Present address: ]Department of Physics, Princeton University, Princeton, New
Jersey 08544, USA

# Dissipationless flow in a Bose-Fermi mixture

Zoe Z. Yan1 [  Yiqi Ni1 Alexander Chuang1 Pavel E. Dolgirev2,3 Kushal
Seetharam2,4 Eugene Demler3 Carsten Robens1 Martin Zwierlein1 1MIT-Harvard
Center for Ultracold Atoms, Research Laboratory of Electronics, and Department
of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA,
2Department of Physics, Harvard University, Cambridge, Massachusetts 02138,
USA,
3Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland,
4Department of Electrical Engineering, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA

###### Abstract

Interacting mixtures of bosons and fermions are ubiquitous in nature. They
form the backbone of the standard model of physics, provide a framework for
understanding quantum materials such as unconventional superconductors [1, 2]
and two-dimensional electronic systems [3], and are of technological
importance in 3He/4He dilution refrigerators [4]. Bose-Fermi mixtures are
predicted to exhibit an intricate phase diagram featuring coexisting liquids
[5], supersolids [6], composite fermions [7], coupled superfluids [8], and
quantum phase transitions in between [9]. However, their coupled
thermodynamics and collective behavior challenge our understanding, in
particular for strong boson-fermion interactions. Clean realizations of fully
controllable systems are scarce. Ultracold atomic gases offer an ideal
platform to experimentally investigate Bose-Fermi mixtures, as the species
concentration and interaction strengths can be freely tuned [10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Here, we study the collective
oscillations of a spin-polarized Fermi gas immersed in a Bose-Einstein
condensate (BEC) as a function of the boson-fermion interaction strength and
temperature. Remarkably, for strong interspecies interactions the fermionic
collective excitations evolve to perfectly mimic the bosonic superfluid
collective modes, and fermion flow becomes dissipationless. With increasing
number of thermal excitations in the Bose gas, the fermions’ dynamics exhibit
a crossover from the collisionless to the hydrodynamic regime, reminiscent of
the emergence of hydrodynamics in two-dimensional electron fluids [24, 25,
26]. Our findings open the door towards understanding non-equilibrium dynamics
of strongly interacting Bose-Fermi mixtures.

###### pacs:

Valid PACS appear here

The paradigmatic example of Bose-Fermi mixtures are itinerant electrons moving
through the crystal of ions. The coupling to the ionic lattice vibrations
endows the electrons with a shifted energy and mass, as they become dressed
into polarons [27, 28], the original example of the quasiparticle concept. We
also encounter Bose-Fermi mixtures as dilute solutions of fermionic 3He in
bosonic superfluid 4He [4], in quark-meson models in high-energy physics [29]
and in 2D electronic materials, where interactions between excitons and
electrons can be controlled [30, 31, 3].

Ultracold atomic gases provide arguably the purest realizations of Bose-Fermi
mixtures, featuring precisely understood, tunable short-range interactions and
a high degree of experimental control, offering a direct comparison to
theoretical models. In recent years, atomic Bose-Fermi mixtures enabled the
study of dual superfluids [32, 33, 34, 35, 36], the onset of phase separation
[14, 15, 21, 37], and the observation of strong-coupling Bose polarons [38,
39].

The general dynamics of fermions strongly interacting with a partially
condensed Bose gas at finite temperature are challenging to describe, as
interactions between fermions and bosons come in two flavors. On the one hand,
fermions can incoherently scatter with thermal bosons, leading to momentum-
changing collisions, while on the other hand, fermions also experience
momentum-preserving interactions, in particular with the BEC, in the form of
an effective potential [40, 5, 41, 42, 43]. The interplay between the two
types of interactions dictates the dynamics of the whole system [44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54]. Most challenging is the regime of strong
boson-fermion interactions, when already the non-superfluid system crosses
over from collisionless to the collisionally hydrodynamic regime. Such a
regime is observed in electron-phonon mixtures in the context of high-
temperature superconductivity [55, 56]. With such strong interactions and with
bosons partially condensed, the question arises whether the system indeed
remains superfluid, with fermions flowing without dissipation through the Bose
gas. Also, it is unclear how thermal bosonic excitations alter the transport
properties of fermions. Here, we directly study these questions, observing
evidence for dissipationless flow of fermions through the condensate despite
strong interactions. With increasing temperature we observe a crossover from
dissipationless to collision-dominated hydrodynamic flow.

Figure 1:  Collective oscillations in a Bose-Fermi mixture. a. Illustration of
a Fermi gas (blue) immersed in a Bose-Einstein condensate (red), both trapped
in an optical potential. In-situ absorption images of the fermionic 40K and
bosonic 23Na are shown beneath. Radial collective oscillations are induced by
periodically modulating the depth of the optical potential. b. The doubly
integrated line densities along the transverse (y) direction of the boson
(left) and fermion (right) clouds as a function of modulation frequency
$\omega$, at zero interspecies interaction ($a_{\mathrm{BF}}\,{=}\,0$). c.
Spectra of the uncoupled Bose-Fermi system, showing the boson (left, red
circles) and fermion (right, blue circles) relative widths, reveal the
superfluid hydrodynamic response of the BEC and the response of the non-
interacting Fermi gas. The solid black lines show predictions from the
hydrodynamic scaling ansatz and a phenomenological Gaussian fit, respectively.

Collective excitations are excellent probes to reveal the role of
interparticle scattering. They have been used to demonstrate the superfluid
hydrodynamic flow of BECs [57, 58, 59] and collisional hydrodynamics in
interacting Fermi gases [60, 61, 62]. Collective oscillations have also been
measured in coupled Bose-Fermi superfluids [32, 33, 35, 36] and in Bose-Fermi
mixtures [63, 64, 22, 37, 23], revealing phenomena such as the transition
between collisionless to collisionally hydrodynamic flow in coupled thermal
gases [63], and sound propagation [23]. Here we study a novel regime – the
limit of a dilute Fermi gas immersed in a BEC – as relevant for the physics of
Bose polarons [38, 39] and unconventional superconductors with low carrier
densities [65].

We probe collective excitations of 40K fermions and a 23Na BEC as a function
of drive frequency and across a range of Bose-Fermi interactions, revealing
the energy and spectral width of low-lying excitations. The experiment starts
with an ultracold Fermi gas immersed in the BEC, both held in a
$1064\text{\,}\mathrm{nm}$ optical trap with near-cylindrical symmetry (see
Fig. 1a). For our coldest samples we evaporatively cool both species to
$T{\approx}$ 30 nK, corresponding to $T/T_{\mathrm{c}}\,{\lesssim}\,0.2$,
where $T_{\mathrm{c}}$ is the condensate’s critical temperature. Both species
are in their respective hyperfine ground state ($\ket{F=1,m_{F}=1}$ for 23Na
and $\ket{F=9/2,m_{F}=-9/2}$ for 40K). We control the interspecies
interactions by ramping the magnetic field near Feshbach resonances, allowing
us to continuously tune the $s$-wave scattering length, $a_{\mathrm{BF}}$. The
typical peak boson density is $n_{\mathrm{B}}\,{=}\,7\times 10^{13}$ cm-3, and
the typical fermion impurity concentration is between
$n_{\mathrm{F}}/n_{\mathrm{B}}\,{\approx}\,0.003-0.02$.

We characterize the low-energy radial excitations of the mixture at varying
interspecies interaction strengths. Modulating the radial trapping potential
depth at frequency $\omega$ for 10 cycles, we measure the in situ width of the
clouds (see Fig. 1a.) The number of cycles $N$ allows for spectral resolution
${\sim}\,\omega/N$ while the probe time is kept short compared to the
mixture’s lifetime. When the modulation excites a resonant mode of either
species, the cloud’s width expands radially. Fig. 1b-c depicts the bosonic and
fermionic spectrograms for a decoupled mixture at $a_{\rm BF}{=}\,0$. A
fermionic resonance is excited at $2\,\omega_{y}^{\text{F}}$, and two
resonances of the BEC are apparent at $\sqrt{2}\omega_{\perp}^{\text{B}}$ and
$2\omega_{\perp}^{\text{B}}$. Here, $\omega_{y}^{\text{F}}$ is the fermions’
radial trapping frequency, and
$\omega^{\mathrm{B}}_{\perp}\,{=}\,(\omega^{\mathrm{B}}_{x}\omega^{\mathrm{B}}_{y})^{1/2}$
is the bosons’ geometric mean radial trapping frequency, with the two
frequencies related by $\omega_{y}^{\text{F}}=1.16\,\omega_{\perp}^{\text{B}}$
[66]. The spin-polarized fermions are a collisionless gas [67], which in a
purely harmonic trap has its lowest parametric resonance for motion along the
y-axis at $2\omega_{y}^{\text{F}}$. The BEC obeys superfluid hydrodynamics,
which couples the two collisionless radial modes, giving rise to a quadrupole
(out of phase) and a breathing (in phase) resonance at
$\sqrt{2}\omega_{\perp}^{\text{B}}$ and $2\omega_{\perp}^{\text{B}}$,
respectively [68, 69]. Across all frequencies, the bosonic spectral response
is well captured using a hydrodynamic scaling ansatz [70, 66] (see red shaded
area in Fig. 1c):

$\displaystyle\ddot{b}_{i}$
$\displaystyle+{\omega_{i}^{B}}(t)^{2}b_{i}-\frac{{\omega_{i}^{B}}(0)^{2}}{b_{i}\prod_{j}b_{j}}=0$
(1)

where $i\,{\in}\,(x,y,x)$ and $b_{i}$ is the dimensionless scaling parameter
of the BEC’s Thomas-Fermi radius in the $i$-th direction. The energy and
spectral width of the fermions’ spectral response is obtained using a fit to a
phenomenological function [66] (see Fig. 1c). The broad fan below
$2\omega_{y}^{\text{F}}$ visible in the fermionic response (Fig. 1b) arises
from strong driving in an anharmonic trap [66].

Figure 2:  Evolution of Bose-Fermi collective modes across varying interaction
strength. Illustrations depict the oscillations of the cloud’s transverse
($x$-$y$) cross-section. The boson quadrupole and breathing modes (shown in
red) lie at $\sqrt{2}\omega^{\mathrm{B}}_{\perp}$ and
$2\omega^{\mathrm{B}}_{\perp}$, respectively, whereas the fermions’ transverse
resonance in the collisionless regime lies at $2\omega^{\mathrm{F}}_{y}$
(shown in blue). The spectra depict bosonic and fermionic cloud widths
(circles) for varying modulation frequencies. As $a_{\rm BF}$ increases toward
either repulsive or attractive interactions, the fermion spectra evolve to
mode lock to the BEC’s superfluid hydrodynamic modes. An extended data set can
be found in [66].

Fig. 2 shows a selection of the bosons’ and fermions’ spectral response for
various interspecies coupling strengths. The resonances of the BEC are in all
cases well described by the hydrodynamic scaling ansatz from Eq. 1. The BEC
spectrogram remains unaffected by the interactions with the fermionic
impurities because the Fermi gas is much more dilute. By contrast, the
response of the Fermi gas shows a strong dependence on the interspecies
coupling strength. At weak couplings, the collisionless mode of the fermions
shifts linearly with $a_{\text{BF}}$. With increasing attractive interactions
strengths, the fermionic response changes drastically, revealing two
additional modes in the their spectral response that coincide with the BEC’s
hydrodynamic superfluid modes at $2\omega_{\perp}^{\text{B}}$ and
$\sqrt{2}\omega_{\perp}^{\text{B}}$. All three modes are spectrally well
resolved and show no broadening beyond the Fourier limit, in contrast to the
broadened profiles that would arise from momentum-relaxing collisions [66].
This is remarkable, given that the mean-free path for collisions changes from
infinity at zero interaction strength to $l_{\rm mfp}=(4\pi
a_{\mathrm{BF}}^{2}n_{\mathrm{B}})^{-1}\,\sim\,0.6\,\mu$m at the strongest
measured interactions, so much shorter than the radial system size $L\sim
10\,\mu$m. A thermal mixture would thus cross over from collisionless to
collisionally hydrodynamic behavior, through an intermediate regime of strong
damping. Here, instead, the fermions continue to flow through the condensate
without dissipation, and, for the strongest interactions, even “copy” the
BEC’s superfluid collective modes, not unlike dye particles in water.

The fermion collective modes are summarized in Fig. 3. For strong
interactions, regardless of their sign ($|a_{\text{BF}}|\,{>}\,350\,a_{0}$,
with $a_{0}$ being the Bohr radius), the fermions exclusively respond at the
BEC’s hydrodynamic superfluid modes and show no signal of their own
collisionless mode. For weaker interactions, we observe a mean-field like
shift of the fermion frequency proportional to the sign of the interaction,
which for strong repulsive interactions merges with the BEC’s breathing mode
at $2\omega_{\perp}^{\text{B}}$ and becomes spectrally indistinguishable. We
note that at strong repulsive interactions we observe a dispersive - rather
than absorptive - feature in the fermionic response at the BEC quadrupole mode
(Fig. 2). This Fano-type behavior can be understood from coherent coupling of
the fermionic and bosonic mode [71].

Figure 3:  Fermionic mode frequencies versus interspecies interaction. The
color density plot displays the power spectra of cloud widths. White markers
denote the peak frequencies. The arrows on the right side indicate the three
modes of Fig. 2. The dashed line shows the naive mean-field prediction, and
the red line is the scaling ansatz solution of the mean-field model. The black
lines show the dominant modes of the Boltzmann-Vlasov solution accounting for
finite system size and trap anharmonicity [66].

To understand the dynamics across all interaction strengths, we compare
various numerical models to the data. The linear dependence of the fermionic
collisionless mode at weak interaction strengths qualitatively agrees with a
mean-field description, considering the effective potential experienced by the
fermions immersed in the Bose gas. The BEC in the Thomas-Fermi approximation
takes on the shape of the inverted optical potential. The fermions thus
experience a joint effective potential comprising the optical trap and the
mean-field potential of the BEC, where attractive (repulsive) interspecies
interactions provide a steeper (shallower) potential that shifts the trapping
frequencies according to
$\tilde{\omega}^{2}{=}\,{\omega^{F}_{y}}^{2}\left(1\,{-}\,\frac{g_{\rm
BF}\alpha_{\rm B}}{g_{\rm BB}\alpha_{\rm F}}\right)$ [66]. Here,
$g_{\mathrm{BF}}\,{=}\,2\pi\hbar^{2}a_{\mathrm{BF}}/\mu$ is the Bose-Fermi
coupling strength,
$\mu\,{=}\,m_{\mathrm{F}}m_{\mathrm{B}}/(m_{\mathrm{F}}+m_{\mathrm{B}})$ is
the reduced mass,
$g_{\mathrm{BB}}\,{=}\,4\pi\hbar^{2}a_{\mathrm{BB}}/m_{\mathrm{B}}$ is the
Bose-Bose coupling, and $\alpha_{\rm B(F)}$ is the boson (fermion) optical
polarizability. Qualitatively, this mean-field model (black dashed line in
Fig. 3) shares the trend of the measurements for small $a_{\text{BF}}$, but
with a different slope $\omega$ versus $a_{\rm BF}$. It also fails to predict
the appearance of additional modes in the fermionic spectral response. To
capture the resonances that fermions inherit from the BEC’s superfluid
hydrodynamic modes, the mean-field potential itself must properly incorporate
the bosons’ response given by the scaling ansatz Eq. 1.

We therefore turn to the full dynamics of the fermions as described by the
Boltzmann-Vlasov equation,

$\frac{\partial f}{\partial t}+\frac{d\textbf{r}}{dt}\frac{\partial
f}{\partial\textbf{r}}+\frac{d\textbf{p}}{dt}\frac{\partial
f}{\partial\textbf{p}}=I_{\rm coll}$ (2)

where $f$ is the fermion distribution at momentum p and position r, and
$I_{\rm coll}$ is the collision integral. Anticipating the absence of
collisions for fermions only interacting with the BEC, in the absence of
thermal excitations, we may set $I_{\rm coll}=0$. Then, to first order in
$g_{\rm BF}$, we derive a scaling ansatz for the fermions’ width, assuming a
harmonic trap and fermionic impurities that are deeply immersed in the BEC
[70, 72, 47]:

$\displaystyle\ddot{c}_{i}$ $\displaystyle+{{\omega^{F}_{\rm
i,eff}}}(t)^{2}c_{i}-\left(1-\frac{g_{\rm BF}\alpha_{\rm B}}{g_{\rm
BB}\alpha_{\rm F}}\right)\frac{{\omega^{F}_{y}}(0)^{2}}{c_{i}^{3}}=0$
$\displaystyle{\mathrm{with~{}}{\omega^{F}_{\rm i,eff}}}$
$\displaystyle(t)^{2}=\left(1-\frac{g_{\rm BF}\alpha_{\rm B}}{g_{\rm
BB}\alpha_{\rm F}}\right){\omega^{F}_{i}}(t)^{2}-\frac{g_{\rm BF}}{g_{\rm
BB}}\frac{\ddot{b_{i}}}{b_{i}}$ (3)

Here, $c_{i}$ is the dimensionless scaling parameter of the width of the Fermi
gas, with $c_{i}(t{=}0)\,{=}\,1$. This ansatz captures the fermionic response
to the BEC’s superfluid mode on the attractive side [66], which can thus
indeed be explained as the dissipationless flow of fermions experiencing the
coherent interactions with the BEC. The simple ansatz fails for repulsive
interactions above $a_{\text{BF}}\,{>}\,100\,a_{0}$ where the mean-field
potential is strong enough to repel fermions from the BEC. A full numerical
simulation of the dissipationless Boltzmann-Vlasov equation [66] – including
the temperature-dependent fermionic cloud size and the trap anharmonicity – is
shown as the black solid line in Fig. 3, which accurately captures all of the
observed modes across all interaction strengths, validating our neglect of the
collisional term in Eq. 2.

Figure 4:  Temperature dependence of the bosonic and fermionic collective
modes at $\mathbf{a_{\mathrm{BF}}\,{=}\,{-}400a_{0}}$. a. Spectrograms of the
boson (left) and fermion (right) widths, as a function of the bosons’ reduced
temperature $T/T_{\mathrm{c}}$. Top cartoons illustrate the bosonic
hydrodynamic (red), collisionless (yellow), and the fermionic collisionless
(blue) modes. Peak boson densities for increasing $T/T_{\mathrm{c}}$ are
$n_{\rm B}{=}(7.3,5.9,5.4,2.3,1.4,0.07){\times}10^{13}$ cm-3, respectively.
The fermion spectrograms indicate hydrodynamic response at the BEC’s
quadrupole and breathing mode frequencies for the lowest temperatures. Dashed
lines in the upmost panels for $T/T_{\mathrm{c}}\approx 0.1$ and
$T/T_{\mathrm{c}}\approx 0.2$ represent the theoretical prediction based on
the scaling ansatz for the BEC dynamics and Boltzmann-Vlasov equation for the
dynamics of fermions. Dashed lines for $T>T_{\mathrm{c}}$ show the result of
solving coupled Bose-Fermi Boltzmann equations [66]. b. Summary of the
obtained peak frequencies for bosons (left) and fermions (right), with color
indicating spectral response.

Indeed, dissipationless flow is expected for the Fermi gas well below the
bosons’ superfluid transition temperature, as fermions slower than the
condensate’s speed of sound ($\sim\,5\mu$m/ms) can only dissipate energy
through collisions with thermal bosons, which are essentially absent at low
temperatures $T/T_{\mathrm{c}}\approx 0.2$. To measure the impact of
collisions of fermions with thermal bosons, we now probe the mixture at
increasing temperatures across the BEC phase transition. The physics is
complex already for bosons alone, as the BEC becomes immersed in a cloud of
thermal excitations, and the thermal cloud’s collective modes couple to the
superfluid modes [69]. We employ the same protocol as before, using a drive
that couples strongly to the BEC’s quadrupole mode [66]. The relative
temperature $T/T_{\mathrm{c}}$ is varied at a fixed interaction strength by
reducing the number of bosons while fixing the same final temperature. The
radial trap ellipticity [66] allows us to distinguish the collisionless mode
of the bosons’ thermal component at $2\omega_{y}^{\text{B}}$ from the
superfluid breathing mode at $2\omega_{\perp}^{\text{B}}$, as depicted in Fig.
4a. Fig. 4b displays the change of the bosons’ and fermions’ spectral
responses with temperature. At low temperatures, the bosons exhibit resonances
at the BEC quadrupole ($\sqrt{2}\omega^{\mathrm{B}}_{\perp}$) and breathing
($2\omega^{\mathrm{B}}_{\perp}$) modes. At $T\,{>}\,0.5\,T_{\mathrm{c}}$ we
observe an additional peak at $2\omega_{y}^{\text{B}}$, corresponding to the
intraspecies collisionless mode of the bosonic thermal component [58, 57].
With increasing $T/T_{\mathrm{c}}$ the bosonic response at the BEC superfluid
modes is reduced while the response at the intraspecies collisionless mode
grows, until above $T_{\mathrm{c}}$ only the collisionless mode persists.

The fermions can be viewed as a highly sensitive probe for the complex
crossover of modes in the Bose gas. At the coldest temperatures, they respond
exclusively at the frequencies of the BEC’s hydrodynamic modes, giving rise to
the dissipationless flow. Remarkably, starting at
$T/T_{\mathrm{c}}\,{\approx}\,0.4$, two additional modes appear in the
fermionic response. The lower resonance at $2\omega^{\mathrm{B}}_{y}$
coincides with the intraspecies collisionless mode of the bosonic thermal
component whereas the higher one at $2\omega^{\mathrm{F}}_{y}$ coincides with
the native collisionless mode of the fermions. Above $T_{\mathrm{c}}$, the
fermions respond only at their collisionless mode at
$2\omega^{\mathrm{F}}_{y}$, and this regime is well captured by solving the
full coupled Boltzmann equations for the mixture with nonvanishing $I_{\rm
coll}$ [66, 71]. We note that while the thermal components of the bosons and
the spin-polarized fermions are both in the collisionless regime by
themselves, strong interspecies interactions bring this mixture into local
equilibrium [63]. We interpret the fermionic response at the bosonic
collisionless mode ($0.3\,T_{\rm c}\lesssim T\lesssim T_{\rm c}$) as an
indication of the mixture’s crossover from the collisionless to the
collisionally hydrodynamic regime. Above $T_{\mathrm{c}}$, the mixture reverts
to collisionless flow for both species due to the lowered density of bosons.

Our results demonstrate dissipationless flow of fermions immersed in a Bose
condensate, i.e. the absence of momentum-changing collisions even for strong
interspecies interactions. As the temperature is increased, incoherent
collisions between the thermal bosons and fermions cause a crossover into the
collision-dominated hydrodynamic regime, in direct analogy to 2D electron
gases, where the electron mean-free path is tuned with the density and
temperature. At temperatures lower than those achieved in this study, induced
fermion-fermion interactions [73, 74] are predicted to arise within the Bose-
Fermi mixture, a precursor of the long-sought $p$-wave superfluidity of
fermions mediated by bosons [75, 76, 77].

Acknowledgements: We acknowledge Eric Wolf for helpful discussions. This work
was supported by the NSF, AFOSR, ARO, AFOSR MURI on “Exotic Phases of Matter,”
the David and Lucile Packard Foundation, the Vannevar Bush Faculty Fellowship,
and the Gordon and Betty Moore Foundation through grant GBMF5279. Z. Z. Y. and
A. C. acknowledge support from the NSF GRFP. C. R. acknowledges support from
the Deutsche Forschungsgemeinschaft (DFG) Germany research fellowship
(421987027). K. S. acknowledges funding from NSF EAGER-QAC-QCH award No.
2037687. P. D. and E. D. were supported by ARO grant number W911NF-20-1-0163,
the SNSF project 200021-212899. E. D. acknowledges support from the Swiss
National Science Foundation under Division II.

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Supplemental Information

##  1 Methods

Here we describe details of the collective oscillation measurements. The
optical potential is approximately cylindrically symmetric with trap
frequencies of
$\omega^{\mathrm{B}}_{x,y,z}/2\pi{=}[103(3),94(2),12.2(0.3)]\,$Hz and
$\omega^{\mathrm{F}}_{x,y,z}/2\pi{=}[125(2),114(2),15(1)]$ Hz for bosons and
fermions. The fermions are moderately degenerate with $T/T_{\mathrm{F}}$
ranging from 0.6 to 2, where $T_{\mathrm{F}}$ is the Fermi temperature. The
BEC is weakly interacting with a Bose-Bose scattering length of $a_{\rm
BB}\,{=}\,52\,a_{0}$, where $a_{\text{0}}$ is the Bohr radius. Our lowest
temperatures are $T/T_{\mathrm{c}}\,{\lesssim}\,0.2$ with respect to the BEC
critical temperature $T_{\mathrm{c}}$. To create an interacting Bose-Fermi
mixture, we ramp the magnetic field in between two interspecies Feshbach
resonances, allowing continuous tuning of the interspecies $s$-wave scattering
length, $a_{\mathrm{BF}}$.

We study the collective modes versus interaction strength at a fixed
$T/T_{\mathrm{C}}\,{\approx}\,0.2$ (see main text Fig. 2) by applying a
sinusoidal intensity perturbation on the $z$-axis $1064\text{\,}\mathrm{nm}$
trap beam with a modulation amplitude of 20% and a variable drive frequency.
This method primarily excites the transverse breathing mode. We study the
collective modes versus temperature at a fixed interaction strength (set by
the scattering length $a_{\rm BF}\,{=}\,-400\,a_{0}$) by applying intensity
modulations to both the $z$\- and $y$-directional $1064\text{\,}\mathrm{nm}$
beams (see main text Fig. 4). The modulation in each direction is
$180^{\circ}$ out of phase in order to better couple to the transverse
quadrupole mode. The modulation depths are 15% for both beams. In both cases,
the clouds are imaged after a duration lasting ten oscillation cycles.

Fitting of the fermion spectrograms, i.e. in Fig. 1 of the main text, is
performed with a phenomenological asymmetric function of the form

$\displaystyle c(\omega,p_{i})=\frac{p_{1}}{f(\omega,p_{2},p_{3},p_{4})}$
$\displaystyle e^{-(\omega-p_{2})^{2}/f(p_{2},p_{3},p_{4})^{2}}$
$\displaystyle\text{with}~{}f(\omega,p_{2},p_{3},p_{4})$
$\displaystyle=p_{3}(1+e^{p_{4}(\omega-p_{2})})^{-1}$ (S1)

with $p_{i}$ as free fitting parameters. This function reduces to a Gaussian
in the limit $p_{4}\,{=}\,0$, and only the peak location and width are used in
data analysis. The asymmetry is due to strong driving of the fermions in an
anharmonic trap.

In Fig. 1-4 of the main text, the modulation frequencies are shown as
normalized to the boson mean radial trap frequency, $\omega_{\perp}^{\rm B}$,
which is $2\pi\times$ 98 Hz on average. However, long-term drifts of the trap
depth over many experimental repetitions necessitated a different
normalization for each interaction strength in Fig. 2-3. The exact
normalization value was determined by fitting a Gaussian to the boson
spectrograms at frequencies near $\omega_{\perp}^{\rm B}$, extracting the
frequency of the largest response, and assigning that as the normalization for
the boson and fermion spectograms for that particular $a_{\rm BF}$. This
procedure necessarily forces the BEC breathing mode in Fig. 2 to fall at
$2\omega_{\perp}^{\rm B}$, and falls to the same absolute frequency to within
2% error for all driving conditions.

##  2 Boson hydrodynamic collective modes at $T=0$

To understand the collective modes of the majority bosons, we utilize the
approach of [70] based on a scaling ansatz. For BECs in a time-dependent
harmonic potential, the spatial density evolves as a dilation, characterized
by $x$-,$y$-, and $z$-scaling factors $\lambda_{j}(t)$ that satisfy

$\displaystyle\ddot{\lambda}_{i}$
$\displaystyle+\omega_{i}^{2}(t)\lambda_{i}-\frac{\omega_{i}^{2}(0)}{\lambda_{i}\prod_{j}\lambda_{j}}=0$
(S2) $\displaystyle\omega^{2}_{x}(t)$
$\displaystyle=\omega^{2}_{x}(0)\big{(}1+f\alpha_{2}\sin(\omega_{\mathrm{mod}}t)$
$\displaystyle+(1-f)\alpha_{1}\sin(\omega_{\rm mod}t+\phi_{1})\big{)}$
$\displaystyle\omega_{y}^{2}(t)$
$\displaystyle=\omega_{y}^{2}(0)\big{(}1+\alpha_{2}\sin(\omega_{\mathrm{mod}}t)\big{)}$
$\displaystyle\omega_{z}^{2}(t)$
$\displaystyle=\omega_{z}^{2}(0)\big{(}1+\alpha_{1}\sin(\omega_{\mathrm{mod}}t+\phi_{1})\big{)}$

for $j=\\{x,y,z\\}$. Here, $f$ depends on the trapping geometry and was taken
as a fitting parameter for the data in main text Fig. 2a, held constant over
all $a_{\rm BF}$. $\alpha_{1},\alpha_{2}$ are the modulation amplitudes and
$\phi_{1}$ is the relative phase shift. Numerical solutions to these equations
are shown in the boson spectra of Fig. 1 and 2 of the main text.

##  3 Mean field shift of fermion collective mode

For weak interactions and small perturbations, a single fermion impurity feels
a combined potential from the optical trap and the mean-field energy of the
bosons. Its trapping frequency is shifted from the interaction-free frequency
$\omega_{i}$ by

$\tilde{\omega}^{2}_{i}=\omega^{2}_{i}\left(1-\frac{g_{\mathrm{BF}}\alpha_{\mathrm{B}}}{g_{\mathrm{BB}}\alpha_{\mathrm{F}}}\right)$
(S3)

where, $g_{\mathrm{BF}}\,{=}\,2\pi\hbar^{2}a_{\mathrm{BF}}/\mu$ is the Bose-
Fermi coupling,
$\mu\,{=}\,m_{\mathrm{B}}m_{\mathrm{F}}/(m_{\mathrm{B}}+m_{\mathrm{F}})$ is
the reduced mass,
$g_{\mathrm{BB}}\,{=}\,4\pi\hbar^{2}a_{\mathrm{BB}}/m_{\mathrm{B}}$ is the
Bose-Bose coupling, and $\alpha_{\mathrm{B(F)}}$ is the boson’s (fermion’s)
optical polarizability. Physically, this represents the fermion seeing an
effectively higher potential when attracted to the boson reservoir, which
occupies the bottom of the optical trap, and vice versa for repulsive
interactions. As can be seen in the dashed black line of Fig. 3 of the main
text, Eq. S3 qualitatively captures the trend of the breathing response in the
weak-interaction regime but overpredicts the magnitude of the trend. The
deviation arises because of anharmonicity and because the Fermi cloud has a
similar width compared to the boson size, overlapping with regions of zero
condensate density, thereby invalidating the assumption of Eq. S3.

Therefore we employ a model that takes into account the equilibrium atom
distributions. The fermions are given by a Maxwell-Boltzmann distribution at
$T{=}80$ nK, initially in equilibrium in the optical potential with $U_{\rm
ODT}(y)=-U_{0}e^{-\frac{2y^{2}}{w^{2}}}$, where the trap depth is
$U_{0}=h\times 155$ kHz and the waist is 110$\,\mu$m. The bosons are described
by a zero-temperature Thomas-Fermi (TF) approximation [69].

Figure S1:  Spectra of fermion widths at various interaction strengths,
comparing (a) the experiment to (b) the simulation of the Boltzmann mean field
model with a Gaussian trapping potential and finite-sized BEC, as described in
the text. The color density plot shows the power spectra of fermion widths
relative to their equilibrium widths. (c) The solution of the Boltzmann mean
field model using the scaling ansatz, as captured in Eq. S9. This solution is
only valid for $g_{\rm BF}\alpha_{B}/g_{\rm BB}\alpha_{F}<1$. All spectra
neglect the presence of thermal bosons.

The fermion trajectories are solved numerically from the Boltzmann equation
with zero collisions.

$\displaystyle\frac{\partial f}{\partial t}+\sum_{i}v_{i}\frac{\partial
f}{\partial x_{i}}-\frac{1}{m}\frac{dU}{dx_{i}}\frac{\partial f}{\partial
v_{i}}$ $\displaystyle=0$ (S4)

Here, $U(\mathbf{x},t)=U_{\rm trap}(\mathbf{x},t)+U_{mf}(\mathbf{x},t)$, where
the mean-field potential is $g_{\mathrm{BF}}n_{\mathrm{B}}(\mathbf{x},t)$.
Results of the numerical simulation are shown in Fig. S1(b), accurately
reproducing the features of the data, and shown in Fig. S1(a).

Finally, we can derive a simple solution to Eq. S4 using a scaling ansatz, as
employed by [70, 72, 47]. Here we neglect the anharmonicity of the trap as
well as the possibility that the fermions leave the BEC spatial region. Let
$b_{i}\,(c_{i})$ be the dimensionless scaling parameter for the boson Thomas-
Fermi radius (fermion Gaussian width) in the $i$th direction,
$x_{\mathrm{TF,i}}$. Without interactions, both species admit scaling
solutions:

$\displaystyle\ddot{b}_{i}+\omega_{iB}^{2}(t)b_{i}-\frac{\omega_{iB}^{2}(0)}{b_{i}\prod_{j}b_{j}}$
$\displaystyle=0$ (S5)
$\displaystyle\ddot{c}_{i}+\omega_{i}^{2}(t)c_{i}-\frac{\omega_{i}^{2}(0)}{c_{i}^{3}}$
$\displaystyle=0.$ (S6)

for bosons and fermions, respectively. This simple scaling is no longer
satisfied in the case of a finite Bose-Fermi (BF) interaction, but is still a
useful approximation to first order in $g_{\mathrm{BF}}$. The BF interaction
adds terms of the following form to Eq. S6

$\displaystyle-\frac{c_{i}}{mx_{i}}\partial_{x_{i}}(g_{\mathrm{BF}}n_{\mathrm{B}})$
$\displaystyle=$ (S7)
$\displaystyle\frac{c_{i}~{}g_{\mathrm{BF}}~{}\omega_{iB}^{2}(0)}{g_{\mathrm{BB}}~{}\Pi_{j}b_{j}b_{i}^{2}}.$
(S8)

With appropriate substitutions, the scaling solution can therefore be
approximated as

$\displaystyle\ddot{c}_{i}+{{\omega^{F}_{\rm i,eff}}}(t)^{2}c_{i}$
$\displaystyle-\left(1-\frac{g_{\rm BF}\alpha_{\rm B}}{g_{\rm BB}\alpha_{\rm
F}}\right)\frac{{\omega^{F}_{y}}(0)^{2}}{c_{i}^{3}}=0$
$\displaystyle{\mathrm{with~{}}{\omega^{F}_{\rm i,eff}}}(t)^{2}$
$\displaystyle=\left(1-\frac{g_{\rm BF}\alpha_{\rm B}}{g_{\rm BB}\alpha_{\rm
F}}\right){\omega^{F}_{i}}(t)^{2}-\frac{g_{\rm BF}}{g_{\rm
BB}}\frac{\ddot{b_{i}}}{b_{i}}$ (S9)

for small displacements within the BEC TF radius. This equation describes a
harmonic oscillator driven by both the direct variation of the fermion
frequency and by the changing condensate shape. As such, one observes
resonances, apart from the mean-field solution, also where the BEC is strongly
modulated. These resonances couple strongly when they equal an integer
fraction of the dressed fermion frequency (see Fig. S1c).

##  4 Equilibrium line densities of the Bose-Fermi mixture

We report the equilibrium distributions of the Bose-Fermi mixture across
different interaction strengths and temperatures. Fig. S2 shows the line
densities of the fermions in the $y$\- and $z$\- directions, obtained by
integrating the absorption image in the $x$ and $y$ directions, respectively.
These examples were taken at a modulation frequency that was not resonant for
any interaction strength. Therefore, they represent the equilibrium
distribution of the fermions prior to any excitations. Fig. S3 shows the line
densities of the equilibrium Bose-Fermi mixture corresponding to the data
shown in Fig. 4 of the main text.

Figure S2:  Equilibrium line density distributions for the fermions versus
interaction strength. Line densities showing the optical density versus
position for the fermions (in blue) and the bosons (in red), for the $y$\- and
$z$\- directions. Figure S3:  Equilibrium line density distributions for the
fermions versus reduced temperature. Line densities showing the optical
density versus position for the fermions (in blue) and the bosons (in red),
for the $y$\- and $z$\- directions.

##  5 Collective oscillations in the time domain

The collective oscillations can also be studied in the time domain rather than
in the frequency domain, revealing coherent, in-phase oscillations between the
bosons and fermions when interactions are tuned to where the fermions respond
to the superfluid bosons’ hydrodynamic modes. Fig. S4 shows the evolution of
the cloud size versus time, revealing how the fermion cloud width follows the
bosonic behavior across multiple cycles without apparent damping or dephasing.
Both clouds oscillate at the boson breathing mode near $2\omega^{B}_{\perp}$.

Figure S4:  Evolution of cloud size versus time.  Collective oscillations of
transverse width of bosons (top) and fermions (bottom) in the hydrodynamic
regime. The traces were taken at a modulation frequency of
$\omega\,{=}\,1.93\,\omega^{B}_{\perp}$ at an interaction strength of $a_{\rm
BF}=-250\,a_{0}$. The oscillation was excited by modulating the $x$-direction
ODT intensity at 5% for ten cycles. The transverse $y$\- widths are shown
relative to the equilibrium widths as a function of time. Dashed lines are
sinusoidal fits to the data, with the fitted frequencies to the bosons and
fermions as $1.9(1)$ and $2.0(1)\,\omega^{B}_{\perp}$, respectively.

##  6 Evolution of collective modes with temperature

Upon tuning the temperature $T$ at $a_{\rm BF}=-400\,a_{0}$, the experimental
data in Fig. 4 of the main text exhibits three regimes that we discuss here in
greater detail.

At the lowest temperatures $T\lesssim 0.2\,T_{\rm c}$, the thermal cloud of
normal bosons is suppressed, and, as such, the bosonic response to trap
perturbations is fully characterized by the superfluid hydrodynamics of the
BEC. It is noteworthy that in our discussion of the bosonic response, we
neglect the fermions as the fermionic density is at least two orders of
magnitude smaller compared to the bosonic density. As mentioned above, the
BEC’s evolution is well captured by the scaling ansatz. At the same time, the
dynamics of fermions follow the Boltzmann-Vlasov equation, where the BEC
enters as a time-dependent mean-field potential and the incoherent collisions
are suppressed. Our findings, both in Fig. 3 and Fig. 4 of the main text,
indicate that this combined approach captures well the frequencies and
lifetimes of the collective modes of the mixture. One discrepancy between this
theory and the experiment is that the fermionic amplitudes (for instance, of
the quadrupole mode response) are slightly different – this perhaps indicates
that the quantum aspects of the Bose polaron dynamics are important to
describe the full fermionic response. Based on this discussion, the intrinsic
collective response of the mixture at the lowest temperatures is therefore
coherent and obeys superfluid hydrodynamics.

Above $T_{\rm c}$, the analysis is also simplified as the bosons behave as a
weakly interacting thermal gas. The dynamics of the mixture is then expected
to be governed by the coupled Boltzmann equations for the bosonic $n_{\rm
B}(\bm{p},\bm{r},t)$ and fermionic $n_{\rm F}(\bm{p},\bm{r},t)$ distribution
functions:

$\displaystyle\Big{(}\partial_{t}+\frac{\bm{p}}{m_{\rm
B}}\cdot\partial_{\bm{r}}-\partial_{\bm{r}}U_{\rm
eff,B}\cdot\partial_{\bm{p}}\Big{)}n_{\rm B}=I_{\rm BB}+I_{\rm BF},$ (S10)
$\displaystyle\Big{(}\partial_{t}+\frac{\bm{p}}{m_{\rm
F}}\cdot\partial_{\bm{r}}-\partial_{\bm{r}}U_{\rm
eff,F}\cdot\partial_{\bm{p}}\Big{)}n_{\rm F}=I_{\rm FB}.$ (S11)

For the left-hand sides, we assume the Hartree-Fock approximation:

$\displaystyle U_{\rm eff,B}=U_{\rm trap,B}+2g_{\rm B}n_{\rm
B}(\bm{r},t)+g_{\rm BF}n_{\rm F}(\bm{r},t),$ (S12) $\displaystyle U_{\rm
eff,F}=U_{\rm trap,F}+g_{\rm BF}n_{\rm B}(\bm{r},t),$ (S13)

where $n_{\rm
B/F}(\bm{r},t)=\displaystyle\int\frac{d^{3}\bm{p}}{(2\pi)^{3}}n_{\rm
B/F}(\bm{p},\bm{r},t)$ represents the bosonic/fermionic real-space density.
One can neglect the fermionic density in the effective potentials since it is
significantly lower than the bosonic density. At temperatures above $T_{\rm
c}$, one is required to take into account the Bose-Bose, Bose-Fermi, and
Fermi-Bose collision integrals that read:

$\displaystyle I_{\rm BB}(\bm{p},\bm{r},t)$ $\displaystyle=2g_{\rm
B}^{2}\int\frac{d^{3}\bm{p}^{\prime}}{(2\pi)^{3}}\frac{d^{3}\bm{p}_{1}}{(2\pi)^{3}}\frac{d^{3}\bm{p}^{\prime}_{1}}{(2\pi)^{3}}\times(2\pi)^{3}\delta(\bm{p}+\bm{p}^{\prime}-\bm{p}_{1}-\bm{p}_{1}^{\prime})\times(2\pi)\delta(\varepsilon_{\rm
B}(\bm{p})+\varepsilon_{\rm B}(\bm{p}^{\prime})-\varepsilon_{\rm
B}(\bm{p}_{1})-\varepsilon_{\rm B}(\bm{p}_{1}^{\prime}))$
$\displaystyle\qquad\qquad\qquad\times[(1+n_{\rm B}(\bm{p}))(1+n_{\rm
B}(\bm{p}^{\prime}))n_{\rm B}(\bm{p}_{1})n_{\rm B}(\bm{p}_{1}^{\prime})-n_{\rm
B}(\bm{p})n_{\rm B}(\bm{p}^{\prime})(1+n_{\rm B}(\bm{p}_{1}))(1+n_{\rm
B}(\bm{p}_{1}^{\prime}))],$ (S14) $\displaystyle I_{\rm BF}(\bm{p},\bm{r},t)$
$\displaystyle=g_{\rm
BF}^{2}\int\frac{d^{3}\bm{p}^{\prime}}{(2\pi)^{3}}\frac{d^{3}\bm{p}_{1}}{(2\pi)^{3}}\frac{d^{3}\bm{p}^{\prime}_{1}}{(2\pi)^{3}}\times(2\pi)^{3}\delta(\bm{p}+\bm{p}^{\prime}-\bm{p}_{1}-\bm{p}_{1}^{\prime})\times(2\pi)\delta(\varepsilon_{\rm
B}(\bm{p})+\varepsilon_{\rm F}(\bm{p}^{\prime})-\varepsilon_{\rm
B}(\bm{p}_{1})-\varepsilon_{\rm F}(\bm{p}_{1}^{\prime}))$
$\displaystyle\qquad\qquad\qquad\times[(1+n_{\rm B}(\bm{p}))(1-n_{\rm
F}(\bm{p}^{\prime}))n_{\rm B}(\bm{p}_{1})n_{\rm F}(\bm{p}_{1}^{\prime})-n_{\rm
B}(\bm{p})n_{\rm F}(\bm{p}^{\prime})(1+n_{\rm B}(\bm{p}_{1}))(1-n_{\rm
F}(\bm{p}_{1}^{\prime}))],$ (S15) $\displaystyle I_{\rm FB}(\bm{p},\bm{r},t)$
$\displaystyle=g_{\rm
BF}^{2}\int\frac{d^{3}\bm{p}^{\prime}}{(2\pi)^{3}}\frac{d^{3}\bm{p}_{1}}{(2\pi)^{3}}\frac{d^{3}\bm{p}^{\prime}_{1}}{(2\pi)^{3}}\times(2\pi)^{3}\delta(\bm{p}+\bm{p}^{\prime}-\bm{p}_{1}-\bm{p}_{1}^{\prime})\times(2\pi)\delta(\varepsilon_{\rm
F}(\bm{p})+\varepsilon_{\rm B}(\bm{p}^{\prime})-\varepsilon_{\rm
F}(\bm{p}_{1})-\varepsilon_{\rm B}(\bm{p}_{1}^{\prime}))$
$\displaystyle\qquad\qquad\qquad\times[(1-n_{\rm F}(\bm{p}))(1+n_{\rm
B}(\bm{p}^{\prime}))n_{\rm F}(\bm{p}_{1})n_{\rm B}(\bm{p}_{1}^{\prime})-n_{\rm
F}(\bm{p})n_{\rm B}(\bm{p}^{\prime})(1-n_{\rm F}(\bm{p}_{1}))(1+n_{\rm
B}(\bm{p}_{1}^{\prime}))],$ (S16)

where $\varepsilon_{\rm B/F}(\bm{p})=p^{2}/(2m_{\rm B/F})$.

Figure S5:  Thermal cloud responses at $T/T_{\rm c}=0.9$. Here the modulation
of the optical trap, the observables, and the choice of the simulation
parameters follow the experiment (see text). The bosonic thermal cloud
response (a) exhibits a peak at $2\omega_{\rm B}$, while the fermionic one (b)
at $2\omega_{\rm F}$. Besides, the fermions exhibit a weak feature at
$2\omega_{\rm B}$ indicating they are sensitive to the bosonic collective
modes. The trap modulation also perturbs the BEC, which in turn then gives an
additional drive to the two thermal clouds via the time-dependent mean-field
potentials. (c) and (d) show the resulting responses to this perturbation. In
particular, the bosons (c) show a notable enhancement at $2\omega_{\rm B}$, as
in our modeling, the BEC gives rise to a resonant driving of the bosonic
cloud. The fermions (d) now also exhibit much stronger peak at $2\omega_{\rm
B}$ – this contribution comes from both the BEC oscillations and from the
resonantly driven thermal bosonic cloud.

One can then analyze collective modes, and linear response of the mixture in
general, using the method-of-moments (for specific examples of this approach
we refer the reader to Refs. [78, 79]). In particular, we first linearize the
above equations of motion on top of the equilibrium state: $n_{\rm
B/F}(\bm{r},\bm{p},t)=n_{\rm B/F,\rm eq}(\bm{r},\bm{p})+\Delta_{\rm
B/F}(\bm{r},\bm{p})\Phi_{\rm B/F}(\bm{r},\bm{p},t)$, where $\Delta_{\rm
B/F}(\bm{r},\bm{p})=n_{\rm B/F,\rm eq}(\bm{r},\bm{p})[1\pm n_{\rm B/F,\rm
eq}(\bm{r},\bm{p})]$. We then expand the small fluctuations $\Phi_{\rm
B/F}(\bm{r},\bm{p},t)$ in some suitable basis set, which in turn reduces the
infinite-dimensional linear problem to a finite-dimensional matrix equation.
By solving this equation numerically, one obtains linear response properties
of the mixture and, thus, can compare this theory and the experiment. Because
of the complexity of the collision integrals, the actual calculations are
tedious and, for this reason, relegated to a separate manuscript [71] – below
we simply outline the results of such an analysis. In our analysis we also
neglect the small transverse asymmetry of the traps and focus only on the
monopole breathing mode that is now decoupled from the quadrupole one. As
indicated in the bottom panels of Fig. 4b of the main text, this approach
gives an excellent agreement with the experiment. Because the bosonic density
above $T_{\rm c}$ in the experiment is relatively small, the intrinsic
dynamics we observe above $T_{\rm c}$ are best described as collisionless.
Indeed, we find that the bosonic response peaks at $2\omega_{\rm B}$, while
the fermionic one is at $2\omega_{\rm F}$, and the two quantities seem to be
decoupled from each other. A signature of a collision-dominated regime would
be if either of these responses exhibited a notable admixture of both bosonic
and fermionic resonances – this situation is expected to emerge for larger
bosonic densities [71].

Figure S6:  Thermal cloud responses at $T/T_{\rm c}=1.3$ to a broadband
optical quench. Here we plot slightly different quantities than in Fig. S5,
namely: the bosonic and fermionic spectral functions ${\cal
A}_{\rho^{2}}(\omega)$ encode the time-dependent changes of the transverse
cloud sizes to a $\delta(t)$-like modulations of the transverse optical traps.
The bosonic response (left) exhibits a resonance at $2\omega_{\rm B}$, while
the fermions (right) are peaked at $2\omega_{\rm F}$. The inset shows that the
fermionic response shows a weak feature at $2\omega_{\rm B}$, indicating that
the fermions can be viewed as a sensor of the bosonic subsystem.

The most challenging, yet perhaps the most interesting regime is when
$0.3T_{\rm c}\lesssim T\lesssim T_{\rm c}$. We point out that a theory for
this situation is not yet even fully established. Nonetheless, we can combine
the low and high temperature approaches to attempt to understand the
intermediate temperatures. In this intermediate temperature regime, both the
BEC and the bosonic thermal cloud are important. As for the case of the lowest
temperatures, we approximate the BEC dynamics within the scaling ansatz.
Similar to the case of high temperatures, we take into account the collision
processes and describe the mixture of the thermal bosons and fermions within
the Boltzmann framework. The BEC creates time-dependent Vlasov-like potentials
to the thermal clouds, but we neglect the back-action from the clouds to the
condensate dynamics; in particular, we neglect incoherent scatterings off the
condensate. In other words, the BEC learns about the normal bosons only at the
level of the equilibrium wave function that is still well-described within the
Thomas-Fermi approximation.

Figure S5 summarizes the results of this approach. The response of either of
the thermal clouds can be viewed as a sum of two terms – the first term comes
from the direct optical driving of the trap potentials; the second one is due
to the time-dependent BEC, which actually also acquires its dynamics through
this trap modulation. As shown in Fig. S5a,b, the first term results in
responses similar to the ones we had for $T>T_{\rm c}$, i.e., the bosonic
cloud exhibits a peak at $2\omega_{\rm B}$, while the fermionic one – at
$2\omega_{\rm F}$ [see also Fig. S6]. Now, since in our approach the small
transverse anisotropy is neglected, the BEC and the bosonic cloud both exhibit
a resonance at the same frequency $2\omega_{\rm B}$. This in turn implies that
the second term drives the bosonic cloud resonantly, and our calculations in
Fig. S5c indeed show a notable enhancement of the bosonic signal at
$2\omega_{\rm B}$. This should be contrasted with the experiment in Fig. 4 of
the main text, where the bosonic cloud is driven off-resonance because of the
small trap anisotropy and, as such, no such dramatic enhancement is observed.
In describing the bosonic thermal cloud, scattering off the condensate should
smoothen out the signal and therefore may be important to take into account.
In addition, our method-of-moments in its present form overestimates the BEC
driving to the thermal clouds in situations of small condensates, i.e., for
$T\simeq T_{\rm c}$; one way to circumvent this is a better choice of the
basis functions [71] that have a finite support of the BEC size.

Our calculations in Fig. S5d indicate that the time-dependent BEC also gives
rise to the development of a resonance at $2\omega_{\rm B}$ in the fermionic
response. To certain degree, this agrees with the experiment, where the
fermions were observed to exhibit peaks from both the BEC and normal bosons.
In our approach, as we already discussed, these two peaks are not
distinguishable, and, as such, it is not entirely clear which of the bosonic
modes the fermions are more sensitive to. As such, a model that extends our
framework to include the anisotropy should resolve this issue. We note that,
as our analysis at the lowest temperatures indicates, we do expect the
fermions to exhibit a peak at the BEC resonance. However, to get a similar
peak but at the bosonic thermal cloud resonance, one is required to consider
not only the incoherent scatterings between normal bosons and fermions, but
also mean-field corrections to the effective fermionic potential, cf. Eq.
(S13). To further appreciate this point, in Fig. S6(right), we show that even
above $T_{\rm c}$, the fermions do couple to the normal bosons and show a
clear (but weak) feature at the bosonic resonance – we explore this further in
Ref. [71]. It is also interesting that the bosons in turn show very little
response at the fermionic resonance, for the reason that the boson density is
much larger compared to the fermionic one. A possible conclusion from this is
that the less dense subsystem could be used as a good sensor of the collective
dynamics of the more dense one.