Tunable Coupling of a Double Quantum Dot Spin System to a

Aug 7, 2019 - The interaction of quantum systems with mechanical resonators is of practical interest for applications in quantum information and sensi...
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Tunable coupling of a double quantum dot spin system to a mechanical resonator Samuel George Carter, Allan S Bracker, Michael K. Yakes, Maxim K. Zalalutdinov, Mijin Kim, Chul Soo Kim, Bumsu Lee, and Daniel Gammon Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.9b02207 • Publication Date (Web): 07 Aug 2019 Downloaded from pubs.acs.org on August 8, 2019

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Tunable coupling of a double quantum dot spin system to a mechanical resonator Samuel G. Carter1,* , Allan S. Bracker1, Michael K. Yakes1, Maxim K. Zalalutdinov1, Mijin Kim2, Chul Soo Kim1, Bumsu Lee,3 and Daniel Gammon1 1Naval

2

Research Laboratory, Washington, DC 20375

KeyW corporation, 7740 Milestone Parkway, Suite 150, Hanover, Maryland 21076 3NRC

Research Associate at the Naval Research Laboratory, Washington, DC

ABSTRACT: The interaction of quantum systems with mechanical resonators is of practical interest for applications in quantum information and sensing and also of fundamental interest as hybrid quantum systems. Achieving a large and tunable interaction strength is of great importance in this field as it enables controlled access to the quantum limit of motion and coherent interactions between different quantum systems. This has been challenging with solid state spins, where typically the coupling is weak and cannot be tuned. Here we use pairs of coupled quantum dots embedded within cantilevers to achieve a high coupling strength of the singlet-triplet spin system to mechanical motion through strain. Two methods of achieving strong, tunable coupling are demonstrated. The first is through different strain-induced energy shifts for the two QDs when the cantilever vibrates, resulting in changes to the exchange interaction. The second is through a laser1 ACS Paragon Plus Environment

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driven AC Stark shift that is sensitive to strain-induced shifts of the optical transitions. Both of these mechanisms can be tuned to zero with electrical bias or laser power, respectively, and give large spin-mechanical coupling strengths.

KEYWORDS: Double quantum dots, hybrid quantum systems, spin, optomechanics, cantilever, strain

Spin is typically the most coherent degree of freedom of any quantum system due to its weak coupling to the environment. While this makes spin a good choice for storing quantum information, it is often difficult to controllably couple a spin quantum bit (qubit) to other degrees of freedom and other qubits, which poses a problem for applications in quantum sensing and computation. In recent years, there has been significant effort to couple solid state quantum systems to mechanical resonators,1–10 as mechanical motion is seen as a universal transducer between disparate quantum systems.11–15 While this area of hybrid quantum systems has seen dramatic progress in achieving strong coupling to mechanical motion and cooling to the quantum ground state of the mechanical resonator,2,5,15–17 most of this work has not taken advantage of spin, due to relatively weak spin-mechanical coupling. Some early efforts to couple spin to motion have focused on magnetic coupling by placing the spin in a strong magnetic field gradient, such that displacement changes the field at the position of the spin. The coupling strength tends to be limited by the strength of the magnetic field gradient, and so far relatively small single phonon couplings have been achieved.1,3 More recent efforts have made use of intrinsic strain coupling, in which strain produced within the mechanical structure modifies the electronic states.4,6–8,10,15,18–27 This coupling is typically quite weak for electron spins, 2 ACS Paragon Plus Environment

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but recent results on hole spins in InGaAs quantum dots (QDs) and the silicon vacancy in diamond have shown significantly better couplings,10,25 due to the stronger spin-orbit interaction. The use of surface acoustic wave (SAW) devices that operate at high mechanical frequencies is also promising for improving single phonon coupling to spin14,23,27. In all of these cases, the spinmechanical coupling strength is difficult to control in situ and obtaining strong coupling is quite challenging. Here we present results on pairs of interacting InGaAs QDs embedded in cantilevers, in which two new spin-mechanical coupling mechanisms are demonstrated, which are much stronger than other mechanisms and are tunable to zero coupling. Each QD is charged with a single electron, using a built-in diode, and tunnel coupling between QDs results in a singlet-triplet ground state spin system. These quantum dot molecules (QDMs) have the advantage of a large exchange splitting even at zero magnetic field,28–30 tunable interactions and optical transitions,31–33 and a weaker hyperfine interaction.34 The first physical mechanism is due to the time-dependent strain gradient within the vibrating cantilever, where the strain varies through the thickness of the cantilever (see Fig. 1). Instead of coupling to the individual spins, the strain gradient affects the exchange splitting between the singlet and triplet with a coupling strength at least three orders of magnitude larger than for an electron spin in a single QD. The coupling is also strongly dependent on the electrical bias applied to the QDs and tunes through zero at the minimum exchange splitting. The second physical mechanism uses a near-resonant laser to induce an AC Stark shift that changes the exchange splitting. The magnitude of the change depends on the laser detuning from the optical transitions, which in turn depends on strain. The coupling strength can be tuned with the laser power and detuning and is measured through Raman spin flip emission driven by the near-resonant laser. 3 ACS Paragon Plus Environment

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Both of these mechanisms are very promising for improving quantum sensing of motion and reaching sensitivity to a single quantum of motion.

Figure 1. Quantum dot molecule in a cantilever. Finite element model of a 5.6 m long cantilever, showing the in plane strain (𝜖𝑥𝑥 + 𝜖𝑦𝑦) for a tip displacement of 1 nm. The color scale labels are in 10-6 fractional change in length. An illustration of a quantum dot molecule at the actual depth is superimposed on the expanded view of the strain depth profile. In the lower right, the reflectivity response (black line) of the QDM1 cantilever vs. mechanical drive frequency is plotted, along with a fit (red line) to the square root of a Lorentzian.

Pairs of InGaAs QDs are grown by molecular beam epitaxy, separated by a 9 nm tunnel barrier of GaAs/AlGaAs. They are grown within a 200 nm thick n-i-n-i-p diode structure that allows charging each QD with one electron, and this diode structure is grown on top of a 950 nm AlGaAs layer.30 The sample is processed into cantilevers oriented along a 〈110〉 direction by etching a pattern through the diode structure and then selectively etching away the sacrificial AlGaAs 4 ACS Paragon Plus Environment

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underneath. This forms cantilevers that are 5.6 m long and 2 or 3 m wide, with the lowest vibrational frequencies of 3.68 MHz and 3.57MHz, respectively. As shown in Fig. 1, these modes are excited with a mechanical drive laser at 900 nm focused near the base of the cantilever, amplitude modulated at the resonance frequency. The displacement response using the reflectivity of a probe laser focused on the tip is plotted as a function of mechanical drive modulation frequency, showing the resonance at 3.5695 MHz with a Q of 16,900. Displacement (i.e. bending) of the cantilever results in an in-plane (𝜖𝑥𝑥 + 𝜖𝑦𝑦) strain profile shown in Fig. 1, calculated using a finite element model. The strain is concentrated near the base of the cantilever, where curvature is greatest, and varies linearly from compressive to tensile strain as a function of depth. The QDM layers are grown below the center plane of the cantilever to increase the bending-driven strain for the QDs, and the vertical distance between QDs results in the bottom QD having 1.5 times the strain of the top QD. The system of a doubly charged QDM has been examined in detail in previous studies.35,36 A model of the system, based on tunneling between dots, is used to plot the energy levels in the ground state and optically excited state in Fig. 2(a). More detail on this model is found in the Supporting Information. The ground state spin system can be understood in terms of two anticrossings involving the three charge configurations shown in the lower half of Fig. 2(a). Taking the energy of the charge configuration with one electron in each QD as zero, the energies of configurations with both electrons in the bottom or top QD vary strongly with electrical bias and have opposite slopes. Due to the Pauli exclusion principle, tunneling between QDs can only occur for the singlet spin state, giving rise to two anticrossings for the singlet state and three degenerate triplet states with a constant energy with bias. This gives rise to an exchange splitting Δex between singlet and triplets that depends on bias and tunneling strength. 5 ACS Paragon Plus Environment

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Figure 2. Energy levels of the 2e system. (a) Model of the ground state and excited state (X2-) energy levels vs. bias, showing the two singlet anticrossings between charge configurations in the ground state and the X2- anticrossing. Red, vertical arrows represent the two optical transitions. The model parameters are chosen to fit QDM2. (b) Measured singlet and triplet transition photoluminescence vs. emission energy and bias. (c) Exchange splitting 𝛥ex vs. the relative bias for QDM1 and QDM2 (solid circles) and the model (lines).

The spin states are measured through optical transitions to charged exciton states X2-, shown in the upper half of Fig. 2(a), in which an electron-hole pair is excited into the top QD. Because the top QD is intentionally grown thicker than the bottom QD, this is the lowest exciton state. There is also an anticrossing between charged exciton states at lower biases, shifted from ground state anticrossings due to Coulomb energies. Optical transitions to X2- are possible for both the singlet and triplet and form a  energy level system.28,29 6 ACS Paragon Plus Environment

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Two QDMs in two different cantilevers, QDM1 and QDM2, are measured in this study. Figure 2(b) displays the photoluminescence (PL) of the singlet and triplet transitions for QDM2 as a function of bias and emission energy, measured using a tunable Fabry-Perot cavity in conjunction with a grating spectrometer to obtain a high spectral resolution of 2.6 eV. Since the singlet and triplet transitions share a common excited state, the difference between them is Δex, which is plotted in Fig. 2(c) for both QDMs. The bias Δ𝑉 is relative to the exchange splitting minimum. The singlet-triplet emission can only be measured over a finite bias range that is different for each QDM, outside of which the system is more stable in the one electron (lower bias) or three electron (higher bias) charge configuration. A model of the singlet-triplet system matches the experimental data quite well (see Supporting Information), with the two QDMs having different values for the tunnel coupling and Coulomb repulsion energy.

We first examine how motion-induced strain affects the spin system of QDMs by driving the mechanical resonance and measuring PL of the singlet and triplet transitions, synchronized to the modulation of the mechanical drive laser. Figure 3(a) plots the PL vs. time and emission energy, relative to the undriven triplet, for QDM1 at a bias of Δ𝑉 = ―60 mV, away from the minimum Δex. Both transitions oscillate in emission energy, but the shift amplitude of the singlet is about 2.5 times larger than for the triplet. The difference between transitions gives Δex as a function of time, which is plotted in Fig. 3(b). The oscillation amplitude of the exchange splitting is 45 eV, which is larger than oscillations of the triplet transition by 150%. This is quite different than observations in single QDs where no change in the electron spin splitting was observed, and the shift in the hole spin splitting was at most 4% of the shift in the optical transitions.10 Under the driving conditions here, we estimate the tip displacement to be ±25 nm, with positive displacement 7 ACS Paragon Plus Environment

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Figure 3. Effects of motion-induced strain. (a) Emission from QDM1 at V = -60 mV as a function of time, measured synchronously with the mechanical drive laser, modulated at the cantilever resonance of 3.5693 MHz. The mechanical drive (PL excitation) laser is 900 (923.87) nm with an average power of 10 (8) W. (b) Exchange splitting 𝛥ex extracted from fits to panel (a) plotted vs. time. (c) The extrema of 𝛥ex plotted vs. V for QDM2, obtained by taking time-dependent spectra while driving the mechanical resonance for each bias and fitting the emission energies. The undriven value of 𝛥ex is also plotted. (d) Amplitude of the spin-strain coupling vs. relative bias V for both QDM1 and QDM2.

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(away from substrate) at 40 ns and negative displacement at 180 ns (see Supporting Information). From the finite element model, this corresponds to strain amplitudes of 𝜖B = 3.88 × 10-5 and 𝜖T = 2.54 × 10-5 for the bottom and top dots, respectively. This gives a spin-strain coupling, 𝐺ex = 𝑑∆ex 𝑑𝜖, of -1.4 eV/strain (-340 THz/strain), using the average strain for the dots. This coupling is more than 300 times larger than for a single electron spin (based on the noise level in Ref. 10) and about 30 times larger than for a single hole spin. The strain coupling of the triplet transition is -0.91 eV/strain (-220 THz/strain), comparable to optical transition coupling in single QDs. The spin-strain coupling is strongly dependent on bias and helps identify the mechanism. We have performed time-dependent measurements of the singlet-triplet system for a series of biases for QDM1 and QDM2 with a constant mechanical drive. The mechanical drive conditions are somewhat different than in Fig. 3(a,b), and the estimated strain amplitude is 2.6 (1.7) × 10-5 for the bottom (top) QDs in QDM1 and 5.2 (3.4) × 10-5 for QDM2. Figure 3(c) plots Δex at the maximum strain (𝜖 + ), minimum strain (𝜖 ― ), and no applied strain (𝜖0) measured at each bias for QDM2. Some points are missing at the ends of the 2e bias stability range for 𝜖 + or 𝜖 ― , indicating that the PL lines have become very weak, presumably due to strain changing the charge state. The strain shifts Δex(𝑉) toward reverse bias for positive strain and toward forward bias for negative strain. The measured strain coupling (the oscillation amplitude of Δex divided by strain) is plotted for both QDM1 and QDM2 in Fig. 3(d), showing zero coupling at Δ𝑉 = 0 and increasing away from zero, with the sign of the coupling the same as the sign of Δ𝑉. The coupling follows the derivative of Δex with respect to bias, as expected for a small, linear shift in bias with strain. There is a nonlinear

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increase in coupling away from Δ𝑉 = 0 as the system approaches the singlet anticrossings [see Fig. 2(a)] because the magnitude of 𝑑∆ex 𝑑𝑉 increases nonlinearly. This bias-dependent coupling can be explained by the strain shifting the energy of the electrons in the two QDs by different amounts [see inset of Fig. 3(d)]. The phenomenological model for the strain coupling (see Supporting Information) shifts the energy of each electron by 𝜒B𝜖B and 𝜒T𝜖T for the bottom and top QDs, respectively, with 𝜒B = 𝜒T assumed for simplicity. The microscopic mechanisms for these shifts are not explored here, but they could give rise to changes in the InGaAs conduction band, confinement energy, or Coulomb energies. We also note that strain-induced changes to the electric field within the device heterostructure may contribute. These could occur due to the piezoelectric effect or changes to the GaAs bandgap, as discussed further in the Supporting Information. The effect of strain on the top and bottom QDs is similar to the effect of a change in bias, which also gives rise to a relative change in energies. The simple model fits the experimental data very well with 𝜒 = 41 eV/strain for QDM1 and 𝜒 = 36 eV/strain for QDM2, which gives confidence in our model. These values are comparable to those estimated for single QDs (24 eV/strain) strained using a piezoelectric actuator.37 There is also the possibility of strain changing the tunneling rate, which has been observed for hole tunneling in QDMs,38 in which strain from a piezoelectric actuator statically modified the heavy hole/light hole confinement potentials. This effect would likely be weaker with electrons, and it would give a significant coupling at Δ𝑉 = 0, which we do not observe here. Based on the bias-dependent spin-strain coupling presented in Fig. 3, the linear coupling vanishes at Δ𝑉 = 0, the minimum of Δex. This bias is sometimes called the “sweet spot” as it is protected from electric field fluctuations, which lead to fluctuations in Δex for a finite slope, 𝑑Δex 𝑑𝑉.34 At the sweet spot, there should still be quadratic coupling to strain through this 10 ACS Paragon Plus Environment

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differential strain mechanism, but it may be desirable to have strong, linear coupling even at the sweet spot. Next we will demonstrate a different coupling mechanism that should be biasindependent, with significant coupling even at the sweet spot. To demonstrate the second coupling mechanism, we operate at Δ𝑉 = 0 in QDM2, where the differential strain coupling is negligible, and we use a laser near-resonant with one of the optical transitions, which in this study is the triplet transition. The near-resonant laser serves two purposes. First, the laser gives rise to dressed states for the triplet and X2- that modify Δex, depending on the strain-dependent detuning. Second, it results in coherent Raman spin flip emission, which provides a direct measure of the spin splitting.39–41 As illustrated in Fig 4(a), coherent Raman is separated from the QD drive laser by Δex. Figure 4(b) displays the emission spectrum near the singlet as the laser is tuned through the triplet. At the largest negative detuning 𝛿 = ―24 eV, there is a sharp (3.6 eV FWHM) emission peak 158 eV above the triplet that corresponds to anti-Stokes Raman spin flip emission and a weaker, broader (5.5 eV FWHM) peak 186 eV above the triplet that corresponds to incoherent singlet PL. The Raman follows the tuning of the QD drive laser, and the singlet PL stays roughly constant until the laser is near the triplet transition, at which point the states are strongly dressed by the laser. On resonance, Autler-Townes splitting42,43 is observed with peaks separated by 10.6 eV that corresponds to the Rabi frequency Ω. At this point the two lines have equal linewidths of 4.4 eV. Even when the QD drive laser detuning 𝛿 is larger than Ω, the states are weakly dressed, and the effect can be described as an AC Stark shift of the triplet and X2- by ± Ω2 (4𝛿), respectively. (An AC Stark shift of the singlet will also occur, but it will be much smaller due to the larger detuning.) This shift gives rise to a modified value of the singlettriplet splitting Δex = Δex + Ω2 (4𝛿) that depends on strain through shifts in the optical transitions. 11 ACS Paragon Plus Environment

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A strain-induced shift of the triplet transition by ∆𝜔𝑇 gives 𝛿 = 𝛿0 ― ∆𝜔𝑇, where 𝛿0 is the detuning without any applied strain. The linear coupling of the modified exchange splitting to strain is then 𝐺ex = 𝑑Δex 𝑑𝜖 = 𝐺transΩ

2

(4𝛿02), where 𝐺trans is the coupling of the triplet transition to strain.

Essentially, the dressed triplet state contains a small component of the exciton state, which is sensitive to strain. The coupling of the dressed triplet state to strain is determined by this component, which has a probability amplitude of Ω (2𝛿0).

Figure 4. AC Stark-mediated strain coupling. (a) Energy level diagram of the singlet-triplet system, showing Raman spin flip emission with a detuned QD drive laser and energy level shifts from the AC Stark effect. (b) Emission spectrum from QDM2 at V = 0 with a 5 W QD drive laser tuned through the triplet, with the laser and triplet energies indicated by red and black lines to the left. (c) Time-dependent emission from QDM2 at V = 0 while mechanically driving the 12 ACS Paragon Plus Environment

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cantilever with 1 W average power at 900 nm. The QD drive laser is detuned above the triplet transition ( = +17 eV) at a power of 20 W. Modelled Raman and singlet PL emission energies are superimposed on the map as yellow, dashed lines, using a Rabi frequency of 21.9 eV and an oscillation amplitude of the transitions of 9 eV. (d) Shift in the Raman and singlet PL emission lines (solid circles) vs. QD drive laser power for an average mechanical drive laser power of 1 W at 900 nm and  = +25 eV. The model shifts (lines) are for a maximum Rabi frequency of 23.4 eV at 20 W and a transition oscillation amplitude of 7 eV. We demonstrate the laser-induced coupling of strain to spin in Fig. 4(c) by detuning the QD drive laser 17 eV above the triplet, weakly mechanically driving the cantilever, and measuring the singlet PL and Raman emission. The energy of the Raman emission relative to the QD drive laser is a direct measurement of Δex, which shows a clear oscillation due to the motion-induced strain. In Fig. 4(d), the amplitude of this oscillation is plotted as a function of the near-resonant laser power for similar conditions, also showing the oscillation amplitude of the singlet PL, which is modified by the AC Stark effect. The maximum shift of Δex is ±0.73 eV, which corresponds to a strain coupling of -136 meV/strain (33 THz/strain). The dependence on laser power should be linear for ∆𝜔𝑇, Ω ≪ δ, but this condition is not met at the higher powers where Ω~δ. Clearly the coupling can be tuned to zero using this technique by increasing δ or decreasing Ω, and the sign of the coupling can be reversed by changing the sign of δ. In the experiment of Fig. 4(c) the QD drive laser was sufficiently detuned from the triplet such that only small mixing between the spin ground states and the exciton states occurred. In Fig. 5(a), we center the laser on the triplet transition and increase the mechanical drive amplitude to explore the strongly dressed state regime. Instead of tuning the QD drive laser through the triplet 13 ACS Paragon Plus Environment

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transitions, the time dependent strain is tuning the triplet transition through the laser. Likewise the Raman is centered on the singlet transition, which then moves in and out of resonance with mechanical strain. Emission near the singlet and triplet transitions is measured, with most laser light eliminated by collecting emission cross-polarized with respect to the laser. The QD drive laser is on resonance at 75 ns and 215 ns, showing a clear Mollow triplet44,45 centered on the laser and Autler-Townes splitting of the Raman near the singlet, transitioning to the AC Stark regime at significant detunings. A dressed state level diagram in Fig. 5(b) illustrates the emission under resonant excitation. A dressed state model of the emission energies (see Supporting Information), plotted as dashed lines in Fig. 5(a), provides excellent agreement with the emission energy lines. In this strongly dressed regime, the QD drive laser has a dramatic effect on the spin system which is fully controlled by strain.

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Figure 5. Mechanically-tuned dressed state emission. (a) Time-dependent emission map of QDM2 while mechanically driving with 5 W average power at 900 nm, showing the Mollow triplet on the left and the singlet PL (S) and Raman emission on the right. The QD drive laser power is 8 W and is tuned to the undriven triplet transition. Modelled dressed state emission energies are superimposed on the map as yellow, dashed lines, using a Rabi frequency of 12.65 eV and an oscillation amplitude of the transitions of 33 eV. (b) Dressed state energy level diagram for resonant driving of the triplet transition, with dashed lines indicating dressed states.

The spin-strain coupling demonstrated here using the differential strain mechanism can be particularly strong, at least 3 orders of magnitude larger than for a single electron spin in an InGaAs QD and larger than in any other known system. The coupling should be increased further by increasing the distance between QDs, reducing the cantilever thickness, and increasing the 15 ACS Paragon Plus Environment

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tunneling rate. The large coupling is a result of the strong connection between Δex and the orbital energies of the individual electrons, which are very sensitive to strain. The downside of this strong connection is that Δex can also be very sensitive to electric field fluctuations, as discussed previously. At the sweet spot, Δex is insensitive to these fluctuation to first order, with a 𝑇2∗ of hundreds of ns measured,34 but away from this bias 𝑇2∗ decreases according to the slope.29 Spin echo pulses can be used to eliminate dephasing from these slow fluctuations to take advantage of the high strain sensitivity away from the sweet spot. We also note that the quadratic coupling to strain expected at the sweet spot has special application in measuring and controlling motion of mechanical resonators.13,46 The spin strain coupling mediated by the AC Stark effect is somewhat weaker than possible with the differential strain mechanism, but it should be bias-independent and thus effective at the sweet spot, where 𝑇2∗ should be the longest. In fact, the AC Stark effect mechanism should work equally well for other optically active spin systems, including an electron or hole in a single QD, as long as there is spin selectivity in the AC Stark effect. This selectivity can be achieved through differences in transition energies, as demonstrated here, or through polarization. The effects of the near-resonant laser on the spin dephasing should also be considered. The AC Stark effect may couple electric field fluctuations to spin in the same way it couples strain to spin. Spin dephasing and Raman linewidth broadening can also occur when the laser power is too high, so the condition of Ω ≪ δ should be maintained. This may mean reducing the coupling in order to preserve spin coherence. To take full advantage of the spin coherence, higher resolution Raman spectroscopy or pulsed Ramsey fringe/spin echo experiments29,47,48 can be performed. In the current 2 m wide cantilever, a single quantum of motion produces a strain 𝜖0 of 8.9 × 10-11 at the midpoint of the QDs at the base of the cantilever. This gives a single phonon coupling 16 ACS Paragon Plus Environment

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𝑔0 of 30 kHz for the differential strain coupling at Δ𝑉 = ―60 mV and 3 kHz for the AC Starkmediated coupling. Reducing the length and width of the cantilevers by a factor of 10 or using other small resonators could perhaps increase the coupling by a factor of 100 and lead to 𝑔0 greater than the spin decoherence rate. The fact that the coupling can be rapidly turned on and off with bias or laser field could allow additional functionality, including controlled gates between multiple quantum systems embedded in a mechanical resonator.

Methods. The sample is grown by molecular beam epitaxy on an n-doped GaAs substrate with the QDs grown within an n-i-n-i-p diode for electron injection. The diode has the following structure: 25 nm Si-doped GaAs, 40 nm undoped GaAs spacer barrier, two QDs separated by a 3/2/4 nm GaAs/AlGaAs/GaAs tunnel barrier, 71 nm undoped GaAs, 10 nm Si-doped GaAs, 10 nm undoped GaAs, and 30 nm Be-doped GaAs. The intermediate n-doped layer near the Be-doped layer is used to reduce the electric field at zero bias.30 Without this layer, the diode must be forward biased at higher voltages, leading to high currents and heating in the device.41,49 The 2 nm of AlGaAs in the tunnel barrier between QDs is used to reduce the exchange splitting. The sample temperature is about 6 K for all measurements. PL measurements are performed by focusing a laser either near the QDM position or at the tip of the cantilever. When focused on the tip with the polarization oriented perpendicular to the length, the laser couples into the cantilever, travels down its length, and excites the QDMs. The emission is bright at the edge of the cantilever near the QDM position and is polarized primarily parallel to the cantilever. This technique is also used for near-resonant spectroscopy, where it strongly reduces the collection of scattered laser light. A polarizer orthogonal to the laser polarization is used to further suppress the laser.

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Emission is collected with a single mode fiber and sent through a temperature-tunable fiber Fabry-Perot interferometer (FFPI) with a resolution of 2.6 eV and free spectral range of 333 eV. The FFPI is calibrated with a tunable laser and wavemeter. For PL measurements, emission is next sent through a 750 mm grating spectrometer to a CCD detector or silicon single photon counting module. For near resonant Raman measurements, emission through the FFPI is sent directly to the single photon counting module. Time-resolved measurements are performed using time-correlated photon counting (TCPC), in which the photon detection time is measured relative to the modulation signal of the mechanical drive laser. For the time-resolved emission spectra, the FFPI temperature is stepped through a range of values with TCPC performed at each photon transmission energy, building up a map of PL vs. time and energy.

ASSOCIATED CONTENT Supporting Information. The Supporting Information is available free of charge on the ACS Publications website at DOI: Cantilever displacement calibration, the QDM model, PL bias maps, the dressed state model, and single phonon coupling strength (PDF) AUTHOR INFORMATION Corresponding Author * E-mail: [email protected]

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Notes The authors declare no competing financial interest. ACKNOWLEDGEMENTS This work was supported by the U.S. Office of Naval Research, the Defense Threat Reduction Agency (Grant No. HDTRA1-15-1-0011), and the OSD Quantum Sciences and Engineering Program. REFERENCES (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

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Rugar, D.; Budakian, R.; Mamin, H. J.; Chui, B. W. Single Spin Detection by Magnetic Resonance Force Microscopy. Nature 2004, 430, 329–332. O’Connell, A. D.; Hofheinz, M.; Ansmann, M.; Bialczak, R. C.; Lenander, M.; Lucero, E.; Neeley, M.; Sank, D.; Wang, H.; Weides, M.; et al. Quantum Ground State and SinglePhonon Control of a Mechanical Resonator. Nature 2010, 464, 697–703. Arcizet, O.; Jacques, V.; Siria, A.; Poncharal, P.; Vincent, P.; Seidelin, S. A Single Nitrogen-Vacancy Defect Coupled to a Nanomechanical Oscillator. Nat. Phys. 2011, 7, 879–883. MacQuarrie, E. R.; Gosavi, T. A.; Jungwirth, N. R.; Bhave, S. A.; Fuchs, G. D. Mechanical Spin Control of Nitrogen-Vacancy Centers in Diamond. Phys. Rev. Lett. 2013, 111, 227602. Treutlein, P.; Genes, C.; Hammerer, K.; Poggio, M.; Rabl, P. Hybrid Mechanical Systems. In Cavity Optomechanics; Aspelmeyer M., Kippenberg T., M. F., Ed.; Springer, Berlin, Heidelberg: Berlin, 2014; pp 327–351. Teissier, J.; Barfuss, A.; Appel, P.; Neu, E.; Maletinsky, P. Strain Coupling of a NitrogenVacancy Center Spin to a Diamond Mechanical Oscillator. Phys. Rev. Lett. 2014, 113, 020503. Yeo, I.; de Assis, P.-L.; Gloppe, A.; Dupont-Ferrier, E.; Verlot, P.; Malik, N. S.; Dupuy, E.; Claudon, J.; Gérard, J.-M.; Auffèves, A.; et al. Strain-Mediated Coupling in a Quantum Dot-Mechanical Oscillator Hybrid System. Nat. Nanotechnol. 2014, 9, 106–110. Munsch, M.; Kuhlmann, A. V.; Cadeddu, D.; Gérard, J.-M.; Claudon, J.; Poggio, M.; Warburton, R. J. Resonant Driving of a Single Photon Emitter Embedded in a Mechanical Oscillator. Nat. Commun. 2017, 8, 76. Satzinger, K. J.; Zhong, Y. P.; Chang, H.-S.; Peairs, G. A.; Bienfait, A.; Chou, M.-H.; Cleland, A. Y.; Conner, C. R.; Dumur, E.; Grebel, J.; et al. Quantum Control of Surface Acoustic Wave Phonons. Nature 2018, 563, 661–665. Carter, S. G.; Bracker, A. S.; Bryant, G. W.; Kim, M.; Kim, C. S.; Zalalutdinov, M. K.; Yakes, M. K.; Czarnocki, C.; Casara, J.; Scheibner, M.; et al. Spin-Mechanical Coupling of an InAs Quantum Dot Embedded in a Mechanical Resonator. Phys. Rev. Lett. 2018, 121, 246801. Rabl, P.; Kolkowitz, S. J.; Koppens, F. H. L.; Harris, J. G. E.; Zoller, P.; Lukin, M. D. A 19 ACS Paragon Plus Environment

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(12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24)

(25) (26) (27)

Quantum Spin Transducer Based on Nanoelectromechanical Resonator Arrays. Nat. Phys. 2010, 6, 602–608. Aspelmeyer, M.; Meystre, P.; Schwab, K. Quantum Optomechanics. Phys. Today 2012, 65, 29. Aspelmeyer, M.; Kippenberg, T. J.; Marquardt, F. Cavity Optomechanics. Rev. Mod. Phys. 2014, 86, 1391–1452. Schuetz, M. J. A.; Kessler, E. M.; Giedke, G.; Vandersypen, L. M. K.; Lukin, M. D.; Cirac, J. I. Universal Quantum Transducers Based on Surface Acoustic Waves. Phys. Rev. X 2015, 5, 031031. Lee, D.; Lee, K. W.; Cady, J. V.; Ovartchaiyapong, P.; Bleszynski Jayich, A. C. Topical Review: Spins and Mechanics in Diamond. J. Opt. 2017, 19, 033001. Teufel, J. D.; Donner, T.; Li, D.; Harlow, J. W.; Allman, M. S.; Cicak, K.; Sirois, A. J.; Whittaker, J. D.; Lehnert, K. W.; Simmonds, R. W. Sideband Cooling of Micromechanical Motion to the Quantum Ground State. Nature 2011, 475, 359–363. Chan, J.; Alegre, T. P. M.; Safavi-Naeini, A. H.; Hill, J. T.; Krause, A.; Gröblacher, S.; Aspelmeyer, M.; Painter, O. Laser Cooling of a Nanomechanical Oscillator into Its Quantum Ground State. Nature 2011, 478, 89–92. Metcalfe, M.; Carr, S. M.; Muller, A.; Solomon, G. S.; Lawall, J. Resolved Sideband Emission of InAs/GaAs Quantum Dots Strained by Surface Acoustic Waves. Phys. Rev. Lett. 2010, 105, 037401. Ovartchaiyapong, P.; Lee, K. W.; Myers, B. A.; Jayich, A. C. B. Dynamic Strain-Mediated Coupling of a Single Diamond Spin to a Mechanical Resonator. Nat. Commun. 2014, 5, 4429. Montinaro, M.; Wüst, G.; Munsch, M.; Fontana, Y.; Russo-Averchi, E.; Heiss, M.; Morral, A. F. i; Warburton, R. J.; Poggio, M. Quantum Dot Opto-Mechanics in a Fully SelfAssembled Nanowire. Nano Lett. 2014, 14, 4454–4460. Schülein, F. J. R.; Zallo, E.; Atkinson, P.; Schmidt, O. G.; Trotta, R.; Rastelli, A.; Wixforth, A.; Krenner, H. J. Fourier Synthesis of Radiofrequency Nanomechanical Pulses with Different Shapes. Nat. Nanotechnol. 2015, 10, 512–516. Weiss, M.; Kapfinger, S.; Reichert, T.; Finley, J. J.; Wixforth, A.; Kaniber, M.; Krenner, H. J. Surface Acoustic Wave Regulated Single Photon Emission from a Coupled Quantum Dot – Nanocavity System. Appl. Phys. Lett. 2016, 109, 033105. Golter, D. A.; Oo, T.; Amezcua, M.; Lekavicius, I.; Stewart, K. A.; Wang, H. Coupling a Surface Acoustic Wave to an Electron Spin in Diamond via a Dark State. Phys. Rev. X 2016, 6, 041060. Villa, B.; Bennett, A. J.; Ellis, D. J. P.; Lee, J. P.; Skiba-Szymanska, J.; Mitchell, T. A.; Griffiths, J. P.; Farrer, I.; Ritchie, D. A.; Ford, C. J. B.; et al. Surface Acoustic Wave Modulation of a Coherently Driven Quantum Dot in a Pillar Microcavity. Appl. Phys. Lett. 2017, 111, 011103. Meesala, S.; Sohn, Y. I.; Pingault, B.; Shao, L.; Atikian, H. A.; Holzgrafe, J.; Gündoǧan, M.; Stavrakas, C.; Sipahigil, A.; Chia, C.; et al. Strain Engineering of the Silicon-Vacancy Center in Diamond. Phys. Rev. B 2018, 97, 205444. Sohn, Y.-I.; Meesala, S.; Pingault, B.; Atikian, H. A.; Holzgrafe, J.; Gundogan, M.; Stavrakas, C.; Stanley, M. J.; Sipahigil, A.; Choi, J.; et al. Controlling the Coherence of a Diamond Spin Qubit through Strain Engineering. Nat. Commun. 2018, 9, 2012. Whiteley, S. J.; Wolfowicz, G.; Anderson, C. P.; Bourassa, A.; Ma, H.; Ye, M.; Koolstra, 20 ACS Paragon Plus Environment

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(28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (43)

G.; Satzinger, K. J.; Holt, M. V.; Heremans, F. J.; et al. Spin–Phonon Interactions in Silicon Carbide Addressed by Gaussian Acoustics. Nat. Phys. 2019, 15, 490–495. Elzerman, J. M.; Weiss, K. M.; Miguel-Sanchez, J.; Imamoǧlu, A. Optical Amplification Using Raman Transitions between Spin-Singlet and Spin-Triplet States of a Pair of Coupled In-GaAs Quantum Dots. Phys. Rev. Lett. 2011, 107, 017401. Kim, D.; Carter, S. G.; Greilich, A.; Bracker, A. S.; Gammon, D. Ultrafast Optical Control of Entanglement between Two Quantum-Dot Spins. Nat. Phys. 2011, 7, 223–229. Vora, P. M.; Bracker, A. S.; Carter, S. G.; Sweeney, T. M.; Kim, M.; Kim, C. S.; Yang, L.; Brereton, P. G.; Economou, S. E.; Gammon, D. Spin–Cavity Interactions between a Quantum Dot Molecule and a Photonic Crystal Cavity. Nat. Commun. 2015, 6, 7665. Bayer, M.; Hawrylak, P.; Hinzer, K.; Fafard, S.; Korkusinski, M.; Wasilewski, Z. R.; Stern, O.; Forchel, A. Coupling and Entangling of Quantum States in Quantum Dot Molecules. Science 2001, 291, 451–453. Krenner, H. J.; Sabathil, M.; Clark, E. C.; Kress, A.; Schuh, D.; Bichler, M.; Abstreiter, G.; Finley, J. J. Direct Observation of Controlled Coupling in an Individual Quantum Dot Molecule. Phys. Rev. Lett. 2005, 94, 057402. Stinaff, E. A.; Scheibner, M.; Bracker, A. S.; Ponomarev, I. V; Korenev, V. L.; Ware, M. E.; Doty, M. F.; Reinecke, T. L.; Gammon, D. Optical Signatures of Coupled Quantum Dots. Science 2006, 311, 636–639. Weiss, K. M.; Elzerman, J. M.; Delley, Y. L.; Miguel-Sanchez, J.; Imamoğlu, A. Coherent Two-Electron Spin Qubits in an Optically Active Pair of Coupled InGaAs Quantum Dots. Phys. Rev. Lett. 2012, 109, 107401. Doty, M.; Scheibner, M.; Bracker, A.; Ponomarev, I.; Reinecke, T.; Gammon, D. Optical Spectra of Doubly Charged Quantum Dot Molecules in Electric and Magnetic Fields. Phys. Rev. B 2008, 78, 115316. Scheibner, M.; Bracker, A. S.; Kim, D.; Gammon, D. Essential Concepts in the Optical Properties of Quantum Dot Molecules. Solid State Commun. 2009, 149, 1427–1435. Kuklewicz, C. E.; Malein, R. N. E.; Petroff, P. M.; Gerardot, B. D. Electro-Elastic Tuning of Single Particles in Individual Self-Assembled Quantum Dots. Nano Lett. 2012, 12, 3761– 3765. Zallo, E.; Trotta, R.; Křápek, V.; Huo, Y. H.; Atkinson, P.; Ding, F.; Šikola, T.; Rastelli, A.; Schmidt, O. G. Strain-Induced Active Tuning of the Coherent Tunneling in Quantum Dot Molecules. Phys. Rev. B 2014, 89, 241303(R). Fernandez, G.; Volz, T.; Desbuquois, R.; Badolato, A.; Imamoglu, A. Optically Tunable Spontaneous Raman Fluorescence from a Single Self-Assembled InGaAs Quantum Dot. Phys. Rev. Lett. 2009, 103, 087406. He, Y.; He, Y.-M.; Wei, Y.-J.; Jiang, X.; Chen, M.-C.; Xiong, F.-L.; Zhao, Y.; Schneider, C.; Kamp, M.; Höfling, S.; et al. Indistinguishable Tunable Single Photons Emitted by SpinFlip Raman Transitions in InGaAs Quantum Dots. Phys. Rev. Lett. 2013, 111, 237403. Sweeney, T. M.; Carter, S. G.; Bracker, A. S.; Kim, M.; Kim, C. S.; Yang, L.; Vora, P. M.; Brereton, P. G.; Cleveland, E. R.; Gammon, D. Cavity-Stimulated Raman Emission from a Single Quantum Dot Spin. Nat. Photonics 2014, 8, 442–447. Autler, S. H.; Townes, C. H. Stark Effect in Rapidly Varying Fields. Phys. Rev. 1955, 100, 703. Xu, X.; Sun, B.; Berman, P. R.; Steel, D. G.; Bracker, A. S.; Gammon, D.; Sham, L. J. Coherent Optical Spectroscopy of a Strongly Driven Quantum Dot. Science (80-. ). 2007, 21 ACS Paragon Plus Environment

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(44) (45) (46) (47) (48) (49)

317, 929–932. Mollow, B. R. Power Spectrum of Light Scattered by Two-Level Systems. Phys. Rev. 1969, 188, 1969–1975. Flagg, E. B.; Muller, A.; Robertson, J. W.; Founta, S.; Deppe, D. G.; Xiao, M.; Ma, W.; Salamo, G. J.; Shih, C. K. Resonantly Driven Coherent Oscillations in a Solid-State Quantum Emitter. Nat. Phys. 2009, 5, 203–207. Thompson, J. D.; Zwickl, B. M.; Jayich, A. M.; Marquardt, F.; Girvin, S. M.; Harris, J. G. E. Strong Dispersive Coupling of a High-Finesse Cavity to a Micromechanical Membrane. Nature 2008, 452, 72–76. Greilich, A.; Carter, S. G.; Kim, D.; Bracker, A. S.; Gammon, D. Optical Control of One and Two Hole Spins in Interacting Quantum Dots. Nat. Photonics 2011, 5, 702–708. Carter, S. G.; Economou, S. E.; Greilich, A.; Barnes, E.; Sweeney, T.; Bracker, A. S.; Gammon, D. Strong Hyperfine-Induced Modulation of an Optically-Driven Hole Spin in an InAs Quantum Dot. Phys. Rev. B 2014, 89, 075316. Carter, S. G.; Sweeney, T. M.; Kim, M.; Kim, C. S.; Solenov, D.; Economou, S. E.; Reinecke, T. L.; Yang, L.; Bracker, A. S.; Gammon, D. Quantum Control of a Spin Qubit Coupled to a Photonic Crystal Cavity. Nat. Photonics 2013, 7, 329–334.

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