$(\hat{x},\hat{y},\hat{z},\hat{w})$ vanish to orders $(2,3,1,4)$ at a
particular locus. As an example, we can consider the Weierstrass model of Eq.
183 and the locus $\\{\sigma=b_{0,0}=0\\}$. We must account for the fact that
the $\hat{z}$ component of the generating section, which is given by
$b_{0,0}$, vanishes to order 1 at the $\text{II}^{*}$ locus. We therefore
construct the $f$ and $g$ of the $p$-adic elliptic curve by plugging in
appropriate values of the parameters into $fb_{0,0}^{4}$ and $gb_{0,0}^{6}$
from Eq. 183; at the end of the calculation, we will account for the fact that
these extra factors of $b_{0,0}$ actually come from $\hat{z}$. We set both
$\sigma$ and $b_{0,0}$ to a prime integer $p$ and set the other parameters to
numbers such that $f$ and $g$ are integers. Performing this procedure with a
particular choice of numbers leads to an elliptic curve $E$ of the form

$\displaystyle y^{2}$
$\displaystyle=x^{3}-p^{8}\mathopen{}\mathclose{{}\left(1-12p+p^{2}+24p^{3}+12p^{6}}\right)x$
(385)
$\displaystyle\qquad\quad+p^{11}\mathopen{}\mathclose{{}\left(-6+p+24p^{2}-p^{3}-26p^{4}+24p^{5}-2p^{6}-48p^{7}-16p^{10}}\right)\,.$

The discriminant $\Delta_{E}$ for this elliptic curve is proportional to
$p^{22}$. After plugging the same numbers into the $[\hat{x}:\hat{y}:\hat{z}]$
section components, we see that $E$ admits an integral point $P$ with
coordinates

$P\colon(x,y)=\mathopen{}\mathclose{{}\left(p^{2}(1-2p^{2}-2p^{5}),p^{3}(-1+3p^{2}-p^{4})}\right)\,.$
(386)

The elliptic curve $E$ has additive reduction modulo $p$. We therefore extend
the field with $p^{\prime}=p^{1/6}$, let $x=x^{\prime}{p^{\prime}}^{22}$ and
$y=y^{\prime}{p^{\prime}}^{33}$, and divide both sides of the equation for the
elliptic curve by ${p^{\prime}}^{66}$. This gives a new elliptic curve
$E^{\prime}$ given by the equation

$\displaystyle{y^{\prime}}^{2}$
$\displaystyle={x^{\prime}}^{3}-{p^{\prime}}^{4}\mathopen{}\mathclose{{}\left(1-12{p^{\prime}}^{6}+{p^{\prime}}^{12}+24{p^{\prime}}^{18}+12{p^{\prime}}^{36}}\right){x^{\prime}}$
(387)
$\displaystyle\qquad+\mathopen{}\mathclose{{}\left(-6+{p^{\prime}}^{6}+24{p^{\prime}}^{12}-{p^{\prime}}^{18}-26{p^{\prime}}^{24}+24{p^{\prime}}^{30}-2{p^{\prime}}^{36}-48{p^{\prime}}^{42}-16{p^{\prime}}^{60}}\right)\,,$

which admits a rational point $P^{\prime}$ with coordinates

$P^{\prime}\colon(x^{\prime},y^{\prime})=\mathopen{}\mathclose{{}\left({p^{\prime}}^{-10}(1-2{p^{\prime}}^{12}-2{p^{\prime}}^{30}),{p^{\prime}}^{-15}(-1+3{p^{\prime}}^{12}-{p^{\prime}}^{24})}\right)\,.$
(388)

The elliptic curve $E^{\prime}$ has good reduction modulo $p^{\prime}$, and
the original elliptic curve $E$ has potential good reduction modulo $p$. Since
$v(\Delta_{E})$ is $22$ and $p^{\prime}$ has ramification degree $d=6$, Eq.
155, which comes from Theorem 19 of StangeEllTrouble , states that the EDS
valuations associated with $(E,P)$ should satisfy the formula

$v(W_{m})=\frac{22}{12}(m^{2}-1)+\frac{1}{6}v_{1}(W_{m}^{\prime})\,,$ (389)

where $v_{1}(W_{m}^{\prime})$ represents the EDS valuations for $E^{\prime}$
and $P^{\prime}$. Note that $P^{\prime}$ reduces to the identity point on
$E^{\prime}$ modulo $p^{\prime}$, so according to Eq. 156 and Theorem 14 of
StangeEllTrouble ,

$\displaystyle v_{1}(W_{m}^{\prime})$
$\displaystyle=\frac{v_{1}(x^{\prime}_{P^{\prime}})}{2}m^{2}+S_{m}\mathopen{}\mathclose{{}\left(p^{\prime},p^{\prime},v_{1}(p^{\prime}),0,s,w}\right)$
(390)
$\displaystyle=-5m^{2}+S_{m}\mathopen{}\mathclose{{}\left(p^{\prime},p^{\prime},v_{1}(p^{\prime}),0,s,w}\right)\,.$

According to Appendix B, the $S$ term can be replaced with a constant $s_{P}$.
Combining everything together, we find that

$v(W_{m})=\frac{22}{12}(m^{2}-1)+\frac{1}{6}\mathopen{}\mathclose{{}\left(-5m^{2}+s_{P}}\right)=m^{2}+\frac{1}{6}\mathopen{}\mathclose{{}\left(s_{P}-11}\right)\,.$
(391)

Since the $\hat{z}$ component of the generating section vanishes to order 1 at
the matter locus, $v(W_{m})$ should morally be 1 for our purposes. We
therefore set $s_{P}$ to 11, giving us

$v(W_{m})=m^{2}\,.$ (392)

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# Collective modes of nanoparticles levitated in a cavity: mechanical mode
combs

M. Rademacher  Department of Physics and Astronomy, University College London,
Gower Street, WC1E 6BT London, United Kingdom.  T.S. Monteiro
<EMAIL_ADDRESS>Department of Physics and Astronomy, University College
London, Gower Street, WC1E 6BT London, United Kingdom.

###### Abstract

Levitated nanoparticles are being investigated as ultrasensitive sensors of
forces and accelerations, with applications ranging from fundamental physics
phenomena such as dark matter or quantum gravity to real world applications.
Quantum cooling in up to 2 motional modes was achieved recently, and attention
is now turning to multimode and multiparticle regimes. We investigate here the
spectral characteristics of collective motion of
$\operatorname*{N^{tot}}=2-20$ nanoparticle modes interacting via the optical
light of a cavity. We find output spectra of collective motion typically
exhibit two dominant generic features: a broad spectral feature, the
collective bright mode (CBM) as expected; but we introduce also the mechanical
mode comb (MC) a - not previously studied- spectral feature. We investigate
numerically, and can also precisely reproduce all the intricate details of the
MC spectra, including ponderomotive squeezing, with a simple closed-form
expression. We show the mechanical comb can autonomously repair loss of teeth
due to nanoparticle loss, a feature that may offer robustness to a levitated
sensor of external forces.

Levitated nanoparticles in high vacuum [1] offer near-perfect decoupling from
environmental noise and decoherence. They offer exceptional sensitivity to
weak forces, and experiments are undergoing rapid development, driven by
applications such the search for dark matter and fundamental physics [2, 3, 4,
5, 6] but also real world applications such as for instance communication with
under sea vessels [7]. Sensing of forces as low as the zeptoNewton scale has
already been demonstrated [8, 9, 10, 11, 12, 13]. The nanoparticles can be
passively cooled and controlled in cavities, or by active feedback. Cooling to
quantum (or near) ground state of a single mode of the motion was achieved
recently both by active control [14, 15] or via the optical mode of a cavity
[16, 17, 18].

Figure 1:  (a) We consider an array of levitated nanoparticles trapped in
tweezers in a cavity. Cavity-mediated coupling leads to interparticle as well
as intraparticle mode hybridisation (b) Shows the heterodyne-detected cavity
output. When the modes are degenerate, there is only a very broad spectral
feature, the collective bright mode (CBM). In a realistic setting there will
be a spread of frequencies; the mechanical comb (MC) is then a distinctive
spectral feature. The MC collective mode spectra rearrange to fill the
frequency gaps in the “teeth” of the unperturbed mode comb. We illustrate for
the cases where the unperturbed modes are (i) equal spaced; unperturbed modes
are shown in red and blue (ii) pairs of near degenerate modes: an almost
equidistant pattern of alternating broad/narrow features arises (iii) randomly
distributed unperturbed frequencies. Figure 2:  Robustness of comb to particle
loss (loss of comb teeth). Upper comb shows the signal from a 16 particle
regular comb; lower comb (reflected about zero) shows the spectrum if the two
nanoparticles corresponding to the black arrows are knocked out of the array.
The neighbouring ‘teeth’ of the comb displace and broaden so as to largely
repair the gap. An independent particle array might leave a near zero-signal
gap at the missing frequencies. $g/2\pi=30$ kHz, $\Delta\omega=45$kHz.

Multimode and multiparticle dynamics and cooling is of growing interest [19,
20, 21, 22, 18, 23, 24, 25]. The enhanced light-matter interaction effects of
collective dynamics are well-established in optical and atomic physics.
Collective modes have been investigated in optomechanics [26] and briefly
reviewed in [27], see references therein. A known phenomenon is that for an
ensemble with $\operatorname*{N^{tot}}$ individual, subsystems $\hat{q}_{j}$,
$j=1,2...\operatorname*{N^{tot}}$, without direct interaction, but that all
couple to a common ‘bus’ mode via the interaction term
$g\sum_{j}\hat{q}_{j}(\hat{a}+\hat{a}^{\dagger})$, then there results an
effective enhancement of the coupling strength to mode $\hat{a}$, which
couples strongly to a center of mass mode $\hat{Q}=\sum_{j}\hat{q}_{j}$, with
strength $g\sqrt{\operatorname*{N^{tot}}}$. In optical and atomic physics, it
is usual to assume a system of $j$ identical modes as all photons and atoms or
molecules of the same species may be considered identical. Levitation-based
force sensing within this framework has been analysed [28].

However, in realistic experimental scenarios, no two loaded levitated
nanoparticles loaded into optical traps are ever precisely identical. Although
adjustments in trap power and so forth are in principle possible in order to
control this, we argue that these apparent imperfections could be turned to
advantage, rather than fully suppressed. In Fig.(1a), we illustrate the
proposal: an array of nanoparticles in tweezer traps, with
$\operatorname*{N^{tot}}$ motional modes, within a cavity. The
$\operatorname*{N^{tot}}$ modes span a frequency range $\Delta\omega$ about a
mean, so $\omega_{j}\in[\omega_{m}\pm\Delta\omega/2]$, and the frequencies
have mean spacing $\delta=\Delta\omega/\operatorname*{N^{tot}}$. There is no
requirement for precisely equal couplings, just that the couplings $g_{j}\sim
g$ are of similar order . We investigate the regime $g\sim\Delta\omega$.

Considering 3D center of mass motion, in principle,
$\operatorname*{N^{tot}}=3N_{p}$. where $N_{p}$ is the number of trapped
particles. More recent experiments [25] on two particles trapped in two
tweezers found only cavity-mediated coupling was significant. With some
caveats ( including the assumption that relative tweezer phases are
controled), intraparticle and interparticle coupling takes the same form, so
whether we consider $\operatorname*{N^{tot}}=3N_{p},2N_{p}$ or
$\operatorname*{N^{tot}}=N_{p}$ does not affect our results. For the
illustrations, we mainly focus on parameters relevant to recent experimental
set-ups that that allowed quantum cooling in one and two modes [16, 17, 18] of
a single particle. Here, the $x_{j},y_{j}$ degrees of freedom are typically at
much higher frequency, and much more strongly coupled than $z_{j}$. Without
loss of generality, here we mainly illustrate behavior
$\operatorname*{N^{tot}}=2N_{p}$ or $\operatorname*{N^{tot}}=N_{p}$ as we have
investigated 2D mode dynamics and force sensing for $N_{p}=1$ in the $x-y$
plane experimentally, in weak coupled regimes or free space [19, 20, 22, 23].

We calculate the spectra by full numerics with the linear amplifier model
(LAM) and well as an effective single mode analytical model that gives
indistinguishable results thus offers a versatile tool for experimental
analysis. In Fig.(1b) we show that the cavity output exhibits the very broad
spectral feature (collective bright mode or CBM) that is also seen in the
degenerate case [28]. But we introduce and investigate a feature unique to
these not-perfectly-identical levitated nanoparticle arrays, the mechanical
mode comb (MC), that is the central focus of this work.

The comb shows orders of magnitude variability in the peak widths. Calculation
of the variances of $\hat{q}_{j}(\omega)$ for individual modes (only possible
with the LAM), shows however that the modes are mostly thermalised , with
phonon occupancies $n_{j}=k_{B}T_{j}/(\hbar\omega_{j})\sim n$ that are of
similar order. Because of hybridisation, there is not a one to one mapping
between each peak with a single mode. The comb also shows the interesting
feature that peaks move to mostly fill ‘gaps’ in the teeth of the unperturbed
comb. This is clearest for pairs (middle panel): one sees that the peaks
broaden and are displaced so as to fill the spectral gap. In Fig.1(b),
$g/2\pi=40$ kHz while $\Delta\omega=100$kHz and the (red) detuning
$-\Delta=1.5\omega_{0}$. We consider the sideband resolved regime and take
$\omega_{0}\simeq 1.5\kappa$.

Figure 3: (a) Illustration showing that the CBM/MC features in
$\operatorname*{N^{tot}}>2$ systems are the multi-particle analogue of the
bright-dark mode state spectra analysed in [20] and investigated in recent
experiments [21, 18] with $N_{p}=1$. Compares the behavior between
$\operatorname*{N^{tot}}=2$ and $\operatorname*{N^{tot}}=6$. The CBM is the
lower frequency branch of an avoided crossing of width
$2\sqrt{N^{\text{t}ot}}g$. The MC in turn are a feature that results from
slightly lifting the degeneracy of dark modes. (b) The phonon occupancies
depend on the unperturbed comb spacing. For the one-particle 2D $x-y$ case,
quantum cooling was shown to only be possible in a Goldilocks zone
$\sqrt{\kappa\Gamma/8}\apprle
g\apprle\sqrt{\kappa/(8\Gamma)}|\omega_{y}-\omega_{x}|$ [20] where the
coupling $g$ is strong enough for sufficient back-action cooling, but weak
enough for dark modes to be suppressed by mixing with the bright mode, shown
experimentally in [18]. The multimode case shows similar behavior. We set the
optomechanical coupling $g_{j}\equiv g=40$ kHz for all modes
$j=1,{\textrm{N}^{\text{t}ot}}$, pressure $P=10^{-7}$ mbar. $n$ denotes the
mean phonon occupancies of the comb.

Theoretical model: The linearised Hamiltonian gives rise to a set of
$\operatorname*{N^{tot}}+2$ Langevin equations. For the modes of interest:

$\displaystyle\hat{a}$
$\displaystyle=\textrm{i}\chi_{c}(\omega)[\sum_{j}g_{j}\hat{q}_{j}(\omega)+\sqrt{\kappa}\tilde{a}_{\textrm{in}}]$
(1) $\displaystyle\hat{q}$
${}_{j}=\textrm{i}g_{j}[\chi_{j}(\omega)-\chi^{*}_{j}(-\omega)](\hat{a}(\omega)+\hat{a}^{\dagger}(\omega))+\sqrt{\Gamma}\hat{X}^{th}$

where $X^{th}$ are thermal noise operators (see SuppInfo) corresponding to the
$T=300$K Gaussian bath due to Brownian motion of background gas molecules. We
can take the corresponding mechanical damping $\Gamma_{j}\equiv\Gamma$,
related to gas pressure $P$ as $\Gamma\sim 15PR^{2}/M{\bar{v}}$ with $R,M$ the
nanoparticle radius, mass and ${\bar{v}}$ the mean gas velocity. The Brownian
bath has phonon occupancies of $n_{B,j}=kT_{B}/(\hbar\omega_{j})$.

The linear amplifier model (LAM) is nothing more than the full numerical
solution of the above Langevin equations by matrix inversion. Above we have
introduced the optical (cavity mode) susceptibility
$\chi_{c}(\omega)=\left[-i(\omega+\Delta)+\frac{\kappa}{2}\right]^{-1}$ and
mechanical susceptibilities
$\chi_{j}(\omega)=\left[-i(\omega-\omega_{j})+\frac{\Gamma}{2}\right]^{-1}$.

For the effective 1D model, we take $\hat{Q}\equiv\sum_{j}\hat{q}_{j}$. We can
then show that the power spectral density (PSD) for the collective motion is
written:

$S_{\hat{Q}\hat{Q}}(\omega)=|M_{tot}(\omega)|^{-2}\left(\Gamma
S_{tot}^{\text{th}}+\kappa
g_{j}^{2}|\eta_{tot}(\omega)|^{-2}|\chi_{c}(\omega)|^{2}\right)$ (2)

where
$S_{tot}^{\text{th}}=\sum_{j}\left[|\chi_{j}(\omega)|^{2}(n_{B,j}+1)+|\chi^{*}_{j}(-\omega)|^{2}n_{B,j}\right]$
and where $M_{tot}=1+g^{2}\eta_{tot}(\omega)\eta_{c}(\omega)$ , with
$\eta_{tot}(\omega)=\sum_{j}\chi_{j}(\omega)-\chi^{*}_{j}(-\omega)$. We can
readily write the exact corresponding expression for a heterodyne PSD. The
experimentally measured displacement PSD of the collective coordinate will be
given by $S^{het}(\omega)\approx
g^{2}|\chi_{c}(\omega)|^{2}S^{meas}_{\hat{Q}\hat{Q}}(\omega)$.

By inspection, we can see that both the CBM feature as well as all the
complexity and orders of magnitude variability of the widths of the Mechanical
comb (MC) plotted in Fig.1 are fully determined the rather simple analytical
prefactors $|M_{tot}(\omega)|^{-2}|\chi_{j}(\omega)|^{2}$. But while
$|\chi_{j}|$ are simply narrow functions (width $\Gamma$), sharply peaked
about $\omega=\omega_{j}$, it is $|M_{tot}(\omega)|^{-2}$ that plays the
significant role: it is the collective-mode equivalent of the optical back-
action prefactor that accounts for cavity cooling and optical spring effects
in single-particle optomechanics (further details in the SuppInfo).

One caveat is that we assume that the relative phase of the different tweezer
beams is controllable and hence neglect this effect. While the geometric modes
of a single nanoparticle are unaffected, arbitrary dephasing between different
tweezers has not been investigated.

Sensing external forces with the comb: it is of interest to consider the
differences between using the MC, for detection of an external force centered
on an unknown frequency $\omega\approx\omega_{ext}$ and (say) an array of
independent nanoparticles in $\operatorname*{N^{tot}}$ tweezers, in free
space, also in an interval of frequencies
$\omega_{j}\in[\omega_{m}\pm\Delta\omega/2]$ containing
$\omega\approx\omega_{ext}$; and with similar damping from feedback cooling.
One notable difference is illustrated in Fig.2, which is the remarkable
tendency of the MC to autonomously repair the loss of some of its teeth;
particle loss is a perennial problem in these set-ups and the problem of
efficient reloading remains to be fully addressed. In this respect, the
mechanical mode comb, in the $g\sim\Delta\omega$ regime, may add some
robustness to particle loss.

A full study of the relative sensitivity remains for the future, but here we
discuss potential sensitivity enhancements. A key point to consider will be
whether the external unknown force acts acts mainly or exclusively on a single
particle, or all particles equally. The former case might arise because the
particles are naturally charged, so unless this is controlled experimentally,
the most highly charged nanoparticles would be most sensitive to an external
electromagnetic force. The latter case might arise from, say, a gravitational
force, as all nanoparticles have comparable masses. We note there is, in
general, no one to one mapping between a single frequency and a single
particle or mode of the MC. In [23] the case where the force acts on more than
one mode of a single particle was investigated experimentally, including for
the case of a stochastic force, in order to measure the orientation of an
external force. For now, here we focus on the $\operatorname*{N^{tot}}=N_{p}$
case.

For the independent particle array, it is usual to consider the relation
$S_{q_{j}q_{j}}(\omega)=|\chi^{(j)}(\omega)|^{2}S^{ext}_{FF,j}(\omega)$, the
relation between the displacement PSD and the force spectrum, in terms of the
appropriate single-particle susceptibility. The latter is broadened by a
damping $\Gamma_{j}$ that includes the effect of feedback cooling, for
instance, but remains peaked at $\omega=\omega_{j}$. For the MC we can write:

$|\chi^{(j)}_{eff}(\omega)|^{2}\equiv|M_{tot}(\omega)|^{-2}|\chi_{j}(\omega)|^{2}$
(3)

where $\chi^{(j)}_{eff}(\omega)|^{2}$ includes the dampings and frequency
shifts due to the optical back action function $M_{tot}$. In both cases the
effect of many particles on the detected heterodyne signal detecting single-
particle forces is still additive and
$S^{het}(\omega)/(g^{2}|\chi_{c}(\omega)|^{2})\approx\sum_{j}|\chi^{(j)}_{eff}(\omega)|^{2}S^{ext}_{FF,j}(\omega)$.
A harmonic force restricted to a single particle selects a single component in
the sum. A harmonic force acting on several $j$ will select mode components
that have a significant amplitude near $\omega\approx\omega_{ext}$.

A force acting on all particles equally is a different matter. It would have
$S^{het}(\omega)/(g^{2}|\chi_{c}(\omega)|^{2})\approx
S^{ext}_{FF}(\omega)|\left(\sum_{j}\chi^{(j)}_{eff}(\omega)\right)|^{2}$ so
there is interference between the $\operatorname*{N^{tot}}$ components, and a
resulting sensitivity enhancement relative to the $\operatorname*{N^{tot}}$
independent particle case is expected, for overlapping components, with
constructive interference, in the collective mode case.

Figure 4: Homodyne signal, illustrating that the width of the squeezing region
scales as ${\text{N}^{tot}}$, and for certain quadratures, the mechanical comb
can sink below the quantum shot noise noise floor. Parameters of Fig.1b(ii)
and homodyne angle $\pi/4$, but lower pressure, so the effect of the quantum
shot noise, as opposed to the thermal noise, is predominant.

CBM and dark-bright modes  In [28] , the sensitivity enhancement of the CBM
was analysed theoretically, for force sensing in the fully degenerate case
(with no comb). The CBM is still present for the fully degenerate case, and an
$\operatorname*{N^{tot}}$ enhancement for forces acting on all particles
equally was reported. In Fig.3 we show that the CBM is the multiple-mode
generalisation of the bright mode that was analysed in [20] then
experimentally investigated in [21] for $N_{p}=1$. In strong-coupling regimes
the CBM then appears, above resonance ($-\Delta\gtrsim\omega_{j}$), as the
low-frequency branch of a wide avoided crossing of width
$2g\sqrt{\operatorname*{N^{tot}}}$.

Figure 5:  (a) Shows that collective bright mode (CBM) formation and the MC
may already be investigated with current $N_{p}=2$, 2D experiments with modes
$x_{1},y_{1},x_{2},y_{2}$. As $\Delta\to\infty$, 4 narrow peaks at the tweezer
frequencies are seen. As $-\Delta$ is lowered, the lower frequency peak peels
off to form the CBM. The MC is also cooled and for sufficient $\delta$
(unperturbed frequency spacing) can reach quantum regimes. (b) The tendency of
the MC to fill frequency gaps is also shown using paired unperturbed
frequencies.

In [19] , it was shown that hybridisation in the $N_{p}=1$ case could be
described in terms of a cavity-mediated effective coupling term. For trapping
at the node of the cavity light, there is no direct coupling between two modes
$j,k\equiv x_{p},y_{p}$ for particle $p$, 2D, and the cavity-mediated coupling
takes the form
$\mathcal{G}_{jk}(\omega)=\frac{i\eta_{j}(\omega)}{M_{j}(\omega)}\left[i\eta_{c}(\omega)g_{j}g_{k}\right]$
[19] . We can then write
$\hat{q}_{j}\equiv\hat{q}^{(1D)}_{j}+\mathcal{G}_{jk}(\omega)\hat{q}_{k}$
where $\hat{q}^{(1D)}_{j}$ is the independent particle solution. For centre of
mass motion, hybridisation has then a geometric interpretation in terms of a
spatial rotation [19, 20] of the modes. In [22] , the spatial rotation angle
was measured experimentally as this allows for detection of the direction, as
well as magnitude of an external force. The multimode case will be more
complex, but the cavity-mediated couplings take precisely the same form for
coupling between modes of different particles.

In Fig.4 we plot the homodyne spectra, with the quantum shot noise normalised
to 1. The figure shows that the ponderomotive squeezing effect associated with
the CBM has a width that increases linearly with $\operatorname*{N^{tot}}$.
For some homodyne angles it forms a basin that encompasses the MC, partly
lowering the comb below the quantum shot noise floor.

Finally, in Fig.5 we show that the CBM and MC features, including the filling
of gaps are already demonstrated in a 2D, $N_{p}=2$ system with two tweezers,
which is already accessible to state of the art experiments with the coherent
scattering set-up.

In conclusion, we presented an analysis of collective effects and spectral
features of arrays of nanoparticles levitated in a cavity. We introduce a not
previously investigated spectral feature, the mechanical mode comb (MC) and
explore its potential for sensing external forces over a band of frequencies
rather than a single resonant frequency.

Acknowledgements: The authors acknowledge funding from the Engineering and
Physical Sciences Research Council (EPSRC) Grant No. EP/W029626/1 and an
EPSRC-funded Quantum technology fellowship (grant number EP/S021582/1) for MR.

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