Optical coupling between quantum dots and photonic crystal cavities and waveguides has been studied for many years in order to explore interesting physics and to advance quantum technologies. Here, we demonstrate strain-based coupling between mechanical motion of a photonic crystal membrane and embedded single InGaAs quantum dots. The response to high frequency mechanical vibration is measured for a series of quantum dots along the length of a photonic crystal waveguide for several flexural modes by optically driving the membrane while measuring high resolution time-resolved photoluminescence. The position-dependent response is similar to the measured and calculated displacement profile of the membrane but falls off less rapidly at higher frequencies. These results indicate potential for nanoscale strain sensing with high bandwidth and sensitivity.

There is currently a strong interest in coupling quantum systems to mechanical degrees of freedom in order to dynamically change the properties of the quantum system, to sense motion or acceleration, to access the quantized nature of mechanical systems, or to mediate interactions between distant or disparate quantum systems.1–6 For solid state quantum systems, this coupling can often be accomplished through motion-induced strain that modifies band structure, crystal symmetry, and the spin-orbit interaction.

For self-assembled quantum dots (QDs), there have been many reports of static strain tuning of their optical properties7–15 and a few in which the effects of time-dependent strain were studied.16–19 One motivation for this previous work with QDs has been to tune their emission energies and also to eliminate fine structure splitting of excitons to improve entangled photon emission. These strain-induced changes also enable the use of a QD as an optomechanical transducer. For nanomechanical structures (RF mechanical resonators in particular) the exceedingly small footprint of a QD and parasitics-free optical readout provide key advantages in sensing mechanical motion over a wide range of frequencies, at least up to the inverse of the QD emission time of ∼1 ns. The integrated QDs can be placed at a position of maximum strain with nanometer-scale precision to maximize the responsivity of the transducer. By engineering the size, shape, charge state, and even coupling two QDs together, the sensitivity can be greatly enhanced.

QDs have been incorporated into optical resonators for many years, but there have been few studies with QDs coupled to mechanical resonators. In particular, the optical coupling of QDs to photonic crystal cavities and waveguides (WGs) has been studied for many years in order to enhance photon emission and collection, enable qubit-photon interfaces, and study cavity quantum electrodynamics.20–26 The optomechanical coupling of photonic crystal cavities has also been characterized27 and surface acoustic waves have been used to modulate the optical properties of photonic crystal cavities28 and QDs coupled to them.19 Here, we experimentally investigate the strain-based coupling of flexural vibrations of a photonic crystal membrane to a single embedded InGaAs QD. We demonstrate temporal and spatial resolution of QD-based optomechanical readout by mapping out the strain-based coupling of single QDs to three different flexural modes of a photonic crystal membrane. The mapping is accomplished by optically driving motion of the membrane and examining in the time domain the response of a series of QDs embedded at different positions within a photonic crystal WG. The results are consistent with the calculated flexural modes and demonstrate that the strain response of QDs falls off less rapidly at higher frequencies than the displacement readout. These results demonstrate the ability to sense motion using QDs and also the ability to perform fast modulation of QD energies in photonic structures.

The InGaAs QDs are grown by molecular beam epitaxy on a GaAs substrate within a diode structure that allows injection of electrons into the QDs.24 The QDs are randomly distributed in the plane normal to the growth direction with a density of ∼30/μm2. The structure consists of 950 nm Al0.7Ga0.3As, 30 nm Si-doped n-type GaAs; 30 nm undoped GaAs; InGaAs QDs followed by partial cap and indium flush; 70 nm undoped GaAs; 10 nm n-type Si-doped GaAs; 10 nm undoped GaAs; and 30 nm p-type Be-doped GaAs. The intermediate n-type layer in this n-i-n-i-p diode reduces the forward bias required to charge the QD, avoiding high currents through the device.

The photonic crystal pattern (15.2 μm × 8.6 μm) is produced using electron beam lithography and a SiCl4-based inductively coupled plasma to etch a triangular lattice of holes (64 nm radii) with a lattice constant of 244 nm through the diode. The sacrificial layer of Al0.7Ga0.3As is then undercut with hydrofluoric acid. There are six circular drain holes (1 μm diameter) on the edge of the pattern that extend the membrane beyond the photonic crystal pattern, resulting in a suspended membrane with dimensions of about 18 μm × 12 μm. This structure intentionally places QDs 30 nm from the center of the 180 nm thick membrane since strain from flexural motion is zero at the center. Ohmic contact is made to the top p-type layer and to the n-doped substrate. As shown in the scanning electron micrograph in Fig. 1(a), a WG consisting of a row of missing holes is patterned, and two L3 cavities (3 missing holes) are included just above the WG that are not used in the experiments of this letter. One end of the WG is terminated in the photonic crystal, and the other is terminated with a circular outcoupler23 to enhance collection efficiency.

FIG. 1.

(a) Scanning electron micrograph of a photonic crystal membrane with a WG, two cavities, and an outcoupler. (b) Measured mechanical spectrum of the membrane using reflected laser light at 939 nm to probe displacement, with the 920 nm drive laser at an average power of 300 μW. The four resonances, f0-f3, are at 4.5, 7.75, 12.3, and 12.6 MHz. The inset displays a higher resolution spectrum of the fundamental mode f0, probing with 40 μW at 956 nm and driving with 150 μW average power at 920 nm. (c), (d) Calculated displacement and volumetric strain of the membrane 30 nm from the center for the first 4 flexural modes. The displacement scale is in nm, and the strain scale is in 10−5 fractional change in volume. The calculated resonance frequencies are 4.75, 7.28, 10.76, and 10.98 MHz. Black circles in (d) indicate the positions of QDs measured in Fig. 3. (e)–(g) Calculated strain tensor components εxx, εyy, and εzz for f3, zoomed in on the waveguide, with the strain scale in 10−5 fractional change in length.

FIG. 1.

(a) Scanning electron micrograph of a photonic crystal membrane with a WG, two cavities, and an outcoupler. (b) Measured mechanical spectrum of the membrane using reflected laser light at 939 nm to probe displacement, with the 920 nm drive laser at an average power of 300 μW. The four resonances, f0-f3, are at 4.5, 7.75, 12.3, and 12.6 MHz. The inset displays a higher resolution spectrum of the fundamental mode f0, probing with 40 μW at 956 nm and driving with 150 μW average power at 920 nm. (c), (d) Calculated displacement and volumetric strain of the membrane 30 nm from the center for the first 4 flexural modes. The displacement scale is in nm, and the strain scale is in 10−5 fractional change in volume. The calculated resonance frequencies are 4.75, 7.28, 10.76, and 10.98 MHz. Black circles in (d) indicate the positions of QDs measured in Fig. 3. (e)–(g) Calculated strain tensor components εxx, εyy, and εzz for f3, zoomed in on the waveguide, with the strain scale in 10−5 fractional change in length.

Close modal

The mechanical resonances of the membrane are identified by optically driving the membrane with an amplitude modulated laser at 920 nm, focused near a center drain hole, and simultaneously measuring the displacement with a probe laser focused at different locations on the membrane. The modulation depth for the drive laser intensity is roughly 50% and results in ∼nanometer amplitude membrane vibrations when the modulation frequency is resonant with mechanical modes. The drive laser wavelength is longer than the GaAs bandgap to avoid strong excitation of carriers in the membrane and provides mechanical excitation through a local optically induced stress. The configuration used in our experiments ensures that the driving source is well-separated spatially from the QD readout. The reflectivity of a separate cw probe laser (at 939 or 956 nm) is modulated by displacement due to interference between reflection off the membrane and the substrate. The resonance spectrum in Fig. 1(b) was obtained at a temperature of ∼5 K by sending the reflected probe laser beam to a detector and spectrum analyzer. The tracking generator output of the spectrum analyzer was used to modulate the drive laser.

The four resonances f0–f3 in Fig. 1(b) correspond to the four lowest flexural modes. The fundamental mode f0 (with no nodes except the clamping lines) exhibits a displacement signal about an order of magnitude larger than the higher order modes, which all have one or more node lines. The linewidth of f0 is 2.7 kHz (Q-factor of 2250) although the lineshape is slightly asymmetric at this drive power (150 μW average power), which we attribute to nonlinearity in the response. For low vibration amplitudes the spatial distribution of the strain is expected to largely follow the local curvature of the membrane and have the same symmetry as the displacement map. The calculated displacement and volumetric strain are displayed in Figs. 1(c) and 1(d), respectively. The finite element model used for the modal analysis is based on shell elements (COMSOL) and takes into account the holes in the photonic crystal membrane, the WG, the size and shape of the membrane, and gives resonant frequencies close to those experimentally observed. Figures 1(e)–1(g) display the components of the strain tensor εxx, εyy, and εzz for the mode f3, which show that significant strain anisotropy induced by the photonic crystal holes gives rise to enhanced εxx and relaxed εyy within the WG. Similar results are found for f0 and f1.

In our experiment, the response of the QD to mechanical motion is measured by optically driving the mechanical resonances with a laser focused near a center drain hole and taking high resolution photoluminescence (PL) spectra as a function of time, synchronized to the modulation of the drive laser. The low temperature (5 K) PL is excited with 20–40 μW of laser power at 910 nm focused on particular QDs along the length of the WG, with emission collected from the outcoupler. The position of the QD is determined by the position of the laser spot on the image of the WG. The PL passes through a scanning Fabry-Perot interferometer (FPI) with a resolution of 9 μeV and then through a 750 mm grating spectrometer, before going to a silicon single photon counting module (SPCM). The FPI is locked to a tunable diode laser at ∼978 nm in order to eliminate drift and provide calibrated scans of the FPI. At each spectral position, time-correlated photon counting is performed, measuring relative to the drive laser modulation. Figures 2(a)–2(c) plot the response of a QD at about 7.5 μm from the end of the WG when f0, f1, or f3 is driven. No response is displayed for f2 since there is a node line along the WG for this mode, and no response is observed for several QDs along the WG when driven at this frequency. The QD is charged with a single electron in order to observe the charged exciton X, which has only one emission line at zero magnetic field for simplicity. Fits to the emission line as a function of time give shifts of ±27, ±4.7, and ±7.9 μeV for f0, f1, and f3. To illustrate that these shifts only occur at a mechanical resonance, we set the FPI to the side of the emission line of a different QD positioned at 6 μm, as displayed in Fig. 2(d), and vary the drive frequency. For a small drive laser power, the change in the emission should be sinusoidal in time, and the amplitude and phase of the response are plotted vs. drive frequency in Fig. 2(e). There is a strong resonance at f0 and a phase change of 180° going across the resonance, as expected for a driven resonator.

FIG. 2.

(a)–(c) Photoluminescence of a QD at the center of the waveguide (positioned 7.5 μm from the outcoupler) as a function of time and emission energy. The membrane is optically driven at 150 μW, modulated at the flexural resonances f0, f1, and f3. (d) High resolution PL of a QD positioned at ∼6 μm, with no mechanical drive, illustrating the Fabry-Perot transmission tuned to the emission edge. (e) Amplitude and phase of the modulated PL signal from (d) under a 20 μW mechanical drive laser near f0.

FIG. 2.

(a)–(c) Photoluminescence of a QD at the center of the waveguide (positioned 7.5 μm from the outcoupler) as a function of time and emission energy. The membrane is optically driven at 150 μW, modulated at the flexural resonances f0, f1, and f3. (d) High resolution PL of a QD positioned at ∼6 μm, with no mechanical drive, illustrating the Fabry-Perot transmission tuned to the emission edge. (e) Amplitude and phase of the modulated PL signal from (d) under a 20 μW mechanical drive laser near f0.

Close modal

The response of eight QDs along the length of the waveguide is measured when driving at f0, f1, and f3, providing a spatial map of the modes. The response for each QD while driving at f1 is displayed in Fig. 3(a). While each of these QDs has a different emission energy, from 1287 meV to 1316 meV, there is a clear pattern. At f1, the response near the center at 6 μm is very weak (±0.4 μeV) and increases away from this point. There is also a change in phase on opposite sides of the QD at 6 μm. This behavior is consistent with the f1 mode, which has a node at the center of the WG, with the displacement and strain reversing sign across the node. The phase change in Fig. 3(a) is due to the sign reversal of strain across the node although there are significant variations in the exact phase (standard deviation of 24°). We suspect these variations are due to small changes in the resonance frequency between measurements that, based on Fig. 2(e), result in phase changes. The responses of individual QDs to mechanical driving were often measured on different days, with each full map, e.g., Fig. 2(a), typically taking about 10 min. The amplitude of the QD shift is plotted in Fig. 3(b) as a function of QD position for all three drive frequencies, with the sign of the shift determined by the phase. The spatial dependence qualitatively matches the expected dependence for all three modes, with 0, 1, and 2 nodes for f0, f1, and f3. These measurements are similar to the visualization of flexural modes in Chladni plate experiments,29 although on a much smaller scale with stationary crystallites and yielding additional phase information.

FIG. 3.

(a) PL colormaps as a function of time and emission energy for a series of QDs along the WG, with 150 μW mechanical drive laser at f1. (b) Solid markers represent the amplitude of the QD shift as a function of QD position for f0, f1, and f3. A negative amplitude represents shifts that are roughly 180° out of phase. Curves are sinusoidal fits to the data. (c) Solid markers plot the amplitude of the reflectivity modulation as a function of position of the laser probe, which is at 956 nm. Data at f1 and f3 are multiplied by 10. No phase information is obtained from this measurement so all values are positive. Curves are fits to the data with the absolute value of a cosine. The uncertainty in the QD position is estimated at the laser spot size of about 1 μm.

FIG. 3.

(a) PL colormaps as a function of time and emission energy for a series of QDs along the WG, with 150 μW mechanical drive laser at f1. (b) Solid markers represent the amplitude of the QD shift as a function of QD position for f0, f1, and f3. A negative amplitude represents shifts that are roughly 180° out of phase. Curves are sinusoidal fits to the data. (c) Solid markers plot the amplitude of the reflectivity modulation as a function of position of the laser probe, which is at 956 nm. Data at f1 and f3 are multiplied by 10. No phase information is obtained from this measurement so all values are positive. Curves are fits to the data with the absolute value of a cosine. The uncertainty in the QD position is estimated at the laser spot size of about 1 μm.

Close modal

For comparison, we also perform spatial mapping of the displacement response along the length of the WG using the interferometry-based cw probe reflectivity, which is plotted in Fig. 3(c). The spectrum analyzer gives no phase information, so the data should represent the absolute value of the displacement. We expect that each point where the displacement goes to zero there should be a sign change in the displacement. Taking this into account, the symmetry of the displacement dependence and QD dependence match reasonably well although there appears to be a spatial shift of about 1 μm between the two, which we attribute to the imperfect reproducibility of spatial mapping. A big difference between the QD readout and displacement responses is that the higher order modes, f1 and f3, have displacement responses over an order of magnitude weaker compared to f0, while they are only 2–3 times weaker for the QD response.

We attribute this difference between the QD readout and displacement response to the relation between the strain and the wavelength λ of the flexural modes of thin plate vibrations. For a one dimensional flexural wave of wavelength λ and displacement amplitude d, the in-plane strain amplitude ε is proportional to d/λ2, assuming strain is proportional to curvature.30 For f1 and f3, which have a λ roughly 1/2 and 1/3 that of f0, we expect the strain to be 4 and 9 times larger for equal values of d. This consideration ignores strain from curvature along the smaller dimension of the membrane, which is partly justified by the model in Fig. 1(f) that shows significant strain relaxation for εyy. Based on Fig. 3(c), d is reduced by a factor of ∼30 for f1 and f3 compared to f0, which is likely due to enhanced stiffness and less efficient coupling of the drive laser to these modes. However, given that the strain for higher overtones is enhanced quadratically by the reduced flexural wavelength, we expect the strain to be reduced only by 30/47 and 30/93 for f1 and f3 compared to f0. The QD response at f1 actually exceeds this expected ratio by a factor of 2 for reasons that are not yet understood, but it is clear that the QD response has advantages over displacement at higher frequencies and smaller structures.

From reports in the literature on InGaAs QDs subject to static strain along [110],10 the QD emission energy changes by about 7 μeV/MPa or 600 meV/strain, with large variations from dot to dot (roughly ±7 μeV/MPa). The maximum QD shift measured here of ±27 μeV should correspond to a strain of about 5 × 10−5. The minimum measured QD shift, comparable to the noise, is about 0.3 μeV, which corresponds to a strain of about 5 × 10−7. An interesting point of comparison is to examine the sensitivity of photonic crystal cavities to strain. The strain expected to shift the QD by a typical emission linewidth of 3 μeV is about 5 × 10−6, whereas the strain required to shift a photonic crystal cavity resonance by its linewidth can be higher by an order of magnitude or more.12,28,31 The emission lines measured here are 15–40 μeV, in some cases partly limited by the FPI resolution, but similar photonic crystal samples have shown QD linewidths of a few μeV with resonant spectroscopy. Making use of spin transition linewidths, which can be 10 s of neV,32,33 may enable orders of magnitude improvement in sensitivity.

In this letter, we have demonstrated strain-based coupling of single InGaAs QDs to the flexural modes of photonic crystal membranes. Mapping the response of QDs along the length of a photonic crystal waveguide for different flexural modes shows excellent agreement with the expected spatial profile of strain for each mode, considering the inhomogeneity of the QDs. This result can be used to rapidly modulate the emission energy of QDs embedded in photonic structures. For application in nanomechanics (e.g., force, mass sensors, etc.) the QD-based strain readout might lead to a novel design approach where the mechanical motion invokes highly inhomogeneous stress patterns with QDs positioned34 at the stress concentrators. For such devices the lateral dimensions of QDs will enable a level of miniaturization unattainable with electrostatic or interferometric readout. While the resonant motion (i.e., frequency response) of the suspended photonic crystal has been explored in this manuscript, the time-domain nature of the implemented QD readout opens possibilities for registering fast transients in the mechanical response, in an arrangement akin to an optical pump-probe setup. Sensing applications that require high bandwidth and small structures should be particularly attractive, due to the better scaling of strain with smaller sizes and higher frequencies. Significant improvements to sensitivity can be made by taking advantage of more advanced QD structures such as QD molecules as well as the spin degree of freedom.

This work was supported by the U.S. Office of Naval Research, the Defense Threat Reduction Agency (Award No. HDTRA1-15-1-0011), and the OSD Quantum Sciences and Engineering Program.

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