11institutetext: Instituto de Física Fundamental (IFF), CSIC. Calle Serrano
121-123, 28006, Madrid, Spain. 22institutetext: LERMA, Observatoire de Paris,
PSL University, CNRS, Sorbonne Université, 92190 Meudon, France.

# Multi-line Observations, Models, and Data Needed to Understand the Nature of
UV-irradiated Interstellar Matter

Javier R. Goicoechea 11<EMAIL_ADDRESS>Sara Cuadrado 11 Franck Le
Petit 22

###### Abstract

Far-ultraviolet photons from OB-type massive stars regulate the heating,
ionization, and chemistry of much of the neutral interstellar gas in star-
forming galaxies. The interaction of FUV radiation and interstellar matter
takes place in environments broadly known as photodissociation regions (PDRs).
PDR line diagnostics are the smoking gun of the radiative feedback from
massive stars. Improving our understanding of stellar feedback in the ISM
requires quantifying the energy budget, gas dynamics, and chemical composition
of PDR environments. This goal demands astronomical instrumentation able to
deliver multi-line spectroscopic images of the ISM (of the Milky Way and
nearby galaxies). It also requires interdisciplinary collaborations to obtain
the rate coefficients and cross sections of the many microphysical processes
that occur in the ISM and that are included in models such as the Meudon PDR
code.

##  1 Introduction

Far-UV (FUV: $E$ $<$ 13.6 eV) photons emitted by massive O- and B-stars
govern, or at least greatly influence, the heating, ionization, and chemistry
of the neutral interstellar gas: everywhere hydrogen atoms are in
predominantly neutral form. The interaction of stellar FUV radiation and
interstellar baryonic matter (atoms, molecules, and dust grains) takes place
in so-called photodissociation regions (PDRs; Hollenbach & Tielens (1999)).
This interaction occurs for different doses of FUV radiation ($G_{0}$) and at
very different spatial scales: from the illuminated rims of star-forming
clouds to kpc scales in starburst galaxies. The emission from PDRs reflects
the radiative feedback from massive stars, a collection of processes that
establish the phases and pressures of the ISM, and also regulate star
formation and molecular cloud destruction Pabst et al. (2020).

The physical state of the interstellar gas depends on a plethora of detailed
microphysical processes that determine its chemical composition and how the
gas is heated and cooled. Therefore, in addition to understanding the (macro)
astrophysical processes driving the dynamics and evolution of the ISM, it is
also mandatory to understand the subtle microprocesses that form, destroy, and
excite its basic constituents. Accompanied by new developments in astronomical
instrumentation, last years on PDR research have reinforced closer
collaborations with molecular physicists and laboratory experimentalists able
to determine the rate coefficients and cross sections of the relevant
microprocesses.

Bright molecular cloud rims, such as the iconic Orion Bar or the Horsehead,
have always been excellent laboratories to study the molecular content of FUV-
irradiated interstellar gas and the wealth of physical and chemical processes
that occur at microscopic (molecular) level. The number of molecules, atoms,
and ions observed in these dense PDRs are not so easily detected toward
fainter environments also influenced by stellar FUV radiation (e.g, the
surface layers of protoplanetary disks and distant star-forming galaxies).

In this invited paper we summarize some of the latest developments in
observations and modeling of PDRs. Due to the lack of space, we mostly limit
to our own recent work. Readers are referred to Wolfire et al. Wolfire et al.
(2022) for a modern review on PDRs. Here we strengthen the interdisciplinary
aspects of PDR research, and we emphasize the need of precise atomic and
molecular data to correctly interpret future observations of the ISM in
galaxies.

Figure 1: Structure of a strongly irradiated molecular cloud near massive
stars (e.g., the Orion Bar).

##  2 PDRs everywhere

PDRs host the critical conversion from atomic to molecular gas (the H / H2 and
C+/ C / CO transition zones, Fig. 1). This stratification occurs as the column
density of gas and dust increases deeper inside molecular clouds, and as the
flux of FUV photons is attenuated by dust (for high $G_{0}$ / $n_{\rm H}$
ratios) and by H2 absorption lines (for low $G_{0}$ / $n_{\rm H}$ ratios)
Sternberg et al. (2014). The penetration of FUV radiation depends on
metallicity and grain properties. FUV photons reach larger depths in lower
metallicity environments (lower dust-to-mass ratios) and in regions affected
by grain growth (flatter extinction curves). Both effects shift the C+/ C / CO
transition to higher $A_{V}$ and leave a larger mass fraction of H2 gas that
is “CO-dark” (or too faint).

PDRs emit most of the IR radiation arising from the ISM of star-forming
galaxies: FUV-pumped IR bands from polycyclic aromatic hydrocarbons (PAHs), H2
ro-vibrational, [C ii]158 $\mu$m and [O i]63, 145 $\mu$m fine-structure, and
mid-$J$ CO lines, as well as warm dust continuum – grains are heated by FUV
photons and reemit FIR contimuum. Photoelectrons ejected from small grains and
PAHs heat the gas. In dense PDRs, collisional de-excitation of FUV-pumped H2
is an important gas heating mechanism too. The presence of vibrationally
excited H2($v$ $\geq$ 1) overcomes the endoergicity of the initiating
reactions H2 \+ (C+, S+, O, N, …) Agúndez et al. (2010) and leads to the
formation of CH+, SH+, OH, and NH Goicoechea & Roncero (2022).

In their most general definition – neutral gas regulated by FUV radiation –,
PDRs represent the dominant fraction of the neutral atomic and molecular gas
in the ISM of star-forming galaxies Hollenbach & Tielens (1999); Wolfire et
al. (2022). Indeed, except for the cold and dense molecular cores associated
with the first stages of star formation, most of the ISM (in volume and mass)
is effectively at $A_{V}$ $<$ 8 mag. Hence, permeated by stellar FUV photons.

Figure 2: Central parsec ($\sim$ 7.5′ $\times$ 11.5′) of the Orion molecular
core OMC-1 around the Trapezium cluster (right, Goicoechea et al. (2015)) and
1′′ resolution zoom ($\sim$ 50′′ $\times$ 50′′) resolving a high pressure
($P_{\rm th}/k_{\rm B}$ $\sim$ 108 K cm-3) and nearly edge-on H
ii/PDR/molecular cloud interface in the Orion Bar PDR (left, Goicoechea et al.
(2016)).

###  2.1 New-generation observations of the ISM and PDR environments

State-of-the-art observations allow us to access a plethora of multi-line
diagnostics at increasingly higher sensitivity, angular resolution, and field-
of-view. Recent observations challenge some of our previous views of PDRs, for
example, their fundamental small-scale structure(s), the origin of the
observed rich chemistry, and their gas dynamics: propagation of
ionization/dissociation fronts, photoevaporation, gas compression, and
radiation pressure on grains. Last years have seen rapid progress in
heterodyne receiver technology and specific techniques involving (sub)mm
spectral-imaging of the ISM:

i) Increased sensitivity and angular resolution of interferometric mosaics
(with ALMA and NOEMA), providing astonishing sub-arcsecond resolution images
of the fundamental structure of the ISM (e.g., see Fig. 2 for the Orion Bar
PDR Goicoechea et al. (2016)).

ii) Broader instantaneous bandwidth, allowing us to a) map large portions of
star-forming clouds in several molecular lines simultaneously (CO, HCN, HNC,
HCO+, N2H+, … Pety et al. (2017)), and b) to obtain deep line surveys of the
molecular content (through the detection of hundreds of molecular lines
Cuadrado et al. (2015, 2017)) and physical conditions of representative
environments. IRAM, its forefront instrumentation, and the variety of
available observational techniques (from on-the-fly mapping to line surveys)
have played a pivotal role in improving our understanding of the ISM and its
underlying PDR processes.

iii) Development of airborne multi-pixel arrays up to the THz domain, able to
map the main gas cooling lines (i.e., the cloud energetics) at high-spectral
resolution and providing, for example, velocity-resolved square-degree maps of
the [C ii]158 $\mu$m emission Pabst et al. (2020).

As a consequence of these developments, current research on PDRs and
associated processes is not only about the detailed study of small fields in
bright cloud rims such as the Orion Bar – from which we learn so much – but
more generally about the role of stellar feedback and the evolution of the
FUV-illuminated ISM at all relevant spatial scales: from cores and giant
molecular clouds (GMCs) Goicoechea et al. (2015); Pabst et al. (2020) to
distant star-forming galaxies.

Figure 3: Multi-line diagnostics (molecular and recombination lines) observed
toward the Orion Bar PDR and adjacent H ii region using the IRAM 30m Cuadrado
et al. (2015, 2017, 2019) and Yebes 40m Goicoechea & Cuadrado (2021)
radiotelescopes.

###  2.2 Molecular content of PDRs and peculiarities

PDRs were once thought to be too harsh to develop a rich chemistry. However,
our IRAM 30 m and Yebes 40 m line surveys of the Orion Bar, a strongly
irradiated PDR ($G_{0}$ $\simeq$ 104), exemplify that PDRs do host multiple
lines and unique chemical features that are not seen (or are too faint to be
detected) in regions shielded from FUV radiation. These specific lines (Fig.
3) have a great diagnostic power as tracers of FUV radiation from short-lived
massive stars, and thus, of star formation: narrow carbon radio recombination
lines (with line intensities proportional to $n_{\rm e}^{2}T_{\rm e}^{-1.5}$)
tracing the presence of C+ ions and of high ionization fractions ($n_{\rm
e}$/$n_{\rm H}$ $\approx$ C+/H; Cuadrado et al. (2019)), reactive molecular
ions (CH+, SH+, HOC+, and CO+; Nagy et al. (2013); Goicoechea et al. (2017)),
as well as enhanced abundances of certain hydrocarbons (C3H+, $l$-C3H;
Cuadrado et al. (2015)), radicals (NH Goicoechea & Roncero (2022), OH Cuadrado
et al. (2017), and HCO Goicoechea et al. (2011)), and unstable isomers of
organic molecules ($cis-$HCOOH; Cuadrado et al. (2016)). In addition, specific
sulfur-bearing molecules such as H2S show bright emission lines and enhanced
abundances (as revealed by the detection of the rare H${}_{2}^{33}$S
isotopologue; Goicoechea et al. (2021)). Indeed, the detection of sulfur radio
recombination lines Goicoechea & Cuadrado (2021) implies a gas-phase S+/H
abundance of (1.4 $\pm$ 0.4)$\cdot$10-5 at edge of the Orion Bar. This is the
elemental gas-phase sulfur abundance before an undetermined fraction goes into
S-bearing molecules and deplete as grain ice mantles in the cloud interior.
The inferred sulfur abundance in the PDR edge matches the solar abundance
Goicoechea et al. (2021). It thus leaves little room for large depletions of
sulfur onto rocky grains in molecular clouds.

Some of the above "PDR species" are readily detected in diffuse clouds Gerin
et al. (2019) as well as in starburst galaxies Fuente et al. (2008). These
detections prove the interaction of stellar radiation with interstellar matter
in different regimes of FUV flux, gas density, and spatial scale.

##  3 Models, need of precise atomic and molecular data, and outlook

The theoretical study and first thermo-chemical models of dense PDRs started
to develop more than 50 years ago. PDR models solve the penetration of FUV
radiation into molecular clouds. This includes treating the absorption of FUV
photons to electronically excited levels of H, H2, and CO, as well as the
absorption and anisotropic scattering by dust grains Goicoechea & Le Bourlot
(2007). Among other things, the wavelength-dependent FUV ($\lambda>911\AA$)
radiation field at different cloud depths determines the rate at which low-
ionization-potential atoms are ionized (C, S, Si, Fe, …) and at which
molecules are ionized and dissociated. The FUV field, gas density, and
dust/PAH properties set the main heating mechanisms, whereas cooling from [C
ii], [O i], H2, and mid-$J$ CO emission lines Joblin et al. (2018) determines
the cloud-depth variation of the gas temperature. This temperature sets the
rates of a large network of gas-phase ion-neutral and neutral-neutral chemical
reactions forming and destroying molecules. At deeper depths into the
molecular cloud, typically $A_{V}$ $>$ 8 mag, the flux of stellar FUV photons
is almost completely attenuated and gas-phase molecules and atoms freeze-out
on dust grains as temperatures drop. Cosmic-rays drive the ionization of atoms
and molecules, but at much lower rates than in gas directly exposed to stellar
FUV radiation. This leads to low ionization fractions and a slower chemistry.
The catalytic formation of molecules on grain surfaces and subsequent thermal
(sublimation) and nonthermal (photodesorption, cosmic-ray induced, or
chemical) desorption adds a new layer of chemical complexity to model.

Table 1: Microphysical processes relevant to the study of interstellar PDRs Process |  Relevance |  Required parameter |  Methodology
---|---|---|---
Inelastic collisions with $o$-H2, $p$-H2, H, and $e^{-}$ |  Non-LTE excitation. Precise comparison with observed line intensities |  Inelastic collisional rate coefficient $\gamma_{ij}(T)\sim\sigma_{ij}\,\cdot\,v$ (cm3 s-1) |  Scattering calculations. Laboratory.
Chemical reactions |  Formation and destruction of molecules and PAHs |  Reaction rate coefficient $k(T)$ (cm3 s-1) |  Reaction dynamics calculations. Laboratory.
Reactions with vibrationally excited H2 ($v$$\geq$1, $J$) |  Reactions of FUV-pumped H2 with atoms and molecules overcomes reaction endoergicities |  State specific rate coefficient $k_{v,\,J}(T)$ (cm3 s-1) |  Reaction dynamics calculations. Laboratory.
Photo-ionization and dissociation of atoms, molecules, and PAHs |  Photochemistry induced by stellar FUV photons |  Photo cross sections $\sigma_{\rm ion}(\lambda)$, $\sigma_{\rm diss}(\lambda)$ (cm-2) |  Experiments (e.g., synchrotron). Calculations.
Low-$T_{\rm e}$ radiative and dielectronic recombination of S+ (Si+, Fe+, …) |  Defines S+/S transition. Determines recombination line spectrum |  Recombination rate coefficients $k_{\rm RR,\,DR}(T_{\rm e})$ (cm3 s-1) |  Calculations.
Adsorption/desorption of atoms and molecules onto/from dust grains |  Freeze-out, grain surface chemistry and sublimation of atoms/molecules from dust grains |  Binding energies $E_{b}$/$k_{B}$ (K) |  Temperature-programmed desorption experiments. Calculations.
Non-thermal desorption from grains |  Photodesorption and cosmic-ray induced desorption of ice mantles |  Desorption yields $Y$ (molecule photon-1) $k_{\rm CR}$ (s-1) |  FUV, X-ray, and particle irradiation experiments.
FUV extinction Photoelectric effect |  FUV penetration. Gas heating |  Grain/PAH optical props. $Q_{\rm abs}$($\lambda$), $Q_{\rm sca}$($\lambda$), $g$($\lambda$) |  Laboratory.

In order to correctly interpret the amount of information and the many details
provided by state-of-the-art multi-line observations, astrochemical models
need to accurately calculate the rates of the above microphysical processes
(interstellar gas is rarely in thermal or chemical equilibrium). This is the
main goal of the Meudon PDR code Le Petit et al. (2006), a publicly available
and open source model (https://ism.obspm.fr/) that is continuously upgraded
toward more realistic descriptions of these processes Bron et al. (2014) and
as new rate coefficients and cross sections are published in the literature.
Hence, it is mandatory to strengthen and foster interdisciplinary
collaborations aimed to better characterize these processes, and to determine
the precise atomic and molecular data needed in astrochemical models. These
data can be obtained in sophisticated laboratory experiments or through high-
level ab initio calculations. Table 1 summarizes some of the most relevant
microphysical processes taking place in the ISM (with emphasis on PDRs)
together with the physical parameters and rate coefficients typically needed
in accurate thermo-chemical and non-LTE excitation models.

PDR models play a central role to correctly interpret spectroscopic
observations of a very significant fraction of the ISM in star-forming
galaxies (all neutral gas at $A_{V}$ $<$ 8 mag). New instrumentation will
allow mapping several square-degree areas of GMCs (reaching the scales that
dominate the extragalactic emission) in multiple rotational lines, critical to
derive accurate physical conditions and precise chemical abundances. More
automatic statistical data analysis tools – designed to link large grids of
astrochemical models and hydrodynamic PDR simulations with observations – will
be used to extract all the information contained in the million spectra these
velocity-resolved cubes will generate Pety et al. (2017). These cubes provide
access to the gas kinematics, thus to the driving forces in the ISM, and
reveal the dynamical and non-stationary aspects of stellar feedback in the ISM
(e.g., Maillard et al. (2021)). In parallel, interferometric mosaics will
allow us to locally zoom into particular fields/templates at very high angular
resolution. These images reveal the spatial scales at which chemical
abundances and physical conditions abruptly change in clouds (Fig. 2).
Interferometric observations also spatially resolve GMCs in nearby galaxies,
with emission features often dominated by gas at low $A_{V}$. JWST will soon
reveal the evolution of PAHs, warm dust grains, and H2 emission (e.g., Berné
et al. (2022)) in all kind of FUV-irradiated environments. The IR to cm
emission from these “PDRs” is determined by many microphysical processes that
form, excite, and destroy the gas and dust in the ISM. Knowing the rates of
these processes is equally important to correctly understand the many faces of
stellar feedback in the ISM.

## References

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# Search for ultralight dark matter with a frequency adjustable diamagnetic
levitated sensor

Rui Li  CAS Key Laboratory of Microscale Magnetic Resonance and School of
Physical Sciences, University of Science and Technology of China, Hefei
230026, China  CAS Center for Excellence in Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei 230026, China
Shaochun Lin  Liang Zhang  CAS Key Laboratory of Microscale Magnetic Resonance
and School of Physical Sciences, University of Science and Technology of
China, Hefei 230026, China  CAS Center for Excellence in Quantum Information
and Quantum Physics, University of Science and Technology of China, Hefei
230026, China  Changkui Duan  CAS Key Laboratory of Microscale Magnetic
Resonance and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  CAS Center for Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei 230026, China  Pu Huang<EMAIL_ADDRESS>National Laboratory of
Solid State Microstructures and Department of Physics, Nanjing University,
Nanjing, 210093, China  Jiangfeng Du  CAS Key Laboratory of Microscale
Magnetic Resonance and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  CAS Center for Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei 230026, China

###### Abstract

Among several dark matter candidates, bosonic ultra-light (sub-meV) dark
matter is well motivated because it could couple to the Standard Model (SM)
and induce new forces. Previous MICROSCOPE and Eöt-Wash torsion experiments
have achieved high accuracy in the sub-1 Hz region, but at higher frequencies
there is still a lack of relevant experimental research. We propose an
experimental scheme based on the diamagnetic levitated micromechanical
oscillator, one of the most sensitive sensors for acceleration sensitivity
below the kilohertz scale. In order to improve the measurement range, we used
the sensor whose resonance frequency $\omega_{0}$ could be adjusted from 0.1Hz
to 100Hz. The limits of the coupling constant $g_{\scriptscriptstyle B-L}$ are
improved by more than 10 times compared to previous reports, and it may be
possible to achieve higher accuracy by using the array of sensors in the
future.

##  I  INTRODUCTION

There are many astronomical Sofue and Rubin (2001); Massey  _et al._ (2010)
and cosmological observations Markevitch  _et al._ (2004) that prove the
existence of dark matter particlesGianfranco (2010); Salucci (2019), but the
specific parameters of dark matter, especially the quality, are still highly
uncertain Tanabashi  _et al._ (2018). Many direct detection studies have
assumed that dark matter is composed of supersymmetric fermions, but so far
there has not been enough evidence. Now the focus of research is gradually
shifting to ultralight bosons and the quality range is approximately
$10^{-22}\mathrm{eV}\textless m_{\phi}\textless 0.1\mathrm{eV}$ Irastorza and
Redondo (2018); Hui  _et al._ (2017). For ultralight bosons with a mass less
than 1eV, due to their high particle number density, they behave like a
classical field. Due to the viral theorem , if the DM has virialized to the
Galaxy, it will be moving with a typical speed
$v_{\scriptscriptstyle\mathrm{DM}}\approx 10^{5}$m/s Bovy  _et al._ (2012);
O’Hare  _et al._ (2018); Myeong  _et al._ (2017). This corresponds to Compton
frequency $\omega_{s}=m_{\phi}/\hbar$ and De Broglie wavelength
$\lambda_{\scriptscriptstyle\mathrm{DM}}=hc^{2}/(m_{\phi}v_{\scriptscriptstyle\mathrm{DM}})$.

According to the previous reports, such as ADMX Du  _et al._ (2018) can search
for the Peccei-Quinn axion in the mass range $10^{-6}\mathrm{eV}\textless
m_{\phi}\textless 10^{-3}\mathrm{eV}$ Backes  _et al._ (2021); Kwon  _et al._
(2021). And the pseudoscalar axion-like ULMBs with masses between
$10^{-23}\mathrm{eV}$ and $10^{-18}\mathrm{eV}$ Abel  _et al._ (2017); Terrano
_et al._ (2019); Smorra  _et al._ (2019) and scalar dilaton ULMBs with masses
between $10^{-21}$eV and $10^{-5}\mathrm{eV}$ by use ultrastable clocks
Kennedy  _et al._ (2020); Arvanitaki  _et al._ (2015) and gravitation wave
detectors Vermeulen  _et al._ (2021) have recently been reported.

When DM is a vector field couples to a conserved current, corresponding to the
baryon number minus lepton number (B$-$L charge) in the SM. The Lagrangian in
this case can be written as Carney  _et al._ (2021):

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}m_{\phi}^{2}A^{2}+ig_{\scriptscriptstyle
B-L}A_{\mu}\overline{n}\gamma^{\mu}n$ (1)

where $n$ is the neutron field and the DM field couples directly to the number
of neutrons, $g_{\scriptscriptstyle B-L}$ is the coupling strength. Using the
Lorentz gauge and the plane wave approximation, the dark electric field can be
written as:
$E\approx\sqrt{\rho_{\scriptscriptstyle\mathrm{DM}}}\mathrm{sin}(\omega_{s}t-\vec{k}\cdot\vec{x})$,
where $\rho_{\scriptscriptstyle\mathrm{DM}}\approx
0.3\mathrm{GeV}/\mathrm{cm^{3}}$ Read (2014) is the local DM density.

In ground experiments, assume that using a magnet-gravity mechanical
oscillator to measure the ultralight DM field along the Earth’s axis, we can
parameterize the force exerted on the sensor as:

$F_{\mathrm{sig}}(t)=\alpha g_{\scriptscriptstyle
B-L}N_{g}F_{0}\mathrm{sin}(\omega_{s}t)$ (2)

because the De Broglie wavelength of DM is much larger than the size of the
sensor so that we drop the $x$ dependence. In this equation,
$\alpha=\mathrm{sin}\theta_{N}$ denotes the component along the direction of
gravity and $\theta_{N}$ means the latitude of the location of the ground
experiment system. In order to avoid the effects of the Earth’s rotation under
long time measurements and increase the force, experiment system is best
carried out at high latitudes like in the Arctic which $\alpha=1$.
$F_{0}=\sqrt{\rho_{\scriptscriptstyle\mathrm{DM}}}\approx 10^{-15}$N and
$N_{g}$ is the total number of neutrons in the sensor. We can approximate
write it as $N_{g}\approx\frac{1}{2}m/m_{\mathrm{neu}}$ in a sensor with mass
$m$ and $m_{\mathrm{neu}}$ is the neutron mass. The force
$F_{\mathrm{sig}}(t)$ is proportional to the mass of the sensor, so the main
criterion about the sensor is acceleration sensitivity.

Here we propose a experiment scheme to detect DM using a frequency adjustable
diamagnetic levitated sensor. The resonance frequency could be changed by
adjust the magnetic field gradient in a paramagnetic part of the oscillator
and frequency range from 0.1Hz to 100Hz. This means that we have high
detection accuracy to detect DM with mass in the range from $10^{-16}$eV to
$10^{-13}$eV. Compare to previously reported experiments, our experiment
scheme can achieve more than one order of magnitude improvement in the
measurement of the coupling strength $g_{\scriptscriptstyle B-L}$ based on the
results of theoretical calculation.

##  II THEORETICAL CALCULATION

Under the effect of the ultralight DM field, consider thermal noise and
measurement noise, the motion equation of a mechanical oscillator at resonant
frequency $\omega_{0}$ could be written as:

$m\ddot{x}+m\gamma\dot{x}+m\omega_{0}^{2}x=F_{\mathrm{sig}}(t)+F_{\mathrm{th}}+F_{\mathrm{mea}}$
(3)

where $\gamma$ is damp coefficient; the $F_{\mathrm{sig}}(t)$ is the DM field
drive from equation (2); $F_{\mathrm{th}}$ is the environmental thermal noise;
and the $F_{\mathrm{mea}}$ represents the measurement noise which is mainly
composed of the detector imprecision noise and backaction of radiation
pressure fluctuations.

The total acceleration noise of the system is given by:

$S_{\mathrm{aa}}^{\mathrm{tot}}=S_{\mathrm{aa}}^{\mathrm{th}}+(\frac{S_{\mathrm{xx}}^{\mathrm{imp}}}{|\chi_{\mathrm{\scriptscriptstyle
m}}(\omega,\omega_{0})|^{2}}+\frac{S_{\mathrm{ff}}^{\mathrm{ba}}}{m^{2}})$ (4)

where $\chi_{\mathrm{\scriptscriptstyle m}}(\omega,\omega_{0})$ is the
mechanical susceptibility given by $|\chi_{\mathrm{\scriptscriptstyle
m}}(\omega,\omega_{0})|^{2}=1/[(\omega^{2}-\omega_{0}^{2})^{2}+\gamma^{2}\omega^{2}]$,
and $S_{\mathrm{aa}}^{\mathrm{th}}=4\gamma k_{B}\mathrm{T}/m$ is the thermal
noise where $k_{B}$ is Boltzmann constant and T indicates environment
temperature. The detector imprecision noise $S_{\mathrm{xx}}^{\mathrm{imp}}$
and the backaction noise $S_{\mathrm{ff}}^{\mathrm{ba}}$ make up the total
measurement noise
$S_{\mathrm{aa}}^{\mathrm{mea}}=S_{\mathrm{xx}}^{\mathrm{imp}}/|\chi_{\mathrm{\scriptscriptstyle
m}}(\omega,\omega_{0})|^{2}+S_{\mathrm{ff}}^{\mathrm{ba}}/m^{2}$, and
$S_{\mathrm{xx}}^{\mathrm{imp}}\cdot
S_{\mathrm{ff}}^{\mathrm{ba}}=(1/\eta)\hbar^{2}$ meanwhile. Here
$\eta\leqslant 1$ is the measurement efficiency, and $\eta=1$ corresponding to
standard quantum limit (SQL).

The total measurement noise $S_{\mathrm{aa}}^{\mathrm{mea}}$ for the sensor
operating at SQL condition at resonance frequency $\omega_{0}$ could be given
by the simple formula Clark  _et al._ (2016):

$S_{\mathrm{aa}}^{\mathrm{mea,SQL}}=\frac{2\sqrt{(\omega_{0}^{2}-\omega^{2})^{2}+\gamma^{2}\omega^{2}}}{m}$
(5)

And achieving the SQL in a frequency range need to optimize the measurement
parameters frequency by frequency as the range is scanned.

Figure 1: (a) Schematic diagram of the experimental setup. A diamagnetic
sphere of 0.5 mm radius is levitated in the magnetic gravity trap, and a
paramagnetic microsphere of 11 $\mu m$ radius is connected to the upper
diamagnetic sphere by a thin glass rod. A 1550 nm laser is transmitted through
the left fibre to the right fibre, passing the transparent diamagnetic sphere.
(b) The magnetic field gradient $\partial B_{\scriptscriptstyle B}/\partial z$
and the resonance frequency $\omega_{0}^{\prime}$ changes by the relative
distance $d$, expressed by the blue and red lines respectively.

We use the total acceleratioon noise $S_{\mathrm{aa}}^{\mathrm{tot}}$ as the
acceleration measurement sensitivity of the system. From the equations
(2)-(4), consider the optimal case of $\alpha=1$, we obtain the relationship
between coupling strength $g_{\scriptscriptstyle B-L}$ and the acceleration
measurement sensitivity $S_{\mathrm{aa}}^{\mathrm{tot}}$ by:

$g_{\scriptscriptstyle
B-L}=\frac{2m_{neu}}{F_{0}}\sqrt{\frac{S_{\mathrm{aa}}^{\mathrm{tot}}}{T_{\mathrm{tot}}}}$
(6)

where $T_{\mathrm{tot}}$ denotes the effective total integration time. The DM
signal is essentianlly a coherent force and the timescales
$T_{\mathrm{coh}}\approx 10^{6}/\omega_{s}$. When the DM frequency
$\omega_{s}$ is lower to satisfy $T_{\mathrm{coh}}\textgreater
T_{\mathrm{mea}}$, all the measurement time $T_{\mathrm{mea}}$ contributes to
the coherent DM signal. And as the DM frequency $\omega_{s}$ increases, when
$T_{\mathrm{coh}}\textless T_{\mathrm{mea}}$, only the proportion of
$T_{\mathrm{coh}}/T_{\mathrm{mea}}$ in the measurement time contributes to the
coherent signal. So we define the effective integration time:

$T_{\mathrm{tot}}=\left\\{\begin{array}[]{ll}T_{\mathrm{mea}}&\textrm{if
$T_{\mathrm{coh}}<T_{\mathrm{mea}}$}\\\ \sqrt{T_{\mathrm{mea}}\cdot
T_{\mathrm{coh}}}&\textrm{if
$T_{\mathrm{coh}}>T_{\mathrm{mea}}$}\end{array}\right.$

##  III EXPERIMENTAL SCHEME

The levitated micromechanical and nanomechanical oscillators have been
demonstrated as one of the ultrasensitive acceleration sensors due to its
ultralow dissipation Leng  _et al._ (2021); Monteiro  _et al._ (2020). We
propose a reasonable scheme by our calculation as shown in Fig.1(a). A
diamagnetic sphere made by PMMA with radius $r_{1}$=0.5mm(corresponding volume
$V_{1}$), density $\rho_{1}$ and magnetic susceptibility
$\chi_{\scriptscriptstyle 1}$ is levitated in the upper magnet (name as
$Magnet\text{-}A$) center region, and the oscillator signal is detected
through the fibre on both sides. A paramagnetic microsphere made by
$\mathrm{Tb_{2}O_{3}}$ with radius $r_{2}=11\mu$m(corresponding volume
$V_{2}$), density $\rho_{2}$ and magnetic susceptibility
$\chi_{\scriptscriptstyle 2}$ is connected to the upper diamagnetic sphere
through a thin glass rod. And another combined magnets (name as
$Magnet\text{-}B$) is placed under the paramagnetic microsphere. The whole
magnet assembly is placed in a multi-stage suspension system, and uses active
vibration isolation devices to further improve the isolation effectAcernese
_et al._ (2010, 2014).

Figure 2: Acceleration power spectral density $S_{\mathrm{aa}}$ (a) Resonance
frequency $\omega_{0}$=10Hz, the grey dashed line indicates the thermal noise
$S_{\mathrm{aa}}^{\mathrm{th}}$; the red line indicates the acceleration
detection noise $S_{\mathrm{aa}}^{\mathrm{mea,SQL}}$; the blue dashed line
indicates the $S_{\mathrm{aa}}^{\mathrm{mea}}$ with the optimal light
intensity $P_{\mathrm{opt}}(\omega,\omega_{0})$ in each frequency between 8Hz
to 12Hz and the measurement efficiency $\eta$=1; the green dashed line
indicates the same $P_{\mathrm{opt}}(\omega,\omega_{0})$ as the blue dashed
line but the $\eta$=0.1; the purple line indicates the light intensity
$P_{\mathrm{opt}}(\omega_{0},\omega_{0})$ and $\eta$=1; the yellow line
indicates the same $P_{\mathrm{opt}}(\omega_{0},\omega_{0})$ and $\eta$=0.1;
(b) Resonance frequency $\omega_{0}$=100Hz, and the others are the same as
(a); (c) Adjust resonance frequency $\omega_{0}$ from 0.1Hz to 100Hz, the grey
dashed line indicates the thermal noise $S_{\mathrm{aa}}^{\mathrm{th}}$; the
yellow line indicates the acceleration measurement noise
$S_{\mathrm{aa}}^{\mathrm{mea}}$ with $\eta$=0.1, and here the scan step
$\Delta\omega_{s}$=10Hz it is only used to show the measurement scheme; the
green line indicates the envelope of the yellow line in the diagram and write
it as $S_{\mathrm{aa}}^{\mathrm{mea}^{\prime}}$; the red line is the
acceleration measurement sensitivity
$S_{\mathrm{aa}}^{\mathrm{tot}}=S_{\mathrm{aa}}^{\mathrm{th}}+S_{\mathrm{aa}}^{\mathrm{mea}^{\prime}}$.

$Magnet\text{-}A$ is constructed in a similar way to our previous
articlesZheng  _et al._ (2020). And need to use high remanence magnetic
material with two different magnetisation direction to generate enough
magnetic force. The red express the direction point to the centre, and the
blue express the direction out to the centre. In addition, using a less
remanence magnetic material to build the upper layer of $Magnet\text{-}B$ and
high magnetic material to build the lower layer. The combination of two
different remanence magnetic materials allows $Magnet\text{-}B$ to have a
higher magnetic field gradient while reducing the magnetic field strength. And
the direction of magnetisation is also indicated by red and blue colours.

The magnetic field energy of the upper paramagnetic sphere can be written as:

$U_{1}=-\int_{V_{1}}\frac{\chi_{\scriptscriptstyle
1}}{2\mu_{0}}B_{\scriptscriptstyle\mathrm{A}}^{2}dV$ (7)

where $B_{\scriptscriptstyle\mathrm{A}}$ represents the magnetic field created
by $Magnet\text{-}A$. Assuming that the $Magnet\text{-}B$ is far away at
beginning , the $z$ direction equilibrium position $z_{0}$ of the oscillator
in the magnetic-gravity trap satisfies: $\partial U_{1}/\partial
z|_{z=z_{0}}=(\rho_{1}V_{1}+\rho_{2}V_{2})g$. And the resonance frequency in
$z$ direction is:

$\omega_{0}=\sqrt{\frac{1}{\rho_{1}V_{1}+\rho_{2}V_{2}}\cdot\frac{\partial^{2}U_{1}}{\partial
z^{2}}}\bigg{|}_{z=z_{0}}$ (8)

Then we make the $Magnet\text{-}B$ rise, the magnetic field
$B_{\scriptscriptstyle B}$ from $Magnet\text{-}B$ in the lower paramagnetic
microsphere will become larger. And because of $V_{2}\ll V_{1}$, we can
simplify the magnetic field energy of the paramagnetic microspheres as
$U_{2}=-\chi_{\scriptscriptstyle
2}B_{\scriptscriptstyle\mathrm{B}}^{2}V_{2}/2\mu_{0}$. Now the resonance
frequency along $z$ direction of the oscillator change as:

$\omega_{0}^{\prime}=\sqrt{\omega_{0}^{2}-\frac{\chi_{\scriptscriptstyle
2}V_{2}}{\mu_{0}(\rho_{1}V_{1}+\rho_{2}V_{2})}\left(\frac{\partial
B_{\scriptscriptstyle B}}{\partial z}\right)^{2}}\bigg{|}_{z=z_{0}}$ (9)

where $\chi_{\scriptscriptstyle 2}\textgreater 0$ and
$\omega_{0}^{\prime}\textless\omega_{0}$. We ignore the second order gradient
term because of $(\partial B_{\scriptscriptstyle\mathrm{B}}/\partial z)^{2}\gg
B_{\scriptscriptstyle\mathrm{B}}(\partial^{2}B_{\scriptscriptstyle B}/\partial
z^{2})$. And the magnetic force from $Magnet\text{-}B$ on the paramagnetic
microsphere is much lower than the total gravity of oscillator since
$B_{\scriptscriptstyle\mathrm{B}}$ and $V_{2}$ are very small, the equilibrium
position $z_{0}$ will not be changed therefore.

We use finite element method to simulate the magnetic field gradient $\partial
B_{\scriptscriptstyle\mathrm{B}}/\partial z$ changes by the distance between
the paramagnetic microsphere and $Magnet\text{-}B$ expressed by $d$ range from
50$\mu$m to 100 $\mu$m, then use equation (9) to calculate the corresponding
resonance frequency $\omega_{0}^{\prime}$, as shown in Fig.1(b). It is
theoretically possible to bring the resonance frequency $\omega_{0}^{\prime}$
close to zero by reducing the distance $d$. But in order to improve the
stability of the oscillator and reduce the requirement for the isolation
system, we select resonance frequency $\omega_{0}^{\prime}$ variation range
from 0.1Hz to 100Hz.

##  IV EXPERIMENTAL RESULT ESTIMATE

Now we calculate the acceleration measurement sensitivity of this system. In
order to improve the acceleration sensitivity, the whole system was placed in
a low temperature environment which T=30mK, and estimate the damp coefficient
$\gamma=10^{-4}$Hz Leng  _et al._ (2021); Xiong  _et al._ (2021). In the
Supplementary material, we calculate the dependence of the total measurement
noise $S_{\mathrm{aa}}^{\mathrm{mea}}$ on the laser input power
$P_{\mathrm{in}}$ and obtained the optimized laser input power
$P_{\mathrm{opt}}(\omega,\omega_{0})$ to minimised the total measurement
noise.

In the cases of the oscillator resonance frequency $\omega_{0}$ equal to 10Hz
and 100Hz, we calculate the corresponding acceleration noise and the results
are shown in Fig.2(a) and Fig.2(b). When resonance frequency
$\omega_{0}=10$Hz, assuming measurement efficiency $\eta=1$ and we set the
laser input power to optimal laser power for each point as
$P_{\mathrm{opt}}(\omega,\omega_{0})$, the measurement noise
$S_{\mathrm{aa}}^{\mathrm{mea}}$ can almost reach the SQL at this time. With
the measurement efficiency $\eta$ reduce to 0.1, the measurement noise is
slightly increased. But actually, to simplify the experiment, the laser input
power need to choose near the resonance frequency $\omega_{0}$ by
$P_{\mathrm{opt}}(\omega_{0},\omega_{0})$, it will make the measurement noise
$S_{\mathrm{aa}}^{\mathrm{mea}}$ increase rapidly. In Fig.2(a), in the
frequency range from 9Hz to 11Hz, the measurement noise
$S_{\mathrm{aa}}^{\mathrm{mea}}$ is always below the thermal noise
$S_{\mathrm{aa}}^{\mathrm{th}}$ with $\eta=0.1$. When the resonance frequency
$\omega_{0}$ is adjusted to 100Hz, the range of measurement noise
$S_{\mathrm{aa}}^{\mathrm{mea}}$ below thermal noise
$S_{\mathrm{aa}}^{\mathrm{th}}$ is reduced to 99.6Hz to 100.4Hz in Fig.2(b).
We choose the appropriate oscillator resonance frequency scan step
$\Delta\omega_{0}$ from this.

According to the calculation results from Fig.2(a) and Fig.2(b), we choose the
scan step $\Delta\omega_{0}=1$Hz in the region resonance frequency
$\omega_{0}$ range from $0.1$Hz to $100$Hz, each scan cover the frequency
range from $\omega_{0}-\Delta\omega_{0}/2$ to $\omega_{0}+\Delta\omega_{0}/2$,
and fix the laser input power
$P_{\mathrm{in}}=P_{\mathrm{opt}}(\omega_{0},\omega_{0})$ in each scan
meanwhile. We calculate the acceleration measurement noise
$S_{\mathrm{aa}}^{\mathrm{mea}}$ with $\eta=0.1$ in each scan, and calculate
the envelope of these series $S_{\mathrm{aa}}^{\mathrm{mea}}$ writen as
$S_{\mathrm{aa}}^{\mathrm{mea}^{\prime}}$. The acceleration measurement
sensitivity
$S_{\mathrm{aa}}^{\mathrm{tot}}=S_{\mathrm{aa}}^{\mathrm{th}}+S_{\mathrm{aa}}^{\mathrm{mea}^{\prime}}$,
and these results are presented in Fig.2(c).

Figure 3: Ultra-light Dark Matter search range. The top axis represents the DM
mass $m_{\phi}$ corresponding to the frequency $\omega_{s}$. The upper grey
and yellow regions are excluded by E$\mathrm{\ddot{o}}$t-Wash torsion balance
Wagner  _et al._ (2012); Schlamminger  _et al._ (2008); Arvanitaki  _et al._
(2016) and MICROSCOPE experiments Hees  _et al._ (2018); Bergé  _et al._
(2018), and the red region is the range that this system can cover. In torsion
balance system, they use a pair of accelerometers (Beryllium and Titanium)
with a differential neutron/nucleon ratio
$\Delta=\mathrm{N}_{1}/\mathrm{A}_{1}-\mathrm{N}_{2}/\mathrm{A}_{2}=0.037$,
where N and A are the neutron and nucleon numbers of Beryllium and Titanium
respectively. From the equation(2), $N_{g}$ can be approximated as
$N_{g}=\Delta\cdot m/m_{\mathrm{neu}}$ at this time.

According to the previous discussion on the effective integration time
$T_{\mathrm{tot}}$, we fix the measurement time of each scan as
$T_{\mathrm{mea}}=10^{5}$s. When DM frequency $\omega_{s}\textless 10$Hz,
$T_{\mathrm{tot}}=T_{\mathrm{mea}}$; and when $\omega_{s}\textgreater 10$Hz,
$T_{\mathrm{tot}}=\sqrt{T_{\mathrm{mea}}\cdot 10^{6}/\omega_{s}}$. Combining
previous discussion of the scan step, we estimate that about one hundred times
adjustments and measurements will be required in total, corresponding to a
total time of $1\times 10^{7}$ seconds. The final result of coupling strength
$g_{\scriptscriptstyle B-L}$ from equation (6) is shown in Fig.3. In the
region of $\omega_{s}\textless 100$Hz, this system always has high
acceleration sensitivity by adjusting the resonance frequency of the
mechanical oscillator. And we achieve more than an order of magnitude
improvement in the measurement of $g_{\scriptscriptstyle B-L}$ compare to the
MICROSCOPE and the Eöt-Wash torsion experiment. And in the region of
$\omega_{s}\textgreater 100$Hz, the measurement accuracy of
$g_{\scriptscriptstyle B-L}$ decreases rapidly, due to the increase in
measurement noise $S_{\mathrm{aa}}^{\mathrm{mea}}$.

Finally, we estimated the minimum $g_{\scriptscriptstyle B-L}$ that this
system can detect. Assume that the DM frequency $\omega_{s}$ is 1Hz, 10Hz and
100Hz respectively. From the equation (6) and the measurement time
$T_{\mathrm{mea}}$ range from $10^{3}$s to $10^{7}$s, the results are shown in
Fig.4. When $T_{\mathrm{mea}}$ is less than the coherent time
$T_{\mathrm{coh}}$, $g_{\scriptscriptstyle B-L}$ decreases rapidly as
$T_{\mathrm{mea}}$ increases; and when $T_{\mathrm{mea}}$ is greater than
$T_{\mathrm{coh}}$, $g_{\scriptscriptstyle B-L}$ decreases more slowly. If the
final measurement time is about $10^{7}s$, the minimum $g_{\scriptscriptstyle
B-L}$ that can be measured scale is about $10^{-26}$.

##  V CONCLUSION

We propose an experimental scheme to detect ultra-light dark matter using a
frequency adjustable diamagnetic levitated microsphere sensor which can
theoretically approach the standard quantum limit. We change the resonance
frequency by adjusting the distance between the paramagnetic microsphere and
the lower combined magnets, and to obtain a lager range that maintains high
acceleration measurement sensitivity. Compared to the existing system, our
method can achieve at least one order of magnitude improvement in the coupling
constant $g_{\scriptscriptstyle B-L}$, especially in the frequencies from
0.1Hz to 100Hz. And it may be possible to achieve higher accuracy by using the
array of sensors in the future.

Figure 4: The minimum $g_{\scriptscriptstyle B-L}$ can reach in different DM
frequency $\omega_{s}$. The yellow, red and blue line indicates $\omega_{s}$
is 1Hz, 10Hz and 100Hz respectively. The maximum measurement time is $10^{7}$s
(about 115.7 days).

In this article, we consider only the effects of thermal noise and quantum
measurement noise on the acceleration measurement sensitivity of the system.
In fact, there are many low frequency noises such as seismic waves and Earth
tidal forces which also have a great impact on the accuracy of the experiment,
and that cannot be shielded by the suspension system. This poses a great
challenge to the actual measurement. Reducing the frequency scan step
according to the accuracy of the active vibration isolation device may make
the effect of other noise lower than thermal noise, and this needs to be
verified by further experiments.

In general, the current ground-based precision measurement system may have a
broader prospect in terms of dark matter measurement compared to the previous
astronomical observation methods. In the future, with the development of
measurement sensitivity and measurement range of mechanical sensors ,
especially with the improvement quantum sensing technology, the measurement
sensitivity may break through the standard quantum limit. It will open up more
possibilities in terms of dark matter measurement.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(Grants No.12205291, No. 12075115, No. 12075116, No. 11890702 and No.
12150011), the Fundamental Research Funds for the Central Universities, and
Anhui Provincial Natural Science Foundation (Grant No. 2208085QA16).

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## APPENDIX: LIGHT FIELD CALCULATION AND MEASUREMENT NOISE OPTIMIZATION

Optical Calculation. The light emitted from the incident fiber is assumed to
be Gaussian, taking the light propagation direction as the z-axis, the
incident Gaussian light intensity distribution at waist can be written as
Svelto (2010):

$I_{1}(r)=I_{0}\mathrm{exp}\left(-\frac{2r^{2}}{\omega_{01}^{2}}\right)$

And the waist radius of incident Gaussian beam is $\omega_{01}$, which
satisfies relation:

$\omega_{01}=\sqrt{\frac{a_{0}^{2}\lambda^{2}}{\lambda^{2}+\pi^{2}a_{0}^{2}tan^{2}\alpha}}$

where $a_{0}$ is the radius of fiber core, and sin$\alpha$ = N.A, N.A. is the
numerical aperture of the fiber. In there $a_{0}=5\mu$m and N.A.=0.13 for
single-mode fiber. The incident optical power is:

$P_{\mathrm{in}}=\int_{0}^{\infty}I_{1}(r)2\pi
rdr=\frac{\pi}{2}\omega_{01}^{2}I_{0}$

The response of the light to the micro-sphere is calculated using the standard
optical ABCD ray matrix Kogelnik and Li (1966). Under the par-axial
approximation, the transmission matrix $\mathbf{T}$ is:

$\mathbf{T}=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$

which has the equation:

$\begin{pmatrix}r_{f}\\\
\theta_{f}\end{pmatrix}=\mathbf{T}\begin{pmatrix}r_{i}\\\
\theta_{i}\end{pmatrix}$

In calculating the transmission matrix $\mathbf{T}$, we neglected the
reflection of light at the interface and the absorption in the micro-sphere.
Here A, B, C, D are

$A=\frac{2}{n}-1,B=\frac{2R}{n},C=\frac{1-n}{n}\frac{2}{n},D=\frac{2}{n}-1,\beta_{0}=\frac{\lambda}{\pi\omega_{01}^{2}}$

with the parameters $\lambda=1550nm$, n=1.45, the we get the $d_{2}$ and
$\omega_{02}$ satisfy

$d_{2}=\frac{AC/\beta_{0}^{2}+ACd_{1}^{2}+ADd_{1}+BCd_{1}+BD}{C^{2}/\beta_{0}^{2}+C^{2}d_{1}^{2}+2CDd_{1}+D^{2}}$
$\omega_{02}=\omega_{01}\sqrt{(A+Cd_{2})^{2}+\beta_{0}^{2}(Ad_{1}+B+Cd_{1}d_{2}+Dd_{2})^{2}}$

$d_{2}$ and $\omega_{02}$ are functions of $d_{1}$, choose a suitable $d_{1}$
so that $\omega_{02}\approx a_{0}$. The coupling efficiency $\Gamma$, of the
laser beam and the single-mode optical fiber can be written as:

$\Gamma=\Gamma_{0}\mathrm{exp}\bigg{(}-\Gamma_{0}\cdot\frac{x_{\mathrm{fib}}^{2}}{2}(\frac{1}{\omega_{02}^{2}}+\frac{1}{a_{0}^{2}})\bigg{)},\Gamma_{0}=\frac{4\omega_{02}^{2}a_{0}^{2}}{(\omega_{02}^{2}+a_{0}^{2})^{2}}$

$x_{\mathrm{fib}}$ indicate the fiber shift from the $x$ direction, when
$x_{\mathrm{fib}}=0$, $\Gamma=\Gamma_{max}=\Gamma_{0}$. In the experiment, fix
$x_{\mathrm{fib}}$ at the place where $\partial\Gamma/\partial
x_{\mathrm{fib}}$ is the largest. As $x_{\mathrm{fib}}=2.51\mu m$ and
$\Gamma(x_{\mathrm{fib}})=0.604$ in Fig.5(b).

Figure 5: (a) Optical ray of the laser. $\theta_{i}$ ($\theta_{f}$) and
$r_{i}$ ($r_{f}$) are used to characterize the optical ray coming from
incident fiber and reaching the detection fiber, $d_{1}$ ( $d_{2}$) are the
distance between the incident fiber (detection fiber) and the optical axis,
$R$ is the radius of the micro-sphere. (b) Dependence of the light field
distribution with the microsphere position. The position of the image on the
incident fiber core $\delta x^{\prime}$ in x axis depends on the position of
the micro-sphere position $\delta x$. The transmission coefficient $\Gamma$
changes with $\delta x$.

$\delta x$ is the displacement of the micro-sphere vertically to the optical
axis (similar result for y direction), while $\delta x^{\prime}$ is the
projection on the incident fiber surface. Under par-axial approximation,
$\delta x=\zeta\cdot\delta x^{\prime}$ for small displacement $\delta x$ of
the micro-sphere, with the displacement magnification factor:

$\zeta=\frac{d_{1}+d_{2}+2R}{d_{1}+R},\varsigma=\frac{\partial\Gamma}{\partial
x}=\frac{\partial\Gamma}{\partial x^{\prime}}\cdot\frac{\partial
x^{\prime}}{\partial x}=\zeta\cdot\frac{\partial\Gamma}{\partial x^{\prime}}$

Measurement Noise. The relationship between the average power $P$ and the
photon number $N$ is:

$N_{\mathrm{in}}=\frac{P_{\mathrm{in}}T_{\mathrm{mea}}}{\hbar\omega_{\mathrm{op}}},N_{\mathrm{dec}}=\frac{P_{\mathrm{dec}}T_{\mathrm{mea}}}{\hbar\omega_{\mathrm{op}}}$

where $\omega_{\mathrm{op}}$ is the light frequency. The photons satisfy the
Poisson distribution and the corresponding photon number fluctuation is
$\delta N_{\mathrm{in}}=\sqrt{N_{\mathrm{in}}}$ and $\delta
N_{\mathrm{dec}}=\sqrt{N_{\mathrm{dec}}}$. Such fluctuation brings a imprecise
detection noise of displacement $\delta x_{\mathrm{imp}}$:

$\displaystyle\delta x_{\mathrm{imp}}$ $\displaystyle=\frac{\partial
x}{\partial\Gamma}\sqrt{\left(\frac{\partial\Gamma}{\partial
N_{\mathrm{in}}}\delta
N_{\mathrm{in}}\right)^{2}+\left(\frac{\partial\Gamma}{\partial
N_{\mathrm{dec}}}\delta N_{\mathrm{dec}}\right)^{2}}$
$\displaystyle=\frac{1}{\varsigma}\sqrt{\frac{\Gamma+\Gamma^{2}}{N_{\mathrm{in}}}}$

Thus the power density of displacement noise is:

$S_{\mathrm{xx}}^{\mathrm{imp}}=\frac{1}{\varsigma^{2}}\frac{(\Gamma+\Gamma^{2})\hbar\omega_{\mathrm{op}}}{P_{\mathrm{in}}}$

On the other hand, the photon passes through the micro-sphere which changes
the direction and therefore generated a back-action force $\delta
f_{\mathrm{ba}}$ with the strength also proportional to the fluctuation of the
incident photon $\delta N_{\mathrm{in}}$. The back-action force $\delta
f_{\mathrm{ba}}$ can be written as:

$\delta f_{\mathrm{ba}}=\sqrt{N_{\mathrm{in}}}\hbar\Delta k/T_{\mathrm{mea}}$

where $\Delta k$ is the change of the wave vector.

Here we suppose that the direction of light wave vector is along the direction
of the Gaussian light wavefront, and the probability of photon appearing is
proportional to the intensity of Gaussian light. $\Delta k$ is the average
change of light wave vector pass through the micro-sphere. It is calculated by
$\sqrt{(\Delta k_{\mathrm{in}})^{2}+(\Delta k_{\mathrm{out}})^{2}}$, where
$\Delta k_{\mathrm{in}}$ is the average light wave vector go to the micro-
sphere, $\Delta k_{\mathrm{out}}$ is the average light wave vector go out of
the micro-sphere. We obtain

$\displaystyle(\Delta k)^{2}=$ $\displaystyle k^{2}\beta$ $\displaystyle=$
$\displaystyle
k^{2}\int_{0}^{\infty}\frac{k^{2}r^{3}}{k^{2}r^{2}+((1-\frac{z_{r}^{2}}{z_{l}^{2}})\frac{kR^{2}}{2\rho(z_{l})}+\frac{z_{r}}{z}-k\rho(z_{l}))^{2}}\cdot$
$\displaystyle\frac{1}{\omega_{1}^{2}(z_{l})}\mathrm{exp}\left(-\frac{2r^{2}}{\omega_{1}^{2}(z_{l})}\right)dr$

where $k=\omega_{\mathrm{op}}/c$, $z_{l}=d_{1}+R-\sqrt{R^{2}-r^{2}}$,
$\omega_{1}(z_{l})=\omega_{01}\sqrt{1+(z_{l}/z_{r})^{2}}$,
$z_{r}=2\pi\omega_{01}^{2}/\lambda$ and
$\rho(z_{l})=z_{r}(z_{l}/z_{r}+z_{r}/z_{l})$.

The power density of back-action noise is thus:

$S_{\mathrm{ff}}^{\mathrm{ba}}=\frac{P_{\mathrm{in}}\hbar\omega_{\mathrm{op}}\beta}{c^{2}}$

and the product of imprecision noise and back-action noise is:

$S_{\mathrm{xx}}^{\mathrm{imp}}\cdot
S_{\mathrm{ff}}^{\mathrm{ba}}=\frac{1}{\varsigma^{2}}(\Gamma+\Gamma^{2})(\omega_{\mathrm{op}}/c)^{2}\beta^{2}\hbar^{2}$

The quantum efficiency of the measurement is defined as:

$\eta=\frac{\varsigma}{4(\Gamma+\Gamma^{2})\beta k^{2}}$

where $\eta$ = 1 corresponding standard quantum limit (SQL). The total
measurement noise is

$S_{\mathrm{aa}}^{\mathrm{mea}}(\omega)=\frac{S_{\mathrm{xx}}^{\mathrm{imp}}}{|\chi_{\mathrm{\scriptscriptstyle
m}}(\omega,\omega_{0})|^{2}}+\frac{S_{\mathrm{ff}}^{\mathrm{ba}}}{m^{2}}$

$S_{\mathrm{aa}}^{\mathrm{mea}}$ is minimized by tuning the incident laser
power $P_{\mathrm{in}}$ under the product constraint of the imprecision noise
and backaction noise. The optimized power is:

$P_{\mathrm{opt}}(\omega,\omega_{0})=\sqrt{\frac{\Gamma+\Gamma^{2}}{\beta}}\frac{mc}{\varsigma|\chi_{\mathrm{\scriptscriptstyle
m}}(\omega,\omega_{0})|}$

with the minimised total acceleration measurement noise as:

$S_{\mathrm{aa,min}}^{\mathrm{mea}}=\frac{2\hbar\omega_{\mathrm{op}}}{m\varsigma
c|\chi_{\mathrm{\scriptscriptstyle
m}}(\omega,\omega_{0})|}\sqrt{\beta(\Gamma+\Gamma^{2})}$

And in order to simplify the experiment process, we choose
$P_{\mathrm{in}}=P_{\mathrm{opt}}(\omega_{0},\omega_{0})$, with the optimized
acceleration measurement noise at this time:

$S_{\mathrm{aa,opt}}^{\mathrm{mea}}=\frac{\hbar\omega_{\mathrm{op}}\sqrt{\beta(\Gamma+\Gamma^{2})}}{m\varsigma
c\gamma\omega_{0}}\cdot\left(\frac{1}{|\chi_{\mathrm{\scriptscriptstyle
m}}(\omega,\omega_{0})|^{2}}+\gamma^{2}\omega_{0}^{2}\right)$