Electromechanical transduction gain of 21 dB is realized in a micro-cantilever resonant force sensor operated in the unresolved-sideband regime. Strain-dependent kinetic inductance weakly couples cantilever motion to a superconducting nonlinear resonant circuit. A single pump generates motional sidebands and parametrically amplifies them via four-wave mixing. We study the gain and added noise, and we analyze potential benefits of this integrated amplification process in the context force sensitivity.

A resonant electromechanical transducer converts force to motion with a responsivity solely determined by its mechanical design. Cavity optomechanics offers an efficient scheme for measuring motion, but in many cases the transducer's overall sensitivity to force is limited by added noise in the measurement chain. In such cases the addition of a quantum-limited parametric amplifier can enhance sensitivity.1 Here, we use the intrinsic Kerr nonlinearity of a superconducting microwave resonant circuit to achieve transduction gain in a micro-cantilever force sensor. However, for the device studied here, analysis of the noise revealed that this gain did not result in enhanced sensitivity to force.

In comparison to optical cavities, microwave circuits are appealing for their relative ease of integration with micro- and nano-electromechanical systems and their ability to achieve stronger electromechanical coupling. Since their first implementation,1,2 a large variety of superconducting microwave electromechanical devices have been studied and optimized for different goals,3 such as a large single-photon coupling rate g0,4–7 electromechanical cooperativity C,8–10 amplification,11,12 and efficient cooling of the mechanical mode.13–19 Depending on the application, navigation of the design parameter space results in significantly different paths toward the goal.

Optimizing an electromechanical transducer for sensitivity to force constrains the mechanical resonator's shape, size, mass, and resonance frequency. These constraints can conflict with other figures of merit. For example, a larger mass reduces the coupling rate g0 and mechanical resonance frequency Ωm, leading to operation in the unresolved-sideband regime and larger phonon occupation n¯m at a given temperature. Reduced g0 can be compensated for with a stronger pump that increases the intracavity photon number n¯c, but superconducting circuits experience current-induced depairing that sets a limit on n¯c. This depairing results in a nonlinear inductance, which is usually seen as a problem.16 However, recent works exploit this nonlinearity for efficient cooling of the mechanical mode, both in the resolved18 and unresolved20 sideband regimes. The Kerr nonlinearity of kinetic inductance is also used for parametric amplification21 via three-wave or four-wave mixing processes.22,23

In this paper, we report on a microwave resonant circuit patterned from a thin film of niobium titanium nitride (Nb–Ti–N), which is weakly coupled to a micro-cantilever in the unresolved-sideband regime. We exploit the intrinsic Kerr nonlinearity of the superconducting circuit in a four-wave mixing process to realize 21 dB of transduction gain. We define the transduction gain as the ratio of the measured amplitude of the optomechanical motional sideband, normalized to that which would be produced by a linear cavity pumped to the same intracavity photon number.

In the cavity optomechanical transduction scheme, mechanical fluctuations with average phonon number n¯m are imprinted on the optical spectrum leaking out of the cavity, appearing as motional noise sidebands around the cavity pump frequency. In microwave electromechanics, the signal leaking from a resonant circuit into a transmission line is amplified before being demodulated and typically the added noise of this amplifier is the limiting factor degrading force sensitivity. Force sensitivity is expressed as a minimum detectable force Fmin, i.e., the force per unit bandwidth detected at unity signal-to-noise ratio. This definition accounts for actual force noise from the environment and the backaction of measurement as well as added photon shot noise and amplification noise nadd expressed as an equivalent force noise.24 The normalized force sensitivity is given by
FminFSQL=(n¯m+12)Mechanics+|Ceff|Backaction+116|Ceff|Shotnoise+nadd8|Ceff|Amplifiernoise,
(1)
where FSQL=2meffΓΩm is the force sensitivity at the standard quantum limit (SQL), set by the resonator's internal loss rate Γ and effective mass meff. Measurement induces noise, which translates to force through the effective cooperativity Ceff.24 In the unresolved-sideband regime, the effective cooperativity can be approximated to
Ceff(n¯c)=1(12iω/κ)24g02n¯cκΓ4g02n¯cκΓ,
(2)
where κ is the total loss rate of the cavity. Figure 1 shows each contribution of Eq. (1) as a function of input power P and their overall effect on sensitivity. Here, PSQL is the applied microwave power when the measurement is at the standard quantum limit, where shot noise and backaction noise are equal, corresponding to Ceff = 1/4. For input power below PSQL, the added noise of the first stage of amplification limits force sensitivity. For example, in Fig. 1 we show the contribution from a cryogenic low-noise amplifier (LNA) with an equivalent noise temperature of 4 K. At 4.5 GHz, this corresponds to approximately nLNA=19 photons, deteriorating the force sensitivity so that detection at SQL is no longer possible and the optimal sensitivity is achieved for powers P>PSQL.
FIG. 1.

Force sensitivity on mechanical resonance vs input power, both normalized to that at the standard quantum limit. The total force sensitivity (solid lines) is given by the mechanical fluctuations, backaction noise, as well as the force-equivalent shot noise and added amplifier noise. The effect of an additional stage of amplification is shown, either for a thermal state (n¯m=304) or the quantum ground state of the mechanical mode.

FIG. 1.

Force sensitivity on mechanical resonance vs input power, both normalized to that at the standard quantum limit. The total force sensitivity (solid lines) is given by the mechanical fluctuations, backaction noise, as well as the force-equivalent shot noise and added amplifier noise. The effect of an additional stage of amplification is shown, either for a thermal state (n¯m=304) or the quantum ground state of the mechanical mode.

Close modal

Cantilevers operating in cryogenic environments typically have mechanical resonance frequencies that put the mechanical mode in a thermal state with n¯mkBT/Ωm1. Figure 1 shows that, compared to the standard quantum limit, the thermally noise-limited case has an extended interval of power where force sensitivity is nearly constant and solely determined by the properties of the mechanical resonator. However, the nonlinearity of superconducting microwave circuits commonly limits n¯c, which, together with a small coupling rate g0, results in a situation where force sensitivity is instead limited by the added noise of a subsequent LNA. As shown in Fig. 1, one may improve the force sensitivity by adding a quantum-limited amplifier between the resonant circuit, hereafter called “cavity,” and the LNA. For a sufficiently high-gain phase-preserving quantum-limited parametric amplifier, the number of added photons by the amplifier to the measurement is given by nPA=1/2.25 Yet, adding a separate quantum-limited amplifier comes at significant cost in complexity and more complicated operation, including an additional pump, pump-cancellation tone, as well as additional isolators and associated cabling.14 In our work, we investigate a simpler implementation where a nonlinear cavity is not only used for transduction of motion to measured signal but also for amplification, i.e., with one pump both generating and amplifying the motional sidebands.

We demonstrate this implementation with the device shown in Fig. 2(a). The cantilever force sensor is tightly integrated with a compact microwave circuit consisting of an interdigitated capacitor in series with a long meandering nanowire having large kinetic inductance. The nanowire has width 100 nm and thickness 15 nm, and it meanders along the base of the cantilever, as shown in detail in the false-color micrograph of Fig. 2(b). The silicon nitride (Si–N) micro-cantilever has a fundamental bending mode with resonance frequency Ωm/2π=685.387kHz and linewidth Γ/2π=4.3Hz. The microwave cavity with resonance frequency ω0/2π=4.378GHz is strongly overcoupled to the transmission line, having a total linewidth κ/2π=21.18MHz, corresponding to a loaded quality factor of Q=206.7. Cantilever bending generates surface strain, which is maximum at the line where the cantilever meets the silicon substrate. This strain changes the nanowire's kinetic inductance, shifting the cavity resonance and thereby realizing electromechanical mode coupling.26,27 For purely geometric coupling where the kinetic inductance per unit length is assumed constant, simulations of the strain28 result in a single-photon coupling strength g050mHz, giving an effective cooperativity of Ceff104 at intracavity photon number n¯c=106.

FIG. 2.

Scanning electron microscope (SEM) image of (a) the cantilever and (b) the 100 nm-wide meandering nanowire inductor. The cantilever is formed from a 600 nm-thick silicon nitride plate and the nanowire is etched from a 15 nm-thin film of Nb–Ti–N. (c) Phase vs frequency of the microwave resonance for increasing input power Pin. The resonance frequency of the cavity ω0 shifts to lower frequencies with increasing Pin, typical of a resonator with a Kerr-type nonlinearity. (d) The shift in resonance frequency Δω0 as a function of intracavity photon number n¯c for a pump blue-detuned by approximately 50 MHz. For each pump power, ω0 is measured by sweeping a weak probe tone through resonance. The Kerr coefficient is K/2π = −1.57 Hz/photon. (e) Mechanical susceptibility measured from the driven motional sideband with resonant pumping in the linear regime of the cavity. The sideband amplitude is expressed in decibel with respect to the measured response at the carrier pump frequency (dBc).

FIG. 2.

Scanning electron microscope (SEM) image of (a) the cantilever and (b) the 100 nm-wide meandering nanowire inductor. The cantilever is formed from a 600 nm-thick silicon nitride plate and the nanowire is etched from a 15 nm-thin film of Nb–Ti–N. (c) Phase vs frequency of the microwave resonance for increasing input power Pin. The resonance frequency of the cavity ω0 shifts to lower frequencies with increasing Pin, typical of a resonator with a Kerr-type nonlinearity. (d) The shift in resonance frequency Δω0 as a function of intracavity photon number n¯c for a pump blue-detuned by approximately 50 MHz. For each pump power, ω0 is measured by sweeping a weak probe tone through resonance. The Kerr coefficient is K/2π = −1.57 Hz/photon. (e) Mechanical susceptibility measured from the driven motional sideband with resonant pumping in the linear regime of the cavity. The sideband amplitude is expressed in decibel with respect to the measured response at the carrier pump frequency (dBc).

Close modal

Figure 2(c) shows the phase response of the cavity as a function of frequency and power measured in a dilution refrigerator at T=10mK. The phase of the reflected signal changes by 2π when sweeping through resonance, characteristic of an overcoupled resonator measured in reflection. Increasing power shifts the resonance frequency to lower values and the phase response sharpens, as expected from a current-induced pair-breaking nonlinear inductance. Such behavior is approximated to leading order by a Kerr-type nonlinearity, where the Kerr coefficient K describes the strength of the nonlinearity in terms of a frequency shift per photon. We measure the shift of the cavity resonance frequency as a function of intracavity photon number n¯c, by varying the power of a blue-detuned pump while sweeping a much weaker probe tone through resonance. Figure 2(d) shows the result of this measurement and the linear fit to determine a Kerr coefficient K/2π = −1.57 Hz/photon for our device.

We operate the nonlinear cavity as a four-wave mixing parametric amplifier29 and analyze the gain G by injecting a single probe tone at ωs, blue-detuned by Ωm from a strong pump tone at ωp. The amplifier bandwidth is smaller than the cavity linewidth κ and it decreases with increasing gain (pump power). We therefore intentionally designed the cavity with a relatively large κ (low Q) to operate in the unresolved-sideband regime, ensuring high gain at detuning Ωm. Figures 3(a) and 3(b) show the measured gain vs pump frequency for various pump powers, obtained by sweeping both pump and probe frequency with fixed separation ωsωp=Ωm. For increasing pump power, the gain peak shifts to lower frequency, as expected for a negative Kerr coefficient. For our device, we reach G24dB for the largest pump power before the cavity bifurcates and the gain degrades. Figure 3(c) shows the gain and bandwidth for selected pump powers, measured by fixing ωp at the previously determined maximum gain for each power and sweeping the signal tone ωs.

FIG. 3.

(a) Four-wave mixing parametric gain of an injected signal tone ωs as a function of pump frequency ωp and pump power Pin at fixed detuning (ωsωp)/2π = 685.386 kHz. (b) Signal gain vs ωp at various values of pump power. (c) Gain vs ωs for various ωp and Pin along the ridge of maximum gain in (a).

FIG. 3.

(a) Four-wave mixing parametric gain of an injected signal tone ωs as a function of pump frequency ωp and pump power Pin at fixed detuning (ωsωp)/2π = 685.386 kHz. (b) Signal gain vs ωp at various values of pump power. (c) Gain vs ωs for various ωp and Pin along the ridge of maximum gain in (a).

Close modal

To demonstrate the transduction gain, we repeat the sweep over pump power and pump frequency, now driving the cantilever through a piezoelectric shaker. The cantilever motion generates sidebands in the upconverted spectrum, at either side of ωp. Since two sidebands are generated, the gain will be sensitive to their relative phase, which is determined by the properties of the cavity. However, in the unresolved-sideband regime, this relative phase is such that transduction gain is observed.30 To capture the motion spectrum, we use a multifrequency lock-in to excite the cantilever with a tuned frequency comb with equal amplitudes and phases chosen to reduce peak excitation in the time domain (see the supplementary material). The lock-in measures the response at the upconverted comb frequencies, capturing the mechanical motion spectrum in a single measurement time window. Fitting a model composed of a Lorentzian with a frequency-independent added noise to the data, we extract the area under the sideband, proportional to the amplitude spectrum of the drive mechanical displacement. Figure 4(a) shows the integrated sideband response normalized to the input amplitude, from which we identify the frequency of maximum response for each pump power.

FIG. 4.

(a) Parametric amplification of an upconverted motional sideband when the mechanical mode is driven with a frequency comb. The area under the motional sideband is plotted as a function of pump frequency ωp and pump amplitude/power, normalized to the input amplitude. Increasing the pump power shifts the cavity's resonance frequency and amplifies the response at the motional sideband. (Inset) Zoom of the dashed box with finer stepping, highlighting the ridge of maximum response. (b) Transduction gain G and n¯c at multiple pump powers along the ridge of maximum sideband response in (a). (c) Fluctuations of the undriven mechanical mode measured at the upper motional sideband for three pump powers with ωp placed on the ridge of maximum gain. The power spectral density (PSD) is normalized to the intracavity photon number n¯c. (d) Measured nadd and that expected for a quantum-limited parametric amplifier followed by the LNA.

FIG. 4.

(a) Parametric amplification of an upconverted motional sideband when the mechanical mode is driven with a frequency comb. The area under the motional sideband is plotted as a function of pump frequency ωp and pump amplitude/power, normalized to the input amplitude. Increasing the pump power shifts the cavity's resonance frequency and amplifies the response at the motional sideband. (Inset) Zoom of the dashed box with finer stepping, highlighting the ridge of maximum response. (b) Transduction gain G and n¯c at multiple pump powers along the ridge of maximum sideband response in (a). (c) Fluctuations of the undriven mechanical mode measured at the upper motional sideband for three pump powers with ωp placed on the ridge of maximum gain. The power spectral density (PSD) is normalized to the intracavity photon number n¯c. (d) Measured nadd and that expected for a quantum-limited parametric amplifier followed by the LNA.

Close modal

Maximum transduction gain follows a ridge in pump power and frequency, as expected for parametric gain generated close to bifurcation. Figure 4(b) shows the transduction gain G and n¯c vs pump power, where the pump frequency is adjusted to follow the ridge of maximum sideband response. We determine n¯c using methods described in Ref. 18. At each pump power, the amplified signal is the integrated sideband power spectrum, normalized to n¯c. The transduction gain is given by the ratio of this amplified signal, to that in the low power regime, where G1. For a detailed description, see the supplementary material.

Operating the sensor on this ridge of maximum gain, we are able to resolve the amplified motional noise of the undriven cantilever, as shown in Fig. 4(c). The plotted data are an average of ten consecutive Power Spectral Densities (PSDs) measured with resolution bandwidth 1.86 Hz. The solid lines represent the best fit of a model comprised of a Lorentzian plus white added noise. Averaging over 100 consecutive fits, we extract the added noise expressed as equivalent photon number, as shown in Fig. 4(d).

The ideal phase-insensitive parametric amplifier would result in a reduction of the added noise with increasing gain, approaching nadd=1/2 at high gain [dashed curve in Fig. 4(d)] as described by
nadd=nPA+nLNAG.
(3)
However, we observe instead an increase in the added noise. Possible explanations of this additional noise could be nonlinear loss mechanisms in the cavity, for example increased quasiparticle losses associated with the current-induced pair-breaking nonlinearity, or heating of the cavity by the pump.31 

The use of a nonlinear cavity to parametrically amplify the motional sidebands also results in increased backaction noise due to the amplification of the intracavity fluctuations.32 In contrast, a cold isolator screens the cavity from this backaction when using a separate parametric amplifier.1 The potential improvements in the resulting force sensitivity are contingent upon the effective cooperativity. In the case considered here, Ceff1/4, such amplified backaction is negligible at the observed level of gain.

In conclusion, we described a micro-cantilever force sensor with a compact and integrated microwave cavity. The sensor employed kinetic inductive electromechanical coupling to realize force transduction, and the nonlinearity of the superconducting cavity was used for parametric amplification. The measurement configuration required only a single pump to combine these two effects, achieving up to 21 dB transduction gain of the electromechanical motional sidebands. However, noise analysis revealed that this gain did not come with an improvement in signal-to-noise ratio, most likely due to nonlinear loss mechanisms or heating by the pump. It is possible to mitigate these effects by designing for higher Kerr coefficient through a thinner superconducting film (lower critical current) or through a shorter nanowire (larger current for given n¯c). Further study of this sensor concept is required to determine under which circumstances this gain mechanism can be exploited for improved force sensitivity.

See the supplementary material for details on the multifrequency lock-in measurement technique with a tuned frequency comb driving the mechanical oscillator, and determination of the K coefficient with two tone spectroscopy and analysis of the intracavity photon number and transduction gain.

We acknowledge funding from the European Union Horizon 2020 EIC Pathfinder Grant Agreement No. 828966—QAFM and the Swedish SSF Grant No. ITM17-0343 supported this work. We thank the Quantum-Limited Atomic Force Microscopy (QAFM) team for fruitful discussions: T. Glatzel, M. Zutter, E. Tholén, D. Forchheimer, I. Ignat, M. Kwon, and D. Platz. We particularly thank anonymous reviewer 2 whose careful reading and thoughtful comments helped to substantially improve the paper.

The authors have no conflicts to disclose.

Ermes Scarano: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Elisabet K. Arvidsson: Conceptualization (equal); Formal analysis (equal); Investigation (supporting); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (equal). August K. Roos: Conceptualization (equal); Formal analysis (supporting); Investigation (supporting); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (supporting). Erik Holmgren: Supervision (supporting). David B. Haviland: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are openly available in Zenodo at https://doi.org/10.5281/zenodo.10605406, Ref. 33.

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