Abstract
Photonic cavities have emerged as an indispensable tool to control and manipulate harmonic motion in opto/electromechanical systems^{1}. Invariably, in these systems a highqualityfactor photonic mode is parametrically coupled to a highqualityfactor mechanical oscillation mode^{2,3,4,5,6,7,8,9,10,11,12}. This entails the demanding challenges of either combining two physically distinct systems, or else optimizing the same nanostructure for both mechanical and optical properties^{4,5,6,7,8,9,10,11}. In contrast to these approaches, here we show that the cavity can be realized by the second oscillation mode of the same mechanical oscillator^{13,14}. A piezoelectric pump^{15,16} generates straininduced parametric coupling between the first and the second mode at a rate that can exceed their intrinsic relaxation rate. This leads to a mechanically induced transparency in the second mode which plays the role of the phonon cavity^{17,18}, the emergence of parametric normalmode splitting^{19,20} and the ability to cool the first mode^{2,3,4,5,6,7,8,9,10,11}. Thus, the mechanical oscillator can now be completely manipulated by a phonon cavity^{21}.
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Main
The dynamical backaction of a photonic cavity that is parametrically coupled to a mechanical oscillation mode has recently led to the realization of a quantized macroscopic mechanical system^{10,11,22,23,24}, a longstanding goal in solidstate physics^{25}. The backaction of the mechanical motion on the cavity has also resulted in the emergence of opto/electromechanically induced transparency, which has great potential for communications technology and quantum information science^{17,18,20,26,27,28}.
The success of this approach is leveraged on the requirement that the mechanical oscillation completes many cycles before the cavity relaxes, thus enhancing the effectiveness of the dynamical backaction, or in other words the coupled system is operated in the resolved sideband regime^{1,22,23}. The parametric coupling in these systems arises from the harmonic motion of the mechanical element modifying the cavity’s resonance frequency by means of a change in either the cavity’s length or capacitance. This has resulted in exquisitely engineered devices in which the mechanical motion can be manipulated by the parametrically coupled photonic cavity^{1}.
In contrast to a photonic cavity, a phonon cavity operated in the resolved sideband regime should also be able to host dynamical backaction onto the mechanical element. One approach to this goal could be realized by physically coupling an additional mechanical oscillator to the system. Here we show that a more natural method is to simply couple two different colour mechanical oscillation modes in the same mechanical system, where the first mode represents the mechanical oscillation of interest and the second mode affords a phonon cavity. The key to this approach is a geometric intermodal coupling, where the motion of the first (second) mode creates tension that causes a shift in the frequency of the second (first) mode, that can be parametrically manipulated^{29,30}, which enables phononcavity electromechanics to be realized^{1,20}.
The electromechanical resonator used in this study is shown in Fig. 1a and described in detail elsewhere^{15,16}. It hosts flexural oscillation modes that are transduced by the piezoelectric effect, where the first mode ω_{1}/2π=171.3 kHz with damping γ_{1}/2π=ω_{1}/2π Q_{1}=1.1 Hz (Q is the quality factor) and the second mode ω_{2}/2π=470.93 kHz with γ_{2}/2π=ω_{2}/2π Q_{2}=8.5 Hz, which henceforth will be known as the phonon cavity (see Fig. 1b). This data indicates ultradeep resolved sideband operation with an unprecedented ratio ω_{1}/γ_{2}≈2×10^{4}, which is orders of magnitude larger than photonbased counterparts^{1,2,3,4,5,6,7,8,9,10,11,14,22,23}.
In the phononbased system and in contrast to photoncavity electromechanical systems, the motion of the first mode induces tension that can modify the frequency of the phonon cavity, creating sidebands at ω_{2}±ω_{1}, as schematically depicted in Fig. 1c (ref. 1). The coupling strength G=dω_{2}/dX_{1} quantifies the degree of parametric intermodal coupling and is given by the tensioninduced change in the phononcavity frequency for a given displacement of the first mode (see Supplementary Information for an illustrative example)^{20,30}.
On the other hand, application of d.c. bias also enables the frequency of the first mode and the phonon cavity to be manipulated via piezoelectrically induced tension, which yields δ ω_{1}/2π=11 Hz V^{−1} and δ ω_{2}/2π=6 Hz V^{−1} respectively, as shown in Fig. 1b (ref. 15). Consequently, if the system is periodically pumped (ω_{p}) by a sufficiently large bias (V _{p}) it can not only modulate the eigenfrequency of both the first mode and the phonon cavity as δ ω_{n}V _{p}cos(ω_{p}t), where n=1 or 2 respectively, but it can also drive the parametric coupling between them. For example, the pump can induce a change in the displacement of the first mode, which in turn modifies the tension and thus the frequency and energy of the phonon cavity. The relaxation time (1/γ_{2}) delayed response of the phonon cavity results in a backaction force on the first mode via the same coupling mechanism modifying its oscillation dynamics^{1,14}. The parametric coupling and the dynamical backaction effects can be greatly enhanced when ω_{p}^{−}=ω_{2}−ω_{1} (ω_{p}^{+}=ω_{2}+ω_{1}) and pumping on the antiStokes/red sideband (Stokes/blue sideband) can damp (amplify) the first mode (Fig. 1c)^{29}.
In this configuration, the Hamiltonian of the coupled system can be expressed as
where the first two terms are the kinetic and potential energies respectively, with canonical coordinates X_{n} and P_{n} denoting the position and conjugate momentum of the constituent systems with mass m_{n}. Λ is the amplitude of the harmonic probe of the phonon cavity with ω_{s}≈ω_{2}. Δ_{n} is proportional to the piezoelectric pump amplitude, where this term arises from piezoelectric frequency pulling of the mechanical resonator, and it results in parametric resonance when ω_{p}=2ω_{n} (ref. 15). The combination of the terms with Λ and Δ_{2} can also permit degenerate/nondegenerate parametric amplification and squeezing when ω_{p}≈2ω_{2} (refs 16, 31). The last term describes the linearized parametric coupling between the first mode and the phonon cavity^{20,32}, where the coupling rate g∝G V _{p}, and this becomes the dominant term when ω_{p}=ω_{p}^{±}. Thus the term containing Δ_{n} can be neglected in the present investigation.
To probe the phonon cavity, a weak harmonic excitation (3 mV_{r.m.s.}) is applied at ω_{s}≈ω_{2} and the system is pumped on the red sideband at ω_{p}^{−} whilst the pump amplitude is increased. In this configuration, the phonons from the pump and in the phonon cavity are annihilated whilst phonons are created in the first mode to conserve energy^{29}. The numerical solution to equation (1) (Fig. 2d and Methods) confirms that the phonon cavity transfers energy to the first mode, and when it results in a mechanically (that is, phonon) induced transparency in the phonon cavity^{17,18,20,27,28}. As the pump amplitude is increased further, the condition g≥γ_{2}≫γ_{1} can be satisfied, and the system enters the strongcoupling regime. In this limit, the phonon cavity and the first mode can no longer be considered as distinct entities, but rather as a composite of their initial eigenmodes, and will undergo parametric normalmode splitting; a widely observed phenomenon in strongly coupled systems^{19,20,25,32}.
The experimental results shown in Fig. 2a,b confirm this assertion, where the phonon cavity becomes transparent at pump amplitudes of 0.3 V_{r.m.s.}. As the pump amplitude is increased further the system exhibits the emergence of parametric normalmode splitting and the peak separation given by 2g enables the coupling rate to be extracted, as shown in Fig. 2c (ref. 20). This reveals g/π can be linearly increased by the antiStokes pump to 8 Hz, placing the system on the cusp of the strong coupling regime^{19,20}.
Alternatively, detuning the antiStokes pump frequency and probing the phonon cavity reveals the presence of the first mode, as shown in Fig. 2e–g. As the first mode’s sideband approaches the phonon cavity it acquires its damping rate and becomes broader. This effect is enhanced as the antiStokes pump amplitude is increased and results in the avoidedcrossing characteristic of strongly coupled systems, namely the first mode and the phonon cavity. Numerical simulations using equation (1) (see Methods) can easily reproduce the above spectroscopy and are shown in the Supplementary Information.
The phononcavity electromechanical system can also be excited by the Stokes pump, where this process creates phonons in both the phonon cavity and the first mode at the expense of the pump phonons, and is described in detail in the Supplementary Information^{33}.
The intermodal parametric coupling can be engineered to control and manipulate the first mode by means of the phonon cavity. By pumping on the red sideband at ω_{p}^{−}, the first mode is probed with a weak (100 μV_{r.m.s.}) harmonic excitation as a function of the pump amplitude, as shown in Fig. 3a,b. In this configuration, the phonons from the antiStokes pump and in the first mode are annihilated whilst phonons are created in the phonon cavity. Consequently, the energy and hence the area associated with the spectral response of the first mode decreases, as shown in Fig. 4d. Moreover, as the antiStokes pump amplitude is increased, the enhanced coupling between the first mode and the phonon cavity results in Q_{1} being greatly reduced, as shown in Fig. 4c. For the strongest pump amplitudes, the coupling reaches a point where parametric normalmode splitting can even be observed in the first mode and its behaviour can no longer be characterized by a single harmonic oscillator response function^{19}.
On the other hand, if the antiStokes pump is detuned, the phonon cavity’s dynamics can be imprinted onto the first mode, where γ_{1} and ω_{1} trace out its inphase and quadrature components, as shown in Fig. 3c–g. For a pump amplitude of 0.5 V_{r.m.s.}, γ_{1}/2π tends to (γ_{1}+γ_{2})/4π ∼5 Hz; concurrently ω_{1}/2π also undergoes a 5 Hz shift (Fig. 3f,g). From the theory of photonbased opto/electromechanical systems operated in the resolved sideband regime, the change in ω_{1} and γ_{1} can be expressed as 4g^{2}δ/(γ_{2}^{2}+4δ^{2}) and 4g^{2}γ_{2}/(γ_{2}^{2}+4δ^{2}) respectively, where the pump detuning about ω_{p}^{−} is given by δ (refs 20, 22, 23). We find that this formalism is equally applicable to the phonon cavity realized here and it can reproduce the experimental response as shown in Fig. 3f,g approximately using the experimentally determined values for g and γ_{2}.
Invariably, the dynamical backaction of the photonic cavity has been employed to cool a mechanical oscillation mode^{1,2,3,4,5,6,7,8,9,10,11,22,23,24}. Here we show that this concept can also be applied to the phonon cavity. A white noise voltage corresponding to a displacement X′_{1}=0.37 nm (see Methods and Supplementary Information) and temperature T′≫3.6 K is injected into the spectral region around the first mode, as shown in Fig. 4a,b. Activating the antiStokes pump at ω_{p}^{−} transfers phonons corresponding to the displacement noise energy of the first mode to the phonon cavity, where a stronger pump amplitude enhances the transfer rate. For the largest pump amplitude, almost all the displacement noise energy is transferred to the phonon cavity, resulting in the temperature T and hence X_{1} and Q_{1} of the first mode reducing (Fig. 4c,d). The normalized temperature at various pump amplitudes indicates that the temperature of the first mode can be reduced by a factor of ∼2 by means of the dynamical backaction of the phonon cavity, as shown in Fig. 4d. The measured response of T and Q_{1} as a function of the antiStokes pump amplitude can be reproduced through equation (1) and is detailed in the Supplementary Information. The strong pump also results in thermal drift (Fig. 2a) but a corresponding increase in the noise temperature is not observed, as it is obscured by the very large T′ and by the noise floor temperature from the room temperature amplifier.
At an ambient temperature of 3.6 K (that is, without the white noise) the parametric coupling will result in the phonon number, N_{n}=k_{b}T/ℏω_{n}, where k_{B} is the Boltzmann constant, of the first mode and the phonon cavity converging to (N_{1}+N_{2})/2. This will result in a first mode temperature of 2.4 K. To achieve cooling beyond this requires the phonon cavity to be realized at a much higher frequency, namely N_{2}≪1. Consequently, even though the phonon cavity permits dynamical backaction on to the first mode it is illsuited to cool it into its quantum ground state, that is, N_{1}<1. This objective is much more successfully served by photonic cavities which can be initialized in their quantum ground state by virtue of their higher operation frequencies^{10,11}.
However, the novelty of the present approach lies in the unprecedented ability to parametrically manipulate mechanical nonlinearities, as demonstrated previously with the present system^{15,16,31}, to engineer novel Hamiltonians (equation (1)). Ultimately, this will enable phonons to be squeezed^{34} and entangled in two different colour states^{13} when the mechanical oscillation modes are operated in their quantum ground state^{10,11,25}. Indeed, this phenomenon is not limited only to piezoelectric transducers and it can be generalized to any scheme that can modulate the eigenfrequencies of the first mode and the phonon cavity, enabling the intrinsic intermodal coupling to be exploited. Consequently, this work opens up a new direction for opto/electromechanical systems that inherently incorporate phonon cavities, which promises hitherto unconsidered prospects^{34}.
Methods
Experimental.
The electromechanical resonator was fabricated by conventional micromachining processes from a GaAs/AlGaAs modulationdoped heterostructure sustaining a twodimensional electron gas 90 nm below the surface. The sample was mounted in a highvacuum insert which was then placed in a ^{4}He cryostat.
The pump and probe excitations were generated from a single signal generator with two synchronized outputs (NF Wavefactory 1966). A homodyne measurement scheme was employed and the electromechanical oscillator’s response was amplified by an onchip Sinanofieldeffecttransistor with a 25 dB gain, followed by a transimpedance amplifier (Femto DLPCA200) with a gain of 10^{6} VA^{−1}, and measured in a lockin amplifier (Ametek 7265). A spectrum analyser with a builtin random noise source was used for the noise measurements (Agilent 89410A).
Numerical simulations.
The equations of motion of the parametrically coupled modes ω_{2} and ω_{1} can be extracted from equation (1) as ∂ H/∂ P_{n}=Ẋ_{n} and −∂ H/∂ X_{n}=Ṗ_{n}, using the identity P_{n}=m_{n}Ẋ_{n}, giving
where ω_{s}=ω_{2}+δ_{s} and ω_{p}=ω_{p}^{−}+δ_{p}. Dissipation terms parameterized by Q_{n}, the Duffing nonlinearity defined by β_{n}and a generic pump amplitude Γ in lieu of g have also been introduced and, for simplicity, the mass of both modes is set to unity. Equations (2) and (3) are solved in the rotating frames at ω_{2} and ω_{1}respectively by decomposing X_{2} and X_{1} to
where a_{2}(t), b_{2}(t), a_{1}(t) and b_{1}(t)are slowly varying compared with ω_{2} and ω_{1} respectively. Using equations (4) and (5), equations (2) and (3) can thus be expressed as
where all the offresonance coefficients have been neglected. Equations (6)–(9) are simultaneously numerically solved using the Runge–Kutta method, and the response of the phonon cavity and the first mode at steadystate can be extracted from and respectively. The numerical simulations using this formalism can reproduce all the experimental results and are shown in the Supplementary Information.
Noise displacement calibration.
The displacement of the first mode is calibrated by normalizing the output signal amplitude at the onset of nonlinearity to the critical displacement and is shown in the Supplementary Information^{15}. This yields the responsivity, that is, the change in the output signal per unit displacement . In the proximity of the first mode, a current spectral density limited by the roomtemperature amplifier S_{i}^{1/2}=0.9 pA Hz^{−1/2} is measured, which yields a displacement spectral density (ref. 31).
Consequently, white noise corresponding to 1 mV injected into the spectral region around the first mode results in {S}_{{i}_{N}}^{1/2}=14 pA Hz^{−1/2}. This yields a noise displacement .
Change history
26 April 2012
In the version of this Letter originally published, the name of the first author of ref. 21 was incorrect. This error has been corrected in the HTML and PDF versions of the Letter.
References
Kippenberg, T. J. & Vahala, K. J. Cavity optomechanics: Backaction at the mesoscale. Science 321, 1172–1176 (2008).
Metzger, C. H. & Karrai, K. Cavity cooling of a microlever. Nature 432, 1002–1005 (2004).
Arcizet, O., Cohadon, P. F., Briant, T., Pinard, M. & Heidmann, A. Radiationpressure cooling and optomechanical instability of a micromirror. Nature 444, 71–74 (2006).
Gröblacher, S. et al. Demonstration of an ultracold microoptomechanical oscillator in a cryogenic cavity. Nature Phys. 5, 485–488 (2009).
Riviére, R. et al. Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state. Phys. Rev. A 83, 063835 (2011).
Thompson, J. D. et al. Strong dispersive coupling of a highfinesse cavity to a micromechanical membrane. Nature 452, 72–75 (2008).
Teufel, J. D., Harlow, J. W., Regal, C. A. & Lehnert, K. W. Dynamical backaction of microwave fields on a nanomechanical oscillator. Phys. Rev. Lett. 101, 197203 (2008).
Rocheleau, T. et al. Preparation and detection of a mechanical resonator near the ground state of motion. Nature 101, 72–75 (2009).
Park, Y. S. & Wang, H. Resolvedsideband and cryogenic cooling of an optomechanical resonator. Nature Phys. 5, 489–493 (2009).
Teufel, J. D. et al. Sideband cooling of micromechanical motion to the quantum ground state. Nature 475, 359–363 (2011).
Chan, J. et al. Laser cooling of a nanomechanical oscillator into its quantum ground state. Nature 478, 89–92 (2011).
Faust, T., Krenn, P., Manus, S., Kotthaus, J. P. & Weig, E. M. Microwave cavityenhanced transduction for plug and play nanomechanics at room temperature. Nature Commun. 3, 728 (2012).
ZakkaBajjani, E. et al. Quantum superposition of a single microwave photon in two different colour states. Nature Phys. 7, 599–603 (2011).
Venstra, W., Westra, H. & van der Zant, H. Qfactor control of a microcantilever by mechanical sideband excitation. Appl. Phys. Lett. 99, 151904 (2011).
Mahboob, I. & Yamaguchi, H. Bit storage and bit flip operations in an electromechanical oscillator. Nature Nanotech. 3, 275–279 (2008).
Mahboob, I., Flurin, E., Nishiguchi, K., Fujiwara, A. & Yamaguchi, H. Interconnectfree parallel logic circuits in a single mechanical resonator. Nature Commun. 2, 198 (2011).
Weis, S. et al. Optomechanically induced transparency. Science 330, 1520–1523 (2010).
SafaviNaeini, A. H. et al. Electromagnetically induced transparency and slow light with optomechanics. Nature 472, 69–73 (2011).
Gröblacher, S., Hammerer, K., Vanner, M. R. & Aspelmeyer, M. Observation of strong coupling between a micromechanical resonator and an optical cavity field. Nature 460, 724–727 (2009).
Teufel, J. D. et al. Circuit cavity electromechanics in the strongcoupling regime. Nature 471, 204–208 (2011).
De Liberato, S., Lambert, N. & Nori, F. Quantum noise in photothermal cooling. Phys. Rev. A 83, 033809 (2011).
Marquardt, F., Chen, J. P., Clerk, A. A. & Girvin, S. M. Quantum theory of cavityassisted sideband cooling of mechanical motion. Phys. Rev. Lett. 99, 093902 (2007).
WilsonRae, I., Nooshi, N., Zwerger, W. & Kippenberg, T. J. Theory of ground state cooling of a mechanical oscillator using dynamical backaction. Phys. Rev. Lett. 99, 093901 (2007).
Lambert, N., Johansson, R. & Nori, F. A macrorealism inequality for optoelectromechanical systems. Phys. Rev. B 84, 245421 (2011).
O’Connell, A. D. et al. Quantum ground state and singlephonon control of a mechanical resonator. Nature 464, 697–703 (2010).
Rodrigues, D. A. Fanolike antiresonances in nanomechanical and optomechanical systems. Phys. Rev. Lett. 102, 067202 (2009).
Agarwal, G. S. & Huang, S. Electromagnetically induced transparency in mechanical effects of light. Phys. Rev. A 81, 041803 (2010).
Schliesser, A. & Kippenberg, T. J. Cavity optomechanics with whisperinggallery mode optical microresonators. Adv. At. Mol. Opt. Phys. 58, 207–323 (2010).
Mahboob, I., Wilmart, Q., Nishiguchi, K., Fujiwara, A. & Yamaguchi, H. Wideband idler generation in a GaAs electromechanical resonator. Phys. Rev. B 84, 113411 (2011).
Westra, H. J. R., Poot, M., van der Zant, H. S. J. & Venstra, W. J. Nonlinear modal interactions in clampedclamped mechanical resonators. Phys. Rev. Lett. 105, 117205 (2010).
Mahboob, I., Flurin, E., Nishiguchi, K., Fujiwara, A. & Yamaguchi, H. Enhanced force sensitivity and noise squeezing in an electromechanical resonator coupled to a nanotransistor. Appl. Phys. Lett 97, 253105 (2010).
Dobrindt, J. M., WilsonRae, I. & Kippenberg, T. J. Parametric normalmode splitting in cavity optomechanics. Phys. Rev. Lett. 101, 263602 (2008).
Massel, F. et al. Microwave amplification with nanomechanical resonators. Nature 480, 351–354 (2011).
Santamore, D. H., Doherty, A. C. & Cross, M. C. Quantum nondemolition measurement of Fock states of mesoscopic mechanical oscillators. Phys. Rev. B 70, 144301 (2004).
Acknowledgements
The authors are grateful to S. Miyashita for growing the heterostructure, A. Fujiwara for support and N. Lambert, P. D. Nation and F. Nori for discussions and comments. This work was partly supported by Japan Society for the Promotion of Science (JSPS) KAKENHI (20246064) and (23241046).
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I.M. conceived the experiment, designed and fabricated the electromechanical resonator, performed the measurements, analysed the data and wrote the paper. K.N. designed and fabricated the SinanoFET amplifiers. H.Y. planned the project. All authors discussed the results.
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Mahboob, I., Nishiguchi, K., Okamoto, H. et al. Phononcavity electromechanics. Nature Phys 8, 387–392 (2012). https://doi.org/10.1038/nphys2277
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DOI: https://doi.org/10.1038/nphys2277
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