# Decoupled sound and amplitude modes in trapped dipolar supersolids

J. Hertkorn  0000-0002-5377-9981<EMAIL_ADDRESS>5\.
Physikalisches Institut and Center for Integrated Quantum Science and
Technology,Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
P. Stürmer  0000-0002-1447-8545 Division of Mathematical Physics and NanoLund,
LTH, Lund University, Box 118, SE-221 00 Lund, Sweden  K. Mukherjee
0000-0002-1182-2785 Division of Mathematical Physics and NanoLund, LTH, Lund
University, Box 118, SE-221 00 Lund, Sweden  K.S.H. Ng  P. Uerlings  F.
Hellstern  L. Lavoine  5\. Physikalisches Institut and Center for Integrated
Quantum Science and Technology,Universität Stuttgart, Pfaffenwaldring 57,
70569 Stuttgart, Germany  S.M. Reimann  0000-0003-1869-9799 Division of
Mathematical Physics and NanoLund, LTH, Lund University, Box 118, SE-221 00
Lund, Sweden  T. Pfau  R. Klemt  0009-0002-3231-447X<EMAIL_ADDRESS>stuttgart.de 5\. Physikalisches Institut and Center for Integrated Quantum
Science and Technology,Universität Stuttgart, Pfaffenwaldring 57, 70569
Stuttgart, Germany

###### Abstract

We theoretically investigate elementary excitations of dipolar quantum gases
across the superfluid to supersolid phase transition in a toroidal trap. We
show how decoupled first sound, second sound, and Higgs modes emerge by
following their origin from superfluid modes across the transition. The
structure of these excitations reveals the interplay between crystal and
superfluid oscillations. Our results unify previous notions of coupled
Goldstone and Higgs modes in harmonic traps, allowing us to establish a
correspondence between excitations of trapped and infinitely extended
supersolids. We propose protocols for selectively probing these sound and
amplitude modes, accessible to state-of-the-art experiments.

Spontaneous symmetry breaking permeates our understanding of nature and is
central to concepts ranging from the emergence of elementary particles to
phase transitions in condensed matter [1]. These phase transitions are often
signaled by a characteristic spectrum of Goldstone and Higgs collective
excitation modes across criticality. In superfluids at zero temperature, the
Goldstone mode emerges as superfluid sound from breaking the $U(1)$ symmetry
[2, 3]. The coupling to another (non-superfluid) component can then lead to a
second sound mode. This concept historically originated from a hydrodynamic
two-fluid model describing the out-of-phase oscillation between thermal and
superfluid components of Helium [4, 5, 6]. In a supersolid [7, 8, 9], the
coexistence of superfluidity and density modulation results in a second
Goldstone mode at zero temperature and a novel type of second sound.
Corresponding theoretical investigations of sound in Helium have a long
history [10, *Prokofev2007, *Balibar_2008], yet in experiments to date the
supersolid phase of helium remains elusive [13, *Kim2004b, *Balibar2010,
*Chan2013]. Supersolids were first realized in the context of ultracold gases,
using BECs with spin-orbit coupling [17, 18, 19] or in multimode optical
cavities [20], as well as dipolar quantum gases [21, 22, 23]. In the latter
system, indications of a Goldstone mode associated with translational symmetry
breaking were observed [24, 25]. Despite considerable efforts, the
identification of second sound, its relationship with the Goldstone mode and
the development of a consistent picture describing the different sound modes
has proven challenging, both experimentally and theoretically [26, 24, 21, 27,
28, 29, 30, 31]. In addition, hydrodynamic theories, which can identify the
two sound modes, are unable to capture gapped modes such as the Higgs mode
[32, 33, 34, 35], which is yet to be observed experimentally in a supersolid.
The search is typically complicated by harmonic trap confinement, where
elementary excitations couple to center-of-mass (COM) modes. Here, toroidal
potentials [36, 37, 38, 39, 40] are a better choice: The continuous rotational
symmetry and periodic boundary conditions along the azimuthal direction link
experimentally achievable finite-size supersolids with those in the bulk
addressed by mean-field [41, 35] and quantum Monte Carlo methods [42, 43].

In this work, we show how first sound, second sound, and Higgs amplitude modes
emerge in dipolar supersolids by calculating the elementary excitation
spectrum in a toroidal trap across criticality. We classify the spatial
symmetries of the eigenmodes using group theory and show how a quasimomentum
$\tilde{q}$ is assigned to them. This establishes a direct correspondence
between the excitations of finite-sized systems and infinite systems in which
the description using band structures becomes meaningful. We show that the
second (first) sound is an out-of-phase (in-phase) oscillation between
superfluid and crystal components and that both sound modes emerge from the
superfluid sound branch near the roton momentum $q_{\mathrm{rot}}$ which
defines the crystal periodicity. At the critical point, $q_{\mathrm{rot}}$
becomes the $\tilde{q}=0$ point of the emergent Brillouin zone, at which the
superfluid sound mode splits into the $\tilde{q}=0$ instance of the second
sound branch – the zero-energy Goldstone mode – and a separate isolated Higgs
amplitude mode. The analysis of these modes allows us to design experimental
protocols for probing single elementary excitations of the supersolid and more
generally, to define selection rules of the supersolid excitation spectrum.

We numerically calculate the ground state (GS) of our atomic dipolar system
using the extended Gross-Pitaevskii (eGPE) framework [44, 45, 46, 47, 48],
which includes quantum fluctuations to leading order in a local density
approximation [49, 50, 51, 52, 46]. The contact and dipolar two-body
interaction strengths of the $N$ atom system are controlled by the s-wave
scattering length $a_{s}$ and dipolar length
$a_{dd}=\mu_{0}\mu_{d}^{2}M/12\pi\hbar^{2}$, respectively. Here $\mu_{d}$
denotes the magnetic moment and $M$ the atomic mass. We introduce a toroidal
trap geometry by setting the external trapping potential to
$V_{t}=M\omega_{r}^{2}\left[(\rho-\rho_{0})^{2}+\lambda^{2}z^{2}\right]/2$,
where $\rho_{0}$ denotes the radius of the torus, $\omega_{r}$ the radial trap
frequency and $\lambda$ the aspect ratio between transversal and radial
confinement [37]. We perform the 3D GS search using imaginary time evolution
[53] and conjugate gradient [54, 44, 55, 56] techniques. On top of the GS, we
study the elementary excitations within the Bogoliubov-de Gennes (BdG)
formalism described in detail e. g. in Refs. [44, 26]. It is based on a linear
expansion in the small amplitude parameter $\alpha$ of the wavefunction
${\psi(\boldsymbol{r},t)=\left\\{\psi_{0}(\boldsymbol{r})+\alpha\left[u_{i}(\boldsymbol{r})e^{-i\omega
t}+v_{i}^{\ast}(\boldsymbol{r})e^{i\omega t}\right]\right\\}e^{-i\mu
t/\hbar}}$, with $\psi_{0}$ the GS wavefunction at a chemical potential $\mu$
and $i$ labeling the $i^{\text{th}}$ excitation. With this ansatz, we arrive
at a system of linear BdG equations. Solving them numerically, we obtain the
Bogoliubov amplitudes $u_{i},\,v_{i}$ as well as the linear combination
$f_{i}=u_{i}+v_{i}$ which relate to the density fluctuations of the GS density
$n(t)=|\psi_{0}|^{2}+2\alpha\delta n_{i}\cos(\omega_{i}t)$, where $\delta
n_{i}=\psi_{0}f_{i}$. Due to the finite system size, the spectrum of
elementary excitations is discrete.

Figure 1: Elementary excitations across the superfluid-to-supersolid phase
transition. (a) GS density ${n(x,y,z=0)}$ and Bogoliubov mode patterns
${f_{i}(x,y,z=0)}$ in arbitrary units for a selection of modes in the
superfluid (top row) and supersolid (bottom row) regime. Atoms flow in the
direction of the gradient of $f_{i}$, where red indicates a depletion and blue
an increase of atomic density. (b) Energies (in units of the radial trap
frequency $\omega_{r}$) of the BdG excitations as a function of
$\epsilon_{dd}$. We label the modes by their circulation quantum number $m$
and show the mode pattern on the superfluid side for the three lowest $m$. We
identify a branch of superfluid sound modes (dark blue) and roton modes
(purple) on the superfluid side of the transition as well as the first sound
branch (green), second sound branch including Goldstone mode (light blue), and
Higgs modes (orange) on the supersolid side. A lighter color corresponds to a
higher quasimomentum. (c) Speed of sound associated with the first sound as
well as the second sound branches of (b) across the transition in units of the
sound velocity at the phase transition $c_{0}\simeq 2.2\,\mathrm{mm/s}$
(colors as in (b)). The inset shows the superfluid fraction, numerically
calculated based on the non-classical rotational inertia.

For a direct comparison to the experimental platform of strongly dipolar
${}^{162}\mathrm{Dy}$ atoms in optical dipole traps, we choose parameters
yielding solutions with ${n_{d}=7}$ droplets [37]. Scaling relations allow for
a direct transfer of the results to systems with other dipolar strengths, atom
numbers, and trapping strengths [57, 28, 58, 59]. We assume a torus radius
$\rho_{0}=$1\text{\,}\mu\mathrm{m}$$, radial trap frequency
$\omega_{r}/2\pi=$1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$, aspect ratio
$\lambda=1.7$, particle number $N=$5\text{\times}{10}^{3}$$, and dipolar
length $a_{dd}=130a_{0}$. We calculate the GS for s-wave scatting lengths
between $a_{s}=59a_{0}$ and $a_{s}=67a_{0}$, corresponding to relative dipolar
strengths $\epsilon_{dd}=a_{dd}/a_{s}\in\left[1.95,2.20\right]$. As a function
of $\epsilon_{dd}$ we find the GS to be either superfluid (SF)
($\epsilon_{dd}\lesssim 2.09$) or supersolid (SS). Upon further increasing
$\epsilon_{dd}$ the SS eventually develops into a state of $n_{d}$ isolated
droplets.

As shown in the inset to Fig. 1(c), we classify the GSs based on the SF
fraction. It is unity in the SF and continuously drops in the SS towards
higher $\epsilon_{dd}$. The spectrum of the low energy modes across the SF to
SS phase transition is shown in Fig. 1(b), which constitutes one of the main
findings of this work. In the SF, each mode is twice degenerate due to the
cylindric symmetry [44] and the energy of the modes increases monotonically
with the circulation number $m$. For a sufficiently large torus radius, where
the curvature of the torus is much smaller than the typical length scales of
the mode patterns, we can interpret this as a quantization of the one-
dimensional momentum scale $q_{m}=2\pi m/\Theta$, where $\Theta$ is the
circumference of toroidal density distribution 111Note that $\Theta\approx
2\pi(1.1\rho_{0})$ as the dipolar interaction effectively increases the radius
of the SS compared to the actual trap radius [34]. In the long-wavelength
limit, we expect a phononic dispersion relation $\omega_{m}=c_{s}q_{m}$, with
the SF speed of sound $c_{s}$. Due to the influence of the dipolar
interactions (for the given system size) only the lowest mode ($q=q_{1}$)
falls into this linear regime for the lowest $\epsilon_{dd}$ shown here. Upon
increasing $\epsilon_{dd}$, a number of states with higher $m$ decrease in
energy, and can be identified as the angular roton modes [61, 62, 28, 63].

At the quantum critical point ($\epsilon^{crit}_{dd}\approx 2.09$) the two
$m=7$ modes (degenerate before the transition) touch zero excitation energy
and trigger the phase transition from SF to SS, splitting into what we
identify as a Higgs and a Goldstone mode 222For brevity, in the following we
will refer to the $\tilde{q}=0$ excitations of the SS Higgs and Goldstone
branch as Higgs and Goldstone mode, respectively.. The nature of these modes
can be best understood by considering the BdG mode patterns $f_{m}(x,y)$
displayed in Fig. 1 (a). As both modes emerge from the $m=7$ roton mode, the
density fluctuation patterns share the periodicity of the GS. The Goldstone
mode pattern is phase-shifted by half a period with respect to the density,
while the Higgs pattern is in phase. The two modes correspond to a pure phase
and amplitude mode, with striking clarity compared to harmonic 1D systems
where phase and amplitude oscillations are always coupled. Due to the periodic
boundary conditions, the Higgs and the Goldstone modes are gapless at
criticality.

In the SS regime, the GS has developed a periodic density modulation.
Accordingly, we can classify the modes by a quasimomentum $\tilde{q}$ folded
back into the first Brillouin zone and identify the Higgs and Goldstone modes
with $\tilde{q}_{0}=0$. The SS has three distinct excitation branches: The
Higgs branch, with higher-lying $\tilde{q}>0$ modes in addition to the
$\tilde{q}_{0}$ mode, and the first and second sound branches, as indicated in
Fig. 1. The periodic boundary conditions allow us to identify these three
branches predicted for infinitely extended SSs [41, 65] in finite and
experimentally feasible systems. The Higgs branch consists primarily of
amplitude oscillations and rises quickly in energy. The two sound modes show a
weaker dependence on $\epsilon_{dd}$, with the first sound branch rising in
energy while the second sound branch is decreasing, consistent with a smaller
SF fraction [33, 34].

Based on the circulation number $m$ and the structure factor (to be discussed
below), we can attribute the quasimomentum $\tilde{q}_{m}=2\pi m/\Theta$, with
$m=1,2,3$, to the respective modes. This assignment is supported by employing
group theory. The SF GS has the symmetry of the point group $C_{\infty,v}$
describing its invariance under rotation and reflection. At the SF-SS
transition this symmetry is broken, $C_{\infty,v}\rightarrow C_{7,v}$ [66],
where $C_{7,v}$ contains the irreducible representations $A_{1},\ A_{2},\
E_{1},\ E_{2},\ E_{3}$, with symmetries of $A_{1,2}$ corresponding to an $m=7$
angular roton mode, and of $E_{j}$ to one with $m=j$ compatible with
quasimomentum $\tilde{q}_{j}$. The allowed quasimomenta, purely based on
symmetry arguments, are thus $\tilde{q}=0,1,2,3$. This analysis directly
generalizes to an arbitrary droplet number $n_{d}$ through subduction of
$C_{\infty,v}$ to $C_{n_{d},v}$ [67, 68, 66].

As an example, the mode pattern of the $m=6$ and $m=8$ modes before and after
the transition are displayed in Fig. 1 (a). In the SS these modes give rise to
the second and first sound at quasimomentum $\tilde{q}_{1}$ and show a
reflection antisymmetry described by $E_{1}$. We note that in compliance with
our observations, the dispersion relation of the two sound modes is expected
to be linear for low momenta, before flattening off close to the edge of the
Brillouin zone. The Higgs mode energy is a convex function of the
quasimomentum. Albeit similar excitation spectra were previously reported for
finite harmonically trapped systems [26, 30, 21, 23, 24], their interpretation
in terms of different sound branches was hindered by the absence of well
defined quasimomenta and Brillouin zones due to the coupling of linear and COM
motion. In contrast, in the toroidal supersolid investigated here, the Higgs
modes branch keeps its character even relatively far away from the transition
point, as it is not coupled to modes with a different quasimomentum. This
allows us to estimate the effective mass of the Higgs excitation [66]. While
the $\tilde{q}=0$ Higgs mode is exactly zero at the phase transition, even in
the few-droplet limit, the finite system size results in a finite Higgs mass
at criticality.

Using the energies of the the respective $\tilde{q}_{1}$ modes we calculate
the speed of sound $c_{i}=\omega_{i}/\tilde{q}_{1}$, where $i=1,2$ corresponds
to first and second sound, respectively. The results are shown in Fig. 1 (c)
and agree with a recent description in terms of a hydrodynamic model [33, 34,
66]. For our parameters, the $m=8$ modes drop below the $m=1$ modes in the SF,
causing them to evolve into a first sound and excited Higgs mode in the SS.
This reordering is consistent with an avoided crossing of Higgs and first
sound branches described in an infinite system at the same density [41]. The
resulting discontinuity between the SF and first sound at the phase transition
point together with the continuous SF fraction, is consistent with the
expectation of a second-order phase transition at intermediate densities in
the corresponding bulk system.

Figure 2: Spatial and temporal character of sound and amplitude modes (a-c)
Angular spectrum of the dynamic structure factor
${\tilde{S}(k_{\phi},\omega)}$ across the phase transition. We normalize
${\tilde{S}}$ at each $\omega_{i}$ for visual clarity. Lines through the most
prominent peaks of $\tilde{S}$ are guides to the eye. (d-f) The upper panel
shows cuts of the density along the torus $n_{1D}(\phi,t)$ at ${t/T=0,\ 1/2}$
during the time evolution, where ${T=2\pi/\omega}$. The lower panels show the
full density evolution $n_{1D}(\phi,t)$ for two periods $T$ of the Higgs mode,
second sound, and first sound at $\tilde{q}_{1}$ from (c). The Higgs mode is
an amplitude oscillation mediating a periodic change in the superfluid
fraction through superfluid flow into and out of the droplets. The second
(first) sound is characterized by an out-of-phase (in-phase) oscillation
between the crystal structure and the superfluid background.

Our approach allows us to develop a microscopic and intuitive understanding of
the character of these elementary excitations (Fig. 2). To relate our findings
to quantities which can be directly measured in experiments [63], we define
the angular spectrum of the structure factor
${\tilde{S}(k_{\phi},\omega)=\int\\!\mathrm{d}\phi\,e^{ik_{\phi}\phi}\tilde{S}(\phi,\omega)}$,
shown in Fig. 2(a)-(c), where $\tilde{S}(\phi,\omega)$ is readily obtained
from a Fourier transform of the density fluctuation patterns $\delta
n_{i}=f_{i}\psi_{0}$ [27] in cylinder coordinates at $z=0$ along the torus
$r\simeq\rho_{0}$ [66]. A mode with an $m$-fold angular symmetry corresponding
to an excitation at momentum $q_{m}$ shows up as a peak in
$\tilde{S}(k_{\phi},\omega)$ at ${k_{\phi}=2q_{m}/q_{1}=2m}$. In the SF regime
(Fig. 2(a)-(b)), $\tilde{S}$ reproduces a discretized version of the known
Bogoliubov dispersion relation, which is linear at small $k_{\phi}$ and
quadratic towards larger $k_{\phi}$. Near the phase transition, a roton
minimum develops. On the SS side (Fig. 2(c)), the SF branch splits up into the
first sound, second sound, and Higgs branches.

We find that for the first sound branch, the first mode shows peaks at
$k_{\phi}=2,\ 14,\ 16$, the second mode at $k_{\phi}=4,\ 14,\ 18$, and the
third mode at $k_{\phi}=6,\ 8,\ 14$. The peak at $k_{\phi}=14$ corresponds to
a modulation at the spatial frequency of the crystal structure and the other
peaks can be mapped to a quasimomentum of $|\tilde{q}_{m}|$ with $m=1,2,3$.
This assignment is consistent with a $m-$fold rotational symmetry of the
excitation pattern. In contrast, for the second sound branch, the peak at
$k_{\phi}=14$ is absent close to the phase transition for all modes. Instead
the main peaks are at $12$ ($m=1$), $10$ ($m=2$), and $(6,8)$ for $m=3$. While
the character of these modes is strikingly different from the first sound they
can be mapped to the same symmetry and quasimomentum. The structure of the
Higgs modes similarly maps to a quasimomentum of $|\tilde{q}_{m}|=0,1,2$
333Apart from the contributions discussed so far, there is a low-frequency
contribution at $k_{\phi}=1$ which does not correspond to a specific
quasimomentum but rather a slow modulation of the envelope of the excitation
pattern most likely caused by residual coupling between the two degenerate
modes within our numerical framework..

It is instructive to study the linearized time evolution shown for the first
mode on each branch, respectively, in Fig. 2(d)-(f). In Fig. 2(d), the
amplitude character of the Higgs mode becomes apparent. Atoms flow from the SF
background into the droplets and vice versa. In contrast, the second (first)
sound modes are an out-of-phase (in-phase) oscillation between crystal
compression and SF flow (Fig. 2(e)-(f)). As the crystal compresses towards one
node on the torus, the SF density flows towards the opposite (same) direction,
reducing (increasing) the density where the droplet spacing is smaller. Due to
the dipolar repulsion of the droplets, the compression of the crystal is
associated with an energy cost, which is reduced (increased) by the out-of-
phase (in-phase) SF flow, providing an intuitive picture of the second sound
being slower than the first sound. This zero temperature mode structure is
reminiscent of the finite temperature in- and out-of-phase oscillations
between SF and normal fluid flow in a (non-dipolar) SF within the two-fluid
model [5, 6]. This a-posteriori motivates our interpretation of the
corresponding modes in terms of first and second sound.

Figure 3: Spectroscopy of supersolids (a) Dominant PCs after applying an
excitation scheme (i)-(ii) (see main text) to the GS at $\epsilon_{dd}\simeq
2.1$. (b) Fourier spectra $|\mathcal{F}_{w}(\omega_{\mathrm{PC}})|^{2}$ of the
dominant PC weights $w(t)$. Vertical lines in the spectra indicate BdG
energies (Fig. 1(b)). All modes are individually excited with $>90\%$
selectivity. All PCs show above $95\%$ overlap with the density fluctuation
pattern obtained from the BdG analysis (Fig. 1(a)) and oscillate at
frequencies corresponding to the BdG energies.

Understanding the spatial and temporal character of these elementary
excitations as developed in this work helps devise simple and experimentally
feasible schemes to selectively excite individual modes (Fig. 3). We employ
principal component analysis (PCA) [27], to assess our excitation scheme. PCA
is a model-free method giving access to the principal components (PCs) and
their weights $w(t)$ quantifying their contribution to the spatial variance of
the time evolution. We define the selectivity $s$ as the amount of variance
described by the time evolution for a given PC. The overlap between the
strongest PC and the corresponding BdG mode is above $95\%$ in all cases
discussed here.

The Higgs mode (Fig. 3(i)) can be selectively excited by modulating the s-wave
scattering length, which can be done conveniently in state-of-the-art
experiments by modulating a magnetic offset in the vicinity of a Feshbach
resonance. This scheme directly couples to the (symmetric) $A_{1}$ symmetry
and allows us to excite the $\tilde{q}=0$ Higgs mode with a selectivity of
$>99\%$.

The Goldstone mode corresponds to a rotation of the system at zero energy in a
cylindrically symmetric trap. Any breaking of the cylindrical symmetry
elevates this mode to a finite frequency and enables imparting angular
momentum into the system. This can be achieved by deforming and rotating the
trapping potential or by using magnetostirring [70], where the magnetic field
is tilted and rotated. For more details, see [66].

The two sound modes at $\tilde{q}_{1}$ (Fig. 3(ii)) share the symmetry
$E_{1}$. These modes therefore couple to a dipole excitation pattern and we
modulate the trap with ${\Delta V\propto x\sin(\omega t)}$ along a symmetry
axis $x$ of the SS with the frequency $\omega$ matching the energy of the
respective modes. This scheme selectively excites the resonantly driven sound
with only a few percent weight in the other sound mode. This selectivity can
be further increased by optimizing duration, amplitude, and other details of
the modulation scheme.

Importantly, this scheme cannot couple to the modes at $\tilde{q}_{2}$ as
their symmetry $E_{2}$ has no overlap with the symmetry $E_{1}$, while
coupling to the $E_{3}$ mode is possible in principle, but suppressed due to
the frequency detuning. Similarly, a quadrupole (hexapole) excitation scheme
can be used to selectively excite the $\tilde{q}_{2}$ ($\tilde{q}_{3}$) sound
modes with no coupling to modes at $\tilde{q}_{1,3}$ (small coupling to modes
at $\tilde{q}_{1}$) [66]. Based on the symmetry considerations, one can
generalize these observations to a set of selection rules, establishing a
complete toolbox for the spectroscopy of supersolids. The absence of linear
and COM motion coupling ensures that a mode retains its character over time
and a range of interaction strengths and is the basis of such spectroscopy.

In conclusion, we have uncovered the origin of second sound and Higgs
amplitude modes in dipolar supersolids by employing a toroidal trap,
establishing a correspondence between elementary excitations of finite and
bulk supersolid systems. Strikingly, toroidal supersolids in the low-momentum
limit realize an isolated and massive Higgs excitation that should propagate
as a massive quasiparticle in the self-generated periodic structure of the
supersolid. Scattering of the Higgs excitation at an obstacle, such as a weak
link on the torus, and its dispersion in larger tori is the starting point for
studies of massive quasiparticles in supersolids. Our work highlights the
analogy between the second sound in supersolids at zero temperature and second
sound in superfluids at finite temperature, which are out-of-phase
oscillations between superfluid and non-superfluid components. We believe that
future studies considering finite temperature supersolids [71, 72] in toroidal
traps could provide insight into the origin of further sound modes arising
through the presence of two non-superfluid (crystal and thermal) components.

## Acknowledgments

We acknowledge discussions and input in particular from T. Langen, as well as
M. Nilsson Tengstrand at an earlier stage of the project, and also thank T.
Arnone Cardinale, L. Chergui and H. P. Büchler for discussions. P.S., K.M. and
S.M.R. gratefully acknowledge financial support from the Knut and Alice
Wallenberg Foundation (KAW 2018.0217) and the Swedish Research Council (grant
no. 2022-03654). J.H., K.S.H.N., P.U., F.H., L.L., T.P. and R.K. acknowledge
support from the European Research Council (ERC) (grant agreement No.
101019739). J.H. gratefully acknowledges financial support by the Vector
Stiftung (project no. P2021-0114). L.L. acknowledges funding from the
Alexander von Humboldt Foundation. Part of the computations were enabled by
resources provided by the National Academic Infrastructure for Supercomputing
in Sweden (NAISS), partially funded by the Swedish Research Council through
grant agreement no. 2022-06725.

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## Supplementary Material

###  .1 Details on the numerically calculated excitation spectrum

We calculate the GS of the dipolar toroidal system using the eGPE framework
previously described e. g. in [47, 26] and gain access to the excitation
spectrum using the BdG framework described in more detail in e. g. [24, 26,
27]. We have chosen the parameters used in this study to ensure that the
$n_{d}=7$ droplet solution is the ground state for any $\epsilon_{dd}$
considered. However, for larger $\epsilon_{dd}\gtrsim 2.15$, the $n_{d}=8$
solutions become very similar in energy, such that numerically a bistability
between $n_{d}=7$ and $n_{d}=8$ solutions develops. In this work, we restrict
ourselves to $n_{d}=7$. In the spectrum, we focus on the low-energy behavior
of longitudinal excitations along the torus ($\omega<\omega_{r}$). The
Goldstone mode numerically shows a finite energy $\omega/\omega_{r}<0.01$ for
the spatial grid size employed here ($96^{3}$ grid points). We have confirmed
that this energy decreases for finer spatial grids and tighter convergence
criteria, consistent with a gapless Goldstone mode.
We also note that for the torus radius of $\rho=$1\text{\,}\mu\mathrm{m}$$
used in this work, there is still a small residual dipolar coupling across the
torus. We have however explicitly checked by calculating spectra with a scaled
torus size and atom number, that such scaling does not qualitatively change
any of the results presented in this work.

###  .2 Superfluid speed of sound and compressibility

In a hydrodynamic description [2] the speed of sound

$c_{\kappa}=\frac{1}{\sqrt{M\bar{n}\kappa}}$ (1)

is defined based on the compressibility

$\kappa=\frac{1}{\bar{n}^{2}}\frac{\partial\bar{n}}{\partial\mu}$ (2)

where $\mu$ is the chemical potential and $\bar{n}$ is the average density.
The compressibility is a GS property, providing a definition of this speed of
sound independent of the BdG analysis. In the SF, $c_{\kappa}$ coincides with
the SF speed of sound $c_{s}$ [2].

For comparison to our BdG analysis we calculate $c_{\kappa}$ in the SF. To
this end we note that $\bar{n}\kappa$ is independent of the unit of $\bar{n}$,
and so is $c_{\kappa}$. Therefore we can write
${\bar{n}\kappa=\mathrm{d}\ln\bar{n}/\mathrm{d}\mu}$. The average azimuthal
density for a torus with circumference $\Theta$ is defined by
$\bar{n}=N/\Theta$ which is independent of the detailed spatial dependence of
the density $n(x,y,z)$. We find that $\Theta/2\pi$ varies by less than
$0.01\,\mathrm{\mu m}$ in the SF for our parameters, allowing to a good
approximation to calculate $c_{\kappa}$ using
${\bar{n}\kappa\approx\mathrm{d}\ln N/\mathrm{d}\mu}$. To estimate
$\bar{n}\kappa$ we vary $N$ by a few percent, calculate the GS, and obtain
$\mu(\bar{n})$. Using either method we obtain reasonable agreement with the
BdG description, with $c_{\kappa}$ typically $\simeq 20\%$ larger compared to
the BdG speed of sound. Similar deviations have been reported [34] and are
likely due to the finite torus size. Very recent developments on the
hydrodynamic description of supersolids also provide expressions for the
speeds of sound in the SS [32, 33, 34, 35] depending on $c_{\kappa}$, the SF
fraction $f_{s}$, and the layer compressibility modulus $B$.

###  .3 Higgs effective mass

Figure 4: Effective mass of the Higgs mode. The effective mass of the
amplitude mode, calculated based on the lowest two energy levels of the
corresponding Higgs branch, is plotted as a function of $\epsilon_{dd}$. We
observe a finite mass even at criticality and an increase in the effective
mass, when the superfluid fraction decreases towards the regime of isolated
droplets. The vertical gray line indicates the phase transition point.

In previous studies on 1D supersolids in harmonic traps, where the linear
quasimomentum couples to the center of mass momentum, the Higgs character of
the amplitude mode was maintained only very close to the phase transition
[26]. In stark contrast, in the toroidal geometry studied here, we observe the
Higgs modes at $\tilde{q}_{m}=2\pi m/\Theta$ with $m=1,2,3$ to be isolated
over the whole $\epsilon_{dd}$ regime studied in this work. This isolation
allows us to define an effective mass $M^{*}$ of the excitation. The effective
mass describes the dispersion of a localized Higgs quasiparticle wavepacket
excited at some position on the torus. To estimate the mass we define the
energy difference between the $\tilde{q}_{0}$ and $\tilde{q}_{1}$ Higgs mode
as $\hbar\Delta\omega_{H}$ and calculate the effective mass via
$M^{*}=\hbar^{2}\tilde{q}_{1}^{2}/2\hbar\Delta\omega_{H}$. Here, we only use
the lowest two Higgs energy levels as the $\tilde{q}_{2}$ mode is already
close to the edge of the Brillouin zone. The Higgs effective mass is shown as
a function of $\epsilon_{dd}$ in Fig 4. In accordance to its behaviour in bulk
systems, we observe the effective mass to increase when reducing the SF
fraction. In the limit of large system sizes, we expect the effective mass to
approach zero at the phase transition point. Due to the finite size of the
system and correspondingly the finite sampling of allowed quasimomenta, we
observe a

Table 1: Compatible irreducible representations between the angular roton
modes of the superfluid to elementary excitations of the supersolid with
$n_{d}$ droplets when the rotational symmetry is spontaneously broken
$C_{\infty,v}\rightarrow C_{n_{d},v}$. A section of the full table which
continues indefinitely to the right and bottom is shown. The row for the
symmetry $C_{7,v}$ of the ground state in the main text is highlighted. The
pattern emerging in the body of the table towards higher $n_{d}$ along the
rows continues predictably and has a physical interpretation, see main text.

$m$ ($C_{\infty,v}$) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$C_{1,v}$ | $A_{1}$ | $A_{1}+A_{2}$ | $A_{1}+A_{2}$ | $A_{1}+A_{2}$ | $A_{1}+A_{2}$ | $A_{1}+A_{2}$ | $A_{1}+A_{2}$ | $A_{1}+A_{2}$ | $A_{1}+A_{2}$ | $A_{1}+A_{2}$ | $A_{1}+A_{2}$ | $A_{1}+A_{2}$ | $A_{1}+A_{2}$ | $A_{1}+A_{2}$
$C_{2,v}$ | $A_{1}$ | $B_{1}+B_{2}$ | $A_{1}+A_{2}$ | $B_{1}+B_{2}$ | $A_{1}+A_{2}$ | $B_{1}+B_{2}$ | $A_{1}+A_{2}$ | $B_{1}+B_{2}$ | $A_{1}+A_{2}$ | $B_{1}+B_{2}$ | $A_{1}+A_{2}$ | $B_{1}+B_{2}$ | $A_{1}+A_{2}$ | $B_{1}+B_{2}$
$C_{3,v}$ | $A_{1}$ | $E$ | $E$ | $A_{1}+A_{2}$ | $E$ | $E$ | $A_{1}+A_{2}$ | $E$ | $E$ | $A_{1}+A_{2}$ | $E$ | $E$ | $A_{1}+A_{2}$ | $E$
$C_{4,v}$ | $A_{1}$ | $E$ | $B_{1}+B_{2}$ | $E$ | $A_{1}+A_{2}$ | $E$ | $B_{1}+B_{2}$ | $E$ | $A_{1}+A_{2}$ | $E$ | $B_{1}+B_{2}$ | $E$ | $A_{1}+A_{2}$ | $E$
$C_{5,v}$ | $A_{1}$ | $E_{1}$ | $E_{2}$ | $E_{2}$ | $E_{1}$ | $A_{1}+A_{2}$ | $E_{1}$ | $E_{2}$ | $E_{2}$ | $E_{1}$ | $A_{1}+A_{2}$ | $E_{1}$ | $E_{2}$ | $E_{2}$
$C_{6,v}$ | $A_{1}$ | $E_{1}$ | $E_{2}$ | $B_{1}+B_{2}$ | $E_{2}$ | $E_{1}$ | $A_{1}+A_{2}$ | $E_{1}$ | $E_{2}$ | $B_{1}+B_{2}$ | $E_{2}$ | $E_{1}$ | $A_{1}+A_{2}$ | $E_{1}$
$\boldsymbol{C_{7,v}}$ | $\boldsymbol{A_{1}}$ | $\boldsymbol{E_{1}}$ | $\boldsymbol{E_{2}}$ | $\boldsymbol{E_{3}}$ | $\boldsymbol{E_{3}}$ | $\boldsymbol{E_{2}}$ | $\boldsymbol{E_{1}}$ | $\boldsymbol{A_{1}+A_{2}}$ | $\boldsymbol{E_{1}}$ | $\boldsymbol{E_{2}}$ | $\boldsymbol{E_{3}}$ | $\boldsymbol{E_{3}}$ | $\boldsymbol{E_{2}}$ | $\boldsymbol{E_{1}}$
$C_{8,v}$ | $A_{1}$ | $E_{1}$ | $E_{2}$ | $E_{3}$ | $B_{1}+B_{2}$ | $E_{3}$ | $E_{2}$ | $E_{1}$ | $A_{1}+A_{2}$ | $E_{1}$ | $E_{2}$ | $E_{3}$ | $B_{1}+B_{2}$ | $E_{3}$
$C_{9,v}$ | $A_{1}$ | $E_{1}$ | $E_{2}$ | $E_{3}$ | $E_{4}$ | $E_{4}$ | $E_{3}$ | $E_{2}$ | $E_{1}$ | $A_{1}+A_{2}$ | $E_{1}$ | $E_{2}$ | $E_{3}$ | $E_{4}$
$C_{10,v}$ | $A_{1}$ | $E_{1}$ | $E_{2}$ | $E_{3}$ | $E_{4}$ | $B_{1}+B_{2}$ | $E_{4}$ | $E_{3}$ | $E_{2}$ | $E_{1}$ | $A_{1}+A_{2}$ | $E_{1}$ | $E_{2}$ | $E_{3}$
$C_{11,v}$ | $A_{1}$ | $E_{1}$ | $E_{2}$ | $E_{3}$ | $E_{4}$ | $E_{5}$ | $E_{5}$ | $E_{4}$ | $E_{3}$ | $E_{2}$ | $E_{1}$ | $A_{1}+A_{2}$ | $E_{1}$ | $E_{2}$
$C_{12,v}$ | $A_{1}$ | $E_{1}$ | $E_{2}$ | $E_{3}$ | $E_{4}$ | $E_{5}$ | $B_{1}+B_{2}$ | $E_{5}$ | $E_{4}$ | $E_{3}$ | $E_{2}$ | $E_{1}$ | $A_{1}+A_{2}$ | $E_{1}$
$C_{13,v}$ | $A_{1}$ | $E_{1}$ | $E_{2}$ | $E_{3}$ | $E_{4}$ | $E_{5}$ | $E_{6}$ | $E_{6}$ | $E_{5}$ | $E_{4}$ | $E_{3}$ | $E_{2}$ | $E_{1}$ | $A_{1}+A_{2}$

small finite Higgs mass given by ${M^{*}\simeq 0.104M(^{162}\mathrm{Dy})\simeq
0.58M_{0}}$ even at the phase transition, where
${M_{0}=\hbar\tilde{q}_{1}/c_{0}\simeq 29\,\mathrm{u}}$. Here, $M_{0}$ is
defined via the SF speed of sound $c_{0}$ at the critical point resulting in
the convenient dimensionless expression
$M^{*}/M_{0}=\omega_{0}/(2\Delta\omega_{H})$ with
$\omega_{0}=c_{0}\tilde{q}_{1}$.

###  .4 Superfluid fraction

The spontaneous breaking of the continuous rotational symmetry, in addition to
the already broken $U(1)$ symmetry, results in periodic density modulations
and a supersolid phase. A common way to measure the interplay between solid
and SF ’components’ is via the SF fraction $f_{s}$. While frequently
calculated via an upper and and lower bound $f_{s}^{\pm}$ introduced by Legget
[73], our annular geometry permits calculation of the SF fraction via the
system’s moment-of-inertia and response to external rotation as [74, 37, 40,
34]

$\displaystyle f_{s}=1-\lim_{\Omega\to
0}\frac{\langle\hat{L}_{z}\rangle}{NM\langle\rho^{2}\rangle\Omega},$ (3)

where $\langle\rho^{2}\rangle$ is evaluated numerically. $\Omega$ denotes the
angular frequency of rotation (herein, $\Omega=10^{-7}\omega_{r}$). We have
explicitly checked, that the SF fraction obtained this way (shown in Fig. 1 of
the main text) is consistent with the $f_{s}^{\pm}$ Legget bounds.

###  .5 Group theory

Here, we provide some additional details on the group $C_{n_{d},v}$,
describing the modes of a supersolid with $n_{d}$ droplets in a cylindrically
symmetric trap. Our discussion concerning groups closely follows standard
group theory texts [68], to which the interested reader is referred for
further information. Our description extends and generalizes the understanding
obtained for one-dimensional and two-dimensional supersolids in harmonic traps
with small droplet numbers [24, 26, 28].
The $m=n_{d}$ angular roton modes drive the SF to SS phase transition breaking
the continuous rotational symmetry $C_{\infty,v}\rightarrow C_{n_{d},v}$. In
Tab. (1) we tabulate which $m-$angular roton modes are compatible with which
irreducible representation across every point group (up to $m=n_{d}=13$). We
obtain this table by considering that $E_{i}$ transform as $m=i$ angular roton
modes. The periodicity in the pattern in Tab. 1 has a physical interpretation.
When the symmetry is broken $C_{\infty,v}\rightarrow C_{n_{d},v}$ momenta that
were continuously increasing corresponding to the angular roton modes with $m$
up to $\infty$ can be folded back to the first Brillouin zone in which only
momenta up to $m=n_{d}/2$ for even $n_{d}$ and $m=(n_{d}-1)/2$ for odd $n_{d}$
are available. Therefore angular roton modes are assigned to $E_{i}$ with
increasing $i$ for $i<m/2$, to $B_{1}+B_{2}$ right at the edge of the
Brillouin zone $i=m/2$, and to $E_{i}$ with decreasing $i$ for $i>m/2$. For
example in $C_{4,v}$ modes compatible with $E$ transform as $m=1$ angular
roton modes and modes compatible with $B_{1},B_{2}$ transform as $m=2$ angular
roton modes. Angular roton modes with $m=1,3,5,...$ remain degenerate ($E$)
and those with $m=2,6,10,...$ split up into two levels ($B_{1},B_{2}$). To
discuss the symmetries in a specific group we exemplify $n_{d}=7$ case. The
symmetries of the resulting supersolid GS can be described by the point group
$C_{7,v}$. The character table of the group $C_{7,v}$ is reproduced in Tab.
(2), where classes are arranged along the columns, irreducible representations
along the rows and the body of the table shows their character. The supersolid
is symmetric with respect to the identity ($E$), rotation ($C_{7}$) around
$\pm 2\pi/7$, rotation ($C_{7}^{2}$) around $\pm 4\pi/7$, rotation
($C_{7}^{3}$) around $\pm 6\pi/7$ around the $z$-axis, and reflection
$(\sigma_{v})$ through the seven vertical planes intersecting the $z$-axis and
the droplets. These five types of symmetries divide the group $C_{7,v}$ into
the five classes $E$, $2C_{7}$, $2C_{7}^{2}$, $2C_{7}^{3}$, $7\sigma_{v}$ and
yield by the orthogonality theorem five irreducible representations $A_{1}$,
$A_{2}$, $E_{1}$, $E_{2}$, and $E_{3}$. $A_{i}$ and $E_{i}$ are one-
dimensional and two-dimensional irreducible representaitons, respectively. The
notations used for the symmetries and irreducible representations are known as
Schoenflies and Mulliken symbols, respectively. Defining the character as the
trace of the matrix of a representation and using that the character for each
element in a class is the same, essential information of a group can be
summarized in a character table. The statements can be easily adapted for any
point group $C_{n_{d},v}$ once the character table of the point group is
known. The character tables for many point groups are tabulated [67, 68].

Table 2: Character table for the group $C_{7,v}$. The irrotational character values are ${\chi_{n}=2\cos(2\pi n/7)}$. $C_{7,v}$ |  $E$ |  $2C_{7}$ |  $2C_{7}^{2}$ |  $2C_{7}^{3}$ |  $7\sigma_{v}$ |  example basis
---|---|---|---|---|---|---
$A_{1}$ |  1 |  1 |  1 |  1 |  1 |  $z$
$A_{2}$ |  1 |  1 |  1 |  1 |  -1 |  $R_{z}$
$E_{1}$ |  2 |  $\chi_{1}$ |  $\chi_{2}$ |  $\chi_{3}$ |  0 |  $\\{x,y\\}$
$E_{2}$ |  2 |  $\chi_{2}$ |  $\chi_{3}$ |  $\chi_{1}$ |  0 |  $\\{x^{2}-y^{2},xy\\}$
$E_{3}$ |  2 |  $\chi_{3}$ |  $\chi_{1}$ |  $\chi_{2}$ |  0 |  $\\{x(x^{2}-3y^{2}),y(3x^{2}-y^{2})\\}$

To get an intuition on the symmetries compatible with each of the irreducible
representations, we consider the example basis functions given in Tab. 2.
Modes compatible with $A_{1}$ transform as $z$ – symmetric with respect to all
rotations and reflections in the group. Modes compatible under $A_{2}$
transform as $R_{z}$ (rotation around the $z$-axis) – antisymmetric with
respect to $7\sigma_{v}$ and symmetric otherwise. The $m=7$ angular roton
modes therefore split into a zero energy Goldstone mode compatible with
$A_{2}$ and a Higgs amplitude mode compatible with $A_{1}$. The Goldstone mode
corresponds to the $q=0$ limit of the second sound branch and is a rotation of
the system. An alternative point of view is that this mode is not physical
because it lies at $\omega=q=0$. We find that the identification of the zero
energy Goldstone mode as the first instance on the second sound branch is
nonetheless useful, in particular since this interpretation is consistent with
the lowest energy Goldstone mode being second sound in harmonic traps, lifted
to a finite energy due to the already broken translational symmetry. The Higgs
mode is an amplitude modulation of the supersolid order parameter, where the
SF and crystal part exchange density in phase. The two-dimensional
representations $E_{i}$ share the same symmetry with the $m=i$ angular roton
modes. For example, modes compatible with $E_{1}$ are either antisymmetric
with respect to $x$ or $y$, exactly like the two degenerate $m=1$ angular
roton modes. This identification allows generally to think of the $m=i$
angular roton modes as basis functions, representing the symmetry of modes
compatible with $E_{i}$. $E$ in $C_{3,v}$ and $C_{4,v}$ transforms just as
$E_{1}$ in the point groups of higher symmetry.

Finally, we comment on the correspondence to the one-dimensional droplet
chains in harmonic traps [24, 26]. The group $C_{1,v}$ is identical to $C_{s}$
which is isomorphic to $C_{2}$ and the symmetric group $S_{2}$. Modes can
therefore be classified according to their reflection symmetry around the
center of the trap and will be either symmetric and compatible to $A_{1}$ or
antisymmetric and compatible to $A_{2}$. Whether specific modes correspond to
first or second sound can be identified by the eigenenergies increasing (first
sound) or decreasing (second sound) towards larger $\epsilon_{dd}$. Avoided
crossings are expected whenever modes of the same symmetry are close in
energy. This holds in particular in systems with low symmetry such as small
droplet numbers in one-dimensional and two-dimensional harmonic traps, which
explains previous observations showing many avoided crossings [26, 24, 28].

###  .6 Structure factor

The dynamic structure factor [27] is given by

$S(\boldsymbol{k},\omega)=\sum_{i}|\delta
n_{i}(\boldsymbol{k})|^{2}\delta(\omega-\omega_{i})$ (4)

where $\delta n_{i}(\boldsymbol{k})$ is the Fourier transform of the density
fluctuation pattern $\delta
n_{i}(\boldsymbol{r})=f_{i}^{*}(\boldsymbol{r})\psi_{0}(\boldsymbol{r})$ with
the Bogoliubov mode pattern $f_{i}=u_{i}+v_{i}$ and the GS wavefunction
$\psi_{0}$. As we focus on angular excitations and thus the $x-y$ plane, we
transform the structure factor to polar coordinates ${S(x,y,\omega)\rightarrow
S(k,\phi,\omega)}$. We then fix $k$ to the radial momentum at which $S$
displays a maximum and are left with the angular distribution of the structure
factor $\tilde{S}(\phi,\omega)=S(\phi,\omega)-\overline{S}$ where
$\overline{S}$ indicates the average of $S(\phi,\omega)$ along $\phi$. An
angular roton mode with $m$ nodes will result in $2m$ maxima or equivalently
$2m$ full modulations in $\tilde{S}(\phi,\omega)$. To assign a quasimomentum
to these modes and classify the more complicated shapes of $f_{i}$ in the
supersolid phase, we Fourier-transform $\tilde{S}(\phi,\omega)$ along the
angular direction

$\tilde{S}(k_{\phi},\omega)=\int\\!\mathrm{d}\phi\,e^{ik_{\phi}\phi}\tilde{S}(\phi,\omega)$
(5)

where $k_{\phi}$ is the angular frequency. In this way, $2m$ maxima along
$S(\phi,\omega)$ for $\phi$ from $0$ to $2\pi$ correspond to an angular
frequency of $k_{\phi}=2m$. In conclusion, a peak in
$\tilde{S}(k_{\phi},\omega)$ at $k_{\phi}=2m$ corresponds to $2m$ angular
modulations in the structure factor $S(\boldsymbol{k},\omega)$ and indicates
an excitation with an angular symmetry compatible with $m$ modulations, such
as a $m$-angular roton mode or a mode compatible with $E_{m}$ in the group
$C_{n_{d},v}$, see Sec. .5.

In Fig. 1 of the main text, we broaden $\tilde{S}(k_{\phi},\omega)$ along
$\omega$ and normalize at each $\omega_{i}$ by the maximum of $\tilde{S}$
along $k_{\phi}$ for visual clarity. We use $\tilde{S}$ to assign
quasimomenta, follow the dispersion relation across the phase transition, and
relate the spectrum to the dispersion relation of infinitely extended
supersolids. The amplitude of $\tilde{S}$ plays a role in evaluating sum rules
based on $S(\boldsymbol{k},\omega)$ [2, 27, 34], but is not considered any
further here.

###  .7 Spectroscopy of supersolids

Figure 5: Exciting the Goldstone mode. (a) Dominant PCs after magnetostirring
the GS at $\epsilon_{dd}\simeq 2.1$. (b) Fourier spectrum
$|\mathcal{F}_{w}(\omega_{\mathrm{PC}})|^{2}$ of the dominant PC weights
$w(t)$.

The toroidal trap protects the individual modes from coupling to each other,
allowing spectroscopic methods to selectively excite individual modes of the
supersolids. Here, we provide additional details on the dynamic protocols used
in the main text. Starting with the ground state at $\epsilon_{dd}\simeq 2.1$,
we apply a modulation scheme for a certain period of time and then time-evolve
the state for typically $100\,\mathrm{ms}$ (or longer for the low-energy
Goldstone mode).

To excite the Higgs amplitude mode at $\tilde{q}_{0}$ we modulate the
scattering length ${a_{s}(t)=a_{s,mod}\sin(\omega_{mod}t)}$ with an amplitude
of ${a_{s,mod}=0.05\,a_{0}}$ for a duration ${t_{mod}=4T_{mod}}$ with
$T_{mod}=2\pi/\omega_{mod}$ and the driving frequency
$\omega_{mod}/2\pi=50\,\mathrm{Hz}$. We have chosen a frequency that does not
match the Higgs frequency, highlighting that the excitation of the
$\tilde{q}_{0}$ mode does not require resonant driving. With further
simulations using this modulation scheme with different parameters, we
confirmed that the similar selectivity and spectral purity of the Higgs mode
is reached with frequencies below $\omega_{mod}/2\pi=100\,\mathrm{Hz}$.
Eventually at sufficiently high modulation frequencies ($\gtrsim
200\,\mathrm{Hz}$), further modes are excited as well. In these cases, the PCs
become superpositions of BdG modes and the corresponding (multiple) BdG
frequencies show up in the spectra of the PC weights. A similar effect is
observed towards higher amplitudes $a_{s,mod}\gg 0.1\,a_{0}$.

The Goldstone mode (Fig. 5) corresponds to a rotation of the system at zero
energy in a cylindrically symmetric trap. Breaking the cylindrical symmetry
elevates this mode to a finite frequency and enables imparting angular
momentum into the system. This can be done for example by deforming and
rotating the trapping potential or by using magnetostirring [70] where the
magnetic offset field is tilted and rotated. Using the latter, PCA captures
the resulting slow rotation by two PCs, identical up to rotation of $2\pi/14$,
oscillating in time $\pi/2$ out of phase. Each PC matches the Goldstone mode
pattern and the symmetry $A_{2}$. In order to further validate this approach,
we have explicitly calculated the excitation spectrum, for a tilt of the
magnetic field axis of $10^{\circ}$ with respect to the axial direction. This
is the tilt employed in the magnetostirring in Fig. 5. For these small tilt
angles, the Goldstone mode is still well isolated for all values of
$\epsilon_{dd}$ and the overall structure of the spectrum remains intact. The
main qualitative difference is that twofold degenerate modes split up as the
rotational symmetry is broken with a tilted magnetic field.

The first and second sound modes at $\tilde{q}_{1}$ are excited by modulating
a linear gradient ${\Delta V=px\sin(\omega t)}$ on top of the toroidal trap
where $x$ corresponds to the direction of a symmetry axis of the SS. The
perturbation $x$ is chosen since it shares the same symmetry with the basis
functions of $E_{1}$. We choose an amplitude of ${p\simeq 6.3\times
10^{-3}\omega_{r}/\rho_{0}}$ ${(p/2\pi\hbar=6.3\,\mathrm{Hz}/\mathrm{\mu m})}$
and modulate at the resonance frequency $\omega_{i}=c_{i}\tilde{q}_{1}$ for a
time $t_{mod}=4T_{mod}$ for first sound $(i=1,\ \omega_{1}/2\pi\simeq
212\,\mathrm{Hz})$ and second sound $(i=2,\ \omega_{2}/2\pi\simeq
315\,\mathrm{Hz})$. The scheme is robust with respect to the specific values
chosen for $t_{mod}$, $\omega$, and $p$. Deviations of $\omega$ from
$\omega_{i}$, of $x$ from the symmetry axis, and towards large $p$ result
increasingly in the excitation of superpositions of first and second sound.
With a misalignment of $x$ from the symmetry axis it is also possible to set
the SS into rotation (excitation of the Goldstone mode). We find similar
behavior, although slightly worse selectivity ($60$-$90\%$ depending on
modulation parameter details), when modulating a quadrupole field ${\Delta
V\propto xy\sin(\omega t)}$ on top of the toroidal trap in order to excite the
sound modes at $\tilde{q}_{2}$.