# Supplemental Material: Accurate modeling and characterization of
photothermal forces in optomechanics

André G. Primo1,2,∗ Cauê M. Kersul1,2,∗ Rodrigo Benevides1,2 Natália C.
Carvalho1,2 Michaël Ménard3 Newton C. Frateschi1,2 Pierre-Louis de Assis1
Gustavo S. Wiederhecker1,2 Thiago P. Mayer Alegre1,2,† 1 - Applied Physics
Department, Gleb Wataghin Physics Institute, University of Campinas, Campinas,
SP, Brazil
2 - Photonics Research Center, University of Campinas, Campinas, SP, Brazil
3 - Department of Computer Science, Université du Québec à Montréal, Montréal,
Canada
* These authors contributed equally for this work.
$\dagger$ Corresponding author<EMAIL_ADDRESS>
###### Contents

  1. S1 Derivation of the photothermal coupling
    1. S1.1 Thermodynamic derivation
    2. S1.2 Derivation from the mechanical equation of motion
  2. S2 Thermal mode analysis
  3. S3 Frequency domain description of the photothermal coupling
    1. S3.1 Single-mode Photothermal coupling
    2. S3.2 Multimode Photothermal coupling
    3. S3.3 Nonlinear Multimode Photothermal coupling
  4. S4 Localized Absorption
  5. S5 Experimental Setup
  6. S6 Fitting of the nonlinear optical response
    1. S6.1 Data collection
    2. S6.2 Extracting lumped parameters from the optical transmission spectra
    3. S6.3 Fitting of the polynomial model
  7. S7 Optical and mechanical mode simulations - Fabricated device
    1. S7.1 Optical mode identification
    2. S7.2 Mechanical anisotropy
  8. S8 Optomechanical response
    1. S8.1 Data collection
    2. S8.2 Mechanical linewidth variation
    3. S8.3 Mechanical frequency shift

## Table Summary

Tab.1 summarizes all experimental and simulated parameters used throughout the
main and supplemental text. Each parameter used on our model is also presented
with its value, source (the way they were obtained), and the section where it
is discussed.

Parameter description | Symbol | Value |  Source | Section
---|---|---|---|---
First resonance’s optical frequency | $\omega_{c}/(2\pi)$ | $191.5418\pm 0.0002\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$ |  Experiment | S6.2
Initial extrinsic loss | $\kappa_{e}/(2\pi)$ | $4.197\pm 0.002\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ |  Experiment | S6.2
Linear intrinsic loss | $\kappa_{i}/(2\pi)$ | $7.33\pm 0.01\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ |  Experiment | S6.2
Doublet splitting | $\delta\omega/(2\pi)$ | $11.314\pm 0.007\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ |  Experiment | S6.2
Correction to the linear losses | $\kappa_{\text{nl,0}}/(2\pi)$ | $-0.16\pm 0.02\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ |  Experiment | S6.3
First order nonlinear loss coefficient (TPA) | $\kappa^{\prime}_{1}/(2\pi)$ | $20\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{f}\mathrm{J}$ |  FEM simulationsa | S6.3
Second order nonlinear loss coefficient | $\kappa^{\prime}_{2}/(2\pi)$ | $31\pm 3\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{f}\mathrm{J}^{2}$ |  Experiment | S6.3
Correction to the linear dispersion | $\delta\omega_{nl,0}/(2\pi)$ | $0.93\pm 0.06\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ |  Experiment | S6.3
Thermo-optic shift per heating power (static source)b | $G^{\theta}_{s}R^{\theta}_{s}/(2\pi)$ | $1695\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{m}\mathrm{W}$ |  FEM simulationsa | S6.3
Absorptive nonlinear losses ratio | $\eta_{nl}$ | $0.66\pm 0.01\text{\,}$ |  Experiment | S6.3
Cold cavity mechanical frequency | $\Omega$ | $274.184\pm 0.001\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ |  Experiment and FEM simulationsa | S7.2
Cold cavity mechanical loss | $\Gamma$ | $175.0\pm 0.2\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ |  PSD | S8.2
Optomechanical coupling | $G_{x}\sqrt{\frac{\hbar}{2m_{\text{eff}}\Omega}}$ | $29.2\pm 0.5\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ |  Experiment Gorodetsky  _et al._ (2010), and FEM simulationsa | S7.2
Response time of the different thermal modes | $\tau_{k}$ | - |  FEM simulationsa | S3.1
Thermal resistances | $R_{\text{k,j}}^{\theta}$ | - |  FEM simulationsa | S3.1
Thermo-elastic couplings | $\Lambda_{\text{k}}^{\theta}$ | - |  FEM simulationsa | S3.1
Thermo-optic couplings | $G_{\text{k}}^{\theta}$ | - |  FEM simulationsa | S3.1
a In the FEM simulations the material properties were taken from references,
Krishnamurthy  _et al._ (2011); Adachi (1985); DellaCorte2000Temperaturem;
Balram  _et al._ (2014); Soma1982ThermalInP
b The static thermal response of our system is largely dominated by the
fundamental thermal mode.
Table 1: All parameters used in our model, indicating both its value and how
they were obtained. The value of parameters dependent on the different thermal
modes are not shown here.

##  S1 Derivation of the photothermal coupling

We discuss two different approaches to derive the photothermal coupling; the
first is built from thermodynamic arguments and the second directly from the
mechanical equation of motion. Our analysis assumes that all transport
processes are quasi-static, in the sense that local equilibrium is assumed at
each step. This hypothesis allows us to safely use definitions of quantities
such as temperature and is valid within the mechanical frequencies considered
in this work. Local heat transport is ultimately related to thermal phonons
relaxation times which range around $10^{-11}$\text{\,}\mathrm{s}$$, much
faster than the mechanical oscillations with periods around
$10^{-9}$\text{\,}\mathrm{s}$$.

###  S1.1 Thermodynamic derivation

The total strain tensor ($\bm{S}$) in a thermoelastic body is given by:

$\bm{S}=\bm{S}^{x}+\bm{S}^{\theta}=\hat{\bm{\nabla}}\vec{U},$ (S1)

where $\bm{S}^{\theta}$ and $\bm{S}^{x}$ are the thermal and elastic
components of the strain, respectively. The symbol
$\hat{\bm{\nabla}}=\frac{1}{2}(\nabla+\nabla^{T})$ denotes the symmetric
gradient tensor operation and $\vec{U}$ is the displacement field. The thermal
strain tensor is defined as:

$\bm{S}^{\theta}=\bm{\alpha}\delta T,$ (S2)

where $\bm{\alpha}$ is the rank 2 thermal expansion tensor and $\delta T$ is
the temperature field of the body. In an isotropic medium $\bm{\alpha}$ can be
simplified to, $\alpha\mathcal{I}$, where $\alpha$ is the thermal expansion
coefficient and $\mathcal{I}$ is the $3\times 3$ identity matrix.

Denoting $\bm{c}$ as the stiffness tensor, the stress tensor is obtained
through the following constitutive relation:

$\bm{\sigma}=\bm{c}:\bm{S}^{x}=\bm{c}:\bm{S}-\bm{c}:\bm{S}^{\theta}.$ (S3)

We argue that the additional stresses related to photothermal forces are
necessarily related to the term $-\,\bm{c}:\bm{S}^{\theta}$. Evaluating the
work per unit volume $\delta w$ done during an arbitrary deformation
$\delta\vec{U}$ we get:

$\delta w=\frac{\partial\sigma_{ij}}{\partial X_{j}}\delta U_{i},$ (S4)

where $X_{j}$ ($j=1,2,3$) denotes each of the cartesian coordinates and
summation over repeated indexes is used. Considering the work in the entire
volume where the deformation takes place we have:

$\delta W=\int\delta w\,dV=\int\frac{\partial\sigma_{ij}}{\partial
X_{j}}\delta U_{i}\,dV.$ (S5)

Using Gauss’s theorem and integration by parts:

$\int\frac{\partial\sigma_{ij}}{\partial X_{j}}\delta
U_{i}dV=-\int\sigma_{ij}\delta\frac{\partial U_{i}}{\partial
X_{j}}dV+\int\sigma_{ij}\delta U_{i}dA_{j}.$ (S6)

For a free surface ($\bm{\sigma}\cdot\hat{n}=0$), the second integral on the
RHS vanishes and we get:

$\delta w=-\sigma_{ij}\delta\frac{\partial U_{i}}{\partial
X_{j}}=-\sigma_{ij}\delta S_{ij}.$ (S7)

where we have identified $\frac{\partial U_{i}}{\partial X_{j}}$ with the
total strain.

We are only interested in forces that effectively transfer energy from the
thermal to the mechanical domain over a stationary cycle 111Here, we define
the stationary cycle period as the time necessary for the strain to return to
its initial state., specifically, in the form of work. Substituting Eq. S3 in
Eq. S7 we get:

$\delta w=-c_{ijkl}S_{kl}\delta S_{ij}+c_{ijkl}S^{\theta}_{kl}\delta S_{ij}.$
(S8)

Integrating this expression over a mechanical cycle and noting that the first
term on the right-hand side is an exact differential – and therefore vanishes
–, we conclude that $c_{ijkl}S^{\theta}_{kl}dS_{ij}$ is responsible for
feeding energy into the mechanical domain. Aiming at finding a linear theory,
underpinned by modal analysis, we keep only terms to first order in
temperature. The final expression for the work per unit volume, done by
photothermal forces, is given by:

$\delta w^{\theta}=c_{ijkl}S^{\theta}_{kl}\delta S^{x}_{ij}.$ (S9)

Integrating over the entire volume we obtain the total photothermal work:

$\delta W^{\theta}=\int{(\mathbf{c}\mkern 1.0mu{:}\mathbf{S}^{\theta})\mkern
1.0mu{:}\,\delta\mathbf{S}^{x}dV}.$ (S10)

The elastic strain can be decomposed in the mechanical modes
$\vec{u}_{n}(\vec{r})$ of the system as
$\mathbf{S}^{x}=\sum_{n}x_{n}(t)\bm{S}^{x}_{n}$, where $x_{n}(t)$ is the
$n$-th mode amplitude (i.e. the $\vec{u}_{n}(\vec{r})$ are normalized such
that its global maximum is set to be $1$) and
$\bm{S}^{x}_{n}=\hat{\bm{\nabla}}u_{n}(\vec{r})$ . Using the modal amplitudes
$x_{n}$ as generalized coordinates in the sense of analytical mechanics, the
generalized force on the $n$-th acoustic mode is:

$\frac{\partial W^{\theta}}{\partial
x_{n}}=F^{\theta}_{n}(t)=\int{\frac{\partial\mathbf{S}^{x}}{\partial
x_{n}}\mkern 1.0mu{:}(\mathbf{c}\mkern 1.0mu{:}\mathbf{S}^{\theta})dV.}$ (S11)

Or simplifying:

$F^{\theta}_{n}(t)=\int{\mathbf{S}^{x}_{n}\mkern 1.0mu{:}(\mathbf{c}\mkern
1.0mu{:}\mathbf{S}^{\theta})\,dV},$ (S12)

and explicitly in terms of mechanical modes and the temperature field:

$F^{\theta}_{n}(t)=\int{\hat{\bm{\nabla}}u_{n}(\vec{r})\mkern
1.0mu{:}(\mathbf{c}\mkern 1.0mu{:}\bm{\alpha})\,\delta T(\vec{r},t)dV}.$ (S13)

###  S1.2 Derivation from the mechanical equation of motion

We start with the mechanical equation of motion:

$\rho\ddot{\vec{U}}=\bm{\nabla}\cdot\bm{\sigma},$ (S14)

with the boundary condition:

$\bm{\sigma}\cdot\hat{n}=0.$ (S15)

In view of Eq. S3, these equations can be recast as:

$\rho\ddot{\vec{U}}=\bm{\nabla}\cdot\big{(}\bm{c}\mkern
1.0mu{:}\bm{S}\big{)}-\bm{\nabla}\cdot\big{(}\bm{c}\mkern
1.0mu{:}\bm{S}^{\theta}\big{)},$ (S16)

with the boundary condition:

$\big{(}\bm{c}\mkern 1.0mu{:}\bm{S}\big{)}\cdot\hat{n}=\big{(}\bm{c}\mkern
1.0mu{:}\bm{S}^{\theta}\big{)}\cdot\hat{n}.$ (S17)

The thermal strain in the above equations can be pictured to induce both an
external volume load $\vec{f}_{\theta}=-\bm{\nabla}\cdot\big{(}\bm{c}\mkern
1.0mu{:}\bm{S}^{\theta}\big{)}$, and a surface traction
$\vec{t}_{\theta}=\big{(}\bm{c}\mkern
1.0mu{:}\bm{S}^{\theta}\big{)}\cdot\hat{n}$.

In order to perform the modal decomposition of the photothermal force it is
important to consider the mechanical eigenmodes of our device. These are
solutions of:

$-\rho\Omega_{n}^{2}\vec{u}_{n}=\bm{\nabla}\cdot\big{(}\bm{c}\mkern
1.0mu{:}\bm{S}^{x}_{n}\big{)},$ (S18)

with free (untractioned) boundary conditions:

$\bm{S}^{x}_{n}\cdot\hat{n}=0,$ (S19)

where the strain is given by $\bm{S}^{x}_{n}=\hat{\bm{\nabla}}\vec{u}_{n}$,
and $\Omega_{n}$ is the angular mechanical frequency of the $n$-th mechanical
mode.

Projecting Eq. S16 in a mechanical mode $\vec{u}_{n}$ and performing a volume
integration, we get:

$\int\rho\vec{u}_{n}\cdot\ddot{\vec{U}}\,dV=\int\vec{u}_{n}\cdot\Big{(}\bm{\nabla}\cdot\big{(}\bm{c}\mkern
1.0mu{:}\bm{S}\big{)}\Big{)}\,dV-\int\vec{u}_{n}\cdot\Big{(}\bm{\nabla}\cdot\big{(}\bm{c}\mkern
1.0mu{:}\bm{S}^{\theta}\big{)}\Big{)}\,dV.$ (S20)

Integration by parts of the first term on the right-hand side yields:

$\int\vec{u}_{n}\cdot\Big{(}\bm{\nabla}\cdot\big{(}\bm{c}\mkern
1.0mu{:}\bm{S}\big{)}\Big{)}\,dV=\int\vec{u}_{n}\cdot\Big{(}\bm{c}\mkern
1.0mu{:}\bm{S}\Big{)}\cdot\hat{n}\,dS-\int\vec{U}\cdot\Big{(}\bm{c}\mkern
1.0mu{:}\bm{S}^{x}_{n}\Big{)}\cdot\hat{n}\,dS+\int\vec{U}\cdot\Big{(}\bm{\nabla}\cdot\big{(}\bm{c}\mkern
1.0mu{:}\bm{S}^{x}_{n}\big{)}\Big{)}\,dV.$ (S21)

Using Eqs. S17 and S19 to rewrite the surface integrals and also using Eq. S18
to recast the volume integral on the right-hand side, Eq. S21 reads:

$\int\vec{u}_{n}\cdot\Big{(}\bm{\nabla}\cdot(\bm{c}\mkern
1.0mu{:}\bm{S})\Big{)}\,dV=\int\vec{u}_{n}\cdot\big{(}\bm{c}\mkern
1.0mu{:}\bm{S}^{\theta}\big{)}\cdot\hat{n}\,dS-\Omega_{n}^{2}\int\rho\vec{U}\cdot\vec{u}_{n}\,dV.$
(S22)

Substituting Eq. S22 back in Eq. S20:

$\int\rho\vec{u}_{n}\cdot\ddot{\vec{U}}\,dV=-\Omega_{n}^{2}\int\rho\vec{U}\cdot\vec{u}_{n}\,dV+\int\vec{u}_{n}\cdot\big{(}\bm{c}\mkern
1.0mu{:}\bm{S}^{\theta}\big{)}\cdot\hat{n}\,dS-\int\vec{u}_{n}\cdot\Big{(}\bm{\nabla}\cdot\big{(}\bm{c}\mkern
1.0mu{:}\bm{S}^{\theta}\big{)}\Big{)}\,dV.$ (S23)

Finally, since the first term in the RHS is only defined inside the device
volume, we can then perform a modal expansion of the displacement field as
$\vec{U}=\sum_{n}x_{n}(t)\vec{u}_{n}(\vec{r})$. Using the orthogonality
relations for the mechanical modes,
$\int\rho\vec{u}_{n}\cdot\vec{u}_{m}\,dV=m_{\text{eff},n}\delta_{n,m}$, where
$m_{\text{eff}}$ is effective mass, we get:

$m_{\text{eff,n}}\big{(}\ddot{x}_{n}+\Omega_{n}^{2}\,x_{n}\big{)}=\int\vec{u}_{n}\cdot\big{(}\bm{c}\mkern
1.0mu{:}\bm{S}^{\theta}\big{)}\cdot\hat{n}\,dS-\int\vec{u}_{n}\cdot\Big{(}\bm{\nabla}\cdot\big{(}\bm{c}\mkern
1.0mu{:}\bm{S}^{\theta}\big{)}\Big{)}\,dV,$ (S24)

Usually, a damping term $\Gamma_{n}\dot{x}_{n}$ is heuristically introduced on
the left hand side, but will be omitted for the moment.

The surface and the volume contributions to the photothermal force can be
assembled in a single volume integral:

$m_{\text{eff,n}}\big{(}\ddot{x}_{n}+\Omega_{n}^{2}x_{n}\big{)}=\int\bm{S}^{x}_{n}\mkern
1.0mu{:}\,\big{(}\bm{c}\mkern 1.0mu{:}\bm{S}^{\theta}\big{)}\,dV$ (S25)

Which agrees with the thermodynamic treatment previously discussed. In
previous works the surface load, which is evident in our analysis, was
ignored.

##  S2 Thermal mode analysis

The heat equation for diffusive heat transport reads:

$c_{p}\rho\partial_{t}\delta
T(\vec{r},t)=\nabla\cdot\big{(}k_{\text{th}}\nabla\delta
T(\vec{r},t)\big{)}+\dot{Q}(\vec{r},t),$ (S26)

where $c_{p}$, $\rho$ and $k_{\text{th}}$ are respectively the specific heat,
mass density and thermal conductivity of the medium. $\dot{Q}$ is the heat
generation in the structure, which will be associated with optical absorption.

In the absence of heat sources, this equation is separable and admits
solutions of the form $\delta
T(\vec{r},t)=\delta\tilde{T}(\vec{r})e^{-t/\tau}$ leading to a generalized
eigenvalue equation for the thermal modes:

$\nabla\cdot\big{(}k_{\text{th}}\nabla\delta\tilde{T}(\vec{r})\big{)}=-\frac{1}{\tau}c_{p}\rho\delta\tilde{T}(\vec{r}).$
(S27)

Since the operators $\nabla\cdot\big{(}k_{\text{th}}\nabla)$ and $c_{p}\rho$
are both Hermitian and the latter is positive definite, the equation above is
associated with a complete set of eigenmodes, $\delta
T(\vec{r},t)=\sum_{k}\theta_{k}(t)\delta\tilde{T}_{k}(\vec{r})$. Applying this
expansion to Eq. S26 yields:

$c_{p}\rho\sum_{k}\dot{\theta}_{k}(t)\delta\tilde{T}_{k}(\vec{r})=\sum_{k}[\theta_{k}\nabla\cdot\big{(}k_{\text{th}}\nabla\delta\tilde{T}_{k}(\vec{r})\big{)}]+\dot{Q}(\vec{r},t).$
(S28)

Assuming fixed temperature or insulating boundary conditions, one derives
orthogonality relations for the thermal eigenmodes:

$\int{c_{p}\rho\delta\tilde{T}_{k}(\vec{r})\delta\tilde{T}_{l}(\vec{r})dV=C_{k}\delta_{k,l}},\quad\text{with}\,\,C_{k}\neq
0,$ (S29)

which allows the time evolution of any thermal mode to be written as:

$\dot{\theta}_{k}=-\frac{1}{\tau_{k}}\theta_{k}+\frac{\int{\dot{Q}(\vec{r},t)\delta\tilde{T}_{k}(\vec{r})dV}}{\int{c_{p}\rho(\delta\tilde{T}_{k})^{2}dV}}.$
(S30)

Applying the thermal mode expansion over the expression for the lumped
photothermal force, Eq. S13, allows us to write
$F^{\theta}_{n}(t)=\sum_{k}F^{\theta}_{k,n}(t)$, where:

$F^{\theta}_{k,n}(t)=\big{[}\int{\hat{\bm{\nabla}}\vec{u}_{n}\mkern
1.0mu{:}\big{(}\mathbf{c}\mkern
1.0mu{:}\bm{\alpha}\big{)}\delta\tilde{T}_{k}(\vec{r})dV}\big{]}\theta_{k}(t)=\Lambda_{k,n}^{\theta}\theta_{k}(t),$
(S31)

where the coefficient $\Lambda_{k,n}^{\theta}$ has units of N/K.

We are interested in heating due to optical absorption. Assuming that only one
optical mode is excited and denoting the local absorptive (optical) loss rate
as $\kappa_{\text{loc}}(\vec{r},t)$, the total power dissipated per unit
volume is given by:

$\dot{Q}(\vec{r},t)=2\kappa_{\text{loc}}(\vec{r},t)\varepsilon_{0}\varepsilon_{r}(\vec{r})\vec{e}(\vec{r})\cdot\vec{e}^{\,*}(\vec{r})|a(t)|^{2},$
(S32)

where $|a(t)|^{2}$ is the number of photons circulating in the optical cavity,
$\vec{e}(\vec{r})$ is the electromagnetic mode profile, $\varepsilon_{0}$ and
$\varepsilon_{r}$ are the vacuum permittivity and dielectric function,
respectively.

We would like to express Eq. S30 in terms of the experimentally accessible
total absorptive loss rate $\kappa_{\text{abs}}(t)$ (the time dependency
arises if nonlinear absorption is considered) instead of
$\kappa_{\text{loc}}(\vec{r},t)$. These quantities are related by:

$\kappa_{\text{abs}}(t)=\frac{\int\kappa_{\text{loc}}(\vec{r},t)\varepsilon_{r}(\vec{r})|\vec{e}(\vec{r})|^{2}dV}{\int\varepsilon_{r}(\vec{r})|\vec{e}(\vec{r})|^{2}dV}.$
(S33)

Since $\kappa_{\text{abs}}(t)$ is part of a lumped model, it is clear that
with its knowledge alone, one cannot solve for
$\kappa_{\text{loc}}(\vec{r},t)$. Instead, if linear absorption is considered,
we simplify matters by assuming $\kappa_{\text{loc}}(\vec{r},t)$ is a constant
$\kappa_{\text{loc,}0}$ within an absorbing volume $V_{\text{abs}}$. In that
case, we may solve for $\kappa_{\text{loc,}0}$ in terms of the total linear
absorption $\kappa_{\text{abs}}=\eta_{0}\kappa_{0}$, where $\kappa_{0}$
denotes the cold-cavity (linear) intrinsic losses of the device, and
$\eta_{0}$ its absorptive fraction. We have:

$\kappa_{\text{loc},0}=\eta_{0}\kappa_{0}\frac{\int\varepsilon_{r}(\vec{r})|\vec{e}(\vec{r})|^{2}dV}{\int_{V_{\text{abs}}}\varepsilon_{r}(\vec{r})|\vec{e}(\vec{r})|^{2}dV}.$
(S34)

Although assuming homogeneous absorption within the absorbing volume is
certainly an approximation, it is generally sufficient to correctly capture
optical absorption even when it takes place near the surface. For example, in
case the main mechanism of absorption is mediated by surface imperfections, we
should expect an exponential decay in $\kappa_{\text{loc}}(\vec{r})$ as we
move away from the surface. In that case, we could simply treat
$\kappa_{\text{loc}}$ and $V_{\text{abs}}$ as effective quantities.

In order to incorporate nonlinear absorption mechanisms such as two-photon
(TPA) and free-carrier (FCA) absorption, the spatio-temporal dependency of
$\kappa_{\text{loc}}(\vec{r},t)$ must be related to the fields in the
resonator. For phenomena that scale linearly with the electromagnetic energy
in the resonator, e.g. TPA Barclay  _et al._ (2005), we expect
$\kappa_{\text{loc}}(\vec{r},t)$ to have the following form:

$\kappa_{\text{loc,}1}(\vec{r},t)=C_{1}\frac{\varepsilon_{r}(\vec{r})|\vec{e}(\vec{r})|^{2}}{\int\varepsilon_{r}(\vec{r})|\vec{e}(\vec{r})|^{2}dV}\times|a(t)|^{2},$
(S35)

since the fields themselves generate absorption. Analogously, for phenomena
that scale quadratically with the electromagnetic energy in the resonator, e.g
FCA:

$\kappa_{\text{loc,}2}(\vec{r},t)=C_{\text{2}}\frac{\varepsilon_{r}(\vec{r})^{2}|\vec{e}(\vec{r})|^{4}}{(\int\varepsilon_{r}(\vec{r})|\vec{e}(\vec{r})|^{2}dV)^{2}}\times|a(t)|^{4},$
(S36)

where nonlocal effects such as carrier diffusion are neglected.

In both cases above, $C_{2}$ and $C_{1}$ are approximated as material-
dependent constants which are only appreciable inside the volume of the
cavity, defined by $V_{\text{cav}}$. They can be expressed in terms of the
experimentally accessible quantities such as the first
($\kappa_{\text{abs},2}(t)$) and second ($\kappa_{\text{abs},2}(t)$) order
absorption rates:

$C_{\text{1}}=\frac{\kappa_{1}(t)}{|a(t)|^{2}}\frac{(\int\varepsilon_{r}(\vec{r})|\vec{e}(\vec{r})|^{2}dV)^{2}}{\int_{V_{\text{cav}}}\varepsilon_{r}^{2}(\vec{r})|\vec{e}(\vec{r})|^{4}dV},$
(S37)

$C_{\text{2}}=\frac{\kappa_{2}(t)}{|a(t)|^{4}}\frac{(\int\varepsilon_{r}(\vec{r})|\vec{e}(\vec{r})|^{2}dV)^{3}}{\int_{V_{\text{cav}}}\varepsilon_{r}^{3}(\vec{r})|\vec{e}(\vec{r})|^{6}dV}.$
(S38)

Writing
$\kappa_{\text{loc}}(\vec{r},t)=\kappa_{\text{loc,}0}+\kappa_{\text{loc,}1}(\vec{r},t)+\kappa_{\text{loc,}2}(\vec{r},t)$
in Eq. S32 and using the definitions of $C_{\text{1}}$ and $C_{\text{2}}$
above, equation S30 can be written in terms of the photon energy
$\hbar\omega_{l}$ and the zeroth, first, and second order thermal resistances,
yielding:

$\dot{\theta}_{k}=-\frac{1}{\tau_{k}}\theta_{k}+\frac{\hbar\omega_{l}}{\tau_{k}}(\eta_{0}\kappa_{0}R^{\theta}_{k,0}+\kappa_{1}(t)R^{\theta}_{k,1}+\kappa_{2}(t)R^{\theta}_{k,2})\times|a(t)|^{2},$
(S39)

with:

$R^{\theta}_{k,0}=\frac{2\tau_{k}\varepsilon_{0}}{\hbar\omega_{l}}\frac{\int_{V_{\text{abs}}}{\varepsilon_{r}(\vec{r})|\vec{e}(\vec{r})|^{2}\delta\tilde{T}_{k}(\vec{r})dV}}{\int{c_{p}\rho(\delta\tilde{T}_{k})^{2}dV}}\frac{\int\varepsilon_{r}(\vec{r})|\vec{e}(\vec{r})|^{2}dV}{\int_{V_{\text{abs}}}\varepsilon_{r}(\vec{r})|\vec{e}(\vec{r})|^{2}dV},$
(S40)

$R^{\theta}_{k,1}=\frac{2\tau_{k}\varepsilon_{0}}{\hbar\omega_{l}}\frac{\int_{V_{\text{cav}}}{\varepsilon_{r}^{2}(\vec{r})|\vec{e}(\vec{r})|^{4}\delta\tilde{T}_{k}(\vec{r})dV}}{\int{c_{p}\rho(\delta\tilde{T}_{k})^{2}dV}}\frac{\int\varepsilon_{r}(\vec{r})|\vec{e}(\vec{r})|^{2}dV}{\int_{V_{\text{cav}}}\varepsilon_{r}^{2}(\vec{r})|\vec{e}(\vec{r})|^{4}dV},$
(S41)

$R^{\theta}_{k,2}=\frac{2\tau_{k}\varepsilon_{0}}{\hbar\omega_{l}}\frac{\int_{V_{\text{cav}}}{\varepsilon_{r}^{3}(\vec{r})|\vec{e}(\vec{r})|^{6}\delta\tilde{T}_{k}(\vec{r})dV}}{\int{c_{p}\rho(\delta\tilde{T}_{k})^{2}dV}}\frac{\int\varepsilon_{r}(\vec{r})|\vec{e}(\vec{r})|^{2}dV}{\int_{V_{\text{cav}}}\varepsilon_{r}^{3}(\vec{r})|\vec{e}(\vec{r})|^{6}dV}.$
(S42)

Eqs. S40, S41, S42 can be evaluated in a straightforward way by any
electromagnetic/thermal solver. Mechanical vibrations may also lead to heating
through thermoelastic damping, however, due to its negligible dynamical
contributions in the devices discussed here, this effect will be neglected.
Importantly, if $\delta\tilde{T}_{k}(\vec{r})$ is approximately constant in
the region of overlap with the optical mode, all three expressions above yield
similar values. This approximation is well justified for the first few low-
order thermal modes, which turn out to be the most relevant for the case
treated in the main text.

In practice, not all of the nonlinear absorptive losses ($\kappa_{1}$ and
$\kappa_{2}$) are necessarily converted to heat, as implied by Eq. S39.
Nonlinear absorption is strongly tied to carrier dynamics inside the device.
As we are dealing with a direct band-gap semiconductor, we expect that
carriers generated through virtual processes such as TPA to display both
radiative and non-radiative recombination routes. That is, only a fraction of
the nonlinear loss rate acts as a heat source in our resonator. As a
consequence, we are led to define the ratios $\eta_{i}$ of the $i$th order
nonlinear loss rate that contributes to the heating of the resonator. With
that in mind, the generalized evolution of the amplitude $\theta_{k}(t)$ is
given by:

$\dot{\theta}_{k}=-\frac{1}{\tau_{k}}\theta_{k}+\frac{\hbar\omega_{l}}{\tau_{k}}(\eta_{0}\kappa_{0}R^{\theta}_{k,0}+\eta_{1}\kappa_{1}(t)R^{\theta}_{k,1}+\eta_{2}\kappa_{2}(t)R^{\theta}_{k,2})\times|a(t)|^{2},$
(S43)

##  S3 Frequency domain description of the photothermal coupling

###  S3.1 Single-mode Photothermal coupling

We start from the coupled equations for the temporal evolution of the optical,
mechanical and temperature fields. In this subsection, we assume that the
dynamics is dominated by one mode in each of the previously cited domains,
therefore, we drop all mode-related indices. For simplicity, in this and in
the next subsection we treat the case of linear absorption, i.e.
$\kappa_{\text{abs},1}(t)=\kappa_{\text{abs},2}(t)=0$. The equations read:

$\begin{split}\dot{a}=i\Delta(t)a-\frac{\kappa}{2}a+\sqrt{\kappa_{e}}\alpha_{in},\\\
m_{\text{eff}}(\ddot{x}+\Gamma\dot{x}+\Omega^{2}x)=F^{L}(t)+F^{\text{RP}}(t)+F^{\theta}(t),\\\
\dot{\theta}=-\frac{1}{\tau}\theta+\frac{\hbar\omega_{l}\kappa_{\text{abs}}R^{\theta}}{\tau}|a|^{2},\end{split}$
(S44)

where $\Delta(t)=\omega_{l}-\omega_{c}+G^{\theta}\theta+G^{x}x$, is the
relative detuning between the cavity and the laser, including optomechanical
and thermal frequency pullings, $\omega_{c}$ is the (optical) resonance
frequency and $\omega_{l}$ the external laser frequency. $F^{\text{L}}$,
$F^{\text{RP}}$ and $F^{\theta}$ are respectively the thermal Langevin, the
radiation pressure and the photothermal (bolometric) forces, $G^{x}$ and
$G^{\theta}$ are respectively the optomechanical and thermo-optical coupling
rates. We also choose a normalization for the electromagnetic modes such that
$|a|^{2}$ is the number of circulating photons in the cavity, consequently,
$\alpha_{in}$ is the rate of incident photons on the resonator, driven by an
external source.

We may derive expressions for $G^{\theta}$ and $G^{x}$ using first-order
perturbation theory on Maxwell’s equations Johnson  _et al._ (2002); Balram
_et al._ (2014):

$\displaystyle
G^{\theta}=\omega_{c}\frac{\int{|~{}\vec{e}~{}|^{2}n\frac{dn}{dT}\delta\tilde{T}dV}}{\int{|~{}\vec{e}~{}|^{2}\varepsilon
dV}},$ (S45a) $\displaystyle
G^{x}=-\frac{\omega_{c}}{2}\frac{\int{\vec{e}\cdot(\bm{p}\mkern
1.0mu{:}\bm{S}^{x})\cdot\vec{e}\,^{*}dV}}{\int{|~{}\vec{e}~{}|^{2}\varepsilon
dV}}+\frac{\omega_{c}}{2}\frac{\int
d\vec{A}{\cdot}\vec{u}\left(\Delta\bm{\varepsilon}|\vec{e}_{\parallel}|^{2}-\Delta\bm{\varepsilon}^{-1}|\vec{d}_{\perp}|^{2}\right)}{\int|~{}\vec{e}~{}|^{2}\varepsilon
dV},$ (S45b)

where Eq. S45a is related to the thermo-optic effect, related to temperature
induced changes in the dielectric response of the medium. The integrals in Eq.
S45b model the photoelastic and moving boundary optomechanical couplings
related to a mechanical mode $\vec{u}$, where $\bm{p}$ is the photoelastic
tensor. The moving boundary contribution is a function of
$\Delta\bm{\varepsilon}=\bm{\varepsilon}_{1}-\bm{\varepsilon}_{2}$ and
$\Delta\bm{\varepsilon}^{-1}~{}=~{}(\bm{\varepsilon}_{1})^{-1}-(\bm{\varepsilon}_{2})^{-1}$,
which are related to the permittivities of the guiding
($\bm{\varepsilon}_{1}$) and surrounding ($\bm{\varepsilon}_{2}$) materials.

In the spirit of deriving an entirely linear theory, we look for linearized
solutions to Eqs. S44 as $a(t)=a_{0}+\delta a(t)$, $x(t)=x_{0}+\delta x(t)$,
and $\theta(t)=\theta_{0}+\delta\theta(t)$. Using $F^{\text{RP}}=\hbar
G^{x}|a|^{2}$, $F^{\theta}=\Lambda^{\theta}\theta$ and collecting the
fluctuating terms up to first order we obtain the following dynamical
equations for thermo-optomechanics (initially disregarding the Langevin term):

$\begin{split}\delta\dot{a}=i(G^{x}\delta
x+G^{\theta}\delta\theta)a_{0}+(i\Delta_{0}-\frac{\kappa}{2})\delta a,\\\
\delta\ddot{x}+\Gamma\delta\dot{x}+\Omega^{2}\delta x=\frac{\hbar
G^{x}}{m_{\text{eff}}}(a_{0}^{*}\delta a+a_{0}\delta
a^{*})+\frac{\Lambda^{\theta}}{m_{\text{eff}}}\delta\theta,\\\
\delta\dot{\theta}=-\frac{1}{\tau}\delta\theta+\frac{\hbar\omega_{l}\kappa_{\text{abs}}R^{\theta}}{\tau}(a_{0}^{*}\delta
a+a_{0}\delta a^{*}),\end{split}$ (S46)

where $\Delta_{0}$ absorbs the static frequency shifts given by
$G^{\theta}\theta_{0}+G^{x}x_{0}$. In frequency space:

$\begin{split}[(\Delta_{0}+\omega)+i\frac{\kappa}{2}]\delta
a(\omega)=-\big{[}G^{x}\delta
x(\omega)+G^{\theta}\delta\theta(\omega)\big{]}a_{0},\\\
\big{[}(\Omega^{2}-\omega^{2})-i\omega\Gamma\big{]}\delta
x(\omega)=\frac{\hbar G^{x}}{m_{\text{eff}}}\big{[}a_{0}^{*}\delta
a(\omega)+a_{0}[\delta
a]^{*}(\omega)\big{]}+\frac{\Lambda^{\theta}}{m_{\text{eff}}}\delta\theta(\omega),\\\
-i\omega\delta\theta(\omega)=-\frac{1}{\tau}\delta\theta(\omega)+\frac{\hbar\omega_{l}\kappa_{\text{abs}}R^{\theta}}{\tau}\big{[}a_{0}^{*}\delta
a(\omega)+a_{0}[\delta a]^{*}(\omega)\big{]},\end{split}$ (S47)

since $[\delta a]^{*}(\omega)=[\delta a(-\omega)]^{*}$ and for real variables
such as $\delta\theta$ and $\delta x$, $\delta x(\omega)=[\delta
x(-\omega)]^{*}$, we may eliminate $\delta a(\omega)$ and $[\delta
a]^{*}(\omega)$, obtaining:

$\displaystyle\delta\theta(\omega)=-\frac{\hbar\omega_{l}\kappa_{\text{abs}}R^{\theta}\chi^{\theta}(\omega)\Psi(\omega,\Delta_{0})|a_{0}|^{2}G^{x}}{\big{[}\tau+\hbar\omega_{l}\kappa_{\text{abs}}R^{\theta}G^{\theta}\chi^{\theta}(\omega)\Psi(\omega,\Delta_{0})|a_{0}|^{2}\big{]}}\delta
x(\omega),$ (S48a)
$\displaystyle\big{[}(\Omega^{2}-\omega^{2})-i\omega\Gamma\big{]}\delta
x(\omega)=-\frac{\hbar
G^{x}}{m_{\text{eff}}}\Psi(\omega,\Delta_{0})|a_{0}|^{2}(G^{\theta}\delta\theta(\omega)+G^{x}\delta
x(\omega))+\frac{\Lambda^{\theta}}{m_{\text{eff}}}\delta\theta(\omega),$
(S48b)

where
$\Psi(\omega,\Delta_{0})=[\frac{1}{(\Delta_{0}+\omega)+i\kappa/2}+\frac{1}{(\Delta_{0}-\omega)-i\kappa/2}]$,
a combination of the cavity’s optical susceptibility and its conjugate, and
$\chi^{\theta}(\omega)=\frac{1}{1/\tau-i\omega}$ is the thermal
susceptibility.

Eliminating $\delta\theta(\omega)$ in Eq. S48b, we readily obtain the dressed
optomechanical self-energy. The term on the RHS which is being multiplied by
$G^{\theta}$ yields the correction acting on the radiation pressure feedback,
therefore the total $RP$ modification to the linear response of the system is:

$\Sigma_{\text{eff}}^{\text{RP}}(\omega)=\frac{\Sigma^{\text{RP}}(\omega)}{\big{[}1+\frac{\hbar\omega_{l}\kappa_{\text{abs}}R^{\theta}G^{\theta}\chi^{\theta}(\omega)\Psi(\omega,\Delta)|a_{0}|^{2}}{\tau}\big{]}},$
(S49)

where we defined
$\Sigma^{\text{RP}}(\omega)=\hbar(G^{x})^{2}\Psi(\omega,\Delta_{0})|a_{0}|^{2}$,
the optically induced mechanical inverse susceptibility in the absence of
thermal feedback. The correction above is negligible for the GaAs microdisks
considered in the main text, however it might become relevant in other systems
where mechanical frequencies are comparable to or smaller than the inverse of
thermal relaxation times. In that case, thermal dispersion affects the optical
spectrum within the mechanical timescale, impacting optomechanical
transduction.

The term proportional to $\Lambda^{\theta}$ in Eq. S48b yields the dressed
bolometric force, which is this work’s object of study. The photothermal
contribution to the inverse mechanical susceptibility is given by:

$\Sigma^{\theta}_{\text{eff}}(\omega)=\frac{\Sigma^{\theta}(\omega)}{\big{[}1+\frac{\hbar\omega_{l}\kappa_{\text{abs}}R^{\theta}G^{\theta}\chi^{\theta}(\omega)\Psi(\omega,\Delta)|a_{0}|^{2}}{\tau}\big{]}},$
(S50)

where we defined the bare bolometric contribution to the inverse mechanical
susceptibility as:

$\Sigma^{\theta}(\omega)=\frac{\hbar\omega_{l}\kappa_{\text{abs}}R^{\theta}\Lambda^{\theta}G^{x}\chi^{\theta}(\omega)\Psi(\omega,\Delta)|a_{0}|^{2}}{\tau}.$
(S51)

The dressed inverse mechanical susceptibility
$\chi_{m,\mathrm{eff}}^{-1}(\omega)$, including RP and photothermal
contributions is finally given by:

$\chi_{m,\mathrm{eff}}^{-1}(\omega)=\chi_{m}^{-1}(\omega)+\Sigma_{\text{eff}}^{\text{RP}}(\omega)+\Sigma^{\theta}_{\text{eff}}(\omega),$
(S52)

where
$\chi_{m}^{-1}(\omega)=m_{\mathrm{eff}}\left[\left(\Omega^{2}-\omega^{2}\right)-i\Gamma\omega\right]$
is the inverse of the bare mechanical susceptibility.

The expressions above for $\Sigma^{\text{RP}}_{\text{eff}}$ and
$\Sigma^{\theta}_{\text{eff}}$ may be simplified in many practical scenarios.
In agreement with the experimental case reported in the main text, we assume
the bad-cavity regime ($\Omega\ll\kappa/2$) and $\tau\gg 1/\Omega$. Evaluating
all frequency responses at $\Omega$ , which is the driving frequency given by
the mechanical oscillations, and also taking the detuning
$\Delta_{0}=\kappa/2$, we note that $\chi^{\theta}\approx i/\Omega$, and
$\Psi\approx 2/\kappa$. For typical microdisks, $\tau/(2\pi)\approx 10$
$\mu$s, $\Omega/(2\pi)\approx 250$ MHz, $\kappa_{\text{abs}}/(2\pi)\approx 1$
GHz, $\omega_{l}/(2\pi)\approx 200$ THz, $R^{\theta}\approx 4\times 10^{4}$,
$G^{\theta}/(2\pi)\approx 10$ GHz, $\kappa/(2\pi)\approx 10$ GHz and
$|a_{0}|^{2}\approx 10^{4}$. With these numbers we see that:

$\frac{\hbar\omega_{l}\kappa_{\text{abs}}R^{\theta}G^{\theta}\chi^{\theta}(\omega)\Psi(\omega,\Delta)|a_{0}|^{2}}{\tau}<<1.$
(S53)

Confirming that the correction due to the thermo-optical dispersion is
negligible in our case.

One last approximation can be made in order to compare $\Sigma^{\theta}$ and
$\Sigma^{\text{RP}}$. Considering only cooling/amplification under the same
approximations as before, the ratio of photothermal ($\delta\Gamma^{\theta}$)
and RP ($\delta\Gamma^{\text{RP}}$) mechanical amplification, evaluated at
$\Delta_{0}=\kappa/2$ is given by:

$\frac{\delta\Gamma^{\theta}}{\delta\Gamma^{\text{RP}}}(\Delta=\kappa/2)=\frac{1}{2}\frac{\omega_{l}\kappa_{\text{abs}}R^{\theta}\Lambda^{\theta}\kappa}{\tau\Omega^{2}G^{x}},$
(S54)

which can be used as a simple test for the relevance of photothermal forces in
devices that operate within the regimes above.

###  S3.2 Multimode Photothermal coupling

We now generalize our discussion to the case where several thermal modes
couple relevantly to optical and mechanical modes. The appropriate equations
in frequency space are given by:

$\begin{split}[(\Delta_{0}+\omega)+i\frac{\kappa}{2}]\delta
a(\omega)=-\big{[}G^{x}\delta
x(\omega)+\sum_{k}G^{\theta}_{k}\delta\theta_{k}(\omega)\big{]}a_{0},\\\
\big{[}(\Omega^{2}-\omega^{2})-i\omega\Gamma\big{]}\delta
x(\omega)=\frac{\hbar G^{x}}{m_{\text{eff}}}\big{[}a_{0}^{*}\delta
a(\omega)+a_{0}[\delta
a]^{*}(\omega)\big{]}+\sum_{k}\frac{\Lambda^{\theta}_{k}}{m_{\text{eff}}}\delta\theta_{k}(\omega),\\\
-i\omega\delta\theta_{k}(\omega)=-\frac{1}{\tau_{k}}\delta\theta_{k}(\omega)+\frac{\hbar\omega_{l}\kappa_{\text{abs}}R^{\theta}_{k}}{\tau_{k}}\big{[}a_{0}^{*}\delta
a(\omega)+a_{0}[\delta a]^{*}(\omega)\big{]},\,k=1,2,3....\end{split}$ (S55)

Eliminating $\delta a(\omega)$ from the equations for the thermal modes we
get:

$\delta\theta_{k}(\omega)=\frac{A^{\theta}_{k}}{(1-A^{\theta}_{k}G^{\theta}_{k})}\big{(}G^{x}\delta
x(\omega)+\sum_{l\neq k}G^{\theta}_{l}\delta\theta_{l}(\omega)\big{)},$ (S56)

where we defined:

$A^{\theta}_{k}(\omega,\Delta_{0})=-\frac{\hbar\omega_{l}\kappa_{\text{abs}}R^{\theta}_{k}\chi^{\theta}_{k}(\omega)\Psi(\omega,\Delta_{0})|a_{0}|^{2}}{\tau_{k}}.$
(S57)

By inspection, one may show that
$\frac{\delta\theta_{l}(\omega)}{A^{\theta}_{l}(\omega,\Delta_{0})}=\frac{\delta\theta_{k}(\omega)}{A^{\theta}_{k}(\omega,\Delta_{0})}$.
With that result we get:

$\delta\theta_{k}(\omega)=\frac{A^{\theta}_{k}(\omega,\Delta_{0})G^{x}}{1-\sum_{k}A^{\theta}_{k}(\omega,\Delta_{0})G^{\theta}_{k}}\delta
x(\omega),$ (S58)

and the total photothermal modification to the inverse mechanical
susceptibility is:

$\Sigma^{\theta}_{\text{eff}}(\omega)=\frac{\Sigma^{\theta}}{1-\sum_{k}A^{\theta}_{k}(\omega,\Delta_{0})G^{\theta}_{k}},$
(S59)

where the bare photothermal contribution is given by:

$\Sigma^{\theta}(\omega)=\sum_{k}A^{\theta}_{k}(\omega,\Delta_{0})\Lambda^{\theta}_{k}G^{x},$
(S60)

In analogy with the single thermal mode case, the corrected RP induced inverse
susceptibility is given by:

$\Sigma^{\text{RP}}_{\text{eff}}(\omega)=\frac{\Sigma^{\text{RP}}(\omega)}{1-\sum_{k}A^{\theta}_{k}(\omega,\Delta_{0})G^{\theta}_{k}}.$
(S61)

From the symmetries of the equations above, it is useful to define “effective
thermal response” functions, at a given frequency $\omega$:

$\begin{split}h_{1}^{\theta}(\omega)=\kappa_{\text{abs}}\sum_{k}\frac{R^{\theta}_{k}\Lambda^{\theta}_{k}\chi^{\theta}_{k}(\omega)}{\tau_{k}},\\\
h_{2}^{\theta}(\omega)=\kappa_{\text{abs}}\sum_{k}\frac{R^{\theta}_{k}G^{\theta}_{k}\chi^{\theta}_{k}(\omega)}{\tau_{k}},\end{split}$
(S62)

in terms of which, expressions S59 and S61 can be recast as:

$\begin{split}\Sigma^{\theta}_{\text{eff}}(\omega)=\frac{\hbar\omega_{l}\Psi(\omega,\Delta_{0})|a_{0}|^{2}G^{x}h_{1}^{\theta}(\omega)}{1+\hbar\omega_{l}\Psi(\omega,\Delta_{0})|a_{0}|^{2}h_{2}^{\theta}(\omega)},\\\
\Sigma^{\text{RP}}_{\text{eff}}(\omega)=\frac{\hbar\Psi(\omega,\Delta_{0})|a_{0}|^{2}(G^{x})^{2}}{1+\hbar\omega_{l}\Psi(\omega,\Delta_{0})|a_{0}|^{2}h_{2}^{\theta}(\omega)},\end{split}$
(S63)

which share the same form as their single mode counterparts. We note that the
summations for $h_{1}^{\theta}(\omega)$ and $h_{2}^{\theta}(\omega)$ are well
behaved since $R^{\theta}_{k}/\tau_{k}$ is independent of $\tau_{k}$ and
$\chi^{\theta}_{k}$ is a monotonically decreasing function of $\tau_{k}$,
guaranteeing gradually decreasing importance of high-order thermal modes
($\tau_{k}\to 0$).

From the equations above it is straightforward to compare the photothermal and
radiation pressure forces amplitudes. The ratio
$\Sigma^{\theta}/\Sigma^{\text{RP}}$ yields:

$\frac{\Sigma^{\theta}_{\text{eff}}(\omega)}{\Sigma^{\text{RP}}_{\text{eff}}(\omega)}=\frac{\hbar\omega_{l}h_{1}^{\theta}(\omega)}{\hbar
G^{x}}.$ (S64)

Since the denominator of the RHS essentially gives the RP force per photon,
the numerator may be interpreted likewise, but regarding the photothermal
force. Notice however, that due to non-zero relaxation times, the two forces
display a non-trivial phase relation, here captured by the complex part of
$h^{\theta}_{1}(\omega)$.

The approximation obtained in the previous section can be generalized to the
multimode case. Again, considering the bad-cavity limit and neglecting thermal
dispersion, i.e. $h^{\theta}_{2}(\Omega)\ll 1$:

$\frac{\delta\Gamma^{\theta}}{\delta\Gamma^{\text{RP}}}(\Delta=\kappa/2)=\frac{1}{2}\frac{\omega_{l}\kappa\text{Im}[h_{1}^{\theta}]}{\Omega
G^{x}},$ (S65)

###  S3.3 Nonlinear Multimode Photothermal coupling

In this subsection, we summarize the main results concerning the usage of Eq.
S55, without neglecting the nonlinear losses in general. That is, we use Eq.
S39 and make $\kappa\rightarrow\kappa(|a|^{2})$ in the evolution of $a$. That
is, the total nonlinear intrinsic loss, $\kappa_{i}(t)$, is given by
$\kappa_{i}(t)=\kappa_{0}+\kappa_{1}(t)+\kappa_{2}(t)$. The coupled equations
of the optical, mechanical and thermal responses read:

$\begin{split}\dot{a}=i\Delta(t)a-\frac{\kappa_{e}+\kappa_{0}+\kappa_{1}(t)+\kappa_{2}(t)}{2}a+\sqrt{\kappa_{e}}\alpha_{in},\\\
m_{\text{eff}}(\ddot{x}+\Gamma\dot{x}+\Omega^{2}x)=F^{L}(t)+F^{\text{RP}}(t)+F^{\theta}(t),\\\
\dot{\theta}_{k}=-\frac{1}{\tau_{k}}\theta_{k}+\frac{\hbar\omega_{l}}{\tau_{k}}(\eta_{0}\kappa_{0}R^{\theta}_{k,0}+\eta_{1}\kappa_{1}(t)R^{\theta}_{k,1}+\eta_{2}\kappa_{2}(t)R^{\theta}_{k,2})\times|a(t)|^{2},\,k=1,2,3...,\end{split}$
(S66)

where
$\Delta(t)=\omega_{l}-\omega_{c}+\sum_{k}G_{k}^{\theta}\theta(t)+G^{x}x(t)$.

A fundamental difference between the present case and the analysis in past
sections is that, when considering the frequency domain version of the above
equations, the Fourier transform of the nonlinear losses must also be
performed, since those also display a time dependency, substantially modifying
the final results.

In our model for the nonlinear losses we neglected all but the dynamics of the
energy in the resonator. That is, the only time dependency of the first and
second order dissipation is given through the number of photons $|a(t)|^{2}$.
Since those losses are intrinsically related to carrier dynamics, this is
explicitly an approximation. Its validity requires that any timescale
associated to carrier relaxation (i.e. its recombination rate) is must faster
than the period of mechanical oscillations, in such a way that we may assume
that their population responds instantly to energy variations. As stated in
past work on this subject, typical recombination rates for GaAs range around
$10\text{\,}\mathrm{ps}$ Guha  _et al._ (2017), much faster than the
$\approx$4\text{\,}\mathrm{ns}$$ associated with mechanical oscillations.

Under the aforementioned hypothesis, we write
$\kappa_{1}(t)=\kappa^{\prime}_{1}|a(t)|^{2}$ and
$\kappa_{2}(t)=\kappa^{\prime}_{2}|a(t)|^{4}$, where the $\kappa^{\prime}_{i}$
are constants. This form allows us to linearize the losses according to
$a(t)\rightarrow a_{0}+\delta a(t)$. After linearizing and writing Eqs. S66 in
frequency space, they read:

$\begin{split}\left[i\Delta_{0}-\frac{\kappa_{0}+2\bar{\kappa}_{1}+3\bar{\kappa}_{2}}{2}\right]\delta
a(\omega)-\frac{a_{0}}{a_{0}^{*}}\left[\frac{\bar{\kappa}_{1}+2\bar{\kappa}_{2}}{2}\right][\delta
a]^{*}(\omega)=-i[\sum_{k}G^{\theta}_{k}\delta\theta_{k}(\omega)+a_{0}G^{x}\delta
x(\omega)]a_{0},\\\ \big{[}(\Omega^{2}-\omega^{2})-i\omega\Gamma\big{]}\delta
x(\omega)=\frac{\hbar G^{x}}{m_{\text{eff}}}\big{[}a_{0}^{*}\delta
a(\omega)+a_{0}[\delta
a]^{*}(\omega)\big{]}+\sum_{k}\frac{\Lambda^{\theta}_{k}}{m_{\text{eff}}}\delta\theta_{k}(\omega),\\\
-i\omega\theta_{k}(\omega)=-\frac{1}{\tau_{k}}\theta_{k}(\omega)+\frac{\hbar\omega_{l}}{\tau_{k}}(\eta_{0}\kappa_{0}R^{\theta}_{k,0}+2\eta_{1}\bar{\kappa}_{1}R^{\theta}_{k,1}+3\eta_{2}\bar{\kappa}_{2}R^{\theta}_{k,2})\times\left(a_{0}[\delta
a]^{*}(\omega)+a_{0}^{*}\delta a(\omega)\right),\,k=1,2,3...,\end{split}$
(S67)

where we defined the average losses
$\bar{\kappa}_{j}=\kappa^{\prime}_{j}|a_{0}|^{2j}$ and
$\bar{\kappa}_{0}=\kappa_{0}$

We follow the same procedure as before: we eliminate $\theta(\omega)$ and
$\delta a(\omega)$ from the equation for the mechanical response, from which
we recover a dressed susceptibility for the mechanical mode. The PT-induced
and the RP-induced modifications to the mechanical susceptibility are:

$\displaystyle\Sigma^{\theta}_{\text{eff}}(\omega)$ $\displaystyle=$
$\displaystyle\frac{\hbar\omega_{l}\Psi_{\text{dyn}}(\omega,\Delta_{0})|a_{0}|^{2}G^{x}h_{1,\text{dyn}}^{\theta}(\omega)}{1+\hbar\omega_{l}\Psi_{\text{dyn}}(\omega,\Delta_{0})|a_{0}|^{2}h_{2,\text{dyn}}^{\theta}(\omega)}$
(S68) $\displaystyle\Sigma^{\text{RP}}_{\text{eff}}(\omega)$ $\displaystyle=$
$\displaystyle\frac{\hbar\Psi_{\text{dyn}}(\omega,\Delta_{0})|a_{0}|^{2}(G^{x})^{2}}{1+\hbar\omega_{l}\Psi_{\text{dyn}}(\omega,\Delta_{0})|a_{0}|^{2}h_{2,\text{dyn}}^{\theta}(\omega)},$
(S69)

where $h_{1,\text{dyn}}^{\theta}(\omega)$ and
$h_{2,\text{dyn}}^{\theta}(\omega)$ are, respectively, the effective responses
for the photothermal force and for the thermo-optic dispersion:

$\begin{split}h_{1,\text{dyn}}^{\theta}(\omega)=\sum^{2}_{j=0}(j+1)\eta_{j}\bar{\kappa}_{j}\sum_{k}\frac{R^{\theta}_{k,j}\Lambda^{\theta}_{k}\chi^{\theta}_{k}(\omega)}{\tau_{k}},\\\
h_{2,\text{dyn}}^{\theta}(\omega)=\sum^{2}_{j=0}(j+1)\eta_{j}\bar{\kappa}_{j}\sum_{k}\frac{R^{\theta}_{k,j}G^{\theta}_{k}\chi^{\theta}_{k}(\omega)}{\tau_{k}},\end{split}$
(S70)

and $\Psi_{\text{dyn}}$ is the modified optical response due the dynamic of
the nonlinear losses:

$\Psi_{\text{dyn}}(\Delta_{0},\omega)=\frac{8\bar{\Delta}_{0}}{4\Delta_{0}^{2}+\Big{(}\sum^{2}_{j=0}(2j+1)\bar{\kappa}_{j}-2i\omega\Big{)}\Big{(}\sum^{2}_{j=0}\bar{\kappa}_{j}-2i\omega\Big{)}}.$
(S71)

The equations above were used in the analysis of the experiment reported in
the main text.

##  S4 Localized Absorption

In this section we use FEM simulations to investigate how surface localized
absorption affects the photothermal effects on our microdisk devices. An
absorptive layer with thickness $t_{\text{abs}}$ is defined within the GaAs
membrane composing the microdisk illustrated in Fig. S1 a). Importantly, the
absorption is taken to be constant over this layer. The radius of the disk is
$6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and the pedestal diameter is
$1.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$, the latter is measured in the
region of contact of the AlGaAs/GaAs layers. The geometry of this device
agrees with the one used in Fig. 2 in the main text. In the following analysis
we consider the first order TE optical mode and the first order mechanical
breathing ($\Omega/(2\pi)=$230\text{\,}\mathrm{MHz}$$) mode of the microdisk.

In the main text we study the case of an homogeneously absorbing GaAs membrane
(no absorption layer), in which the total PTh force per mode tends to become
negligibly small for higher order thermal modes ($k>5$), this happens due to
the self-cancellation of the surface and volume contributions. This is shown
in Fig. S1 b) where each thermal mode’s contribution to the imaginary and real
parts total PTh force per photon was evaluated. As the thermal frequency –
defined as $\tau^{-1}/(2\pi)$ – increases, the red T-shaped bars in the
imaginary part of such force become negligibly small. Nevertheless, we verify
appreciable – contrary – surface and volume forces for frequencies as high as
$1\text{\,}\mathrm{GHz}$, in agreement with the discussion in the main text.
The total real component of the force (horizontal dashed line) is evaluated to
be approximately two orders of magnitude smaller than its quadrature and its
related red bars are barely seen when compared to the dark blue and cyan bars.

In Fig. S1 c1) and c2) we evaluate the consequences of PTh backaction on the
mechanical mode’s linewidth and frequency, respectively, for several different
absorption layers, with
$t_{\text{abs}}=$15\text{\,}\mathrm{,}$\,$62.5\text{\,}\mathrm{,}$\,$110\text{\,}\mathrm{nm}$$
and also for the case of homogeneously distributed absorption in the GaAs
membrane. The total surface and volume contributions are appreciably altered
by the absorptive layer thickness, remarkably the overall response is
essentially unaffected. This is understood under the light of the results of
Fig. S1 b). Since only low-order thermal modes are important for the device’s
overall response and those are essentially homogeneous over the
electromagnetic mode’s confinement region, their response is unaffected due to
the presence of an absorption layer. As the surface and volume forces are
still appreciable for high-order modes – and those display rapid spatial
oscillations –, their response is significantly modified by localized
absorption. We conclude that the differences verified in the navy/cyan curves
in Fig. S1 c1) and c2) are due to high-frequency thermal modes that barely
contribute to the total response of the system.

Figure S1: (a) Schematic of absorbing and non-absorbing regions of the GaAs
microdisk. The variable $t_{\text{abs}}$ defines the thickness of the
absorbing layer. (b) PTh force per photon (real and imaginary components) for
each of the first 200 thermal modes, here separated by frequency. (c1)
Mechanical linewidth and (c2) frequency variations as a function of the
normalized detuning $\Delta/\kappa$ for different $t_{\text{abs}}$. From
lighter to darker colors: $t_{\text{abs}}=$15\text{\,}$$, $62.5\text{\,}$,
$110\text{\,}$, and $125\text{\,}\mathrm{nm}$ (entire GaAs membrane).

##  S5 Experimental Setup

Here we discuss the experimental setup used in our two experiments: the
characterization of static optical nonlinearities (Section S7), and the
measurement of optomechanical effects (Section S8).

Light emitted from a tunable laser source is sent into a (99/1) beam splitter.
The 99% output is directed to the main measurement setup, while the remaining
power is used in an auxiliary frequency calibration setup, shown in Fig. S2,
composed of a Mach-Zehnder interferometer and a referenced HCN gas cell. In
the main measurement setup, light passes through a phase modulator (PM), used
in the calibration of the optomechanical coupling rate; a polarization
controller (PC) and a variable attenuator (ATT1) used respectively to control
the polarization and intensity of the light that will be coupled to the
cavity.

Light is coupled in and out the microdisk resonator through a tapered fiber
loop. The loop is supported by a carefully designed parking lot surrounding
the cavity, allowing us to probe the microdisks response without ever touching
it, which would degrade both optical and mechanical quality factors. The
tapered fiber output is split between two photodetectors: a slow one, linked
to the DAQ (Data AcQuisition system) and used to probe the device’s static
optical response; and a fast one, linked to the ESA (Electrical Spectrum
Analyzer), for the measurement of high frequency fluctuations of the optical
signal related to the brownian mechanical motion in the optomechanical
resonator. The output variable attenuator, ATT2, placed right before the slow
photodetector, is used to compensate attenuation changes given by ATT1,
allowing us to maintain the electronic gain settings and incident power on the
slow detector fixed, avoiding both issues of signal saturation and detection
nonlinearities.

Figure S2: Experimental setup: PM (Phase Modulator), ATT (Attenuator), MZ
(Mach-Zehnder), HCN (HCN gas cell), PC (Polarization Controller), DAQ (Data
AcQuisition system) and ESA (Electrical Spectrum Analyzer). The gray
photodetectors have slow responses (response cut-off at
$10\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$), while the white one has a fast
response (cut-off at $800\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$).

In the frequency calibration branch of our setup, light is evenly split
between a $4\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ FSR Mach-Zehnder
interferometer (a relative frequency reference), and the HCN cell (an absolute
frequency reference), see Fig. S2. The transmission spectra of both references
are captured with the DAQ. Its features, such as the distance between peaks in
the Mach-Zehnder and the extinctions of the HCN spectrum, are used as
frequency references to construct a reliable frequency axis for the cavity’s
transmission spectrum.

The optical power is measured at the input and output of the tapered fiber
loop, allowing us to characterize the propagation losses of the fiber taper,
$-3.86\text{\,}\mathrm{d}\mathrm{B}$, which are both due to the taper non-
adiabaticity, and due to the scattering at the region coupled to the cavity.
Considering the loss is approximately symmetric in the tapered fiber, that is,
the losses between the coupling region and the input and output regions are
approximately equal, we can estimate the incident power at the cavity region
from the input power in the fiber.

##  S6 Fitting of the nonlinear optical response

Our model predicts the photothermal effects given the absorption rate is known
as a function of the cavity internal energy (number of circulating photons).
Multiple linear and nonlinear processes contribute to absorption in typical
semiconductor platforms, most of them are hard to determine theoretically,
such as surface induced linear/two-photon absorption. Given such difficulty,
we experimentally characterize the absorption rate. It is important to stress
that we are not interested in carefully discriminating between the
contributions of different nonlinear optical effects, but are rather focused
on obtaining the heating intensity as a function of the cavity internal
energy. In that sense, a phenomenological interpretation of our fitting is
appropriate.

We perform measurements of the optical transmission spectrum as a function of
the incident power on the cavity. For each incident power, we extract data
points containing the optical frequency shift, the optical loss variation, and
the cavity internal energy at resonance. Lastly, we concurrently fit such data
points using a polynomial model that accounts for different kinds of optical
nonlinearities.

All data and scripts discussed in this section can be accessed in Ref. Primo
_et al._ (2021).

###  S6.1 Data collection

Once the fiber loop is coupled to the cavity and propagation losses are
calibrated, the measurements are automated. The attenuation in ATT1 is varied
from $28\text{\,}\mathrm{d}\mathrm{B}$ to $8\text{\,}\mathrm{d}\mathrm{B}$ in
steps of $1\text{\,}\mathrm{d}\mathrm{B}$. For each attenuation, the laser
frequency is swept $5\text{\,}\mathrm{n}\mathrm{m}$ around the optical
resonances, which are centered about $1565\text{\,}\mathrm{n}\mathrm{m}$; the
transmission is then recorded and saved. The entire experiment took
approximately $7\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$, or
$19\text{\,}\mathrm{s}$ per power level.

###  S6.2 Extracting lumped parameters from the optical transmission spectra

In Fig. S3 a) we display the normalized transmission spectrum of our microdisk
for the second lowest incident power, it is centered around the optical
doublet used in all the experiments reported in the main text. The frequency
axis is the same as in Fig. 4 in the main text, its definition is
$\Delta^{\prime}/(2\pi)=\omega/(2\pi)-\omega_{0}/(2\pi)$, where $\omega$ is
the absolute frequency and
$\omega_{0}=$191.5418\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$$ is
approximately the frequency of the higher frequency local minimum of the
transmission for the low incident power data, as seen in Fig. S3 a) .

In microdisks, doublets are present due to a scattering-induced coupling
between degenerate clockwise and counter-clockwise propagating modes. We fit
this transmission spectrum to a model given by two equal lorentzian extinction
curves:

$T(\Delta)=1-\frac{2\kappa_{i}\kappa_{e}}{4\Delta^{2}+\kappa^{2}}-\frac{2\kappa_{i}\kappa_{e}}{4(\Delta+\delta)^{2}+\kappa^{2}}.$
(S72)

From the fit we can obtain the cold-cavity parameters describing the doublet:
the splitting between the resonances
($\delta/(2\pi)=$11.3\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$), the intrinsic
losses ($\kappa_{i}/(2\pi)=$7.3\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$), and
the extrinsic losses
($\kappa_{e}/(2\pi)=$4.2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$, related to
the coupling to the tapered fiber). It is also useful to define the total
loss, $\kappa=\kappa_{i}+\kappa_{e}$.

Figure S3: (a) Fit of the cold-cavity spectrum,
$P_{in}=$10\text{\,}\mathrm{\SIUnitSymbolMicro W}$$. (b) Comparison of the
optical transmission between $P_{in}=$10\text{\,}\mathrm{\SIUnitSymbolMicro
W}$$ (light red curve) and $P_{in}=$399\text{\,}\mathrm{\SIUnitSymbolMicro
W}$$ (dark red curve), the markers indicate the transmission minima, the
frequency shift and the extinction variation are indicated in the graph.
Inset: Zoom-out of the same graph. (c) Measured frequency shift as a function
of incident power for the first (circles) and second (squares) resonances of
the doublet. (d) Measured extinctions as a function of incident power for the
first (circles) and second (squares) resonances of the doublet.

As the incident power increases, nonlinear effects change both the frequency
and the linewidth of the optical doublet, giving rise to transmission spectra
such as the red curve in Fig. S3 b). Results from the fit of such high power
transmission spectra are unreliable, even for a simple model where nonlinear
effects are polynomial functions of the internal energy. For that reason, we
follow a method similar to the one used in Barclay  _et al._ (2005) where the
nonlinear effects are extracted from the transmission local minima, indicated
by the circle and square markers in Fig. S3 b). For our analysis we collect
the extinction (depth), $E(P_{in})$, and the frequency,
$\delta\omega_{E}(P_{in})$, of the local minima for all transmission spectra.
In Fig. S3 c) we plot the variation of $\delta\omega_{E}(P_{in})$ while in
Fig. S3 d) the extinction.

In order to extract information regarding the nonlinear effects we make some
assumptions for the case of an optical doublet: 1) the frequency splitting
between the two resonances and the extrinsic loss, are independent of the
cavity internal energy and 2) nonlinear effects are approximately equal for
both resonances. Using such assumptions and the cold cavity parameters, the
transmission becomes:

$T(\Delta)=1-\frac{2(\kappa_{i}+\kappa_{nl})\kappa_{e}}{4(\Delta+\delta\omega)^{2}+(\kappa+\kappa_{nl})^{2}}-\frac{2(\kappa_{i}+\kappa_{nl})\kappa_{e}}{4(\Delta+\delta+\delta\omega)^{2}+(\kappa+\kappa_{nl})^{2}},$
(S73)

where $\kappa_{nl}$ and $\delta\omega$ are the, yet unknown, nonlinear loss
and nonlinear dispersion.

Considering that the cold-cavity parameters are known from the fit and that
$\delta\omega$ simply shifts the frequency axis, the extinction of the local
minima, $E$, in Eq. S73 are a function only of the nonlinear loss,
$E(\kappa_{nl})$. As the cavity-taper coupling remains in the undercoupled
regime throughout all measurements, we can restrict our analysis to a range in
which this function is bijective. Inverting it we use the extinction data in
Fig. S3 d) to obtain the nonlinear losses.

With the nonlinear losses we can determine the internal energy of the cavity
at the extinction minima. In order to simplify our calculations we consider
that the energy at the extinction minima is approximately equal the energy at
the resonance points ($\Delta=0$ and $\Delta=-\delta$):

$\displaystyle U_{1}$ $\displaystyle=$ $\displaystyle
U_{2}=\frac{2\kappa_{e}}{(\kappa_{i}+\kappa_{nl})^{2}}P_{in}+\frac{2\kappa_{e}}{4\delta^{2}+(\kappa_{i}+\kappa_{nl})^{2}}P_{in}$
(S74)

As our doublet is nearly resolved this approximation is good, the difference
between such points is of approximately
$\Delta_{E}-\delta\omega~{}$1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$, much
smaller than both the linewidth and the splitting of the resonances, rendering
an error smaller than $1\%$ at the final cavity energy.

For the determination of the nonlinear dispersion we approximate the variation
of the extinction minima, $\Delta_{E}(P_{in})$, Fig. S3 d), by the variation
of the resonance frequency itself, $\delta\omega$ . Our approximation is good,
rendering an error smaller than $250\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$
for the higher incident power. This happen because of two reasons, 1) we
assumed $\delta$ is fixed and 2) the nonlinear loss variation is small when
compared to $\kappa$ and $\delta$.

Since we assume that the linear losses and the nonlinear effects are equal for
both modes of the doublet, data used in our analysis is obtained through an
averaging process of the information collected for each of the modes in the
doublet given an incident power.

It is worth mentioning that in our determination of the nonlinear effects we
took the lowest power measurement as a frequency reference, leading to two
issues: 1) despite being small, the nonlinear dispersion/losses are different
from zero in our reference, and 2) the extinction data for low optical powers
is considerably noisy, see Fig.S3c), such that we obtain negative values for
the nonlinear loss for some of the experimental points, indicating an
uncertainty at the cold-cavity losses. Both these issues can be solved with
the introduction of offsets, $\delta\omega_{\text{nl,0}}$ and
$\kappa_{\text{nl,0}}$ in our nonlinear model.

###  S6.3 Fitting of the polynomial model

In the fitting process we consider a polynomial model for both the nonlinear
losses and the nonlinear dispersion. First we fit the nonlinear losses, and
then we use this information in the fitting of the nonlinear dispersion. This
method is efficient since in our model we consider only the nonlinear
dispersion due to the thermo-optic effect.

In our model we consider nonlinear losses scaling both linearly and
quadratically with the internal energy, $U$:

$\kappa_{nl}(U)=\kappa_{\text{nl,0}}+\kappa^{\prime}_{1}U+\kappa^{\prime}_{2}U^{2}.$
(S75)

The free parameter, $\kappa_{\text{nl,0}}$, is an offset to the nonlinear
losses used to correct the cold cavity intrinsic loss. The first order
nonlinear loss coefficient, $\kappa^{\prime}_{1}$, is related to two-photon
absorption (TPA). It is calculated through FEM simulations where the material
properties in Krishnamurthy  _et al._ (2011) are applied to the model present
in Barclay  _et al._ (2005). The second order nonlinear loss coefficient,
$\kappa^{\prime}_{2}$, is added as a phenomenological free parameter.

First order nonlinear effects (related to TPA) are reliably negligible with
respect to the quadratic effects, as seen in Fig. 4d) where the nonlinear loss
displays a parabolic shape. This has also been verified by allowing the
coefficient $\kappa^{\prime}_{1}$ to vary up to one order of magnitude during
the fitting process, yielding remarkably similar results to those obtained
using the FEM-estimated $\kappa^{\prime}_{1}$.

With the knowledge of the nonlinear losses, we proceed to fit the nonlinear
dispersion, given by:

$\delta\omega_{nl}=\delta\omega_{\text{nl,0}}+G^{\theta}_{s}R^{\theta}_{s}\Big{(}\kappa_{\text{abs}}+\eta_{nl}\big{(}\kappa^{\prime}_{1}U+B\kappa^{\prime}_{2}U^{2}\big{)}\Big{)}U.$
(S76)

The free parameters in this step are $\delta\omega_{\text{nl,0}}$, the
nonlinear dispersion offset, $\kappa_{abs}$, the absorption loss rate, and
$\eta_{nl}$, the absorptive fraction of the nonlinear losses. The coefficient
$G^{\theta}_{s}R^{\theta}_{s}$ is the product of the thermo-optic dispersion
by the thermal resistance, for static heating sources, it is calculated with
FEM simulations. We choose to neglect the change on the thermal resistance due
the nonlinear losses, because simulations show that for the static case it is
negligible.

As seen in Fig. 4 c) and d), our model fits well the experimental data, with
$\kappa_{\text{nl,0}}=$-1\pm 0.1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$,
$B=$1\pm 0.1\text{\,}\mathrm{n}\mathrm{s}$$,
$\delta\omega_{\text{nl,0}}=$-5.8\pm
0.4\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$, $\kappa_{\text{abs}}=$5.05\pm
0.006\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ and $\eta_{nl}=$0.83\pm
0.01\text{\,}$$. As a final test of the validity of our model we ”simulate”
our experiment by solving the transcendental equation for the internal energy
of the cavity. We use the cold cavity parameters and our nonlinear model,
obtaining the transmission as a function of the laser detuning. The agreement
between the experimental transmission spectra and its ”simulated” counterpart,
Fig. 4 b), is nearly perfect.

##  S7 Optical and mechanical mode simulations - Fabricated device

###  S7.1 Optical mode identification

In order to identify the optical mode that was excited in our experiment, we
analysed the optical spectrum of our microdisk as shown in Fig. S4 a). Due to
the symmetry of our device, modes can be classified through an azimuthal modal
number $m$. Using a reference mode with frequency $\omega_{0}$ and modal
number $m_{0}$, optical frequencies may be conveniently described as Fujii
_et al._ (2020):

$\omega_{\mu}=\omega_{0}+\mu
D_{1}+\frac{1}{2}\mu^{2}D_{2}+\frac{1}{6}\mu^{3}D_{3}+\ldots,$ (S77)

where $\mu=m-m_{0}$, $D_{1}/(2\pi)$ is the Free Spectral Range (FSR) and the
$D_{n}/(2\pi)$, $n>1$, give the FSR dependency on $\mu^{n-1}$.

Figure S4: (a) Measured optical spectrum for the fabricated GaAs microdisk.
(b) Schematic diagram of the device, with
$R=$5.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$R_{\text{ped}}=$0.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The measured
$6$-th order TE optical mode is shown. (c1) Simulation for the optical mode’s
frequencies as a function of the modal number $m$ for TE modes. (c2) Residual
dispersion for TE modes. Analogous results for the TM counterparts are shown
in (d1) and (d2).

For the measured optical modes FSR $\approx$3\text{\,}\mathrm{THz}$$. We
compare this value with simulations that incorporate both geometric and
material dispersions for our device. Our measurements are found to be
consistent with the $6$-th radial order TE optical mode shown in Fig. S4 b).
In Fig. S4 c1) (d1) the $m$ dependency with the frequency is plotted for TE
(TM) optical modes between $180$ and $210\text{\,}\mathrm{THz}$. For
completeness, in Fig. S4 c2) (d2) we also show the residual dispersion
$D_{\text{int}}=\omega_{\mu}-\omega_{0}-\mu D_{1}$ for the simulated TE (TM)
modes. Results are summarized in Table 2.

Mode |  $D_{1}/(2\pi)$ (THz) |  $D_{2}/(2\pi)$ (GHz)
---|---|---
TE1 | 2.61 | -8.0
TE2 | 2.70 | -10.0
TE3 | 2.78 | -12.0
TE4 | 2.85 | -13.0
TE5 | 2.92 | -16.0
TE6 | 2.99 | -18.0
TE7 | 3.07 | -22.0
TM1 | 1.95 | -9.0
TM2 | 2.05 | -15.0
TM3 | 2.16 | -21.0
Table 2: $D_{1}$ and $D_{2}$ for each of the optical modes in Figs. S4 (c2)
and (d2).

###  S7.2 Mechanical anisotropy

We incorporate the known mechanical anisotropy of GaAs in a 3D numerical model
of the microdisk. This is done by correctly prescribing values for the
stiffness tensor coefficients used in our simulations: (using Voigt notation)
$c_{11}=$119\text{\,}\mathrm{GPa}$$, $c_{12}=$53.4\text{\,}\mathrm{GPa}$$, and
$c_{44}=$59.6\text{\,}\mathrm{GPa}$$ Balram  _et al._ (2014). The symmetries
of the system allow for correct calculations of the acoustic eigenmodes by
considering only 1/4 of the full structure. All overlap integrals between
optical, thermal, and mechanical modes were carefully evaluated by considering
isotropic models for the thermal and optical responses. The former are
calculated from 2D axisymmetric simulations for the device, which are then
extruded into fully 3D fields. Only by taking in account the GaAs anisotropy
we were able to reach an agreement between the simulation and the experimental
data for the single photon optomechanical coupling,
$g_{0}/2\pi=$29\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$.

##  S8 Optomechanical response

In this section we discuss in detail how the experimental data on the
optomechanical response was collected, analyzed and compared with the results
expected from our model.

Figure S5: (a) Extinction of the optical transmission as a function of the
incident power, for both the optical nonlinear characterization, or fast
experiment (red dots); and for the optomechanical characterization, or slow
experiment (blue dots). (b) Extrinsic loss during the slow experiment. (c) PSD
with fit. (d) Optical transmission collected during the measurement of the PSD
map. (e) PSD peak as a function of the optical detuning. (f) Mechanical
linewidth variation as a function of the optical detuning. In
graphs(c),(d),(e) and (f) the incident power is
$126\text{\,}\mathrm{\SIUnitSymbolMicro W}$. The brown shade in the graphs
indicates the higher optomechanical transduction.

###  S8.1 Data collection

For greater precision, all optomechanical measurements were performed by fine-
tuning the laser frequency with a piezoelectric actuator. This choice limits
the measurable optical frequency span to $30\text{\,}\mathrm{GHz}$ around a
center frequency, which is chosen to render the peak modification on the
mechanical linewidth within the measured range. Furthermore, the thermally-
induced optical bistability restricts our experiments to the blue side of the
optical resonance for the higher incident powers used in this work.

In this experiment we use a fast photodetector, with a
$800\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ bandwidth, to measure the
mechanical fluctuations induced in the optical signal. The electrical signal
of the photodetector is monitored with an Electrical Spectrum Analizer (ESA),
which returns the Power Spectral Density (PSD) of its input signal. Fig.4 f)
in the main text is an example of a PSD where the fluctuations due to the
fundamental breathing mode of our microdisk clearly rise above the detector’s
noise floor. As the optical fluctuations are very weak, each ESA measurement
must be integrated for some seconds.

For each attenuation a regular, static, optical transmission spectrum is
measured using the standard laser frequency sweep – as described in section S6
–, and a PSD map varying the laser frequency with the piezoelectric actuator,
Fig.4e). A PSD map is composed of 100 PSD taken at equally spaced laser
frequencies inside the $30\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ piezo
range. The laser spends approximately $3\text{\,}\mathrm{s}$ in each
frequency, in such a way that building a PSD map for one attenuation takes
approximately $5\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$ and
$30\text{\,}\mathrm{s}$. The attenuation is varied from
$20\text{\,}\mathrm{d}\mathrm{B}$ to $12\text{\,}\mathrm{d}\mathrm{B}$ in
steps of $1\text{\,}\mathrm{d}\mathrm{B}$. The entire experiment takes around
$1\text{\,}\mathrm{h}$.

As this experiment is considerably longer than the previous one, the tapered
fiber position drifts moderately with respect to the microdisk, changing the
extrinsic loss of our optical resonances. This is clearly seen in Fig. S5 a)
where the extinctions of the optical spectra taken in the optomechanical
characterization (slow experiment) and in the optical nonlinear
characterization (fast experiment) are compared. For lower incident powers the
extinctions of both are similar, but as the $P_{in}$ increases the extinction
of the slow experiment becomes smaller than the one in the fast experiment. As
the slow experiment is performed just after the fast one, this indicates that
the taper position slowly drifts closer to the cavity as a function of time,
increasing its optical coupling to the cavity.

We use the extinction information to correct for the extrinsic loss. Since the
static optical transmission is recorded for each of the input powers in the
fast experiment, we are able to recover its resonance frequency shift and
compare it to the data obtained in the slow experiment. This gives us direct
knowledge of the energy (at the extinction minima) in the resonator during the
fast experiment. We use this information to determine the nonlinear loss, in
such a way that the extinction becomes now a function only of the extrinsic
loss. In a similar way to what was previously done for the determination of
the nonlinear loss, we derive a numerical bijective relation between the
extrinsic loss and the extinction, which we can invert to obtain the extrinsic
loss, as shown in Fig. S5b). This estimate of the extrinsic loss during the
optomechanical characterization is applied to our model for the photothermal
effects.

###  S8.2 Mechanical linewidth variation

Fitting the PSD to a model taking into account the mechanical susceptibility
and a constant noise background, as in Fig. S5 c), we can determine the
mechanical frequency and linewidth as a function of the laser frequency.

In order to study photothermal effects we analyzed the mechanical linewidth
variation. In Fig. S5 f) two distinct behaviors are verified. For
$\Delta^{\prime}$ between $10\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ and
$-12\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ (clear region), the mechanical
linewidth varies smoothly and presents very small uncertainty bars, while for
$\Delta^{\prime}<$-12\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ (shaded
region), uncertainty increases considerably. As seen in Fig. S5 d), the clear
region is found to match the blue detuned lateral of the optical resonance,
where the high optomechanical transduction, shown in Fig. S5 e), leads to a
large signal to noise ratio (SNR) in the measurement of the mechanical
linewidth. Due to its larger precision, we focus our analysis on the clear
region. The shaded region is related to the transmission minima, where the
optomechanical transduction approaches zero, leading to a low SNR in the PSD.

Feeding our model for the photothermal effects with the optical
nonlinearities, and with the extrinsic loss estimate, we can predict effects
such as the mechanical linewidth variation. Fig.4g) and h) of the main text
show our predictions as a function of both the laser detuning and the incident
power at the cavity, matching very well with the experimental data. The value
of the parameters fed to our model are listed in Tab.1 as well as a brief
explanation of how they were obtained. Here we must stress that, differently
from other works, we do not fit the optomechanical data, instead we predict it
independently using FEM simulations and the characterization of the optical
nonlinearities in the resonator. As discussed in the main text, the use of
multiple thermal modes is crucial to the level of accuracy obtained here.

The geometrical dimensions ($R=$5.1\pm 0.1\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ and $R_{\text{ped}}=$0.6\pm 0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$)
of our device are roughly determined from optical microscopy data. The value
for the radius used in all simulations is fine-tuned matching the mechanical
frequency from FEM simulations with the experimental mechanical frequency.

###  S8.3 Mechanical frequency shift

As mentioned in the main text, the mechanical frequency modification is
dominated by thermalization of the cavity. Qualitatively, the heat generated
by the optical absorption softens the GaAs membrane, which in turn downshifts
the mechanical frequencies. This competes with the positive mechanical
frequency shift predicted due to optomechanical backaction (for blue
detunings), as shown in Fig. 3 (a) of the main text. Plots for the measured
transmission spectrum and mechanical frequency shift $\delta\Omega$ are found
in Figs. S6 (a) and (b), where one readily observes that the trend followed by
$\delta\Omega$ strikingly contrasts with the one followed by $\delta\Gamma$
(Fig. 4 (g), main text). In fact, the transmission and $\delta\Omega$ display
similar patterns. This is a direct consequence of the increase in the
circulating power when the laser approaches the optical resonance, increasing
the overall temperature in the resonator.

Figure S6: (a) Measured optical spectrum for the fabricated GaAs microdisk
under an excitation of $126\text{\,}\mathrm{\SIUnitSymbolMicro W}$. (b)
Mechanical frequency modification for the same incident power.

Dataset:FEM and scripts files for generating each figure are available at Ref.
Primo  _et al._ (2021).

## References

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  * Note (1) Here, we define the stationary cycle period as the time necessary for the strain to return to its initial state.
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# Search for Supersymmetry in $\textrm{p}\overline{\textrm{p}}$ Collisions at
$\sqrt{\textrm{s}}$ =1.96 TeV Using the Trilepton Signature of Chargino-
Neutralino Production

R. Forrest  Department of Physics, University of California, Davis, Davis, CA
95616, USA

###### Abstract

The production of chargino-neutralino pairs and their subsequent leptonic
decays is one of the most promising supersymmetry (SUSY) signatures at the
Tevatron $p\bar{p}$ collider. We present here the most recent results on the
search for the three-lepton and missing-transverse-energy SUSY signature using
data collected with the CDF II detector. The results are interpreted within
the minimal supergravity (mSugra) scenario.

##  I Introduction

In the search for new phenomena, one well-motivated extension to the Standard
Model (SM) is supersymmetry (SUSY). SUSY particles (sparticles) contribute to
the Higgs mass squared with opposite sign relative to the contributions of SM
particles, and thus protect the weak mass scale, $M_{W}$, from divergences.
SUSY is a broken symmetry since the sparticles obviously do not have the same
mass as their SM partners, but the breaking must be ‘soft’ to allow the
divergence canceling to remain effective. If $R_{p}$ parity is
conserved111$R_{p}=(-1)^{3B+L+2S}$, where $B$ is baryon number, $L$ is lepton
number, and $S$ is spin., the lightest SUSY particle (LSP) is absolutely
stable and provides a viable candidate for cosmological dark matter
susy_primer . We use as a reference the mSugra model of SUSY breaking. This
model has the virtue of containing only five free parameters to specify.
However, our search is signature-based; we do not modify our selection to
follow the details of mSugra.

One very promising mode for SUSY discovery at hadron colliders is that of
chargino-neutralino associated production with decay into three leptons.
Charginos decay into a single lepton through a slepton

$\tilde{\chi}_{1}^{\pm}\rightarrow~{}\tilde{l}^{(*)}~{}\nu_{l}\rightarrow\tilde{\chi}_{1}^{0}~{}l^{\pm}~{}\nu_{l}$

and neutralinos similarly decay into two detectable leptons

$\tilde{\chi}_{2}^{0}\rightarrow~{}\tilde{l}^{\pm(*)}~{}l^{\mp}\rightarrow\tilde{\chi}_{1}^{0}~{}l^{\pm}~{}l^{\mp}.$

The decays can also proceed via $W$ and $Z$ bosons. The detector signature is
thus three SM leptons with associated missing energy from the undetected
neutrinos and lightest neutralinos, $\tilde{\chi}_{1}^{0}$ (LSP), in the
event. Due to its electroweak production, this is one of the few ‘jet-free’
SUSY signatures.

##  II Detector, Data and Analysis Overview

This analysis is preformed with the CDF II detector at the Tevatron with
$p\bar{p}$ collisions at $\sqrt{s}=1.96$ TeV. The CDF II detector is a mostly
cylindrical particle detector composed of cylindrical sub-detectors. From the
beam axis outwards there is a silicon strip vertex detector, and a gas filled
drift chamber. This tracking system is surrounded by a solenoid providing a
1.4 T magnetic field, followed by electromagnetic and hadronic calorimeters.
The outermost detectors are wire chambers used to detect muons that escape the
inner detectors.

For this analysis we use two categories of event triggers. The first is the
high $P_{t}$ inclusive lepton trigger, which consist of single lepton objects,
the second is the SUSY dilepton trigger witch allows two lower $P_{t}$
leptons. These data are combined and in the analysis overlapping trigger
effects and efficiencies are accounted for. These data were collected up until
1 Jul, 2008, totaling $3.23~{}\textrm{fb}^{-1}$ for the unprescaled triggers.

We follow the same analysis strategy and implementation used in the previous
CDF II search rut_note . From the outset, we define lepton categories and
event level trilepton channels. Each lepton and category is exclusive and
selected based on expected purity. This channel independence allows easy
statistical combination of the final results.

The general procedure is as follows. For each event, we select muons,
electrons and tracks of some quality. Each of these objects, except the tracks
(T), have tight (t) and loose (l) categories. We then define event level
exclusive trilepton channels composed of combinations of these objects and
arrange them sequentially by expected signal sensitivity. There are several
virtues of this approach. The largest advantage is that we perform several
lepton flavor, channel-specific searches simultaneously, without the need to
account for overlapping results.

We define two selection stages to test our background estimations against
data. The first stage is the dilepton selection, which consists of the first
two objects of the trilepton selection. The second stage is the final
trilepton selection, with some event cuts applied. Once we are satisfied with
the agreement in the control regions, we apply SUSY specific cuts and look at
signal region data to compare against background.

##  III Object Selection, Event Categories and Cuts

We define both tight and loose lepton categories as well as a track object.
All of these objects are central to the detector, meaning that generally
$|\eta|<1.0$ and they are isolated from nearby objects. Tight muons are
objects that have tracks, deposit a minimum amount of ionizing energy in the
calorimeter system, and are detected in the outer muon systems. Loose muons
are similar, but the requirement of the muon system detection is relaxed; they
compensate for gaps in the muon detector coverage. Tight electrons are
similarly again required to leave a good track, but they are expected to
deposit a majority of their energy in the electromagnetic calorimeter. Loose
electrons have slightly fewer requirements on the matching between objects in
the sub-detectors. We also include one type of track object in the analysis as
a possible third object. This greatly increases our sensitivity by allowing
detection of leptons that failed selection cuts, as well as single pronged
hadronic tau decays. The track object is a single, isolated track in the
tracking chamber. It differs from loose muons in that, it can have an
arbitrary amount of energy deposition.

After the object selection, we categorize the trilepton events. We first look
for three tight leptons. If the event does not qualify, we look for two tight
and one loose lepton. If the event still does not qualify, we look for one
tight and two loose leptons. Events that do not make it into the trilepton
selection are tested for two tight leptons and a track and finally one tight,
one loose and one track object. The complete list is shown in Table 1 along
with the $E_{t}$ (electrons) or $P_{t}$ (muons) requirements on these objects
222Some selections differ slightly to increase sensitivity or accommodate
standard object definitions, see rut_note ..

Channel | Selection |  $E_{t}$ or $P_{t}$
---|---|---
ttt | 3 tight leptons | 15, 5, 5
ttC | 2 tight and 1 loose lepton | 15, 5, 5
tll | 1 tight and 2 loose leptons | 20, 8, 5
tt$T$ | 2 tight leptons and 1 track | 15, 5, 5
tl$T$ | 1 tight lepton, 1 loose lepton and 1 track | 20, 8, 5

Table 1: Trilepton selection event categories.

At this stage we apply additional event level cleaning cuts. We require that
every analysis level object (leptons, tracks and jets) be separated from each
other by $\Delta R>0.4$. Events with a mismeasured jet can have false
$\,/\\!\\!\\!\\!E_{T}$. We remove events with $\,/\\!\\!\\!\\!E_{T}$ and any
jet separated by less than $\Delta\phi<0.35$. We also make invariant mass cuts
at this stage. The highest opposite signed object pair invariant mass is
required to be above $20\ \textrm{GeV}/c^{2}$ and the second highest
oppositely-charged object pair is required to be above $13\
\textrm{GeV}/c^{2}$. This cut helps eliminate heavy flavor backgrounds.

Additional backgrounds due to mismeasurement are removed by cutting events
that have $\,/\\!\\!\\!\\!E_{T}$ and leptons aligned, requiring
$\Delta\phi>0.17$ for each of the leading two leptons.

To further clean up events, we require the third lepton in trilepton events to
be isolated. We also require that there not be more than three leptons or
tracks in the event above 10 GeV and that the three objects’ charges sum to
$\pm 1$.

###  III.1 Backgrounds

The standard model background estimation for the analysis differs slightly
between the lepton+track channels and the trilepton channels. Generally, Monte
Carlo is used to estimate the backgrounds, but isolated track, fake lepton and
gamma conversion rates are determined from data.

###  III.2 Trilepton Backgrounds

Backgrounds are treated differently based on the underlying process. Those
that give three real leptons (WZ , ZZ, $t\bar{t}$) are estimated with Monte
Carlo by simply taking them through the analysis.

The remaining background processes have two real leptons (Z, WW) and require a
third object from elsewhere in the event. This can happen, for example, with
FSR photon conversion where a photon radiated off a charged particle hits
matter in the detector and converts to an $e\bar{e}$ pair. For these
processes, we estimate this 2 lepton plus conversion rate from Monte Carlo.

The final contribution to the trilepton background is from objects in the
underlying event faking a third lepton in an event that has two genuine
leptons. This fake contribution is estimated in the trilepton channels by
selecting two well identified leptons and a third fakeable object from data
events. Fake rates have been measured for jets faking electrons and for tracks
faking muons of both tight and loose quality. These jets and tracks are the
fakeable objects selected. The event is then carried through the analysis and
weighted by the appropriate fake rate.

###  III.3 Dilepton + Track Backgrounds

For channels with tracks, backgrounds are handled slightly differently.
Background processes that give three real leptons (WZ , ZZ, $t\bar{t}$) are
still estimated from Monte Carlo as previously described.

As for fakes in dilepton + track channels, we account for fake leptons, and
separately estimate the rate of isolated tracks in dilepton backgrounds.

For fake leptons, we use a method similar to the trilepton method but
calculated from data by selecting lepton + track events containing a fakeable
object. As was done with trilepton fakes, we carry the event through the
analysis, and apply the appropriate fake rate to the event.

The remaining background in the dilepton + track channels is that of dilepton
events with an isolated track from the underlying event. We measure the rate
of extraneous isolated tracks from data, and apply this rate to dilepton Monte
Carlo. This procedure gives very good agreement in our dilepton + track
control regions.

##  IV Control Regions

We inspect both our dilepton selection and our trilepton selection for
agreement against predictions. The control region parameter space is
$\,/\\!\\!\\!\\!E_{T}$ vs. Invariant mass, and for easy reference is coded
according to Figure 1.

Figure 1: Control regions and codes used to refer to the control regions.

We select the first two leptons in the event and check agreement against
backgrounds. See Figure 2 for a complete listing of all the dilepton control
regions. A dilepton kinematic plot is displayed and described in Figure 3.

Figure 2: Summary of dilepton control regions. (Observed - Expected) /
Expected number of events for each control region. Figure 3: Invariant mass of
the first two tight leptons in events with low $\,/\\!\\!\\!\\!E_{T}$.

After we are satisfied with the dilepton control region agreement, the
trilepton selection is applied to an event. We check trilepton plots and
tables to ensure good agreement between background and predictions. The total
trilepton background and prediction comparison is shown in Figure 4.

Figure 4: Trilepton Control Region Summary. (Observed - Expected)/Expected
number of events.

We again look at distributions comparing data and predictions in control
regions. Trilepton control region plots are shown in Figure 5.

Figure 5: Energy of leading lepton in events with low $\,/\\!\\!\\!\\!E_{T}$
in the ttt channel.

##  V Results and Limits

For an mSugra reference point we use
$\textrm{M}_{0}=60,\textrm{M}_{1/2}=190,\textrm{tan}\beta=3,\textrm{A}_{0}=0$;
the results of background and expected signal are shown in Table 2. After
looking at the signal region in the data, we see a total of seven signal
events on an expected background of $10.84\pm 1.34$ events.

CDF II Preliminary, 3.2 $\textrm{fb}^{-1}$
---
Channel | Total Background $\pm$ (stat) $\pm$ (sys) | Signal Point $\pm$ (stat) $\pm$ (sys) | Observed
ttt | 0.83 $\pm$ 0.14 $\pm$ 0.11 | 3.64 $\pm$ 0.22 $\pm$ 0.49 | 1
ttC | 0.39 $\pm$ 0.07 $\pm$ 0.04 | 2.62 $\pm$ 0.18 $\pm$ 0.35 | 0
tll | 0.25 $\pm$ 0.08 $\pm$ 0.03 | 1.12 $\pm$ 0.12 $\pm$ 0.15 | 0
ttT | 5.85 $\pm$ 0.57 $\pm$ 1.11 | 7.15 $\pm$ 0.31$\pm$ 0.91 | 4
tlT | 3.53 $\pm$ 0.52 $\pm$ 0.5 | 4.06 $\pm$ 0.23 $\pm$ 0.53 | 2
mSugra Signal point:
$\textrm{M}_{0}=60,\textrm{M}_{1/2}=190,\textrm{tan}\beta=3,\textrm{A}_{0}=0$

Table 2: Expected background and signal, errors are statistical and full
systematic.

To extract a 1-D 95% confidence level limit, we set $M_{0}=60$ and vary
$M_{1/2}$ which has the direct effect of varying the chargino mass. For each
point we scan, we get the expected limit based on the acceptance of our
analysis to the signal at that point. If we plot this against the theoretical
$\sigma\times BR$ of the signal mSugra point as a function of chargino mass,
we expect to exclude regions where our analysis’s $\sigma\times BR$ is less
than the theoretical value. Our expected limit is about 156 GeV/$c^{2}$ Figure
6, while we observe a limit of 164 GeV/$c^{2}$.

Figure 6: Expected and observed limit for the mSugra model
$M_{0}=60,tan\beta=3,A_{0}=0,(\mu)>0$. In red is the theoretical $\sigma\times
BR$ and in black is our expected limit with one and two $\sigma$ errors. We
expect to set a limit of about 156 GeV/$c^{2}$, and observe a limit of 164
GeV/$c^{2}$.

To explore a broader parameter space it is useful to scan both $M_{0}$ and
$M_{1/2}$ simultaneously. We calculate NLO cross section of the process as a
function of $M_{0}$ and $M_{1/2}$. We then calculate branching ratio to three
leptons in this same range. This gives us a plot of $\sigma\times BR$. We
generate signal Monte Carlo to test the expected and observed sensitivity at
many points in $M_{0}$ and $M_{1/2}$ space.

Figure 7: Expected and observed limit contours for the mSugra model
$tan\beta=3,A_{0}=0,(\mu)>0$ in $M_{1/2}$ vs $M_{0}$ space.

We calculate (Expected - Theory $\sigma\times$ BR) /(Theory $\sigma\times$ BR)
for both the expected and observed limits. The final exclusion contains both
of these contours which can be seen in Figure 7.

Our observed 1-D limit excludes chargino masses of less than $164\
\textrm{GeV}/c^{2}$, an improvement over the expectation due to the deficit of
data events in the lepton + track channels.

## References

  * (1) A Supersymmetry Primer, Stephen P. Martin, hep-ph/9709356.
  * (2) Search for Supersymmetry in $p\bar{p}$ Collisions at $\sqrt{s}$ = 1.96 TeV Using the Trilepton Signature for Chargino-Neutralino Production, CDF Collaboration, Phys. Rev. Lett. 101, 251801 (2008), DOI:10.1103/PhysRevLett.101.251801