Coupled nanomechanical resonators have recently attracted great attention for both practical applications and fundamental studies owing to their sensitive sympathetic oscillation dynamics1,2,3,4,5,6,7,8,9,10. 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 GaAs-based 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 oscillations11,12,13. The nature of the dynamic coupling can also be tuned from a linear first-order interaction to a nonlinear higher-order process in which more than one pump phonon mediates the coherent oscillations (that is, multi-pump phonon mixing). This coherent manipulation is not only useful for controlling classical oscillations14 but can also be extended to the quantum regime11,12,13, opening up the prospect of entangling two distinct macroscopic mechanical objects15,16.
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 mid-seventeenth century and were extensively studied for their novel sympathetic oscillation dynamics17,18. 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 sensors1,2,3, high-quality-factor band-pass filters4, signal amplifiers5 and logic gates6. 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 doubly-clamped 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 frequency-different 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 GaAs-based mechanical beams, as shown in Fig. 1a, in which the piezoelectric effect is exploited to mediate all-electrical displacement transduction19,20. 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 4He 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 (ω1 = 2π×293.93 kHz) is dominated by the vibration of beam 1 while mode 2 (ω2 = 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 4×109 (ref. 21). The major contribution to the energy dissipation (1/Q) arises from the clamping losses22, while the thermoelastic dissipation is negligible at cryogenic temperatures23.
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 ωp at around the frequency difference between the two modes, Δω≡ω2−ω1 (Fig. 1c). The resulting dynamics can then be expressed by the following equations of motion:
where xi (i = 1,2) is the displacement of the i-th mode, ωi is the mode frequency, γi ( = ωi/Qi) is the energy dissipation rate, Fi is the drive force (F1≫F2), ωd is the drive frequency and Γi and Λ are the intra- and inter-modal coupling coefficients respectively. When the frequency mismatch between mode 1 and mode 2 is compensated by activating the pump at ωp≃Δω, the terms containing Λ transfer phonons (oscillations) from one mode to the other (Fig. 1d). This inter-modal coupling can also be explained by the mixing of mode 1 (2) and the Stokes sideband, ω2−ωp (the anti-Stokes sideband, ω1+ωp) of mode 2 (1) leading to normal-mode splitting in the strong-coupling regime19,24,25,26 (Fig. 1d). The equations also include terms proportional to Γi. These terms lead to intra-modal coupling, which becomes significant for the higher-order couplings shown later. The above model can reproduce all the experimental results and is described in detail in Supplementary Information.
The normal-mode splitting for modes 1 and 2 was experimentally confirmed by applying a pump with voltage V p at ωp in addition to the weak harmonic driving at ωd 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 ωp≃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 ωd+ωp (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 one-pump phonon absorption process, ℏω1+ℏωp→ℏω2 (Figs 1d and 3h). The strong coupling in the large pump regime also results in the reverse emission process, ℏω2→ℏω1+ℏωp (Fig. 1d).
The V p dependence of the mode splitting at ωp = Δω shows that the coupling strength is highly controllable (Fig. 2i). The linear V p dependence is due to the fact that the inter-modal 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 (γ1≃γ2 = 2π×21 Hz) by more than a factor of four (g = 2π×90 Hz for V p = 1.0 Vp−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 ωp≃Δω/2 = 2π×0.22 kHz for both modes 1 and 2 (Fig. 2c). This splitting is caused by a second-order coupling via a two-pump phonon absorption/emission process, that is, , which leads to the coupling between mode 1 (2) and the second Stokes sideband, ω2−2ωp (the second anti-Stokes sideband, ω1+2ωp) 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 second-order process requires a two-step phonon excitation path from mode 1 to mode 2, and vice versa, through the intermediary energy level (ω1+ω2)/2 via both the intra-modal coupling (ΓiV p) and the inter-modal coupling (Λ;V p), therefore, [Γi×Λ];V p2, 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 time-domain 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 time-domain response of beam 2 at ω2 clearly shows the periodic amplitude oscillations at ωp≃Δω for the first-order (n = 1) coupling (Fig. 3b), which corresponds to a one-step phonon process (Fig. 3h). The V p dependence at ωp = 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 Vp−p (Fig. 3d). Coherent energy exchange for the second-order (n = 2) coupling can also be observed at ωp≃Δω/2 (Fig. 3b), where up to five oscillation periods are observed in the range of V p≤1.0 Vp−p (Fig. 3e). The coupling rate, g, extracted from the Fourier transforms of the time-domain response, is proportional to V p for the first-order coupling and V p2 for the second-order coupling (Fig. 4). These coupling rates correspond perfectly to the mode splitting observed in the frequency response measurements.
The time-domain measurements also enable us to observe higher-order coupling, that is, n≥3. Figure 3f,g show the coherent energy exchange between the two modes for ωp = Δω/3 = 2π×0.147 kHz and ωp = Δω/4 = 2π×0.11 kHz, respectively. These coherent oscillations are caused by the n-pump phonon absorption/emission processes, that is, , through the intermediary energy levels via intra- and inter-modal coupling, for example, [Γ1×Γ1×Λ] and [Γ1×Γ1×Γ1×Λ], as shown in Fig. 3j,k. The Fourier transforms of the time-domain response reveal that the coupling rate exhibits a V pn 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 two-level systems11,27, can also be applied to coherently control mechanical systems. By tuning the parametric-pump frequency, multi-wave phonon mixing involving an arbitrary number of pump phonons can be achieved in an analogous fashion to multi-wave photon mixing28. The parametric pumping allows highly controllable time-domain manipulation of phonon populations in the two modes simply by the adjustment of the pump-pulse duration, thus permitting π and π/2-pulse operations on the Bloch sphere11 (see also independent experiments at LMU with a single mechanical resonator29). This coherent control further expands the applications of mechanical resonators, including the high-speed operation of high- Q mechanical resonators14 and mechanical logic operations30. Although the system demonstrated here is in the classical regime with large mode occupation, where the decoherence is governed by the energy relaxation29, these techniques could also be extended to the quantum regime with vibration modes at sufficiently high frequency11. This in turn leads to the exciting possibility of quantum-coherent coupling and entanglement between two distinct macroscopic mechanical objects15,16,27.
The sample was fabricated by photolithography from a heterostructure consisting of 300-nm-thick Al0.25Ga0.75As, 100-nm-thick Si-doped n-GaAs, 400-nm-thick undoped i-GaAs and 2-μm-thick Al0.65Ga0.35As 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 n-GaAs layer, while 60-nm-thick 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 40-μm-separated 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 (δΩ1/δ V p = 2π×0.69 kHz V−1), which determines the relation between V p and Λ (Γi), and all the remaining parameters were experimentally determined.
Spletzer, M., Raman, A., Wu, A. Q., Xu, X. & Reifenberger, R. Ultrasensitive mass sensing using mode localization in coupled microcantilevers. Appl. Phys. Lett. 88, 254102 (2006).
Gil-Santos, E. et al. Mass sensing based on deterministic and stochastic resonances of elastically coupled nanocantilevers. Nano Lett. 9, 4122–4127 (2009).
Okamoto, H. et al. High-sensitivity charge detection using antisymmetric vibration in coupled micromechanical oscillators. Appl. Phys. Lett. 98, 014103 (2011).
Bannon, F. D., Clark, J. R. & Nguyen, C. T.-C. High-Q HF microelectromechanical filters. IEEE J. Solid-State Circuits 35, 512–526 (2000).
Karabalin, R. B. et al. Signal amplification by sensitive control of bifurcation topology. Phys. Rev. Lett. 106, 094102 (2011).
Masmanidis, S. C. et al. Multifunctional nanomechanical systems via tunably coupled piezoelectric actuation. Science 317, 780–783 (2007).
Shim, S-Bo, Imboden, M. & Mohanty, P. Synchronized oscillation in coupled nanomechanical oscillators. Science 316, 95–99 (2007).
Karabalin, R. B., Cross, M. C. & Roukes, M. L. Nonlinear dynamics and chaos in two coupled nanomechanical resonators. Phys. Rev. B 79, 165309 (2009).
Faust, T. et al. Nonadiabatic dynamics of two strongly coupled nanomechanical resonator modes. Phys. Rev. Lett. 109, 037205 (2012).
Okamoto, H., Kamada, T., Onomitsu, K., Mahboob, I. & Yamaguchi, H. Optical tuning of coupled micromechanical resonators. Appl. Phys. Exp. 2, 062202 (2009).
O’Connel, A. D. et al. Quantum ground state and single-phonon control of a mechanical resonator. Nature 464, 697–703 (2010).
Verhagen, E., Deléglise, S., Weis, S., Schliesser, A. & Kippenberg, T. J. Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode. Nature 482, 63–67 (2012).
Palomaki, T. A., Harlow, J. W., Teufel, J. D., Simmonds, R. W. & Lehnert, K. W. Coherent state transfer between itinerant microwave fields and a mechanical oscillator. Nature 495, 210–214 (2013).
Yamaguchi, H., Okamoto, H. & Mahboob, I. Coherent control of micro/nanomechanical oscillation using parametric mode mixing. Appl. Phys. Exp. 5, 014001 (2012).
Brown, K. R. et al. Coupled quantized mechanical oscillators. Nature 471, 196–199 (2011).
Harlander, M., Lechner, R., Brownnutt, M., Blatt, R. & Hänsel, W. Trapped-ion antennae for the transmission of quantum information. Nature 471, 200–203 (2011).
Huygens, C. in Horologium Oscillatorium [The Pendulum Clock] (Apud F. Muguet, 1673).
Pikovsky, A., Rosenblum, M. & Kurths, J. Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge Univ. Press, 2001).
Mahboob, I., Nishiguchi, K., Okamoto, H. & Yamaguchi, H. Phonon-cavity electromechanics. Nature Phys. 8, 387–392 (2012).
Yamaguchi, H. & Mahboob, I. Parametric mode mixing in asymmetric doubly clamped beam resonators. New J. Phys. 15, 015023 (2013).
Ding, L. et al. High frequency GaAs nano-optomechanical disk resonator. Phys. Rev. Lett. 105, 263903 (2010).
Cole, G. D., Wilson-Rae, I., Werbach, K., Vanner, M. R. & Aspelmeyer, M. Phonon-tunnelling dissipation in mechanical resonators. Nature Commun. 2, 231 (2011).
Okamoto, H., Ito, D., Onomitsu, K. & Yamaguchi, H. Thermoelasticdamping in GaAs micromechanical resonators. Phys. Stat. Sol. C 5, 2920–2922 (2008).
Weis, S. et al. Optomechanically induced transparency. Science 330, 1520–1523 (2010).
Safavi-Naeini, A. H. et al. Electromagnetically induced transparency and slow light with optomechanics. Nature 472, 69–73 (2011).
Teufel, J. et al. Circuit cavity electromechanics in the strong-coupling regime. Nature 471, 204–208 (2011).
Ansmann, M. et al. Violation of Bell’s inequality in Josephson phase qubits. Nature 461, 504–506 (2009).
Kippenberg, T. J., Holzwarth, R. & Diddams, S. A. Microresonator-based optical frequency combs. Science 332, 555–559 (2011).
Faust, T., Rieger, J., Seitner, M. J., Kotthaus, J. P. & Weig, E. M. Coherent control of a classical nanomechanical two-level system. Nature Phys. http://dx.doi.org/10.1038/nphys2666 (2013).
Mahboob, I., Flurin, E., Nishiguchi, K., Fujiwara, A. & Yamaguchi, H. Interconnect-free parallel logic circuits in a single mechanical resonator. Nature Commun. 2, 198 (2011).
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).
The authors declare no competing financial interests.
About this article
Cite this article
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
Microsystems & Nanoengineering (2021)
Nature Communications (2021)
Nano Research (2021)
Nature Nanotechnology (2020)
International Journal of Theoretical Physics (2020)