Cavity quantum electrodynamics with charge-controlled quantum dots coupled to a fiber Fabry-Perot cavity
We demonstrate non-perturbative coupling between a single self-assembled InGaAs quantum dot and an external fiber-mirror based microcavity. Our results extend the previous realizations of tunable microcavities while ensuring spatial and spectral overlap between the cavity-mode and the emitter by simultaneously allowing for deterministic charge control of the quantum dots. Using resonant spectroscopy, we show that the coupled quantum dot cavity system is at the onset of strong coupling, with a cooperativity parameter of 2. Our results constitute a milestone towards the realization of a high efficiency solid-state spin-photon interface.
The interaction between a quantum emitter and a single optical cavity mode, termed cavity quantum electrodynamics (QED), has enabled a number of key experimental advances in quantum optics, including the observation of enhancement of spontaneous emission , demonstration of the photon blockade effect  and vacuum induced transparency . The key requirement for the observation of the heretofore mentioned phenomena is a large cooperativity parameter (),  which is attained if the square of the coupling strength () between the emitter and the mode exceeds the product of the emitter () and the mode () energy decay rates. Minimizing the ratio of the cavity mode volume () to its quality factor () allows for maximizing , provided that the emitter is located at a maximum of the cavity electric field (spatial overlap) and the resonance frequency of the emitter and the mode are identical (spectral overlap).
The condition has been achieved for a number of different emitters and cavity designs, ranging from single atoms coupled to Fabry-Perot cavities [2, 5] or on-chip micro-toroids  to super-conducting qubits in coplanar waveguide resonators working in the microwave domain . A technologically very relevant all-solid-state cavity QED platform in the optical domain consists of quantum dots (QD) coupled to nano-fabricated cavities. However, for these integrated devices achieving spectral and spatial overlap has been a major challenge. Even though techniques that overcome these limitations using state-of-the-art nanotechnology methods have been demonstrated, a flexible cavity design where large can be achieved for every QD would greatly improve the prospects for novel solid-state cavity-QED experiments.
In this Article, we demonstrate by coupling excitonic transitions of single self-assembled QDs to a hybrid cavity structure which consists of a GaAs/AlAs-based distributed Bragg reflector (DBR) mirror below the QD layer, and a curved fiber-end mirror approached from the top. Spectral and spatial overlap in this structure is achieved by moving the sample with respect to the fixed top mirror. Previous attempts with QDs in similar hybrid structures realized -values well below one [8, 9, 10]. Besides the significantly enhanced cooperativity value in the present setup, the main novel feature is the fact that our QDs are embedded in a structure: by separately contacting the and layers electrically and applying a gate voltage, we achieve full charge control of the QDs. This in turn opens up the path to perform cavity-QED experiments where optical transitions address given QD spin states. We thus demonstrate a fully tunable spin-cavity-QED system requiring a minimum of technological steps, together with fiber-coupled optical output, that can in principle satisfy the high collection efficiency requirement of quantum information processing protocols.
2 Cavity QD coupling
The dynamics of a two-level emitter, e.g. an excitonic QD transition, coupled to a single cavity mode is accurately described by the so-called Jaynes-Cummings model . The Hamiltonian includes a coupling term between the cavity and the emitter, which is characterized by a pulsation , also called coupling strength. This coupling is proportional to the scalar product of emitter dipole and the intra-cavity electric field generated by a single photon 
Here, is the oscillator strength of the emitter transition, the electron mass, and the effective mode volume which is defined as
is the dielectric function, whereas denotes the intracavity electric field. Equation (1) gives the maximal value of the coupling parameter assuming that the emitter is located at the maximum of the electric field. A small cavity volume enhances the coupling. For strong enough coupling, the transmission and reflection of the coupled system are significantly modified compared to the bare cavity at the condition that the emitter and the cavity can be brought to resonance.[13, 14] The effects of the cavity-emitter coupling can be quantified by the cooperativity . In the large cooperativity limit, a significant portion of the emitted light ends in the cavity mode which can be efficiently extracted, making a desirable operating range for many applications including quantum information processing. It is also well known that is required for obtaining strong photon-photon interactions. Last but not least, the fidelity of cavity mediated qubit-qubit interactions typically scale with . To achieve is challenging and generally requires careful engineering of the coupling. A common technique to increase is by reducing the cavity volume and positioning the QD at the cavity field maximum.
Usually, epitaxial QDs are randomly distributed on the wafer surface. Experiments have tried to achieve spatial matching by defining a nano-fabricated array of cavities on top of the randomly distributed QDs. Even though strong coupling could be observed this way the chance for close-to-optimal coupling is rather low. More sophisticated methods for relative alignment of QD and cavity mode have been reported in literature. The first approach controls the position of the QDs on the wafer during growth by nucleation sites, [15, 16] but the QDs grown this way still lack the close-to-transform-limited linewidths of their randomly distributed counterparts by about one order of magnitude. In the second approach the cavity is written around single pre-selected (randomly distributed) QDs which are precisely located on the wafer by SEM metrology  or optical spectroscopy [18, 19, 20].
Even though these approaches enable excellent spatial overlap, the resonance frequencies of cavity and QD are generally different which implies the need of post-processing tuning mechanisms to bring the system into resonance. Besides irreversible fine-tuning by digital etching  an in-situ reversible tuning mechanism for photonic-crystal cavities by means of adsorbingdesorbing gas molecules was demonstrated  and is now widely used in many laboratories. The hybrid cavity-QED approach we detail here stands in stark contrast to these earlier approaches, since it allows for a much more straightforward spatial and spectral alignment with any of the QDs on a given device.
3 Experimental setup
Our semi-integrated hybrid cavity system consists of a sample-based DBR mirror below an active QD layer and a curved fiber-end mirror (Figure 1a) which is approached from the top. The sample is mounted on a stack of piezoelectric nanopositioners for precise positioning in all three spatial dimensions. The fiber in turn is fixed above the sample surface as illustrated in figure 1b). For performing photoluminescence spectroscopy, an intensity-stabilised pulsed Ti:Sapphire laser at 785nm is used. The system is excited through the DBR-coated fiber mirror close to a reflectivity minimum of the mirror. The photoluminescence is collected through the same fiber and sent to a high-resolution grating spectrometer with a nitrogen-cooled CCD camera for recording the spectrum. For resonant spectroscopy, a mode-hop free intensity-stabilized diode laser, tunable from 890 nm to 910 nm, is sent through the fiber, and the transmitted light is collected on a silicon detector mounted at the bottom of the sample (referred to as transmission detector henceforth). The transmission signal is expected to be orders of magnitude larger than the resonant reflection signal due to the considerably higher reflectivity of the top fiber mirror (Transmission and losses for this mirror are 26 ppm and 13.5 ppm, respectively). The transmission signal is directly amplified by a high-gain low-noise amplifier. Sample and fiber are part of a home-built cage system that is inserted into a buffer-gas filled dipstick which in turn sits in a liquid He dewar and is kept cold at 4K.
4 Fiber mirror cavity
4.1 Cavity modes
To a good approximation, the cavity can be considered as a planar-concave Fabry-Perot cavity whose properties are determined by the radius of curvature of the concave mirror at the tip of the fiber and by the length of the gap between the fiber and the substrate. A given cavity mode is characterized by its polarization and by three integers and , which label the longitudinal and the transverse mode structure, respectively. For a curved mirror with rotational symmetry, one would expect the transverse modes with equal values of to be degenerate. As figure 2a illustrates, in our system modes with the identical values of are non degenerate, because of the slight ellipticity of the fiber mirror. In addition, all the modes exhibit a polarization splitting as demonstrated for the TEM mode in the inset of figure 2a. In the remainder of this paper, we label the two orthogonally polarized modes as TEM and TEM.
These effects are summarized by the following equation which gives the resonance frequency of a mode
The and symbols distinguishes the two eigenpolarizations which correspond to two orthogonal linearly polarized modes. The associated phase for the mode is a consequence of the small birefringence present in both DBR mirrors and is on the order of 10 mrad. The two curvature radii and account for the ellipticity of the fiber mirror (the two values typically differ by a few percents), and the phase is the sum of the phases acquired through the reflection on each DBR. This phase varies slowly around the common central wavelength of the DBRs and can be considered as constant to first approximation ( for our sample), allowing a direct determination of the mode frequency. In order to calculate resonance frequencies beyond this approximation, equation 3 must be solved with the frequency dependence of the phase included explicitly which can be determined using numerical methods, such as the transfer matrix method.
We expect the transverse distribution of the electric field in one mode to be given by the corresponding Hermite-Gauss function. This dependence can be observed by laterally scanning the fiber above the sample and monitoring the emitted fluorescence collected through the fiber on a spectrometer (see figure 2b). As the QD is much smaller in size compared to the wavelength of light, and the fiber only collects light in the cavity mode at the cavity resonance, scanning QDs in space while exciting with a non-resonant laser power above saturation gives a very accurate measurement of the cavity mode profile. Plotting the integrated fluorescence in a narrow frequency window versus the lateral fiber position consequently gives a cut of the intensity profile of this mode assuming only one QD to be present in the narrow frequency window. Figure 2c-2e show three profiles obtained from such measurements. As expected for the fundamental mode, the profile fits quite accurately to a Gaussian (figure 2c).
The tunability of the cavity is assessed in figure 2f). By collecting photoluminescence while slowly decreasing the cavity length we observe how we can access smoothly many FSRs to couple the QDs to any cavity energy for many different lengths. Emission from cavity modes are observed in a wide range due to the fact that this particular scan was performed in a sample region with high QD density. The large variation of QD sizes is responsible for the broad band emission. Being able to choose the cavity length while working with the same QD also opens up the possibility to build cavities with very long lifetimes. For example, the value of the quality factor reported in the inset of figure 2a can be significantly increased (we have been able to measure ) if it is needed in a specific experiment.
4.2 Cavity mode volume
Considering a TEM mode, the effective mode volume is given by where is the mode waist on the substrate, is the effective length of the mode including the penetration depth of the cavity field into the DBRs, and is the ratio of the field maximum in the vacuum to the maximum in the whole cavity (for our sample ). This length can be precisely estimated using a transfer matrix calculation. For our sample, we obtain m. The waist can be estimated from Gaussian optics to be
In order to minimize the mode volume and thus maximize the cavity/emitter coupling, both and should be minimized. Using the CO laser ablation technique, radii down to 10 m have been reported. Here, we use a fiber mirror with a radius of curvature on the order of 75 m . The smallest that can be achieved is often limited by geometrical aspects such as the depth of the mirror structure at the fiber tip (on the order of a few m) and/or surface defects at the fiber tip. These are however not fundamental reasons and cavities with lengths on the order of a few wavelengths could in principle be fabricated. This sets the minimal waist for this type of cavity to lie between 1 and 1.5 m for a design wavelength close to 1 m. In the following, we minimize by moving the fiber down until it touches the substrate and we step back by a few hundred nanometers. From the coupling strength to the quantum dot that we measured (see section 8), we can estimate the effective mode volume to be smaller than 150 for our current setup (900 nm).
5 The device: Charge controlled QDs
A schematic of the sample is shown in figure 3a). It was grown by molecular beam epitaxy (MBE) on a GaAs (100) substrate. The epitaxial structure growth starts with a 300 nm GaAs buffer, followed by 28 pairs of AlAs/GaAs that form the bottom mirror of the cavity, which sets the timescale for the photon lifetime in the cavity since it has lower reflectivity than the top fiber mirror. The reflectivity of the bottom mirror was measured over the relevant wavelength range at room temperature and is plotted in figure 2c). The interference pattern below 860 nm is slightly smoothed out due to absorption by the GaAs layers as was also confirmed by a transfer matrix calculation.
A key feature of our system is the tunability of the exciton energy and the control of the QD charging state. This is achieved by a p-i-n structure on top of the AlAs/GaAs DBR. The n-layer consists of a 40nm Si-doped GaAs layer with a carrier concentration of 1 cm, while the top p-layer is a 35 nm wide GaAs layer doped with C atoms. The quantum dots are sandwiched between the two conductive layers and were grown without rotating the substrate to ensure a smooth gradient in QD density across the wafer. The QD layer is separated from the conductive n-layer through a 40 nm tunnel barrier of undoped GaAs. An additional AlGaAs blocking barrier was introduced between the QD and the p-doped region. In the experiments reported here, we used two different samples, with the QD emission wavelengths centered at 900 nm and 970 nm, respectively. The emission energy of the quantum dots was controlled by using the partially covered island method . The overall thickness of all the layers on top of the DBR mirror amounts to , i.e. one optical wavelength. This ensures that the QD layer will be at an antinode of the cavity field along the growth direction. The p- and the n-layers result in an intrinsic electrical field that can be modulated by applying an additional external bias voltage. For contacting the sample, the processing procedure was as follows: In a small portion of the sample, the top 80 nm were removed by wet etching with HSO:HO:HO. In a second step, an Ohmic contact to the n-doped layer was formed by annealing some Indium on the sample surface for 360 s at an oven temperature of around 350C. The highly doped p-layer was contacted using silver paint deposited on the sample surface. The I-V curve of the final device is displayed in figure 3b). The deviations from the ideal diode curve arise from the very simple processing protocol and unavoidable imperfections in the sample due to microscopic structural defects. Due to the p-i-n structure of the device, the application of a bias voltage induces current flow of both electrons and holes which - through relaxation into the QDs and the wetting layer - leads to spontaneous light emission. Part of this electroluminescence signal was recorded on our bottom detector when recording the I-V characteristics and is plotted in the same figure 3b) (blue circles).
Recently, experiments based on photonic-crystal and micropillar technology demonstrated charge control of QD excitons [25, 26, 27, 28], also in combination with resonant spectroscopy . While all of these approaches require rather sophisticated fabrication procedures, it turns out that the simple processing steps listed above are sufficient to obtain charge control in our fiber-cavity setting. Charge control is demonstrated in figure 3d: Here, photoluminescence emission was recorded as a function of applied electric field. Different charging states can clearly be identified in the spectrum, with the emission lines originating from the neutral exciton (X), biexciton (XX), and the positively (X) and negatively (X) charged excitons (trions) indicated in the plot. In addition, emission involving other multiply-charged states are visible. The capability to deterministically charge the QD allows us to selectively address trionic QD states and therefore make use of the spin degrees of freedom. The spin properties of the present sample were investigated (without fiber mirror), and complete spin pumping in Faraday geometry  was found. The inset of figure 3d) displays the energy shift of the neutral exciton line X due to the quantum confined Stark effect as a function of applied electric field. The significant Stark shift opens up the possibility of electrically tuning excitonic states into resonance with the cavity mode as will be demonstrated later in the article.
6 Photoluminescence spectroscopy
To demonstrate coupling of the cavity mode to single QD transitions, we perform photoluminescence (PL) spectroscopy with an above-bandgap pulsed laser at 785 nm. The emitted photons from the coupled QD-cavity system are analyzed on a grating spectrometer. In a first experiment, we fix the QD gate voltage such that the X exciton is stable. We continuously collect PL spectra while tuning the cavity length and thereby scanning the cavity resonance across the X transition. The result of this measurement is shown in Figure 4a. The cavity resonance is detectable for all cavity lengths within the scanning range even at very low powers due to off-resonant cavity feeding [31, 32]. When the cavity resonance wavelengths for the two non-degenerate orthogonally linearly polarized modes TEM and TEM match the X emission wavelengths around 977.3 nm, there is a clear increase in the detected intensity. A careful analysis, shown in Figure 4c, indicates that the detected intensity follows a Lorentzian line shape as a function of cavity length which is to be expected for an emitter weakly coupled to a cavity.
As Figure 4b illustrates, for other QD exciton emission line, the wavelength at which the maximum intensity is detected differs for the TEM and TEM mode. We attribute this difference to the X-Y splitting  of the transitions of the charge state. For a single QD, the coupling strength of a particular transition to a given cavity mode depends on the relative angle between the QD axis and the direction of linear polarization of the mode. Thus a rotator in the system would allow for maximizing the QD-cavity coupling by aligning the QD dipole emission along the cavity polarization axis.
A complementary PL spectrum can be obtained by fixing the cavity length and tuning the QD bias voltage, which in turn tunes the QD transition wavelengths via the quantum confined Stark effect. Such a spectrum is shown in figure 4d). In this case, two spectrally close excitonic emissions, which we tentatively attribute to and are tuned in resonance with one of the cavity modes. Again, the detected intensity increases by an order of magnitude when the QD transitions are tuned to the cavity resonances.
We also performed PL saturation measurements (pulsed excitation) with fixed cavity length and bias voltage, with the cavity wavelength tuned to a single QD excitonic transition. The intensity of the emitted light from the QD-cavity system as a function of excitation power is displayed in the inset of figure 4a). The clear saturation behaviour above a certain excitation power provides evidence that one and only one QD is coupled to the cavity mode for the particular wavelength detected here.
A real door-opener for a multitude of future experiments with our new system is its flexibility, reversibility and speed of tuning by changing either the cavity length or controlling the gate voltage, in particular when compared to more established techniques in other systems such as gas deposition or temperature tuning [22, 34]. Another advantage of the present system is the built-in fiber coupling which allows for straightforward efficient probing using more sophisticated techniques such as high-resolution resonant spectroscopy.
7 Resonant spectroscopy
We performed resonant laser spectroscopy by measuring the transmission of a tunable diode laser through the coupled QD-cavity system using the transmission detector. Since the top fiber mirror has a higher reflectivity than the one grown on the sample, most of the intra-cavity photons escape through that bottom mirror onto the photodiode. The transmission signal can be directly detected with a low-noise amplifier without the need for a lock-in technique.
Figure 5a) shows the transmission (normalized to the maximum transmission) spectra as a function of resonant laser wavelength (horizontal axis) and cavity length (vertical axis). The QD modifies the Lorentzian transmission of the unperturbed cavity (top and bottom of the scan) by introducing a dispersive response fixed at the QD resonance. As we performed all the experiments well below saturation, we are able to extract the coupling parameters directly from a fit to the data using 
where is the cavity angular frequency, is the QD transition angular frequency and is the bare cavity transmission. From the experimental data we extract =11.7eV, =11.4eV, and =78eV. The inset in figure 5a & 5b shows a calculated spectral map using these parameters and the above expression 5.
In figure 5b, the cavity length is kept constant while the bias voltage tunes the QD resonance through the cavity resonance. Note that for each spectrum the background electroluminescence (fig 2b) seen by the transmission detector was subtracted and finally normalized to the peak transmission. Again as in figure 5a, the QD significantly modifies the transmission spectrum through the cavity.
We plot the equation 5 using the the previously extracted parameters in the inset of figure 5b). Figure 5c shows a horizontal line cut taken from data presented in Figure 5b at a bias voltage of 1.458 V (blue line), and its corresponding fit (red line). The dip appearing in transmission in Figures 5a-c when the laser is on resonance with the exciton is a spectacular signature of quantum interference effect. 
The great flexibility of our system is best illustrated by the fact that one can demonstrate the QD-cavity coupling with an almost randomly chosen laser wavelength by adjusting cavity length and QD bias voltage simultaneously. We demonstrate this by parking the laser at a wavelength of 903.865 nm and scanning both cavity length and gate voltage. The resulting 2D map is displayed in figure 5d) with the resonance condition indicated by the black circle.
8 Onset of unity cooperativity and strong coupling
A particulary interesting regime for experiments in cavity quantum electrodynamics is the regime of strong coupling where the photon exchange between emitter and cavity mode is as fast or faster than the photon decay from the system. In this regime, new eigenstates, so-called polaritons, form. In general, the eigenenergies of the coupled QD-cavity system can be determined from 
By carefully positioning QD and cavity relative to each other, the coupling strength can be optimized by reading out the PL counts. Once an optimally coupled QD was found, we recorded resonant transmission spectra as a function of cavity length as displayed in figure 6a). Clearly, when crossing the QD resonance, the cavity mode splits into two distinct peaks which form an avoided level crossing. Fits to the spectra (see e.g. figures 6 b) and c)) yield =12.32.5eV, =50.02.7eV, and =3.10.7eV which implies that for this particular QD with a narrow linewidth the system is at the onset of strong coupling with .  We fit the peak positions for each spectrum in figure 6a) and plot the results in figure 6d) which clearly yields an avoided level crossing that is well described by equation 6 and the given value for .
The present QD-cavity system has a cooperativity of C2. In order to enlarge this number significantly, different strategies can be adopted: increasing the number of layers of the DBR semiconductor stack while simultaneously reducing the doping density in the and layers should decrease significantly. In addition, a smaller radius of curvature of the fiber mirror would reduce the effective mode volume of the cavity and hence increase . On the emitter side, other QD systems such as quantum-well monolayer fluctuations with oscillator strengths up to one order of magnitude larger , could push the system much deeper into the strong coupling regime, thereby increasing the cooperativity up to an order of magnitude.
9 Conclusions and outlook
In this article, we have presented a very versatile QD-microcavity platform for performing state-of-the-art cavity QED experiments. The system is fully tunable, i.e. both cavity length and QD energy can be controlled at will. The high of our system together with the moderate mode volume brings us into the high-cooperativity regime where the coherent interaction starts to dominate the system dynamics. The onset of strong coupling was demonstrated through the observation of an avoided level crossing in resonant transmission spectroscopy. We anticipate that with some simple improvements on both the cavity and emitter side, the system can enter deeply into the strong coupling regime. The ability to control the charge state of the QD by means of the structure will allow us to perform experiments on quantum information processing with a first step being the demonstration of an efficient fiber-coupled spin-photon interface . Our QD-fiber-cavity system might then serve as a node in a future solid-state based quantum network similar to what has been demonstrated with atom-cavity interfaces .
Appendix A Cavity modulation for QD spectroscopy
Sample is not always perfectly mounted flat due to fixing imperfections, and it can have a slight angle with respect to the fiber. Hence when performing horizontal scans with the fiber cavity, the resonance of the cavity changes with horizontal position (for a fixed vertical piezo position) due to the change in length of the cavity. This makes some of the experiments requiring horizontal scanning (determination of the mode profiles for example) more involved, since the resonance of the cavity easily shifts away from QD resonance. Here, we overcome this difficulty by modulating the cavity length, , with a triangular waveform at a frequency . The amplitude f(t) thereby varies between and . This way, small perturbations in cavity length do not matter anymore, provided the the optical signal is aquired with a time constant . On the spectrometer, the narrow cavity modes are transformed into broad rectangular resonances with a width given by
An example of such a spectrum is shown in figure A1 c). There we can see the PL of a spectrum of the static cavity. As schematically shown in figures A1 a) and A1 b), by applying a zigzag (triangular) modulation of 4 Hz in frequency and amplitude of only 58nm, we obtain the spectra seen in figure A1 d). There we can observe the broad window through which we can detect the luminescence. As seen in figure 3b) still by scanning horizontally the cavity the central frequency of the cavity will shift with position (dashed lines in figure 3b), but any QD measured within the modulated cavity window can be analyzed (QD1 in figure 3b). An example spectrum of QDs through the TEM is shown in figure A e). Additionally, from the slope of the dashed lines in figure 3b we could extract the sample-fiber tip tilt of 0.15.
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