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# Intrinsic Kerr amplification for microwave electromechanics

Ermes Scarano  Department of Applied Physics, KTH Royal Institute of
Technology, Hannes Alfvéns väg 12, SE-114 19 Stockholm, Sweden  Elisabet K.
Arvidsson  Department of Applied Physics, KTH Royal Institute of Technology,
Hannes Alfvéns väg 12, SE-114 19 Stockholm, Sweden  August K. Roos  Department
of Applied Physics, KTH Royal Institute of Technology, Hannes Alfvéns väg 12,
SE-114 19 Stockholm, Sweden  Erik Holmgren  Department of Applied Physics, KTH
Royal Institute of Technology, Hannes Alfvéns väg 12, SE-114 19 Stockholm,
Sweden  David B. Haviland<EMAIL_ADDRESS>Department of Applied Physics, KTH
Royal Institute of Technology, Hannes Alfvéns väg 12, SE-114 19 Stockholm,
Sweden

###### Abstract

Electromechanical transduction gain of $24\text{\,}\mathrm{dB}\text{/}$ 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.

kinetic inductance; parametric amplification; four-wave mixing; optomechanics;
force sensing; strain

††preprint: arXiv:0000.00000

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 intrinsic nonlinearity of the cavity can be
exploited to parametrically amplify the detected signal and, in theory,
enhance force sensitivity. 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 $\mathcal{C}$ [8, 9, 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
$\Omega_{m}$, leading to operation in the unresolved sideband regime and
larger phonon occupation $\bar{n}_{\mathrm{m}}$ at a given temperature.
Reduced $g_{0}$ can be compensated for with a stronger pump that increases the
intracavity photon number $\bar{n}_{\mathrm{c}}$, but superconducting circuits
experience current-induced depairing that sets a limit on
$\bar{n}_{\mathrm{c}}$. 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
resolved [18] and unresolved [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 and use a four-wave mixing
process to realize $24\text{\,}\mathrm{dB}\text{/}$ of force transduction gain
of the optomechanical motional sideband.

In the cavity optomechanical transduction scheme, mechanical fluctuations with
average phonon number $\bar{n}_{\mathrm{m}}$ are imprinted on the 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 $F_{\mathrm{min}}$, 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 back action of measurement, as well
as added photon shot noise and amplification noise $n_{\mathrm{add}}$
expressed as an equivalent force noise [24]. The normalized force sensitivity
is given by

$\frac{F_{\mathrm{min}}}{F_{\mathrm{SQL}}}=\sqrt{\underbrace{\left(\bar{n}_{\mathrm{m}}+\frac{1}{2}\right)}_{\text{Mechanics}}+\underbrace{\lvert\mathcal{C}_{\mathrm{eff}}\rvert}_{\text{Back
action}}+\underbrace{\frac{1}{16\lvert\mathcal{C}_{\mathrm{eff}}\rvert}}_{\text{Shot
noise}}+\underbrace{\frac{n_{\mathrm{add}}}{8\lvert\mathcal{C}_{\mathrm{eff}}\rvert}}_{\text{Amplifier
noise}}},$ (1)

where $F_{\mathrm{SQL}}=\sqrt{2m_{\mathrm{eff}}\Gamma\hbar\Omega_{m}}$ is the
force sensitivity at the standard quantum limit (SQL), set by the resonators
internal loss rate $\Gamma$ and effective mass $m_{\mathrm{eff}}$. Measurement
induces noise, which translates to force through the effective cooperativity
$\mathcal{C}_{\mathrm{eff}}$ [24]. In the unresolved sideband regime,

$\mathcal{C}_{\mathrm{eff}}(\bar{n}_{\mathrm{c}})\simeq\frac{4g_{0}^{2}\bar{n}_{\mathrm{c}}}{\kappa\Gamma},$
(2)

where $\kappa$ is total loss rate of the cavity.

Figure 1:  Force sensitivity on-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, back-action noise, and the
force-equivalent shot noise and added amplifier noise.

Figure 1 shows each contribution of Eqn. (1) as a function of input power $P$
and their overall effect on sensitivity. Here $P_{\mathrm{SQL}}$ is the
applied microwave power when the measurement is at the standard quantum limit,
where shot noise and back-action noise are equal, corresponding to
$\mathcal{C}_{\mathrm{eff}}$ = 1/4. Below $P_{\mathrm{SQL}}$ the added noise
of the first stage of amplification limits force sensitivity. For example, in
Fig. 1 we show the contribution from a cryogenic low-noise amplifier (LNA)
with an equivalent noise temperature of $4\text{\,}\mathrm{K}\text{/}$. At
$4.5\text{\,}\mathrm{GHz}\text{/}$ this correspond to approximately 19
photons, deteriorating the force sensitivity so that detection at SQL is no
longer possible and the optimal sensitivity is achieved for powers
$P>P_{\mathrm{SQL}}$.

Figure 2:  Scanning electron microscope (SEM) image of (a) the cantilever and
(b) the $100\text{\,}\mathrm{nm}\text{/}$-wide meandering nanowire inductor.
The cantilever is formed from a $600\text{\,}\mathrm{nm}\text{/}$-thick
silicon nitride plate and the nanowire, is etched from a
$15\text{\,}\mathrm{nm}\text{/}$-thin film of Nb-Ti-N. (c) Phase versus
frequency of the microwave resonance for increasing probe power
$P_{\mathrm{in}}$. The resonance frequency of the cavity $\omega_{0}$ shifts
to lower frequencies with increasing $P_{\mathrm{in}}$, typical of a resonator
with a Kerr-type nonlinearity. (d) The shift in resonance frequency as a
function of intracavity photon number $\bar{n}_{\mathrm{c}}$ for a pump blue-
detuned by approximately $50\text{\,}\mathrm{MHz}\text{/}$. For each pump
power, $\omega_{0}$ is measured by sweeping a weak probe tone through
resonance. The Kerr coefficient $\mathcal{K}/2\pi$ =
$-0.99\text{\,}\mathrm{Hz}\mathrm{/}\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}$.

Cantilevers operating in cryogenic environments typically have mechanical
resonance frequencies that put the mechanical mode in a thermal state with
$\bar{n}_{\mathrm{m}}\approx k_{\mathrm{B}}T/\hbar\Omega_{m}\gg 1$. 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 $\bar{n}_{\mathrm{c}}$, 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 an additional, quantum-limited amplifier
between the resonant circuit, hereafter called “cavity”, and the LNA. For a
sufficiently high-gain phase-preserving quantum-limited amplifier, the number
of added photons by the amplifier to the measurement is given by
$n_{\mathrm{add}}=1/2$ [25]. Yet adding a separate quantum-limited amplifier
comes at significant cost in complexity, such as a more complicated operation,
including an additional pump, pump-cancellation tone, and associated cabling
[14]. In our work, we investigate a more straightforward implementation where
the nonlinearity of cavity itself 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.

Figure 3:  (a) Four-wave mixing parametric gain of an injected signal tone
$\omega_{s}$ as a function of pump frequency $\omega_{p}$ and pump power
$P_{\mathrm{pump}}$ at fixed detuning $(\omega_{s}-\omega_{p})/2\pi$ =
$685.386\text{\,}\mathrm{kHz}\text{/}$. (b) Signal gain versus $\omega_{p}$ at
various pump power. (c) Gain versus $\omega_{s}$ for various $\omega_{p}$ and
$P_{\mathrm{pump}}$ along the ridge of maximum gain in (a).

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 of width $100\text{\,}\mathrm{nm}\text{/}$ having large kinetic
inductance, shown in detail in Fig. 2(b). The silicon nitride (Si-N) micro-
cantilever has a fundamental bending mode with resonance frequency
$\Omega_{m}/2\pi$ = $685.39\text{\,}\mathrm{kHz}\text{/}$. The microwave
cavity with resonance frequency $\omega_{0}/2\pi$ =
$4.378\text{\,}\mathrm{GHz}\text{/}$ is strongly overcoupled to the
transmission line, having total linewidth $\kappa/2\pi$ =
$21.18\text{\,}\mathrm{MHz}\text{/}$, 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 kinetic inductance, shifting the cavity resonance and thereby
realizing electromechanical coupling between the motion of the cantilever and
the resonance frequency of the cavity [26, 27].

Figure 2(c) shows the nonlinear phase response of the cavity as a function of
frequency and power. The phase of the reflected signal changes by $2\pi$ when
sweeping through resonance frequency $\omega_{0}$, 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 described by a Kerr-type nonlinearity, where the Kerr coefficient
$\mathcal{K}$ captures 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 $\bar{n}_{\mathrm{c}}$ 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 $\mathcal{K}/2\pi$ =
$-0.99\text{\,}\mathrm{Hz}\mathrm{/}\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}$
for our device.

We operate our nonlinear cavity as a four-wave mixing parametric amplifier
[28] and analyze the gain $\mathcal{G}$ by injecting a single probe tone at
$\omega_{s}$, blue-detuned by $\Omega_{m}$ from a strong pump tone at
$\omega_{p}$. The amplifier bandwidth is smaller than the cavity linewidth
$\kappa$ and it decreases with increasing gain (pump power). We intentionally
designed the cavity with a relatively low $Q$ to operate in the unresolved-
sideband regime of the overcoupled microwave mode, ensuring high gain for this
detuning. 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 $\omega_{s}-\omega_{p}=\Omega_{m}$. For increasing pump
power the gain peak shifts to lower frequency, as expected for a negative Kerr
coefficient. For our device, we reach
$\mathcal{G}\approx$24\text{\,}\mathrm{dB}\text{/}$$ 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
$\omega_{p}$ at the previously determined maximum gain for each power and
sweeping the signal tone $\omega_{s}$.

Figure 4:  (a) Parametric amplification of an upconverted motional sideband,
showing the area under the sideband as a function of pump frequency
$\omega_{p}$ and pump voltage. Increasing the pump power shifts the cavity’s
resonance frequency and linewidth due to the Kerr-type nonlinearity,
increasing the transduction of the mechanical motion to cavity frequency
shift. (Inset) Zoom of the dashed box with finer stepping, showing the ridge
of maximum gain. (b) Mechanical fluctuations measured at the upper motional
sideband for three pump powers with $\omega_{p}$ placed on the ridge of
maximum gain. The power spectral density (PSD) is normalized to the
intracavity photon number $\bar{n}_{\mathrm{c}}$. (c) Transduction gain
$\mathcal{G}$ and $\bar{n}_{\mathrm{c}}$ versus pump power. (d) Measured
$n_{\mathrm{add}}$ and that expected for a quantum limited parametric
amplifier followed by the LNA.

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 $\omega_{p}$. 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 [29]. We capture the motion spectrum
with a multifrequency lock-in while exciting the cantilever with a tuned
frequency comb having phases chosen to reduce peak excitation in the time
domain [30], significantly increasing measurement speed. 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 mechanical signal
power. Normalizing this mechanical power to the pump power, we map out the
gain in mechanical transduction as a function of pump power and frequency, as
shown in Fig. 4(a).

Maximum transduction gain follows a ridge in pump power and frequency, as
expected for parametric gain generated close to bifurcation. 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(b). Normalizing
to $\bar{n}_{\mathrm{c}}$, determined using methods described in Ref. [18], we
extract a quantity proportional to the motional noise. To determine the
transduction gain we take the ratio of the area under the measured Lorentzian
to that measured in the linear regime where $\mathcal{G}=1$. Figure 4(c) shows
the transduction gain $\mathcal{G}$ and $\bar{n}_{\mathrm{c}}$ versus pump
power, where pump frequency is adjusted to follow the ridge of maximum gain.
Figure 4(c) also shows that $\bar{n}_{\mathrm{c}}$ saturates at high power, as
expected.

We also analyzed the added noise of the amplification chain and its dependence
on transduction gain, shown in Figure 4(d) where we plot the total added noise
referred to the input of the parametric amplifier,

$n_{\mathrm{add}}=n_{\mathrm{PA}}+\frac{n_{\mathrm{LNA}}}{\mathcal{G}}.$ (3)

The ideal phase-insensitive parametric amplifier would result in a reduction
of the added noise with increasing gain, approaching $n_{\mathrm{add}}=1/2$ at
high gain. However, we observe instead an increase in the added noise,
implying that the pumped nonlinear cavity is adding additional 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].

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 $24\text{\,}\mathrm{dB}\text{/}$ 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 $\bar{n}_{\mathrm{c}}$). Further study of
this sensor concept is required to determine under which circumstances this
gain mechanism can be exploited for improved force sensitivity.

The European Union Horizon 2020 Future and Emerging Technologies (FET) 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.

## Author Declarations

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

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

## Data Availability

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

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