Abstract
Coupled nanomechanical resonators have recently attracted great attention for both practical applications and fundamental studies owing to their sensitive sympathetic oscillation dynamics^{1,2,3,4,5,6,7,8,9,10}. A challenge to the further development of this architecture is the coherent manipulation of the coupled oscillations. Here, we demonstrate strong dynamic coupling between two GaAsbased mechanical resonators by periodically modulating (pumping) the stress using a piezoelectric transducer. This strong coupling enables coherent transfer of phonon populations between the resonators, namely phonon Rabi oscillations^{11,12,13}. The nature of the dynamic coupling can also be tuned from a linear firstorder interaction to a nonlinear higherorder process in which more than one pump phonon mediates the coherent oscillations (that is, multipump phonon mixing). This coherent manipulation is not only useful for controlling classical oscillations^{14} but can also be extended to the quantum regime^{11,12,13}, opening up the prospect of entangling two distinct macroscopic mechanical objects^{15,16}.
Main
The mechanical motion of two harmonic oscillators can interact if the oscillators are geometrically interconnected. Such coupled oscillations were first observed in paired pendulum clocks in the midseventeenth century and were extensively studied for their novel sympathetic oscillation dynamics^{17,18}. In this era of nanotechnologies, coupled oscillations have again emerged as subjects of interest when realized in nanomechanical resonators owing to their potential applications in highly precise sensors^{1,2,3}, highqualityfactor bandpass filters^{4}, signal amplifiers^{5} and logic gates^{6}. However, a key obstacle to the further development of this platform is the ability to coherently manipulate the coupling between different mechanical oscillations. This limitation arises as a consequence of the usually weak vibration coupling between the constituent nanomechanical elements. In this Letter, we demonstrate dynamic coupling between two geometrically interconnected GaAs doublyclamped beams by using piezoelectrically induced parametric mode mixing (pumping)^{19,20}. This technique enables coherent manipulation of phonon populations as well as strong vibration coupling in frequencydifferent mechanical resonators in which the energy exchange between the two resonators is intrinsically inefficient owing to the frequency mismatch.
The dynamic coupling is developed in paired GaAsbased mechanical beams, as shown in Fig. 1a, in which the piezoelectric effect is exploited to mediate allelectrical displacement transduction^{19,20}. Application of the gate voltage causes piezoelectric stress along the beam, resulting in modulation of its tension and the creation of a bending moment. This effect enables both harmonic driving and parametric pumping, where the mechanical motion can be detected by the voltage induced by the piezoelectric effect. All the measurements were done by setting the sample in a vacuum (5×10^{−5} Pa) and cooling it to 1.5 K with a ^{4}He cryostat.
The frequency response of beam 1 measured by harmonically driving it while the parametric pump is deactivated exhibits two coupled vibration modes (Fig. 1b), where mode 1 (ω_{1} = 2π×293.93 kHz) is dominated by the vibration of beam 1 while mode 2 (ω_{2} = 2π×294.37 kHz) is dominated by the vibration of beam 2. The amplitude of mode 2 is much smaller than that of mode 1, reflecting the energy exchange due to the structural coupling via the overhang is inefficient because of the eigenfrequency difference between the two beams. The quality factor of these modes is Q∼14,000 with a frequency Q product of 4×10^{9} (ref. 21). The major contribution to the energy dissipation (1/Q) arises from the clamping losses^{22}, while the thermoelastic dissipation is negligible at cryogenic temperatures^{23}.
The frequency difference between the two modes can be compensated by activating the parametric pump, which results in mixing between the two vibration modes. This is induced by piezoelectrically modulating the spring constant of beam 1 with the pump frequency ω_{p} at around the frequency difference between the two modes, Δω≡ω_{2}−ω_{1} (Fig. 1c). The resulting dynamics can then be expressed by the following equations of motion:
where x_{i} (i = 1,2) is the displacement of the ith mode, ω_{i} is the mode frequency, γ_{i} ( = ω_{i}/Q_{i}) is the energy dissipation rate, F_{i} is the drive force (F_{1}≫F_{2}), ω_{d} is the drive frequency and Γ_{i} and Λ are the intra and intermodal coupling coefficients respectively. When the frequency mismatch between mode 1 and mode 2 is compensated by activating the pump at ω_{p}≃Δω, the terms containing Λ transfer phonons (oscillations) from one mode to the other (Fig. 1d). This intermodal coupling can also be explained by the mixing of mode 1 (2) and the Stokes sideband, ω_{2}−ω_{p} (the antiStokes sideband, ω_{1}+ω_{p}) of mode 2 (1) leading to normalmode splitting in the strongcoupling regime^{19,24,25,26} (Fig. 1d). The equations also include terms proportional to Γ_{i}. These terms lead to intramodal coupling, which becomes significant for the higherorder couplings shown later. The above model can reproduce all the experimental results and is described in detail in Supplementary Information.
The normalmode splitting for modes 1 and 2 was experimentally confirmed by applying a pump with voltage V _{p} at ω_{p} in addition to the weak harmonic driving at ω_{d} to beam 1 (Fig. 1a). Probing the modes via beam 1 while the parametric pump is activated shows that mode 1 splits into two when ω_{p}≃Δω = 2π×0.44 kHz, clearly demonstrating strong coupling (Fig. 2a–c). The transfer of phonons from mode 1 to mode 2 can also be confirmed in the response of beam 2 which was detected at frequency ω_{d}+ω_{p} (Fig. 2d–f). This measurement indicates the creation of a vibration in mode 2 that is excited by the parametric pump. A phonon reaction picture can help to understand this elementary process, where phonons are created in mode 2 at the expense of probe phonons in mode 1 and pump phonons, that is, via the onepump phonon absorption process, ℏω_{1}+ℏω_{p}→ℏω_{2} (Figs 1d and 3h). The strong coupling in the large pump regime also results in the reverse emission process, ℏω_{2}→ℏω_{1}+ℏω_{p} (Fig. 1d).
The V _{p} dependence of the mode splitting at ω_{p} = Δω shows that the coupling strength is highly controllable (Fig. 2i). The linear V _{p} dependence is due to the fact that the intermodal coupling coefficient, Λ, is proportional to V _{p}, which can be theoretically reproduced by equations (1a) and (1b) (Fig. 2k). The separation between the split peaks provides the coupling rate, g, which can become so large that it can exceed the intrinsic energy dissipation rate of the two modes (γ_{1}≃γ_{2} = 2π×21 Hz) by more than a factor of four (g = 2π×90 Hz for V _{p} = 1.0 V_{p−p}).
More remarkably, additional mode splittings in which the pump frequency does not correspond to the frequency difference between the two modes can also be observed. For example, the splitting occurs when ω_{p}≃Δω/2 = 2π×0.22 kHz for both modes 1 and 2 (Fig. 2c). This splitting is caused by a secondorder coupling via a twopump phonon absorption/emission process, that is, , which leads to the coupling between mode 1 (2) and the second Stokes sideband, ω_{2}−2ω_{p} (the second antiStokes sideband, ω_{1}+2ω_{p}) of mode 2 (1). The V _{p} dependence of this mode splitting indicates that it has a parabolic dependence (Fig. 2j). This is because the secondorder process requires a twostep phonon excitation path from mode 1 to mode 2, and vice versa, through the intermediary energy level (ω_{1}+ω_{2})/2 via both the intramodal coupling (Γ_{i}V _{p}) and the intermodal coupling (Λ;V _{p}), therefore, [Γ_{i}×Λ];V _{p}^{2}, as shown in Fig. 3i. The corresponding mode splitting shows good agreement with the theoretical simulations (Fig. 2l). Consequently, the additional mode splittings observed in Fig. 2c correspond to even higher order coupling processes, requiring more than two pump phonons.
The strong dynamic coupling between the two mechanical resonators opens up a path to coherent control of the coupled mechanical oscillations. The timedomain measurements using the pulse sequence shown in Fig. 3a enable us to observe coherent and periodic energy exchange between the two beams/modes. The pump frequency dependence of the timedomain response of beam 2 at ω_{2} clearly shows the periodic amplitude oscillations at ω_{p}≃Δω for the firstorder (n = 1) coupling (Fig. 3b), which corresponds to a onestep phonon process (Fig. 3h). The V _{p} dependence at ω_{p} = Δω shows that the vibration energy of mode 1 (beam 1) can be transferred to mode 2 (beam 2) and back eight times before energy relaxation at V _{p} = 1.0 V_{p−p} (Fig. 3d). Coherent energy exchange for the secondorder (n = 2) coupling can also be observed at ω_{p}≃Δω/2 (Fig. 3b), where up to five oscillation periods are observed in the range of V _{p}≤1.0 V_{p−p} (Fig. 3e). The coupling rate, g, extracted from the Fourier transforms of the timedomain response, is proportional to V _{p} for the firstorder coupling and V _{p}^{2} for the secondorder coupling (Fig. 4). These coupling rates correspond perfectly to the mode splitting observed in the frequency response measurements.
The timedomain measurements also enable us to observe higherorder coupling, that is, n≥3. Figure 3f,g show the coherent energy exchange between the two modes for ω_{p} = Δω/3 = 2π×0.147 kHz and ω_{p} = Δω/4 = 2π×0.11 kHz, respectively. These coherent oscillations are caused by the npump phonon absorption/emission processes, that is, , through the intermediary energy levels via intra and intermodal coupling, for example, [Γ_{1}×Γ_{1}×Λ] and [Γ_{1}×Γ_{1}×Γ_{1}×Λ], as shown in Fig. 3j,k. The Fourier transforms of the timedomain response reveal that the coupling rate exhibits a V _{p}^{n} dependence even for n≥3 (Fig. 4), which again shows good agreement with the theoretical model.
The present results show that electromagnetic pulse techniques, which are commonly used to coherently manipulate quantum twolevel systems^{11,27}, can also be applied to coherently control mechanical systems. By tuning the parametricpump frequency, multiwave phonon mixing involving an arbitrary number of pump phonons can be achieved in an analogous fashion to multiwave photon mixing^{28}. The parametric pumping allows highly controllable timedomain manipulation of phonon populations in the two modes simply by the adjustment of the pumppulse duration, thus permitting π and π/2pulse operations on the Bloch sphere^{11} (see also independent experiments at LMU with a single mechanical resonator^{29}). This coherent control further expands the applications of mechanical resonators, including the highspeed operation of high Q mechanical resonators^{14} and mechanical logic operations^{30}. Although the system demonstrated here is in the classical regime with large mode occupation, where the decoherence is governed by the energy relaxation^{29}, these techniques could also be extended to the quantum regime with vibration modes at sufficiently high frequency^{11}. This in turn leads to the exciting possibility of quantumcoherent coupling and entanglement between two distinct macroscopic mechanical objects^{15,16,27}.
Methods
The sample was fabricated by photolithography from a heterostructure consisting of 300nmthick Al_{0.25}Ga_{0.75}As, 100nmthick Sidoped nGaAs, 400nmthick undoped iGaAs and 2μmthick Al_{0.65}Ga_{0.35}As sacrificial layers grown on a GaAs(001) substrate by molecular beam epitaxy. AuGeNi was deposited on the supporting part to obtain an ohmic contact to the conductive nGaAs layer, while 60nmthick Au gates were formed on the top of the beams. The suspended structure was completed by a deep mesa and isotropic sacrificial layer etching, where the 40μmseparated beams were electrically isolated by the shallow mesa etch. Details of the measurements are described in Supplementary Information.
The simulations were carried out by solving equations (1a) and (1b) with Mathematica 8.0 (Wolfram Research). The derivation of equations (1a) and (1b) and the simulation details are described in Supplementary Information. In the simulation, the only adjustable parameter was the piezoelectric detuning coefficient (δΩ_{1}/δ V _{p} = 2π×0.69 kHz V^{−1}), which determines the relation between V _{p} and Λ (Γ_{i}), and all the remaining parameters were experimentally determined.
Change history
19 August 2013
In the version of this Letter originally published, the second author affiliation should have read "The Department of Materials Science and Engineering, National ChiaoTung University, Hsinchu 9808578, Taiwan". This error has now been corrected in the HTML and PDF versions of the Letter.
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Acknowledgements
H.O. thanks A. Taspinar for supporting the data analysis. The authors acknowledge T. Faust, J. P. Kotthaus and E. M. Weig for their critical reading of the manuscript. This work was partly supported by JSPS KAKENHI (23241046 & 20246064).
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H.O. designed and fabricated the sample with support from K.O., CY.C. and E.Y.C. The measurements and data analysis were performed by H.O. and A.G. The simulation was performed by H.Y. The paper was written by H.O., based on the discussions with I.M. and H.Y. The project was planned by H.Y.
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Okamoto, H., Gourgout, A., Chang, CY. et al. Coherent phonon manipulation in coupled mechanical resonators. Nature Phys 9, 480–484 (2013). https://doi.org/10.1038/nphys2665
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