# Beating ringdowns of near-degenerate mechanical resonances

Matthijs H. J. de Jong  Department of Precision and Microsystems Engineering,
Delft University of Technology, Mekelweg 2, 2628CD Delft, The Netherlands
Kavli Institute of Nanoscience, Department of Quantum Nanoscience, Delft
University of Technology, Lorentzweg 1, 2628CJ Delft, The Netherlands  Andrea
Cupertino  Department of Precision and Microsystems Engineering, Delft
University of Technology, Mekelweg 2, 2628CD Delft, The Netherlands  Dongil
Shin  Department of Precision and Microsystems Engineering, Delft University
of Technology, Mekelweg 2, 2628CD Delft, The Netherlands  Department of
Materials Science and Engineering, Delft University of Technology, Mekelweg 2,
2628 CD Delft, The Netherlands  Simon Gröblacher  Kavli Institute of
Nanoscience, Department of Quantum Nanoscience, Delft University of
Technology, Lorentzweg 1, 2628CJ Delft, The Netherlands  Farbod Alijani
Department of Precision and Microsystems Engineering, Delft University of
Technology, Mekelweg 2, 2628CD Delft, The Netherlands  Peter G. Steeneken
Department of Precision and Microsystems Engineering, Delft University of
Technology, Mekelweg 2, 2628CD Delft, The Netherlands  Kavli Institute of
Nanoscience, Department of Quantum Nanoscience, Delft University of
Technology, Lorentzweg 1, 2628CJ Delft, The Netherlands  Richard A. Norte
<EMAIL_ADDRESS>Department of Precision and Microsystems Engineering,
Delft University of Technology, Mekelweg 2, 2628CD Delft, The Netherlands
Kavli Institute of Nanoscience, Department of Quantum Nanoscience, Delft
University of Technology, Lorentzweg 1, 2628CJ Delft, The Netherlands

###### Abstract

Mechanical resonators that possess coupled modes with harmonic frequency
relations have recently sparked interest due to their suitability for
controllable energy transfer and non-Hermitian dynamics. Here, we show
coupling between high Q-factor ($>\\!\\!10^{4}$) resonances with a nearly 1:1
frequency relation in spatially-symmetric microresonators. We develop and
demonstrate a method to analyze their dynamical behavior based on the
simultaneous and resonant detection of both spectral peaks, and validate this
with experimental results. The frequency difference between the peaks
modulates their ringdown, and creates a beat pattern in the linear decay. This
method applies both to the externally driven and the Brownian motion (thermal)
regime, and allows characterization of both linear and nonlinear parameters.
The mechanism behind this method renders it broadly applicable to both optical
and electrical readout, as well as to different mechanical systems. This will
aid studies using near-degenerate mechanical modes, for e.g. optomechanical
energy transfer, synchronization and gyroscopic sensors.

##  I Introduction

The dynamics of coupled resonators have been intensely studied from numerous
perspectives over the last centuries. With the advent of ultra-high Q
mechanical resonators ([1, 2, 3, 4] as recent examples), the regime of linear
coupling between resonances [5, 6, 7] has seen renewed interest, due to the
sensitivity to small perturbations. High-Q resonators can be utilized for
their long coherence- and lifetimes [8, 9, 10], but they often feature spatial
symmetries that naturally predisposes them to have (near-) degenerate or
harmonically related eigenmodes (e.g. [11, 12, 13, 14]). In the case of
trampoline membranes, the near degenerate eigenmodes make them ideal
candidates for various schemes in optomechanics, such as synchronization and
phonon lasing [15, 16, 17, 18, 19], or heat and energy transport [20, 21, 22].
Other effects, such as mechanical squeezing [23, 24, 25, 26] and noise
cancellation [27, 28, 29, 30] have been shown in a variety of geometries.
Mechanical systems with near-degenerate eigenmodes are an attractive platform
for studying exceptional points, non-reciprocal coupling and other phenomena
of non-Hermitian (open) systems [31, 32, 33, 34, 35]. An exceptional point,
for example, requires the frequencies of the eigenmode to be degenerate while
the decay rates are opposite (e.g. one mode is driven, the other decays).
Additionally, near-degenerate mechanical resonances feature direct
applications to sensors such as gyroscopes [36, 37].

Due to the development of suspended micro- and nano-scale resonators, some
recent studies have focused on the nonlinear behavior of coupled mechanical
modes. Coupled nonlinear modes [38, 39, 40] can lead to stabilization [41] and
low-noise oscillators. Strong coupling [42] can be used in mechanical signal
processing, while coherent [43, 44] and nonlinear decay paths [45, 46, 47]
could be leveraged for controlling energy transfer between modes [48].

In this work, we leverage our fabrication precision to design and study high
Q-factor mechanical modes with a 1:1 frequency relation. Based on structures
with spatial symmetries and modelling by finite element methods, we find pairs
of modes whose shapes are identical except for a rotation by
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$ in the plane of the suspended
structures. Measurements of these modes show two peaks in the displacement
spectrum that are separated in frequency by less than 6 parts per million
(ppm). The energy decay shows two different decay rates, which we can extract
without resolving the two spectral peaks. There is evidence that energy is
exchanged between these two resonances, which indicates a coupling mechanism.
We distinguish between the frequency splitting of the peaks due to fabrication
imperfections and the frequency splitting due to this coupling using a
characterization method based on the simultaneous, resonant detection of both
peaks. The detector has a bandwidth larger than the total frequency
difference, such that we measure the superposition of both peaks. Based on the
frequency difference and relative amplitudes of the peaks, we observe a
characteristic beating pattern by performing ringdown measurements. From these
ringdowns, we can extract damping rates $\gamma_{1,2}$ of the seemingly
uncoupled peaks (i.e. decay follows the expected exponential trend). Due to
the difference between the peak frequencies $\omega_{1,2}$, a characteristic
beat pattern appears, with an amplitude inversely proportional to the relative
amplitude difference between the two spectral peaks. Modelling of this effect
provides evidence of a small coupling between the two (unhybridized) modes
with rate $J$, of a frequency shift due to a small Duffing nonlinearity, and
it allows us to investigate the resonator phase decoherence in the thermal
(Brownian) motion regime.

##  II Results

Figure 1: a: Schematic of the two homodyne detection setups used to measure
the mechanical motion of membranes (left) or spiderwebs (right). SA: Spectrum
analyzer, $\phi$ phase shifter. The sketched laser positions match
experimental conditions. b: Ringdowns of first three membrane modes (inset:
mode shape, red indicates maximum displacement). They are aligned to the time
where the driving is stopped, at $t=10$ $\mathrm{s}$. The fundamental (purple)
and third (brown) mode can be fit with single linear slopes (black), while the
second mode shows two distinctly different linear slopes (red, orange, with
fitted Q-factors) and a characteristic beating pattern. The spectrum of
measured around this frequency shows two near-degenerate peaks that we
associate with the symmetric (simulated) mode shapes indicated in the inset.
c: Simulated spectrum of two modes that exist as distinct spectral peaks
separated by less than the detection bandwidth (black bar). d: The frequency
difference $\Delta f$ leads to a characteristic beat in the ringdown signal.
The relative amplitudes of the modes (gold, pink horizontal lines) determine
the amplitude of the beating pattern. e: Simulated amplitudes of the near-
degenerate peaks 1 and 2 based on extracted decay rates from b, reconstructed
signal with the beating pattern (black) and the measured data (blue).
Amplitudes are vertically offset for clarity. Inset shows the total power in
the simulation, ignoring the detection efficiency. This is to verify that the
energy monotonically decreases, as expected from a system with decay.

#### Coupling and beating —

We study the dynamics of near-degenerate mechanical modes using optical
interferometers, shown schematically in Fig. 1a. Two types of resonators are
studied, trampoline membranes [1] and spiderweb resonators [3]. A free-space
readout system is used for trampoline membranes (Fig. 1a, left), while the
narrow beams of the spiderwebs necessitate a lensed-fiber readout (Fig. 1a,
right). Applying a resonant drive to a piezo shaker mounted on the sample
holder excites mechanical modes of the resonator. By stopping the drive and
letting the amplitude decay, we perform a ringdown experiment that allows
precise measurement of the decay rate.

The amplitude of a linear harmonic oscillator decays exponentially, which we
observe in the motion of the fundamental and third modes of the trampoline
membranes (Fig. 1b, purple and brown lines). These fit well to a straight line
(black) when plotted in log-scale. The ringdown trace at
$240\text{\,}\mathrm{kHz}$ shows significantly different behavior,
corresponding to two different slopes, two different decay rates. The
extracted Q-factors (red, orange lines indicate fits) differ by almost an
order of magnitude. We will show that this peculiar ringdown behavior is due
to the near-degeneracy of two membrane modes (simulated shapes inset in Fig.
1b). This near-degeneracy is present for the second mode of the membranes due
to symmetry, but absent for the fundamental and third modes. In the following,
we refer to the two near-degenerate (second) modes as peaks with indices 1,2.

To model the ringdown with the near-degeneracy, we start with two
(unhybridized) modes, $X_{1}=\phi_{1}(\vec{r})x_{1}(t)$ and
$X_{2}=\phi_{2}(\vec{r})x_{2}(t)$, where mode shapes
$\phi_{1}(\vec{r}),\phi_{2}(\vec{r})$ describe the spacial form of the mode
through position vector $\vec{r}$, and generalized coordinates
$x_{1}(t),x_{2}(t)$ describe the oscillating behavior in time. We obtain the
mode shapes from finite-element simulations of our structure, and based on the
symmetry of the structure and similarity of the two mode shapes, the effective
masses of the unhybridized modes should be equal. We add to our model the
decay rates $\gamma_{1},\gamma_{2}$, resonance frequencies
$\omega_{1},\omega_{2}$ and a linear coupling with rate $J$ [5]. This coupling
leads to hybridization, such that the eigenmodes of the system of equations
are formed by linear superpositions of $x_{1}$ and $x_{2}$. This coupling $J$
could for instance occur via the substrate to which the resonator is anchored
[49, 50, 51]. In a ringdown measurement, we only observe the decay so our
model does not need driving terms or noise sources; the initial amplitudes
$x_{1}(t=0),x_{2}(t=0)$ of both unhybridized modes are non-zero. Thus we
obtain the following set of equations of motion,

$\displaystyle\ddot{x}_{1}$
$\displaystyle+\gamma_{1}\dot{x}_{1}+\omega_{1}^{2}x_{1}+J^{2}x_{2}=0$ (1)
$\displaystyle\ddot{x}_{2}$
$\displaystyle+\gamma_{2}\dot{x}_{2}+\omega_{2}^{2}x_{2}+J^{2}x_{1}=0.$

It bears mention that the system of Eq. (1) can always be diagonalized to
obtain the eigenmodes with coordinates $y_{1},y_{2}$ and frequencies
$\omega_{1}^{\prime},\omega_{2}^{\prime}$, which are decoupled (see also the
supplementary information (SI) [52], Sec. A). Both descriptions yield the same
dynamics if their parameters are properly matched. We will not diagonalize Eq.
(1) but keep coordinates $x_{1},x_{2}$. This allows us to distinguish between
the frequency difference $\omega_{2}-\omega_{1}$ that comes from fabrication
imperfections and the frequency difference due to coupling $J$. This will be
required to separate the two contributions to the beating effect of Eq. (2),
one from energy transfer and the other from the frequency difference.
Furthermore, using the amplitudes of the unhybridized modes $x_{1},x_{2}$
allows us to add a difference in detection efficiency to $x_{1},x_{2}$ based
on the mode shapes and position where the displacement is measured. Finally,
we work in the weakly-coupled regime, so $x_{1},x_{2}$ are not too different
from $y_{1},y_{2}$.

The spectrum of our resonator around $240\text{\,}\mathrm{kHz}$ shows two
distinct peaks separated by $9\text{\,}\mathrm{Hz}$, as schematically shown in
Fig. 1c (see also [52], Sec. B). They can be distinguished and fitted with a
sum of two Lorentzians without overlap of the confidence interval of center
frequency, so we refer to them as ’near-degenerate’. However, this frequency
difference is smaller than the detection bandwidth of our spectrum analyzer
during the ringdown measurement (SA in Fig. 1a), such that both peaks are
captured in the single ringdown trace. This condition explains the observation
of two different slopes in the same ringdown, as the two modes we observe have
different decay rates. The slopes are each fitted to a single exponential
decay. The mode with the fastest decay rate is (typically) more susceptible to
the drive, such that its amplitude dominates the signal at the start of the
ringdown measurement. For short timescales, we thus observe mainly the fast-
decaying mode. At longer timescales, the slow-decaying mode dominates the
signal, as the fast-decaying mode has lost most of its amplitude. This means
that for long timescales, we observe mainly the slow-decaying mode. This leads
to a single ringdown measurement showing two different slopes, and allows us
to extract the decay rates of the two modes without spectrally resolving them.

The difference in Q-factor between the near-degenerate modes appears similar
to the spread in Q-factors of fundamental modes of nominally identical devices
(e.g. [56, 51]). This spread is commonly attributed to local fabrication
imperfections or material variations. Optical microscope images cannot always
corroborate these explanations, since local material impurities (C, O, etc.)
are too small to resolve optically. Both fabrication imperfections and
material variations are local, and the two near-degenerate modes exist in the
same device. However, the deflection profile is different, so the modes can be
affected differently. This way, these effects could cause the difference in
Q-factor between near-degenerate modes.

In the kink between the two slopes (red, orange), we observe a particular
beating pattern. Such patterns commonly indicate energy exchange [5, 6].
However, the coupling rate necessary to create a beating pattern with the
frequency we observe would mean that we are in the strong coupling regime.
This regime is incompatible with the observation of the two (different)
slopes, since strong coupling allows the energy to decay via the fastest-
decaying mode, i.e. one would only see the steepest slope (see [52] Sec. C).
Furthermore, the observed beating pattern is particular because it starts
small in amplitude (at $t=0$ $\mathrm{s}$ in Fig. 2a), grows until some point
($t=10.5$ $\mathrm{s}$) but then decreases in amplitude again ($t=18$
$\mathrm{s}$). Beating patterns in ringdowns due to energy exchange typically
show only a decrease in amplitude of the beat pattern [46, 47].

Instead, we propose a different effect that contributes to the beating
pattern. The signal measured in a ringdown measurement is the total
displacement of the resonator, which is the sum of the displacement of both
coordinates $x_{1}$ and $x_{2}$ at the detection spot (Fig. 1a). The total
detected displacement power is thus of the form

$x^{2}=\lvert
e^{i\omega_{1}t}\bar{x}_{1}+e^{i\omega_{2}t}d\bar{x}_{2}\rvert^{2},$ (2)

where we have split the coordinates $x_{1},x_{2}$ into envelopes
$\bar{x}_{1},\bar{x}_{2}$, and fast-oscillating terms
$e^{i\omega_{1}t},e^{i\omega_{2}t}$. The bandwidth of our detector
($100\text{\,}\mathrm{Hz}$) makes it much slower than the frequencies
$\omega_{1},\omega_{2}$ ($\simeq 240$ $\mathrm{kHz}$). We have assumed equal
effective masses in Eq. (1), but in principle the effective masses are
dependent on the position where the displacement is measured and the shape of
the mode [55]. To ensure energy conservation when matching our model with
experimental observations, we must introduce a relative detection efficiency
$d$. This constant factor is a function of the position where we measure the
resonator displacement, and the two simulated mode shapes.

The frequency difference $\Delta_{f}=\omega_{2}-\omega_{1}$ (gold, pink in
Fig. 1c,d) creates a beat with period $1/\Delta f$, which is slow enough to be
detected. This means the detected displacement power $x^{2}$ gains a periodic
modulation, a beating which is proportional only to the frequency difference
between the peaks and which does not imply energy transfer between them. This
effect is well-understood as a beat between signals (see e.g. Ref. [40]), and
it applies to different detection mechanisms (optical, electrical) since it
only requires the bandwidth of the resonant detector to encompass both peaks.
It is analogous to electronically or optically down-mixing a signal by sending
in a different tone and observing the beat pattern. In this instance, it is
passively obtained based on the small frequency difference between the peaks
and the high Q-factor of the involved resonances.

To correctly fit the measured ringdown (Fig. 1e), we require both the linear
coupling and the beating effect. Without the latter, the frequency of the
beating pattern would indicate strong coupling, but this is incompatible with
the presence of two slopes [5]. Without the former, the beating pattern would
only briefly appear in the ringdown as the peak amplitudes cross ([52] Sec.
C), whereas it extends much further in Fig. 1e. A nonlinear coupling ($\propto
x^{3}$) would lead to a beating pattern that is qualitatively different to the
one observed: the beating would slow down in frequency as the amplitude decays
(see [52] Sec. C for a comparison of the linear and nonlinear models).

We obtain the individual center frequencies $\omega_{1}=2\pi\times
240,331.6\pm 0.1$ $\mathrm{Hz}$ and $\omega_{2}=2\pi\times 240,341.4\pm 0.1$
$\mathrm{Hz}$ of the peaks from a separate spectrum measurement ([52], Sec. B)
which also allows us to estimate the detection efficiencies ($d\simeq 20$ in
Fig. 1e). From the linear parts of the ringdown, we extract decay rates
$\gamma_{1}=2\pi\times 6.0\pm 0.1$ $\mathrm{Hz}$ and $\gamma_{2}=2\pi\times
0.707\pm 0.001$ $\mathrm{Hz}$. We can simulate the ringdown using only three
fit parameters: the initial positions $x_{10},~{}x_{20}$ and coupling rate
$J$. The optimized fit (black) to the data (blue) is shown in Fig. 1e, with
dimensionless initial positions $x_{10}=0.089\pm 0.001$, $x_{20}=0.071\pm
0.001$ and coupling $J/(2\pi)=320\pm 5$ $\mathrm{Hz}$. The frequency shift due
to the linear coupling, $J^{2}/\omega_{1}$, would correspond to
$0.4\text{\,}\mathrm{Hz}$ if $\omega_{1}=\omega_{2}$, so we are in the weakly-
coupled regime. Nonetheless, there is sufficient coupling to observe energy
exchange between the reconstructed powers of the unhybridized modes (red,
orange lines in Fig. 1e). For example, between $11.3\text{\,}\mathrm{s}$ and
$11.5\text{\,}\mathrm{s}$, $x_{1}$ gains energy from $x_{2}$, shown by the
increase of the reconstructed power. The power in $x_{2}$ correspondingly
decreases, but this is not clearly visible in Fig. 1e due to the difference in
detection efficiency. We numerically verify that the reconstructed power
always decreases over time, as shown by plotting the total power in the inset
of Fig. 1e. Our measurement thus marks the observation of purely linear
coupling between two resonances with a near-degenerate frequency relation,
without showing nonlinear power decay.

Figure 2: a: Ringdown with a prominent beating pattern, measured (blue) and
simulated (red, orange, black). This experiment is performed on a different
device than those of Fig. 1, but in the same experimental setup. Two separate,
linear slopes can be distinguished, but they are similar since the decay rates
of both peaks are similar. b: Frequency difference between the two near-
degenerate peaks as the ringdown progresses, determined from the distance
between subsequent minima of a. The fit (orange) is an exponential described
in the main text.

#### Frequency shift —

The beating pattern is periodic with the inverse of the frequency difference
between the two near-degenerate peaks. Careful observation of Fig. 1e shows
that this period is not constant; it changes with time. This means that the
frequency difference is not constant, and that either one or both peaks change
in frequency over time, similar to Ref. [40]. In the previous section, the
energy decay was observed to be linear, sufficiently so that no nonlinearity
is necessary to describe the ringdown completely. However, a small
nonlinearity might still be present and lead to a frequency shift without
measurably affecting the energy decay. Since the period of the beating is
inversely proportional to the frequency difference, analyzing this periodicity
provides a measure sensitive to small nonlinearities.

From the ringdown shown in Fig. 2a, we extract the frequency difference by
finding the minima of the beating pattern and taking the inverse of their time
differences. We plot the frequency difference in Fig. 2b (blue), which shows
an exponential trend towards $\Delta f=5.0$ $\mathrm{Hz}$. It is likely that
the drive pulled the frequencies of the two peaks together [57], which
indicates the presence of a nonlinearity. Assuming a small Duffing nonlinear
term ($\propto x^{3}$), the frequency shift due to frequency pulling is of the
form [57]

$\omega_{nl}=\omega_{0}+\frac{3}{8}\frac{\alpha}{m_{\mathrm{eff}}\omega_{0}}\tilde{x}^{2},$
(3)

with Duffing coefficient $\alpha$, effective mass $m_{\mathrm{eff}}$ and
displacement amplitude $\tilde{x}$.

Figure 3: a: Mechanical spectrum of near-degenerate modes of a spiderweb
resonator (blue), with Lorentzian fits (orange). b: Two consecutive ringdowns
performed on the same device, one showing the decay from the driven state
(blue), the other showing the last part of the ringdown and subsequent
transition towards a Brownian motion driven state (green). Insets highlight
the beating behavior (dotted line drawn as a guide to the eye), which
continues even in the Brownian-motion regime. Grey-colored region denotes the
cut between the two ringdowns. The two simulated mode shapes are shown
schematically. c Autocorrelation functions of the signal in the ringdown
(blue) and Brownian-motion regime (green), showing a coherence time of $24\pm
2$ $\mathrm{s}$. Points indicate the experimental data, solid lines indicate
the fit.

By observing the change in the frequency difference, we can extract which of
the two peaks behaves nonlinearly, and whether it is softening or hardening.
The linear parameters are extracted by the same procedure as before (the
frequencies follow from a spectrum measurement, the decay rates are fits to
the linear parts of the ringdown and their coupling is modelled via Eq. 1). We
find $\omega_{1}=2\pi\times 240,302.4\pm 0.2$ $\mathrm{Hz}$,
$\gamma_{1}=2\pi\times 0.71\pm 0.01$ $\mathrm{Hz}$, $\omega_{2}=2\pi\times
240,307.4\pm 0.1$ $\mathrm{Hz}$, $\gamma_{2}=2\pi\times 0.218\pm 0.001$
$\mathrm{Hz}$, and coupling $J=2\pi\times 20\pm 1$ $\mathrm{Hz}$ (independent
of the value of $\alpha$) with relative detection efficiency $d\simeq 50$. We
calculate the frequency shift as the peak amplitude decays from Eq. (3). The
decay rates are different, which means that their frequency shift should
change over time at different rates, illustrated by the dashed lines in Fig.
2b. The measured frequency shift is consistent with peak 2 being frequency-
pulled by the drive. This implies a softening nonlinearity ($\alpha<0$), which
is usually associated with an external (e.g. electrical or optical) force
source (geometric nonlinearity tends to yield hardening behavior [40, 58]).
However, curvature of the membrane in the out-of-plane direction could also
lead to a softening nonlinearity. Although the suspended membrane is nominally
flat due to the tensile stress, some curving at the membrane edges can be
observed under microscope (cf. the supplementary of Ref. [59]). In the absence
of any external sources of (softening) nonlinearity, this curvature could be
responsible for the observed change in periodicity of the beating pattern.
Buckling[60] of the whole membrane is unlikely given the high
($1\text{\,}\mathrm{GPa}$) fabrication pre-stress. An estimate for $\alpha$
based on the onset of the Brownian motion regime yields a value
$\alpha_{2}=-3\times 10^{-21}$ ${\mathrm{m}}^{-2}\text{\,}{\mathrm{s}}^{-2}$,
but a more accurate value should be obtained after calibration of the
displacement and detection efficiency. The nonlinear term is sufficiently
small compared to the coupling $J$ that it is negligible for the measured
ringdowns. Despite this, we can observe and measure the frequency shift due to
the nonlinearity. Comparison to literature shows that we are three orders of
magnitude more sensitive than Ref. [40], who utilize a similar technique but
for GaAs nanowires.

#### Resonator decoherence —

Frequency shifts of resonators due to external factors are associated with
resonator phase decoherence [9] or dephasing. This is detrimental for coherent
control [8, 10], which is important for many quantum mechanical applications.
Classically, one can observe the spectral width integrated over sufficient
time to obtain the phase decoherence time [9]. We will describe how the
beating effect provides another mechanism to evaluate the phase decoherence
time by utilizing the second peak as a frequency reference.

In spiderweb resonators [3], there is a set of modes extremely close to
degeneracy, which we observe as two peaks at frequencies
$\omega_{1}=2\pi\times 120,725.517\pm 0.002$ $\mathrm{Hz}$,
$\omega_{2}=2\pi\times 120,726.141\pm 0.001$ $\mathrm{Hz}$ ($<6$ parts per
million difference), as shown in Figs. 1a and 3a. These peaks have linewidths
below the resolution of the spectrum analyzer in this setup
($0.01\text{\,}\mathrm{Hz}$). We estimate their spectral linewidths to be
$2.5\text{\,}\mathrm{mHz}$ and $2.3\text{\,}\mathrm{mHz}$ respectively (full
width at half maximum, corresponding to $Q\simeq 25\times 10^{6}$), which
matches to the Q-factor extracted from a linear fit to the ringdown
measurement, $Q=21.1\times 10^{6}$.

The ringdown measurement (Fig. 3b), shows the beating pattern both in the
strongly driven regime (blue) and in the Brownian motion regime (green). The
dynamics of $x_{1},x_{2}$ are much faster than our detection bandwidth, and we
work at room temperature, so we cannot directly resolve the thermal
decoherence rate [10]. However, the beating pattern is sensitive to frequency
fluctuations. A phase-shift in the beating pattern corresponds to a frequency
shift between the two peaks. Thus monitoring the beating pattern phase allows
us to observe frequency shifts, which correspond to phase decoherence between
the two observed modes. We can do so by calculating the autocorrelation of the
beating signal of Fig. 3b, which we plot in in Fig. 3c. It shows behavior
similar to that of a single, underdamped particle on a spring undergoing
Brownian motion, which has an autocorrelation given by [61]

$C(t)=\frac{\Gamma k_{\mathrm{B}}T}{m_{\mathrm{eff}}}e^{-\Gamma
t}\left(\frac{1}{\Gamma}\cos(\delta t)-\frac{1}{\delta}\sin(\delta t)\right),$
(4)

with $\delta=\sqrt{\Omega^{2}-\Gamma^{2}}$. Here the frequency $\Omega$ is the
frequency of the beating pattern, and decay $\Gamma$ is related to the phase
decoherence time, i.e. when the amplitude of the autocorrelation drops by
half. We fit a $1/\Gamma=24.5\pm 1.8$ $\mathrm{s}$ ($24.8\pm 2.8$
$\mathrm{s}$) in the driven (thermal) regime respectively. This is close to
the linear (energy) decay time ($Q/\omega_{1,2}\simeq 27.8$ $\mathrm{s}$)
extracted from the ringdown directly. This is the expected behavior for a
linear harmonic oscillator: the phase decoherence time (measured via the
beating pattern) should be similar to the energy decay time (measured directly
from the ringdown). Additionally, we fit $\Omega=0.195\pm 0.001$ $\mathrm{Hz}$
($\Omega=0.185\pm 0.002$ $\mathrm{Hz}$) in the driven (thermal) regime, which
matches well with the frequency difference between the two peaks
$|\omega_{1}-\omega_{2}|/\pi$. This illustrates that the beating effect can
provide a measurement of the phase decoherence of (near-) degenerate modes of
a mechanical resonator.

##  III Conclusion

We have experimentally studied high Q-factor mechanical modes of spatially-
symmetric microresonators that are nearly degenerate. We provide evidence of a
linear coupling between the two resonances, present without driving the
resonator to nonlinearity. This is in contrast to previous studies, that
generally require a quadratic or cubic term for coupling. To detect this
linear coupling, we developed a characterization method based on a single,
resonant detector. This detector has a detection bandwidth that encompasses
both spectral peaks, such that we can see two decay rates in a single ringdown
measurement. When the spectral peaks are sufficiently close in frequency, an
interference effect occurs that leads to a particular beating pattern. The
relative amplitudes of the individual peaks control the beating pattern
amplitude, while their frequency difference controls the beating period. From
the beating pattern, we can monitor the amplitudes of the individual spectral
peaks and find their coupling rate. The beating period provides a sensitive
measure to the relative frequency difference between the two peaks. This
allows us to observe slight frequency shifts due to nonlinearities, which
would be difficult to observe using other methods. Our Si3N4 trampoline
membranes feature a softening nonlinearity, which we attribute to out-of-plane
curving. The beating period can additionally be used to monitor frequency
shifts associated with phase decoherence of the modes. In Si3N4 spiderweb
resonators, we find the phase decoherence time is similar to the energy decay
time, as expected for linear harmonic oscillators.

Our characterization method is applicable to both optical and electrical
readout schemes, and is particularly suited to high Q-factor mechanical
resonators. This type of resonators is highly relevant for e.g. sensing,
optomechanics and forms an attractive platform to study non-Hermitian systems.
We have thus developed a broadly applicable method to characterize near-
degenerate resonators, which can be immediately applied to studies in
different fields.

#### Data availability

All data, simulations, measurement and analysis scripts in this work are
available at https://doi.org/10.4121/21428517.v1.

#### Acknowledgments

R.N. would like to acknowledge support from the Limitless Space Institute’s I2
Grant. This publication is part of the project, Probing the physics of exotic
superconductors with microchip Casimir experiments (740.018.020) of the
research programme NWO Start-up which was partly financed by the Dutch
Research Council (NWO). M.J., A.C., and R.N. acknowledge valuable support from
the Kavli Nanolab Delft and from the Technical Support Staff at PME, 3mE
Delft.

##  IV Supplementary Information

###  IV.1 Decoupling the EOM

It is convenient to obtain the relation between the parameters of the coupled
and decoupled equations of motion for our system of near-degenerate
resonances. We start from the Fourier-transform of Eq. (1), where we have
simplified $\omega_{1}=\omega_{0}$, $\omega_{2}=\omega_{0}+\nu$ and
$(\omega_{0}+\nu)^{2}\simeq\omega_{0}^{2}+2\nu\omega_{0}$ (thus assuming small
$\nu$). Thus we start the equation of motion

$\begin{bmatrix}\omega_{0}^{2}+i\gamma_{1}-\omega^{2}&-J^{2}\\\
-J^{2}&\omega_{0}^{2}+2\nu\omega_{0}+i\gamma_{2}-\omega^{2}\end{bmatrix}\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix}=\begin{bmatrix}0\\\ 0\end{bmatrix},$ (S1)

which is of the form $\bm{\mathrm{A}}\vec{x}=\vec{0}$. It is straightforward
to obtain the eigenvectors $\vec{e}_{1},~{}\vec{e}_{2}$ of matrix
$\bm{\mathrm{A}}$, and we put them in a $2\times 2$ matrix
$\bm{\mathrm{U}}=[\vec{e}_{1}~{}\vec{e}_{2}]$. Then we can obtain the
diagonalized matrix $\bm{\mathrm{B}}$ using
$\bm{\mathrm{B}}=\bm{\mathrm{U}}^{-1}\bm{\mathrm{AU}}$. The coordinates in
which the equations are decoupled are then given by
$\vec{y}=\bm{\mathrm{U}}^{-1}\vec{x}$. The eigenvalues (diagonal elements) of
$\bm{\mathrm{B}}$ are then given by

$\displaystyle(\omega_{1,2}^{\prime})^{2}$
$\displaystyle=\omega_{0}^{2}+\nu\omega_{0}+\frac{i}{2}(\gamma_{1}+\gamma_{2})\pm\frac{\tau}{2},$
(S2) $\displaystyle\tau$
$\displaystyle=\sqrt{4J^{4}+4\nu^{2}\omega_{1}^{2}-\gamma_{1}^{2}-\gamma_{2}^{2}+2\gamma_{1}\gamma_{2}+4i\nu\omega_{0}(\gamma_{1}+\gamma_{2})}.$

These $\omega_{1,2}^{\prime}$ are the frequencies of the eigenmodes of the
system, which are the frequencies we observe in the spectrum. If we measure
the splitting $2\pi\Delta
f=\Delta\omega=|\omega_{1}^{\prime}-\omega_{2}^{\prime}|$ and want to know the
detuning of the original coordinates $x_{1}$ and $x_{2}$,
$\nu=|\omega_{1}-\omega_{2}|$ for a given coupling $J$, we use

$\nu=\frac{1}{2}\left(\Delta\omega^{2}+i\gamma_{1}-i\gamma_{2}\right)-\sqrt{\Delta\omega^{2}(\omega_{0}^{2}+i\gamma_{1})-J^{4}}.$
(S3)

which allows us to find the frequencies of the coupled equations,
$\omega_{1,2}$, from the decoupled ones and coupling rate $J$.

###  IV.2 Spectrum and detection efficiency

Figure S1: a: Spectrum of a typical mechanical resonator, showing the first
three modes ($114.9\text{\,}\mathrm{kHz}$, $240.3\text{\,}\mathrm{kHz}$ and
$360.5\text{\,}\mathrm{kHz}$ identified by their growth under white noise
driving (blue) versus thermal driving (green). b: Spectrum around the near-
degenerate peaks (vertically offset for clarity). Rounded peaks are the result
of the $1\text{\,}\mathrm{Hz}$ measurement bandwidth used. Black line
indicates Lorentzian fit of the sum of the peaks, dotted for equal detection
efficiency and solid for the factor 20 less efficient detection of the lower-
frequency mode. Pink lines indicate the individual Lorentzian components.

A typical membrane device spectrum is shown in Fig. S1a. The the region of
interest is around $240.3\text{\,}\mathrm{kHz}$, where two very closely spaced
peaks appear, separated by $9\text{\,}\mathrm{Hz}$, as in Fig. S1b. Their
frequency separation is well-resolved within the $1\text{\,}\mathrm{Hz}$
bandwidth of our spectrum analyser. The center frequencies of the peaks
reported in the main text are extracted by fitting two summed Lorentzians to
the spectrum, using the curve_fit function from Scipy [53]. The laser spot of
our interferometer is large enough to capture the majority of the membrane pad
and thus detect both the near-degenerate mode shapes that we obtain from
finite element method simulations. In the high-Q limit, the expected PSD for a
harmonic oscillator is a Lorentzian around its resonance [54]. The mechanical
mode shapes are identical except rotated by 90$\mathrm{\SIUnitSymbolDegree}$,
so their effective mass $m_{\mathrm{eff}}$ is the same (barring fabrication
imperfections) and their response to thermal or white-noise driving should
result in equal displacement (power) in both modes [55]. Modelling this
requires the linear decay rates, which we extract from the piecewise linear
parts of the ringdown measurement, Fig. 1b. We obtain decay rates of
$\gamma_{1}=2\pi\times 6.0$ $\mathrm{Hz}$ and $\gamma_{2}=2\pi\times 0.7$
$\mathrm{Hz}$ respectively for the peaks observed at $\omega_{1}=2\pi\times
240331.9$ $\mathrm{Hz}$ and $\omega_{2}=2\pi\times 240341.1$ $\mathrm{Hz}$.

Reconstructing the expected power spectrum results in the black lines of Fig.
S1b. These suggest that instead of the detection efficiency (DE) being equal
(dotted line), the lower-frequency peak is detected a factor 20 less
efficiently (solid line). This is likely determined by the precise position of
the laser beam on the membrane. From the (room-temperature) thermally driven
amplitude ($\lesssim$$2\text{\,}\mathrm{pm}$), we deduce that the maximum
driven amplitude $\simeq$$2\text{\,}\mathrm{nm}$, well within the linear
regime of the interferometer. In that same comparison, no frequency shift of
the peak is visible which confirms that we are in the linear regime of
mechanical motion and the Duffing nonlinearity is small. The rounded shape of
the peaks is determined by the minimum detection bandwidth of the spectrum
analyzer, $1\text{\,}\mathrm{Hz}$.

###  IV.3 Linear and nonlinear coupling

Figure S2: Simulated ringdowns with (solid) and without (dashed) the beating
pattern due to downmixing, for different values of coupling strength $J$. All
simulations share the same initial values as Fig. 1e in the main text, the
dashed lines are vertically offset by $10\text{\,}\mathrm{dB}$ for clarity.
Figure S3: Simulated ringdowns using the nonlinear coupled model of Ref. [46].
Top row (a,b,c) include the beating, equivalent to the detuning in the model
of Ref. [46], while the bottom row (d,e,f) excludes this effect. a,d:
Ringdowns for zero coupling, replicating the result in Fig. S2. b,e: Ringdowns
in the strong, linear coupling regime for different values of coupling rate
$J$. e matches to Fig. S2, but b has an enhanced beat pattern due to the
frequency splitting of the strong coupling. c,f: Ringdowns for nonlinear
resonators, showing a beating pattern that slows down as the amplitudes decay.
All simulations share the same initial values as Fig. 1e in the main text.

To distinguish between the coupling and beating effects, we simulate and show
ringdowns in Fig. S2 for various values of the coupling rate $J$. All
simulations share the same initial condition as Fig. 1e in the main text. In
Fig. S2a, there is no coupling, $J=0$. In the absence of the beating effect
(dashed line), there is a smooth transition between two slopes. With the
presence of beating effect (solid line), the kink between the slopes gains the
characteristic pattern where the beat first grows in amplitude until the
detected powers of the two peaks are equal, and then decreases in amplitude
until it disappears.

In Fig. S2b, we replicate the ringdown of Fig. 1e. Without the beating effect
(dashed line), there is some oscillation present that is indicative of energy
exchange between the peaks. It appear superimposed on the beating pattern
(solid line), illustrating that the effects are independent and both are
necessary to describe the observations.

In Figs. S2c-e, the coupling rate is increased until the strong-coupling
regime is reached ($J=5000$ $\mathrm{Hz}$ corresponds to a splitting of
$104\text{\,}\mathrm{Hz}$ if $\omega_{1}=\omega_{2}$). The oscillations due to
the energy exchanged via the coupling increase in frequency, while the beating
effect decreases in amplitude. The two different slopes also transform into
one slope in the strong coupling regime. Here, energy predominantly leaves the
system via the fast-decaying resonator. This illustrates that for strong
enough coupling, the two slopes merge into one single slope.

The beating observed in ringdowns could also be related to energy exchange
between nonlinear resonators with a 1:1 resonance [46]. The equations
describing this are [46]

$\displaystyle\dot{a}_{1}$
$\displaystyle=-i\left(\delta_{1}a_{1}-\frac{3}{2}\alpha_{1}a_{1}|a_{1}|^{2}+Ja_{2}\right)-\frac{\gamma_{1}}{2}a_{1}$
(S4) $\displaystyle\dot{a}_{2}$
$\displaystyle=-i\left(\delta_{2}a_{2}-\frac{3}{2}\alpha_{2}a_{2}|a_{2}|^{2}+Ja_{1}\right)-\frac{\gamma_{2}}{2}a_{2}.$

Here, $a_{1},a_{2}$ are the complex mode amplitudes, $\delta_{1},\delta_{2}$
describes the detuning of $a_{1},a_{2}$ with respect to the rotating frame we
choose (we pick $\delta_{1}=0,\delta_{2}=9$ $\mathrm{Hz}$ to match the
$9\text{\,}\mathrm{Hz}$ difference between our two peaks),
$\alpha_{1},\alpha_{2}$ are the Duffing nonlinearities, and $J$ and
$\gamma_{1},\gamma_{2}$ are the coupling and linear decay rates as in the main
text.

We simulate our system using Eq. (S4), and show the results in Fig. S3. In the
absence of coupling or nonlinearities ($J=0,\alpha_{1}=\alpha_{2}=0$), Eq.
(S4) gives identical results to the model introduced in the main text. The
detuning $\delta$ takes the role of the beating effect, if we plot
$|a_{1}+a_{2}|^{2}$. This illustrates the generality of the effect, as
$a_{1},a_{2}$ exist in a rotating frame (similar to the center frequency of
our detector bandwidth). Figs. S3a,d thus match closely with simulated results
of Fig. S2a. In the strong coupling regime (Figs. S3b,e), there is a
difference due to the large frequency split. The model introduced in the main
text explicitly contains the frequencies observed from a measurement of the
spectrum ($\omega_{1},\omega_{2}$), and thus constrains the frequency
difference to this value. In contrast, Eq. (S4) does not constrain this
frequency splitting. Nonetheless, both models predict a single slope with
fast, small oscillations.

In Figs. S3c,f, we introduce nonlinearity in the resonators ($\alpha_{1}\neq
0,\alpha_{2}\neq 0$). This creates an oscillating pattern, where the frequency
of oscillation decreases as the resonators decay. This trend is general (i.e.
not dependent on the sign or values of $\alpha$), and different from the
trends observed in the main text. In Fig. 1e, the beating pattern stays
constant in frequency, while in Fig. 2a, it increases in frequency. Thus these
observations are inconsistent with a model whereby the beating originates from
the nonlinearity of the resonators.

Figure S4: Simulated ringdown (orange) for a resonator with nonlinear damping,
with the observed ringdown (blue) for comparison.

Finally, the two linear slopes observed in the ringdowns could alternatively
be explained by a single resonator that has nonlinear damping [45], of the
form

$\ddot{x}+2(\gamma_{\mathrm{l}}+\gamma_{\mathrm{nl}}x^{2})\dot{x}+\omega^{2}x=0.$
(S5)

Here, $\gamma_{\mathrm{l}}$ the linear damping used in the main text and
$\gamma_{\mathrm{nl}}$ the nonlinear damping. This term causes a fast decay
directly after the driving stops, and returns the slow decay for lower
amplitude. Qualitatively, the kink that results from this transition could
have the same shape as the observed kink. However, the fast decay of the
nonlinear damping region is much steeper than the fast decay we observe as
show in Fig. S4, which rules out this nonlinear damping as an alternative
explanation.

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