated with those of the zero-phonon line that have not undergone phonon scattering. On the other hand, while filtering out the sideband revives the indistinguishability, it comes with the cost of a reduction in efficiency, corresponding to the fraction of photons contained in the sideband. By placing the emitter in a photonic cavity with a narrow resonance line, the emission can be preferentially funneled into the zero-phonon line via the Purcell effect, thereby simultaneously improving the indistinguishability and efficiency. However, this strategy only continues to improve the performance as long as the emitter–cavity system is in the Purcell regime. As soon as the system enters the strong coupling regime, where the cavity lifetime is long enough for the emitter to re-absorb an emitted photon and the system undergoes vacuum Rabi oscillations, the phonon environment will drive incoherent transitions from the upper polariton state to the lower one [63]. This polaritonic relaxation process drastically degrades the coherence of the emitted light and in turn sets the upper limit for the performance, as illustrated in Fig. 5, which shows the indistinguishability, efficiency and their product as a function of cavity linewidth (blue solid lines).

In recent reports on broadband ($\hbar \kappa \sim$ 10 meV) sources of quantum light based on circular Bragg gratings, the measured indistinguishabilities of 0.90 [4] and 0.95 [5] are higher than that predicted in Fig. 5(a). However, here 150 GHz [4] and 5 GHz [5] etalons were used, leading to improved indistinguishability at the cost of efficiency as discussed above. While the indistinguishability measured in the absence of a filter is the relevant figure of merit describing the device performance in the context of scalable optical quantum information technology, it is only rarely [3] reported.

In the weak light–matter coupling regime, the indistinguishability and effiency can be approximated by the analytical expressions [6,38]

If one assumes that the emitter–cavity coupling is weak and that phonon effects can be described by a constant pure dephasing rate, $\gamma$, one ends up with a simple model leading to the result [27]

A typical experimental strategy for tuning the emitter into resonance with an optical cavity is to change the temperature [2]. Figures 6(a)–6(c) presents the single-photon source performance as a function of cavity linewidth for temperatures in the range $5-55\mathrm {\; K}$. With increased temperature, the thermal phonon population will enhance the fraction of emission into the phonon sideband ($1-B^2$) and increase the phonon-induced pure dephasing rate [23,25]. These two effects are quantified in Figs. 6(d)–6(e), with coloured circles indicating the temperature of the corresponding lines in Fig. 6(a)–6(c).

The size of the QD also has a strong impact on the emission properties. This is illustrated in Fig. 7, where the indistinguishability, efficiency and their product is shown for a range of QD sizes, $d$. The QD size has an impact on the FC factor, the energy splitting between the excitonic states and the phonon cutoff-frequency. The FC factor increases monotonically with QD size, because the cutoff-frequency is inversely proportional with the size. This means that for larger QDs, the power in the sideband is decreased relative to the zero-phonon line. The splitting between the $s$- and $p$-shell excitons scales as $d^{-2}$. For this reason, the rate of pure dephasing induced by virtual scattering of thermal phonons increases as the QD becomes larger, but only for relatively small QDs. As seen in Eq. (18), the pure-dephasing rate involves a frequency integral over $e^{-(\nu /\nu _c)^2}$. As the size is increased, the cutoff-frequency, $\nu _c$, will decrease as $d^{-1}$, and thus eventually the pure dephasing rate will decrease with increasing size. It is worth mentioning that these calculations have been performed under the assumption of a spherically symmetric QD, and more intricate features might be revealed when the confinement potential mimicks the shape of a realistic QD more closely. Furthermore, we have taken the spontaneous emission rates, $\Gamma _0$ and $\Gamma _{\mathrm {R}}$, to be independent of the QD size. Strictly speaking, these rates are size-dependent, although experiments have shown a very modest variation over the sizes used here [76]. We do, however, note that care should be taken when going to the large-dot limit, where the light-matter interaction may be strongly influenced by the QD size [77–79].

## 5. Outlook

Although this article focuses on intrinsic limitations of quantum dots as single-photon sources, quantum dots remain among the most promising solid-state emitters for on-demand high-rate generation of single photons with simultaneous high efficiency and indistinguishability. In comparison to other leading solid-state quantum emitters, the electron-phonon interaction strength of QDs is relatively weak, with $\sim 90$% of light emitted through the zero phonon line in bulk at cryogenic temperatures. For example, in contrast, Nitrogen Vacancy centers diamond emit only $\sim 3$% of photons through the zero phonon line [80], while displaying better spin coherence properties than quantum dots. Furthermore, the natural integrability of QDs in photonic structures allows to engineer emission properties and may vastly improve the coherence of emitted photons.

Many important properties of quantum dots and their interaction with a photonic environment remain to be answered. Recent studies investigate the dynamics and limitations of the optical excitation process [10] and demonstrate how the phonon sideband can be used to increase the pumping efficiency without sacrificing the photon indistinguishability [11]. Other recent investigations focus on the properties of light scattered from a solid-state quantum emitter [81,82]. Furthermore, a larger parameter space and a range of new phenomena arise when the Fabry-Perot cavity is replaced with a non-Lorenzian cavity, which opens up a rich class of dynamics [83,84]. We also note that the topic of phonon effects in quantum dots has recently been reviewed with focus on optical excitation schemes [85] and in a broader scope, presenting a range of different phonon-induced phenomena and theoretical modeling techniques [86].

## Funding

Det Frie Forskningsråd (DFF-4181-00416); Royal Commission for the Exhibition of 1851; Villum Fonden (8692); QuantERA ERA-NET Cofund (HYPER-U-P-S).

## Disclosures

The authors declare no conflicts of interest.

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