Enhanced emission from a single quantum dot in a microdisk at a deterministic diabolical point

We report on controllable cavity modes through controlling the backscattering by two identical scatterers. Periodic changes of the backscattering coupling between two degenerate cavity modes are observed with the angle between two scatterers and elucidated by a theoretical model using two-mode approximation and numerical simulations. The periodically appearing single-peak cavity modes indicate mode degeneracy at diabolical points. Then interactions between single quantum dots and cavity modes are investigated. Enhanced emission of a quantum dot with a six-fold intensity increase is obtained in a microdisk at a diabolical point. This method to control cavity modes allows large-scale integration, high reproducibility and fexible design of the size, location, quantity and shape for scatterers, which can be applied for integrated photonic structures with scatterer-modified light-matter interaction.


INTRODUCTION
Optical whispering-gallery microcavities with high quality (Q) factors and small mode volumes can enhance the interactions between light and matter [1], and have found widespread applications in nonlinear optics [2], low-threshold lasers [3,4], dynamic filters and switches [5], single-photon sources [6], and cavity quantum electrodynamics (CQED) [7][8][9]. In a whisperinggallery microcavity, the counter-propagating degenerate eigenmodes with the same polarization and resonance frequency correspond to a diabolical point (DP) in the presence of mirror symmetry. However, intrinsic scatterers in microcavities will lift the mode degeneracy and cause the symmetry breaking. Embedded quantum dots (QDs) [10,11], material inhomogeneity [12,13], attached particles or fabrication imperfections [14][15][16] can serve as intrinsic Rayleigh scatterers in microcavities and cause backscattering of the light into the counter-propagating mode and couple two counter-propagating modes with each other, lifting the mode degeneracy. Two new standing-wave modes will form and split in frequency, propagating with a phase π/2 between their spatial field distributions. The backscattering from the uncontrollable intrinsic scatterers will bring unwanted disturbances to the applications which in principle use the counter-propagating degenerate modes, such as particle sensors [17,18], dual frequency microcombs [19], and optical gyroscopes [20].
Moreover, the backscattering can be precisely controlled by coupling nanotips to a microcavity [21][22][23], which have been used to realize exceptional points for chiral lasing [24], highsensitivity particle detection [25] and electromagnetically induced transparency [26]. However, the above mentioned controlling method requires high-precision piezo positioners, which is challenging for large-scale integration. Instead, by introducing two subwavelengthscale perturbations as scatterers on the microcavity edge and adjusting the size, location, quantity and shape of the perturbations, backscattering can be controlled in an integrable way [27]. Introduced designed scatterers have been used for lasing mode selection [28] and unidirectional lasing [29] by tuning the wave propagations and mode field distributions in microcavities.
QDs, also called the artificial atoms, have great potential for realizing ideal single-photon or entangled-photon sources [30][31][32][33][34]. To improve the indistinguishability of photons and enhance the photon extraction efficiency [6,35,36], QDs are usually weakly coupled to a cavity with high Q factors. Since spontaneous emission arises due to interactions between matter and its local electromagnetic environment [37,38], the interactions between a single QD and cavity modes depend on their spatial and frequency overlaps. The randomlydistributed intrinsic scatterers can cause the mode field redistribution of two newly formed modes in space and frequency including the mode field distribution surrounding QDs [39].
But these intrinsic scatterers affect the cavity modes in an uncontronllable way. In addition, split modes will bring unwanted mode competition and prevent the single-mode lasing in a microcavity [40,41] with embedded QDs as gain medium for ultra-low threshold lasers [42,43]. Therefore, to realize controlled backscattering is highly desired to control the cavity modes in microcavities with embedded QDs.
Here we realize controllable cavity modes through controlling backscattering in GaAs microdisks with embedded QDs. We design microdisks with two identical nanoscale cuts serving as Rayleigh scatterers on the cavity perimeter and control the backscattering through changing the relative angle between two cuts. Periodic changes of the backscattering coupling characteristics of cavity modes in fabricated microdisks are obtained. A theoretical model based on two-mode approximation and three-dimensional numerical simulations are used to explain the periodic behaviours [44], in particular, the periodic mode degeneracy and difference in frequency and linewidth of split modes. Split cavity modes periodically merge into single-peak cavity modes, indicating DPs. Multiple cavity modes are also simultaneously controlled by scatterers in a periodical way. To investigate the interactions between single QDs and cavity modes, we calculate optimal Purcell factors and experimentally obtain enhanced photoluminescence (PL) intensity of QDs. We obtain enhanced PL intensity of a single QD with an approximately six-fold intensity increase in a microdisk at a deterministic DP.

THEORY
For an unperturbed whispering-gallery microcavity, light can be confined as degenerate travelling-wave modes propagating along clockwise (CW) and counter-clockwise (CCW) directions, which correspond to two orthogonal states with coalescent eigenvalues. This indicates that the microcavity is at a DP. The Hamiltonian of an unperturbed microcavity in the travelling-wave basis can be written as where Ω − iΓ/2 is the eigenvalues of H 0 , Ω is the resonance frequency and Γ/2 is the decay rate. When adding two identical scatterers with size much smaller than the mode wavelength, backscattering of the mode fields from the scatterers will result in the coupling between the two degenerate modes and then two new modes are formed as standing-wave modes. The effective Hamiltonian of the whole system using the two-mode approximation can be represented by [44] with where ω ′ ≃ 2ǫ, A = ǫ + ǫe −i2mβ , and B = ǫ + ǫe i2mβ . m is the azimuthal mode number, β is the angle between two scatterers on a microcavity, and ǫ is a complex frequency splitting resulting from the perturbation by a scatterer, defined as ǫ = g − iΓ 1 /2 [17]. g is the resonance frequency shift and Γ 1 /2 is the additional decay rate caused by the scatterer.
The offdiagonal elements A and B represent two light backscattering processes from CW to CCW and CCW to CW propagation directions, respectively. Obviously, A and B have periodic components with a calculated period of P cal = π/m. Here |A| = |B| indicates the symmetric backscattering and no chirality for the cavity [24]. Therefore, we have the total effective Hamiltonian in the travelling-wave basis as and the new eigenvalues of this cavity are ω ± = Ω − iΓ/2 + 2ǫ ± 2ǫ cos(mβ).
To analyse the effects from two scatterers on the cavity, three parameters describing the backscattering coupling characteristics of the newly formed modes can be defined as Q sp = 2g cos(mβ) Γ/2 + Γ 1 .
The linewidth distributions of the new modes with the corresponding relative wavelengths can be specifically inferred according to Equation (6) and (7). The sign of g depends on the dielectric permittivity difference ε m − ε p between the surrounding media (ε m ) and the scatterers (ε p ) [17]. When ε m < ε p , then g < 0. ∆ω and ∆λ dif f will exhibit opposite signs, which means the new mode with a longer wavelength will exhibit a wider linewidth. When ε m > ε p , then g > 0. ∆ω and ∆λ dif f will exhibit the same signs, which means the new mode with a shorter wavelength will exhibit a wider linewidth. Here we assume ε m > ε p since the effective dielectric permittivity ε p of the designed scatterers in our devices is 1. Therefore, for β = Nπ/m, one of the new eigenvalue will exhibit a smaller imaginary part than that of the above coalescent eigenvalues and corresponds to a longer resonance wavelength as shown in Fig. 4(a) and (b).
Additionally, when A = 0, B = 0 or B = 0, A = 0 is satisfied, exceptional points appear where both eigenvalues and eigenvectors of the system coalesce with asymmetric backscattering [24]. Different from the identical scatterers to achieve a DP with symmetric backscattering, the two scatterers to achieve an exceptional point should have different sizes and distances from cavity rim to break the mirror symmetry and achieve a high asymmetry in backscattering at β = (2N + 1)π/2m [44].

DESIGN AND METHODS
The sample grown by molecular beam epitaxy is made of three layers, namely GaAs substrate, a 1-µm-thick AlGaAs sacrifice layer and a 250-nm-thick GaAs slab with InAs QDs grown in the middle. To investigate the interactions between single QDs and cavity, the QDs are intentionally grown in an appropriately low density of approximately 30 µm −2 for distinguishing the discrete spectral emissions of single QDs. In order to investigate how the backscattering is controlled by the angle β between two scatterers on the microdisk perimeter and guarantee the cavity mirror symmetry to achieve DPs [27,44], we design two identical arc cuts to serve as the scatterers through removing two circles off the cavity with the circle centers are on the microdisk perimeter as shown in the inset in Fig. 1 Fig. 1(b) and (c). The split modes from the microdisk with cuts in Fig. 1(b) exhibit two unequal linewidths according to Equation (7). That the left branch of the double peaks exhibits a wider linewidth than that of the right branch is due to a smaller effective dielectric permittivity of the scatterers compared to that of the microdisk [17], as discussed above in T heory section. Due to the intrinsic scatterers, split modes exhibiting randomness were also observed in the microdisks without cuts as shown in Fig. 1(c). By comparison, the split peaks exhibit two almost equal linewidths resulted from a large amount of randomly distributed scatterers [16].
Compared with the random and uncontrollable intrinsic backscattering, periodically controlled backscattering coupling characteristics including ∆ω, ∆λ dif f and Q sp are achieved by two designed cuts. Taking the mode with wavelength of approximately 1043 nm (T E 2,10 ) as an example in microdisks with R cut =40 nm and β from 90 • to 180 • with 5 • as one angle step, split cavity modes periodically merge into single-peak cavity modes marked by black triangles, as shown in Fig. 2(a). The average fitted experimental results (red) from identically designed microdisks including ∆ω, ∆λ dif f and Q sp are shown in Fig. 2(b), (c) and (d), respectively. Synchronously periodic changes of experimental ∆ω, ∆λ dif f and Q sp can be observed where they reach their extreme values at the same β, agreeing with the above conclusion that the backscattering characteristics are periodically controlled by cos(mβ).
Therefore, those single-peak cavity modes in Fig. 2(a) correspond to the reappearing DPs.
Such periodic behaviours also happen to other modes in Fig. 1(a) because the two scatterers affect all the cavity modes. Their experimental periods P exp are listed in Table I  To further analyse the backscattering controlled by the scatterers, we have done threedimensional numerical simulations based on finite element method on a microdisk with two cuts. Specific modes are confirmed by comparing mode wavelength λ exp in Fig. 1(a) with simulated wavelength λ sim in Table I. For more relations between backscattering and mode field distribution, we focused on the mode T E 2,10 . Synchronously periodic changes of ∆ω, ∆λ dif f and Q sp as a function of β are obtained as shown in Fig. 2

(b) and (d) labelled
by blue geometries which exhibit almost the same periodicity as the experimental results, respectively. In comparison with simulation results, a smaller experimental Q sp is due to a lower Q factor caused by fabrication imperfection. The experimental ∆ω are also smaller than simulation results, which can be attributed to that the size of etched cuts on the microdisks is smaller than designed. Instead, the cut size will decrease from top to bottom at the cavity perimeter by inductively coupled plasma. The smaller etched cuts can cause smaller splitting as shown in Fig. 4(d). The simulated period is equal to the calculated period P cal = 180 • /10 = 18.0 • . P cal for other T E 1,m and T E 2,m are listed in Table I Fig. 4(b). The fitted linewidths of the cavity modes are shown in the insets accordingly.
It can be seen that the right branches of the split modes exhibit smaller linewidths than those of the reappearing single-peak modes as theoretically predicted above. Hence, the spectral shapes of modes from identically designed devices are highly consistent, proving the high reproducibility of the devices. The minor wavelength differences are attributed to the fabrication imperfection.
To study the effects of R cut on the backscattering, we measure microdisks with the same β of 45 • and 110 • but different R cut (from 30 to 60 nm). Figure 4(c) and (d) show that the increasing R cut barely affects the relative spectral shape but does lower Q factors. Obviously, the linewidths of left branches of the split modes increase with R cut while the linewidths of the right branches are barely affected, as exhibited by the fitted spectral linewidths in the insets. This is due to the unbalanced effects from the scatterers on two modes, as visualized in Fig. 3(c) and (d). The frequency splitting between two split modes also increases with R cut because a bigger scatterer can cause more backscattering and further enhance the coupling between two degenerate modes [14].
Though the backscattering is well controlled by two cuts to modify cavity modes, it is at the cost of lowering Q factors. Therefore, to estimate the potentials of this control method, we calculate optimal Purcell factors in microdisks with different R cut and two different β based on Q factors of our fabricated and ideal microdisks. Here we assume the QD is positioned at the maximum mode field of one cavity mode and the exciton dipole in the QD is parallel to the local mode electric field with much narrower linewidth than that of the cavity mode [47]. Then the optimal Purcell factor is given by [1] where V is the mode volume and n m is the refractive index of cavity. For Purcell factors of the mode T E 2,10 , we take λ = 1043 nm, n m = 3.46 and V ≈ 3.4 × λ 3 /n 3 m calculated by finite element method. Figure 4 The right branch of split modes shows a slower decline than the left branch.
Through controlling the backscattering by two identical scatterers, we obtain microdisks at periodically deterministic DPs, where split cavity modes merge into single peaks. Then we focus on the interactions between single QDs and the cavity modes with backscattering control. At a high excitation power, PL of the ensemble of QDs can be observed in a wide spectrum range as shown in the background spectra of the cavity modes in Fig. 1(a).
To observe well-isolated discrete spectral lines from single QDs, a low excitation power is required. Due to the growth technology of QDs, QDs show randomness in its size and position [48] which means the QD may be blue-detuned or red-detuned from the cavity. For these two detuning cases, we use two methods by controlling the device temperature and a magnetic field to bring a single QD and a cavity mode into resonance in our experiment.
When temperature increases, the emission energy of a QD redshifts, mainly ascribing to a shrinkage of the band gap of QDs [49]. The energy of a cavity mode also redshifts but at a much slower speed than QDs due to an increase in the refractive index of the cavity [50].
When a magnetic field is applied, the emission energy of a QD blueshifts due to diamagnetic effect while cavity modes are unaffected [51][52][53][54]. Therefore, for the QD blue-detuned from the cavity, we increase the temperature to tune the QD emission for resonance, while for the QD red-detuned from the cavity we apply a magnetic field to change the QD emission. Furthermore, we investigate the interactions between a single QD and cavity at a DP. As shown in Fig. 5(b), a single QD labelled by QD 3 in Fig. 5(d) is tuned into resonance with a single-peak mode at a deterministic DP with R cut =40 nm by increasing the temperature.
Both of the emission wavelengths of the single QD and the cavity mode red shift while the previous one shifts faster. A spectral crossing for the QD and the cavity mode can be observed, where two peaks merge into one peak with an obvious emission enhancement region and then separate with further increasing the temperature. This indicates weak coupling between the QD and the cavity mode. The merged peak reaches its maximum intensity at approximately 52.2 K. By comparing the off-resonant spectrum at 40 K and the resonant spectrum at 52.2 K, Purcell-enhanced PL intensity of the single QD can be more apparently observed as shown in Fig. 5(d). By Lorentz fitting the peaks, the PL intensity of the single QD is enhanced by approximately six times. It should be noted that the factor of the emission intensity enhancement is not sufficient to reveal the specific Purcell factor without lifetime measurement. The enhancement is strongly limited by the spatial overlap between the single QD and the mode field. Therefore, a small spatial overlap will result in a much weaker emission enhancement compared to the optimal Purcell enhancement where the QD is assumed optimally position in the mode field. Two discrete PL peaks from other two single QDs labelled by QD 1 and QD 2 which are uncoupled to cavity modes can also be observed in Fig. 5(d) for comparison. With a temperature increase, emissions of two QDs shift to red as expected.

DISCUSSION
Previously, researchers have been focused only on the original mode degeneracy but with inevitable split cavity modes [8,10,46] or backscattering control by nanotips requiring highprecision piezo positioners. Here, the backscattering is periodically controlled by the designed scatterers in microdisks with the changing angle β. Periodic mode degeneracy at DPs and Purcell-enhanced emission from a single QD coupled to a microdisk at a deterministic DP are obtained. Designed scatterers can control the mode field distribution,such as mode antinodes, indicating that whispering-gallery cavity with quantum emitters and controlled scatterers can provide a good platform to study scatterer-modified light-matter interactions.
In addtion, designed scatterers can even be used to achieve exceptional points, which will extend studying light-matter interactions into a non-Hermitian system. The scatterers can be designed flexibly in size, location, quantity and shape. Smaller increasing angle steps for β can also be designed for more precise controlled backscattering. Meanwhile, with further optimization of the fabrication, a cavity both with higher Q factors and with sufficiently controlled backscattering can be obtained by smaller scatterers. As the QD growth and imaging technology being improved [55][56][57], pre-selected and accurately aligned QDs could be beneficial to optimize the spatial overlap between QDs and controlled cavity mode field. Therefore, interactions between light and QDs with backscattering control can be greatly enhanced in the future.

CONCLUSION
In summary, we demonstrate the control of cavity modes through controlling the backscattering by two identical designed scatterers integrated into optical microdisks. Periodic control of backscattering coupling characteristics of cavity modes are obtained when the angle between the two scatterers is changed. The periodically appearing single peaks indicate the periodic mode degeneracy at DPs. Theoretical discussion and numerical simulations are presented for understanding the periodic backscattering coupling characteristics. In particular, mode degeneracy, linewidth difference and frequency splitting at two extreme cases are discussed in theory part and visualized by the simulated mode field distributions. Meantime, multiple cavity modes are simultaneously controlled by scatterers but with different periods, which agree well with theory. To prove the potential application in light-matter interactions with cavity modes controlled by scatterers, optimal Purcell factors are calculated. Purcell-enhanced PL emission from a single QD with a six-fold intensity increase is experimentally obtained in a cavity at a deterministic DP. The enhancement can be further improved with increasing the spatial overlapping between QDs and cavity mode and the Q factors. We believe that whispering-gallery cavities with designed and integrated scatterers provide a good platform to study the light and matter interaction in a controllable way, which have great potentials for implementing optical quantum information processing and cavity quantum electrodynamics [58][59][60].