Resolving the temporal evolution of line broadening in quantum emitters
Light emission from solid-state quantum emitters is inherently prone to environmental decoherence, which results in an inhomogeneous line broadening and in the deterioration of photon indistinguishability. Here we employ photon correlation Fourier spectroscopy (PCFS) to study the temporal evolution of such a broadening for the biexciton and exciton emission in resonantly driven GaAs quantum dots. Differently from previous experiments, the time scales we probe range from a few nanoseconds to milliseconds and, simultaneously, the spectral resolution we achieve can be as small as . We find pronounced differences in the temporal evolution of the two lines, which we attribute to differences in their homogeneous linewidth and sensitivity to charge noise. We then analyze the effect of irradiation with additional white light, which reduces blinking at the cost of enhanced charge noise. Due to its robustness against experimental imperfections and its high temporal resolution and bandwidth, PCFS outperforms established spectroscopy techniques, such as Michelson interferometry. We discuss its practical implementation, its limitations, and the possibility to use it to estimate the indistinguishability of consecutively emitted single photons for applications in quantum communication and photonic-based quantum information processing.
Semiconductor quantum dots are excellent resources of non-classical light for modern photonics. They enable pivotal single photon applications in the field of quantum communication and quantum information processing, such as quantum teleportation  and Boson sampling [2, 3]. A key quantity in all these applications is the indistinguishability of consecutively emitted photons, as it determines the interference visibility of the overlapping photon wave packets . In turn, this directly impacts the error rate in envisioned applications. The photon indistinguishability of solid state light emitters is predominantly reduced by coupling to the phonon bath  and by a fluctuating charge environment , both leading to a dephasing between the single photon wavefunctions. In semiconductors, charge noise can stem from ionisation of impurities or from photogenerated carriers, which produce shifts of the central emission energy of a nearby quantum dot due to the quantum confined Stark effect . This results in inhomogeneously broadened spectral lines, usually observed in time-averaged micro-photo-luminescence (PL) spectroscopy. The 1/f noise frequency spectrum determines the indistinguishability at different timescales, which can be important up to hundreds of nanoseconds, as in time-to-path demultiplexing [8, 3].
The temporal dynamics of the dephasing mechanisms usually elude established spectroscopy methods, such as PL and Michelson interferometry, due to relatively long integration times required. Scanning resonance fluorescence techniques have been used to probe timescales down to the microseconds range , shedding light on noise sources, but cutting off crucial parts of the frequency spectrum. The most reliable way of quantifying the indistinguishability so far is a direct measurement of the interference visibility in a Hong-Ou-Mandel (HOM) experiment , using varying delay times to probe photons from different excitation cycles . However, this approach has limited temporal resolution due to the inefficient way of varying the delay time and the maximum achievable delay is limited by optical fiber losses.
Photon Correlation Fourier Spectroscopy (PCFS)  can provide quantitative information about the emission line broadening at time scales down to the source lifetime (usually sub-nanoseconds) and up to several hours, all in one measurement. It therefore allows an estimation of the trend of the HOM visibility with increasing photon delay. As a Fourier method, it benefits from a high energy resolution compared to diffraction spectroscopy, similar to Michelson interferometry, but without the high requirements in mechanical stability of the optical components.
In this article we employ PCFS to investigate line broadening in single GaAs quantum dots (QDs). Such systems exhibit intrinsically short lifetimes , a near-zero multi-photon emission probability , and a high indistinguishability of consecutively emitted photons  , despite the non-lifetime-limited time-averaged linewidth . In addition, these QDs can be used to generate polarization-entangled photon pairs with near-unity entanglement fidelity using the biexciton-exciton recombination cascade . These properties recently allowed performing entanglement swapping between subsequently emitted photon pairs , where the success rate is mostly determined by the indistinguishability of the photons used for the Bell state measurement. We therefore investigate line broadening for the emission lines XX and X corresponding to the biexciton-to-exciton and the exciton-to-groundstate decays for different experimental conditions.
The GaAs QDs studied here were grown by molecular beam epitaxy in an AlGaAs matrix and embedded in a simple planar cavity. GaAs nanostructures providing three-dimensional confinement of excitons were obtained by the local droplet etching method . The sample structure is similar to that used in recent publications [15, 19, 1, 17, 20].
Figure 1 illustrates the experimental arrangement for PCFS. The QDs are cooled to in a He flow cryostat and excited by a pulsed TiSa laser with a pulse duration of about and a repetition period of . To access both the XX and X lines (see spectrum in figure 2 (a)) we employ a two-photon excitation scheme (TPE) [23, 24], with the laser energy tuned precisely to the half of the biexciton energy. To decrease the total acquisition time, we double the average repetition rate of the laser by an unbalanced Mach-Zehnder interferometer (MZ), yielding two pulses with a time delay of per laser pulse. In addition to the laser, white light from an LED with variable intensity can be coupled into the same excitation path.
The spectral separation between X and XX lines (about / ), allows a selection of the lines via standard diffraction monochromators (M) before detection, see figure 1. The scattered laser light, located between the X and XX lines, is reflected by tunable notch filters (NF) with a bandwidth of . Since the exciton energy level is split by a fine structure splitting (FSS), the X and XX lines actually consist of doublets with orthogonal linear polarization . In the setup we can select one fine structure component of each doublet by rotating the polarization plane of the emission signal by a half wave plate (HWP), until one component is cancelled by a subsequent polarizer.
The interferometer of the PCFS setup consists of a 50:50 beam splitter (BS) and two retroreflectors (RR), where one RR is mounted on a motorized linear stage. The position of the stage defines the relative optical path difference between the two arms of the interferometer. In contrast to standard Michelson interferometry, PCFS detects the signal of both BS outputs via two avalanche photodiodes (APDs), which are connected to correlation hardware. In our experiment we use APDs with a time resolution of .
The PCFS measurement is performed by positioning the linear stage at several equidistant positions, yielding a discrete set of different optical path differences , , with a constant spacing of . In Michelson interferometry, usually the intensity contrast (or ”visibility”) between the interference maxima and minima at one BS output is determined. In PCFS the interference fringes are averaged out by slowly moving the linear stage with a constant velocity . To illustrate the working principle of PCFS we first consider a monochromatic plane wave with a time-dependent angular frequency . In the classical picture, the resulting intensities at the respective BS outputs A and B are then proportional to
In this picture, the frequency of the emitted light would be well defined at any time. We know from Heisenberg’s principle that this is impossible for light stemming from spontaneous emission and that the variable has to follow a probability distribution with a finite natural linewidth, depending on the lifetime of the emitter. Dephasing processes lead to further broadening of this distribution.
Fluctuations of in semiconductor structures usually occur on timescales much smaller than the measurement times of commonly-used spectroscopy methods. Those techniques only reflect the time average of the frequency distribution, concealing its temporal evolution. We recover the time dependency by exploiting the time resolution of our single photon detectors (APDs) and determining the second order correlation function
where denotes the time average over the integration time and is the time difference between detection events.
To avoid confusion we stress at this point that the optical path difference gives us access to the spectral distribution, similar to conventional Michelson interferometry. It is in fact evident from equation 1 that interference will produce a drop in the coincidence counts between the two detectors and that a relatively low value of is thus expected as long as is small compared to the coherence length of the source, i.e., as long as interference fringes are detectable. The high temporal resolution and large temporal bandwidth on the determination of adds to the desired temporal information.
The velocity in PCFS is particularly relevant and should satisfy the following conditions: 1) The travelling distance of the stage within has to be small, so that . This condition is equivalent to asserting that the interference visibility has a well defined value for a given value of path difference . 2) The travelling distance must be sufficiently long in order to average over several interference fringes, so that , where is the mean frequency of the emitted signal. 3) The change in within the maximum time delay of interest has to be substantially lower than the coherence length of the emitter, i.e. (see  for details).
Figure 3 (a) shows the unnormalized of the X signal for a fixed . The triple-peak pattern originates from the pulse sequence of the excitation laser, doubled by the MZ. The peak at is absent due to the single photon emission characteristics of the QD . The single photons arrive within a minimum time delay of at the two detectors, yielding the two peaks around and thus defining the time resolution . This threshold solely arises from the specific measurement arrangement and can, in principle, be pushed down to the emitter lifetime by further reducing the delay of the MZ or by utilizing a CW pump laser. The next value of accessible with the present configuration corresponds to the delay between two subsequent laser pulses, i.e. .
We evaluate the for time delays up to , which turned out to be sufficiently high to cover the slowest frequency fluctuations in the used QDs. The total observable noise frequency range in this case is thus from to , which extends well beyond , previously accessed by resonant fluorescence . After repeating the measurement for all different , a two-dimensional map of values as a function of and can be constructed, as shown in figure 3 (b). Provided that the aforementioned conditions regarding are fulfilled, the value of always drops to (ideally) at . The measured value of about is a result of experimental imperfections, such as a non-perfect overlap of the interfering beams at the BS. For and fixed the value increases towards (when corrected for the ”blinking” of the emitter, explained in section 3.3), following a functional behaviour, which depends on the spectral broadening mechanisms acting within a time delay : the stronger the broadening, the narrower the dip of . From the 2D map we already qualitatively see that the spectral fluctuations of the emitters increase with increasing delay time . The maximum path delay induced by the linear stage therefore has to be well above the maximum coherence length of the emitted signal, in order not to lose information.
where describes a random frequency shift between and , with the frequency deviation from . For sufficiently long , all possible frequency fluctuations will occur, hence we can apply the ergodic theory to substitute the time average with the ensemble average:
where the normalized probability distribution over the random variable contains the desired information about the time-dependent spectral properties of the measured signal. We identify the integral in (4) as Fourier transform . Applying the discrete inverse Fourier transform yields the values for ,
with and . The value of is evaluated by integrating over an appropriate time bin (see supplementary).
The maximum induced optical path difference defines the frequency resolution by . For our linear stage with a maximum travel range of , the energy resolution results in . The measurable spectral range is given by . For , the energy range then results in .
It should be noted that the distribution of spectral shifts can not directly be identified as the spectral distribution of the measured light emission. To elaborate this statement, we take a closer look at the formal definition:
In (6) the emission frequency is generalized to the time-dependent homogeneous distribution . In the case of QDs, it ideally corresponds to the natural Lorentzian spectral distribution with a linewidth of , often referred to as the Fourier limit. The integral in (6) represents the cross-correlation function between the spectral distributions at the time and at , which could differ because of a spectral shift within . The ensemble average () over all possible realisations of frequency fluctuations leads to .
For the limiting case of no frequency fluctuations have occurred and . Consequently, corresponds to the auto-correlation function of the Fourier-limited line shape, which is again a Lorentzian distribution, but with a FWHM of . For the second limiting case of , all possible frequency shifts have happened between and . This situation corresponds to the regime accessed by standard Michelson interferometry and represents the auto-correlation function of the time averaged spectral distribution. In the case of a Gaussian broadening with a FWHM of , the resulting observed is again a Gaussian distribution with a FWHM of .
3 Measurements and results
3.1 Resolving the fine-structure of inhomogeneously broadened lines with PCFS
To illustrate the high spectral resolution of PCFS and gain further insight in the information provided by the technique, we focus here on the fine-structure-split X emission of a GaAs QD.
Figure 4 shows an excerpt of a PCFS measurement, evaluated at two different time delays (a,b) and (c,d) . Instead of using the angular frequency shift , we plot the results as a function of the energy shift . The panels on the left show measurements of both fine structure components (FSCs). These components represent a static doublet in the spectral distribution, separated by the FSS. The value of the FSS was determined from polarization-resolved photo-luminescence (PL) spectra as (see supplementary for details). According to (6), a static doublet of homogeneously broadened lines results in a peak triplet in , as seen in panel (a), corresponding to all different combination of spectral shifts: The peak at represents the absence of a spectral shift (if two photons stem from the same FSC). The two identical side peaks correspond to an equally probable positive or negative energy shift of (if two photons stem from different FSCs). The triplet is fitted by the sum of three Pseudo Voigt functions. The FWHM of each peak is and the mean energies of the two side peaks yield , which is in good agreement with the value measured by PL. Figure 4 (b) corresponds to the same X emission signal, when selecting only one FSC. In that case, shows only one line with fitted FWHM of . The expected FWHM of a Fourier limited Lorentzian spectral line is . The discrepancy indicates the presence of a broadening mechanism, already active at , presumably dominated by interaction with lattice vibrations .
As previously stated when discussing figure 3 (b), the spectral broadening is expected to increase with delay time . In figures 4 (c) and (d) the same two measurement configurations (both and one FSC) are evaluated at . In these cases, the inhomogeneous broadening due to charge noise  in the QD surrounding dominates the line shapes. The triplet shown in (a) is completely obscured and the fitted FWHM of (c) and (d) of and are found to be similar. A Michelson measurement of one FSC yields a linewidth of . We recapitulate that corresponds to an auto-correlation function of a spectral distribution ((6)). Therefore, the real linewidth can be deduced from , if the original lineshape is known. If not, we can still assume the Lorentzian and Gaussian distribution as limiting cases to estimate the original linewidth. Assuming a predominantly Lorentzian lineshape, the results of PCFS and Michelson interferometry are in sufficiently good agreement.
From the above discussion it is clear that PCFS allows to overcome inhomogenous broadening and thus to access spectrally close lines even without resorting to polarization selection.
3.2 Temporal evolution of the X and XX spectral line broadening from a GaAs QD
We focus now on the temporal evolution of the line broadening of a GaAs QD displaying a relatively broad emission linewidth () in time-averaged µPL.
Figures 5 (a-c) present for the X (magenta) and XX (blue) emission at three relevant time regimes. The corresponding FWHM of for all evaluated values of is shown in figure 5 (d). The dashed lines in the inset represent the X and and XX Fourier limits and , calculated from the measured lifetimes (see figure 2 (b)). Note that the actual Fourier limits can slightly vary among different QDs. At the lowest accessible time delay , X (XX) emission exhibits a FWHM (fitted by a Pseudo-Voigt function) of and . Because of the substantially different lifetimes of XX and X lines, it is not surprising to see that the XX line is broader than the X line. We attribute the discrepancy between the measured values and the respective Fourier limits predominantly to phonon coupling .
Starting at around , a monotonically increasing broadening of for both the XX and the X emission becomes apparent (due to decreasing statistical error, the increase becomes significant from approximately ). While for short time delays the X line remains narrower than the XX counterpart, a crossover of the linewidths is observed at (see figure 5 (d)). All QDs measured so far showed the same behaviour. Since it is known that the XX emission is less sensitive to electric fields than the X emission , the observation is consistent with a broadening stemming from fluctuating electric fields (e.g. charge noise).
Finally, at the millisecond timescale both FWHM converge to (for X) and (for XX), which correspond to the inhomogeneously broadened spectral lines observed by a spectrometer or by Michelson interferometry.
As mentioned before, line broadening beyond the Fourier limit affects the interference visibility in a Hong-Ou-Mandel (HOM)  measurement between two consecutively emitted photons. For the X line and a time delay of we find an interference visibility of under TPE, without correction for imperfections of BS and (see supplementary for details). In the XX-X decay cascade, we expect the visibility of both the XX and the X photons to be inherently limited by time-correlation between XX and X photons  to . In contrast to that, resonant excitation schemes directly addressing the exciton state yield HOM visibilities above on the very same QDs . The measured HOM visibility for X photons with is . The decreasing HOM visibility with increasing photon delay, which is reproducible also for other QDs, is consistent with the increasing line broadening measured by PCFS and is in accordance with the work of other groups .
We stress that a direct measurement of the HOM visibility by employing delay lines with variable length   certainly is the most precise approach, but likewise impractical and time consuming if intended for frequent characterization. PCFS inherently includes the full range of delay times in the measured raw data, which can be post-processed at will. Furthermore, PCFS is robust against signal intensity fluctuations, stray-light, imperfections of the transmission/reflection ratio of the BS and the spatial overlap of the beam (if constant over varying ), as all of these factors only lower the signal-to-noise ratio (SNR), but do not falsify the evaluated energy distribution. The implementation of the optical components is straightforward compared to a HOM setup, which demands high precision in temporal and spatial mode overlap.
Before concluding this section, we point out that the results shown in Figure 5 (d) clearly show that XX is more suitable than X for HOM measurements among remote QDs , such as those necessary for a Bell-state measurement used in entanglement swapping. In this case, spectral broadenings for separate emitters are uncorrelated and the deviation from the Fourier limit at long delay times is most important.
3.3 Influence of white light illumination on the charge dynamics
In the context of QDs embedded in a semiconductor host structure, additional weak white light illumination is well known to potentially enhance their brightness, especially under resonant excitation and in absence of intentional doping [32, 33]. This effect is attributed to a reduction of the ”blinking”, which is a on-off-modulation (telegraph noise) of the QD emission, presumably due to diffusion of charges from QD into the vicinity or vice versa. Although the impact of white light on the steady-state emission conditions are well explored, little is known about actual variations in the charge dynamics.
Figure 6 (a-c) show the distribution for three different , without and with additional white light. The intensity of the white light was adjusted in order to achieve a maximum enhancement of the brightness. At short time scales (see figure 6 (a)), the lineshape is close to a Lorentzian and the FWHM of remains similar for both cases, as seen in figure 6 (d). Accordingly, the HOM visibility at this timescale is largely unaffected by the white light, albeit a slight trend towards higher visibility (a few percent) was observed by us on similar QDs. The increase in line broadening discussed in the previous section is however different in the two cases: white light clearly accelerates the linewidth deterioration and substantial differences are observed at timescales of about . This observation indicates that white light accelerates fluctuations of the charge environment, leading to a more significant line broadening starting at lower values of . This is clearly visible by inspecting the lineshape with and without white light (see figure 6 (b)). When fitting with a Pseudo-Voigt function, the line shape of exhibits a more Lorentzian character in the absence of white light, while a more Gaussian character is apparent when using white light.
A previous study reported a similar transition from a Lorentzian to a Gaussian broadening of the X emission line with increasing CW above-band laser pumping power (using InAs QDs) . The effect was explained by ”unconventional” motional narrowing, since the behaviour is opposite from the expected one in bulk semiconductors or quantum wells. Another study reported a different situation, where the emission linewidth from a resonantly excited GaAs QD becomes narrower with higher additional CW laser power . We believe that further research, exploiting the time resolution of PCFS, will help to shine light on the connections between charge dynamics and motional narrowing.
Eventually, at large in the millisecond range, the FWHM of converge to similar value values for both cases (figure 6 (c) and (d)).
More information on the effects produced by white light illumination can be gathered by inspecting the shown in figure 6 (e) for (A large value of was chosen to exclude any correlation effects stemming from interference at the BS). In absence of blinking, we would expect a flat distribution with (Note that the antibunching at due to the single photon emission characteristics is irrelevant due to the large histogram time bin of ). In presence of blinking, has a value of for close to 0 and then decays exponentially with a characteristic time constant, which we denote as correlation time. The value of can be interpreted as the on- to off-time ratio of the QD emission . We see that white light increases from about to , which is in good agreement with the observed increase in brightness. Simultaneously, white light reduces the correlation time from to , corresponding to a faster on/off modulation of the emission. By comparing the extracted correlation time with the time at which the line broadening starts to increase in figure 6 (d) we conclude that the increased intensity produced by white light comes at the expense of an increased charge noise at intermediate timescales .
We demonstrated the implementation of a time resolved Fourier spectroscopy technique - Photon Correlation Fourier Spectroscopy (PCFS)  - to study the optical emission dynamics of GaAs quantum dots under resonant two-photon excitation. The method stands out for its robustness against imperfection of the optical components and the tremendous information content obtained per measurement. The high achievable energy- and time resolution make PCFS a viable comprehensive characterisation method for various photonic structures, outperforming established techniques such as Michelson interferometry, standard diffraction spectroscopy, or even Fabry-Perot interferometry (due to the additional time resolution). We thoroughly explained the experimental implementation and interpretation of obtained data. The experiments gave access to the time evolution of the emission line broadening of GaAs quantum dots. The acquired information allow an estimation of the shape and width of the spectral distribution at different time scales, supporting a better understanding of the underlying broadening mechanisms acting in the solid state environment, such as electron-phonon coupling  and charge noise . The charge-induced inhomogenious broadening lowers the HOM interference visibility , thus the probing of the temporal behaviour of the fluctuating charges allows us to deduce a trend for the indistinguishability of photons emitted at different time delays. Because full time scale is covered by one measurement, PCFS is clearly a viable alternative to direct HOM measurements with varying delay lines  . Moreover, we investigated the effect of additional weak white light illumination and observed an acceleration of the charge dynamics, while enhancing the brightness due to suppressed blinking [32, 33].
This work was financially supported by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (SPQRel, Grant agreement No. 679183), the Austrian Science Fund (FWF): P 29603, the Linz Institute of Technology (LIT), the LIT Secure and Correct Systems Lab, the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 601126 (HANAS), the Central European Institute of Technology (CEITEC) mobility programme (7AMB17AT044), the MEYS, the European Union’s Horizon 2020 (2014-2020) research and innovation framework programme under grant agreement No731473, the project EMPIR 17FUN06 SIQUST, and the Austrian Federal Ministry of Science, Research and Economy under the OEAD project CZ 07/2017.
We thank H. Huang, K.D. Jöns, F. Basso Basset and M. Rota for fruitful discussions.
See Supplement 1 for supporting content.
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