Exciton-polariton dynamics of the single site-controlled quantum dot-nanocavity in the coexisting strong-weak coupling regime

Deterministic positioning single site-controlled high symmetric InGaAs quantum dots (QDs) in (111)B-oriented GaAs photonic crystal cavities with nanometer-scale accuracy provides an idea component for building integrated quantum photonic circuits. However, it has been a long-standing challenge of improving cavity Q-factors in such systems. Here, by optimizing the trade-off between the cavity loss and QD spectral quality, we demonstrate our site-controlled QD-nanocavity system operating in the intermediate coupling regime mediated by phonon scattering, with the dynamic coexistence of strong and weak coupling. The cavity-exciton detuning-dependent micro-photoluminescence spectrum reveals concurrence of a trend of exciton-polariton mode avoided crossing, as a signature of Rabi doublet of the strongly coupled system. Meanwhile, a trend of keeping constant or slight blue shift of coupled exciton–cavity mode(CM) energy across zero-detuning is ascribed to the formation of collective states mediated by phonon-assisted coupling, and their rare partial out-of-synchronization linewidth-narrowing is linked to their coexisting strong-weak coupling regime. We further reveal the pump power-dependent anti-bunching photon statistical dynamics of this coexisting strong-weak coupled system and the optical features of strongly confined exciton-polaritons, and dark-exciton-like states. These observations demonstrate the potential capabilities of site-controlled QD-cavity systems as deterministic quantum nodes for on-chip quantum information processing and provide guidelines for future device optimization for achieving the strong coupling regime.

Moreover, recent study on the quantum network protocol based on deterministic cluster state generation [22,[25][26][27] and solid-state quantum memory enabled single-photon switch [9] received considerable interests.In particular, cavity-enhanced biexciton-dark-exciton cascaded system in GaAs-based QD [28,29] has always served as a promising platform to realize such protocol.
Crucially, pure dephasing of two-level system, defined as any disruption of quantum state without causing population relaxation but introducing random phase evolution, plays a fundamental role in the error-tolerant network [30] and frequency-stabilized on-chip scalable indistinguishable single photon emitter [31,32].In the GaAs-based QDs, the main pure dephasing mechanisms, thereby their spectral broadening, are related to rapidly fluctuating electrical charges [33] and carrier-phonon interactions [13,34,35].In the weak coupling regime, pure dephasing mediated Purcell effect assists a large fraction of the QD emission being channeled into the cavity mode, leading to an efficient off-resonance cavity feeding up to a few meV detuning [33,36,37] At the same time, finite coupling between QD exciton and cavity can also manifest the emission of cavity photon at the QD exciton energy such as in a dissipative cavity where cavity loss photon at a rate much faster than QD [36,38].Alternatively, in the strong coupling regime, the vacuum Rabi splitting and polariton state is modified by the damped coherent Rabi oscillations induced by pure dephasing [39].Apart from the clear boundary between strong and weak coupling, recent studies indicate the possibility of a unique intermediate coupling regime, which shares the coexisting features of both strong and weak coupling in the self-assembled GaAs-based QDs in micropillar [40,41] and photonic crystal (PhC) cavities [42].Specifically, recent theoretical studies suggests phonon-assisted QD-cavity interaction play an important role in the intermediate coupling regime [43,44].It suggests the importance of understanding pure dephasing in a cavity quantum electrodynamical (cQED) system for both fundamental science and practical quantum applications.
Particularly, the intrinsic pure dephasing of two-level system in QDs and its interaction with cavity remain to be difficult to access.It is because most of the current studies rely on the selfassembled Stranski-Krastanov (SK) QDs, which induces spurious cavity feedings related to wetting layer [45].We note that the hybridization of localized SK QD states with delocalized wetting layer states results in a broadband quasi-continuum of states that give rise to a background emission [46,47].In addition, the random SK QDs nucleation leads to unfavorable presence of parasitic QDs in the vicinity and brings ambiguity of the exact location of the presumed single QD inside the cavity, which results from misalignment of QD and cavity electric field, preventing clear identification of pure dephasing effects.Alternatively, without introducing wetting layer, nanoscale site-controlled high symmetric pyramidal QD grown on GaAs substrates patterned with inverted pyramidal recesses ensures deterministic single QD-cavity overlapping and strong suppression of the interaction between confined excitons with external environments [48,49], an ideal platform for uncovering the complex cQED in concurrent strong-weak intermediate coupling regime.In this study, based on our unique singular site-controlled pyramidal InGaAs/GaAs QDphotonic crystal (PhC) cavities hybrid structure, we systematically investigate the intermediate coupling regime and related carrier recombination dynamics, using multiple spectroscopic modalities including high-resolution polarized micro-photoluminescence (µPL) time-resolved PL, and second-order photon correlation measurements.Moreover, we reveal insights into the optical features of exciton-photon complexes strongly confined in our site-controlled QD and the bunching/anti-bunching and collective state photon statistics of coupled exciton-cavity system, towards cavity-enhanced multi-partite quantum information processing and cluster state quantum network implementations.
Our singular site-controlled QD-cavity system is studied via high-resolution microphotoluminescence (µPL).The sample is mounted on the cold finger of a liquid helium flow cryostat, positioned accurately with piezoelectric actuators in xy-direction.The sample is excited by either a continuous wave (CW) low-noise diode-pumped solid-state (DPSS) laser at 532 nm, or a 900 nm pulsed diode laser with ≈ 100 ps pulse duration.A 100× microscope objective with numerical aperture of 0.7 allows an ≈ 1 µm diffraction-limited pump beam spot size on the sample.
The µPL signals are collected and collimated by the same objective and refocused onto the entrance slit of a 1-meter spectrometer with 1200 grooves/cm, enabling a high spectral resolution of 8 pm (10 µeV).At one port of the spectrometer dual exit, a liquid nitrogen-cooled chargecoupled device (CCD) acquires the PL spectrum.The other output is attached to a free-space Hanbury-Brown and Twiss (HBT) setup comprising of a 50:50 non-polarizing beamsplitter and two aligned Si avalanche single-photon detectors (APDs) with 25-Hz dark counts.Time-correlated single-photon counting (TCSPC) is implemented for the second-order correlation (g 2 ) and timeresolved PL (TRPL) measurements.Polarization-resolved µPL is performed by implementing a half-wave plate and a linear polarizer in front of spectrometer entrance.The linear polarizer is fixed at the polarization orientation in which the grating spectrometer has the maximum reflectivity.The measurement setup schematic, description of sample structure, and modeling of PhC cavity are detailed in Supplementary Section I.
Pure dephasing and exciton-photon complexes in the singular site-controlled InGaAs QD-

PhC cavity
Figure 1a presents the resulting sharp excitonic emissions from a single pyramidal InGaAs QD and the L3-PhC cavity mode (CM) at 25 K with X-CM detuning ≈ 2 meV, excited non-resonantly by the 532 nm laser.As labeled in Figure 1a, the s-state excitonic emission [50,51] consists of a complex of negatively-charged exciton (X -), neutral exciton (X), biexciton (BX), and likely an excited light hole (LH) state.The relative binding energies [52] of the BX and X -with respect to X is -1.06 meV and 3.41 meV respectively.The pronounced X -population even at low excitation of 10 nW is mainly due to background-doping donors, incorporated during growth.Moreover, as a contribution factor, the significant asymmetric mobility between electron and hole further favors the formation of X -, rather than the positively charged exciton (X + ) [53].At the low excitation regime (<150 nW), the full-width half-maximum (FWHM) linewidths of the X -, X, BX and LH transitions are nearly constant with  − = 173 ± 9 µeV,   = 143 ± 10 µeV,   = 117 ± 42 µeV, and   = 202 ± 5 µeV respectively, extracted by Lorentzian fitting.It is apparent that the linewidth broadening is larger than the Fourier transform limits of the typical QD nanosecond lifetime (≈ 7 µeV), suggesting the contribution of pure dephasing via fluctuating environmental charges.
Though phonon scattering can contribute to the broadening, it is not expected to be dominant below 50 K [54].The larger linewidth observed at low excitation for X -compared to X can be understood by its additional charge carriers which has increased Coulomb interactions with fluctuating environmental electric fields.We note that  − monolithically decreases by ≈ 30 µeV as excitation increases from 200 to 700 nW, which is due to the saturation of the local charge states.Meanwhile, the significantly larger   is caused by a more delocalized LH state, experiencing stronger surrounding charge fluctuations.Apart from the excitonic emission, the CM line has a FWHM of  CM of 296 ± 12 µeV (Q factor ≈ 4,230), which is in the dissipative cavity regime of  CM ≫   , where   and   are the cavity loss and QD radiative decay rates respectively.
It is worth pointing out that the absence of a 2D wetting layer in our single pyramidal QD growth rules out far-off-resonance cavity feeding by a spurious emission background.Furthermore, though the pyramidal QD nucleates in the vicinity of three 1-dimensional ridge quantum wires at the wedges of the pyramid, the quantum wire influence is shown to be negligible for low QD excitation powers, owing to the large energy difference and absence of hybridization between the localized QD states and 1D delocalized state and the lower mobility of charges in the disordered 1D barrier.To further verify the above assignment of each sharp transition line, Figure 1b presents the pump power-dependent µPL at 25 K.The pump power-dependent integrated µPL intensity of X, X -and LH in logarithmic scale [51] reveals a sublinear slope of 0.6 ± 0.1, 0.8 ± 0.1, and 0.6 ± 0.1 respectively, correlated with the CM increase.In contrast, the far-off resonance BX has the distinctive 2.0 ± 0.2 slope and features a saturation pump power larger than other excitonic emissions.In addition, the off-resonance CM as a function of pump power shows a slope of 1.1 ± 0.1.
By further increasing the pump power, ultrasharp lines appear next to the BX and X transitions (labeled with arrows in the magnified Figure 1c), which are likely due to the charged excitonphoton complexes and possible recombination of ground state QD electron with higher-order QD hole states, which have small (< meV) energy spacings.Particularly the sharp emission slightly below the BX line exhibits a super-linear power dependent behavior, which suggests the feature of a negatively-charged biexciton (BX -) [55] or excited biexcitonic states [56].Meanwhile, the rich lines ≈ 1 meV blue-shifted from BX resembles the positively-charged complexes [57].This can be attributed to the fact that the QD hole capture rate is proportional to the excitation power [58]: the increased direct hole captured by the QD at stronger pump excitation leads to a population from negatively-charged complex to neutral X and, with further hole capture, even to a X + subsequently.Moreover, it is interesting to observe an additional sharp shoulder redshifted from X with sub-barrier pumping at 900 nm, noted in Figure 1c.This is further detailed in the highresolution spectrum of Figure 1d at 6 K. Particularly it reveals an excitonic species at an transition energy ≈ 300 µeV below the X line (Figure 1d), which matches the reported exchange interaction induced splitting between the bright and dark excitons (DX) [28].This is detailed in Supplementary Section II.We note that the residual optical activity of the DX is attributed to mixing of the heavy hole (HH) ground state with the LH component through slightly-reduced symmetry in the QD [28,59,60], which results in a small in-plane polarized dipole moment and partially relieving the spin conservation.
To gain further insights on the interaction between the cavity and QD, as shown in Figure 2a, we carefully tune the X-CM coupling with fine-sweeping down to ≈ 300 mK temperature steps in our cryostat, while driving the sub-barrier excitation at 900 nm which suppresses charge fluctuation induced linewidth broadening of X from the above GaAs bandgap excitation.Figure 2b presents the X and CM peak energy as a function of temperature extracted from the spectral Lorentzian fit, with the X-CM coupling around 47 K.We witness that the slope of temperaturedependent CM energy experiences dramatic changes from 44.8 K to 46.3 K which corresponds to a detuning range 75 µeV <  < 120 µeV, where  =  0 −   with  0 and   as the X transition and CM energies.In the negative detuning side, such energy shift appears from 47.5 K to 47.1 K (−87 µeV <  < −83 µeV).The manner how its slopes change could manifest a trend of an avoided crossing of CM from X.Meanwhile, near zero-detuning, the CM shows a visible blueshift crossing X with increasing temperature (data marked by blue arrow in Figure 2b).The temperature-dependent CM energy recovers its slope at far detuning.As shown in Figure 2c, the CM and X linewidths show an averaging tendency when approaching the zero-detuning temperature, albeit the CM linewidth minima does not appear at the same detuning as the X linewidth maxima.It is worth pointing out that the detuning range with CM linewidth minima corresponds to its dramatic slope changing at positive detuning.Meanwhile, the X linewidth maximum occurs at the position where the CM energy is crossing the X, with a clear blue-shift of the CM line near zero-detuning.Figure 2d further presents the ratio of integrated PL intensity of CM (and X) to the total intensity, indicating their inversion with each other as QD-CM coupling is controlled from far-detuning to resonance.We note that the observed slope distortion of temperature-dependent CM energy appears in small detuning range (< 120 µeV) from 44 K to 46 K. Importantly, the observed avoided crossing of CM is fundamentally different from the mode pulling effect (namely, a tendency of CM energy shifting towards X near resonance), the latter of which results from the spectral overlaps of a low-Q CM and the X phonon sideband [40], in the weak coupling regime.Moreover, both the avoided crossing and visible blue-shift of the CM crossing X near zero-detuning cannot be reproduced by the simple spectral overlaps between CM and X with bulk phonon dispersion as in weak coupling regime.Instead, the co-occurrence of QD-CM avoided crossing, the distinct blue-shift of the CM crossing X, and their partial out-of-synchronization linewidth averaging points to the effect of intermediate coupling mediated by phonon scattering.In this intermediate coupling regime [43], both strong and weak coupling features coexist, as studied recently theoretically.It indicates that avoided crossing can be attributed to the Rabi doublet in the canonical strong coupling regime, with the CM blue-shifted near zero-detuning as a clear signature of the phonon-mediated collective states emission in the intermediate coupling regime [43,44].Specifically, in this concurrent strong and weak coupling regimes, with increased phonon scattering as the bath temperature increases, each eigenstate derived by Jaynes-Cumming (JC) model experiences a different increase of the dephasing rate.The optical transition of lower and upper polaritons in the first rung of the JC ladder experiences less dephasing rate, which resembles the strong coupling feature, and causes the observed avoided crossing of CM with X in our system.
Simultaneously, the higher-order eigenstates experience larger dephasing and form a collective state [43,44], located around the center of zero-detuning, with the resultant blue shift of the coupled QD-CM emission around zero-detuning.Furthermore, we also observe an abrupt change of CM energy when approaching zero-detuning (marked by blue arrows in Figure 2b) -this is a signature of the discrete phonon modes confined in the QD structure [61].Linewidth averaging is a canonical signature of strong coupling regime, where the CM and X linewidths collapse to the averaged value due to their photon-matter polariton nature.As to the observed partially out-ofsynchronization QD-CM linewidth averaging, it indicates the mixture of the collective state and fundamental Rabi doublet [43,44], which perturbs the exciton polariton linewidth averaging in the strong coupling regime.
From the point of view of the effective decay rate and cavity Q [62], the effective CM linewidth can be renormalized by an effective cavity pumping   mainly resulting from phonon-mediated energy transfer from X to CM by Γ   =   −   .This resembles the CM linewidth narrowing when approaching the resonance.The reversed process applies to the effective linewidth broadening of X near the cavity resonance.We note here that our site-controlled single QD-cavity geometry safely excludes the spurious pumping terms from the parasitic QD and wetting layer in the   , which commonly occurs in SK ensemble QDs.In addition, for the same cavity-QD with occasional CM blue-shift as a result of the PhC cavity surface state changes, the avoided crossing can occur at lower temperatures such as 25 K.Here the cavity Q factor is up to ≈ 9,270 (135 ± 11 µeV), resulting in solely the QD-CM linewidth averaging on resonance (not the concurrent strong-weak coupling regimes), in the canonical strong coupling regime.This is detailed in Supplementary Section III.

Span of degree-of-linear polarizations in the concurrent strong-weak intermediate coupling regime
To further investigate phonon-assisted coupling in the intermediate coupling regime, we performed the polarization-resolved detuning-dependent µPL on the same cavity-QD with controlled polarizers and bath temperature, as illustrated in Figure 3a.Here, the degree-of-linear polarization (DOLP) [36,49], defined as DOLP = Importantly, the detuning range for the pronounced co-polarization (DOLP > 0.5) corresponds to the detuning range (≈-110 µeV to +170 µeV) where dramatic slope distortion appears.At such detuning, the vertical-polarized components of QD excitation can be transferred into the cavity decay channel as a result of the phonon-assisted coupling between the CM and QD [36,38], leading to a relative depletion of the vertical-polarized QD µPL energies.In succession, such depletion subsequently enhances the vertical-polarized decay channel of QD itself by the Purcell enhanced cavity photon channeling to the QD µPL energy, resulting in the significant co-polarized QD µPL of DOLP > 0.5 with the CM.Because the cavity loss rate is generally much larger than the QD radiative decay rate, we observe the overall co-polarization effect at detunings where QD and CM sufficiently interact.And the significant co-polarization is in roughly the same detuning range where the dramatic slope changes appear in Figure 2b, indicating the important contribution of the phonon-mediated Purcell enhancement in our observed concurrent strong-weak coupling regime.The fact that the CM modifies the QD polarization only in relatively small detuning range strongly suggest the absence of far-off-resonant coupling induced by wetting layers or background emissions.
We also observed a negative DOLP at much larger detuning range, which manifest an overall S-shape DOLP curve [36,51], unique to these site-controlled QD-cavity system.The polarizationresolved measurements also enable the determination of the fine-structure splitting of X with orthogonal polarization, complying with the spin conservation selection rules.The inset of Figure 3b shows this polarization dependence, with a resulting extracted X fine-structure splitting of ≈ 30 µeV.

Anti-bunching, bunching, and collective state photon statistics in the concurrent strong-weak intermediate coupling regime
To study the radiative recombination dynamics of our coupled QD-cavity system, we measure the detuning-dependent decay time of X -by TRPL, as shown in Figure 4a.When X -is near resonance with the cavity under 160 µW excitation, Purcell enhancement results in a 1.2 ns decay time, extracted by the single-exponential decay convolved with an ≈ 500 ps instrument response function.To examine the µPL decay of the X -far-detuned from resonance, the excitation power is doubled to obtain adequate signal-to-noise ratio.The µPL rise time is clearly time-delayed, which arises from the finite p-state occupation under higher-power excitation [63].The extracted X -offresonance decay time increases up to 3.0 ns.We subsequently derive the Purcell factor [63] via: , where   − () = 1.2 ns is the X -near-resonance lifetime,   = 3.0 ns is the off-resonance X -lifetime,  0 = 1 ns is the typical bulk exciton lifetime, δ = 130 µeV is the near-resonance detuning, Q = 5,500 is the near-resonance Q factor,   = 227 µeV is the near-resonance cavity linewidth, and   = 9.5×10 -5 is inverse X -quality factor.f is a dimensionless constant, which depends on the spatial alignment between the site-controlled QD and the cavity field maxima, and the orientation matching between the QD dipole and cavity field.
The corresponding Purcell factor   is estimated to be between 2.7 to 10.8, with the associated reasonable range of f between 100% to 50%.
Photon statistics of the neutral exciton X coupled to the cavity is further investigated by measuring second-order correlation function  2 ().As shown in Figure 4b, at low excitation of 150 nW, we obtain a clear anti-bunching of 0.2, which supports that the coupled X-CM is at the single-photon single-exciton level.The time-bin width is chosen at ≈ 2 ns, as a consequence of trading off the temporal resolution versus the signal-to-noise ratio.We fit the anti-bunching by using a time-bin-convolved second-order autocorrelation function of two-level system  2 () = (1 −  −||/  )*boxcar(), where boxcar() is the boxcar function accounting for the 2 ns time bin, and   = 1.2 ns is the on-resonance decay time of X.With such fitting parameters, the fit well reproduces the observed anti-bunching signature of the coupled X-CM emission.The non-zero antibunching can be due to limited cavity feeding by X -or LH, leading to an occasional uncorrelated photon.The fluctuation of the  2 () baseline is attributed to a limited counting rate of the HBT setup.With doubling of the pump power to 300 nW, we witness a pronounced bunching, implying a higher conditional probability of a finding a pair of photons in our coupled QD-cavity system.The fit for bunching uses  2 () = (1 +  −||/  )*boxcar(), where 1+A is the amplitude of photon bunching.It is worth pointing out that such super-Poisson statistics has been observed in single self-assembled SK growth QD coupled with a PhC microcavity.It is postulated that the cascaded cavity photon emission can arise from an excitonic continuum induced by intermixing QD and wetting layer states [45,46].Those samples have a wetting layer where, driving the subsystem below saturation, detection of a cavity photon heralds an increase of probability finding the QD in an excited continuum, resulting in a second-photon emission event.
However, the absence of the wetting layer and parasite QDs in our single pyramidal QD rules out such cascaded photon cavity feeding from the excitonic continuum or background.
Instead one can tentatively ascribe the observed bunching to a purely photon-statistical effect: during Purcell enhanced emission of the first photon in the coupled QD-CM system, the transient decrease of X means the population probability allows the subsystem to enter into the sub-Poisson statistical regime, resulting in a temporal-correlated second single-photon emission [65].Here, since the dynamics of the X repopulation determines the difference in mean population probability between the zero and longer delays [66], a proper moderate excitation aids in the visibility of this bunching feature.One can also consider that an increased pump power induces the elevated photon number accumulation in the cavity Fock states.The resulting increased collective state [67,68] emission dominates the photon statistics as a bunching signature.

IV. Description of dynamic phase change and intermediate coupling regime in the Lindblad master equation formalism
The total Hamiltonian governing the QD two-level system (TLS) and a single localized CM is given in the Jaynes-Cummings (JC) form as (ℏ = 1) In the above expression,  0 and   are the TLS transition energy and CM energy, respectively.
+ ( − ) is the creation (annihilation) operator of the TLS and  † () is the creation (annihilation) operator of the CM.TLS and CM are coupled with a coupling strength .

S-7
Considering the irreversible dephasing processes in the semiconducting environment, the finite coupling of the hybrid TLS-CM system can be described in the framework of Lindblad master equation formalism written [S11, S12] as where the Lindblad operator with a dephasing operator  is defined as With above equation, we can define the detuning between TLS and CM as  =  0 −   , TLS exciton spontaneous decay rate as 2  , cavity loss rate as 2  , incoherent QD pumping rate as 2  , and exciton dephasing rate as 2Γ  .Note that the dephasing term is introduced as the annihilation of QD exciton and creation of cavity photon and the reversed process is ignored which is reasonable in the case that   ≫   [S13].Previous experiments reveal that Γ  incorporates the phonon scattering rate which is a S-shape function of temperature [S14].It shows a plateau at low temperature but features a sudden increase at increasing temperature [S14].
To gain insights into the dynamic phase transition in the cQED, previous theory proposes the with  = −Γ  + 4 and each subspace (rungs in the JC ladder), which is closely related to the emission spectrum of the system, is spanned by excitations  and  with eigenvalue problem ℒ ,  , =  ,  , with eigenvalues  , and associated eigenfunctions  , [S12].In the case of finite Γ  , diagonalizing ℒ ,−1 produces four eigenvalues  ±,± ,−1 but only two contribute to the emission spectrum of the optical transition in upper and lower polariton states of the JC −    + where   and   are the vertical-and horizontal-polarized µPL components align with TE and TM cavity modes respectively, is examined.The DOLP of different exciton species at various CM detuning serves as a probe to understand the pure dephasing mediated QD-CM coupling.Figure3bsummarizes the DOLP of X and excited LH at the detuning range covering the range in Figure2.It shows that the X and LH exciton species are co-polarized (positive DOLP) with CM near resonance.It gradually loses the CM-like polarization (DOLP from +0.6 to 0) when detuned from resonance to ≈ ± 1.2 meV detuning where the subsystem approaches the zero-DOLP (grey bar region).

Figs- 1 Figure 1 |Figs- 2 Figure 2 |Figs- 3 Figure 3 |Figs- 4 Figure 4 |
photon number statistics indicates the collective state due to phonon-mediated coupling in the intermediate coupling regime, shedding light on the concurrent strong-weak coupling regimes form different rungs of the Jaynes-Cummings ladder in our singular site-controlled cavity-QD system.

Figure S3- 2 |
Figure S3-2 | QD-cavity detuning by varying temperature in the case of occasional blue-shift of CM due to PhC cavity surface state.a, µPL spectrum of another singular pyramidal InGaAs QD-L3 PhC cavity system at 30 K with 532 nm non-resonant pumping at 10 nW.b, Zoom-in of the X and CM spectrum and their Lorentzian fits at 25 K. c to e, Emission energies, integrated µPL intensities, and linewidths of X and CM emission as a function of temperature.