We present both experimental and theoretical investigations of a laser-driven quantum dot (QD) in the dressed-state regime of resonance fluorescence. We explore the role of phonon scattering and pure dephasing on the detuning-dependence of the Mollow triplet and show that the triplet sidebands may spectrally broaden or narrow with increasing detuning. Based on a polaron master equation approach, which includes electron-phonon interaction nonperturbatively, we derive a fully analytical expression for the spectrum. With respect to detuning dependence, we identify a crossover between the regimes of spectral sideband narrowing or broadening. We also predict regimes of phonon-induced squeezing and anti-squeezing of the spectral resonances. A comparison of the theoretical predictions to detailed experimental studies on the laser detuning-dependence of Mollow triplet resonance emission from single In(Ga)As QDs reveals excellent agreement.
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Resonant excitation of single QDs has recently gained significant interest [1–3], in part because this type of coherent excitation is promising for the generation of single photons with excellent coherence properties . The techniques developed for effective laser stray light suppression have enabled the collection of resonance fluorescence from a single QD with high signal-to-noise ratio. Resonance fluorescence (RF) emission below saturation of the quantum emitter has revealed close-to-Fourier transform limited single photons with record-high emission coherence and two-photon interference visibility . Recent experiments have even been able to beat the Fourier limit for single-photon emission coherence in the Heitler regime, e.g., with excitation strengths well below the saturation of the quantum emitter [6,7]. Another major achievement with respect to RF is the demonstration of single- and cascaded photon emission between the Mollow sidebands above saturation of the QD .
Recent investigations of single QD RF have revealed distinct differences of their emission coherence properties  which need to be theoretically treated beyond a simple two-level description usually used for atoms. One of the main consequences of the solid-state character of these quantum emitters is the consideration of specific dephasing channels primarily caused by carrier-phonon scattering. Dephasing of a resonantly driven QD system has been theoretically studied in detail with respect to effects of electron-phonon interaction on the dynamics of an optically driven system [10–12]. These studies anticipated excitation-induced dephasing (EID) for moderate Rabi frequencies. However, non-monotonic behavior was predicted for Rabi frequencies larger than a cut-off frequency defined by the material parameters and the QD size [13,14]. Non-monotonic behavior is also predicted for cavity structures with suitable microcavity coupling [15, 16]. Experimental evidence of EID effects has recently been observed as oscillation damping in pulsed photocurrent measurements on a resonantly driven QD . This damping was found to have a clear quadratic dependence on the effective Rabi frequency Ω. The effect of EID has also been observed under strictly resonant continuous wave excitation of a QD in a microcavity in terms of Mollow-triplet sideband broadening . These experiments reveal good agreement with a theoretical description based on a polaron master equation approach to multi-phonon and multi-photon effects in a cavity-QED system . In the work of Ulrich et al., the phenomenon of spectral Mollow sideband narrowing in dependence of laser-excitation detuning from the bare emitter resonance had to be left open for further in-depth theoretical analysis.
Motivated by these findings and with the aim of fundamental interpretation, the focus of the current work lies on a detailed study of the detuning-dependent dressed state emission of a single QD without cavity coupling. Our theory is based on a polaron master equation approach from which we develop a fully analytical description of the emission spectrum. Consequently, we are able to clearly distinguish between different regimes of spectral broadening or narrowing of the Mollow sidebands, under strong influence by pure dephasing and phonon-induced scattering. The comparison of detuning-dependent resonance fluorescence data reveals very good agreement with the theoretical model.
2. Sample structure and experimental procedure
The planar sample employed for the measurements in this work is grown by metal-organic vapor epitaxy (MOVPE). The self-assembled In(Ga)As QDs are embedded in a GaAs λ -cavity, sandwiched between 29 (4) periods of λ /4-thick AlAs/GaAs layers as the bottom (top) distributed Bragg reflectors (DBRs). For our experimental investigations, the sample is kept in a Helium flow cryostat providing highly stable temperature of T = 6.0 ± 0.5 K. Suppression of parasitic laser stray-light is achieved by use of an orthogonal geometry between QD excitation and emission detection. In addition, polarization suppression and spatial filtering via a pinhole is applied in the detection path. Resonant (tunable) QD excitation is achieved by a narrow-band (≈ 500 kHz) continuous-wave (cw) Ti:Sapphire ring laser. For high-resolution spectroscopy (HRPL) of micro-photoluminescence (μ-PL) we employ a scanning Fabry-Pérot interferometer with < 1μeV as described earlier [5, 8, 18].
3. Experimental results: Detuning-dependent resonance fluorescence
In our experiments we apply pump powers well above the saturation of the quantum emitters. In this high-field regime, the excitation-induced Rabi rotation of the two-level emitter system becomes much faster than the spontaneous decay rate γ. The incoherent spectrum of the resulting dressed state is the characteristic Mollow triplet . Under strictly resonant excitation, i.e. for a laser detuning Δ = ωL – ωx = 0 from the QD exciton resonance (see Fig. 1(a), green center trace), the spectrum is composed of the central Rayleigh line “R” at the bare emitter energy ω0 and two symmetric satellite peaks, i.e. the Three-Photon Line “T” and the Fluorescence Line “F ” at ω0 ± Ωr, respectively. Ωr denotes the effective Rabi frequency including renormalization effects from the phonon bath as discussed in the theory section below.
Laser detuning (Δ)-dependent Mollow triplet spectra taken at a constant excitation strength of P0 = 500 μW (Ωr ∝ (P0)1/2 = const.) are depicted in Fig. 1(a). According to theory (see, e.g., Ref. ), the laser-detuning between the driving field and the bare emitter resonance ω0 modifies the dressed emission. Besides the center transition at ω0 + Δ the two sideband frequencies become ω0 + Δ ± Ω, where denotes the generalized Rabi frequency at a given excitation strength. The extracted spectral positions for the red- and blue-shifted Mollow sidebands and the central Rayleigh line are depicted in Fig. 1(b) with a corresponding fit to the data. In addition, observations on the detuning-dependent Mollow triplet series depicted in Fig. 1(a) reveal distinct broadening of the Mollow sidebands with increasing Δ, accompanied by a change in the relative sideband intensities. In order to explain these observations, we develop a theoretical description in terms of a polaron master equation formalism and derive an analytical expression for the incoherent spectrum. The main theoretical findings used to interpret our experimental observations are described in the following section.
4.1. Hamiltonian, polaron master equation and phonon-induced scattering rates
We model the QD as an effective two-level system interacting with a coherent pump field and an acoustic phonon reservoir. In a frame rotating with respect to the laser pump frequency ωL, the model Hamiltonian (excluding QD zero-phonon line decay mechanisms) is9, 15, 20, 21] which is particularly well suited for studying quantum optical phenomena such as resonance fluorescence spectra. In the following we will closely follow (and extend where necessary) the theoretical formalisms described in Refs. [9, 15], except we can safely neglect cavity coupling terms.
Defining , the polaron transformed Hamiltonian, H′ → ePHe−P, consists of a system part, reservoir part, and an interaction part, respectively:22] , with 〈B〉 = 〈B+〉 = 〈B−〉, at a bath temperature T = 1/kbβ. For convenience, we will assume that the polaron shift is implicitly included in our definition of ωx below. The operators Xg and Xu are defined through Xg = h̄ηx(σ− + σ+) and Xu = ih̄ηx(σ+ − σ−).
We next present the time-local (or time-convolutionless) ME for the reduced density operator ρ(t) of the QD-bath system in the second-order Born approximation of the system-reservoir coupling. In the interaction picture, we consider the exciton-photon-phonon coupling H′I in the Born approximation and trace over the phonon degrees of freedom. The full polaron ME takes the following form [9, 15, 20, 21]:20, 22], with the phonon correlation function . The Lindblad operators [O] = 2OρO† − O†Oρ − ρO†O describe dissipation through zero-phonon line (ZPL), radiative decay (γ) and ZPL pure dephasing (γ′), where the latter process is known to increase as a function of temperature [23–27]. Without the coherent pump term and the residual ZPL broadenings, this ME formally recovers the independent boson model [22, 28, 29].
For continuous wave (cw) excitation, the integration appearing in Eq. (3) can have the upper time limit t → ∞, resulting in a Markovian ME where the scattering rates are computed as a function of H′S. Such an approach is valid since the acoustic phonon lifetimes are very fast, i.e., on a few ps timescale and much faster than the characteristic time scales of the system dynamics. As was shown elsewhere , for the pump strengths we consider in this work, one can neglect the pump-dependence of H′S appearing in the exponential phase terms above (which we will further justify below) to derive an effective phonon ME as follows:32]. As might be expected, the excitation-dependent rates depend upon the phonon correlation function, the coherent pump rate, and the laser-exciton detuning. The process corresponds to an enhanced radiative decay, while the process represents an incoherent excitation process . We stress that these mechanisms are quite different to simple pure dephasing models, which are frequently used to describe weak (i.e., perturbative) electron-phonon scattering . The phonon-mediated incoherent excitation process has recently been confirmed experimentally . Note that Ωr can be significantly smaller than Ω0 (for suitably large electron-phonon coupling), even at low temperatures. For example, using InAs QD parameters that closely represent our experimental samples  and a phonon bath temperature T ∼ 6 K, then 〈B〉 ≈ 0.75, and this value decreases (increases) with increasing (decreasing) temperature.
For a pump field strength of Ωr = 50 μeV, example phonon scattering rates are shown in Fig. 2. Within the zoomed region of laser detunings |Δ| < 100 μeV, the relevant phonon scattering rates can clearly be assumed to be constant. Therefore, these values will be treated as constant in the following to compute the analytical Mollow triplets—though this is not a model requirement).
4.2. Mollow triplet simulations: Full polaron versus effective phonon ME
To gain better insight into the underlying physics of a QD driven Mollow triplet with a finite detuning, it is desirable to derive an analytical form for the Mollow triplet spectrum which we can derive from the effective phonon ME. Thus we will first investigate how good the approximation is to replace the phonon scattering terms in the full polaron ME [Eq. (3)] by the ones appearing in the effective phonon ME [Eq. (4)]. In Fig. 3, a direct comparison between the numerically calculated Mollow triplet based on the full polaron and the effective phonon ME is shown, revealing excellent agreement even for large detunings, Δ = 30 μeV, and high field strengths of Ω0 = 50 μeV. The main reason that one can neglect the pump-dependence of the phase terms in Eq. (3) is that—for the pump values we consider—phonon correlation times are much faster than the inverse Rabi oscillation.
We highlight that a cw Rabi field of Ωr = 50 μeV is already close to the highest achievable experiments to date, and for our purposes can be considered the high-field regime. However, we note that the polaron approach, although nonperturbative, can break down if extremely high field strengths are used such that Ωr becomes comparable to (or greater than) the phonon cut-off frequency. In this case, other approaches exists such as a variational ME approach  and path integral techniques . Since our maximum Rabi field strengths are much less than ωb (the characteristic phonon cut off frequency), as shown by McCutcheon et al., the polaron ME should be rigorously valid for the field strengths that we model.
4.3. Optical Bloch equations and analytical fluorescence spectrum
One of our main theoretical goals is to derive a useful analytical expression that will allow one to fit the experimental Mollow triplets over a wide range of parameters, explicitly including the laser-exciton detuning. From the effective phonon ME [Eq. (4)] and 〈Ȯ〉 = tr[ρ̇O] , we obtain the following optical Bloch equations:37]: 30], as the emitted spectrum is detected via a weakly-coupled planar cavity mode, in which case 〈δa†(t)δa(t + τ)〉 ∝ 〈δσ+(t)δσ−(t + τ)〉; so we are actually obtaining the cavity emission which requires no change in the aforementioned correlation functions when coming out of the polaron frame . The coherent spectrum, if needed, can be derived in a similar way. By exploiting the quantum regression theorem and Eqs. (7a)–(7c), it is possible to derive the spectrum analytically, e.g., using Laplace transform techniques. We first define the steady-state expectation values f(0) ≡ 〈δσ+δσ−〉ss, g(0) ≡ 〈δσ+δσ+〉ss, and h(0) ≡ 〈δσ+δσz〉ss, and keep the explicit laser-exciton detuning dependence in the solution. Using the frequency detuning δω = ω −ωL, we can obtain the spectrum lineshape,
The corresponding steady-state inversion and polarization components are calculated to beEq. (4)]. The full-width at half-maximum (FWHM) of spectral resonances can be obtained from Eq. (9), though these are too complicated to write down analytically. However, as we have verified, one can simply fit the analytical spectrum to a sum of Lorentzian line shapes (see discussion of Fig. 4) and easily extract the broadening parameters. In the high-field limit, the on-resonance (Δ = 0) FWHM values are and γcenter ≈ γ0 + γph +γ′ + γcd for the sideband and center resonances, respectively; these show that the cross dephasing term acts to squeeze the sidebands while broadening the center line or vice versa.
4.4. Off-resonant Mollow triplet: Regimes of spectral sideband broadening and narrowing
From the analytical spectra above, we can discern when the Mollow sidebands will become asymmetric and whether the detuning dependence will exhibit broadening or narrowing of the sideband resonances. For convenience, we introduce the following ratio:Figs. 4(a) and 4(b) we plot the Mollow triplet as a function of Δ, and extract the FWHM of the sidebands for three values of r. As can be seen, r < 1 (for a suitably small γ′) leads to spectral sideband narrowing, whereas for r > 1 the effect of spectral sideband broadening occurs. Interestingly, the reverse trend occurs for the center resonance (not shown), namely when the sidebands broaden (narrow) then the center line narrows (broadens); so depending on the r value, one can observe squeezing or anti-squeezing of the spectral resonances with increasing Δ (in addition to the squeezing that already occurs from a finite γcd).
5. Comparison between experiment and theory
In the following, we show a detailed modeling of experimentally derived results from detuning-dependent high-resolution PL measurements using the above presented theory. The measurements are performed on a QD in a planar sample structure with negligible cavity coupling, though the planar cavity acts to separate out the QD emission from the pump field.
To reproduce the experimental results with the theoretical model we derive the characteristic values for the calculation via independent measurements or analysis. In the experiments, the sample temperature has been measured to be T = 6.0 ± 0.5 K. The QDs in the planar sample are found to have rather similar radiative lifetimes due to no Purcell-like enhancement. The radiative decay rate γ is extracted from time-correlated photon counting measurements that reveal a typical radiative lifetime of (750–850 ps), yielding γ ≈ (0.77 − 0.88)μeV. The Rabi field Ωr = 22.7μeV is derived from the Mollow center-to-sideband splitting at zero laser-detuning Δ = 0. The cut-off frequency is set to ωb = 1 meV.
To carefully identify the pure dephasing rate γ′, and electron-phonon coupling strength αp, the spectra of a power-dependent Mollow triplet series of the QD under investigation at Δ = 0 have been modeled with γ′ and αp as free parameters. The extracted FWHM can be well reproduced with a pure dephasing rate of γ′ = 4.08 γ = 3.43μeV (equivalent to a pure dephasing time of 192 ps) and αp/(2π)2 = 0.15 ± 0.01 ps2. A direct comparison between the extracted FWHM of the experimental data and the theoretical model is shown in Fig. 5: The expected linear increase [slope: 9.3 × 10−4 (μeV)−1] in the FWHM with shows very good agreement with the experiment [slope: 9.8 × 10−4 (μeV)−1].
The deformation potential constant is somewhat higher compared to the value used in Refs. [38, 39]. However, the value for αp and the dimensionless Huang-Rhys parameter (with cl the speed of sound and le/h the electron/hole confinement length) reported in the literature (i.e. SHR = 0.01 − 0.5) covers a large range and there are no well-accepted numbers to date. Additionally, SHR has been shown to be enhanced in zero-dimensional QDs compared to bulk material, for which different explanations are proposed, e.g., in terms of non-adiabatic effects or the influence of defects .
The sum of the main phonon scattering rates μeV, which is constant in the detuning range accessible in our measurements, is determined by the derived value for αp and and has been calculated according to Fig. 2. The cross-dephasing term has been extracted from the same graph as γcd = 0.13 μeV.
With all parameters at hand, the experimentally measured detuning-dependent Mollow triplet series is compared with the theoretical expectations in terms of sideband broadening and the change in the relative sideband areas Ared/blue = Ired/blue/(Ired + Iblue). Figure 6(a) shows a direct comparison of the Mollow triplet spectra for increasing negative detuning Δ < 0, from which the FWHM and relative intensities are extracted. The discrepancy between the expected and measured central Mollow line intensity results from contributions of scattered laser stray-light to the true QD emission that can experimentally not be differentiated due to the equal emission frequency. For the detuning Δ ≠ 0 the spectral resolution of the high-resolution spectroscopy is not sufficient to distinguish between laser-excitation and QD Rayleigh line emission. The gray shaded peaks in Fig. 6(a) (lower panel) belong to a higher order interference of the Fabry Pérot interferometer. The extracted FWHM values are depicted in Fig. 6(b). For the system under investigation, r is calculated to be around 2.01, and therefore an increase in the sidebands’ width is expected according to the theoretical model. Indeed, we observe a systematic increase with increasing negative detuning Δ < 0. Moreover, we observe spectral narrowing (squeezing) of the center line though we do not attempt to fit this resonance as it has a large contribution from coherent scattering. Additionally, the relative sideband areas Ared/blue in dependence on Δ are plotted in Fig. 6(c). As becomes already visible from the Mollow spectra, for positive detunings the blue sideband gains intensity whereas the red sideband area decreases, and vice versa. The crossing between relative intensities is expected to occur at Δ = 0. Interestingly, we observe crossings at moderate negative laser-detuning values for all different QDs under study. A detailed interpretation of the physics behind this effect has to be left for further on-going analysis and may involve the inclusion of more excitons. The high value of pure dephasing which causes r > 1 in our sample comes from the fact that the samples were manufactured using metal organic chemical vapor depositions, which are supposed to incorporate more impurities compared to sample grown by molecular beam epitaxy (MBE) . The higher magnitude of pure dephasing results in broadening of Mollow sidebands.
The regime of distinct sideband narrowing has also been experimentally observed by Ulrich et al. (see Fig. 3(d) of their paper) on a QD embedded in a micropillar cavity structure, grown by MBE. Even though the effect can be qualitatively understood from the theoretical model discussed above, a direct comparison to theory requires the inclusion of QED-cavity coupling into the polaron ME approach. Previous numerical studies (using a polaron ME approach with cavity coupling) were performed in this regime , but did not obtain this behaviour, suggesting that further work is likely required to explain the significant narrowing effects that are observed in the experiment.
In conclusion, we have presented a combined theoretical-experimental study on the impact of pure dephasing and phonon-induced scattering on the excitation detuning-dependence of Mollow triplet sidebands. Based on a polaron ME approach, supplemented by a useful analytical solution for the Mollow triplet spectrum, it is possible to distinguish different regimes of spectral broadening or narrowing, which depend on the ratio of different phonon-mediated scattering rates. Regimes of spectral squeezing and anti-squeezing have also been identified. For the case of experimentally observed distinct sideband broadening, we have found excellent agreement with the predictions of theory.
During the final preparation of this work we became aware of similar results obtained independently for off-resonant Mollow sideband narrowing . We would like to thank Dara McCutcheon and Ahsan Nazir for bringing these to our attention and for useful discussions. S. Weiler acknowledges financial support by the Carl-Zeiss-Stiftung. S. Hughes thanks Rong-Chun Ge for useful comments and the National Sciences and Engineering Research Council of Canada for research funding. We thank the German Research Foundation (DFG) for financial support via grant DFG MI 500/23-1. This work was also supported by the DFG within the funding program Open Access Publishing.
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