Ultrafast polariton-phonon dynamics of strongly coupled quantum dot-nanocavity systems

We investigate the influence of exciton-phonon coupling on the dynamics of a strongly coupled quantum dot-photonic crystal cavity system and explore the effects of this interaction on different schemes for non-classical light generation. By performing time-resolved measurements, we map out the detuning-dependent polariton lifetime and extract the spectrum of the polariton-to-phonon coupling with unprecedented precision. Photon-blockade experiments for different pulse-length and detuning conditions (supported by quantum optical simulations) reveal that achieving high-fidelity photon blockade requires an intricate understanding of the phonons' influence on the system dynamics. Finally, we achieve direct coherent control of the polariton states of a strongly coupled system and demonstrate that their efficient coupling to phonons can be exploited for novel concepts in high-fidelity single photon generation.

The strong coupling between a single photon and a single quantum emitter is of substantial interest for both investigations of the fundamentals of quantum optics as well as potential applications in optical computing, quantum metrology, and quantum cryptography [1][2][3]. This universality is reflected by the diversity of associated experimental realizations, ranging from those in atomic physics [4][5][6] to superconducting systems [7,8] and semiconductor devices [9]. In the solid state, self-assembled quantum dots (QDs) are the most investigated quantum emitters due to their strong interaction with light as well as their nearly transform-limited linewidth [10][11][12]. Both micropillar structures [9] and photonic crystal cavities [13] are optical resonators widely used to enhance this light-matter interaction. Photonic crystal cavities are especially promising for on-chip integration of quantum optical circuits due to the convenient fabrication of integrated waveguide and detector structures [14]. In contrast to other systems, semiconductor quantum emitters are usually embedded in a crystalline host matrix, resulting in strong interactions with phonons. For QDcavity systems the coupling to phonons and its impact on applications has been extensively studied in the weakly coupled regime [15,16]. However, for strongly coupled systems these effects remain largely unexplored experimentally.
In this paper, we investigate the interaction of a strongly coupled QD-cavity system with phonons and explore the impact of this coupling on different schemes for non-classical light generation. By performing temporally resolved spectroscopy experiments with resonant excitation, and with the QD and cavity both on-and nearresonance, we map out the detuning-dependent polariton lifetime and extract the spectrum of the polaritonto-phonon coupling. Photon-blockade experiments under different pulse length and detuning conditions reveal that the pulse length has to be optimized for a given detuning in order to obtain high-fidelity photon blockade. We also present quantum-optical simulations that are in very good agreement with the experiments. In particular, the impact of coupling to phonons is found to be more pronounced for the case of detuned photon blockade, which is the condition that yields the highest fidelity and efficiency [17]. Finally, we present coherent control of the polariton populations of the strongly coupled system and demonstrate that the efficient coupling to phonons can be utilized for novel concepts in high-fidelity single photon generation. To this end, we present measurements of the degree of second-order coherence yielding values as low as g (2) (0) = 0.03 ± 0.01.
The sample investigated consists of a layer of low density InAs QDs grown by molecular beam epitaxy and embedded in a photonic crystal L3 cavity [18]. The energy structure of a QD strongly coupled to an optical cavity is usually described by the Jaynes-Cummings (JC) Hamiltonian where ω a denotes the frequency of the cavity, a the cavity mode operator, σ the lowering operator of the quantum emitter, ∆ the detuning between quantum emitter and cavity, and g the emitter-cavity field coupling strength.
After introducing dissipation into the JC system we obtain a quantum Liouville equation with complex eigenenergies [19]: where E n ± corresponds to the nth rung of the system and κ and γ to the cavity and QD energy decay rates, respectively. The real parts of E n ± yield the energies of the states whereas the imaginary parts yield their linewidths. Note that all four parameters κ, γ, ∆ and g contribute to the polariton splittings and linewidths. The resulting eigenenergies, the Jaynes-Cummingsladder dressed states, are presented in figure 1a. They consist of a series of anticrossing branches that are labelled UPn (LPn) for the upper (lower) polariton where n is the index of the rung. For n photons in the cavity the energy is nω a (dashed red lines), and the energy of the quantum emitter (dotted green line) varies with a detuning parameter. Due to the coupling, the resulting energy eigenstates are the anticrossing polariton branches. At resonance, the splitting is given by 2g √ n. Experimentally, we can observe the splitting of the lowest-manifold polaritons in cross-polarized reflectivity [20] through control of the crystal lattice temperature. The result of such an experiment is presented in figure 1b and shows a clear anticrossing. A fit to the data reveals values of g = 12.3·2π GHz and κ = 18.4·2π GHz. The radiative lifetime of a QD in a bulk photonic environment is known to be ∼ 1 ns and thus, for QDs in photonic crystal cavities γ is much smaller than κ. Due to the photonic bandgap γ is further suppressed and reported values for the continuum mode lifetimes are around 4−12 ns [21,22] corresponding to γ ∼ 0.01 · 2π GHz.
Using the values above and equation 2, we plot the decay rates and radiative lifetimes of the polariton branches of an ideal Jaynes-Cummings system as a function of the detuning in figure 2a and 2b, respectively. In these plots, the red (blue) curve corresponds to UP1 (LP1), which is more cavity-like (QD-like) for negative and QD-like (cavity-like) for positive detunings. The decay rates and thus radiative lifetimes exchange over a range that is determined by g and κ.
In contrast to ideal JC-systems, however, for solidstate systems a phonon-assisted population transfer between QD excitons and cavity photons is known to exist [15,[23][24][25]. Thus, we sought to determine the true state lifetimes through time-resolved measurements. The fast cavity decay rate and strong interaction with phonons are expected to result in significantly shortened lifetimes. Hence, we performed the measurements using pulsed resonant excitation and time-resolved detection on a streak camera with sub-5 ps resolution. In this way, the resonant excitation suppresses the effect of slow carrier relaxation into the QDs that would influence non-resonant excitation [26]. To avoid exciting higher rungs of the JC ladder, we pumped the more QD-like polariton branch UP1 (as schematically illustrated by the blue arrow on the right side of figure 1a) with a pulse length that was carefully chosen to be shorter than the state lifetime but spectrally sharp enough not to overlap with the other polariton branch. Luminescence was then observed either from this state directly (red arrow in figure 1a) or from the more cavity-like polariton branch after a phononassisted transfer (purple and red arrows in figure 1a). The dynamics of this process can be simulated using a rate equation model with four rates: radiative recombination from UP1 (Γ r U P 1 ), radiative recombination from LP1 (Γ r LP 1 ), the phonon-assisted transfer rate from UP1 to LP1 (Γ nr f ), and vice versa (Γ nr r ) (for details see supplemental material). As shown in figure 2c, the spectrally integrated photoluminescence intensity of a typical time-resolved measurement at a QD-cavity detuning of ∆ = 2.8 g (including contributions from both decay paths) shows a mono-exponential decay, as expected from the rate equation model (see supplemental material). A fit to the data (red line in figure 2c) reveals a decay time of 34.9 ± 2 ps.
To map out the polariton lifetimes we repeated the time-resolved measurement presented in figure 2c for different detunings in the range ∆ = 0 − 11 g. As shown in figure 2d, the resulting lifetime of UP1 increases with increasing detuning. However, the increase occurs much more slowly than indicated by the rates calculated from the ideal JC model presented in figure 2b. For the detunings investigated here, LP1 is more cavity-like and, thus, has a much shorter radiative lifetime, ranging from 16 ps down to 8 ps as the detuning is increased. Meanwhile, the ideal JC lifetime for UP1 varies from 16 ps up to almost 1000 ps, while the measured lifetime of UP1 only increases up to roughly 80 ps. Therefore, a strong phonon-assisted population transfer from UP1 to LP1 significantly shortens the observed lifetime of UP1. We model this phonon-assisted population transfer using an effective master equation derived in a polaron frame with respect to the phonon interaction [25]. A fit to the data including this model is presented as a red line in figure 2d and produces very good overall agreement with all of the measured data (see supplemental material for details). Since the detuning between cavity and QD was controlled by the lattice temperature, the relative temperature of each detuning was taken into account for these simulations. From the fit we can extract the phonon-assisted population transfer rates Γ nr f and Γ nr r . They are presented in figure 2e as red and blue lines, respectively.
Near resonance the two rates are very close due to the elevated temperature of T = 31 K. With increasing detuning both rates decrease, but Γ nr r decreases faster than Γ nr f due to spontaneous dephasing events. In order to further corroborate this model of phononassisted population transfer we performed measurements with simultaneous spectral and temporal resolution. The result of a typical measurement recorded at ∆ = 7 g is presented in figure 2f. Two distinct luminescence peaks are visible. Most importantly the onset of luminescence from LP1 is delayed with respect to the onset of luminescence from UP1, as population must be transferred before photons can be emitted. At this detuning, the radiative recombination rate Γ r U P 1 is small compared to the strong phonon-assisted population transfer and subsequent radiative recombination of LP1 (Γ r LP 1 ). Hence, the strongest luminescence intensity is observed from LP1 even though UP1 is resonantly excited (for details see supplemental material).
Having mapped out the detuning-dependent polariton lifetimes of the strongly coupled QD-cavity system we can now apply this knowledge to applications where the lifetime is critical. One of the most remarkable applications of strongly coupled systems is photon blockade [1,27], in which a laser pulse is tuned to the first rung of the JCladder and (due to the anharmonicity) is not resonant with higher transitions up the ladder. Therefore, the probability of coupling a single photon through the system is enhanced over that of multi-photon states, which results in a transmitted light beam that has a strong non-classical character. The fidelity of this process for the generation of single photons is inherently limited due to the linewidth of the polariton branches. Nevertheless, it was recently shown that in a strongly coupled system that is detuned by a few g, not only the purity but also the efficiency of single photon generation increases [17]. As we will see, photon blockade depends crucially on the polariton lifetimes, especially in the detuned case.
To quantify the non-classical character of the photocount distribution, we use its measured degree of secondorder coherence for zero time delay [28,29] g (2) computed from expectations of the measured photocount distribution, where n signifies a number of detections.
Only non-classical light sources may have a second-order coherence g (2) (0) < 1, and g (2) (0) = 0 is measured exclusively for pulses with single-photon-like character. Owing to a cavity lifetime much shorter than the timing jitter of the single photon counters, measurements of g (2) (0) on our system can only be performed under pulsed excitation; this experimental configuration is anyways required for on-demand applications. Hence, the choice of pulse length forces a compromise between frequency resolution (reducing the overlap of different rungs) and the likelihood of re-excitation of the system. In other words, if the laser pulses are too long the system will be re-excited during the interaction with a single pulse, reducing the non-classical character of the transmitted light. On the other hand, pulses with a shorter duration are spectrally broader, resulting in a larger overlap with higher rungs. With increasing detuning between QD and cavity resonances, the lifetime of the emitter-like (cavity-like) polariton branch increases (decreases). Therefore, in order to obtain the strongest photon blockade the pulse length has to be chosen according to the detuning-dependent polariton lifetime.
To test this hypothesis we performed measurements of g (2) (0) using a Hanbury Brown and Twiss (HBT) type experiment [30] for different QD-cavity detunings (and thus polariton lifetimes) and pulse lengths. Figure 3a shows g (2) (0) as a function of the laser detuning for pulse lengths [31] of 17 ps on the left and 55 ps on the right, and detunings of ∆ = 0 g, ∆ = 2 g and ∆ = 4 g from top to bottom. At ∆ = 0 both pulse lengths result in a symmetric curve with photon-blockade dips of g (2) (0) < 1 for laser detunings of 1 − 2 g. We measure a minimum g (2) (0) = 0.88 ± 0.03 for 17 ps pulses and g (2) (0) = 0.93 ± 0.03 for 55 ps pulses. Due to the short polariton lifetime of 16 ps at ∆ = 0 the longer pulses result in an increased likelihood of re-excitation during the presence of a single pulse, which leads to a higher value of g (2) (0) in photon blockade. We note that since the radiative recombination rates are very fast when the QD and the cavity are in resonance, phonons have only a minor effect in this case (see supplemental material for simulations and details).
With increasing detuning the fidelity of photon blockade increases as expected [17], and for ∆ = 4 g (bottom plots of figure 3a) we measure values as low as g (2) (0) = 0.44 ± 0.07 for 17 ps pulses and g (2) (0) = 0.34 ± 0.07 for 55 ps pulses. In particular, due to the longer polariton lifetime at ∆ = 4 g, we observe a lower value of g (2) (0) for the longer pulses. Here, re-excitation limits the minimum of the second-order coherence for the long pulses, while spectral overlap with transitions to higher rungs limits the minimum for the short pulses.
To explain this finding we performed quantum optical simulations using the Quantum Optics Toolbox in Python (QuTiP) [32] based on the Quantum Regression Theorem [17]. The results of these simulations are presented in figure 3b-c. The figures show g (2) (0) as a function of the laser detuning and pulse length for ∆ = 4 g, both including (figure 3b) and excluding (figure 3c) phonon-assisted population transfer. In both cases, with increasing pulse length the photon-blockade dip narrows down (as seen in the experiment) due to the improved spectral resolution of longer pulses. Moreover, the minimum value of g (2) (0) decreases due to the decreased overlap with higher rungs (as well as the other polaritonic branch) before increasing due to re-excitation for  ( 2 ) ( 0 ) ∆= 0 g 1 7 p s ( a ) g ( 2 ) ( 0 ) g ( 2 ) ( 0 ) L a s e r d e t u n i n g ( g ) 5  overall agreement with the measurements presented in figure 3a, and they highlight the importance of taking phonon-assisted transfer into account when selecting the optimal pulse length.
While we have seen above that for detuned photon blockade the presence of exciton-phonon coupling slightly decreases the fidelity, this coupling also allows us to investigate more sophisticated schemes for on-demand singlephoton generation. In particular, the population of UP1 can be coherently controlled if the pulse length is chosen shorter than the state lifetime but spectrally narrow enough to avoid exciting higher transitions up the JC ladder. Due to coherent scattering of the laser and imperfect suppression of the laser reflection from the sample surface, it is difficult to observe Rabi oscillations directly. However, phonon-assisted emission from LP1 following excitation of UP1 occurs at a different frequency. In addition, for detunings in the range of ∆ = 5−10 g, Γ r U P 1 is strongly Purcell-suppressed while Γ r LP 1 is very fast. Most strikingly, Γ nr f is proportional to g 2 [25] and, thus, very efficient for a strongly coupled system. Therefore, even when resonantly exciting UP1 for these detunings most luminescence is emitted from LP1. The result of such an experiment is presented in figure 4a, which shows the emission intensity from LP1 for a resonant excitation of UP1 with 16 ps pulses at a detuning of ∆ = 4.8 g. Clear Rabi oscillations are observed and fitted (red line in figure 4a) using a damped oscillation superimposed with a very weak background that is linear with the excitation power, due to a finite leakage of the excitation laser into the detection channel.
To investigate single photon generation in this configuration we measure g (2) (0) while exciting UP1 with a π pulse and detecting LP1 emission. The result of this experiment is presented in figure 4b and shows almost perfect antibunching. By integrating the area of the peaks we obtain a value of g (2) (0) = 0.04 ± 0.01. Fitting the data with a series of Gaussian peaks is a little less sensitive to noise from the dark counts of the de-tectors and yields g (2) (0) = 0.03 ± 0.01. These findings clearly demonstrate the potential of exploiting the efficient exciton-phonon coupling of strongly coupled QDcavity systems for single photon generation.
In summary, we mapped out the detuning-dependent polariton lifetime of a strongly coupled QD-cavity system and extracted the spectrum of the polariton-to-phonon coupling for QD-cavity detunings up to 11 g. We demonstrated that in order to obtain high-fidelity photon blockade the pulse length has to be chosen depending on the QD-cavity detuning. Additionally, we showed that the detuned photon blockade, which has a higher fidelity and efficiency than resonant photon blockade, is also affected more strongly by phonons. Nevertheless, high-fidelity photon blockade can still be achieved through selection of a proper pulse length. Finally, we demonstrated coherent control of polariton states and showed that for strongly coupled systems the efficient coupling to phonons can be exploited for high-fidelity single photon generation. We expect that our contribution towards understanding strongly coupled systems in the solid state will play a vital role in utilizing QD-cavity platforms for novel physics such as higher-order non-classical light generation [3].

Sample fabrication
The MBE grown structure consists of a ∼ 900 nm thick Al 0.8 Ga 0.2 As sacrificial layer followed by a 145 nm thick GaAs layer that contains a single layer of InAs QDs. Our growth conditions result in a typical QD density of 60 − 80 µm −2 . The photonic crystals were fabricated using 100 keV e-beam lithography with ZEP resist, followed by reactive ion etching and HF removal of the sacrificial layer. The photonic crystal lattice constant was a = 246 nm and the hole radius r ≈ 60 nm. The cavity fabricated is a linear three-hole defect (L3) cavity. To improve the cavity quality factor, holes adjacent to the cavity were shifted [33,34].
Optical spectroscopy All optical measurements were performed with a liquid helium flow cryostat at temperatures in the range 10 − 40 K. For excitation and detection a microscope objective with a numeric aperture of N A = 0.75 was used. Cross-polarised measurements were performed using a polarising beam splitter. To further enhance the extinction ratio, additional thin film linear polarisers were placed in the excitation/detection pathways and a single mode fibre was used to spatially filter the detection signal. Furthermore, two waveplates were placed between the beamsplitter and microscope objective: a half-wave plate to rotate the polarisation relative to the cavity and a quarter-wave plate to correct for birefringence of the optics and sample itself.

Autocorrelation measurements
Second-order autocorrelation measurements were performed using a Hanbury Brown and Twiss (HBT) setup consisting of one fibre beamsplitter and two single photon avalanche diodes. The detected photons were correlated with a PicoHarp300 time counting module.

RATE EQUATION MODEL FOR DESCRIBING THE SYSTEM DYNAMICS
As discussed in the main manuscript we use a rate equation model with two states and four rates to describe the system dynamics: radiative recombination from UP1 (Γ r U P 1 ), radiative recombination from LP1 (Γ r LP 1 ), the phonon-assisted transfer rate from the UP1 to LP1 (Γ nr f ), and vice versa (Γ nr r ). The resulting rate equation model then reads: where P U P 1 (t) and P LP 1 (t) describe the populations of the states UP1 and LP1, respectively. From solving these equations we obtain the time dependence of the state populations.
An example is presented in figure 5a that shows the populations of UP1 and LP1 at a detuning of ∆ = 3g in red and blue for excitation of UP1 and using the values for the four rates as obtained in the main manuscript for this detuning. After a short time the populations quasithermalize and decay with an almost perfect exponential shape. The resulting intensity of emission from these two states is presented in figure 5b using the same colour cod-ing as figure 5a and shows the same behaviour. The spectrally integrated emission intensity is presented in figure  5c. Importantly, after the fast quasi-thermalization the emission intensities from the two states (figure 5b) decay with the same time constant. Therefore, the spectrally integrated intensity (figure 5c) decays with the same time constant. This allows extraction of the lifetime of UP1 from temporally analysing the spectrally integrated luminescence intensity. In particular, the streak camera that was used to record the temporally resolved spectra has a lower spectral resolution than the spectrometer and CCD that was used to record the time integrated spectra as presented in figure 1 of the main manuscript. Therefore, for detunings smaller than ∆ ≈ 5 g analysing the spectrally integrated intensity is the only option as the two time traces cannot be spectrally resolved. We note here that these finding hold for all investigated sets of parameters.

PHONON-MODEL
To model the polariton-phonon coupling we used a model that utilizes an effective master equation derived in a polaron frame with respect to the phonon interaction [25]. In this model a characteristic phonon spectral function is used which is given by and describes the interaction between electrons and the logitudinal acoustic phonons via deformation potential coupling. This function uses two parameters α p and ω b . The latter one corresponds to a high frequency cutoff due to the electronic localization length. The fit of the detuning dependent polariton lifetime presented in figure 2 of the main manuscript resulted in the values α p = 3ps 2 and ω b = 0.22 meV. However, we note that these values vary quite a lot throughout literature. This can be understood from the fact that J(ω) is derived for a spherical quantum dot with similar extents for electron and hole wavefunctions. Moreover, experiments typically cover only a limited frequency range and therefore can be fitted with a range of parameters. We also note here that due to the elevated phonon occupancies at our experimental bath temperatures, there is very little asymmetry in the forward and reverse rates at zero detuning. Therefore, the use of a model that is first order is the phonon-assisted cavity-exciton incoherent transfer process is justified.

SIMULATED PHOTON-BLOCKADE WITH QD AND CAVITY IN RESONANCE
In the main manuscript we presented simulations of g 2 (0) as a function of the pulse length and laser detuning for ∆ = 4 g with and without phonons. As mentioned there, we also performed similar simulations for ∆ = 0 g. The result of these simulations is presented in figure 6a with phonons and figure 6b without phonons. In both cases the best photon-blockade is observed for relatively short pulses around 20 ps which can be understood from the short polariton lifetime at resonance which results in strong re-excitation of the system for longer pulses. Moreover, compared to the situation at ∆ = 4 g (main manuscript figure 3b and 3c) the difference in the optimum pulse length as well as observed value of g 2 (0) for the cases with and without phonons is much smaller. This results from the fact that the radiative decay rate of the polariton branches at resonance is much faster than the phonon mediated population transfer. Thus, the effect of phonons is much less pronounced. Regardless, since photon-blockade has a much higher efficiency at a detuning of a few g compared to the resonant case, the impact of phonons on the polariton lifetime has to be considered carefully in order to optimize the experimental conditions for single photon generation by photon blockade.