Light Scattering from Solid-State Quantum Emitters: Beyond the Atomic Picture

TL;DR

This paper investigates how solid-state quantum emitters, specifically quantum dots, scatter light differently from atomic systems due to phonon interactions, revealing unique effects like phonon sidebands and non-atomic behavior.

Contribution

It develops a comprehensive model of light scattering from solid-state emitters that includes phonon interactions, extending beyond the traditional atomic picture.

Findings

Phonon coupling causes a sideband insensitive to excitation conditions.

Non-monotonic relationship between laser detuning and coherent scattering.

Deviations from atom-like behavior in solid-state quantum emitters.

Abstract

Coherent scattering of light by a single quantum emitter is a fundamental process at the heart of many proposed quantum technologies. Unlike atomic systems, solid-state emitters couple to their host lattice by phonons. Using a quantum dot in an optical nanocavity, we resolve these interactions in both time and frequency domains, going beyond the atomic picture to develop a comprehensive model of light scattering from solid-state emitters. We find that even in the presence of a cavity, phonon coupling leads to a sideband that is completely insensitive to excitation conditions, and to a non-monotonic relationship between laser detuning and coherent fraction, both major deviations from atom-like behaviour.

Equations

$${\mathcal{F}}_{CS}=\frac{T_2}{2T_1}\frac{1}{1+\mathcal{S}},$$

$$g_D^{(1)}(\tau )={\int }_{-\infty }^{\infty }\tilde{h}(t-\tau )g_{\mathrm{pol}}^{(1)}(t)dt,$$

$$S_{\mathrm{opt}}(\omega )=B^2{\int }_{-\infty }^{\infty }{g}_{\mathrm{opt}}(\tau )e^{i\omega \tau }d\tau ,$$

$$S_{\mathrm{SB}}(\omega )={\int }_{-\infty }^{\infty }\left(\mathcal{G}(\tau )-B^2\right)g_{\mathrm{opt}}(\tau )e^{i\omega \tau }d\tau ,$$

Full text

Scattering of light by a single quantum emitter is one of the fundamental processes of quantum optics. First observed in atomic systems Gibbs and Venkatesan (1976); Volz et al. (2007), and more recently studied extensively in self-assembled quantum dots (QDs) Nguyen et al. (2011); Matthiesen et al. (2012); Proux et al. (2015); Bennett et al. (2016a), coherent scattering attracts interest as the scattered light retains the coherence of the laser rather than the emitter. As such, their coherence may exceed the conventional radiative limit whilst still exhibiting antibunching on the timescale of the emitter lifetime Nguyen et al. (2011); Matthiesen et al. (2012); Proux et al. (2015); Bennett et al. (2016a). Exploiting this behaviour gives rise to exciting possibilities for quantum technologies such as generating tuneable single photons Matthiesen et al. (2013); He et al. (2013); Sweeney et al. (2014), realising single photon non-linearities Javadi et al. (2015); Sipahigil et al. (2016); Bennett et al. (2016b); De Santis et al. (2017); Hallett et al. (2018), and constructing entangled states between photonic Denning et al. (2017); Scerri et al. (2018) or spin Delteil et al. (2015); Stockill et al. (2017) degrees of freedom.

For a continuously driven emitter, coherent scattering occurs in the weak excitation regime where absorption and emission become a single coherent event. For a simple two-level “atomic picture” with only spontaneous emission and pure dephasing, the coherently scattered fraction ($\mathcal{F}_{CS}$) of the total emission is given by *[Eq.\ref{eq:CoherentFracRabi}generalisesexpressionsfrom][toincludepuredephasing.]CohenTannoudhiAtom:

[TABLE]

where $\mathcal{S}=(\Omega^{2}T_{1}T_{2})/(1+\Delta_{LX}^{2}T_{2}^{2})$ is a generalized saturation parameter, $\Omega$ is the Rabi frequency, $\Delta_{LX}=\omega_{L}-\omega_{X}$ is the detuning between the laser ($\omega_{L}$) and the emitter ($\omega_{X}$) and $T_{1}$ and $T_{2}$ are the emitter life- and coherence times respectively. It is clear from this expression that the fraction of coherently scattered light reaches unity in the limit of driving well below saturation ($\mathcal{S}\ll 1$) and transform-limited coherence ($T_{2}=2T_{1}$).

Solid-state emitters (SSEs), particularly self-assembled QDs, are an attractive system with which to realise such schemes owing to their high brightness and ease of integration with nanophotonic structures. However, unlike atoms, SSEs can experience significant dephasing from fluctuating charges Houel et al. (2012); Kuhlmann et al. (2013) and coupling to vibrational modes of the host material Muljarov and Zimmermann (2004); Reigue et al. (2017). Despite this, state-of-the-art InGaAs QD single photon sources have demonstrated essentially transform-limited photons emitted via the zero phonon line (ZPL) Somaschi et al. (2016); Wang et al. (2016); Liu et al. (2018) through careful sample optimisation, exploitation of photonic structures and by using resonant $\pi$-pulse excitation at cryogenic temperatures. Although these results show ZPL broadening can be effectively suppressed, coupling to vibrational modes also leads to the emergence of a phonon sideband (PSB) in the emission spectrum Krummheuer et al. (2002); Kaer and Mørk (2014); McCutcheon (2016); Iles-Smith et al. (2017a, b); Reigue et al. (2017). This is attributed to a rapid change in lattice configuration of the host material during exciton recombination, leading to the simultaneous emission or absorption of longitudinal acoustic (LA) phonons with the emission of a photon Besombes et al. (2001); Krummheuer et al. (2002); Kaer and Mørk (2014). Therefore, to obtain perfectly indistinguishable photons the PSB must be filtered out, naturally limiting the efficiency of the device, even when using an optical cavity to Purcell enhance emission into the ZPL Iles-Smith et al. (2017b); Grange et al. (2017).

The aforementioned works Krummheuer et al. (2002); Kaer and Mørk (2014); McCutcheon (2016); Iles-Smith et al. (2017a, b); Reigue et al. (2017); Grange et al. (2017) have revealed the importance of phonon coupling in the incoherent regime, where there is a definite change of charge configuration in the QD, such that incoherently scattered resonance fluorescence dominates the spectrum. It is perhaps natural to presume that phonon coupling may be eliminated by operating in the coherent scattering regime, since there is vanishing exciton population and therefore no change in charge configuration. This suggests that, in accordance with most works in the literature Nguyen et al. (2011); Matthiesen et al. (2012); Proux et al. (2015), one may adopt the atom-like picture of Eq. 1, where the coherent fraction tends towards unity for excitation far below saturation and transform-limited coherence. However, a recent theoretical study predicted that PSBs occur even for vanishingly weak resonant driving Iles-Smith et al. (2017a).

Here, we experimentally verify that PSBs persist in the coherent scattering regime and demonstrate additionally that phonon processes also cause large deviations from atom-like physics when driving off-resonance. An extended theoretical model fully describes our solid-state nanocavity system, providing an intuitive picture that attributes the PSB to phonon dressing of the optical dipole moment. This leads to a finite probability that the vibrational environment changes state during an optical scattering event, implying that all optical spectral features will have an associated PSB. Whilst a self-assembled QD is studied here, we emphasize that the physics and methods apply equally to a diverse range of SSEs, including vacancy centers in diamond Faraon et al. (2011); Neu et al. (2011), defects in hexagonal boron nitride Jungwirth and Fuchs (2017), monolayer transition metal dichalcogenides Christiansen et al. (2017) and single carbon nanotubes Jeantet et al. (2016).

To study this phonon coupling experimentally, we investigate a neutral exciton state ($\ket{X}$) of a self-assembled InGaAs QD with dipole moment $|\vec{\mu}|=27.2~{}\mathrm{D}$, weakly coupled ($\hbar g=135~{}\upmu\mathrm{eV}$) to a H1 photonic crystal cavity (linewidth $\hbar\kappa=2.51~{}\mathrm{meV}$) with Purcell factor $F_{P}=43$ (see Ref. Liu et al. (2018) for full details). As well as Purcell enhancing the ZPL, the cavity also acts as a weak spectral filter; this combination can reduce the PSB component of the emission Iles-Smith et al. (2017b); Grange et al. (2017), motivating the coupling of SSEs to cavities. Fig. 1(a) illustrates the experiment; the sample is held in a liquid helium bath cryostat at $T=4.2~{}\mathrm{K}$ and excited by a tuneable laser that is rejected from the detection path by cross-polarisation (typical signal-to-background $>$100:1). The coherence of the scattered light is studied either in the time domain by measuring the fringe contrast $v(\tau)$ in a Mach-Zehnder interferometer or in the frequency domain using a spectrometer or a Fabry-Perot interferometer (FPI) for higher resolution (details in supplemental material (SM) Sup ).

It is instructive to begin with a high resolution time-domain measurement, exciting resonantly below saturation ($\mathcal{S}=0.25$) where coherent scattering is expected to dominate the emission. The measured fringe contrast $v(\tau)$ is proportional to the first order correlation function $g^{(1)}(\tau)$ Sup . The result in Fig. 1(b) departs significantly from the mono-exponential radiative decay predicted by atomic theory (dashed line); a rapid decay of coherence occurs in the first few picoseconds, comparable to phonon dynamics observed in pulsed four-wave mixing measurements of InGaAs QDs in the incoherent regime Jakubczyk et al. (2016), suggesting that the rapid loss of coherence we observe originates from electron-phonon interactions.

In order to describe such behaviour accurately, we must account for the microscopic nature of the QD-phonon coupling Nazir and McCutcheon (2016). This is achieved by applying the polaron transformation to the full system-environment Hamiltonian, dressing the excitonic states of the system with modes of the phonon environment. We may then derive a master equation (ME) that is non-perturbative in the electron-phonon coupling strength Breuer et al. (2002); McCutcheon and Nazir (2010); Roy and Hughes (2011); Iles-Smith et al. (2017a) to describe the evolution of the reduced state of the QD Sup . In the polaron frame, the first-order correlation function is $g^{(1)}_{\mathrm{pol}}(\tau)=\mathcal{G}(\tau)g^{(1)}_{opt}(\tau)$ Iles-Smith et al. (2017a), where $g^{(1)}_{opt}(\tau)$ is the purely optical contribution found using the polaron frame ME, while $\mathcal{G}(\tau)=B^{2}\exp(\varphi(\tau))$ is the correlation function of the phonon environment, which accounts for non-Markovian phonon relaxation. Here we have defined the phonon propagator $\varphi(\tau)=\alpha\int_{0}^{\infty}\nu e^{-\nu^{2}/\nu_{c}^{2}}(\cos(\nu\tau)\coth(\nu/2k_{B}T)-i\sin(\nu\tau))d\nu$, and the Franck-Condon factor $B=\exp(-\varphi(0)/2)$. The coupling of the QD to the phonon environment is thus specified by its thermal energy $k_{B}T$, the deformation potential coupling strength $\alpha$, and cut-off frequency $\nu_{c}$ Krummheuer et al. (2002); Glässl et al. (2011); Nazir and McCutcheon (2016). The cavity leads both to Purcell enhancement of the exciton transition (included within the ME) and spectral filtering of the emission. We incorporate cavity filtering by solving the Heisenberg equations of motion for the cavity field operators, leading to the detected function:

[TABLE]

where $\tilde{h}(t)=\exp(-i\Delta_{XC}t-\kappa|t|/2)$ is the cavity filter function and $\Delta_{XC}$ is the exciton-cavity detuning Sup .

By fitting the phonon bath correlation function contained within Eq. 2 to the first few picoseconds of the measurement, we extract phonon parameters $\alpha=0.0447~{}\mathrm{ps^{2}}$ and $\nu_{c}=1.28~{}\mathrm{ps^{-1}}$, comparable to values previously found for InGaAs QDs Ramsay et al. (2010). Using experimentally determined values for all other parameters, we accurately reproduce the full dynamics of the experimental data, as shown by the solid line in Fig. 1. After phonon relaxation, radiative decay associated with incoherent resonance fluorescence occurs between $\tau=20-200~{}\mathrm{ps}$. Finally, at $\tau\gg 200~{}\mathrm{ps}$, $v(\tau)$ plateaus, corresponding to the coherent fraction of the emission. As the laser coherence time is much greater than the measured delays, no decay of the coherent scattering is observed. From the $v(\tau)$ amplitudes, we extract $\mathcal{F}_{\mathrm{PSB}}=0.06$, $\mathcal{F}_{\mathrm{INC}}=0.14$ and $\mathcal{F}_{\mathrm{CS}}=0.80$ respectively for the PSB, incoherent and coherent fractions of the total emission ($\mathcal{F}$). Crucially, a finite $\mathcal{F}_{\mathrm{PSB}}$ under weak driving indicates that Eq. 1 does not fully describe the scattering dynamics of the system.

To check the accuracy of the extracted parameters we now move to the frequency domain. The theoretical spectrum is calculated by Fourier transforming $g^{(1)}_{D}(\tau)$ and may be written as: $S(\omega)=H(\omega)(S_{\mathrm{opt}}(\omega)+S_{\mathrm{SB}}(\omega)),$ where $H(\omega)=(\kappa/2)/[(\omega-\Delta_{XC})^{2}+(\kappa/2)^{2}]$ is the frequency domain cavity filter function Roy-Choudhury and Hughes (2015); Iles-Smith et al. (2017b); Denning et al. (2018). The spectrum consists of two principal components: a purely optical part

[TABLE]

containing both coherent and incoherent contributions to the spectrum, and a second incoherent component

[TABLE]

which gives rise to the PSB Iles-Smith et al. (2017a, b). The ZPL contribution is thus reduced by the square of the constant Franck-Condon factor $B^{2}$, with the missing fraction emitted through the PSB.

The inset to Fig. 1 illustrates that the parameters extracted from the time domain dynamics lead to excellent agreement between the experimental (blue triangles) and theoretical ($S(\omega)$ - solid line) spectra, with a broad PSB observed in accordance with the short timescale of the phonon processes. These combined time and frequency domain measurements provide critical insight into the nature of electron-phonon interactions in driven QDs: even well below saturation, where the excited state population is small and coherent scattering dominates, a PSB is present, comprising $\sim 6$% of the emission.

To investigate to what extent the PSB persists in the coherent scattering regime, we measure the resonance fluorescence spectrum as a function of saturation by varying $\Omega$. Fig. 2(a) shows a spectrum taken well above saturation ($\mathcal{S}=10$) that exhibits a ZPL (yellow fit) and a PSB ($S_{SB}(\omega)$ - red fit). Performing high resolution spectroscopy of the ZPL with the FPI results in the inset to Fig. 2(a) which exhibits a broad contribution from incoherent resonance fluorescence (blue fit) and a narrow feature from coherent scattering (green fit). As in the ${g}^{(1)}(\tau)$ of Fig. 1(b), the total spectrum thus comprises three components whose fraction of the total emission can be evaluated from their areas (details in Sup ).

Fig. 2(b) shows the evolution of the components of the resonant ($\Delta_{LX}=0$) scattering spectrum as a function of $\mathcal{S}$. The polaron model agrees well with the experiment and produces a curve for $\mathcal{F}_{CS}$ (green dashed line) that is proportional to $(1+\mathcal{S})^{-1}$ like Eq. 1. However, as previously predicted Iles-Smith et al. (2017a), $\mathcal{F}_{CS}$ does not reach unity for vanishing $\mathcal{S}$, a surprising result that may be explained by observing that the PSB fraction $\mathcal{F}_{PSB}$ (red diamonds) is constant and independent of $\Omega$. This contrasts with excitation induced dephasing (EID) Ramsay et al. (2010); Ulhaq et al. (2013) which is also mediated by LA phonons and captured within our model, but is proportional to $(\Omega^{2}+\Delta_{LX}^{2})$ and is thus negligible for resonant driving below saturation.

The results of Fig. 2(b) can be understood by considering the possible scattering channels illustrated in Figs. 2(c,d). The bare transition $\ket{0}\rightarrow\ket{X}$ (solid black levels) is broadened by the presence of a continuum of states corresponding to emission or absorption of an LA phonon (grey shading), dressing the optical transition with vibronic bands. In the simplest case (Fig. 2(c)), a photon in the driving field coherently (Rayleigh) scatters directly from the single exciton transition. However, in the presence of the phonon environment, the dressing of the optical transition results in non-zero overlaps between vibronic states in the ground and excited state manifolds, such that a scattering event can end in a different vibrational state within the ground-state manifold (Fig. 2(d)). This corresponds to inelastic Stokes (anti-Stokes) scattering of a lower (higher) energy photon accompanied by the emission (absorption) of an LA phonon, leading to the emergence of a PSB. At low bath temperatures, phonon absorption is suppressed, resulting in the characteristic asymmetry of the PSB. From Eqs. 3 and 4, the branching ratio between phonon-mediated inelastic and elastic scattering is determined solely by the constant $B^{2}$. Outside the Mollow triplet regime, the coherent ($\mathcal{S}\ll 1$) and incoherent ($\mathcal{S}\gtrsim 1$) resonant scattering spectra of a SSE thus differ only in the width of the ZPL. As such, whilst coherent scattering is often cited as a route to highly coherent single photons, it cannot negate the PSB.

To gain further insight into phonon interactions in the scattering picture, the effect of detuning the laser from the emitter is now considered. Fig. 3(a) shows semi-log plots of spectra taken at constant $\Omega$ with laser detuning $\hbar\Delta_{LX}=\pm 0.27~{}\mathrm{meV}$. The coherent peaks at $\hbar\Delta_{LX}$ are separated from the ZPL and dominate the spectrum. For positive detuning (blue spectrum), it is immediately noticeable that the high-energy edge of the sideband is shifted by $\sim\hbar\Delta_{LX}$. The origins of this behaviour can be seen in Eq. 4, where the product between $g^{(1)}_{\mathrm{opt}}(\tau)$ and $(\mathcal{G}(\tau)-B^{2})$ in the time-domain implies a convolution in frequency between the purely optical spectrum and the frequency-space phonon correlation function. As such, all optical features in $S_{\mathrm{opt}}$ have an associated PSB; the coherent peak (and thus its PSB) shifts with $\Delta_{LX}$. Theoretically (Fig. 3(a) inset), the low-energy edge of the sideband would also be expected to shift for negative detuning (red spectrum); experimentally this is obscured by weak incoherent backgrounds owing to the low count-rate at large $\Delta_{LX}$. The total PSB fraction is still governed by $B^{2}$: since sideband processes arise from phonon dressing of the optical transition, they apply equally to both coherent and incoherent peaks, irrespective of $\omega_{L}$.

Further deviations from the conventional atomic picture can be seen in the balance of coherent and incoherent scattering when driving off-resonance. Compared to the experiment, both the atomic and polaron theories significantly over-estimate the coherent fraction away from resonance (details in SM Sup ). We attribute this to the Lorentzian reduction in QD absorption with $|\Delta_{LX}|$, allowing laser light to instead be absorbed in the doped bulk material Casey et al. (1975), leading to charge noise. To capture the associated pure dephasing in both the atomic and polaron models, we include a Lorentzian detuning-dependent dephasing rate $\gamma(\Delta_{LX})$ with $\gamma(0)=0$, $\gamma(\Delta_{LX}\to\infty)=\gamma_{max}$ and width fixed to the QD linewidth Sup . By fitting the polaron theory to the data we find $\gamma_{\mathrm{max}}=21\pm 0.1~{}\upmu$eV. Spectral wandering is then accounted for by convolving with a Gaussian noise function with width deduced from the incoherent peak observed in detuned spectra (Fig. 3(a)) Sup .

In Fig. 3(b), upper and lower bounds (from uncertainty in $\Omega$) of the atomic (green curves) and polaron (red curves) models are plotted. Experimental values of $\mathcal{F}_{CS}$ (grey circles) are evaluated as in Fig. 1(b). In stark contrast to the atomic theory, where Eq. 1 predicts $\mathcal{F}_{CS}$ will only ever increase with $|\Delta_{LX}|$, the measured data only increases close to resonant driving where EID Ramsay et al. (2010); Ulhaq et al. (2013) is small. For $|\Delta_{LX}|$ between 0.1 and 0.4 meV, this EID becomes significant and the coherent fraction decreases with a noticeable asymmetry, as predicted by the polaron model. This asymmetry originates from the phonon-dressing of the optical transition: when driving above resonance ($\Delta_{LX}>0$) as in Fig. 3(c), $\ket{X}$ can be populated through the emission of an LA phonon Glässl et al. (2013); Hughes and Carmichael (2013); Quilter et al. (2015) (purple arrow), increasing the probability of incoherent scattering (orange arrow). When $\Delta_{LX}<0$ (Fig. 3(d)), populating $\ket{X}$ is inhibited at $T=4.2~{}\mathrm{K}$ as it requires phonon absorption Liu et al. (2016); Brash et al. (2016), inhibiting incoherent scattering. For $\Delta_{LX}<-0.5~{}\mathrm{meV}$, the probability of phonon absorption becomes sufficiently low that $\mathcal{F}_{CS}$ begins to increase again towards the limiting atomic case. This behaviour deviates strongly from the atomic model and requires careful consideration for schemes involving detuned coherent scattering, such as generating single He et al. (2013); Sweeney et al. (2014) or entangled Denning et al. (2017); Scerri et al. (2018) photons.

In conclusion, we have shown that a fixed fraction of light scattered from a solid-state emitter is always lost through a phonon sideband, irrespective of excitation conditions such as Rabi frequency or detuning. We have also demonstrated that the detuning dependence of the coherent fraction is strongly modified by the presence of phonon coupling, contradicting the atomic prediction that the coherent fraction will increase monotonically with detuning. Both processes can be intuitively understood by considering phonon-dressing of the optical transition of the QD. Taken together, these results illustrate the importance of employing an appropriate model of phonon coupling rather than assuming atom-like physics when driving weakly or off-resonance. For example, treating phonons in a crude pure-dephasing approximation (e.g. Eq. 1), suggests they may be suppressed simply by increasing the Purcell factor. This is directly contradicted by the clear separation of phonon and radiative timescales in Fig. 1(b), with the phonon sideband persisting despite a large Purcell enhancement. The methods developed here can be used to optimise quantum information protocols such as spin-photon entanglement schemes for realistic solid-state emitters.

This work was funded by the EPSRC (UK) EP/N031776/1, A.N. is supported by the EPSRC (UK) EP/N008154/1 and J.I.S. acknowledges support from the Royal Commission for the Exhibition of 1851. Note: After the completion of our experiments, we became aware of related results Koong et al. (2019). We thank B.D. Gerardot for bringing these to our attention.