# Laser cooling of traveling wave phonons in an optical fiber

Joel N. Johnson  Department of Applied Physics and Materials Science, Northern
Arizona University, Flagstaff, AZ 86011, USA  Center for Materials Interfaces
in Research and Applications, Flagstaff, AZ, USA  Danielle R. Haverkamp
Department of Applied Physics and Materials Science, Northern Arizona
University, Flagstaff, AZ 86011, USA  Center for Materials Interfaces in
Research and Applications, Flagstaff, AZ, USA  Yi-Hsin Ou  College of Optical
Sciences, University of Arizona, Tucson, AZ, USA  Khanh Kieu  College of
Optical Sciences, University of Arizona, Tucson, AZ, USA  Nils T. Otterstrom
Photonic and Phononic Microsystems, Sandia National Laboratories, Albuquerque,
New Mexico, USA  Peter T. Rakich  Department of Applied Physics, Yale
University, New Haven, CT, USA  Ryan O. Behunin<EMAIL_ADDRESS>Department of Applied Physics and Materials Science, Northern Arizona
University, Flagstaff, AZ 86011, USA  Center for Materials Interfaces in
Research and Applications, Flagstaff, AZ, USA

###### Abstract

In recent years, optical control of mechanical oscillators has emerged as a
critical tool for everything from information processing to laser cooling.
While traditional forms of optomechanical cooling utilize systems comprised of
discrete optical and mechanical modes, it has recently been shown that cooling
can be achieved in a chip-based system that possesses a continuum of modes.
Through Brillouin-mediated phonon-photon interactions, cooling of a band of
traveling acoustic waves can occur when anti-Stokes scattered photons exit the
system more rapidly than the relaxation rate of the mechanical waves—to a
degree determined by the acousto-optic coupling. Here, we demonstrate that a
continuum of traveling wave phonons can be cooled within an optical fiber,
extending this physics to macroscopic length scales. Leveraging the large
acousto-optic coupling permitted within a liquid-core optical fiber,
heterodyne spectroscopy reveals power-dependent changes in spontaneous
Brillouin scattering spectra that indicate a reduction of the thermal phonon
population by 21K using 120 mW of injected laser power.

##  I Introduction

Laser cooling brought about a revolution in atomic, molecular and optical
(AMO) physics, permitting exquisite control of the motion of ions, atoms and
molecules, precision measurements of time, and the creation of new states of
matter Hänsch and Schawlow (1975); Ashkin (1978); Wineland  _et al._ (1978);
Phillips and Metcalf (1982); Anderson  _et al._ (1995); Davis  _et al._
(1995); Ludlow  _et al._ (2015); Epstein  _et al._ (1995); Sheik-Bahae and
Epstein (2007). Beyond these AMO applications, laser cooling has led to
impressive developments in solid-state systems where optical refrigeration in
rare earth doped glasses is rapidly approaching cryogenic operation, and
sideband cooling of individual mesoscopic mechanical oscillators has opened
new windows to the foundations of physics and enabled the generation of novel
quantum states Chan  _et al._ (2011); Marshall  _et al._ (2003); Hong  _et
al._ (2017); Aspelmeyer  _et al._ (2014).

Laser cooling of mechanical oscillators conventionally utilize an optical
cavity with a movable mirror where radiation pressure mediates a parametric
coupling between cavity resonance and a discrete phononic mode Aspelmeyer  _et
al._ (2014). This configuration is critical to optical sideband cooling, where
red-detuned drive laser photons can blue-shift by phonon scattering, rapidly
exit the system, and lower the effective phonon occupancy Aspelmeyer  _et al._
(2014). While optomechanical cooling of traveling wave phonons has been
achieved in whispering gallery mode resonators that support discrete optical
and mechanical modes Bahl  _et al._ (2012), surprisingly, this form of anti-
Stokes cooling can occur in continuous systems—without a optical or mechanical
resonances Otterstrom  _et al._ (2018). Otterstrom et al. showed that cooling
of a continuous band of phonon modes can be achieved when injected laser light
scatters through an anti-Stokes process from traveling wave phonons and exits
the system more rapidly than the phonon bath returns to thermal equilibrium.
This form of cooling simultaneously requires substantial light-sound
couplings, to enable efficient scattering, and phonon lifetimes exceeding the
transit time for light through the waveguide. Owing to these stringent
requirements, optomechanical cooling in continuous systems has only been
observed in silicon waveguides engineered to have large Brillouin coupling and
short $\sim$ cm lengths. While the realization of this physics within optical
fibers may be attractive for low-noise signal processing, high-coherence
lasers, high-fidelity optical squeezing, and variable bandwidth forms of all-
optical slow-light generation, demonstration of cooling of travelling wave
phonons in fibers has remained elusive Shin  _et al._ (2015); Shelby  _et al._
(1986); Okawachi  _et al._ (2005).

Here, we demonstrate cooling of a continuum of phonon modes in optical fiber
for the first time. Through spontaneous Brillouin scattering in a 1m long
CS2-filled liquid-core optical fiber, we achieve cooling (i.e., change in
temperature) of $\sim$21K from room temperature for a band of anti-Stokes
phonons. These dynamics are made possible by the unique features of this fiber
system (Fig. 1)Kieu  _et al._ (2013, 2014); Behunin  _et al._ (2019). The
large refractive index and small sound speed of CS2 enable simultaneous
guidance and tight co-confinement of light and sound (Fig. 1c & d) within
small diameter silica capillaries. This tight acousto-optic overlap, combined
with the large electrostrictive coupling made possible within CS2 Boyd (2020)
permit large light-sound interactions which can be leveraged for
optomechanical cooling in a continuous system.

Figure 1: Illustration of liquid core optical fiber (a) geometry and
scattering processes, (b) cooling signatures in spontaneous light scattering
spectra, and simulated (c) optical and (d) mechanical modes.

Using heterodyne spectroscopy, we show how spontaneously scattered power
spectra evolve with pump power. These spectra reveal the signatures of
cooling, showing changes in the phonon dissipation rate and nonlinear
dependence on the pump power. To confirm depletion of thermal phonons, we
perform a form of pump-probe spontaneous Brillouin scattering where changes in
the phonon population can be directly probed. This pump-probe technique
leverages a unique property of Brillouin scattering, permitting two orthogonal
polarizations of light to couple to the same band of phonons. Using this
feature an intense pump can be used to cool the phonon modes while a weak
probe laser in an orthogonal polarization, that can be isolated from the pump,
measures the phonon population. With fixed probe powers, these results show
that the power of anti-Stokes scattered probe light decreases as the pump
power is increased, indicating that the thermal population of the anti-Stokes
phonons has been reduced in a manner consistent with theory. With the long
lengths and power-handling capabilities of optical fibers, combined with
recent theoretical developments, these results may enable new low-noise fiber
applications, ground state cooling of bands of traveling phonon modes, and new
forms of quantum state synthesis Zhu and Stiller (2022); Behunin and Rakich
(2023).

Figure 2: (a) Heterodyne spectroscopy apparatus for measuring power dependence
of spontaneous Brillouin spectra shown in (b). Lorentzian fits to spectra in
(b) reveal signatures of Brillouin cooling through (c) power-dependent
linewidths and (d) sub- and super-linear growth of the power spectra peaks.
The lines in (c) are obtained through a constrained fit using Eq. 2 providing
an estimated Brillouin gain of 2.3 (Wm)-1. Theory in (d) obtained using the
fitted value of $G_{B}$, Eq. (3) and the measured spectra heights at 4.1 mW
(see Appendix A).(e) Comparison between measured and simulated light
scattering spectra (see Appendix C). Here, EDFA stands for Erbium doped fiber
amplifier, BPF is a tunable band-pass filter, and VOA is variable optical
attenuator.

##  II Laser cooling of traveling wave phonons

Laser cooling of traveling wave phonons occurs through anti-Stokes Brillouin
scattering. During this process an incident pump photon annihilates a counter-
propagating phonon, blue-shifts to higher energy, and rapidly exits the system
(Fig. 1a). In order for cooling to occur, the mechanical degrees of freedom
must return to thermal equilibrium more slowly than the anti-Stokes photons
exit the system. A mean-field analysis of these dynamics, described in
Appendix A, shows that these conditions require $4v_{g}/L>\Gamma_{0}$, where
$v_{g}$ is the optical group velocity, $L$ is the system length and
$\Gamma_{0}$ is the mechanical dissipation rate Otterstrom  _et al._ (2018),
conditions met by the system considered here (Fig. 1(a)) with $4v_{g}/L\approx
0.82$ GHz and $\Gamma_{0}\approx 0.61$ GHz. While these disparate optical and
mechanical timescales allow the phonon population to be driven out of thermal
equilibrium, the degree of cooling requires a relatively large single pass
Brillouin gain $G\equiv G_{B}P_{p}L$ where $G_{B}$ is the Brillouin gain
coefficient [m-1W-1] and $P_{p}$ is the pump power Boyd (2020), yielding a
fractional temperature $\Delta T/T_{0}$ change given approximately by

$\frac{\Delta T}{T_{0}}\approx\frac{G}{G+4}$ (1)

where $T_{0}$ is the equilibrium temperature Otterstrom  _et al._ (2018).
Therefore, we must balance competing requirements, maximizing the single-pass
gain while minimizing the light travel time across the fiber to achieve
efficient cooling. Although the mean-field optical decay rate is marginally
greater than the phonon decay rate, analysis of the coupled envelope dynamics
shows that the key conclusions of the mean-field analysis remain valid in the
limit of small single-pass gains explored here (see Appendix A.1).

Cooling of traveling wave phonons can be identified by power-dependent changes
in the width and height of spontaneous light scattering spectra, revealing the
phonon band temperature and dissipation rate (Fig. 1b). While for low power
($G\ll 1$) the spectrum height increases linearly with pump power, these
increases begin to saturate (grow more rapidly) for the anti-Stokes (Stokes)
spectrum as the single pass gain approaches one and the phonon population is
depleted (increased). Consequently, the anti-Stokes (Stokes) spectrum height
grows sub-linearly (super-linearly), depending on both the pump power and the
phonon population (Figs. 1b & 2d). Owing to additional decay (amplification)
channels opened by spontaneous anti-Stokes (Stokes) Brillouin scattering, the
phonon dissipation rate increases (decreases) with pump power according to

$\Gamma_{\pm}\approx\Gamma_{0}\left(1\pm\frac{1}{4}G\right)$ (2)

where the upper (+) sign indicates an increase in the anti-Stokes phonon decay
rate and the lower (-) sign quantifies how the Stokes spectrum narrows
Otterstrom  _et al._ (2018). As a consequence of these altered decay rates,
detailed balance for thermal equilibrium is broken, reducing (or increasing)
the effective population of the anti-Stokes (Stokes) phonons
$n^{+}_{eff}$($n^{-}_{eff}$) described by

$n^{\pm}_{eff}\approx\frac{\Gamma_{0}}{\Gamma_{\pm}}n_{th},$ (3)

where $n_{th}\approx 2700$, given by the Bose distribution, is the occupation
number for the relevant phonons in our liquid-core optical fiber (LCOF) in
thermal equilibrium Otterstrom  _et al._ (2018). See Appendix A for more
details of the mean-field analysis of the nonlinear optics in this system.

To demonstrate this physics, we utilize a 1m long CS2-filled silica capillary
(Fig. 1), possessing an array of properties that are ideal for laser cooling
of travelling wave phonons. The materials properties and geometry of this
liquid-filled capillary permit guided optical (single-mode at 1.55 $\mu$m)
(Fig. 1c) and acoustic waves (Fig. 1d) within the fiber core and enable large
Brillouin gain. Furthermore, macroscopic meter-scale lengths make large
single-pass gain accessible at modest sub-Watt powers. See Refs. Kieu  _et
al._ (2013, 2014); Behunin  _et al._ (2019) for details on Brillouin
scattering in LCOFs.

##  III Methods & Results

We measure Brillouin cooling in this LCOF using the apparatus shown in Fig.
2a. The output of a laser is split in two, a variable power arm is injected
into the sample for Brillouin cooling, and a fixed-power arm is used to
synthesize a local oscillator (LO) for heterodyne detection. Spontaneously
backscattered light is filtered, to remove stray pump light and isolate the
anti-Stokes (or Stokes) sideband, photomixed with the LO on a high-speed
receiver, and detected on a radio-frequency spectrum analyzer (RFSA) as a
function of pump power. These measurements yield spontaneous Brillouin
scattering spectra of the type shown in Fig. 2b.

Figure 3: Pump-probe measurements of Brillouin cooling. (a) Pump-probe
spectrometer. Orthogonally polarized pump and fixed-power probe light are
combined and sent into the sample using a polarizing beamsplitter (PBS).
Backscattered pump and probe light exit distinct ports of the PBS, and anti-
Stokes sideband is isolated using a circulator and tunable bandpass filter
(BPF). Orthonally polarized pump and probe beams (b) and (c) couple to the
same band of phonons. The scattered probe can be isolated from the scattered
pump using a PBS. (d) Backscattered anti-Stokes power spectra for the probe
plotted for various pump powers. For clarity the data is averaged over 1.67
MHz bins. (e) Comparison of measured anti-Stokes spectra and the power
spectrum predicted by mean-field theory (see Appendix B).

The data displayed in Fig. 2 exhibit key signatures of Brillouin cooling. With
increasing pump power, the linewidth of the anti-Stokes spectrum increases and
the scattered anti-Stokes power increases sub-linearly (Fig. 2c & d). The
extracted linewidths of the Stokes and anti-Stokes spectra exhibit the power
dependence described by Eq. (2), yielding a Brillouin gain coefficient
$G_{B}\sim 2.3$(Wm)-1, qualitatively described by finite-element simulations
shown in Fig. 2e predicting $G_{B}\approx 2.9$ (Wm)-1. Using Eq. (1), these
measurements show that the temperature of the band of traveling anti-Stokes
phonons has been reduced by 21K from room temperature for 120mW of pump power.

The apparatus and data shown in Fig. 2 directly demonstrate the signatures of
laser cooling of a continuous band of phonons. However, in these measurements,
the amplitude of the power spectrum is proportional to the product of phonon
population and the pump power. To directly detect changes to the phonon
population, we utilize a form of pump-probe spectroscopy shown in Fig. 3. In
contrast with the heterodyne measurements depicted in Fig. 2, the output of a
laser is divided into a variable power pump, and a fixed power probe.
Orthogonally polarized pump and probe beams are combined using a polarizing
beam splitter (PBS) and sent into the LCOF where they interact with the same
phonon mode (Fig. 3 b & c). Owing to this polarization multiplexing,
spontaneously back-scattered probe light is separated from the backscattered
pump by the PBS, filtered, and photomixed with the LO for heterodyne
detection, yielding the spectra shown in Fig. 3d. Consistent with the
depopulation of phonons, these results show a decrease in the anti-Stokes
scattering rate with increasing in pump power. Furthermore, Fig. 3c shows that
these spectra are well-described by a simple model of the pump-probe dynamics
with parameters obtained from independent measurements shown in Fig. 2. See
Appendix B for further details of the pump-probe theory.

##  IV Conclusions

We have demonstrated laser cooling of travelling wave phonons in an optical
fiber for the first time. Using spontaneous Brillouin light scattering
spectroscopy of a CS2-filled LCOF, we probe the populations of travelling wave
phonons, revealing power-dependent changes in the phonon dynamics consistent
with a mean-field model and finite-element simulations. Key to accessing this
regime of light matter interactions is the large acousto-optic coupling and
short length of our sample, where pump photons readily scatter from counter-
propagating phonons and rapidly exit the fiber before the phonons can return
to thermal equilibrium. The combination of these properties enables 21K of
phonon cooling for 120mW of pump power.

Building on these results, drastic enhancements in our ability to cool
traveling wave phonons may be achievable with changes in the fiber geometry.
With larger core diameters, the Brillouin gain can be dramatically increased
through improved acousto-optic overlap. Working toward the limit of single-
mode operation (for 1.55 $\mu$m optical wavelengths), finite-element
simulations predict Brillouin gains of $\sim$ 11 (Wm)-1 for a LCOF with a 1.6
$\mu$m core diameter, exceeding the acousto-optic coupling in our LCOF by
nearly a factor of five. Under the same experimental conditions as this study,
such a fiber could achieve 119 K of cooling for 250 mW of pump power,
approaching cryogenic temperatures. Because thermal noise produced by
traveling wave phonons is a critical limitation in a variety of fiber-based
information processing, lasers, and quantum optics experiments, these results
may inspire new ways to achieve improved performance.

##  V Acknowledgements

J. J. and R. B. acknowledge funding from NSF Award No. 2145724. This article
has been authored by an employee of National Technology & Engineering
Solutions of Sandia, LLC under Contract No. DE-NA0003525 with the U.S.
Department of Energy (DOE). The employee owns all right, title and interest in
and to the article and is solely responsible for its contents. The United
States Government retains and the publisher, by accepting the article for
publication, acknowledges that the United States Government retains a non-
exclusive, paid-up, irrevocable, world-wide license to publish or reproduce
the published form of this article or allow others to do so, for United States
Government purposes. The DOE will provide public access to these results of
federally sponsored research in accordance with the DOE Public Access Plan
https://www.energy.gov/downloads/doe-public-access-plan.

##  Appendix A Mean-field analysis of Brillouin cooling

The key features of Brillouin cooling can be understood from a mean-field
analysis of the slowly-varying envelope equations. For an undepleted pump and
ignoring the propagation of phonons, spontaneous Brillouin scattering can be
described by the following coupled stochastic envelope equations

$\displaystyle\dot{A}_{S}+v_{g}\partial_{z}A_{S}=-igA_{p}B_{S}^{\dagger}$ (4)
$\displaystyle\dot{B}_{S}+\frac{\Gamma_{0}}{2}B_{S}=-igA_{p}A_{S}^{\dagger}+\xi_{S}$
(5) $\displaystyle\dot{A}_{aS}+v_{g}\partial_{z}A_{aS}=-igA_{p}B_{aS}$ (6)
$\displaystyle\dot{B}_{aS}+\frac{\Gamma_{0}}{2}B_{aS}=-igA_{aS}A_{p}^{\dagger}+\xi_{aS}$
(7)

expressed in a frame rotating at the resonance frequency for each field Kharel
_et al._ (2016). Here, $A_{S}$ ($A_{aS}$) and $B_{S}$ ($B_{aS}$) are the
respective (right-moving) optical and mechanical envelopes for the Stokes
(anti-Stokes) process, $A_{p}$ is the envelope for a constant undepleted pump,
$g$ is the Brillouin coupling, and $v_{g}$ is the optical group velocity.
Thermal fluctuations of the mechanical field are modeled using the locally
correlated white noise Langevin forces $\xi_{j}$ ($j=S$ or $aS$) with
correlation properties

$\displaystyle\langle\xi_{j}(t,z)\xi_{j^{\prime}}^{\dagger}(t^{\prime},z^{\prime})\rangle=\Gamma_{0}(n_{th}+1)\delta_{jj^{\prime}}\delta(t-t^{\prime})\delta(z-z^{\prime})\quad\quad$
(8)
$\displaystyle\langle\xi_{j}^{\dagger}(t,z)\xi_{j^{\prime}}(t^{\prime},z^{\prime})\rangle=\Gamma_{0}n_{th}\delta_{jj^{\prime}}\delta(t-t^{\prime})\delta(z-z^{\prime})$
(9)

and $\Gamma_{0}$ is the mechanical dissipation rate. Put in terms of readily
measurable quantities, the power in an optical mode is given by
$P_{S/aS/p}=\hbar\omega_{S/aS/p}v_{g}A^{\dagger}_{S/aS/p}A_{S/aS/p}$ and the
Brillouin gain can be expressed as

$G_{B}=\frac{4|g|^{2}}{\hbar\omega_{p}v_{g}^{2}\Gamma_{0}}.$ (10)

We define the mean fields for the respective optical and mechanical fields
$a_{j}$ and $b_{j}$ as the average of the envelope over the length of the
waveguide

$\displaystyle a_{j}=\frac{1}{L}\int_{0}^{L}dz\ A_{j}(z)$ (11) $\displaystyle
b_{j}=\frac{1}{L}\int_{0}^{L}dz\ B_{j}(z).$ (12)

The mean field equations can be obtained by averaging Eqs. (4)-(7) over the
waveguide length, giving

$\displaystyle\dot{a}_{S}+\frac{\gamma}{2}a_{S}=-igA_{p}b_{S}^{\dagger}$ (13)
$\displaystyle\dot{b}_{S}+\frac{\Gamma_{0}}{2}b_{S}=-igA_{p}a_{S}^{\dagger}+\bar{\xi}_{S}$
(14) $\displaystyle\dot{a}_{aS}+\frac{\gamma}{2}a_{aS}=-igA_{p}b_{aS}$ (15)
$\displaystyle\dot{b}_{aS}+\frac{\Gamma_{0}}{2}b_{aS}=-iga_{aS}A_{p}^{\dagger}+\bar{\xi}_{aS}$
(16)

where $\bar{\xi}_{j}$ is the spatial average of the Langevin force. Here, the
mean-field optical decay rate $\gamma=4v_{g}/L$ is obtained from the spatial
average of the derivative terms, assuming $a_{j}\approx(A_{j}(L)+A_{j}(0))/2$
and using that the scattered optical fields vanish at the input face, i.e.,
$A_{j}(0)=0$, according to

$\displaystyle\frac{v_{g}}{L}\int_{0}^{L}dz\ \partial_{z}A_{k}(z)=$
$\displaystyle\frac{v_{g}}{L}(A_{k}(L)-A_{k}(0))$ $\displaystyle=$
$\displaystyle\frac{v_{g}}{L}(A_{k}(L)+A_{k}(0)-2A_{k}(0))$ $\displaystyle=$
$\displaystyle\frac{2v_{g}}{L}a_{k}.$ (17)

Next we analyze the mean field dynamics. When $\gamma>\Gamma_{0}$, the optical
fields respond to changes in the phonon amplitude faster than the phonon
changes, enabling a quasistatic solution to the optical field dynamics given
by

$\displaystyle\frac{\gamma}{2}a_{S}\approx-igA_{p}b_{S}^{\dagger}$ (18)
$\displaystyle\frac{\gamma}{2}a_{aS}\approx-igA_{p}b_{aS}.$ (19)

Inserting Eq. (18) & (19) into Eqs. (14) & (16) we find the effective phonon
dynamics given by

$\displaystyle\dot{b}_{S}+\frac{1}{2}\Gamma_{S}b_{S}=\bar{\xi}_{S}$ (20)
$\displaystyle\dot{b}_{aS}+\frac{1}{2}\Gamma_{aS}b_{aS}=\bar{\xi}_{aS}$ (21)

where $P_{p}=\hbar\omega_{p}v_{g}A^{\dagger}_{p}A_{p}$, Eq. (10) and Eq. (2)
of the main text have been used. In addition, to changing the effective phonon
dynamics, these results show that the power spectra for the optical fields are
proportional to the phonon power spectrum,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{S}[\omega]=\frac{4|g|^{2}|A_{p}|^{2}}{\gamma^{2}}\\!\int_{-\infty}^{\infty}d\tau\
e^{i\omega\tau}\langle b_{S}(t+\tau)b_{S}^{\dagger}(t)\rangle$ (22)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{aS}[\omega]=\frac{4|g|^{2}|A_{p}|^{2}}{\gamma^{2}}\\!\int_{-\infty}^{\infty}d\tau\
e^{i\omega\tau}\langle
b^{\dagger}_{aS}(t\\!+\\!\tau)b_{aS}(t)\rangle\quad\quad$ (23)

illustrating how the optical power spectra, defined by
$S_{j}[\omega]=\int_{-\infty}^{\infty}d\tau\ \exp\\{i\omega\tau\\}\langle
a^{\dagger}_{j}(t+\tau)a_{j}(t)\rangle$, permit a form of nonequilibrium
phonon spectroscopy. Solving Eqs. (20) and using the correlation properties
for the spatially averaged Langevin forces given by

$\displaystyle\langle\bar{\xi}_{j}(t)\bar{\xi}_{j^{\prime}}^{\dagger}(t^{\prime})\rangle=\frac{1}{L}\Gamma_{0}(n_{th}+1)\delta_{jj^{\prime}}\delta(t-t^{\prime})$
(24)
$\displaystyle\langle\bar{\xi}_{j}^{\dagger}(t)\bar{\xi}_{j^{\prime}}(t^{\prime})\rangle=\frac{1}{L}\Gamma_{0}n_{th}\delta_{jj^{\prime}}\delta(t-t^{\prime}),$
(25)

we find

$\displaystyle
S_{S}[\omega]=\frac{\Gamma_{0}G}{4v_{g}}\frac{\Gamma_{0}n_{th}}{\Gamma_{S}}\frac{\Gamma_{S}}{\omega^{2}+\Gamma_{S}^{2}/4}$
(26) $\displaystyle
S_{aS}[\omega]=\frac{\Gamma_{0}G}{4v_{g}}\frac{\Gamma_{0}n_{th}}{\Gamma_{aS}}\frac{\Gamma_{aS}}{\omega^{2}+\Gamma_{aS}^{2}/4}.$
(27)

The peak of the power spectrum (i.e., on resonance, or $\omega=0$), used for
the theory in Fig. 2e, is given by

$\displaystyle
S^{peak}_{S}[P_{p}]=\frac{n_{th}}{v_{g}}\frac{GBP_{p}L}{(1-\frac{1}{4}G_{B}P_{p}L)^{2}}$
(28) $\displaystyle
S^{peak}_{aS}[P_{p}]=\frac{n_{th}}{v_{g}}\frac{G_{B}P_{p}L}{(1+\frac{1}{4}G_{B}P_{p}L)^{2}}.$
(29)

To account for the radio-frequency to electrical conversion, the peak of the
power spectrum at the lowest power ($P_{0}=4.1$ mW) is used to scale the
theoretical curves to have units of $\mu V$

$\displaystyle
S^{peak,RF}_{S}[P_{p}]=\frac{S^{peak,RF}_{S}[P_{0}]}{G_{B}P_{0}L}\frac{G_{B}P_{p}L}{(1-\frac{1}{4}G_{B}P_{p}L)^{2}}\quad\quad$
(30) $\displaystyle
S^{peak,RF}_{aS}[P_{p}]=\frac{S^{peak,RF}_{aS}[P_{0}]}{G_{B}P_{0}L}\frac{G_{B}P_{p}L}{(1+\frac{1}{4}G_{B}P_{p}L)^{2}}.\quad\quad\quad$
(31)

###  A.1 Envelope analysis of cooling dynamics, and validity of the mean-field
model

Here, we show that in the limit of small single-pass gain, the conclusions of
the mean-field model agree with the full envelope dynamics.

In the Fourier domain, Eq. (7) can be readily solved and inserted in Eq. (6),
giving

$\displaystyle(\partial_{z}-i\Lambda)A_{aS}(\omega,z)=-\frac{1}{v_{g}}igA_{p}\hat{B}_{aS}(\omega,z)\quad$
(32)

where

$\displaystyle\Lambda=\frac{1}{v_{g}}\bigg{[}\omega+i\frac{|g|^{2}|A_{p}|^{2}}{-i\omega+\Gamma_{0}/2}\bigg{]},$
(33)

and

$\displaystyle\hat{B}(\omega,z)=\frac{1}{-i\omega+\Gamma_{0}/2}\xi(\omega,z).$
(34)

Using Eqs. (9) and (34), the solution to Eq. (32) at the exit face of the
fiber ($z=L$),

$\displaystyle A_{aS}(\omega,L)=-i\frac{gA_{p}}{v_{g}}\int_{0}^{L}dz\
e^{i\Lambda(L-z)}\hat{B}(\omega,z),$ (35)

can be directly used to obtain the power spectrum of spontaneously scattered
anti-Stokes light

$\displaystyle S_{aS,env}[\omega]$ $\displaystyle\equiv$
$\displaystyle\frac{\langle
A_{aS}^{\dagger}(\omega,L)A_{aS}(\omega^{\prime},L)\rangle}{2\pi\delta(\omega-\omega^{\prime})}$
(36) $\displaystyle=$
$\displaystyle\frac{n_{th}}{v_{g}}\bigg{[}1-e^{-G(\omega)}\bigg{]}$

where

$\displaystyle G(\omega)=\frac{\Gamma^{2}/4}{\omega^{2}+\Gamma^{2}/4}G.$ (37)

Noting that the power spectrum obtained directly from (6) is directly related
to the effective phonon power spectrum $S_{B}[\omega]$

$\displaystyle S_{aS,env}[\omega]\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\left|\frac{gA_{p}}{v_{g}}\right|^{2}\\!\\!\int_{0}^{L}\\!\\!\\!\\!\\!dz\\!\\!\int_{0}^{L}\\!\\!\\!\\!\\!dz^{\prime}e^{i\frac{\omega}{v_{g}}(z\\!-\\!z^{\prime})}\frac{\langle{B}^{\dagger}(\omega,z){B}(\omega^{\prime},z^{\prime})\rangle}{2\pi\delta(\omega-\omega^{\prime})}$
(38) $\displaystyle=$
$\displaystyle\\!\\!\\!\left|\frac{gA_{p}}{v_{g}}\right|^{2}S_{B}[\omega],$

(note the power spectrum of $B$ as opposed to $\hat{B}$) we identify the anti-
Stokes phonon power spectrum including the effects of optomechanical cooling

$\displaystyle S_{B}[\omega]=\frac{4n_{th}L}{G\Gamma_{0}}(1-e^{-G(\omega)}).$
(39)

Using $S_{B}[\omega]$ we can calculate the effective thermal occupation of the
anti-Stokes phonon mode

$\displaystyle n_{eff,env}$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
L}\int_{-\infty}^{\infty}d\omega\ S_{B}[\omega]$ (40) $\displaystyle=$
$\displaystyle n_{th}e^{-G/2}(I_{0}(G/2)+I_{1}(G/2))$ (41)

where the $\omega$-integral can be expressed in terms of modified Bessel
function $I_{0}$ and $I_{1}$. Moreover, the full-width at half maximum
$\Gamma_{aS,env}$ can be calculated from $S_{B}$ using
$S_{B}[\Gamma_{aS,env}/2]=S_{B}[0]/2$ giving effective anti-Stokes phonon
decay rate derived from the envelope dynamics given by

$\displaystyle\Gamma_{aS,env}=\Gamma_{0}\left[\frac{G}{-\ln\left((1+e^{-G})/2\right)}-1\right]^{1/2}.$
(42)

In the limit of small single-pass gain $G$, these results can be compared with
the mean-field model, showing agreement to order $G$ for the effective phonon
decay rate and order $G^{2}$ for the effective thermal occupation

$\displaystyle\\!\\!\\!\\!n_{eff}\approx n_{th}(1-G/4+G^{2}/16-G^{3}/64+...)$
(43) $\displaystyle\\!\\!\\!\\!n_{eff,env}\approx
n_{th}(1-G/4+G^{2}/16-5G^{3}/384+...)$ (44)
$\displaystyle\\!\\!\\!\\!\Gamma_{aS}=\Gamma_{0}(1+G/4)$ (45)
$\displaystyle\\!\\!\\!\\!\Gamma_{aS,env}\approx\Gamma_{0}(1+G/4+G^{2}/32+...).$
(46)

For the maximum single pass gain explored in this study $G<0.3$, the results
of the mean-field and envelope models agree to better than $0.25\%$ in
effective lifewidths and better than $0.007\%$ in effective thermal
occupation. The above analysis is similar for the Stokes process.

##  Appendix B Pump-probe theory

In this section, we develop a mean-field theory to describe the pump-probe
measurements. In these measurements, both pump and probe couple to the same
phonon yielding the coupled envelope equations given by

$\displaystyle\dot{a}_{aS}+\frac{\gamma}{2}a_{aS}=-igA_{p}b_{aS}$ (47)
$\displaystyle\dot{a}_{sig}+\frac{\gamma}{2}a_{sig}=-igA_{pr}b_{aS}$ (48)
$\displaystyle\dot{b}_{aS}+\frac{\Gamma_{0}}{2}b_{aS}=-iga_{aS}A_{p}^{\dagger}-iga_{sig}A_{pr}^{\dagger}+\bar{\xi}_{aS}\quad\quad$
(49)

where $A_{aS}$ is the anti-Stokes light scattered from the pump $A_{p}$ and
$A_{sig}$ is the anti-Stokes light scattered by the probe the undepleted probe
laser $A_{pr}$.

In the quasistatic limit (i.e., $\gamma>\Gamma_{0}$) and assuming the
$|A_{p}|\gg|A_{pr}|$, the effective phonon dynamics is given by Eq. (21) and
the power spectrum of the scattered probe light is given by

$\displaystyle S_{sig}[\omega,P_{p}]$
$\displaystyle\equiv\int_{-\infty}^{\infty}d\tau\ e^{i\omega\tau}\langle
a^{\dagger}_{sig}(t+\tau)a_{sig}(t)\rangle$ (50)
$\displaystyle=\frac{\Gamma_{0}G_{p}LP_{pr}}{4v_{g}}\frac{\Gamma_{0}n_{th}}{\Gamma_{aS}}\frac{\Gamma_{aS}}{\omega^{2}+\Gamma_{aS}^{2}/4}.$

where $P_{pr}$ is the fixed probe power and $\Gamma_{aS}$ is the phonon decay
rate dependent upon the pump power defined in Eq. (2) of the main text. This
result shows that as $\Gamma_{aS}$ increases, i.e., with increased pump power,
the power spectrum broadens and decreases in magnitude. To obtain the
theoretical curve shown in Fig. 2c, we use the peak value of $S_{sig}[0,0]$ to
determine the constant prefactor in Eq. (50) where $\Gamma_{aS}\to\Gamma_{0}$,
including the optical to radiofrequency conversion

$S_{sig}[0,0]=\frac{G_{p}LP_{pr}}{v_{g}}\frac{n_{th}}{\Gamma_{0}^{2}}.$ (51)

Moving out of the rotating frame yields the prediction for the scattered probe
power spectrum is

$\displaystyle
S^{RF}_{sig}[\omega,P_{p}]=\frac{\Gamma_{0}^{2}/4}{(\omega-\omega_{pr}-\Omega)^{2}+\Gamma_{aS}^{2}/4}S^{RF}_{sig}[0,0],$
(52)

and used to plot the theory in Fig. 2e.

##  Appendix C Brillouin gain simulations

We calculate the Brillouin gain spectrum by utilizing finite element
simulations of the optical and acoustic modes of the LCOF. By including
empirically derived material properties (see Tab. 1) and damping for silica
Vacher  _et al._ (1981) and CS2 Coakley and Detenbeck (1975), these
simulations yield the spatial mode profiles for the electric field ${\bf E}$
and the elastic displacement ${\bf u}$ (Fig. 1c & d). We obtain the Brillouin
gain by evaluating the dissipated mechanical power using the equation

$G_{B}=\frac{\omega_{\rm p}}{\Omega}\frac{1}{P_{p}P_{S}}\int_{wg}d^{2}x\
\langle{\bf f}\cdot\dot{\bf u}\rangle$ (53)

where $P_{p}$ and $P_{S}$ are the respective pump and Stokes powers of the
simulated electromagnetic fields, $\Omega$ is the angular frequency of the
mechanical mode, ${\bf f}$ is the electrostrictive force density, $\int_{wg}$
is an integral over the waveguide cross-section, and $\langle{\bf A}\cdot{\bf
B}\rangle$ is the time average of the vector product ${\bf A}\cdot{\bf B}$.
For propagation along the z-axis, the electrostrictive force density ${\bf f}$
is given by

$\displaystyle
f_{j}=\frac{1}{2}\varepsilon_{0}n^{4}p_{ijkl}\partial_{i}(E_{k}E_{l}^{*})\quad{\rm(silica)}$
(54) $\displaystyle{\bf f}=-\frac{1}{4}\varepsilon_{0}\gamma_{e}\nabla|{\bf
E}|^{2}\quad({\rm CS}_{2})$ (55)

where $\varepsilon_{0}$ is the permittivity of vacuum, $n$ is the refractive
index, $p_{ijkl}$ is the photoelastic tensor, $\gamma_{e}$ is the
electrostrictive constant, and the Einstein summation convention is assumed
for repeated indices. In these expressions, we neglect the modal differences
between the pump and the Stokes fields.

Acoustic dissipation critically determines the predicted power spectrum. We
account for dissipation by including empirically obtained acoustic quality
factors for silica and CS2 in our simulations Coakley and Detenbeck (1975);
Vacher  _et al._ (1981). The parameters used in our simulation are summarized
in the table below.

material | parameter | value
---|---|---
CS2 | density | 1260 kg/m3
| refractive index | 1.5885
| speed of sound | 1226 m/s
| electrostrictive constant ($\gamma_{e}$) | 2.297
| acoustic quality factor | 23.5
SiO2 | density | 2203 kg/m3
| refractive index | 1.445
| Young’s modulus | 73.1 GPa
| shear modulus | 31.24 GPa
| phot. elas. tensor (p11,p12,p44) | (0.125,0.27,-0.073)
| acoustic quality factor | 1800
Table 1: Parameters used in simulations of spontaneous Brillouin scattering
spectra.

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