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 system10,11,22,23,24, a long-standing goal in solid-state physics25. 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 science17,18,20,26,27,28.

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 regime1,22,23. 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 cavity1.

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 manipulated29,30, which enables phonon-cavity electromechanics to be realized1,20.

The electromechanical resonator used in this study is shown in Fig. 1a and described in detail elsewhere15,16. It hosts flexural oscillation modes that are transduced by the piezoelectric effect, where the first mode ω1/2π=171.3 kHz with damping γ1/2π=ω1/2π Q1=1.1 Hz (Q is the quality factor) and the second mode ω2/2π=470.93 kHz with γ2/2π=ω2/2π Q2=8.5 Hz, which henceforth will be known as the phonon cavity (see Fig. 1b). This data indicates ultra-deep resolved sideband operation with an unprecedented ratio ω1/γ2≈2×104, which is orders of magnitude larger than photon-based counterparts1,2,3,4,5,6,7,8,9,10,11,14,22,23.

Figure 1: The phonon-cavity electromechanical system.
figure 1

a, A false-colour electron micrograph of the mechanical resonator sustaining a two-dimensional electron gas (purple) above which Schottky-contacted gold electrodes (orange) are located. The piezoelectric effect between them transduces the mechanical motion. Also shown is a simplified measurement circuit, where the harmonic probe is used to investigate the first mode or the phonon cavity and the pump which dynamically couples them. The mechanical-motion-induced piezovoltage is first amplified on-chip then at room temperature and is measured using either a lock-in or a spectrum analyser. b, The phonon-cavity electromechanical system, harmonically probed with an amplitude of 100 μVr.m.s. and 3 mVr.m.s., reveals the first mode and the second mode (that is, the phonon cavity), respectively, as function of d.c. bias instead of the pump as depicted in a (simulated mode profiles are also shown in the insets). The d.c. bias generates piezoelectric strain which modifies their eigenfrequencies and it can be exploited to enhance the parametric intermodal coupling. c, A schematic profile of the phonon-cavity electromechanical system, where the mechanical motion of interest is given by the first mode ω1, the second mode ω2 plays the role of the phonon cavity and the anti-Stokes (Stokes) pump that dynamically couples them when activated at ωp− (ωp+).

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In the phonon-based system and in contrast to photon-cavity electromechanical systems, the motion of the first mode induces tension that can modify the frequency of the phonon cavity, creating sidebands at ω2±ω1, as schematically depicted in Fig. 1c (ref. 1). The coupling strength G=dω2/dX1 quantifies the degree of parametric intermodal coupling and is given by the tension-induced change in the phonon-cavity 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 ω1/2π=11 Hz V−1 and δ ω2/2π=6 Hz V−1 respectively, as shown in Fig. 1b (ref. 15). Consequently, if the system is periodically pumped (ωp) 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 ωnV pcos(ωpt), 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/γ2) delayed response of the phonon cavity results in a backaction force on the first mode via the same coupling mechanism modifying its oscillation dynamics1,14. The parametric coupling and the dynamical backaction effects can be greatly enhanced when ωp−=ω2−ω1 (ωp+=ω2+ω1) and pumping on the anti-Stokes/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 Xn and Pn denoting the position and conjugate momentum of the constituent systems with mass mn. Λ is the amplitude of the harmonic probe of the phonon cavity with ωs≈ω2. Δn 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 ωp=2ωn (ref. 15). The combination of the terms with Λ and Δ2 can also permit degenerate/non-degenerate parametric amplification and squeezing when ωp≈2ω2 (refs 16, 31). The last term describes the linearized parametric coupling between the first mode and the phonon cavity20,32, where the coupling rate g∝G V p, and this becomes the dominant term when ωp=ωp±. Thus the term containing Δn can be neglected in the present investigation.

To probe the phonon cavity, a weak harmonic excitation (3 mVr.m.s.) is applied at ωs≈ω2 and the system is pumped on the red sideband at ωp− 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 energy29. 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 cavity17,18,20,27,28. As the pump amplitude is increased further, the condition g≥γ2≫γ1 can be satisfied, and the system enters the strong-coupling 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 normal-mode splitting; a widely observed phenomenon in strongly coupled systems19,20,25,32.

Figure 2: The phonon-cavity dynamics.
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a,b, The phonon cavity’s response as a function of the anti-Stokes pump amplitude at ωp− where all the spectra are offset for clarity in b. For weak pump amplitudes, phonons are converted from the phonon cavity to the first mode, resulting in a dip in its response (that is, the phonon cavity has a mechanically induced transparency). By increasing the pump amplitude further, the parametric coupling is enhanced and the phonon cavity begins to hybridize with the first mode, undergoing parametric normal-mode splitting (note the phonon cavity undergoes a thermal drift for large pump amplitudes owing to current leakage between the top-gate and the two-dimensional electron gas). c, The coupling rate extracted from a (points) reveals a linear dependence on the anti-Stokes pump amplitude and is confirmed by a least squares fit (line). d, The numerical solution to equation (1) for the conditions described in a confirms the experimental result, albeit without the thermal drift. e–g, The spectroscopy of the parametric coupling via the phonon cavity with the anti-Stokes pump amplitudes V p= 0.125, 0.25 and 0.5 Vr.m.s. respectively. As the coupling between the first mode and the phonon cavity is increased, it results in the characteristic anti-crossing of strongly coupled systems.

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The experimental results shown in Fig. 2a,b confirm this assertion, where the phonon cavity becomes transparent at pump amplitudes of 0.3 Vr.m.s.. As the pump amplitude is increased further the system exhibits the emergence of parametric normal-mode 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 anti-Stokes pump to 8 Hz, placing the system on the cusp of the strong coupling regime19,20.

Alternatively, detuning the anti-Stokes 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 anti-Stokes pump amplitude is increased and results in the avoided-crossing 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 phonon-cavity 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 Information33.

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 ωp−, the first mode is probed with a weak (100 μVr.m.s.) harmonic excitation as a function of the pump amplitude, as shown in Fig. 3a,b. In this configuration, the phonons from the anti-Stokes 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 anti-Stokes pump amplitude is increased, the enhanced coupling between the first mode and the phonon cavity results in Q1 being greatly reduced, as shown in Fig. 4c. For the strongest pump amplitudes, the coupling reaches a point where parametric normal-mode splitting can even be observed in the first mode and its behaviour can no longer be characterized by a single harmonic oscillator response function19.

Figure 3: The backaction of the phonon cavity on the first mode.
figure 3

a,b, The response of the first mode to the anti-Stokes pump at ωp−. As the pump amplitude is increased, phonons are transferred from the first mode to the phonon cavity, resulting in X1 decreasing and γ1 increasing. At the strongest pump amplitudes the parametric coupling is amplified, resulting in the emergence of normal-mode splitting in the response of the first mode. Note all the spectra are offset for clarity in b. c–e, The spectroscopy of the phonon-cavity electromechanical system via the first mode as a function of the anti-Stokes pump detuning and amplitude at 0, 0.25 and 0.5 Vr.m.s. respectively reveals its transformation as it is more strongly coupled to the phonon cavity. f,g, γ1 and ω1 as a function of the anti-Stokes pump frequency (points) clearly reveal signatures of the in-phase and quadrature components of the phonon cavity in the response of the first mode. Also shown are theoretical fits for the dynamical backaction (lines), as described in the main text.

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Figure 4: Cooling the first mode via the phonon cavity.
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a, A white noise displacement X′1corresponding to a temperature T′≫3.6 K is injected into a narrow bandwidth around the first mode. Concurrently, the anti-Stokes pump is activated at ωp−, which results in the displacement noise energy from the first mode being transferred to the phonon cavity, cooling the fundamental mechanical oscillation in the process. b, The power spectral density of the first mode as a function of the pump amplitude clearly reveals its area and, hence, temperature is reduced as it is more strongly coupled to the phonon cavity. At the largest pump amplitudes, the emergence of parametric normal-mode splitting can even be observed in the response of the first mode. Note all the spectra are offset for clarity. c,d, The corresponding Q1 and T/T′ respectively extracted from the above measurement as a function of the anti-Stokes pump amplitude (red points). Also shown are Q1 and the normalized area extracted from Fig. 3a (blue points), confirming the trends observed in the noise measurement. For a pump amplitude of more than 0.3 Vr.m.s. (shaded region) normal-mode splitting can be observed in the response of the first mode, resulting in it no longer being described by a single harmonic oscillator response function.

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On the other hand, if the anti-Stokes pump is detuned, the phonon cavity’s dynamics can be imprinted onto the first mode, where γ1 and ω1 trace out its in-phase and quadrature components, as shown in Fig. 3c–g. For a pump amplitude of 0.5 Vr.m.s., γ1/2π tends to (γ1+γ2)/4π ∼5 Hz; concurrently ω1/2π also undergoes a 5 Hz shift (Fig. 3f,g). From the theory of photon-based opto/electromechanical systems operated in the resolved sideband regime, the change in ω1 and γ1 can be expressed as 4g2δ/(γ22+4δ2) and 4g2γ2/(γ22+4δ2) respectively, where the pump detuning about ωp− 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 γ2.

Invariably, the dynamical backaction of the photonic cavity has been employed to cool a mechanical oscillation mode1,2,3,4,5,6,7,8,9,10,11,22,23,24. 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 anti-Stokes pump at ωp− 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 X1 and Q1 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 Q1 as a function of the anti-Stokes 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, Nn=kbT/ℏωn, where kB is the Boltzmann constant, of the first mode and the phonon cavity converging to (N1+N2)/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 N2≪1. Consequently, even though the phonon cavity permits dynamical backaction on to the first mode it is ill-suited to cool it into its quantum ground state, that is, N1<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 frequencies10,11.

However, the novelty of the present approach lies in the unprecedented ability to parametrically manipulate mechanical nonlinearities, as demonstrated previously with the present system15,16,31, to engineer novel Hamiltonians (equation (1)). Ultimately, this will enable phonons to be squeezed34 and entangled in two different colour states13 when the mechanical oscillation modes are operated in their quantum ground state10,11,25. 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 prospects34.

Methods

Experimental.

The electromechanical resonator was fabricated by conventional micromachining processes from a GaAs/AlGaAs modulation-doped heterostructure sustaining a two-dimensional electron gas 90 nm below the surface. The sample was mounted in a high-vacuum insert which was then placed in a 4He 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 on-chip Si-nano-field-effect-transistor with a 25 dB gain, followed by a transimpedance amplifier (Femto DLPCA-200) with a gain of 106 VA−1, and measured in a lock-in amplifier (Ametek 7265). A spectrum analyser with a built-in random noise source was used for the noise measurements (Agilent 89410A).

Numerical simulations.

The equations of motion of the parametrically coupled modes ω2 and ω1 can be extracted from equation (1) as ∂ H/∂ Pn=Ẋn and −∂ H/∂ Xn=Ṗn, using the identity Pn=mnẊn, giving

where ωs=ω2+δs and ωp=ωp−+δp. Dissipation terms parameterized by Qn, the Duffing nonlinearity defined by βnand 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 ω2 and ω1respectively by decomposing X2 and X1 to

where a2(t), b2(t), a1(t) and b1(t)are slowly varying compared with ω2 and ω1 respectively. Using equations (4) and (5), equations (2) and (3) can thus be expressed as

where all the off-resonance 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 steady-state 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 Information15. 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 room-temperature amplifier Si1/2=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 .