# 2D Li+ ionic hopping in Li3InCl6 as revealed by diffusion-induced nuclear
spin relaxation

Florian Stainer  Graz University of Technology, Institute of Chemistry and
Technology of Materials (NAWI Graz), Stremayrgasse 9, 8010 Graz, Austria  H.
Martin R. Wilkening<EMAIL_ADDRESS>Graz University of Technology,
Institute of Chemistry and Technology of Materials (NAWI Graz), Stremayrgasse
9, 8010 Graz, Austria

(August 28, 2024)

###### Abstract

Ternary Li halides, such as Li3MeX6 with, e.g., Me = In, Sc, Y and X = Cl, Br,
are in the center of attention for battery applications as these materials
might serve as ionic electrolytes. To fulfill their function, such
electrolytes must have an extraordinarily high ionic Li+ conductivity. Layer-
structured Li3InCl6 represents such a candidate; understanding the origin of
the rapid Li+ exchange processes needs, however, further investigation.
Spatially restricted, that is, low-dimensional particle diffusion might offer
an explanation for fast ion dynamics. It is, however, challenging to provide
evidence for 2D diffusion at the atomic scale when dealing with
polycrystalline powder samples. Here, we used purely diffusion-induced 7Li
nuclear magnetic spin relaxation to detect anomalies that unambiguously show
that 2D Li diffusion is chiefly responsible for the dynamic processes in a
Li3InCl6 powder sample. The change of the spin-lattice relaxation rate
$1/T_{1}$ as a function of inverse temperature $1/T$ passes through a rate
peak that is strictly following asymmetric behavior. This feature is in
excellent agreement with the model of P. M. Richards suggesting a logarithmic
spectral density function $J$ to fully describe 2D diffusion. Hence, Li3InCl6
belongs to the very rare examples for which 2D Li+ diffusion has been
immaculately verified. We believe that such information help understand the
dynamic features of ternary Li halides.

##  I Introduction

Materials with low-dimensional features of different types [1, 2, 3, 4, 5, 6]
play a prominent role in many areas of physics and materials science [6]. They
might clear the way for new applications that ground on functional compounds
whose crystal structures give rise to 1D or 2D electronic, magnetic,
mechanical, optical, optoelectronic, chemical and even neuromorphic and
biomedical properties [6].

Restricting ourselves to electric transport properties [7], in literature we
find many studies focussing on the spatially confined movements of any kinds
of electronic charge carriers in, e.g., inorganic and organic layer-structured
compounds, that is, in 2D materials [8, 9, 10, 11, 12, 13, 14, 15, 16, 17].
However, the number of investigations accepting the challenge to unequivocally
identify 2D transport of ionic species is still much lower [18, 19, 20, 21,
22, 23, 24]. Many mixed conductors, i.e., compounds that show both electronic
and ionic conductivity such as TiS2 [25], graphite [26] or LixCoO2 ($0.5<x\leq
1$) [27], act as 2D host structures that can accept and release Li+ in a
highly reversible manner. The rapid insertion and facile de-insertion
processes of small guest species without destroying the rigid host structure
are at the heart of modern electrochemical energy storage systems [28].
Research on this insertion principle finally culminated in the advent of the
now well known lithium-ion batteries for whose development M. S. Whittingham,
J. B. Goodenough, and A. Yoshino were jointly awarded the Nobel prize in
chemistry in 2019 [29, 30].

While some crystal structures provide classical van-der-Waals gaps for 2D
diffusion, such as single phase TiS2 [25], other materials offer geometric
restrictions to guide the ions along the 1D or 2D transport pathways [20, 23,
31]. Examples include the hexagonal form of LiBH4 showing 2D Li+ diffusion
[20] or the binary compound Li12Si7 in which the Li+ ions quickly hop along
the stacked, five-membered Si5 rings, thus performing quasi 1D diffusion [23].

The low-dimensional diffusion pathways of the two examples mentioned have been
studied by nuclear spin relaxation (NSR) [20, 23], which is highly suited to
probe the internal magnetic dipolar or electric quadrupolar field fluctuations
caused by the diffusive hopping processes of the spin-carrying atoms or ions
[32, 33]. Most importantly, while classical orientation-dependent conductivity
measurements require single crystals to probe the anisotropy of the transport
process, for NSR measurements powder samples are, however, sufficient. NSR
probes a spectral density function $J$ that contains the information about the
temporal interactions to which the spins are subjected [25, 34]; the
underlying basics of NSR measurements are provided in the Supporting
Information. Besides information on activation energies and Arrhenius pre-
factors determining a classical over-barrier hopping process, the shape of the
NSR rate peak does also depend on the dimensionality of the diffusive motions
of the spins [20, 32]. The powerfulness of this approach to identify low-
dimensional particle movements has recently been demonstrated on
polycrystalline $\beta$-Li3PS4 [31]. With the help of NSR measurements we were
able to reveal even buried low-dimensional pathways in this thiophosphate that
gives, at first glance, no occasion for 1D (zig-zag) diffusion pathways [31].

The compounds analyzed in this way so far, exhibit rather rapid 2D Li+
diffusion. Hence, enabling 2D ionic transport can be regarded as a valid
design principle to find materials that can serve as powerful solid
electrolytes in devices for electrochemical energy storage. Recently, so-
called ternary halides [35, 36, 37, 38, 39] have re-aroused the interest of
workgroups searching for fast ionic conductors. Lithium indium chloride,
Li3InCl6 (see Figure 1), represents one of the most fascinating
representatives of this class of materials as it provides geometric features
which could indeed lead to facile 2D Li+ diffusion between the In-rich layers
[40, 41, 42]. Explaining its conductivity properties with the help of
dimensionality effects would definitely contribute to disclosing the
structural principles that lead to fast ion transport in these solids.

Figure 1: (a) Crystal structure of monoclinic Li3InCl6 adopting
$C{\textrm{2}}/m$ symmetry; here, it is drawn such that Li1 on 4$h$ and Li2
residing on $2d$ fully occupy the interlayer void between the In-rich layers,
as proposed by single X-ray diffraction analysis by G. Meyer and co-workers.
(b) Illustration to better visualize the (ordered) layered structure of this
ternary halide. Recently, Helm et al. suggested that Li does also regularly
occupy the tetrahedral void indicated as T${}_{\textrm{Li}}$. Moreover, they
suggested that a relatively large number fraction of Li ions does also reside
on the In2 position, which is here either not occupied by Li+ or to a much
lesser extent.

Here, we prepared a phase-pure, polycrystalline Li3InCl6 sample via wet-
chemical synthesis and studied the diffusion-induced 7Li NSR rates $1/T_{1}$
as a function of temperature. Provided data from highly accurate measurements
are at hand, the rates can only be understood in terms of 2D diffusion fully
confirming the semi-empirical 2D spectral density function P. M. Richards has
introduced in 1978 [43, 44].

##  II Experiment

Li3InCl6 has been prepared through a water-assisted dissolution-precipitation
route [45]. For that purpose, the mixture of the starting materials (2 g) was
dissolved in 10 mL of miliQ water. The clear solution was transferred to a
Schlenk flask and the water was evaporated. The remaining white solid was
dried at 100 °C under vacuum for a period of 2 hours resulting in the hydrated
precursor Li3InCl6 $\cdot$ $x$H2O. In a second 4-h drying step carried out at
$\vartheta=$ 200 °C, we obtained the final polycrystalline product Li3InCl6.
After having removed any residual water through this final temperature
treatment, the product was handled strictly under Ar Atmosphere or vacuum to
avoid any further contamination with water.

The final product has been characterized by X-ray powder diffraction. X-ray
powder patterns were recorded on a Rigaku MiniFlex 600 powder diffractometer
operating with Bragg Brentano geometry and Cu-K$\alpha$-radiation. We covered
a 2$\theta$ range of 10° $\leq$ 2$\theta$ $\leq$ 90° (2.5 ° min-1, step size
0.1°). During the measurements, the sample has always been kept under inert
gas atmosphere (Argon) using an airtight sample holder constructed by Rigaku.

7Li NMR spin-lattice relaxation rates were recorded at
$\omega_{0}/2\pi=116\leavevmode\nobreak\ \rm MHz$ and 194 MHz using Bruker
Avance NMR spectrometers equipped with ceramic broadband probe heads (Bruker)
that allowed us to adjust the temperature (Eurotherm controller) inside the
sample chamber with a stream of heated nitrogen gas [46]. We used the
saturation recovery pulse sequence to monitor the buildup curves of the
longitudinal magnetization as a function of waiting time. The 90° excitation
pulse that perturbs the spin population accruing to the Boltzmann term ranged
from 2 to 2.5 $\mu$s depending on temperature $T$. These transients follow
single exponential time behavior and were analyzed to extract the diffusion-
induced rate $1/T_{1}$, which is proportional to the spectral density function
$J(\omega_{0})$ that has components at $\omega_{0}$ [25, 32], see Supporting
Information. Here, diffusion-induced relaxation towards thermodynamic spin
equilibrium is assumed to be controlled by both magnetic dipolar and electric
quadrupolar interactions as 7Li is a spin-3/2 nucleus. Independent of the
exact nature of spin-lattice relaxation, the shape of $J$ allows for the
determination of the dimensionality of the diffusion process, see Supporting
Information containing an introduction into the basics behind the method.

##  III Results and Discussion

In Figure 2 (a) we plotted the purely diffusion-induced 7Li NSR rates
$1/T_{1}$ of Li3InCl6 as a function of the inverse temperature using an
Arrhenius representation. Starting from low temperatures, the rates follow a
linear increase that is characterized by an activation energy $E_{\rm a}$ of
247(5) meV. In this so-called low-temperature regime, where we have to deal
with $\omega_{0}\tau_{\rm c}\gg 1$, the NMR rate is, in general, sensitive to
local jump events of the Li+ ions in Li3InCl6. Hence, we identify this
activation energy as that describing the elementary steps of Li+ diffusion in
the ternary halide.

Figure 2:  (a) Diffusion-induced 7Li NMR spin-lattice relaxation rates of
Li3InCl6 analyzed in the frame of an Arrhenius plot. The rates were recorded
at 116 MHz. At this frequency the low-temperature slope points to an
activation energy of 247(5) meV. While the solid line shows a fit according to
Richards NSR model for 2D diffusion, the dashed line indicates the response
that would be expected for 3D diffusion. $\beta=2$ reflects pure BPP-type
relaxation behavior in the limit $\omega_{0}\tau_{c}\gg 1$ indicating random
instead of correlated ion diffusion. The asymmetric shape of the Richards peak
obtained is diagnostic for low-dimensional diffusion. In the 2D relaxation
model the formal activation energy of 176 meV has no physical parallel as it
does not correspond to a real barrier. This reduced slope (71% $\approx
3/4\,E_{\rm a}$) is a consequence of the spectral density function $J$ of 2D
diffusion; for 1D diffusion it would be reduced to $E_{\rm a}/2$, see
Supporting Information (Table S1). (b) NMR relaxation rate peaks of Li3InCl6
recorded at two different Larmor frequencies as indicated. Here, 2D diffusion
produces a dispersive, frequency-dependent regime governing the rates of the
high-$T$ slope. See text for further details

Increasing the temperature even more, the rates pass through a maximum located
at $T$ $\approx$ 320 K. At this temperature, the mean residence time $\tau$ of
the Li ions equals the correlation time $\tau_{c}$, which is given by
$\omega_{0}\tau_{\rm c}\approx 1$, that is, determined by the angular Larmor
frequency, $\omega_{0}\approx 7.28\times 10^{8}$ rad s-1 used to sample the
rates. The jump rate $\tau^{-1}$ is assumed to depend on temperature following
the Arrhenius law with $k_{\rm B}$ denoting Boltzmann’s constant:

$\tau^{-1}\approx\tau_{\rm c}^{-1}=\tau_{\rm c0}^{-1}\exp(-E_{\rm a}/(k_{\rm
B}T)).$ (1)

Importantly, for 3D diffusion we would expect the $1/T_{1}$ NSR rates of the
peak shown in Figure 2 (a) to pass into a high-$T$ flank that is governed by a
slope either equal or larger than that probed on the low-$T$ side. Here,
however, the diffusion-induced 7Li NSR rates $1/T_{1}$ reveal a weaker-than-
expected temperature dependence in this $T$ regime. Such a behavior shown in
Figure 2 (a) can only be explained by assuming a low-dimensional diffusion
process. Such a process is also suggested by the crystal structure of Li3InCl6
(Figure 1), which favors a 2D process rather than a 3D one that would be
responsible for the (symmetric) $1/T_{1}(1/T)$ NMR peak.

To corroborate this assumption, we used the spectral density function
$J(\omega_{0})$ introduced by Richards [43] for 2D diffusion to analyze our
data; a brief introduction into the basics of NMR relaxation measurements is
given in the Supporting Information. The solid line in Figure 2 (a) shows a
fit according to the following relationship which takes into account a
logarithmic frequency dependence of the rate in the high-$T$ limit
($\omega_{0}\tau_{\rm c}\gg 1$); the constant $C^{\prime\prime}_{0}$ is
proportional to the effective coupling constant and determines the height of
the whole rate peak $1/T_{1}(1/T)$.

$J(\omega_{0})\propto 1/T_{1}=C^{\prime\prime}_{0}\dfrac{\tau_{\rm
c}}{1+(\omega_{0}\tau_{\rm c})^{\beta}}$ (2)

This dependence causes the NMR rate peak to assume an asymmetric shape as also
shown in Figure 2 (a). The excellent agreement between experimental data and
the semi-empirical Richards model [43] proves that Li+ diffusion in Li3InCl6
is of 2D nature. For comparison, the conventional response for 3D diffusion
[20] is also shown, see the dashed line in Figure 2 (a). The clear deviation
of the two models in the high-$T$ limit made it possible to unequivocally
identify Li3InCl6 as a 2D ionic conductor.

In both cases, and especially for the curve representing 2D diffusion, the
best fit is obtained for $\beta=2$. This parameter, if adopting values smaller
than 2, reduces the slope on the low-$T$ side as illustrated in the Supporting
Information. $\beta<2$ [19] is usually used to take into account correlation
effects [34, 33] leading to a sub-quadratic frequency dependence in the limit
$\omega_{0}\tau_{\rm c}\gg 1$ which also manifests in a clear change of the
slope in the limit $\omega_{0}\tau_{\rm c}\gg 1$ (see Figure S1 (a),
Supporting Information).

In the present case, a classical quadratic frequency dependence,
$1/T_{1}\propto\omega_{0}^{-2}$, which follows the idea of the pioneering
model of Bloembergen, Purcell and Pound [47] is fully sufficient to precisely
mirror the NMR response at least at 116 MHz (Figure 2 (a)). $\beta=2$ does not
point to strong correlation effects between the jumping ions which would lead
to a lower slope and thus to a lower (apparent) activation energy. Here, we
interpret the value of 247(5) meV as the elementary barrier the ions need to
surmount to diffuse within the Li-rich layer. In contrast, the same slope
should be seen in the high-$T$ region for the hypothetic case of 3D diffusion,
as indicated by the dashed line in Figure 2 (a). Such a BPP-type relaxation
behavior seen within Richards 2D model [43] suggests that we have to deal with
a symmetric potential landscape in which the ions perform random walks rather
than correlated motions. In such a uniform landscape there will be no
difference between short-range and long-range ion diffusion as earlier
discussed also for another 2D ion donators viz. LixTiS2.

As mentioned above, Richards’s 2D spectral density function predicts a
logarithmic frequency dependence of $1/T_{1}$ on the high-$T$ side (see also
Supporting Information, Figure S1 (b)). Such a behavior is in contrast to what
is expected for 3D diffusion, for which we have $1/T_{1}\neq\omega_{0}$
approaching the limit $\omega_{0}\tau_{\rm c}\ll 1$. Indeed, repeating the 7Li
NMR spin-lattice relaxation measurements at a higher Larmor frequency
$(\omega_{0}/2\pi=194\leavevmode\nobreak\ \rm MHz)$ reveals a frequency
dependence of $1/T_{1}$ on the high-temperature side of the diffusion-induced
peak (cf. see Figure 2 (b)). This feature perfectly corroborates the 2D
dynamic situation in Li3InCl6. In agreement with the maximum condition,
$\omega_{0}\tau_{\rm c}\approx 1$, the rate peak recorded at higher magnetic
field is slightly shifted towards higher temperature, see Figure 2 (b).

We have to note that the NMR measurements carried out at 194 MHz (see Figure 2
(b)) reveal a lower slope on the low-$T$ side than expected. The activation
energy reduces from 247 meV to only 208 meV; $\beta$ changes from 2 to 1.85.
Obviously, at increased Larmor frequency, that is, at a shorter time window,
the effect of correlation effects such as (localized) forward-backward jump
processes begin to influence the slope of the NMR rate peak in this
temperature limit.

Finally, let us briefly note that previous studies [48] also attempted to
tackle the same problem, that is, interpreting NSR rates in terms of 3D or 2D
jump diffusion. However, certain difficulties did not result in such a clear
picture as is obtained here. In the present case, we believe that the high
purity of the crystalline powder sample, for which we also assume a poor
defect density, lead to NMR rates solely induced by the 2D nature of the self-
diffusion process. As a side note, purely diffusion-induced $1/T_{1}$ rates
are an imperative necessity to precisely uncover the 2D dynamic features of
the ternary halide.

##  IV Conclusion

Li3InCl6, crystallizing with a layered structure, turned out to be an optimal
model system to examine the Richards model for its suitability to describe
spatially constrained 2D ionic dynamics in a crystalline matrix. The purely
diffusion-induced 7Li NMR relaxation rates point to an activation energy for
Li+ displacements of 247 meV and fully agree with the Richards’s model
introduced to describe 2D hopping diffusion. This model predicts an asymmetric
$1/T_{1}(1/T)$ rate peak and a logarithmic frequency dependence of the rates
in the high-$T$ limit. Our NSR results build the experimental reflection of
this semi-empirical model. Low dimensional Li+ diffusion is believed to be key
for the understanding of the superior transport properties of materials
related with Li3InCl6.

###### Acknowledgements.

Financial support by the DFG in the frame of the former research unit molife
(FOR1277) is greatly acknowledged.

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# Optomechanical cooling and inertial sensing at low frequencies

Yanqi Zhang  Texas A&M University, Aerospace Engineering & Physics, College
Station, TX 77843  James C. Wyant College of Optical Sciences, University of
Arizona,
1630 E. University Blvd., Tucson, AZ 85721  Adam Hines  Texas A&M University,
Aerospace Engineering & Physics, College Station, TX 77843  Dalziel Wilson
James C. Wyant College of Optical Sciences, University of Arizona,
1630 E. University Blvd., Tucson, AZ 85721  Felipe Guzman<EMAIL_ADDRESS>Texas A&M University, Aerospace Engineering & Physics, College Station, TX
77843

(August 27, 2024)

###### Abstract

An inertial sensor design is proposed in this paper to achieve high
sensitivity and large dynamic range in the sub-Hz frequency regime. High
acceleration sensitivity is obtained by combining optical cavity readout
systems with monolithically fabricated mechanical resonators. A high-
sensitivity heterodyne interferometer simultaneously monitors the test mass
with an extensive dynamic range for low-stiffness resonators. The bandwidth is
tuned by optical feedback cooling to the test mass via radiation pressure
interaction using an intensity-modulated laser. The transfer gain of the
feedback system is analyzed to optimize system parameters towards the minimum
cooling temperature that can be achieved. To practically implement the
inertial sensor, we propose a cascaded cooling mechanism to improve cooling
efficiency while operating at low optical power levels. The overall system
layout presents an integrated design that is compact and lightweight.

##  I Introduction

Acceleration sensing is crucial for tasks such as seismology [1] and inertial
navigation [2], as well as a myriad of applications spanning the automobile
[3], aerospace [4], and consumer electronics industry [5]. In the past decade,
advances in cavity optomechanics [6] have made it possible to develop
accelerometers with ultrahigh sensitivity [7, 8, 9, 10, 11], relevant to tasks
such as geodesy [12, 13], gravitational wave detection [9, 10] and dark matter
searches [11]. Such optomechanical accelerometers typically consist of a
mechanical resonator whose test mass oscillates in response to external
accelerations. They also include an optical cavity to enhance the radiation
pressure interaction between electromagnetic fields of light and modes of the
mechanical resonator. This interaction provides high sensitivity measurements
of the test mass displacement [14]. It also provides the ability to tune the
resonator frequency and damping rate via dynamic radiation pressure back-
action [15, 16], enabling control over the accelerometer bandwidth and dynamic
range.

In the low-frequency regime (below $1\text{\,}\mathrm{H}\mathrm{z}$), the
development of high sensitivity optomechanical accelerometers faces several
challenges. The first challenge is to combine high displacement sensitivity
with a large dynamic range; for resonators with low resonant frequency, the
test mass displacement can reach amplitudes of hundreds to thousands of
microns in applications such as seismometry. In this case, cavity-enhanced
readout systems such as Fabry-Perot interferometers (FPIs) are not suitable,
since their dynamic range is typically lower than the optical wavelength. In
addition, various noise sources become significant in the low-frequency
regime, such as laser frequency and thermo-elastic noise [17, 18]. Alternative
readout systems with high dynamic range have been explored, such as the
heterodyne laser interferometer proposed for the Laser Interferometry Space
Antenna (LISA) and the interferometer launched in LISA Pathfinder,[19, 20, 21,
22]. The LISA Pathfinder interferometer has achieved a displacement
sensitivity of $10\text{\,}\mathrm{p}\mathrm{m}\mathrm{/}\sqrt{Hz}$ at
$1\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$ on ground, and
$30\text{\,}\mathrm{f}\mathrm{m}\mathrm{/}\sqrt{Hz}$ in space. However, its
assembly involves complicated alignment and bonding techniques. Recently we
have developed compact common-mode heterodyne interferometers [23, 24, 25, 26]
for inertial sensing that provide high sensitivity. However, the noise floor
of the interferometer is a few orders of magnitude higher than the thermal
noise of the test mass, pointing toward the need for a hybrid approach.

A second challenge is that lowering the resonant frequency of the mechanical
resonator entails adding mass or reducing stiffness, resulting in a bulky and
likely delicate system. Moreover, radiation pressure back-action damping
becomes less efficient for low-frequency resonators, as it entails operating
in the “bad cavity limit” [27, 28, 29, 30], where the mechanical frequency is
much smaller than the cavity bandwidth. Active radiation pressure feedback
damping, where the test mass motion is suppressed by derivative feedback onto
the laser intensity, has been demonstrated as an effective method for gram-
scale resonators with mechanical frequencies on the order of 100 Hz or above
[31, 32, 33, 34]. However, for large-mass resonators, this entails large
radiation pressure forces and concomitantly high optical power handling
capacity.

In this paper, we propose an optomechanical accelerometer capable of high
performance at sub-Hz frequencies. A key feature is the integration of two
readout systems: an FPI designed for high-sensitivity displacement
measurements and a heterodyne interferometer designed for high dynamic range
displacement measurements. We show how combining these approaches with
radiation pressure enables cascaded feedback cooling, a strategy whereby the
resonator’s motion can be suppressed to within the linewidth of the FPI,
allowing high sensitivity, high dynamic range, and reduced optical power
requirements. The paper is organized as follows: in Section II we present the
overall system design, including detailed analyses of the mechanical
resonator, the optical readout systems, and the feedback cooling system; in
Section III we describe the feedback cooling strategy and how to optimize the
feedback gain to maximize cooling efficiency; Section IV focuses on the
practical implementation of the sensor, designing the system parameters based
on an optimized feedback gain. Finally, we present our cascaded feedback
cooling strategy and show that it relaxes the laser power requirements while
maintaining the lowest effective temperature that can be reached by the
cooling process.

##  II System design

###  II.1 Overall system layout

The design concept for our low-frequency optomechanical inertial sensor is
shown in Figure 1. In our system, the acceleration test mass is suspended from
a rigid frame (representing the inertial reference frame), forming a
mechanical resonator. The acceleration, $a$, is obtained by measuring the
displacement $x$ of the test mass. The transfer function relating these two
quantities is

$\frac{x(\omega)}{a(\omega)}=\frac{-1}{\omega_{0}^{2}-\omega^{2}+i(\omega\gamma_{\text{v}}+\omega_{0}^{2}\phi(\omega))},$
(1)

where $\omega_{0}$ is the resonance frequency of the test mass,
$\gamma_{\text{v}}$ is its velocity damping rate (e.g., due to gas damping),
and $\omega_{0}^{2}\phi(\omega_{0})/\omega$ is its damping rate due to
internal loss in the suspension, characterized by the loss coefficient
$\phi(\omega)$ [35].

Figure 1: Sketch of the low-frequency inertial sensor design. Two optical
readout systems measure the test mass displacement at the same time. The
Fabry-Perot interferometer (FPI) performs displacement measurements with high
sensitivity and serves as the main optical readout system. If the test mass
motion is beyond the measurement range of FPI, the resonator oscillation is
cooled down by the feedback control system via radiation pressure interaction.
The heterodyne laser interferometer (HLI) obtains the test mass motion and
feeds into the feedback control system.

Two optical readout systems monitor the test mass displacement simultaneously.
An FPI with enhanced sensitivity serves as the main readout system, and its
output is converted to acceleration using Equation 1.

To overcome the limited dynamic range of the FPI (smaller than the optical
wavelength, $\lambda$), the test mass is simultaneously monitored over a
significantly larger range by a heterodyne laser interferometer (HLI). The
purpose of the HLI is to determine if the test mass displacement is within the
FPI dynamic range. If not, then the HLI output is imprinted onto the intensity
of an auxiliary laser field to implement radiation pressure feedback cooling.

###  II.2 Mechanical resonator design and characterization

The mechanical resonator, depicted in Figure 2, consists of a
$2.6\text{\,}\mathrm{g}$ test mass supported by a pair of flexures with
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thickness. The overall footprint
is $80\text{\,}\mathrm{mm}$ $\times$ $90\text{\,}\mathrm{mm}$ and the total
mass is $58.2\text{\,}\mathrm{g}$. The design, optimization, and performance
analysis of a mechanical resonator with a similar geometry are discussed in
detail in [35]. The monolithic resonator is fabricated of a single fused
silica wafer to minimize internal losses at room temperature [36].

Figure 2: Geometry of the mechanical resonator design. The
$2.6\text{\,}\mathrm{g}$ test mass is supported by two
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$-thick flexures. The overall
dimension is $80\text{\,}\mathrm{mm}$ $\times$
$90\text{\,}\mathrm{m}\mathrm{m}$ $\times$ $6.6\text{\,}\mathrm{m}\mathrm{m}$
and the total mass is $58.2\text{\,}\mathrm{g}$. The notch on the baseplate is
designed to integrate the optical readout system, and the notch on the test
mass is to mount a plane mirror.

The thermal motion of the test mass fundamentally limits the achievable
acceleration sensitivity to

$\displaystyle a_{\mathrm{th}}(\omega)$
$\displaystyle=\sqrt{\frac{4k_{\mathrm{B}}T}{m}\left(\gamma_{\text{v}}+\frac{\omega_{0}^{2}\phi(\omega)}{\omega}\right)}$
(2)
$\displaystyle=\sqrt{\frac{4k_{\mathrm{B}}T\omega_{0}}{m}\left(\frac{1}{Q_{\text{v}}}+\frac{1}{Q_{\text{int}}}\frac{\omega_{0}}{\omega}\frac{\phi(\omega)}{\phi(\omega_{0})}\right)},$
(3)

where $k_{\mathrm{B}}$ is the Boltzmann constant, $T$ is the device
temperature, $Q_{\text{v}}=\omega_{0}/\gamma_{\text{v}}$ is the mechanical
quality factor ($Q$) due to gas damping and
$Q_{\text{int}}=1/\phi(\omega_{0})$ is the mechanical $Q$ due to internal
damping.

Equation 2 shows that a high value for $mQ/\omega_{0}$ is desirable to achieve
a low thermal noise floor. To determine the resonance frequency $\omega_{0}$
and estimate the internal $Q$, we conducted ringdown measurements at a vacuum
pressure of $10\text{\,}\upmu\mathrm{Torr}$ (to minimize gas damping,
$\gamma_{\text{v}}$). The ringdown shown in Figure 3 is for a device with
$\omega_{0}=2\pi\times$4.72\text{\,}\mathrm{H}\mathrm{z}$$, yielding a
$Q=$4.77\text{\times}{10}^{5}\text{\,}$$, a $mQ$ product of
$1240\text{\,}\mathrm{k}\mathrm{g}$, and an acceleration noise floor of
$a_{\mathrm{th}}(\omega_{0})=$1\text{\times}{10}^{-11}\text{\,}\mathrm{m}\,\mathrm{s}^{-2}\mathrm{/}\sqrt{Hz}$$
near resonance [12].

Figure 3: The envelope of the test mass ringdown measurement under vacuum
conditions of $10\text{\,}\mu\mathrm{Torr}$. By fitting the envelope with an
exponential decay, we obtain the quality factor
$Q=$4.77\text{\times}{10}^{5}\text{\,}$$, corresponding to a $mQ$ product of
$1240\text{\,}\mathrm{k}\mathrm{g}$ [12].

###  II.3 Optical readout systems

####  II.3.1 Fabry-Perot Interferometer (FPI)

For our main optical readout system, we envision a FPI of $L\approx 50$ mm,
formed by mounting one mirror onto the test mass and one onto the resonator
frame. A shorter fiber-cavity approach has been previously demonstrated with
stiffer resonator designs [8]; here the large dynamic range of our sensor
requires a free space approach. The FPI is probed with a
$\lambda=$1064\text{\,}\mathrm{n}\mathrm{m}$$ laser whose frequency is locked
to an FPI resonance using the Pound-Drever-Hall (PDH) technique [37].
Meanwhile, the absolute laser frequency is tracked, for instance, by beating
it against an iodine-referenced laser or another suitable optical frequency
reference. Within the bandwidth of frequency measurement instrument, the
cavity length $L$ and laser frequency $\nu$ fluctuations are related by

$\frac{\nu}{L}=\frac{\bar{\nu}}{\bar{L}}=\frac{c}{\overline{\lambda}\overline{L}}$
(4)

Hence, the readout signal can be expressed as

$\displaystyle\nu(\omega)$
$\displaystyle=\frac{c}{\overline{\lambda}\overline{L}}\frac{a_{\text{ext}}(\omega)+a_{\text{th}}(\omega)}{\omega_{0}^{2}-\omega^{2}+i(\omega\gamma_{\text{v}}+\omega_{0}^{2}\phi(\omega))}+\nu_{\text{n}}(\omega)$
(5a)
$\displaystyle=\frac{c}{\overline{\lambda}\overline{L}}\frac{a_{\text{ext}}(\omega)+a_{\text{th}}(\omega)+a_{\text{n}}(\omega)}{\omega_{0}^{2}-\omega^{2}+i(\omega\gamma_{\text{v}}+\omega_{0}^{2}\phi(\omega))},$
(5b)

where $a_{\text{ext}}$ is the physical acceleration of the resonator (the
desired signal), $a_{\text{th}}$ is the apparent acceleration of the resonator
due to the thermal motion of the test mass (Eq. 2), and $a_{\text{n}}$ is the
apparent acceleration due to readout noise $\nu_{\text{n}}$. The latter arises
from various sources including laser frequency noise, shot noise [38],
technical noise, and cavity mirror noise (e.g. thermo-elastic noise).

For slow changes, the dynamic range $\Delta L$, which is the largest
measurable length change of the optical readout system, is determined by the
laser frequency tuning range $\Delta\nu$:

$\displaystyle\Delta L$
$\displaystyle=\frac{\overline{\lambda}\overline{L}}{c}\Delta\nu$ (6a)
$\displaystyle=$1.8\text{\,}\mathrm{\SIUnitSymbolMicro
m}$\times\frac{\Delta\nu}{$10\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$}\frac{\overline{\lambda}}{$1064\text{\,}\mathrm{n}\mathrm{m}$}\frac{\overline{L}}{$50\text{\,}\mathrm{m}\mathrm{m}$}$
(6b)

In Eq. 6b we estimate a dynamic range of 1.8 $\upmu$m with a typical
commercial Nd:YAG laser with a tuning range of
$10\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. We anticipate that this dynamic
range will be too small for operation in a non-isolated environment since an
equivalent test mass displacement would be produced by an equivalent
acceleration input of only $\Delta L\omega_{0}/\sqrt{Q}\approx
8\;\text{n}g/\sqrt{\text{Hz}}$. With this in mind, in the following sections,
we propose a HLI to monitor the full range of the test mass motion and a
radiation-pressure feedback control system to damp the test mass using the HLI
output as an error signal.

####  II.3.2 Long-range heterodyne laser interferometer (HLI)

Heterodyne interferometry is a common technique that has the potential to
achieve high sensitivity in addition to a large dynamic range and traceable
calibration. Recently we have developed a compact HLI [25] for use with low-
frequency optomechanical inertial sensors. Our HLI employs a common-path
design, as shown in Figure 4, to provide a high rejection ratio to various
forms of environmental noise.

Figure 4: Schematic diagram of the quasi-monolithic interferometer unit in the
isometric view. Two incoming laser beams are split into four beams by the
50-50 non-polarizing splitting surface embedded in the equilateral triangular
prism. The beam pair with different frequencies constructs one interferometer.
The measurement interferometer (MIFO) measures the target displacement. The
reference interferometer (RIFO) measures the systematic noises that share the
common optical paths with the MIFO. The target displacement is calculated from
the differential phase measurement between MIFO and RIFO.

Concretely, a measurement interferometer (MIFO) measures the test mass motion,
and a reference interferometer (RIFO) of common optical paths with MIFO,
monitors ambient noise. The RIFO signal is then subtracted from the MIFO
output to reduce its noise content. The complete system has a footprint of
$$20\text{\,}\mathrm{m}\mathrm{m}$\times$20\text{\,}\mathrm{m}\mathrm{m}$\times$10\text{\,}\mathrm{m}\mathrm{m}$$
and weighs $4.5\text{\,}\mathrm{g}$. To achieve such a compact assembly, all
the optical components are cemented as a quasi-monolithic unit. The small size
and weight enable integration onto the mechanical resonator’s frame.
Preliminary tests have yielded a noise floor of
$2\text{\times}{10}^{-13}\text{\,}\mathrm{m}\mathrm{/}\sqrt{Hz}$ around
$1\text{\,}\mathrm{H}\mathrm{z}$, limited by photodetector noise, as shown in
Figure 5 [25]. If the displacement recorded by the HLI is outside the dynamic
range of the FPI, the HLI data is used as the error signal of the feedback
control system described in the next section.

Figure 5: The linear spectral density (LSD) and its logarithmic average of the
heterodyne laser interferometer (HLI) [25] noise floor. The interferometer
prototype shows a noise floor of
$2\text{\times}{10}^{-13}\text{\,}\mathrm{m}\mathrm{/}\sqrt{Hz}$ above
$1\text{\,}\mathrm{H}\mathrm{z}$ when tested in vacuum.

###  II.4 Feedback control system

The feedback control system includes signal processing modules and digital-
analog conversion interfaces to actuate the test mass. In our case, minimal
contact with the test mass is desirable to reduce surface losses. We therefore
consider radiation pressure as a feedback actuator. A further advantage of
this approach is the traceability of the radiation pressure force through the
laser wavelength [39, 40].

Figure 6: Layout of the feedback control system. The feedback force is
provided by radiation pressure interaction between an intensity-modulated
laser beam and the test mass. The laser intensity is modulated by an electro-
optical amplitude modulator (EOAM). The displacement of the test mass is read
out by a heterodyne interferometer (simplified in the figure), and sent to the
feedback control loop. The controller in the feedback loop calculates and
outputs a corresponding voltage signal to the EOAM to modulate actuating laser
intensity. The laser is stabilized in frequency and amplitude to reduce the
measurement noise.

Figure 6 shows our proposed radiation pressure feedback scheme, where we
launch an auxiliary laser beam onto the test mass, whose intensity is
modulated with the output of an HLI displacement sensor. To control the test
mass position $x$ in a traceable manner, it is necessary to understand the
transduction chain comprising the feedback circuit. To this end we consider
the following model

$\displaystyle x(\omega)$
$\displaystyle=\chi_{\mathrm{m}}(\omega)(F_{\mathrm{th}}(\omega)+F_{\mathrm{ext}}(\omega)+F_{\mathrm{fb}}(\omega))$
(7)
$\displaystyle=\chi_{\mathrm{m}}(\omega)(F_{\mathrm{th}}(\omega)+F_{\mathrm{ext}}(\omega)-\chi_{\mathrm{fb}}(\omega)y(\omega))$
$\displaystyle=\chi_{\mathrm{eff}}(\omega)(F_{\mathrm{th}}(\omega)+F_{\mathrm{ext}}(\omega)-\chi_{\mathrm{fb}}(\omega)x_{n}(\omega))$

where $y=x+x_{n}$ is the apparent position detected by the HLI, with a readout
noise $x_{n}$, $F_{\mathrm{fb}}$ is the radiation pressure feedback force,
$\chi_{\mathrm{fb}}(\omega)\equiv-F_{\mathrm{fb}}(\omega)/y(\omega)$ is the
feedback gain, and

$\chi_{\mathrm{eff}}(\omega)\equiv\frac{\chi_{\mathrm{m}}(\omega)}{1+\chi_{\mathrm{m}}(\omega)\chi_{\mathrm{fb}}(\omega)}$
(8)

is the effective (closed-loop) mechanical susceptibility.

It is evident from Equation 8 that the thermal force sensitivity remains
unchanged by the feedback actuation; however, as shown below, feedback damping
($\chi_{\mathrm{fb}}\propto ix$) is advantageous because it allows the sensor
to operate in the linear regime $x\ll\lambda/\mathcal{F}$, by reducing the
displacement on resonance. The cost of this extended dynamic range is
additional stochastic force due to the feedback of measurement noise,
$\chi_{\mathrm{fb(\omega)}}\,x_{n}(\omega)$.

The phase and magnitude of the feedback gain must be tailored to achieve a
desired closed-loop response and noise figure, which can be tailored in the
digital loop software. We thus consider the following model for the feedback
gain

$\chi_{\mathrm{fb}}(\omega)=G_{FP}\,G_{PV}\,G_{Vx}(\omega),$ (9)

where $G_{FP}$, and $G_{PV}$ are the steady-state response functions of the
radiation pressure actuator and intensity modulator, respectively, and
$G_{Vx}(\omega)$ is the frequency dependent response function of the software-
interfaced HLI that considers the digital processing for phase extraction
(phasemeter).

To model the actuator transfer function, we assume the test mass has perfect
reflectivity for the power $P$ incident on the test mass. Thus, we obtain

$G_{FP}=\frac{dF_{\mathrm{RP}}}{dP}=\frac{2}{c}.$ (10)

The transfer function of the intensity modulator depends on the modulation
method. We envision an intensity modulator based on a polarization-based EOAM,
for which the output power $P$ (referred to the power incident on the test
mass) is related to the voltage $V$ applied across the electrodes as
$P=P_{0}\cos^{2}(\pi V/V_{\pi})$, where $P_{0}$ is the maximum transmitted
power and $V_{\pi}$ is the half-wave voltage of the EOAM. This leads to a
transfer function

$G_{PV}=\frac{dP}{dV}=\frac{\pi P_{0}\sin(V/V_{\pi})}{V_{\pi}}.$ (11)

Finally, the transfer function of the HLI depends on the phase measurement
technique. Common phase extraction algorithms include phase-locked loop (PLL)
[41] and single-bin DFT algorithms [19]. A disadvantage of these methods is
that both involve computationally intensive inverse trigonometric functions,
which lead to a non-linear time-invariant (NLTI) system. Instead, we propose a
digital phasemeter, which only requires a low-pass filter. The phase response
of the entire system is determined by this filter, and has a negligible effect
at low frequencies. With this assumption, the HLI transfer function $G_{Vx}$
can be expressed as

$G_{Vx}(\omega)=\frac{V(\omega)}{x(\omega)}=G_{\mathrm{DAC}}(\omega)\frac{2\pi}{\lambda},$
(12)

where $G_{\mathrm{DAC}}$ is an amplification factor that can be set when
converting the measured phase to the analog voltage output applied to the
EOAM. The significance of $G_{\mathrm{DAC}}$ will be explained in detail in
Section IV.1.

Combining Equations 9-12, the feedback gain of the system shown in Figure 6,
can be expressed as

$\chi_{\mathrm{fb}}(\omega)\approx\frac{4\pi^{2}}{c\lambda}\frac{G_{\mathrm{DAC}}(\omega)P_{0}\sin{2\theta}}{V_{\pi}},$
(13)

where system parameters such as the laser power $P_{0}$ can be optimized to
enhance the feedback cooling efficiency.

##  III Feedback cooling optimization

###  III.1 Optical cooling

In feedback cooling protocols, derivative feedback gain –i.e. a velocity-
proportional feedback force– is used to damp the test mass displacement $x$ to
the measurement noise floor $x_{n}$. In practice, however, the feedback is
bandwidth limited for the low frequencies considered here, such that digital
filtering can provide a good approximation to a purely derivative gain. We
thus consider the following model

$\chi_{\mathrm{fb}}(\omega)=i\,m\,g\,\gamma_{\mathrm{m}}(\omega)\omega,$ (14)

where $g$ is a unitless gain factor and
$\gamma_{\mathrm{m}}(\omega)=\gamma_{\text{v}}+\omega_{0}^{2}\phi(\omega)/\omega$
is a frequency depending damping term including the effects of viscous (gas)
damping and internal damping, as in Equation 1.

Hence, the closed-loop susceptibility of the system is

$\displaystyle\chi_{\mathrm{eff}}(\omega)$
$\displaystyle=\frac{1}{m[\omega_{0}^{2}-\omega^{2}+i(1+g)\gamma_{\mathrm{m}}(\omega)\omega]}$
(15a)
$\displaystyle\equiv\frac{1}{m[\omega_{0}^{2}-\omega^{2}+i\gamma_{\mathrm{eff}}(\omega)\omega]},$
(15b)

which is characterized by an effective damping rate
$\gamma_{\mathrm{eff}}\equiv(1+g)\gamma_{\mathrm{m}}(\omega)$.

Figure 7: Effective susceptibility $\chi_{\mathrm{eff}}(\omega)$ of the test
mass for different gain factors $g=$ 0, 2500, 5000, and 10000. The case of
$g=0$ represents the open loop mechanical susceptibility. Increasing $g$
reduces driven displacement on resonance.

###  III.2 Feedback gain optimization

Figure 7 suggests that increasing the feedback gain $g$ reduces the test mass
displacement at resonance. However, it also increases the feedback of the
measurement noise, expressed by the term $\chi_{\mathrm{fb}}x_{n}$ in Equation
7. As a result, there is an optimal gain at which the total closed-loop
displacement can be minimized. This can be seen by re-writing Equation 7 in
terms of the closed loop displacement power spectral density

$\displaystyle S_{xx}(\omega)$
$\displaystyle=|\chi_{\mathrm{eff}}(\omega)|^{2}\large(S_{FF}^{\mathrm{th}}(\omega)+S_{FF}^{\mathrm{ext}}(\omega)$
$\displaystyle+\left|\chi_{\mathrm{fb}}(\omega)\right|^{2}S_{xx}^{\mathrm{n}}(\omega)\large)$

and integrating over the resonance peak to give the closed loop displacement
variance

$\langle x^{2}\rangle\approx\frac{1}{1+g}\langle
x_{\mathrm{th,0}}^{2}\rangle+\frac{g^{2}}{1+g}\langle
x_{\mathrm{n}}^{2}\rangle+\langle x_{\mathrm{ext}}\rangle^{2}(g)$ (17)

where $\langle
x_{\mathrm{th,0}}^{2}\rangle=k_{B}T/m\omega_{0}^{2}\approx\gamma_{\mathrm{m}}S_{xx}^{\mathrm{th}}(\omega_{0})/4$
is the open loop thermal displacement, $\langle
x_{\mathrm{n}}^{2}\rangle\approx\gamma_{\mathrm{m}}S_{xx}^{\mathrm{n}}(\omega_{0})/4$
is the apparent displacement due to the measurement noise, and $\langle
x_{\mathrm{ext}}^{2}\rangle=\int|\chi_{\mathrm{eff}}(\omega)|^{2}S_{\mathrm{FF}}^{\mathrm{ext}}(\omega)d\omega/2\pi$
is the closed loop displacement due to external forces.

In principle, $g$ can be tailored to minimize $\langle x^{2}\rangle$. However,
the magnitude of this gain depends on the external force spectrum
$S_{FF}^{\mathrm{ext}}(\omega)$. Without knowing
$S_{FF}^{\mathrm{ext}}(\omega)$, it is customary to optimze $g$ to minimize
the test mass displacement at resonance, i.e.

$g_{\mathrm{opt}}\approx\sqrt{\frac{\langle
x_{\mathrm{th,0}}^{2}\rangle}{\langle
x_{\mathrm{n}}^{2}\rangle}}=\sqrt{\frac{4k_{\mathrm{B}}T}{m\omega_{0}^{2}\Gamma_{\mathrm{m}}S_{xx}^{\mathrm{n}}}},$
(18)

yielding

$\langle x^{2}\rangle_{\mathrm{min}}\geq 2\sqrt{\langle
x_{\mathrm{th,0}}^{2}\rangle\langle x_{\mathrm{n}}^{2}\rangle}.$ (19)

Here we assume that $g_{\mathrm{opt}}\gg 1$ corresponds to a large thermal-to-
readout-noise ratio.

Equivalently, the effect of feedback damping is to reduce the effective
temperature of the test mass to

$T_{\mathrm{eff}}=\frac{m\omega_{0}^{2}\langle
x^{2}\rangle}{k_{B}}\geq\frac{1}{1+g}T+\frac{g^{2}}{1+g}T_{n}\geq
2\sqrt{T\,T_{n}},$ (20)

where $T_{n}\equiv m\omega_{0}^{2}\langle x_{n}^{2}\rangle/k_{B}$ is the
apparent temperature of the readout noise and the minimal value is obtained
for $g=g_{\mathrm{opt}}$ in the absence of external forces.

In Figure 8a, we plot the effective temperature of the resonator described in
Section II.2, subject to feedback cooling with a variety of gains and
measurement noise levels, assuming $T=300$ K. In Figure 8b we plot the optimal
gain and associated minimal temperature as a function of measurement noise. In
the next section we consider a parameter space that is practically accessible.

a

b

Figure 8: (a) Relation between effective temperature and gain factor for
different levels of readout imprecision noise; (b) The minimum temperature
that the cooling system can achieve and the corresponding gain factor to
achieve this temperature versus the readout imprecision noise in the system.

##  IV Feedback implementation

In this section, we consider how to practically implement feedback cooling of
our low frequency resonator, described in Section II.2 [12]. We are
particularly interested in the requirements to cool the test mass to the
measurement noise floor, corresponding to a gain factor of
$g=g_{\mathrm{opt}}$ (Equation 18). Building upon the recent development of a
monolithic HLI integrated with a similar test mass [24], we conservatively
assume
$\sqrt{S_{xx}^{\mathrm{n}}}=$5\text{\times}{10}^{-12}\text{\,}\mathrm{m}\mathrm{/}\sqrt{Hz}$$.
Combined with the test mass properties described in Section II.2, this implies
$g_{\mathrm{opt}}=$3.40\text{\times}{10}^{4}\text{\,}$$. We find that
achieving this gain factor is challenging with the long-range (HLI) detection
scheme, due to its (deliberately) small transduction gain. To overcome this
challenge, we propose a cascaded feedback cooling protocol employing both
long-range (HLI) and high sensitivity (FPI) readout systems in series.

###  IV.1 System parameters

Combining Equations 13 and 14, near the resonance frequency, the feedback gain
factor $g$ is related to the system parameters in the control loop by

$g=|\chi_{\mathrm{fb}}(\omega_{0})|\frac{Q}{m\omega_{0}^{2}}=\frac{4\pi^{2}}{c\lambda}\frac{Q}{m\omega_{0}^{2}}\frac{|G_{\mathrm{DAC}}(\omega_{0})|P_{0}\sin{2\theta}}{V_{\pi}}.$
(21)

In practice, the EOAM phase is adjusted to maximize the modulation depth,
corresponding to $\sin{2\theta}=1$. The remaining free parameters are the
optical power $P_{0}$, the half-wave voltage $V_{\pi}$, and the digital
conversion gain $G_{\mathrm{DAC}}$, which is the parameter that provides the
largest tuning range.

As mentioned in Section II.4, the digital gain, $G_{\mathrm{DAC}}$, is a
variable factor in the digital-to-analog conversion (DAC) coefficient that
translates the digital phasemeter output — a phase in radian — into a
corresponding proportional analog voltage that is injected into the EOAM to
control the optical power used for feedback cooling. In practice, the
magnitude of $G_{\mathrm{DAC}}$ is limited by the EOAM $V_{\pi}$ and the
physical displacement range of the test mass $x_{\mathrm{p-p}}$, where

$G_{\mathrm{DAC}}^{\mathrm{max}}=\frac{V_{\pi}\lambda}{2\pi
x_{\mathrm{p-p}}}.$ (22)

Figure 9: Effective temperature of the test mass subject to feedback cooling
with a constant gain factor $g$ = 1, 100, and
$3.40\text{\times}{10}^{4}\text{\,}$, and for an iteratively increased $g$
(cascaded feedback cooling). Each step on the iterative curve corresponds to
the moment when $G_{\mathrm{DAC}}$ changes. The inset shows a zoom-in on the
final state for $g$ = 100 and $3.40\text{\times}{10}^{4}\text{\,}$, and the
iterative $g$. In the cascaded cooling mechanism, the initial gain factor is
$g_{0}=1$. After 6 iterations, its final state reaches the same level as
single-step feedback cooling with
$g_{\mathrm{opt}}=$3.40\text{\times}{10}^{4}\text{\,}$$. For iterative
cooling, the optical power is fixed at
$P_{0}=$1.16\text{\,}\mathrm{m}\mathrm{W}$$.

###  IV.2 Feedback cooling with HLI detector

The open loop displacement of our test mass has been measured to be on the
order of $\pm$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ in our laboratory,
without vibration isolation. Combining Equations 18, 21, and 22, with the
mechanical properties of the resonator in Section II.2, we predict that the
minimum laser power $P_{0}$ needed to achieve a gain factor $g=1$
($T_{\mathrm{eff}}=T/2$) is $1.16\text{\,}\mathrm{m}\mathrm{W}$, and the laser
power to achieve $g_{\mathrm{opt}}=$3.40\text{\times}{10}^{4}\text{\,}$$ is
$34.43\text{\,}\mathrm{W}$. Thus we find that, for our current test mass,
feedback cooling with the long range HLI detector is challenging due to the
required high laser power. In particular, $P_{0}\gtrsim 100$ mW has the
potential to damage the EOAM crystal. It also poses a challenge in terms of
thermal management and handling of other components in the system such as the
fiber couplers and photodetectors. To reduce the power requirements, we
propose a cascaded cooling mechanism employing a dynamical gain
$G_{\mathrm{DAC}}$, as discussed below.

###  IV.3 Cascaded feedback cooling

The critical factor limiting the single stage cooling approach is the readout
conversion coefficient $G_{\mathrm{DAC}}$. We can use the fact that the
maximum level of this gain, $G_{\mathrm{DAC}}^{\mathrm{max}}$, can go higher
as the test mass is cooled, requiring less optical power. This means that we
can _dynamically_ increase $G_{\mathrm{DAC}}$ as the oscillation amplitude of
the test mass goes down. In fact, we can also dynamically switch from the low
sensitivity HLI readout scheme to the high sensitivity FPI readout scheme when
the test mass oscillation is suppressed to $\sqrt{\langle
x^{2}\rangle}<\lambda/\mathcal{F}$, which also reduces backaction due to
readout noise. The implementation of this cascaded feedback cooling approach
from the slow relaxation time of our $\gamma_{\mathrm{m}}\sim 2\pi\cdot
10\,\mu\mathrm{Hz}$ test mass, is given by [42],

$\langle x^{2}(t)\rangle=\frac{\langle
x^{2}(0)\rangle}{1+g}\left(1+g\exp^{-(1+g)\gamma_{\mathrm{m}}t}\right).$ (23)

In Figure 9 we present simulations of single-step and cascaded feedback
cooling of our test mass for various gain settings and optical powers, using
the HLI detector and system parameters described above. Relative to single-
step cooling with $P_{0}\approx 39,\mathrm{W}$, cascaded cooling enables
optimal gain ($g_{\mathrm{opt}}=3.4\cdot 10^{4}$), requiring $10^{4}$-fold
lower optical power ($P_{0}\approx 1\,\mathrm{mW}$), at the expense of a
$10^{4}$-fold higher cooling time (10 days, versus 1 minute). As shown in
Figure 10, the total cooling time for the cascaded approach depends on the
initial gain factor $g_{0}$, which is proportional to optical power, and
scales approximately inversely with $g_{0}$. In practice, however, we can cool
the test mass down to within the FPI capture range at much lower gains than
$g_{\mathrm{opt}}$, which allows us to switch to a FPI readout much faster. In
this case, the required time in our cascaded cooling approach, can be reduced
to a few hours using an optical power feedback of only 11.59 mW, as shown in
Figure 10.

Figure 10: Effective temperature in a cascaded cooling mechanism for different
initial gain factors $g_{0}$ = 1, 5, and 10. A large $g_{0}$ allows the system
to operate much faster and with less iterations to achieve a certain final
state.

##  V Complete system overview

The complete system we propose is shown in Figure 11, including the dual
measurement scheme and radiation pressure feedback circuit. In this figure we
show three lasers for simplicity, however, we note that all these three
sources can be obtained from a single laser unit, following a proper optical
setup and frequency management. The three optical sources are: a) the
frequency-stabilized laser that serves as the optical source for the
heterodyne interferometer to monitor the full-range motion of the test mass,
as well as the frequency standard to obtain the frequency drift of the probe
laser, b) the probe laser that is the optical source for the FPI as the main
optical readout system to perform highly sensitive displacement measurements,
and c) the feedback laser that is interfaced with an EOAM intensity modulator
for feedback cooling.

Figure 11: Overall layout of the optomechanical inertial sensor design,
including details of the mechanical resonator, the optical readout systems,
and the feedback control system. Three individual lasers are used in this
figure due to their different purposes in the system.
Note: In a practical implementation, all three lasers can be replaced with a
single laser source, following a proper optical setup and frequency
management.

The feedback circuit is shown at the top-right of Figure 11. For simplicity,
we omit the fact that the HLI output passes through a near-resonant bandpass
filter and a phase shifter in this Figure, to avoid crosstalk from higher-
order modes as well as to provide a $90\degree$ phase shift for the feedback
force, respectively. The optical components of both the heterodyne
interferometer and the FPI can be mounted on the resonator frame, leading to a
compactly integrated and vacuum-compatible inertial sensor. This design also
reduces the effect of mechanical vibrations and thermal expansion, which are
significant noise sources at low frequencies.

In closed loop, the detected laser frequency fluctuations in Equation 5b can
be rewritten as

$\Delta\nu=\frac{c}{\overline{\lambda}\overline{L}}\frac{a_{\text{ext}}(\omega)+a_{\text{th}}(\omega)+a_{\text{n}}(\omega)}{\omega_{0}^{2}-\omega^{2}+i(1+g)\omega\omega_{0}/Q},$
(24)

where $g$ in the cascaded cooling system can be traced to the final
$G_{\mathrm{DAC}}$ via Equation 21. All the parameters in Equation 24, such as
$\omega_{0}$, $Q$, $\overline{\lambda}$, and $\overline{L}$ can be measured
directly and independently from the displacement sensitivity measurement.

##  VI Conclusions and outlook

We have proposed an optomechanical inertial sensor that features high
displacement sensitivity, to enable thermal-noise-limited performance, and the
ability to perform large-amplitude acceleration measurements in non-isolated
environments via radiation pressure feedback cooling. The design principles
are described in detail for each sub-system, followed by a discussion of a
practical implementation. A novel cascaded cooling approach is presented,
which reduces the laser power requirements and improves the cooling
efficiency.

Cascaded feedback cooling of our relatively large, low-frequency test mass
allows the use of low optical power, at the milliwatt level. However, this
feature comes at the expense of prolonging the time needed for the optical
cooling process, when compared to a single-step method that utilizes high
laser power (tens of Watts). Nonetheless, this extended time in reaching the
minimal operational resonator temperature may likely be acceptable for
applications where the sensor is deployed permanently, and the cascaded
cooling is part of the initial calibration process. Seismic monitoring of a
particular site or space missions are examples of such applications, since
they typically require commissioning of their instrumentation over comparable
time scales. Our design thus introduces a promising optomechanical inertial
sensing technology that is capable of reaching high acceleration sensitivity
at low frequencies (below
$1\text{\times}{10}^{-9}\text{\,}\mathrm{m}\,\mathrm{s}^{-2}\mathrm{/}\sqrt{Hz}$
above 1 mHz [12]). This system is currently under development in our
laboratory.

### Funding

The authors acknowledge financial support from the National Science Foundation
(NSF) through grants PHY-2045579 and ECCS-1945832, and the National
Aeronautics and Space Administration (NASA) through grant 80NSSC20K1723.

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