Spatial quantum-interference landscapes of exciton polaritons with multi-site-controlled quantum dots in extended cavity modes

doi:10.21203/rs.3.rs-4536140/v1

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Spatial quantum-interference landscapes of exciton polaritons with multi-site-controlled quantum dots in extended cavity modes

https://doi.org/10.21203/rs.3.rs-4536140/v1

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Emission properties of quantum light source can be modified through tailored photonic cavities via Purcell effect or strong light-matter interactions with various applications in integrated quantum photonics. The interacting excitonic and photonic states are core elements in the framework of cavity quantum electrodynamics. Successful characterization of subwavelength features of photonic modes from photonic crystal cavities constitutes basic building blocks for engineering the quantum photonic circuits. Potential trapping of polaritonic states has made great progress towards realizing efficient polaritonic devices. However, spatially features of excitonic states are rarely explored because extended wavefunction of quantum well excitons in the conventional quantum well – distributed Bragg reflector cavity system cannot be spatially distinguished from the photonic states. In this work, interactions of site-controlled quantum dots with a high-order cavity mode of an L7-type photonic crystal cavity with extended photonic states are spatially- and spectrally-resolved. We observed the first detuning-dependent spatial avoided crossing of the exciton-polaritons by polarized-imaging of the microphotoluminescence. Interestingly, such phenomenon is observed to be dependent on the position of the quantum dot in the cavity, with our precise control of the four quantum dot sites in the microcavity. The observed effect arises due to a unique quantum interference feature and can facilitate a deeper understanding of the spatial extent of a localized strongly-coupled excitonic state interacting with an extended photonic mode pattern. Based on our results, incorporating site-controlled quantum dots at prescribed locations in a photonic structure with tailored spatial patterns of photonic states can enable new integrated photonic devices with functionalities such as single-photon transport to remote locations for quantum information processing, quantum engineering, and quantum metrology.

Emission properties of quantum emitters can be modified through the Purcell effect by placing them into an optical microcavity. In such a scenario, the light emission rate and spatial profile can be modified according to the local density of states (LDOS) of photons of the optical cavity. Engineering the emission properties of quantum light emitters through the LDOS of the photonic environment helps develop single-photon sources with high purity and brightness. Using high-quality (Q) optical cavities confining light to a small mode volume (V), coherent or strong light-matter interactions can be achieved such that the coupling strength (g) between exciton from the quantum emitter and the cavity photons exceeds the sum of the cavity and exciton decay rates. In such a scenario, quasiparticles called polariton can be formed and its energy dispersion exhibits hybrid features of exciton and photon through the vacuum Rabi splitting (VRS). Strong coupling of a single quantum emitter with light is one of the key pathways toward demonstrating quantum gates operation and polariton manipulation with overcomed decoherence [1–2].

The subwavelength spatial features of confined photonic modes patterns (or LDOS) inherited from optical cavity parameters are intensively investigated using site-controlled pointlike dipole source in the cavity, such as deterministically placed Ge quantum dots (QDs) [3] or DNA origami [4]. Scanning electron beam can also be used as a point dipole source for deep subwavelength imaging of photonic modes [5]. Near-field scanning optical microscopy (NSOM) is also widely used for probing the evanescent contributions of the photonic crystal (PhC) cavity electrical field [6],[7]. Not restricted to surface imaging through NSOM, high-energy resolution electron energy-loss spectroscopy can be used to retrieve photonic modes at the dense core of the cavity for high-resolution tomography of the nanocavities [8]. The spatial distribution of polariton condensates [9–13] in semiconductor microcavity is also studied using various potential traps and the resulting confined polariton wavefunction is measured in momentum and real space [14–23]. The repulsive nature of polariton-exciton interactions also enables trapping polariton wavefunction using an optically generated excition reservoir which gives flexibility for visualizing spatial modulation of the trapped polariton wavefunction in real space [18]. In summary, both intrinsic cavity photonic and the polaritonic spatial distribution have been thoroughly studied. However, the spatial features of excitonic wavefunction modified through the Purcell effect and strong light-matter interactions are less studied experimentally. It is because such features can be hardly distinguished from the photonic part since most studies on spatial features of polaritons are based on 2D materials or quantum wells (QWs) in distributed Bragg reflector (DBR) cavities where the excitonic states are spatially extended. Such features can also subtly depend on the exact position of the dipole source placed in a specific photonic mode pattern, which requires spatial controllability of the dipole. On the other hand, excitonic wavefunctions from QDs that are confined in sub-wavelength zones can be in principle distinguished from the photonic part. Additionally, semiconductor QDs have shown great potential for photonic quantum computing and quantum information technology, most recent advances [24–27] including indistinguishable photon generation using QD cluster states [28–30], highly coherent optically active QD spin qubit [31], entanglement-based quantum key distribution (QKD) [32–35], time-bin entangled single-photon sources [36–37].

In this work, four evenly separated site-controlled pyramidal InGaAs QDs are incorporated along the long axis (x-axis) of the L7 PhC membrane cavity to enable examine the QD-cavity interactions at different locations. Such single site-controlled pyramidal QDs, showing promising properties such as deterministic QD nucleation and position control (< 10 nm) [38–39], narrow spectral inhomogeneity (< 10 meV) [38–40], excellent emission energy control (< 5 meV) [41–43], high symmetry with near-zero fine structure splitting (FSS) [44–45], and absence of cavity feeding from wetting layer or mulitexcitonic background [46–47], are the ideal platform for solid-state cavity electrodynamics (cQED) with applications in integrated quantum photonics and are essential for scaling up the technology [43], [48–50]. Recent works have improved the cavity Q-factor and brought site-controlled QDs into intermediate [51] and onset of strong coupling regimes [52]. Till date, however, direct measurements on spatial features of excitonic state interacting with photonic states are lacking. Based on our device, interactions of QDs with fundamental and higher-order cavity modes are resolved spatially along the x-axis of the L7 cavity by measuring the far-field PL image of the emission. We experimentally observed a spatially avoided crossing in the optical emission of QD excitons at small detuning ranges where the QDs and a spatially extended higher-order cavity mode are efficiently coupled. This is a spatial signature of the polariton state, which otherwise is not detected spectrally due to the strong cavity emission line, is due to an innate quantum interference of QD decay channels in a spatially-extended cavity field. Such spatial features are only present in the y-polarized components of the QD excitons which are parallel with the extended higher-order cavity mode emission, and it is not present when interacting with the fundamental cavity mode which is spatially confined. Additionally, the spatial avoided crossing is only present in the two QDs close to the center of the cavity but not in two side QDs, suggesting that the observed spatial features subtly depend on the position of the QD in the cavity. Using the theory of quantum interference of QD decay channels in a cavity [53], we estimate the coupling strength of the QD-cavity interaction from the observed phenomena. Our experimental demonstration of spatial superposition of excitonic and photonic states in QD-cavity interaction can be important for a deeper understanding of the spatial extent of a single QD exciton interacting with a cavity photonic mode pattern locally with contributions in quantum engineering and metrology. Based on this observed phenomenon, incorporating site-controlled QD at prescribed locations in a photonic structure with tailored extended spatial patterns of photonic states can enable new integrated quantum photonic devices for on-chip quantum information processing.

As shown in the inset in Fig. 1a, four site-controlled pyramidal InGaAs QDs evenly separated by 450 nm are deterministically placed along the x-axis in a modified L7 PhC cavity (hole pitch a = 225 nm) and are arranged symmetric with respect to the cavity center. Figures 1a and 1b present an artistic illustration of our experimental observation of the spatial superposition of the QD excitonic states in an extended cavity photonic state of the 1st order cavity mode (CM1). The total electrical field intensity distribution patterns of the 1st order cavity mode (CM1) calculated using the 3D finite-difference time-domain (FDTD) method is overlapped with the L7 cavity region in Fig. 1 (see also Figure S2 in Supplementary Information for the x- and y-polarized electric field distribution patterns of the fundamental cavity mode (CM0) and CM1). CM1 exhibits a spatially extended pattern with two spatial lobes along the x-axis of the L7 cavity, which are also artistically represented by two yellow clouds. We observe that the photogenerated QD2 exciton (blue balloon) can be coupled with the left or right lobes by controlling the detuning between exciton and CM1 energies as illustrated in Fig. 1. Such spatially distributed coupling of QD2 excitons with two lobes of CM1 can lead to photon emission (blue wavy arrows) at different locations of the L7 cavity.

We first excite four QDs simultaneously using a large laser spot (≈ 3 µm) with high power (5 µW) for each device with various air hole radii. In this case, the saturated QD transitions exhibit a broadband emission. All CMs are sufficiently pumped and can be easily identified (see Figure S3 in Supplementary Information). Specifically, we identify and carefully examine two 4QD-L7 devices, denoted as device 1 and 2, with PhC hole radii r₁ = 32 nm and r₂ = 31 nm, respectively. These two devices exhibit good spectral overlap between QD and CM1. At a temperature T of 10 K, the µPL spectra of devices 1 and 2 are measured with a small laser spot (≈ 1µm) at the center of the cavity as shown in the bottom spectrum of Fig. 2a and 2b respectively. Then we subsequently scan the focused laser spot across QD1 to QD4 along the x-axis of the L7 cavity. The variation in the exciton emission intensity can be used to identify the specific QDs contributing to each emission peak in the µPL spectrum. The results of PL scanning are summarized as intensity bars on top of each exciton peak. For device 1, four exciton lines (X1-X4) are identified in proximity to CM1 and identified to be corresponding to the same QD. For device 2, eight exciton lines are observed and associated with QD1, QD2, and QD3 while QD4 is too weak to be measured. Note that some exciton emissions are contributed from different QDs simultaneously. For example, as shown in Fig. 2b, the exciton line at ≈ 980.46 nm is contributed by both QD1 and QD2, while the exciton line at ≈ 988. 34 nm is contributed by QD2, QD3, and QD4 (contribution from QD4 is weak but is visible with scale that saturates QD2 and QD3 as shown in Figure S4 in the Supplementary Information). Note that for exciton line at ≈ 988.34 nm, the high intensity area roughly corresponding to the QD1 position is from the saturation of CM1 intensity. For simplicity, we associate these two exciton peaks to QD2 which dominates the emission compared to the contribution from other QDs. CM0 is also observed at the longer wavelength side of the µPL spectra (≈ 12 meV below the CM1) for both devices. The cavity Q-factor of device 1 is measured to be ≈ 12,000 for CM1 and ≈ 9,000 for CM0. For device 2, the Q-factor is measured to be ≈ 11,000 for CM1 and ≈ 7,000 for CM0. The observed Q-factor in our study is comparable to the recently reported value for single site-controlled pyramid QD in an L7 PhC cavity operating at the onset of the strong coupling regime [52]. Notably, these Q-factors are more than twice of previously reported values of similar devices operating in the intermediate coupling regime (Q ≈ 4,500) [51] or weak coupling regime (Q < 3,500) [46, 54, 55]. Such improvement in cavity Q-factor is achieved by reducing the pyramid nominal size to approximately 200 nm which leads to red shifting the QD emission energy to ≈ 1.24 eV (≈ 1,000 nm). At this photon energy, the reduced absorption losses from the Urbach tails of GaAs can be achieved, leading to an improved cavity Q-factor [52]. As shown in Figure S5 in Supplementary Information, the second-order photon correlation of the cavity emission which is tuned in between QD2-X2 and QD1-X1 exhibits photon antibunching with g²(0) ≈ 0.9. The large g²(0) value suggests that each QD emits photon individually through the cavity decay channel, and no cooperative spontaneous emission (superradiance) is observed. This lack of superradiance is likely due to the relatively large pure dephasing process, non-resonant pumping, and possible different coupling efficiency of QDs with the CM1.

A linear polarizer and λ/2 waveplate are used to obtain the polarization-resolved µPL spectra. The degree of linear polarization (DOLP) of QD emission is determined using the expression: \(\:DOLP=\left[\right({I}_{y}-{I}_{x})/({I}_{y}+{I}_{x}\left)\right]\), where \(\:{I}_{x}\) and \(\:{I}_{y}\) are the intensity of the QD emissions in the x- and y-polarized directions. Note that CM0 and CM1 are dominant in y-polarization and exhibit large positive DOLP values due to the geometry of L7 cavity. The tuning of QD exciton lines across the CM1 is achieved by adjusting sample temperatures. The temperature-dependent polarization-resolved µPL and the corresponding DOLP of devices 1 and 2 are shown in Figs. 3 and 4 (see Figure S6a and S9a for the accompanying spectral data). For device 1 as shown in Fig. 3b, excitonic transition X2 is tuned through CM1 from T = 6 K to 46 K, reaching a maximum DOLP of approximately 0.96 around T = 34 K when X2 is in resonance with CM1. Other exciton lines exhibit lower but positive DOLP values at all temperatures. For device 2 as shown in Fig. 4b, excitonic transitions QD1-X1, QD2-X2, and QD3-X2 are tuned through CM1 subsequently by adjusting temperatures from T = 6 K to 48 K. Large positive DOLP is observed when they are in resonance with CM1, and other exciton lines exhibit lower but positive DOLP at all temperatures (see also Figure S6b and S9b for the accompanying DOLP spectra). We define the exciton-CM detuning as \(\:{\delta\:}_{X}={E}_{X}-{E}_{CM}\), which represents the relative energy difference between exciton emission (\(\:{E}_{X}\)) and CM (\(\:{E}_{CM}\)). DOLP as a function of \(\:{\delta\:}_{X}\) can serve as an indicator of the QD-cavity coupling which is summarized in Fig. 5b for device 1 and Fig. 6b for device 2. As a result, the DOLP are positive with values > 0.5 for all QDs, due to both phonon scattering and pure dephasing [51, 54, 55]. The co-polarization of all QDs with CM1 suggests they can be simultaneously coupled with the same CM1 for device 1 and 2.

The exciton emissions and CM1 are then resolved spatially on the vertical axis of the CCD. To achieve this measurement, the sample is oriented such that the axis of the PL image corresponding to the x-axis of the L7 cavity is parallel to the spectrometer slit. The far-field image of the higher-order CMs, extending along the x-axis of the cavity can thus be spatially mapped by a proper choice of the lens focal length for focusing the beam on the spectrometer slit. However, the position of four QDs in the studied sample cannot be effectively identified using this method as a result of the limited spatial resolution \(\:{\epsilon\:}\) of the objective with numerical aperature (N.A.) due to the long emission wavelength \(\:\lambda\:\) (\(\:{\epsilon\:}\propto\:\frac{\lambda\:}{N.A.}\)) and low SNR due to the relativly weak exciton emission intensities. In Fig. 5a, the spectrally resolved far-field image of the spatial distribution of X2, X3, and CM1 of device 1 is shown at varying temperatures (see Figure S7 of Supplementary Information for full temperature range data). Note that the vertical axis corresponds to the enlarged spatial distribution along the x-axis of the cavity. A saturated logarithm color scale is used to facilitate the readibility of excitons. From the finite-difference time-domain (FDTD) calculated E-field intensity pattern, CM0 shows a single spatial lobe and CM1 exhibits two spatially extended lobes (Figure S2). With increasing temperatures, X2 is tuned across CM1 and X3 is tuned away from CM1. The vertical positions of QD excitons are extracted by Gaussian fitting their line profile and their relative values to the center of CM1 as a function of detuning \(\:{\delta\:}_{X}\) are then summarized in Fig. 5c for y- and x-polarized components. Interestingly, the y-polarized components of QD excitons tend to coalesce with the lower (upper) lobe of the CM1 when they are brought in resonance with the CM1 from the positive (negative) detuning side. Overall, this spatial behavior exhibits a shape of spatially avoided crossing within a detuning range of approximately ± 3 meV, where the DOLP of QD exhibits large positive values.

In contrast, QD excitons in x-polarized directions do not show such spatial features as shown in the lower panel of Fig. 5c. Note that, in QW systems, the spatial features of the excitonic state from its superposition with the photonic state cannot be distinguished because of the extended feature of QW excitons. In our presented QD system, fortunately the localized QD excitons show spatial coalescence with lobes of the extended photonic states at small detuning (not superimposed with photonic states at large detuning). It reveals how the subwavelength-confined excitonic states can be spatially deviated from its original state as a result of interaction with a spatially cavity extended photonic state. Note that in the polaritonic picture, such deviated excitonic state can be regarded as polaritons with more excitonic components (lower polaritonic arm at negative detuning or upper polaritonic arm at positive detuning). Rabi spectral splitting is not resolved in this case due to the much stronger CM1 emission intensity compared to the QD exciton lines.

To further investigate the effect of QD positions, as shown in Fig. 6, a similar measurement is conducted on device 2 where QD excitons can be identified and associated with different QDs (see Figure S10 in Supplementary Information for full temperature range data). Again CM1 exhibits upper and lower spatial lobes along the x-axis of the cavity but CM0 shows only a single lobe (see Figure S10). With increasing temperatures, QD2-X2 is tuned across the CM1 and QD1-X1 is tuned out of resonance with the CM1. QD2-X1, QD3-X1, and QD3-X2 are tuned towards the CM1 while QD2-X3, QD1-X2, and QD1-X3 are tuned away from the CM1. Vertical positions of y-polarized components of two center QDs (QD2 and QD3) exhibit spatially avoided crossing with the center of the CM1 within approximately ± 3 meV detuning range, corresponding to large positive DOLP values, although QD3 is only at the positive detuning side in the measured temperature range. Interestingly, excitons from side QD (such as QD1) do not show such phenomena. Again, the avoided crossing is not observed in the x-polarized component of all QD excitons as shown in the lower panel of Fig. 6c. A recent study [53] on the DOLP proposes a theory of Fano-like [56–57] quantum interferences between QD decay channels when it is coupled to the CM0 of a L3 cavity: direct QD decay into x- and y-polarized free space modes (FMs) and QD decay into y-polarized FMs through cavity decay channels. In our device, inspired by Ref. [53], with four QDs and two lobes of CM1, for QDj (j = 1, 2, 3, and 4), the total QD emission rate into the y-polarized free space mode mediated by lobe k (k = 1, 2) is given by

\(\:{W}_{yjk}=\frac{\kappa\:+{\gamma\:}_{y}}{2}-Re{[{\left(\frac{\kappa\:-{\gamma\:}_{y}}{2}-i{\delta\:}\right)}^{2}-\left(2\left|{g}_{jk}\right|-i{\chi\:}_{jk}\sqrt{{\gamma\:}_{y}\kappa\:}{e}^{-i{\varphi\:}_{jk}}\right)\times\:\left(2\left|{g}_{jk}\right|-i{\chi\:}_{jk}\sqrt{{\gamma\:}_{y}\kappa\:}{e}^{i{\varphi\:}_{jk}}\right)]}^{1/2}\)

(1.1)

where \(\:\kappa\:\) and \(\:{\gamma\:}_{y}\) are the cavity decay rate and direct QD decay rate into y-polarized FMs. \(\:{g}_{jk}\) is the coupling strength of QDj with lobe k. \(\:{\chi\:}_{jk}\) represents the spatial overlap of the direct emission patterns of QDj and field patterns of lobe k. \(\:{\delta\:}\) is the QD-CM detuning. \(\:{\varphi\:}_{jk}\) is the relative phase difference between decay channels which depends on the QD positions with respect to the lobe. In this framework, observation of detuning dependent spatial features of QD2, for instance, can be due to the dominance of \(\:{W}_{y21}\) over \(\:{W}_{y22}\) at the positive \(\:{\delta\:}\) and vice versa for negavie \(\:{\delta\:}\). Then in the small detuning region, we could phenomenologically use \(\:{W}_{y21}/{W}_{y22}\) to mimic the spatial feature of QD2 excitons which is shown as the blue curve in Fig. 6b, with \(\:\kappa\:\) = 100 µeV, \(\:{\gamma\:}_{y}\) = 0.7 µeV, \(\:{\chi\:}_{21}\) = 0.9 and \(\:{\chi\:}_{22}\) = 0.2. Assume \(\:{g}_{21}={g}_{22}={g}_{2}\) for simplicity, the blue curve with \(\:{g}_{2}=45\:{\mu\:}\text{e}\text{V}\) mimic the QD2 behavior well. Similar analysis can be done on device 1 as shown as the red curve in Fig. 5b with a coupling strength of 45 µeV. Such coupling strength aligns well with similar multi-site-controlled QD systems [58]. It is interesting to note that QD2 and QD3, which are symmetric with respect to the center of the cavity, exhibit the same spatial shape as shown in Fig. 6c. It implies the existence of the symmetry breaking of two spatial lobes of CM1 possibly due to the fabrication disorder of PhC membrane or perturbation induced by the pyramid. This leads to additional strongly-confined localized modes of CM1 spatial pattern [59–60], an additional contributions to the above framework which needs to be further investigated. To further confirm these observations solely occuring with CM1, we extend our measurement to device 3 (PhC hole radii r₃ = 36 nm), where we investigate the interaction of QDs with CM0 (see Section VII in Supplementary Information). Unlike CM1, CM0 is not spatially extended and shows a single lobe. Excitons from one of the central QD (QD3-X2 and QD3-X3) are tuned away from CM0 with increasing temperature. No avoided crossing is observed within a negative detuning range of 4 meV. It implies that the observed spatial avoided crossing is correlated with the extended feature of CM1.

Based on our experimental observation, we propose the concept of a functional spatially-distributed single-photon source using site-controlled InGaAs QDs embedded in a L7 PhC cavity, evanescently coupled with two waveguides (WGs) along the x-axis as shown in Fig. 7. The two side WGs are terminated close to two lobes of the CM1 to facilitate single-photon extraction. As shown in Fig. 7a, detuning of excitons with CM1 can be controlled for QD2 to allow single-photon propagation either to the left (negative detuning) or right (positive detuning) using evescent field coupling. Such operation can also be expressed in the logical bases, as shown in the upper left inset of Fig. 7a. With positive detuning, the output state will be |0_L1_R>; with negative detuning, the output state is |1_L0_R>. Here we define the positive detuning as logic 0 and negative detuninng as logic 1, and similarly define left-propagating photon as 0 and right-propagating photon as 1. To construct a truth table for the outputs, we retrieve the PL intensity of exciton emission when it is tuned close to CM1 from prior spatially- and spectrally-resolved PL measurments. Specifically, the positive and negative detuning values used are + 1.5027 meV and − 0.5857 meV respectively, and the resulting truth table is shown in lower right inset of Fig. 7a. We evaluate the output logic using the expression: \(\:F=\left(1/2\right)Tr\left(\frac{{M}_{exp}{{M}_{ideal}}^{T}}{{{{M}_{ideal}M}_{ideal}}^{T}}\right)\), which calculate the fidelity between the measured truth table \(\:{M}_{exp}\) and the ideal truth table \(\:{M}_{ideal}\), and obtain an estimated fidelity of 70.76 ± 1.87% for the output states [61]. Such logic operation is suitable for building single-photon sources capable of switching operations which control photon propagating along different directions. Particularly, using optical pumping from remote quantum dots [62–64] and dynamic Stark-based ultrafast control of detuning [65], it can open doors for possible low-photon-number all-optical switching [66–70] which are crucial components in linear optical and photonic quantum computers.

Based on the symmetry breaking of the CM1 lobes, Fig. 7b shows a L7 PhC cavity with two central QDs (QD2 and QD3) simulatanously embedded. When the CM1 detuning energy falls between the QD2 and QD3 excitons, single-photons from QD2 and QD3 (with different colors) can propagate to the L and R directions independently. Such behavior facilitates a tunable single-photon source for path encoding. By tuning CM1 to the middle of QD2 and QD3 (lower right inset of Fig. 7b), the single-photons can be emitted through the left or right waveguides, encoded as |L > or |R>. Due to the different energies of the QD2 and QD3 emission (ω₂ and ω₃), the resulting state can be expressed as |1_ω2, _L> or |1_ω3, _R>. Such a two-color tunable single-photon source behaves as a photon switch with different path encodings, which provides possibilities for spatially multiplexed quantum communications [71–73], quantum Boson sampling [74–77], and programmable information processing [78–86].

In summary, we report a direct experimental demonstration of the superposition of subwavelength-confined QD excitonic states from systems of four site-controlled QDs coupled to a spatially extended photonic state from a higher-order photonic mode of an L7 PhC cavity by wavelength-position mapping of QD-cavity interactions at varied detuning. It is observed that the QD excitonic states close to the cavity center coalesce with lobes of CM1 and such coalescence forms a spatial avoided crossing as a function of QD-cavity detuning. This spatial feature is a signature of the polariton state (not detected with spectral avoided crossings due to the strong cavity emission line) and arises from the unique quantum interference of QD decay channels in a spatially-extended cavity field pattern. Furthermore, the symmetry breaking of two lobes of CM1 is observed, and its origin can be further investigated through NSOM or high-energy resolution electron energy-loss spectroscopy. The observed novel phenomena can be important to understand the spatial features of QD excitonic states from a tailored photonic environment during their interactions, with applications in spatial control of single-photon transport to remote positions using detuning, extracted by waveguides for practical photonic logic operations. Using site-controlled QD systems is necessary because the observed phenomena is dependent on the QD position. Furthermore, based on the observed phenomena and the proposed functional all-optical switching, it paves the way towards developing programmable quantum photonic processors [78–86] for scalable quantum information processing, engineering, and metrology.

Sample description and fabrication

Four site-controlled pyramidal QDs are fabricated using metalorganic vapor-phase epitaxy (MOVPE) growth of In_xGa_1−xAs/GaAs (x = 0.25) on a (111)B-oriented GaAs substrate with electron-beam lithography (EBL) inverted pyramidal pits pattern with a triangular lattice of 450 nm pitch. The precision achieved with EBL is within ± 10 nm [87–88]. The resulting QDs are lens-shaped and grow at the apex of highly symmetric inverted pyramids, exhibiting well-defined (111)A Gallium-terminated facets. The nominal pyramid size is approximately 180 nm. Such QD growing process does not involve the formation of 2D wetting layers. Instead, InGaAs/GaAs quantum wires (QWR) are formed on the three wedges of the inverted pyramid during growth under some circumstances.

Photonic crystal structures with membrane thickness t ≈ 265 nm, lattice constant a ≈ 225 nm are lithographically written on top of the QD pattern and all the surrounding QDs are etched away except the four QDs in the L7 cavity. To enhance the cavity Q-factor, the first three side holes on the left and right of the L7 cavity are shifted outwards by 0.23a, 0.15a, and 0.048a. A series of identical 4QD-L7 devices is fabricated where the air hole radii (r) of the photonic crystal is scanned 30 nm to 42 nm in steps of 1 nm to facilitate the fundamental or 1st order cavity mode falls in the inhomogeneous broadened spectrum of 4 QDs.

Photoluminescence measurement

The schematic of the microphotoluminescence (µPL) setup is shown in Figure S1 in the Supplementary Information. A 900 nm laser is focused on the sample with a laser spot size of ≈ 1 µm using a microscope objective with 100× magnification, 0.7 numerical aperture (N.A.), and 10 mm working distance. The position of the focused laser spot on the sample surface is viewed using an imaging camera. The sample is placed in a helium-flow microscopy cryostat (Janis ST-500). The relative position of the focused laser spot on the sample surface can be controlled with high-precision (30 nm) piezo linear actuators. The emission is collected by the same objective and imaged on the slit of a 1-meter-long spectrometer (Horiba 1000M) equipped with a 2D liquid nitrogen-cooled charge coupled device (CCD) with a spectral resolution of 40 µeV. The optical spectra are then obtained by integrating the vertical axis of the PL image. The residual laser is filtered with a long pass (LP) optical filter. A linear polarizer and λ/2 waveplate are place in front of the spectrometer for polarization-resolved µPL measurement. For the second-order photon correlation measurements, the µPL spectra cleaned by tunable long and short pass filters are coupled to a 50:50 single-mode fiber beam splitter. Superconducting nanowire single-photon detectors (SNSPDs, PhotonSpot) and a time-correlated single-photon counting unit (PicoHarp 300) are used for building the histograms [89].

Supplementary Information

Schematics of the experimental setup and additional experimental data analysis can be found in the accompanying Supplementary Information.

Acknowledgment

The authors thank helpful discussions with Murat Can Sarihan, Cody Fan, Jin Ho Kang, and Baolai Liang from UCLA and technical help from Alexey Lyasota and Bruno Rigal from EPFL in the sample fabrication. J.H., W.L., X.C., K-C.C., and C.W.W. acknowledge support from the Army Research Office MURI (W91INF-21-2-0214) and the National Science Foundation (2137984, 1936375, and 1919355). W.L. also acknowledges support from the Swiss National Science Foundation under Project 187963. This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science by Los Alamos National Laboratory (Contract 89233218CNA000001). J.H. led the project and performed the measurements with data analysis. B.D. and E.K. designed the samples. A.M., B. D., and A. R. grew the samples and performed the photonic crystal microcavity fabrication. K-C.C. supports photon correlation measurements, and C.W.W. and E.K. aided in the project. J.H., W.L., X.C., K-C.C., and C.W.W. wrote the manuscript, with contributions from all authors.

Conflict of interest

The authors have no conflicts to disclose.

Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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Spatial quantum-interference landscapes of exciton polaritons with multi-site-controlled quantum dots in extended cavity modes

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