# Amplifying a zeptonewton force with a single-ion nonlinear oscillator

Bo Deng  Institute of Physics, University of Kassel, Heinrich-Plett-Straße 40,
34132 Kassel, Germany  Moritz Göb  Institute of Physics, University of Kassel,
Heinrich-Plett-Straße 40, 34132 Kassel, Germany  Benjamin A. Stickler  Faculty
of Physics, University of Duisburg-Essen, Lotharstraße 1, 47057 Duisburg,
Germany  Max Masuhr  Institute of Physics, University of Kassel, Heinrich-
Plett-Straße 40, 34132 Kassel, Germany  Kilian Singer<EMAIL_ADDRESS>Institute of Physics, University of Kassel, Heinrich-Plett-Straße 40, 34132
Kassel, Germany  Daqing Wang<EMAIL_ADDRESS>Institute of Physics,
University of Kassel, Heinrich-Plett-Straße 40, 34132 Kassel, Germany

###### Abstract

Nonlinear mechanical resonators display rich and complex dynamics and are
important in many areas of fundamental and applied sciences. In this letter,
we show that a particle confined in a funnel-shaped potential features a
Duffing-type nonlinearity due to the coupling between its radial and axial
motion. Employing an ion trap platform, we study the nonlinear oscillation,
bifurcation and hysteresis of a single calcium ion driven by radiation
pressure. Harnessing the bistability of this atomic oscillator, we demonstrate
a 20-fold enhancement of the signal from a zeptonewton-magnitude harmonic
force through the effect of vibrational resonance. Our findings open up a
range of possibilities for controlling and exploiting nonlinear phenomena of
mechanical oscillators close to the quantum regime.

Nonlinear oscillators, whose dynamics go beyond the simple description of
linear equations of motion, are ubiquitous in nature. Driven nonlinear
oscillators can exhibit complex behaviors such as squeezing [1, 2], persistent
resonance [3], phase transitions [4, 5, 6], and chaos [7, 8]. They have a
broad presence in many branches of science and technology, including signal
processing, biomechanics, control systems, and force sensing [9, 10, 11, 12,
13, 14, 15]. Compared to linear oscillators, nonlinear oscillators can be more
sensitive to external forces, especially when they are tuned close to a
bifurcation point. The multistability of nonlinear oscillators provides
opportunities for signal enhancement through the effects of stochastic and
vibrational resonances [16, 17, 18, 19, 20]. Both effects involve the
introduction of an external modulation signal to the oscillator to amplify a
weak signal under detection. Stochastic resonance [16, 17, 18] employs a
broadband noise, whereas a high-frequency harmonic signal is used in
vibrational resonance [19, 20].

The manifestation of these effects in mechanical systems has primarily been
investigated with nano-electromechanical resonators [17, 18, 20]. In this
letter, we show that a particle trapped in a funnel-shaped three-dimensional
potential exhibits an inherent Duffing-type nonlinearity. This nonlinearity
arises from the coupling of radial and axial motion of the particle resulting
from the geometry of the potential [21, 22]. We investigate this model
experimentally using a modified Paul trap that generates a funnel-shaped
secular potential for a single calcium ion. Through radiation pressure forces,
we actuate the radial motion of the ion and observe a bifurcation into two
stable equilibrium states. By introducing an intermediate-frequency parametric
modulation signal analogous to the case of vibrational resonance [19, 20], we
show that a zeptonewton-scale oscillatory force acting on the axial mode can
be enhanced by a factor of 20.

As illustrated in Fig. 1(a), our system consists of a single ion confined to
the potential of a modified Paul trap. In contrast to conventional linear Paul
traps, the two pairs of blade electrodes are inclined with respect to the
axial $z$ axis [21, 22]. We perform numerical field simulations using the
boundary element method [23] to obtain the electrostatic potential resulting
from this geometry. As displayed in Fig. 1(b), this arrangement leads to a
funnel-shaped potential in which the radial confinement becomes stronger with
increasing axial position $z$. To trap a charged particle, the two pairs of
blades are driven by bipolar radiofrequency signals $\pm\tilde{V}_{\rm RF}$.
This creates a secular trapping potential in the radial directions for a
${}^{40}\text{Ca}^{+}$ ion. The blue and red dots in Fig. 1(c) display the
measured secular trapping frequencies along the two radial principal axes $x$
and $y$ as a function of the axial position. Frequencies of both modes can be
well recovered by the linear relation $\omega_{x,y}(1+z/\ell_{0})$, with
$\omega_{x,y}\simeq 2\pi\times\\{1.14\,,1.15\\}$ MHz the trapping frequencies
at $z=0$ and $\ell_{0}=1.81\,{\rm mm}$ a parameter describing the length of
the funnel. In the axial direction, the ion is weakly confined by positive DC
voltages (${V}_{\rm DC}$) applied on the two endcaps, resulting in an axial
trapping frequency of $\Omega\simeq 2\pi\times 100$ kHz.

Figure 1: (a) A laser-cooled ion confined in a funnel-shaped Paul trap.
Intensity modulation of the laser excites the motion along the $x$ direction.
Positive DC bias ($V_{\rm DC}$) and two AC voltages ($\tilde{V}_{\rm s}$,
$\tilde{V}_{\rm e}$) are applied to the endcap electrodes. Inset: A simplified
energy level diagram for the ${}^{40}\text{Ca}^{+}$ ion. The transition at 397
nm is used for Doppler cooling, while the repumping laser at 866 nm ensures a
closed-cycle optical transition. (b) Electrostatic potential obtained from
numerical simulations. (c) Dependence of radial trapping frequencies on the
axial position. The dots represent measured data and the lines display linear
fits. (d) Response of the $x$ mode to a descending frequency sweep. Each
column represents the projection of a fluorescence image along the $z$ axis.
The red lines display the extracted oscillation amplitude. (e) Response of the
axial mode. The red line indicates the axial equilibrium position. (f) The
amplitude of axial displacement plotted against the radial oscillation
amplitude. The solid line shows the predictions following Eq. (5) in the
absence of $F_{z}(t)$.

The motion of the ion is subject to continuous damping in all three directions
achieved through Doppler cooling using a laser red-detuned to its
$4^{2}S_{1/2}-4^{2}P_{1/2}$ transition [see inset of Fig. 1(a)]. A second
laser at $866$ nm repumps the metastable $3^{2}D_{3/2}$-state to facilitate a
closed-cycle transition. In addition to the damping, the $x$\- and $z$-motion
are driven by external forces $F_{x}(t)$ and $F_{z}(t)$. Given that the
frequency of the $y$-mode is sufficiently far from the frequencies of the
other two modes, we consider that the $y$ mode is not driven and set $y=0$ for
all times. In the relevant regime of the experiment, the impact of the funnel
shape is weak, $|z/\ell_{0}|\ll 1$. Keeping the $z$-dependence of the secular
potential in linear order only, we obtain the Hamiltonian for the two-
dimensional oscillation dynamics

$\displaystyle H=$
$\displaystyle\frac{p_{x}^{2}+p_{z}^{2}}{2m}+\frac{1}{2}m\Omega^{2}z^{2}+\frac{1}{2}m\omega_{x}^{2}\left(1+2\frac{z}{\ell_{0}}\right)x^{2}$
$\displaystyle-F_{x}(t)x-F_{z}(t)z\,.$ (1)

Here, $m$ represents the mass of the ${}^{40}\text{Ca}^{+}$ ion. The driving
forces have three distinct contributions: (i) $F_{x}$ is realized by
modulating the intensity of the cooling laser to produce a near-resonant
radiation pressure force of magnitude $F_{0}$ and frequency
$\omega_{0}=\omega_{x}+\Delta\gg\Omega$ (with detuning $\Delta$,
$|\Delta|\ll\Omega$), which drives the $x$-motion of the ion; (ii) a weak
signal force of magnitude $F_{\rm s}$ and frequency $\omega_{\rm
s}\ll\omega_{0}$ acting on the axial mode, which we want to detect; (iii) the
axial enhancement force of strength $F_{\rm e}$ and frequency $\omega_{\rm
e}$, which serves to enhance the detection of the signal force $F_{\rm s}$. In
the experiment, the signal force is introduced by feeding in an AC voltage
$\tilde{V}_{\rm s}$ with an amplitude of $500\,\mu{\rm V}$ on one endcap
electrode. The detection scheme is not limited to electrostatic forces and
should be applicable to any interaction that leads to an axial displacement of
the ion. The enhancement force is applied in the same manner by exerting an AC
voltage $\tilde{V}_{\rm e}$ on the same endcap electrode, as shown in Fig.
1(a). In summary, $F_{x}(t)=F_{0}\cos(\omega_{0}t)$ and

$F_{z}(t)=F_{\rm e}\cos(\omega_{\rm e}t)+F_{\rm s}\cos(\omega_{\rm s}t)\,.$
(2)

Note that the frequencies of these three forces fulfill the relation
$\omega_{\rm s}\ll\omega_{\rm e}\ll\omega_{\rm 0}$, as required for
vibrational resonance [19, 20].

The equations of motion resulting from the Hamiltonian (Amplifying a
zeptonewton force with a single-ion nonlinear oscillator) can be simplified by
exploiting the separation of timescales $\omega_{0}\gg\Omega$. This is
achieved by making the ansatz $x(t)=\alpha(t)e^{-i\omega_{0}t}+{\rm c.c.}$
(with ${\rm c.c.}$ the complex conjugate), where the amplitude $\alpha$
evolves on a timescale much slower than $1/\omega_{0}$, so that
$|\ddot{\alpha}|\ll\omega_{0}|\dot{\alpha}|$ and
$|\dot{\alpha}|\ll\omega_{0}|\alpha|$. Averaging over one drive period
$2\pi/\omega_{0}$ and performing a rotating-wave approximation yields the
approximate equation of motion for the radial mode

$\displaystyle\dot{\alpha}=$ $\displaystyle
i\left(\Delta-\omega_{x}\frac{z}{\ell_{0}}\right)\alpha+if_{0}-\frac{\gamma}{2}\alpha\,,$
(3)

with damping rate $\gamma$ and driving strength $f_{0}=F_{0}/4m\omega_{x}$.
This equation shows that the axial displacement $z$ shifts the resonance
frequency of the radial oscillator by $\omega_{x}z/\ell_{0}$.

The axial motion is described by

$\ddot{z}=-\Omega^{2}z-\frac{2\omega^{2}_{x}}{\ell_{0}}|\alpha|^{2}+\frac{F_{z}(t)}{m}-\gamma\dot{z}\,,$
(4)

which follows from the exact equations of motion via the substitution
$x^{2}=2|\alpha|^{2}$ due to the rotating-wave approximation. This equation
implies that the instantaneous displacement of the axial equilibrium position

$z_{0}=-2\frac{\omega_{x}^{2}}{\Omega^{2}}\frac{|\alpha|^{2}}{\ell_{0}}+\frac{F_{z}(t)}{m\Omega^{2}}$
(5)

is determined by the radial oscillation amplitude as well as by the external
force (2).

The coupling of radial and axial dynamics described by Eqs. (3) and (5) is key
to the emergence of nonlinearity in this system. We verify this coupling in
our experiment by coherently driving the radial oscillator with the force
$F_{x}$ and observing the response in the axial direction. To do so, the
modulation frequency of the cooling laser $\omega_{0}$ is reduced stepwise and
swept across $\omega_{x}$. In each step, a fluorescence image of the ion is
recorded. The integration time of the camera is 400 ms. Each frame therefore
averages over many axial and radial oscillation cycles. The images are then
projected along the radial and axial axes to separate the responses along
these two directions. The magnitude of the radial force is limited by the
saturation of the cooling transition to $\lesssim 30$ zN. The axial force
$F_{z}$ is not applied during this measurement. Figure 1(d) shows the response
of the radial mode as a function of the frequency detuning $\Delta$. As
$\omega_{0}$ approaches the resonance frequency, a coherent oscillation of the
radial mode is excited. Here, only one flank of the oscillation is visible in
the images due to a phase delay between the ion motion and the laser drive
[24]. The red lines in the graph display the extracted coherent oscillation
amplitude $|\alpha|$. As the modulation frequency crosses $\omega_{x}$, the
coherent oscillation persists for an extended range. The increase of radial
oscillation amplitude is accompanied with an axial shift in the negative
direction of the $z$ axis, i.e., towards the open end of the funnel, as
displayed in Fig. 1(e). The red line in this figure shows the extracted axial
center position $z_{0}$. Figure 1(f) displays a quantitative comparison of the
measurement with Eq. (5). Blue and red circles represent the data measured for
the $x$ and $y$ modes, which are in good match with the prediction of Eq. (5)
displayed by the black line.

At further frequency detuning, the oscillator experiences an abrupt change in
both the radial and axial directions and resumes its initial position. Such an
abrupt change is a signature of bistability in the system. Duffing-type
bistability of trapped ions can arise from the intrinsic anharmonicity of
linear Paul traps [25]. Here, the nonlinearity stems from the radial-axial
coupling imposed by the funnel-shaped potential. This can be intuitively
understood as shifting towards the open side of the funnel softens the radial
spring and leads to an amplitude-dependent radial trapping frequency, i.e., a
Duffing-type response [26].

To confirm the intuition, we solve the averaged radial amplitude equation (3).
For this, we use that the axial motion adiabatically follows the radial motion
since $|\Delta|\ll\Omega$. One can thus average over one axial oscillation
period $2\pi/\Omega$, replace $z$ by $z_{0}$ in Eq. (3), and obtain

$\dot{\alpha}=i\left[\Delta-\delta(t)\right]\alpha+i\xi|\alpha|^{2}\alpha+if_{0}-\frac{\gamma}{2}\alpha\,.$
(6)

This equation describes a Duffing oscillator with a linear drive $f_{0}$ and a
parametric drive that modulates the detuning by
$\delta(t)=\omega_{x}F_{z}(t)/m\ell_{0}\Omega^{2}$. Here, the parameter
$\xi=2\omega_{x}^{3}/\Omega^{2}\ell_{0}^{2}$ quantifies the nonlinearity due
to the radial-axial coupling. In our experiment, $\xi/2\pi=9.04\times
10^{13}\,\text{Hz}/\text{m}^{2}$ and significantly exceeds the intrinsic
nonlinearity of the secular trapping potential.

Since the parametric driving frequencies [see Eq. (2)] are both much smaller
than the damping rate, $\omega_{\rm s}\ll\omega_{\rm e}\ll\gamma$, the Duffing
oscillator relaxes for each $\delta(t)$ towards a quasi-stationary state with
the square of the amplitude $|\alpha_{0}(t)|^{2}$ determined by the third-
order polynomial equation

$|\alpha_{0}(t)|^{2}=\frac{f_{0}^{2}}{[\Delta-\delta(t)+\xi|\alpha_{0}(t)|^{2}]^{2}+\gamma^{2}/4}\,.$
(7)

The solutions of this equation as a function of the detuning $\Delta$ are
displayed in Fig. 2(a). Depending on the driving strength, it has either a
single stable solution or three stationary solutions, of which two are stable
and one is unstable [26]. In the latter case, an adiabatic ramping of the
detuning through the bistable region leads to hysteresis [see blue and red
arrows in Fig. 2(a)].

Figure 2: (a) Solutions of the radial amplitude $|\alpha_{0}|$ for
$\gamma=2\pi\times 250$ Hz and driving strengths denoted in the legend. (b)
The emergence of bistability. The red lines show the radial amplitude when
performing descending frequency sweeps and the blue lines represent the
outcome for ascending frequency sweeps.

To verify the emergence of bistability, we measure the response of the radial
oscillator to ascending and descending frequency sweeps. We gradually increase
the modulation depth of the laser intensity, which increases the amplitude of
the driving force. The results are displayed in Fig. 2(b). Under weak driving,
the same Lorentzian-shaped response is observed for ascending (blue lines) and
descending (red lines) sweeps. As the driving force increases, the lineshape
becomes unsymmetric and abrupt changes appear on the side of negative detuning
in both ascending and descending measurements. With stronger driving, the
jumps occur at different frequency detunings for descending and ascending
sweeps, confirming that the oscillation amplitude has more than one stable
solution in this regime. As predicted by theory, the hysteresis region opens
gradually as the driving force further increases.

Having characterized our system, we continue to demonstrate that this
nonlinear oscillator can be utilized for amplifying small signals through a
scheme analogous to vibrational resonance. In typical vibrational resonance
experiments [20], a high-frequency amplitude modulation is introduced to a
one-dimensional forced oscillator and enables switching between its two stable
states. In our system, the three-dimensional feature allows us to explore a
new scheme. Here, the radial mode is forced by $F_{x}$ to the bistable regime,
and a small force $F_{\rm s}$ acting on the axial mode can be amplified by the
axial enhancement force $F_{\rm e}$.

To start, we exert the signal force $F_{\rm s}$ by switching on the AC voltage
$\tilde{V}_{\rm s}$ with an amplitude of 500 $\mu{\rm V}$ at frequency
$\omega_{\rm s}=2\,\pi\times 0.5$ Hz [see Fig. 1(a) and the illustration in
Fig. 3(a)]. In this step, the radial forcing is not applied ($F_{x}=0$). The
resulted signal force has a peak-to-peak amplitude of $2.4$ zN and leads to a
periodic displacement of $\pm 45$ nm of the axial oscillator. This
displacement is comparable to the zero-point fluctuation of the axial mode,
which amounts to $\sqrt{\hbar/2m\Omega}\approx 36$ nm. We determine the axial
trajectory of the ion through super-resolution localization microscopy by
performing a maximum likelihood estimation from the fluorescence images [27].
The resulted axial trajectory is displayed in the inset of Fig. 3(b). Here,
the camera is sampled at about 8 Hz with an exposure time of 100 ms for each
frame. On average $N=240$ photons are detected per frame. The experimental
axial Gaussian spread function has a standard deviation of
$\sigma_{0}=1.64\,\mu{\rm m}$ and sets the lower bound of estimating the axial
center position $\sigma_{0}/\sqrt{N}=106$ nm [27]. Indeed, this small
displacement is not directly resolvable in the trajectory. The harmonic nature
makes it possible to identify this signal in the Fourier domain. As shown in
Fig. 3(b), a small peak at 0.5 Hz is barely visible in the Fourier spectrum.

Next, we force the radial oscillator to the bistable regime by switching on
the radial driving $F_{x}$. The frequency $\omega_{0}$ of the driving force is
tuned to the center of the bistable region. The ion is initially prepared in
the lower stable branch. The signal force $F_{\rm s}$ shifts the ion’s
motional state along the lower branch adiabatically but is too weak to induce
a transition to the upper branch, as illustrated in Fig. 3(c). The measured
axial trajectory and the corresponding Fourier spectrum are displayed in Fig.
3(d). The noise floor is elevated in the Fourier spectrum, but no enhancement
of the signal at 0.5 Hz is observed.

In the last step, we apply a sinusoidal enhancement signal $\tilde{V_{\rm e}}$
at $\omega_{\rm e}=2\pi\times 50$ Hz to one endcap, which exerts a parametric
drive on the radial oscillator. The frequency of the enhancement force
satisfies $\omega_{\rm s}\ll\omega_{\rm e}\ll\omega_{\rm 0}$ as required for
vibrational resonance schemes [19, 20]. By tuning the amplitude of
$\tilde{V_{\rm e}}$ such that the parametric drive brings the oscillator to
the boundaries of the bistable region, the signal force can introduce jumps of
the oscillator to the upper branch and back [see Fig. 3(e)]. The inset of Fig.
3(f) shows the axial trajectory of the ion and clearly evidences the switch
between two branches. In the Fourier spectrum, the strong peak at 0.5 Hz lets
us deduce an enhancement factor of 20 compared to the signals in (b) and (d).
The maximal amplification that can be achieved is determined by the width of
the hysteresis region. In our experiment, this is limited by the saturation of
the cooling transition, which sets an upper bound to the radial driving force.

Figure 3: Amplifying a weak signal force. (a) The weak signal force $F_{s}$ is
applied on the axial oscillator ($F_{x}=0$ and $F_{e}=0$), introducing a small
periodic axial displacement $\Delta z$ which is barely visible in the Fourier
spectrum displayed in (b). The inset shows the extracted axial center position
as a function of time. (c) The radial drive $F_{x}$ is applied. No enhancement
is observed in the Fourier spectrum in (d). (e) The enhancement force $F_{e}$
is switched on. The trajectory in the inset of (f) shows clearly jumps between
the two stable branches. The Fourier spectrum in (f) has a prominent peak at
0.5 Hz.

In summary, we have shown that a particle confined in a funnel-shaped
potential exhibits a Duffing-type nonlinearity that originates from the
coupling between its radial and axial motion. Using an ion trap platform, we
observed the nonlinear oscillation, bifurcation and hysteresis of a laser-
cooled ion oscillator in such a potential. Furthermore, we demonstrated that
the bistability of this system can be exploited to enhance a zeptonewton-scale
oscillatory force using a parametric variant of vibrational resonance. Our
findings open up opportunities for combining the rich nonlinear dynamics with
the precision of atomic physics to study and exploit nonlinear mechanical
phenomena in single- and few-body systems close to their motional ground
state. For instance, future work might assess the role of nonlinearity in
energy transport through oscillator chains [28, 29, 30], study synchronization
of driven nonlinear oscillators in the quantum regime [31, 32], explore the
emergence of limit cycles [33], or investigate thermodynamics with quantum
nonlinear oscillators [34]. The ability to enhance the detection of small
forces could be extended to particles that are co-trapped with the atomic ion,
such as molecules [35, 36, 37], and offers a promising route to detect forces
that have not yet been observed on molecular systems, such as chiral optical
forces [35, 36, 37, 38, 39, 40]. Last but not least, the geometry-induced
nonlinearity could also be exploited to generate nonclassical states of
massive particles that lack sharp internal transitions, such as electrically
levitated nanoparticles [41, 42, 43].

## Acknowledgement

We thank Florian Elsen, David Zionski and Johannes Rossnagel for early-stage
work on the ion trap and Klaus Hornberger for discussions on the theory. This
work was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – projects 499241080, 384846402, 328961117 – through the
QuantERA grant ExTRaQT, the Research Unit Thermal Machines in the Quantum
World (FOR 2724) and the Collaborative Research Center ELCH (SFB 1319).

## References

  * Yang  _et al._ [2021a] F. Yang, M. Fu, B. Bosnjak, R. H. Blick, Y. Jiang, and E. Scheer, Phys. Rev. Lett. 127, 184301 (2021a).
  * Huber  _et al._ [2020] J. S. Huber, G. Rastelli, M. J. Seitner, J. Kölbl, W. Belzig, M. I. Dykman, and E. M. Weig, Phys. Rev. X 10, 021066 (2020).
  * Yang  _et al._ [2021b] F. Yang, F. Hellbach, F. Rochau, W. Belzig, E. M. Weig, G. Rastelli, and E. Scheer, Phys. Rev. Lett. 127, 014304 (2021b).
  * Stambaugh and Chan [2006] C. Stambaugh and H. B. Chan, Phys. Rev. Lett. 97, 110602 (2006).
  * Dolleman  _et al._ [2019] R. J. Dolleman, P. Belardinelli, S. Houri, H. S. van der Zant, F. Alijani, and P. G. Steeneken, Nano Lett. 19, 1282 (2019).
  * Wang and Wu [2020] Q.-W. Wang and S. Wu, Phys. Rev. A 102, 063531 (2020).
  * Testa  _et al._ [1982] J. Testa, J. Pérez, and C. Jeffries, Phys. Rev. Lett. 48, 714 (1982).
  * Lakshmanan and Murali [1996] M. Lakshmanan and K. Murali,  _Chaos In Nonlinear Oscillators: Controlling And Synchronization_ , World Scientific Series On Nonlinear Science Series A, Vol. 13 (World Scientific Publishing Company, Singapore, 1996).
  * Strogatz [2015] S. H. Strogatz,  _Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering_ , 2nd ed. (CRC Press, Boca Raton, 2015).
  * Tadokoro and Tanaka [2021] Y. Tadokoro and H. Tanaka, Phys. Rev. Appl. 15, 024058 (2021).
  * Cleland and Roukes [2002] A. N. Cleland and M. L. Roukes, J. Appl. Phys. 92, 2758 (2002).
  * Gieseler  _et al._ [2013] J. Gieseler, L. Novotny, and R. Quidant, Nat. Phys. 9, 806 (2013).
  * Papariello  _et al._ [2016] L. Papariello, O. Zilberberg, A. Eichler, and R. Chitra, Phys. Rev. E 94, 022201 (2016).
  * Zhang  _et al._ [2020] Y. Zhang, R. Kondo, B. Qiu, X. Liu, and K. Hirakawa, Phys. Rev. Appl. 14, 014019 (2020).
  * Aldana  _et al._ [2014] S. Aldana, C. Bruder, and A. Nunnenkamp, Phys. Rev. A 90, 063810 (2014).
  * Gammaitoni  _et al._ [1998] L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
  * Badzey and Mohanty [2005] R. L. Badzey and P. Mohanty, Nature 437, 995 (2005).
  * Almog  _et al._ [2007] R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, Appl. Phys. Lett. 90, 013508 (2007).
  * Landa and McClintock [2000] P. S. Landa and P. V. E. McClintock, J. Phys. A: Math. Gen. 33, L433 (2000).
  * Chowdhury  _et al._ [2020] A. Chowdhury, M. G. Clerc, S. Barbay, I. Robert-Philip, and R. Braive, Nat. Commun. 11, 2400 (2020).
  * Roßnagel  _et al._ [2016] J. Roßnagel, S. T. Dawkins, K. N. Tolazzi, O. Abah, E. Lutz, F. Schmidt-Kaler, and K. Singer, Science 352, 325 (2016).
  * Levy  _et al._ [2020] A. Levy, M. Göb, B. Deng, K. Singer, E. Torrontegui, and D. Wang, New J. Phys. 22, 093020 (2020).
  * Betcke and Scroggs [2021] T. Betcke and M. W. Scroggs, J. Open Source Softw. 6, 2879 (2021).
  * Drewsen  _et al._ [2004] M. Drewsen, A. Mortensen, R. Martinussen, P. Staanum, and J. L. Sørensen, Phys. Rev. Lett. 93, 243201 (2004).
  * Akerman  _et al._ [2010] N. Akerman, S. Kotler, Y. Glickman, Y. Dallal, A. Keselman, and R. Ozeri, Phys. Rev. A 82, 061402 (2010).
  * Nayfeh and Mook [2008] A. H. Nayfeh and D. T. Mook,  _Nonlinear oscillations_ (John Wiley & Sons, New York, 2008).
  * Lelek  _et al._ [2021] M. Lelek, M. T. Gyparaki, G. Beliu, F. Schueder, J. Griffié, S. Manley, R. Jungmann, M. Sauer, M. Lakadamyali, and C. Zimmer, Nat. Rev. Methods Primers 1, 39 (2021).
  * Zheng  _et al._ [2001] Z. Zheng, G. Hu, and B. Hu, Phys. Rev. Lett. 86, 2273 (2001).
  * Kovaleva and Manevitch [2013] A. Kovaleva and L. I. Manevitch, Phys. Rev. E 88, 022904 (2013).
  * Borlenghi  _et al._ [2015] S. Borlenghi, S. Iubini, S. Lepri, L. Bergqvist, A. Delin, and J. Fransson, Phys. Rev. E 91, 040102 (2015).
  * Walter  _et al._ [2014] S. Walter, A. Nunnenkamp, and C. Bruder, Phys. Rev. Lett. 112, 094102 (2014).
  * Lörch  _et al._ [2016] N. Lörch, E. Amitai, A. Nunnenkamp, and C. Bruder, Phys. Rev. Lett. 117, 073601 (2016).
  * Ben Arosh  _et al._ [2021] L. Ben Arosh, M. C. Cross, and R. Lifshitz, Phys. Rev. Res. 3, 013130 (2021).
  * Mendes  _et al._ [2021] U. C. Mendes, J. S. Sales, and N. G. de Almeida, J. Phys. B: At. Mol. Opt. Phys. 54, 175504 (2021).
  * Wolf  _et al._ [2016] F. Wolf, Y. Wan, J. C. Heip, F. Gebert, C. Shi, and P. O. Schmidt, Nature 530, 457 (2016).
  * Chou  _et al._ [2017] C.-W. Chou, C. Kurz, D. B. Hume, P. N. Plessow, D. R. Leibrandt, and D. Leibfried, Nature 545, 203 (2017).
  * Najafian  _et al._ [2020] K. Najafian, Z. Meir, M. Sinhal, and S. Willitsch, Nat. Commun. 11, 4470 (2020).
  * Cameron  _et al._ [2014] R. P. Cameron, S. M. Barnett, and A. M. Yao, New J. Phys. 16, 013020 (2014).
  * Canaguier-Durand  _et al._ [2013] A. Canaguier-Durand, J. A. Hutchison, C. Genet, and T. W. Ebbesen, New J. Phys. 15, 123037 (2013).
  * Stickler  _et al._ [2021] B. A. Stickler, M. Diekmann, R. Berger, and D. Wang, Phys. Rev. X 11, 031056 (2021).
  * Martinetz  _et al._ [2020] L. Martinetz, K. Hornberger, J. Millen, M. Kim, and B. A. Stickler, npj Quantum Info. 6, 101 (2020).
  * Dania  _et al._ [2021] L. Dania, D. S. Bykov, M. Knoll, P. Mestres, and T. E. Northup, Phys. Rev. Res. 3, 013018 (2021).
  * Ren  _et al._ [2022] Y. Ren, E. Benedetto, H. Borrill, Y. Savchuk, M. Message, K. O’Flynn, M. Rashid, and J. Millen, Appl. Phys. Lett. 121, 113506 (2022).