Quantum dot optomechanics in suspended nanophononic strings

The optomechanical coupling of quantum dots and flexural mechanical modes is studied in suspended nanophononic strings. The investigated devices are designed and monolithically fabricated on an (Al)GaAs heterostructure. Radio frequency elastic waves with frequencies ranging between $f$=250 MHz to 400 MHz are generated as Rayleigh surface acoustic waves on the unpatterned substrate and injected as Lamb waves in the nanophononic string. Quantum dots inside the nanophononic string exhibit a 15-fold enhanced optomechanical modulation compared to those dynamically strained by the Rayleigh surface acoustic wave. Detailed finite element simulations of the phononic mode spectrum of the nanophononic string confirm, that the observed modulation arises from valence band deformation potential coupling via shear strain. The corresponding optomechanical coupling parameter is quantified to $0.15 \mathrm{meV nm^{-1}}$. This value exceeds that reported for vibrating nanorods by approximately one order of magnitude at 100 times higher frequencies. Using this value, a derive vertical displacements in the range of 10 nm is deduced from the experimentally observed modulation. The results represent an important step towards the creation of large scale optomechanical circuits interfacing single optically active quantum dots with optical and mechanical waves.

modulation. The results represent an important step towards the creation of large scale optomechanical circuits interfacing single optically active quantum dots with optical and mechanical waves.
Phonons are of paramount importance for the realization of hybrid quantum architectures. 1 These fundamental excitations of condensed matter couple to almost any quantum system and experience only minute dissipation in crystalline solids.
Surface acoustic waves (SAWs) are one of the very few phononic technologies of industrial relevance 2 and have recently attracted widespread interest in quantum technologies. This interest has been sparked by theoretical [3][4][5] work and hallmark experiments on superconducting qubits. [6][7][8] Semiconductor quantum dots (QDs) enable the direct transduction of the SAW phonons' radio frequencies to the optical frequencies of QD excitonic two-level system 9,10 via deformation potential and Stark effect couplings. [11][12][13] One of the first applications of SAW envisioned and implemented in QD-based quantum technologies was the dynamic acoustic pumping and charge state control via the acousto-electric effect. [14][15][16][17] Moreover, QDs can be integrated in fully suspended photonic crystal membranes with SAW-tunable circuit elements 18,19 enabling the dynamic control of light-matter interactions at gigahertz frequencies. 20 Such suspended systems confine both photons and phonons in the plane. 21 Interestingly, flexural, anti-symmetric Lamb modes excited in these membranes are stress-neutral in the center plane of the membrane, i.e. the volumetric strain vanishes.
Thus, so far experiments focusing on the optomechanical coupling between QDs and mechanical excitations used samples in which the dots were deliberately displaced from the membrane's center. [22][23][24] In contrast, further experiments showed an unexpected pronounced tuning of QDs placed in the center of the membrane 20 indicating strong optomechanical coupling of the QD exciton with modulation amplitudes comparable to that observed of a high-Q cavity mode.
In this communication, we report on the optomechanical coupling of single GaAs/(Al)GaAs QDs to the phononic modes of suspended nanophononic strings. In the studied frequency band from = 250 MHz to 400 MHz, we observe large tuning amplitudes which are enhanced by more than a factor of 15 compared to those of QDs strained by conventional Rayleigh SAWs propagating on the unpatterned surface. Devices were fabricated on a (Al)GaAs heterostructure containing a single layer of GaAs QDs embedded in (Al)GaAs barriers in its center. This type of QD is particularly suited to study strain tuning by elastic waves 25 since charging due to the acoustoelectric effect 16,26 is almost completely suppressed. Figure 1 [27][28][29] It is given by in which ∆ and denote the amplitude of the optomechanical modulation and the unperturbed linewidth of the QD emission line, respectively. Best fits of Equation 1 (full lines in Figure 1 (b) and (c)) faithfully reproduce the experimental data and allow us to quantify ∆ to be 0.23 meV and 1.40 meV for the Rayleigh SAW and on the nanophononic string, respectively.  In conclusion, we demonstrate that QDs can be coupled to flexural modes of a suspended nanophononic string and observe a strong enhancement of the optomechanically induced spectral modulation at radio frequencies exceeding 400 MHz, important to reach the resolved sideband regime. 10 In this regime parametric transduction becomes accessible and enables the implementation of hybrid quantum dot optomechanical transduction and control schemes. Furthermore, our scheme can be directly applied to spin qubits of optically active defect centers, 35-37 for which recent proposals [38][39][40] promise high fidelity quantum control schemes, or QDs forming in nanowires. 12,13,17,41,42 Moreover, our work marks a first important step to interface optomechanical crystals with engineered dispersions of phonons and photons and operation frequencies in the GHz domain. 43,44 Finally, the observed optomechanical coupling arises exclusively from shear strain modulating the valence band of the semiconductor, an rarely studied effect compared to normal strain coupling.

Experimental Section
Sample design: Nanophononic strings were fabricated on an (Al)GaAs heterostructure consisting of the following layer sequence (beginning from the GaAs substrate): a 1.2 µm thick Al0.8Ga0.2As sacrificial layer, a 4 nm thick GaAs layer followed by a 77 nm thick Al0.4Ga0.6As layer, the GaAs QD layer obtained by filling with 2 nm GaAs droplet-etched nanoholes, a 75 nm thick Al0.4Ga0.6As layer, and a 4 nm GaAs think layer. The beam pattern was defined by electron beam lithography and transferred using ICP-RIE using a BCl3/Cl2/Ar process and undercut was obtained by hydrofluoric acid (HF) and critical point drying. After underetching, the QDs are located in the middle of the nanobeam along the growth direction and the two 4 nm thick GaAs layers protect the Al-containing strings from oxidation. Cr/Au (5 nm/ 50 nm) multi-passband IDTs were fabricated in a standard lift-off process. 29 Acousto-optical spectroscopy: QDs are studied by conventional low temperature ( = 10 K) microphotoluminescence (µ-PL) spectroscopy. Rayleigh SAWs are generated on the unpatterned heterostructure by a signal generator connected to the IDTs. The SAW is generated in short pulses to suppress unwanted heating of the sample. 20 Furthermore, the laser exciting the µ-PL is synchronized with the electrically generated SAW pulses by a delay generator to probe the QDs only when the acoustic wave is present. In stroboscopic experiments the laser repetition rate was commensurate to 56 and the relative phase was tuned. 18,45 Numerical simulations: The phononic mode spectrum was calculated using COMSOL Multiphysics using a tetrahedral mesh and bulk mechanical properties of all materials of the heterostructure.
The geometry for simulations comprises in vertical direction the nominal heterostructure and in the plane the shape of the string and the adjacent undercut area as derived from optical microscope and SEM images.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.
Received: ((will be filled in by the editorial staff)) Revised: ((will be filled in by the editorial staff)) Published online: ((will be filled in by the editorial staff))  Figure S1. Stroboscopic photoluminescence spectroscopy of QDs in the unpatterned region (a) and on the nanophononic string (b). Left panels: Emission spectra of the QDs without RF applied. Right Panels: Normalized spectral shift of the QDs with RF applied as a function of delay times between laser and SAW excitation (symbols) and best fit of a sinusoidal modulation (lines). Colors correspond to QD emission line marked in the spectrum by an arrow of the same color. Figure S1 shows stroboscopic photoluminescence spectroscopy 45  shown in the right panel of Figure S1 (a). This is expected as the wavelength of a Rayleigh wave on the unpatterned region is much larger than the diameter of the diffraction-limited laser spot, so that the local phase of the SAW is approximately the same for all QDs at every delay time during the scan. Hence, all QDs exhibit the same phase relationship.

Quantum dot optomechanics in suspended nanophononic strings
On the nanophononic string, the phase dependences of − : of investigated QDs exhibit clear phase shifts as can be seen in the right panel of Figure S1  This observation excludes the exclusive excitation of standing waves and symmetric Lamb waves. For the first, nodes are fixed and do not show any modulation. In the two neighboring regions, points oscillate in phase within each region and relative with a phaseshift. For the latter, the wavelength is greater than that of the Rayleigh wave.