Intrinsic Kerr amplification for microwave electromechanics

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 g 0 [4][5][6][7] , electromechanical cooperativity C 8-10 , amplification 11,12 , and efficient cooling of the mechanical mode [13][14][15][16][17][18][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 g 0 and mechanical resonance frequency Ω m , leading to operation in the unresolved-sideband regime and larger phonon occupation nm at a given temperature.Reduced g 0 can be compensated for with a stronger pump that increases the intracavity photon number nc , but superconducting circuits experience current-induced depairing that sets a limit on nc .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 resolved 18 and a) Electronic mail: haviland@kth.seunresolved 20 sideband regimes.The Kerr nonlinearity of kinetic inductance is also used for parametric amplification 21 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 nm 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 F min , 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 n add expressed as an equivalent force noise 24 .The normalized force sensitivity is given by where F SQL = √ 2m eff ΓℏΩ m is the force sensitivity at the standard quantum limit (SQL), set by the resonators internal loss rate Γ and effective mass m eff .Measurement induces noise, which translates to force through the effective cooperativity C eff 24 .In the unresolved-sideband regime the effective cooperativity can be approximated to, FIG. 1. Force sensitivity on mechanical resonance versus 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 (nm = 304) or the quantum ground state of the mechanical mode.
where κ is total loss rate of the cavity.Figure 1 shows each contribution of Eqn.(1) as a function of input power P and their overall effect on sensitivity.Here P SQL is the applied microwave power when the measurement is at the standard quantum limit, where shot noise and backaction noise are equal, corresponding to C eff = 1/4.For input power below P SQL , the added noise of the first stage of amplification limits force sensitivity.For example, in Fig. 1 we show the contribution from a cryogenic lownoise amplifier (LNA) with an equivalent noise temperature of 4 K.At 4.5 GHz this correspond to approximately n LNA = 19 photons, deteriorating the force sensitivity so that detection at SQL is no longer possible and the optimal sensitivity is achieved for powers P > P SQL .
Cantilevers operating in cryogenic environments typically have mechanical resonance frequencies that put the mechanical mode in a thermal state with nm ≈ k B T /ℏΩ m ≫ 1. 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 limit nc , which, together with a small coupling rate g 0 , result 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 n PA = 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 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 Fig. 2(b).The silicon nitride (Si-N) micro-cantilever has a fundamental bending mode with resonance frequency Ω m /2π = 685.387kHz and linewidth Γ/2π = 4.3 Hz.The microwave cavity with resonance frequency ω 0 /2π = 4.378 GHz is strongly overcoupled to the transmission line, having total linewidth κ/2π = 21.18MHz, corresponding to a loaded quality factor of Q = 206.7.Can-tilever 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 strain 28 result in a single-photon coupling strength g 0 ≈ 50 mHz, giving an effective cooperativity of C eff ≈ 10 −4 at intracavity photon number nc = 10 6 .
Figure 2(c) shows the phase response of the cavity as a function of frequency and power measured in a dilution refrigerator at T = 10 mK.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 nc , 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 = −1.57Hz/photon for our device.
We operate the nonlinear cavity as a four-wave mixing parametric amplifier 29 and analyze the gain G by injecting a single probe tone at ω s , blue-detuned by Ω m from a strong pump tone at ω p .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 Ω m .Figures 3(a) and (b) show the measured gain versus pump frequency for various pump powers, obtained by sweeping both pump and probe frequency with fixed separation ω s − ω p = Ω m .For increasing pump power the gain peak shifts to lower frequency, as expected for a negative Kerr coefficient.For our device, we reach G ≈ 24 dB 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 ω p at the previously determined maximum gain for each power, and sweeping the signal tone ω s .
To demonstrate 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 ω p .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 lockin to excite the cantilever with a tuned frequency comb with equal amplitudes, and phases chosen to reduce peak excitation in the time domain (see Supplementary Material).The lockin 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.Fig. 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.
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 nc versus pump power, where pump frequency is adjusted to follow the ridge of maximum sideband response.We determine nc using methods described in Ref. 18 .At each pump power, the amplified signal is the integrated sideband power spectrum, normalized to nc .The transduction gain is given by the ratio of this amplified signal, to that in the low power regime, where G ≡ 1.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 show 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 n add = 1/2 at high gain [dashed curve in Fig. 4(d)] as described by 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 currentinduced 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 C eff ≪ 1/4 such amplified backaction is negligible at the observed level of gain.
In conclusion, we described a micro-cantilever forcesensor with a compact and integrated microwave cavity.The sensor employed kinetic inductive electromechanical coupling to realize force transduction, and the nonlin-earity 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 nc ).Further study of this sensor concept is required to determine under which circumstances this gain mechanism can be exploited for improved force sensitivity.

SUPPLEMENTARY MATERIAL
See the Supplementary Material for details on the multifrequency lockin measurement technique with a tuned frequency comb driving the mechanical oscillator, determination of the K coefficient with two tone spectroscopy and analysis of the intracavity photon number and transduction gain.We follow the methods outlined in the Supplementary Methods of Ref. 4 to determine the number of circulating photons nc in the cavity with a Kerr-type nonlinearity.The Kerr effect captures the strength of a nonlinearity in terms the Kerr coefficient K which we can interpret as the frequency shift per intracavity photon.The equation of motion for the intracavity field a for a single port cavity measured in reflection, is given by where ω 0 is the bare resonance frequency of the cavity, κ its total linewidth, κ ext its external linewidth, and a in photon field at the input of the cavity.For a single-tone probe at frequency ω p , the input field is given by a in = |α in | exp(iω p t).In this case, the intracavity field may be written as a = α exp(iω p t). Defining a detuning between drive tone and the bare resonance ∆ = ω p − ω 0 , the equation of motion may be written, Multiplying Eqn.(S3) by its complex conjugate forms a third-order polynomial in nc = α * α where the number of incoming photons is given by For each given value of pump power P in and drive frequency ω p , and the values of K, and ω 0 , κ, κ ext previously determined (in the linear regime), we numerically solve the polynomial Eqn.(S4) to arrive at nc .The polynomial has in general three roots, but our input power corresponds to a regime below bifurcation, where only one real root exists.
With the derived value of the Kerr parameter we can reproduce the S 11 (ω) curve for a swept pump tone (Fig. S2(a)).

FIG. 2 .
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 versus 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 ∆ω0 as a function of intracavity photon number nc 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 = −1.57Hz/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. 3 .
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 (ωs − ωp)/2π = 685.386kHz.(b) Signal gain versus ωp at various pump power.(c) Gain versus ωs for various ωp and Pin along the ridge of maximum gain in (a).

FIG. 4 .
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 nc 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 nc.(d) Measured n add and that expected for a quantum limited parametric amplifier followed by the LNA.
FIG. S1.(a) Measured response phase versus frequency of the weak probe tone, plotted for various power of a strong bluedetuned pump tone.The pump is fixed at a frequency detuned by 50 MHz from ω0.The input pump power Pin ranges from −81 dBm to −60 dBm.The inset shows a zoom detailing the small decrease in resonance frequency ω0 for increasing pump power (direction indicated by arrow).(b) The shifted resonance frequency as a function of intracavity photons nc, from which we extract K = (−1.572± 0.017) Hz/photon