# A proposal for detecting the spin of a single electron in superfluid helium

Jinyong Ma  Department of Physics, Yale University, New Haven, Connecticut
06520, USA  Yogesh S. S. Patil  Department of Physics, Yale University, New
Haven, Connecticut 06520, USA  Jiaxin Yu  Department of Applied Physics, Yale
University, New Haven, Connecticut 06520, USA  Yiqi Wang  Department of
Applied Physics, Yale University, New Haven, Connecticut 06520, USA  Jack G.
E. Harris<EMAIL_ADDRESS>Department of Physics, Yale University, New
Haven, Connecticut 06520, USA  Department of Applied Physics, Yale University,
New Haven, Connecticut 06520, USA  Yale Quantum Institute, Yale University,
New Haven, Connecticut 06520, USA

This supplemental material presents detailed calculations for the analyses in
the main paper. Section I provides the theory for acoustic trapping of an
electron bubble. Section II discusses the optomechanical interaction between
an electron bubble and an optical mode. Section III estimates the mechanical
damping of the bubble motion. Section IV gives details of the three protocols
for single-spin detection proposed in the main paper. Section V analyzes the
noise sources influencing the system performance. Section VI lists the various
parameters used in the calculations.

###### Contents

  1. I Sound wave trapping
    1. I.1 Energy and Young’s modulus of electron bubbles
    2. I.2 Potential of an acoustic trap
    3. I.3 Generation of the acoustic trap with a piezoelectric actuator
    4. I.4 Detection of the acoustic trap
  2. II Interaction between electron bubble and cavity optical field
  3. III Damping Mechanism
  4. IV Single spin detection
    1. IV.1 Modulation of magnetic gradient
    2. IV.2 Rabi oscillation
    3. IV.3 Adiabatic spin flip
  5. V Noise analysis
    1. V.1 Optomechanical interaction
    2. V.2 Heterodyne detection
  6. VI Table of parameters

##  I Sound wave trapping

###  I.1 Energy and Young’s modulus of electron bubbles

We consider the following simple model for describing the total energy of an
electron bubble (EB) in liquid helium [1, 2, 3, 4]

$\displaystyle E_{{\rm tol}}$ $\displaystyle=$
$\displaystyle\frac{h^{2}}{8m_{\rm e}R^{2}}+4\pi R^{2}\sigma+\frac{4}{3}\pi
R^{3}P$ (1)

where the first term is the ground state energy of the confined electron, the
second term is the surface tension energy of the bubble, and the last term
represents the energy stored by the surrounding pressure (a more sophisticated
model is described in Ref. [2]). Here the electron mass is $m_{\rm e}$, the
bubble radius is $R$, the surface tension of liquid helium is $\sigma=0.375$
erg/cm2 [5], and the pressure experienced by the EB is $P$. When $P=0$, the EB
radius is found by minimizing the total energy with respect to $R$, giving:

$\displaystyle R_{0}$ $\displaystyle=$ $\displaystyle\left(\frac{h^{2}}{32\pi
m_{\rm e}\sigma}\right)^{1/4}\approx 1.9\ \rm{nm}$ (2)

To find the response of the EB to an acoustic wave in the surrounding liquid,
we rewrite Eq. (1) as

$\displaystyle P$ $\displaystyle=$ $\displaystyle\frac{h^{2}}{16\pi m_{\rm
e}}\left(\frac{3}{4\pi}V_{\rm
EB}\right)^{-5/3}-2\sigma\left(\frac{3}{4\pi}V_{\rm EB}\right)^{-1/3}$ (3)

where $V_{\rm{EB}}=4\pi R^{3}/3$ is the volume of the EB. It is then
straightforward to calculate the Young’s modulus ($E_{{\rm EB}}$) of the EB:

$\displaystyle E_{{\rm EB}}$ $\displaystyle=$ $\displaystyle-V_{\rm
EB}\frac{\partial P}{\partial V_{\rm EB}}\approx 530\ \rm{kPa}$ (4)

###  I.2 Potential of an acoustic trap

An EB in a standing acoustic wave experiences a force exerted by the acoustic
field. As described in the main text, our proposal employs this force to trap
the EB. The potential energy of an EB in an acoustic wave (with wave vector
$k$) can be written as [6]

$\displaystyle U_{\rm ac}$ $\displaystyle=$ $\displaystyle V_{\rm EB}W_{{\rm
ac}}\left[f_{1}\cos^{2}(kz)-\frac{3f_{2}}{2}\sin^{2}(kz)\right]$ (5)
$\displaystyle=$ $\displaystyle V_{\rm EB}W_{{\rm
ac}}\left[f_{1}-\Phi\sin^{2}(kz)\right]$

where $z$ is the coordinate of the EB along the cavity axis (see Fig. 1 of the
main text), and

$\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle 1-\frac{E_{{\rm
He}}}{E_{{\rm EB}}}$ (6) $\displaystyle f_{2}$ $\displaystyle=$
$\displaystyle\frac{2(\rho_{{\rm EB}}/\rho_{\rm He}-1)}{2\rho_{{\rm
EB}}/\rho_{\rm He}+1}$ (7) $\displaystyle\Phi$ $\displaystyle=$ $\displaystyle
f_{1}+\frac{3}{2}f_{2}$ (8)

Here $W_{{\rm ac}}$ is the acoustic energy density (see below), and the
Young’s modulus of the liquid He is $E_{{\rm He}}=\rho_{\rm He}c_{{\rm
He}}^{2}$ where $c_{{\rm He}}=238$ m/s is the velocity of sound in liquid He
and $\rho_{\rm He}=145$ kg/m3 is its mass density. Since the interior of the
EB is close to vacuum, we take its mass density $\rho_{{\rm EB}}=0$. This
gives $f_{1}\approx-14$, $f_{2}\approx-2$, and $\Phi\approx-17$. The quantity
$\Phi$ is known as the acoustophoretic contrast factor, and its sign
determines whether the bubble is trapped at the pressure nodes or antinodes of
the acoustic wave. Here $\Phi<0$, so the EB will be trapped at pressure
antinodes.

The depth of the trapping potential in Eq. (5) is $U_{0}=V_{\rm EB}\Phi W_{\rm
ac}$. Throughout this paper, we assume $U_{0}/k_{{\rm B}}=300$ mK (chosen to
be $10\times$ greater than the assumed operating temperature $T=30$ mK). This
corresponds to an acoustic energy density of $W_{\rm ac}=8.5$
$\text{\,}\mathrm{J}\mathrm{/}\mathrm{m}^{3}$. Generating such an acoustic
wave is discussed in Section I.3.

To find the trap frequency of the EB, we expand $U_{\rm ac}$ to second order
in $z$ around its minimum at $z_{0}$, i.e.

$\displaystyle U(z_{0}+\delta z)$ $\displaystyle\approx$ $\displaystyle
U(z=z_{0})+\frac{1}{2}U^{\prime\prime}(z=z_{0})\delta z^{2}$ (9)
$\displaystyle=$ $\displaystyle V_{\rm EB}W_{\rm ac}f_{1}+\frac{1}{2}k_{{\rm
EB}}\delta z^{2}$

where $k_{{\rm EB}}=-2V_{\rm EB}W_{\rm ac}k^{2}\Phi\cos(2kz_{0})$ is the
spring constant of the trap. This results in a trap frequency $\omega_{{\rm
EB}}=\sqrt{k_{{\rm EB}}/m_{{\rm EB}}}\approx 2\pi\times 2.9$ MHz along the
$z$-direction (here $m_{\rm EB}=1.6\times 10^{-24}$ kg is the bubble’s
effective mass [7]).

###  I.3 Generation of the acoustic trap with a piezoelectric actuator

The acoustic wave is generated by vibrating one of the cavity fibers with a
piezoelectric actuator at frequency $\omega_{\rm ac}$. If the fiber
displacement is $z_{{\rm f}}=z_{{\rm amp}}\sin(\omega_{{\rm ac}}t)$, the power
in the sound wave ($\mathcal{P}_{\rm ac}$) emitted into the liquid helium is
[8]

$\displaystyle\mathcal{P}_{{\rm ac}}$ $\displaystyle=$
$\displaystyle\sigma_{\rm rad}\rho_{\rm He}c_{{\rm He}}A\left\langle
v^{2}\right\rangle$ (10) $\displaystyle=$ $\displaystyle\sigma_{\rm
rad}\rho_{\rm He}c_{{\rm He}}A\left\langle\left(z_{{\rm amp}}\omega_{{\rm
ac}}\cos(\omega_{{\rm ac}}t)\right)^{2}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{\sigma_{\rm rad}\rho_{\rm He}c_{{\rm He}}A\omega_{{\rm
ac}}^{2}z_{{\rm amp}}^{2}}{2}$

where $\sigma_{\rm rad}$ is the radiation ratio (or efficiency), $v$ is the
speed of the fiber tip of area $A$, and $\langle\cdot\rangle$ denotes the time
average. $\mathcal{P}_{\rm ac}$ can also be written in terms of sound pressure
or acoustic energy density

$\displaystyle\mathcal{P}_{{\rm ac}}$ $\displaystyle=2AW_{{\rm ac}}c_{{\rm
He}}$ (11)

We obtain the required vibration amplitude by combining Eqs. (10)-(11).

$\displaystyle z_{{\rm amp}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{4W_{{\rm ac}}}{\sigma_{\rm rad}\rho_{\rm
He}\omega_{{\rm ac}}^{2}}}$ (12)

Thus, to produce a sound wave that results in a trap depth corresponding to
$300$ mK (see Sec I.B), the amplitude of the fiber motion would need to be
$z_{\rm amp}\approx 240$ pm. However if the fiber motion is resonant with an
acoustic mode of the cavity, this is reduced by a factor of the acoustic
mode’s quality factor $Q_{\rm{ac}}$. For the devices demonstrated in Ref. [9,
10], $Q_{\rm{ac}}\approx 10^{5}$. As a result, the required $z_{\rm
amp}\approx 2.4$ fm. This amount of motion at comparable frequency is
demonstrated, for example in [11].

###  I.4 Detection of the acoustic trap

We first look at the acoustic energy stored in the fiber cavity
$\mathcal{E}_{{\rm ac}}$, which is given by [12]

$\displaystyle\mathcal{E}_{{\rm ac}}=\frac{1}{2}E_{{\rm{\rm
He}}}\int\left|\frac{\delta\rho_{\rm He}(r)}{\rho_{\rm
He}}\right|^{2}dV=\frac{1}{2E_{{\rm He}}}\int\left|P(r)\right|^{2}dV$ (13)

where $P(r)$ is the pressure profile of the acoustic mode, and
$\delta\rho_{\rm He}(r)$ is the corresponding density variation. Using
$W_{{\rm ac}}=\frac{P_{0}^{2}}{4E_{{\rm He}}}$ [6] (where $P_{0}$ is the
amplitude of the acoustic mode), and the well-known mode profiles for a Fabry-
Perot cavity, we can write the time-averaged acoustic energy as

$\displaystyle\left\langle\mathcal{E}_{{\rm
ac}}\right\rangle=\frac{1}{2}W_{{\rm ac}}V_{\rm ac}$ (14)

with $V_{{\rm ac}}$ denoting the mode volume of the acoustic cavity.

The acoustic energy can also be written in terms of the phonon number $n_{\rm
ac}$

$\displaystyle\left\langle\mathcal{E}_{{\rm ac}}\right\rangle$
$\displaystyle=$ $\displaystyle\left(n_{\rm
ac}+\frac{1}{2}\right)\hbar\omega_{{\rm ac}}$ (15)

Equations (13)-(15) provide the phonon number $(n_{\rm ac}\thickapprox
1.6\times 10^{11})$ and the corresponding amplitude of the helium density
variation $(\frac{\delta\rho_{\rm He}}{\rho_{{\rm He}}}\thickapprox 2\times
10^{-3})$ required to achieve the trap depth ($300$ mK) for the acoustic mode
with $\omega_{\rm ac}/2\pi=320$ MHz, as considered in the main text.

The acoustic trap can be optically detected using the optomechanical
interaction between the optical and acoustic modes of the fiber cavity. Under
the drive of a monochromatic laser with frequency $\omega_{l}$ and amplitude
$c_{l}$, the equations of motion for the amplitudes $c$ and $d$ of the optical
and acoustic modes (with resonance frequencies $\omega_{\rm c}$ and
$\omega_{\rm ac}$, and dampings $\kappa$ and $\gamma_{\rm ac}$) can be written
in the rotating frame of $\omega_{l}$ as [13, 12],

$\displaystyle\dot{c}$ $\displaystyle=$
$\displaystyle-\left(\frac{\kappa}{2}-i\bar{\Delta}\right)c+\sqrt{\kappa_{\rm
ex}}c_{l},$ (16) $\displaystyle\dot{d}$ $\displaystyle=$
$\displaystyle-\left(\frac{\gamma_{{\rm ac}}}{2}+i\omega_{{\rm
ac}}\right)d-ig_{0,{\rm ac}}c^{\dagger}c+d_{{\rm ext}}e^{-i\omega_{\rm ac}t}$
(17)

where $\Delta_{l}=\omega_{l}-\omega_{\rm c}$. Here we define the effective
cavity detuning as $\bar{\Delta}=\Delta_{l}-ig_{0,{\rm ac}}(d+d^{\dagger})$
and denote $d_{\rm ext}$ as the external acoustic drive applied by the fiber
vibration. The laser is injected into the cavity with input coupling rate
$\kappa_{\rm ex}$, and the single-photon optomechanical coupling rate is
$g_{\rm 0,ac}$. We consider a weak probe beam such that the radiation pressure
on the acoustic wave is ignored. Eq.17 is then

$\displaystyle\dot{d}$ $\displaystyle=$
$\displaystyle-\left(\frac{\gamma_{{\rm ac}}}{2}+i\omega_{{\rm
ac}}\right)d+d_{{\rm ext}}e^{-i\omega_{\rm ac}t}$ (18)

The solution to this equation is simply $d=d_{1}e^{-i\omega_{ac}t}$, where
$d_{1}=2d_{{\rm ext}}/\gamma_{{\rm ac}}$.

Figure 1S: (a) Schematic of the laser frequencies for detecting the acoustic
trap. The trap is formed by a density wave that oscillates with frequency
$\omega_{\rm ac}$ and produces sidebands (at frequency $\omega_{\rm ac}$) on a
“probe” laser tone that drives the optical cavity mode with $n_{\rm opt}=129$.
These sidebands can be detected via standard heterodyne techniques using the
“local oscillator” tone. (b) Schematic of the setup for detecting the acoustic
trap. A laser beam passes through an acousto-optic modulator to produce the
probe tone and the local oscillator. A phase modulator is used to add
sidebands to the probe tone for the Pound-Drever-Hall (PDH) lock (not shown).
Both tones are directed to the cavity via a circulator. One of the cavity
fibers is driven via an ultrasonic actuator at frequency $\omega_{\rm ac}$ to
produce the acoustic trap. The light reflected from the cavity is sent to a
photodiode. Optical path: orange lines. Electrical path: blue lines.

In this analysis, the optical resonance frequency is modulated with a depth
$g_{0,{\rm ac}}(d_{1}+d_{1}^{\dagger})$. With $d_{1}=\sqrt{n_{\rm ac}}\approx
4\times 10^{5}$ and the device parameters considered here ($\kappa/2\pi=15$
MHz, $L_{\rm cav}=100$ $\mu$m, $\mathcal{F}_{\rm opt}=10^{5}$, $g_{\rm
0,ac}/2\pi=3.6$ kHz), the cavity modulation depth $\approx 190\kappa$. The
approach for detecting the acoustic trap is presented in Fig. 1S.

##  II Interaction between electron bubble and cavity optical field

Since the electron bubble radius ($R_{0}\approx 1.9$ nm) is much smaller than
the optical wavelength ($\lambda_{\rm opt}=1550$ nm), the Hamiltonian for a
bubble coupling to the optical field (with frequency $\omega_{\rm c}$) can be
written as [14, 15]

$\displaystyle H=-\frac{1}{2}\alpha\left|E\right|^{2}=-\frac{\hbar\omega_{{\rm
c}}\alpha}{2\varepsilon_{{\rm He}}V_{{\rm opt}}}f(z)a^{\dagger}a$ (19)

where the EB polarizability is

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle 3\varepsilon_{{\rm
He}}V_{{\rm EB}}\text{{\rm Re}}\left\\{\frac{n_{{\rm EB}}^{2}/n_{{\rm
He}}^{2}-1}{n_{{\rm EB}}^{2}/n_{{\rm He}}^{2}+2}\right\\}$ (21)

and $n_{\rm EB}=1$ and $n_{\rm He}=1.028$ are the refractive indices of the EB
and the liquid He, $\epsilon_{\rm He}\approx n_{\rm He}^{2}$ is the
permittivity of liquid He, and $V_{{\rm opt}}$ is the mode volume of the
optical cavity. Here $f(z)$ is the profile of the optical cavity mode along
the $z$ direction, given by

$\displaystyle f(z)$ $\displaystyle=$
$\displaystyle\frac{1}{1+(z_{0}+z)^{2}/z_{{\rm
R}}^{2}}\cos^{2}\left(k_{\rm{opt}}(z_{0}+z)\right)$ (22)
$\displaystyle\approx$
$\displaystyle\cos^{2}\left(k_{\rm{opt}}z_{0}\right)-k_{\rm{opt}}\sin(2k_{\rm{opt}}z_{0})z-k_{\rm{opt}}^{2}\cos(2k_{\rm{opt}}z_{0})z^{2}+\mathcal{O}(z^{3})$

with $k_{\rm{opt}}=2\pi/\lambda_{\rm opt}$. The pre-factor $(1+(z/z_{\rm
R})^{2})^{-1}\approx 1$ for the cavities considered in Ref. [13]. The term
proportional to $z$ leads to the usual linear optomechanical interaction

$\displaystyle H$ $\displaystyle=$ $\displaystyle\hbar
g_{0}a^{\dagger}a(b^{\dagger}+b)$ (23)

with

$\displaystyle g_{0}$ $\displaystyle=$ $\displaystyle\frac{\omega_{{\rm
c}}\alpha k_{\rm{opt}}}{2\varepsilon_{{\rm He}}V_{{\rm
opt}}}\sin(2k_{\rm{opt}}z_{0})z_{{\rm zpf}}$ (24)

where $z_{{\rm zpf}}=\sqrt{\hbar/2m_{\rm EB}\omega_{{\rm EB}}}$ is the zero
point motion of the EB, $z=z_{\rm zpf}(b+b^{\dagger})$, and $b$ denotes the
annihilation operator of the EB motion. As discussed in the main paper, the
equilibrium position of the trapped bubble corresponds to zero optomechanical
coupling of the optical mode used to detect the acoustic trap. To maximize the
coupling rate to the EB motion, one can use a different optical mode. The
phase difference in the expression for $g_{0}$ (see Eq. 24) for two optical
modes of wavenumbers $k_{q+n}$ and $k_{q}$ (i.e. of longitudinal indices $q+n$
and $q$) at the same location $z_{0}$ is

$\displaystyle 2(k_{q+n}-k_{q})z_{0}$ $\displaystyle=$
$\displaystyle\frac{2\pi n}{L_{\rm cav}}z_{0}$ (25)

Considering an adjacent optical mode, i.e., $n=1$, the phase difference is
$\pi/2$ at $z_{0}=L_{\rm cav}/4$, which maximizes the coupling rate.

##  III Damping Mechanism

An EB’s motion may damped by a variety of mechanisms. We begin by noting some
of the damping sources that are not expected to be relevant in the regimes
considered for this study. Then we discuss the two sources that are expected
to be relevant.

When the EB moves at speed (relative to the superfluid) greater than the
critical velocity $v_{L}\approx 50$ m/s it generates vortex rings and
ripplons, leading to strong damping [16, 17]. This damping mechanism is not
expected to be relevant, as even for the largest amplitude of motion
considered in the main text ($z_{\rm{EB}}=100$ nm) the EB’s speed $<2$ m/s.

An EB may also be damped via interactions with remanent vortex lines. To avoid
this effect, the number of vortex lines in the cavity volume will be minimized
by filling the experimental cell via a Rollin film.

The EB may be damped by the thermal excitations in the superfluid. However,
all thermal excitations are frozen out at the temperatures considered here
($T<100$ mK) except for phonons.

Lastly, 3He impurities are inevitably present at some concentration ($\approx
1$ ppm for natural He, and $\approx 10^{-6}$ ppm in isotopically purified He).
In the regimes that are relevant for this work, the 3He atoms may be regarded
as a gas whose collisions with an EB represent a source of damping.

In summary, the EB’s damping is expected to be dominated by two sources:
collisions with thermal phonons, and collisions with 3He impurities. We
estimate the damping rates arising from each of these sources by using the
results of Ref. [18] (for the phonons) and Ref. [19] (for the 3He).

In particular, the EB’s phonon-limited mobility $\mu_{\rm p}$ is given by Eq.
(8) and Eq. (9) of Ref. [18], and the 3He-limited mobility $\mu_{\rm h}$ is
given by the expressions in the first portion of Section III of Ref. [19].
These mobilities are then converted to damping rates via $\gamma_{\rm
p,h}=e/m\mu_{\rm p,h}$.

In Fig. 2S, the blue curve shows the phonon contribution to the damping
(calculated with $R_{0}=1.9$ nm, rather than the slightly smaller value
assumed in Ref. [18]), and the red curves show the 3He contribution for a
range of $x_{3}$ (the concentration of 3He). The total EB damping $\gamma_{\rm
EB}=\gamma_{\rm h}+\gamma_{\rm p}$ is converted to $Q_{\rm EB}=\omega_{\rm
EB}/\gamma_{\rm EB}$ and is shown in Fig. 3 of the main paper (as red curves)
for the same values of $x_{3}$.

Figure 2S: Damping rate of the electron bubble’s motion. The contribution from
thermal phonons (blue) and various concentrations of 3He impurities (red) are
shown, both as a function of temperature.

##  IV Single spin detection

As discussed in the main text, the EB experiences a force along the $z$
direction given by:

$\displaystyle F_{{\rm mag}}=\mu_{\rm S}G$ (26)

where $\mu_{\rm S}=-g\mu_{\rm B}/2$ is the $z$-component of the electron’s
spin magnetic moment, $\mu_{\rm B}$ and $g$ denote the Bohr magneton and
gyromagnetic ratio of the electron, and $G$ is the gradient (with respect to
$z$) of the $z$ component of the magnetic field.

In the following subsections, we will consider three approaches to obtain an
alternating magnetic force that drives the bubble motion on resonance.

###  IV.1 Modulation of magnetic gradient

Modulating the magnetic gradient at the trap frequency is the most
straightforward approach to drive the EB’s motion resonantly. In this case,
the magnetic force is simply

$\displaystyle F_{{\rm mag}}$ $\displaystyle=$ $\displaystyle\mu_{\rm
S}G\cos(\omega_{\rm EB}t+\phi)$ (27)

as described in the main text.

###  IV.2 Rabi oscillation

In this subsection, we consider using Rabi oscillations to produce the
oscillating force. We consider the following magnetic field:

$\displaystyle\vec{B}$ $\displaystyle=$ $\displaystyle
B_{0}\hat{z}+B_{1}\left(\cos(\omega t)\hat{x}-\sin(\omega t)\hat{y}\right)$
(28)

where the static field $B_{0}$ is used to induce the Zeeman splitting and
$B_{1}$ is the amplitude of the microwaves that drive the spin resonance. The
Hamiltonian of the system can be expressed as

$\displaystyle
H=-\vec{\mu}\cdot\vec{B}=-\frac{\hbar}{2}\omega_{0}\sigma_{z}-\frac{\hbar}{2}\omega_{1}\left(\sigma_{x}\cos(\omega
t)-\sigma_{y}\sin(\omega t)\right)$ (29)

with $\omega_{0}=\frac{\mu_{\rm B}g}{\hbar}B_{0}$ and
$\omega_{1}=\frac{\mu_{\rm B}g}{\hbar}B_{1}$. Here
$\vec{\mu}=-\frac{1}{2}g\mu_{\rm
B}(\sigma_{x}\hat{x}+\sigma_{y}\hat{y}+\sigma_{z}\hat{z})$ where
$\sigma_{x,y,z}$ are the Pauli matrices.

If the electron is initially in state $\left|\downarrow\right\rangle$, then
the probability of the electron being found in spin
$\left|\uparrow\right\rangle$ is given by

$\displaystyle P(t)$ $\displaystyle=$
$\displaystyle\left(\frac{\omega_{1}}{\Omega}\right)^{2}\sin^{2}\left(\frac{\Omega
t}{2}\right)$ (30)

Here $\Omega=\sqrt{\Delta^{2}+\omega_{1}^{2}}$ is the Rabi frequency where
$\Delta=\omega-\omega_{0}$ is the detuning of the microwaves from the spin
resonance. Assuming the initial magnetic moment is $\mu_{\rm S,0}$, we have
the equation for magnetic force as follows

$\displaystyle F_{{\rm mag}}$ $\displaystyle=$ $\displaystyle\mu_{\rm
S,0}P(t)G$ (31)

It is straightforward to tune $B_{1}$ and $\Delta$ so that $\Omega=\omega_{\rm
EB}$, producing an alternating force whose frequency is $\omega_{\rm EB}$.

###  IV.3 Adiabatic spin flip

The third approach follows the protocol of Ref. [20, 21], and involves the
modulation of the uniform field (at frequency $\omega_{\rm EB}$ and amplitude
$B_{{\rm mod}}$) about a mean value $\bar{B}_{0}$. The frequency of the
microwave field is $\omega_{0}=\frac{\mu_{\rm B}g}{\hbar}\bar{B}_{0}$, so that
it is resonant with the Zeeman splitting when $|\vec{B}|=\bar{B}_{0}$. Thus,
the overall magnetic field is

$\displaystyle\vec{B}$ $\displaystyle=$
$\displaystyle\left(\bar{B}_{0}+B_{{\rm mod}}\sin(\omega_{\rm
EB}t)\right)\hat{z}+B_{1}\left(\cos(\omega_{0}t)\hat{x}-\sin(\omega_{0}t)\hat{y}\right)$
(32)

This results in the following Hamiltonian:

$\displaystyle H$ $\displaystyle=$
$\displaystyle-\vec{\mu}\cdot\vec{B}=-\frac{\hbar}{2}\left(\omega_{0}+\omega_{{\rm
mod}}\sin(\omega_{\rm
EB}t)\right)\sigma_{z}-\frac{\hbar}{2}\omega_{1}\left(\sigma_{x}\cos(\omega_{0}t)-\sigma_{y}\sin(\omega_{0}t)\right)$

with $\omega_{{\rm mod}}=\frac{\mu_{\rm B}g}{\hbar}B_{{\rm mod}}$. For
convenience, we rewrite this equation in the frame rotating at $\omega_{0}$:

$\displaystyle H^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{\hbar}{2}\omega_{{\rm mod}}\sin(\omega_{\rm
EB}t)\sigma_{z}-\frac{\hbar}{2}\omega_{1}\sigma_{x}$ (33)

If the time-dependence of the Hamiltonian in Eq. (33) is slow enough that the
spin state evolves adiabatically, then the z component of the magnetization is
[22]

$\displaystyle\mu_{\rm s}(t)$ $\displaystyle=$ $\displaystyle\pm\frac{B_{{\rm
mod}}\sin(\omega_{\rm EB}t)}{\sqrt{B_{1}^{2}+B_{{\rm mod}}^{2}\sin(\omega_{\rm
EB}t)^{2}}}\mu_{{\rm eff}}$ (34)

where $\mu_{{\rm eff}}=\frac{1}{2}g\mu_{{\rm B}}$, and the magnetic force
along the $z$ direction is simply

$\displaystyle F_{{\rm mag}}$ $\displaystyle=$ $\displaystyle\mu_{\rm s}(t)G$
(35)

To estimate the probability that the spin evolves adiabatically under the in
Eq. (33), we model each passage through the avoided crossing (which occurs
when $t=n\pi/\omega_{\rm EB}$, with $n$ an integer) using the Landau-Zener
formula. Specifically, we replace $\sin(\omega_{\rm EB}t)$ in Eq. (33) with
$\omega_{\rm EB}t$, so that the resulting Hamiltonian is equivalent to

$\displaystyle H^{\prime}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\alpha
t\sigma_{z}+{\rm Re}(H_{12})\sigma_{x}-{\rm Im}(H_{12})\sigma_{y}$ (36)

where $H_{12}=-\frac{\hbar}{2}\omega_{1}$ and
$\alpha=-\hbar\omega^{\prime}\omega_{{\rm EB}}$. In this case, the probability
to find the system in the state $\left|\uparrow\right\rangle$ at $t=\infty$
given that the system is in the state $\left|\uparrow\right\rangle$ at
$t=-\infty$ is the well-known expression

$\displaystyle P_{{\rm
LZ}}(\uparrow\mid\uparrow)=\exp\left(-2\pi\frac{\left|H_{12}\right|^{2}}{\hbar\left|\alpha\right|}\right)=\exp\left(-\frac{\pi}{2}\frac{\omega_{1}^{2}}{\omega_{\rm
EB}\omega_{{\rm mod}}}\right)$ (37)

As a result, the probability that the spin state evolves adiabatically (and
hence the Eq. (34) holds) is $1-P_{{\rm LZ}}(\uparrow\mid\uparrow)$. This
probability approaches unity when $\omega_{\rm EB}\omega_{{\rm
mod}}\ll\omega_{1}^{2}$.

Equation (37) is used to calculate the probability of adiabatic evolution
given in Fig. 4(c,d) of the main text.

##  V Noise analysis

In this section, we start with the equations of motion for the EB and the
optical mode and analyse the noise sources associated with measuring the EB
displacement.

###  V.1 Optomechanical interaction

The Hamiltonian describing the system can be written as

$\displaystyle H$ $\displaystyle=$
$\displaystyle\hbar\omega_{c}a^{\dagger}a+\hbar\omega_{\rm
EB}b^{\dagger}b+\hbar g_{\rm 0}a^{\dagger}a(b^{\dagger}+b)+F_{{\rm
mag}}x_{{\rm zpf}}(b^{\dagger}+b)$ (38)

The first three terms characterize the optical mode of the fiber cavity, the
EB motion, and their interaction, respectively. The last term describes the
magnetic force imprinted from the electron spin [23]. Adding the optical
($\kappa$) and mechanical ($\gamma_{\rm EB}$) damping terms and environmental
fluctuations, the Heisenberg equations of motion are

$\displaystyle\dot{a}$ $\displaystyle=$
$\displaystyle-\left(\frac{\kappa}{2}+i\omega_{c}\right)a-ig_{\rm
0}(b^{\dagger}+b)a+\sqrt{\kappa_{\rm in}}\xi_{\rm in}+\sqrt{\kappa_{\rm
ex}}(a_{\rm ex}+\xi_{\rm ex})$ (39) $\displaystyle\dot{b}$ $\displaystyle=$
$\displaystyle-\left(\frac{\gamma_{\rm EB}}{2}+i\omega_{\rm
EB}\right)b-ig_{\rm 0}a^{\dagger}a-\frac{i}{\hbar}F_{{\rm mag}}x_{{\rm
zpf}}+\sqrt{\gamma_{\rm EB}}\eta$ (40)

where $\xi_{\rm in}$ and $\xi_{\rm ex}$ denote the optical noises coupled to
the system via internal and external channels with the corresponding coupling
rates $\kappa_{\rm in}$ and $\kappa_{\rm ex}$. We have also included the
mechanical noise $\eta$ that comes from the thermal bath. The noise operators’
correlations are

$\displaystyle\left\langle\xi_{i}(t)\xi_{j}(t^{{}^{\prime}})\right\rangle$
$\displaystyle=$ $\displaystyle 0$ (41)
$\displaystyle\left\langle\xi_{i}^{\dagger}(t)\xi_{j}(t^{{}^{\prime}})\right\rangle$
$\displaystyle=$ $\displaystyle 0$ (42)
$\displaystyle\left\langle\xi_{i}(t)\xi_{j}^{\dagger}(t^{{}^{\prime}})\right\rangle$
$\displaystyle=$ $\displaystyle\delta_{ij}\delta(t-t^{{}^{\prime}})$ (43)
$\displaystyle\left\langle\eta(t)\eta(t^{{}^{\prime}})\right\rangle$
$\displaystyle=$ $\displaystyle 0$ (44)
$\displaystyle\left\langle\eta^{\dagger}(t)\eta(t^{{}^{\prime}})\right\rangle$
$\displaystyle=$ $\displaystyle n_{\rm th}\delta(t-t^{{}^{\prime}})$ (45)
$\displaystyle\left\langle\eta(t)\eta^{\dagger}(t^{{}^{\prime}})\right\rangle$
$\displaystyle=$ $\displaystyle(n_{\rm th}+1)\delta(t-t^{{}^{\prime}})$ (46)

where $n_{\text{th}}$ is the thermal phonon occupancy of the EB’s motion.

We consider a monochromatic optical drive, i.e., $a_{\rm
ex}=a_{l}e^{-i\omega_{l}t}$. In the frame rotating at $\omega_{c}$, we rewrite
Eqs. (39) and (40) as

$\displaystyle\dot{a}$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{2}a-ig_{\rm 0}(b^{\dagger}+b)a+\sqrt{\kappa_{\rm
in}}\xi_{\rm in}+\sqrt{\kappa_{\rm ex}}\xi_{\rm ex}+\sqrt{\kappa_{\rm
ex}}a_{l}e^{-i\Delta_{l}t}$ (47) $\displaystyle\dot{b}$ $\displaystyle=$
$\displaystyle-\left(\frac{\gamma_{\rm EB}}{2}+i\omega_{\rm
EB}\right)b-ig_{\rm 0}a^{\dagger}a-\frac{i}{\hbar}F_{{\rm mag}}x_{{\rm
zpf}}+\sqrt{\gamma_{\rm EB}}\eta$ (48)

where $\Delta_{l}=\omega_{l}-\omega_{c}$. Applying the linearization:
$a\rightarrow a_{0}+a$ and $b\rightarrow b_{0}+b$, we get the zeroth-order
equations

$\displaystyle\dot{a_{0}}$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{2}a_{0}-ig_{\rm
0}(b_{0}^{*}+b_{0})a_{0}+\sqrt{\kappa_{\rm ex}}a_{l}e^{-i\Delta_{l}t}$ (49)
$\displaystyle\dot{b_{0}}$ $\displaystyle=$
$\displaystyle-\left(\frac{\gamma_{\rm EB}}{2}+i\omega_{\rm
EB}\right)b_{0}-iga_{0}^{*}a_{0}$ (50)

and the first-order equations

$\displaystyle\dot{a}$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{2}a-ig_{\rm
0}(a_{0}b^{\dagger}+a_{0}b+b_{0}^{*}a+b_{0}a)+\sqrt{\kappa_{\rm in}}\xi_{\rm
in}+\sqrt{\kappa_{\rm ex}}\xi_{\rm ex}$ (51) $\displaystyle\dot{b}$
$\displaystyle=$ $\displaystyle-\left(\frac{\gamma_{\rm EB}}{2}+i\omega_{\rm
EB}\right)b-ig_{\rm 0}(a_{0}^{*}a+a^{\dagger}a_{0})-\frac{i}{\hbar}F_{{\rm
mag}}x_{{\rm zpf}}+\sqrt{\gamma_{\rm EB}}\eta$ (52)

The zero-order solutions for the optical and mechanical modes are:

$\displaystyle a_{0}$ $\displaystyle=$ $\displaystyle\frac{\sqrt{\kappa_{\rm
ex}}a_{l}}{\kappa/2-i\bar{\Delta}}e^{-i\Delta_{l}t}$ (53) $\displaystyle
b_{0}$ $\displaystyle=$
$\displaystyle\frac{-iga_{0}^{*}a_{0}}{\frac{\gamma_{\rm EB}}{2}+i\omega_{\rm
EB}}$ (54)

where $\bar{\Delta}=\Delta_{l}-g_{0}(b_{0}^{*}+b_{0})$. The reduced first-
order equations are

$\displaystyle\dot{a}$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{2}a-ig_{\rm
0}a_{0}(b^{\dagger}+b)+\sqrt{\kappa}\xi$ (55) $\displaystyle\dot{b}$
$\displaystyle=$ $\displaystyle-\left(\frac{\gamma_{\rm EB}}{2}+i\omega_{\rm
EB}\right)b-ig_{\rm 0}(a_{0}^{*}a+a^{\dagger}a_{0})-\frac{i}{\hbar}F_{{\rm
mag}}x_{{\rm zpf}}+\sqrt{\gamma_{\rm EB}}\eta$ (56)

where we have introduced a combined vacuum noise operator,

$\displaystyle\xi$ $\displaystyle=$ $\displaystyle(\sqrt{\kappa_{\rm
in}}\xi_{\rm in}+\sqrt{\kappa_{\rm ex}}\xi_{\rm ex})/\sqrt{\kappa}$ (57)

Equations (55) and (56) can be written in the frequency domain as follows

$\displaystyle a[\omega]$ $\displaystyle=$
$\displaystyle\chi_{c}[\omega]\left(-ig_{\rm
0}a_{0}(b^{\dagger}[\omega-\Delta_{l}]+b[\omega-\Delta_{l}])+\sqrt{\kappa}\xi[\omega]\right)$
(58) $\displaystyle b[\omega]$ $\displaystyle=$ $\displaystyle\chi_{\rm
m}[\omega]\left(-ig_{\rm
0}(a_{0}^{*}a[\omega+\Delta_{l}]+a_{0}a^{\dagger}[\omega-\Delta_{l}])-\frac{i}{\hbar}x_{{\rm
zpf}}F_{{\rm mag}}[\omega]+\sqrt{\gamma_{\rm EB}}\eta[\omega]\right)$ (59)

Here we use the noise operators’ correlations in the frequency domain:

$\displaystyle\left\langle\xi(\omega)\xi(-\omega)\right\rangle$
$\displaystyle=$ $\displaystyle 0$ (60)
$\displaystyle\left\langle\xi^{\dagger}(\omega)\xi(-\omega)\right\rangle$
$\displaystyle=$ $\displaystyle 0$ (61)
$\displaystyle\left\langle\xi(\omega)\xi^{\dagger}(-\omega)\right\rangle$
$\displaystyle=$ $\displaystyle 1$ (62)
$\displaystyle\left\langle\eta(\omega)\eta(-\omega)\right\rangle$
$\displaystyle=$ $\displaystyle 0$ (63)
$\displaystyle\left\langle\eta^{\dagger}(\omega)\eta(-\omega)\right\rangle$
$\displaystyle=$ $\displaystyle n_{th}$ (64)
$\displaystyle\left\langle\eta(\omega)\eta^{\dagger}(-\omega)\right\rangle$
$\displaystyle=$ $\displaystyle n_{th}+1$ (65)

Note that $x^{\dagger}[\omega]=(x[-\omega])^{\dagger}$. We also define optical
and mechanical susceptibility as

$\displaystyle\chi_{c}[\omega]$ $\displaystyle=$
$\displaystyle\left(\frac{\kappa}{2}-i\omega\right)^{-1}$ (66)
$\displaystyle\chi_{\rm m}[\omega]$ $\displaystyle=$
$\displaystyle\left(\frac{\gamma_{\rm EB}}{2}-i(\omega-\omega_{\rm
EB})\right)^{-1}$ (67)

The solution to (56) is easily obtained as

$\displaystyle b[\omega]$ $\displaystyle=$ $\displaystyle\chi_{{\rm
eff}}[\omega]\left(-i\frac{x_{{\rm zpf}}}{\hbar}F_{{\rm
RPSN}}[\omega]-i\frac{x_{{\rm zpf}}}{\hbar}F_{{\rm
mag}}[\omega]+\sqrt{\gamma_{\rm EB}}\eta[\omega]\right)$ (68) $\displaystyle
b^{\dagger}[\omega]$ $\displaystyle=$ $\displaystyle(\chi_{{\rm
eff}}[-\omega])^{*}\left(i\frac{x_{{\rm zpf}}}{\hbar}F_{{\rm
RPSN}}[\omega]+i\frac{x_{{\rm zpf}}}{\hbar}F_{{\rm
mag}}^{\dagger}[\omega]+\sqrt{\gamma_{\rm EB}}\eta^{\dagger}[\omega]\right)$
(69)

with

$\displaystyle\chi_{{\rm eff}}^{-1}[\omega]$ $\displaystyle=$
$\displaystyle\chi_{\rm m}^{-1}[\omega]+i\Sigma_{\rm opt}[\omega]$ (70)
$\displaystyle\Sigma_{\rm opt}[\omega]$ $\displaystyle=$ $\displaystyle
ig_{0}^{2}\left(\left|a_{0}\right|^{2}\chi_{c}[\omega-\Delta_{l}]-\left|a_{0}\right|^{2}\chi_{c}[\omega+\Delta_{l}]\right)$
(71) $\displaystyle F_{{\rm RPSN}}[\omega]$ $\displaystyle=$
$\displaystyle\frac{\hbar}{x_{{\rm
zpf}}}g_{0}\sqrt{\kappa}\left(a_{0}^{*}\chi_{c}[\omega+\Delta_{l}]\xi[\omega+\Delta_{l}]+a_{0}\chi_{c}[\omega-\Delta_{l}]\xi^{\dagger}[\omega-\Delta_{l}])\right)$
(72)

where $\chi_{{\rm eff}}$ is the effective mechanical susceptibility modified
by the optical backaction. The optical spring $\Omega_{{\rm opt}}$ and optical
damping $\Gamma_{{\rm opt}}$ can be extracted from $\Sigma_{\rm opt}[\omega]$

$\displaystyle\Omega_{{\rm opt}}$ $\displaystyle=$ $\displaystyle{\rm
Re}\left[\Sigma_{\rm opt}[\omega]\right]$ (73) $\displaystyle\Gamma_{{\rm
opt}}$ $\displaystyle=$ $\displaystyle-2{\rm Im}\left[\Sigma_{\rm
opt}[\omega]\right]$ (74)

The radiation pressure shot noise $F_{{\rm RPSN}}$ (see Eq. (72)) originates
from the backaction of the optical fluctuations. Its power spectral density
(PSD) is found using the Wiener-Khinchin theorem

$\displaystyle S_{FF}^{{\rm RPSN}}[\omega]$ $\displaystyle=$
$\displaystyle\left\langle F_{{\rm RPSN}}[\omega]F_{{\rm
RPSN}}[-\omega]\right\rangle$ (75) $\displaystyle=$
$\displaystyle\frac{\hbar^{2}g_{0}^{2}\left|a_{0}\right|^{2}}{x_{{\rm
zpf}}^{2}}\frac{\kappa}{\kappa^{2}/4+(\omega+\Delta_{l})^{2}}$

With the correlations of thermal noise operators given in Eqs. (64)-(65), we
can easily get the PSD of thermal fluctuation

$\displaystyle S_{F^{\dagger}F}^{{\rm th}}[\omega]$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{x_{{\rm zpf}}^{2}}\left\langle\sqrt{\gamma_{\rm
EB}}\eta^{\dagger}[\omega]\sqrt{\gamma_{\rm
EB}}\eta[-\omega]\right\rangle=\frac{\hbar^{2}}{x_{{\rm zpf}}^{2}}\gamma_{\rm
EB}n_{{\rm th}}$ (76) $\displaystyle S_{FF^{\dagger}}^{{\rm th}}[\omega]$
$\displaystyle=$ $\displaystyle\frac{\hbar}{x_{{\rm
zpf}}}\left\langle\sqrt{\gamma_{\rm EB}}\eta[\omega]\sqrt{\gamma_{\rm
EB}}\eta^{\dagger}[-\omega]\right\rangle=\frac{\hbar^{2}}{x_{{\rm
zpf}}^{2}}\gamma_{\rm EB}(n_{{\rm th}}+1)$ (77)

The difference in $S_{F^{\dagger}F}^{{\rm th}}[\omega]$ and
$S_{FF^{\dagger}}^{{\rm th}}[\omega]$ will lead to the well-known quantum
phenomenon called sideband asymmetry [24]. These two equations can be
rewritten as follows for large thermal occupations

Figure 3S: (a) Schematic of the laser frequencies for detecting the EB’s
motion. A “probe” laser tone drives the optical cavity mode with $n_{\rm
opt}=130$. Oscillations of the EB in the acoustic trap produce sidebands at
frequencies $\pm\omega_{\rm{EB}}$. These sidebands are detected via standard
heterodyne techniques using the “local oscillator” tone. (b) Schematic setup
for electron spin detection. An EB produced from the field emission tip is
trapped by the acoustic wave. The electron’s spin is manipulated via the
microwave coil, which is driven by a microwave generator. The spin state is
transduced to the EB’s displacement via the magnetic field gradient, which is
generated by the field coil. A laser beam passes through an acousto-optic
modulator to produce the probe tone and the local oscillator. The probe tone
passes through a phase modulator to add sidebands for the PDH lock (not
shown). Both tones are directed to the cavity via a circulator. The light
reflected from the cavity is sent to a photodiode. Optical path: red lines.
Electrical path: blue lines.

$\displaystyle S_{F^{\dagger}F}^{{\rm th}}[\omega]\approx
S_{FF^{\dagger}}^{{\rm th}}[\omega]\approx 2k_{B}Tm_{\rm EB}\gamma_{\rm EB}$
(78)

Here we have used $n_{{\rm th}}\approx k_{B}T/\hbar\omega_{\rm EB}\gg 1$, as
$T=30$ mK and $\omega_{\rm EB}=2\pi\times 2.9$ MHz corresponds to $n_{{\rm
th}}\approx 215$. Equations (68) and (69) suggest the following PSD for the
mechanical mode

$\displaystyle S_{b^{\dagger}b}[\omega]$ $\displaystyle=$
$\displaystyle\left|\chi_{{\rm eff}}[-\omega]\right|^{2}\frac{x_{{\rm
zpf}}^{2}}{\hbar^{2}}\left(S_{FF}^{{\rm RPSN}}[\omega]+S_{FF}^{{\rm
mag}}[\omega]+S_{F^{\dagger}F}^{{\rm th}}[\omega]\right)$ (79) $\displaystyle
S_{bb^{\dagger}}[\omega]$ $\displaystyle=$ $\displaystyle\left|\chi_{{\rm
eff}}[\omega]\right|^{2}\frac{x_{{\rm zpf}}^{2}}{\hbar^{2}}\left(S_{FF}^{{\rm
RPSN}}[\omega]+S_{FF}^{{\rm mag}}[\omega]+S_{FF^{\dagger}}^{{\rm
th}}[\omega]\right)$ (80)

Because $x[\omega]=x_{{\rm zpf}}(b[\omega]+b^{\dagger}[\omega])$, we obtain
the PSD of the mechanical displacement

$\displaystyle S_{xx}[\omega]$ $\displaystyle=$ $\displaystyle x_{{\rm
zpf}}^{2}\left(S_{bb^{\dagger}}[\omega]+S_{b^{\dagger}b}[\omega]\right)$ (81)
$\displaystyle=$ $\displaystyle\left|\chi_{{\rm
eff}}[-\omega]\right|^{2}\frac{1}{4m^{2}\omega_{\rm EB}^{2}}\left(S_{FF}^{{\rm
RPSN}}[\omega]+S_{FF}^{{\rm mag}}[\omega]+S_{F^{\dagger}F}^{{\rm
th}}[\omega]\right)$ $\displaystyle+\left|\chi_{{\rm
eff}}[\omega]\right|^{2}\frac{1}{4m^{2}\omega_{\rm EB}^{2}}\left(S_{FF}^{{\rm
RPSN}}[\omega]+S_{FF}^{{\rm mag}}[\omega]+S_{FF^{\dagger}}^{{\rm
th}}[\omega]\right)$

Having found the solution for $b[\omega]$, the solution of the intracavity
field in Eq. (55) is obtained.

With the input-output relation $a_{{\rm out}}[\omega]=\sqrt{\kappa_{{\rm
ex}}}a[\omega]$, we find the PSD for the cavity output as

$\displaystyle S_{a^{\dagger}a}^{{\rm out}}[\omega]$ $\displaystyle=$
$\displaystyle\kappa_{{\rm ex}}\left\langle
a^{\dagger}[\omega]a[-\omega]\right\rangle$ (82) $\displaystyle=$
$\displaystyle\kappa_{{\rm ex}}\left|\chi_{c}[-\omega]\right|^{2}g_{\rm
0}^{2}\left|a_{0}\right|^{2}\left(S_{b^{\dagger}b}[\omega+\Delta_{l}]+S_{bb^{\dagger}}[\Delta_{l}+\omega])\right)$
(83)

Figure 4S: The power spectral density of the photodiode current
$S_{ii}[\omega]$, normalized to $S_{ii,\rm{mag}}[0]$ (the magnetic force
signal on resonance), assuming a measurement bandwidth $b=1$ Hz. The
contributions to $S_{ii}$ from RPSN, the EB’s thermal motion, and the detector
shot noise are each shown. The parameters used here are: $Q_{\rm EB}=6\times
10^{5}$, $T=30$ mK, $P=3$ µW, $\mathcal{F}=10^{5}$.

###  V.2 Heterodyne detection

Here we consider optical heterodyne detection. The optical field $a_{{\rm
det}}$ incident on a photodiode includes the cavity output and local
oscillator, as shown in Fig. 3S, which is given by

$\displaystyle a_{{\rm det}}$ $\displaystyle=$ $\displaystyle a_{{\rm
lo}}e^{i\omega_{{\rm lo}}t}+a$ (84)

where $a_{{\rm lo}}$ and $\omega_{{\rm lo}}$ denote the amplitude and
frequency of the local oscillator. The autocorrelation of the photocurrent is

$\displaystyle G_{ii}(t,\tau)$ $\displaystyle=$ $\displaystyle\left\langle
i(t+\tau/2)i(t-\tau/2)\right\rangle$ (85) $\displaystyle=$ $\displaystyle
G^{2}\left\langle:a_{{\rm det}}^{\dagger}(t+\tau/2)a_{{\rm
det}}(t+\tau/2)a_{{\rm det}}^{\dagger}(t-\tau/2)a_{{\rm
det}}(t-\tau/2):\right\rangle+G^{2}\left\langle a_{{\rm
det}}^{\dagger}(t)a_{{\rm det}}(t)\right\rangle\delta(\tau)$
$\displaystyle\approx$ $\displaystyle G^{2}\left|a_{{\rm
lo}}\right|^{4}+G^{2}\left|a_{{\rm lo}}\right|^{2}\left\langle
a^{\dagger}(t+\tau/2)a(t+\tau/2)\right\rangle+G^{2}\left|a_{{\rm
lo}}\right|^{2}\left\langle a^{\dagger}(t-\tau/2)a(t-\tau/2)\right\rangle$
$\displaystyle+G^{2}\left|a_{{\rm lo}}\right|^{2}e^{i\omega_{{\rm
lo}}\tau}\left\langle
a^{\dagger}(t+\tau/2)a(t-\tau/2)\right\rangle+G^{2}\left|a_{{\rm
lo}}\right|^{2}e^{-i\omega_{{\rm lo}}\tau}\left\langle
a^{\dagger}(t-\tau/2)a(t+\tau/2)\right\rangle$ $\displaystyle+G^{2}a_{{\rm
lo}}^{2}e^{2i\omega_{{\rm lo}}t}\left\langle
a^{\dagger}(t+\tau/2)a^{\dagger}(t-\tau/2)\right\rangle+G^{2}\left(a_{{\rm
lo}}^{*}\right)^{2}e^{-2i\omega_{{\rm lo}}t}\left\langle
a(t+\tau/2)a(t-\tau/2)\right\rangle$ $\displaystyle+G^{2}\left|a_{{\rm
lo}}\right|^{2}\delta(\tau)$

where $G$ is the gain of the detector and the last term represents the
detector shot noise. The PSD of the photocurrent is the Fourier transform of
the correlation

$\displaystyle S_{ii}[\omega]$ $\displaystyle=$ $\displaystyle
G^{2}\left|a_{{\rm lo}}\right|^{2}\left(S_{a^{\dagger}a}[\omega_{{\rm
lo}}+\omega]+S_{a^{\dagger}a}[\omega_{{\rm lo}}-\omega]+1\right)$ (86)
$\displaystyle=$ $\displaystyle G^{2}\left|a_{{\rm
lo}}\right|^{2}\bigg{(}g^{2}\left|a_{0}\right|^{2}\left|\chi_{c}[-(\omega_{{\rm
lo}}+\omega)]\right|^{2}\big{(}S_{b^{\dagger}b}[\omega_{{\rm
lo}}+\omega+\Delta_{l}]+S_{bb^{\dagger}}[\Delta_{l}+\omega_{{\rm
lo}}+\omega]\big{)}$
$\displaystyle+g^{2}\left|a_{0}\right|^{2}\left|\chi_{c}[-(\omega_{{\rm
lo}}-\omega)]\right|^{2}\big{(}S_{b^{\dagger}b}[\omega_{{\rm
lo}}-\omega+\Delta_{l}]+S_{bb^{\dagger}}[\Delta_{l}+\omega_{{\rm
lo}}-\omega]\big{)}+1\bigg{)}$

Figure. 4S shows the contribution to $S_{ii}$ from each noise source, as well
as their total.

##  VI Table of parameters

Optical cavity
---
Optical resonant wavelength ($\lambda_{{\rm opt}}$) | 1550 nm
Optical cavity linewidth ($\kappa/2\pi$) | 15 MHz
Optical cavity input coupling rate ($\kappa_{\rm ex}$) | $0.44\kappa$
Cavity Finesse ($\mathcal{F}$) | $10^{5}$
Cavity length ($L_{{\rm cav}}$) | 100 $\mu$m
Cavity waist ($w_{{\rm cav}}$) | 5 $\mu$m
Operating temperature ($T$) | 30 mK
Acoustic trap
Acoustic wavelength ($\lambda_{{\rm ac}}$) | 775 nm
Acoustic resonant frequency ($\omega_{{\rm ac}}/2\pi$) | 320 MHz
Acoustic optomechanical coupling ($g_{0,{\rm ac}}/2\pi$) | 3.6 kHz
Quality factor of acoustic mode ($Q_{{\rm ac}}$) | $10^{5}$
Acoustic trap depth ($U_{0}/k_{{\rm B}}$) | 300 mK
Fiber vibration amplitude ($z_{{\rm amp}}$) | 2.4 fm
Acoustic energy density ($W_{{\rm ac}}$) | 8.5 J/m3
Phonon number ($n_{{\rm ac}}$) | $1.6\times 10^{11}$
Helium density variation ($\delta\rho_{{\rm He}}/\rho_{{\rm He}}$) | $2\times 10^{-3}$
Optical mode for acoustic wave detection ($n_{{\rm opt}}$) | 129
Acoustic mode number ($n_{{\rm ac}}$) | 258
Electron bubble
Electron bubble mass ($m_{{\rm EB}}$) |  $1.6\times 10^{-24}$ kg
Electron bubble radius ($R_{0}$) | 1.9 nm
Trap frequency ($\omega_{{\rm EB}}/2\pi$) | 2.9 MHz
Young’s modulus of the EB ($E_{\rm EB}$) | 530 kPa
Optomechanical coupling rate ($g_{0}/2\pi$) | 0.1 Hz
Cavity mode for EB detection ($n_{{\rm opt}}$) | 130

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