††thanks: Authors contributed equally.††thanks: Authors contributed equally.

# SUPPLEMENTARY INFORMATION:
Cavity-controlled magneto-optical properties of a strongly coupled van der
Waals magnet

Florian Dirnberger§<EMAIL_ADDRESS>Department of Physics, City
College of New York, New York, NY 10031, USA  Jiamin Quan§ Department of
Physics, The Graduate Center, City University of New York, New York, NY 10016,
USA  Photonics Initiative, CUNY Advanced Science Research Center, New York,
NY, 10031, USA  Department of Electrical Engineering, City College of the City
University of New York, New York, NY, 10031, USA  Rezlind Bushati  Department
of Physics, City College of New York, New York, NY 10031, USA  Department of
Physics, The Graduate Center, City University of New York, New York, NY 10016,
USA  Geoffrey Diederich  Intelligence Community Postdoctoral Research
Fellowship Program, University of Washington, Seattle, WA, USA  Department of
Physics, Department of Materials Science and Engineering, University of
Washington, Seattle, WA, USA  Matthias Florian  Department of Electrical and
Computer Engineering, Department of Physics, University of Michigan, Ann
Arbor, Michigan 48109, United States  Julian Klein  Department of Materials
Science and Engineering, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA  Kseniia Mosina  Zdenek Sofer  Department of
Inorganic Chemistry, University of Chemistry and Technology Prague, Technickaá
5, 166 28 Prague 6, Czech Republic  Xiaodong Xu  Department of Physics,
Department of Materials Science and Engineering, University of Washington,
Seattle, WA, USA  Akashdeep Kamra  Francisco J. García-Vidal  Departamento de
Física Teórica de la Materia Condensada and Condensed Matter Physics Center
(IFIMAC), Universidad Autónoma de Madrid, E- 28049 Madrid, Spain.  Andrea Alù
<EMAIL_ADDRESS>Department of Physics, The Graduate Center, City University
of New York, New York, NY 10016, USA  Photonics Initiative, CUNY Advanced
Science Research Center, New York, NY, 10031, USA  Department of Electrical
Engineering, City College of the City University of New York, New York, NY,
10031, USA  Vinod M. Menon<EMAIL_ADDRESS>Department of Physics, City
College of New York, New York, NY 10031, USA  Department of Physics, The
Graduate Center, City University of New York, New York, NY 10016, USA

###### Contents

  1. S1 Crystal structure
  2. S2 Optical properties of CrSBr bulk crystals: Theory and experiment
    1. S2.1 Exciton-polaritons in bulk crystals
      1. S2.1.1 Dielectric function and transfer-matrix calculations
      2. S2.1.2 Quantum model of strong light-matter coupling in bulk crystals
    2. S2.2 Enhancement of exciton-polariton states in crystals with external cavity mirrors
    3. S2.3 Emission and reflectance of crystals with and without external cavity mirrors
    4. S2.4 Polarization-dependent PL emission and reflectance of the 580 nm cavity sample
  3. S3 Theoretical analysis of the effects of coherent and incoherent magnons
    1. S3.1 Coupling of magnetic and electronic structure at zero temperature
      1. S3.1.1 Magnetic ground state with applied magnetic field
      2. S3.1.2 Exciton energy shift in a spin-canting external field
      3. S3.1.3 Exciton energy shift in the presence of coherent magnons
    2. S3.2 Coupling of magnetic and electronic structure at low temperature
      1. S3.2.1 Exciton energy shift in the presence of magnetic fluctuations
      2. S3.2.2 Exciton energy shift in the presence of incoherent magnons
      3. S3.2.3 Exciton energy shift in the presence of incoherent magnons and a saturation magnetic field
      4. S3.2.4 Exciton energy shift induced by changes in sub-lattice magnetizations for a broader temperature range
  4. S4 Experimental and numerical analysis of polariton-magnon coupling
    1. S4.1 Measurement of the coupling between polaritons and coherent magnon
    2. S4.2 Numerical analysis of polariton-magnon coupling: Coherent magnons
    3. S4.3 Measurement of the coupling between polaritons and incoherent magnon
    4. S4.4 Numerical analysis of polariton-magnon coupling: Incoherent magnons

##  S1 Crystal structure

Figure S1: Crystal structure of CrSBr.  a-c Crystal structure of CrSBr viewed
along the $a$–, $b$–, and $c$–direction, respectively. Cr atoms are depicted
in blue, S atoms in yellow, and Br atoms in red.

##  S2 Optical properties of CrSBr bulk crystals: Theory and experiment

###  S2.1 Exciton-polaritons in bulk crystals

To provide a better understanding of the complex optical properties of CrSBr
that arise from strong coupling between excitons and photons, we use a
semiclassical theory as well as a quantum-mechanical approach of the exciton-
radiation interaction to gain access to optical spectra as well as the
underlying exciton-polariton states.

####  S2.1.1 Dielectric function and transfer-matrix calculations

The semiclassical theory of exciton-photon interaction is based on a
susceptibility treatment of the optical response Claudio Andreani (1995)
within the CrSBr crystal. Maxwell’s equations are solved including the
frequency dependent material response using a tranfer-matrix formulation, from
which linear optical spectra for different cavity structures are obtained.

The optical dielectric function for excitons can be derived microscopically
using the semiconductor Bloch equations and written as Haug and Koch (2004)

$\displaystyle\varepsilon_{b}(\omega)=\varepsilon^{\infty}_{b}-\frac{\Delta_{X}}{\hbar\omega-\hbar\omega_{X}+i\gamma}$
(S1)

where $\hbar\omega_{X}$ and $\hbar/\gamma$ are the exciton energy and
lifetime, respectively. Here, we consider only the lowest exciton level, while
contributions due to all other resonances are included in a frequency-
independent background relative permittivity $\varepsilon^{\infty}_{b}$. We
note that the Elliot form (S1) only contains the resonant terms and that it is
closely connected to a Lorentz oscillator model

$\displaystyle\varepsilon_{b}(E)=\varepsilon^{\infty}_{b}-\frac{f_{X}}{E^{2}-E_{X}^{2}+i\Gamma
E},$ (S2)

by identifying $f_{X}=2\sqrt{E_{X}^{2}-\gamma^{2}}\Delta_{X}$,
$E_{X}^{2}=\hbar\omega_{X}-\gamma^{2}$ and $\Gamma=2\gamma$.

The exciton oscillator strength
$\Delta_{X}=2|d_{b}^{cv}|^{2}|\psi_{X}(\mathbf{r}=0)|^{2}$ is determined by
the transition dipole element $d_{b}^{cv}$, which has predominant contribution
along b-direction (c.f. S8). $\Psi(r)$ denotes the electron and hole relative
wave function inducing Coulomb enhancement of the optical transition. Given
the strong dielectric anisotropy, the tensorial character of the dielectric
function needs to be taken into account. Bulk CrSBr crystallizes in an
orthorhombic structure and we consider
$\varepsilon_{a}(\omega)=\varepsilon^{\infty}_{a}$ and
$\varepsilon_{c}(\omega)=\varepsilon^{\infty}_{c}$ for polarizations along a-
and c-direction, respectively.

Figure S2: Differential reflectance spectra of CrSBr bulk crystals on a
$\textrm{SiO}_{2}/\textrm{Si}$ substrate. a Layout of the CrSBr –
$\textrm{SiO}_{2}/\textrm{Si}$ structure used in the simulations. b Calculated
reflectance of a 440 nm-thick flake with (solid line) and without (dotted
line) a second, weak oscillator, $X^{*}$, at $\sim 1.38$ eV. Inclusion of
$X^{*}$ has a negligible effect on the polariton states below 1.37 eV analyzed
in the main manuscript. c–e Simulated and measured reflectance spectra of
three different CrSBr bulk crystals. Crystal thickness $d_{sim}$ is extracted
from the simulations, while $d_{exp}$ is determined by atomic force
microscopy. $E_{X}$ indicates the energy of excitons used in the simulation.

To obtain an estimate for the static dielectric tensor, spin-polarized density
functional theory (DFT) calculations are carried out using Quantum Espresso
Giannozzi  _et al._ (2009, 2017) and considering an antiferromagnetic order
along the stacking direction Wilson  _et al._ (2021). We apply the generalized
gradient approximation (GGA) by Perdew, Burke, and Ernzerhof (PBE) Perdew  _et
al._ (1996, 1997) and use optimized norm-conserving Vanderbilt pseudopotential
van Setten  _et al._ (2018) at a plane-wave cutoff of $80$ Ry. Uniform meshes
with $8\times 6\times 2$ k-points are combined with a Fermi-Dirac smearing of
$5$ mRy. We are using fixed lattice constant of a =
$3.511\text{\,}\mathrm{\SIUnitSymbolAngstrom}\text{/}$, b =
$4.746\text{\,}\mathrm{\SIUnitSymbolAngstrom}\text{/}$ and c =
$7.916\text{\,}\mathrm{\SIUnitSymbolAngstrom}\text{/}$ that are obtained from
synchrotron XRD data of CrSBr López-Paz  _et al._ (2022). Structural
relaxations were performed until the forces were smaller than
$0.005\text{\,}\mathrm{eV}\text{/}\mathrm{\SIUnitSymbolAngstrom}$. The D3
Grimme method Grimme  _et al._ (2011) is used to include van-der-Waals
corrections. We find $\varepsilon_{a}=11.5$, $\varepsilon_{b}=43.1$, and
$\varepsilon_{c}=9.1$.

Reflectance spectra of CrSBr bulk crystals on a $\textrm{SiO}_{2}/\textrm{Si}$
substrate (Fig. S2a) as well as embedded between planar mirrors (Fig. S3a) are
calculated using a general transfer matrix formulation, suitable for the
description of wave propagation through anisotropic media Rumpf (2011).
Frequency dependent refractive indices for SiO2 Gao  _et al._ (2013), Si
Schinke  _et al._ (2015), SiN Luke  _et al._ (2015) and Au Johnson and Christy
(1972) are used together with nominal layer thicknesses. The exciton optical
dielectric function is systematically determined by comparing the polariton
mode structure that is visible in the simulated and measured spectra
(indicated by vertical dashed lines in Fig. S2c-e and Fig. S3b-c) for
different CrSBr bulk crystals.

Figure S3: Differential reflectance and PL emission of CrSBr crystals embedded
between planar mirrors. a Layout of the CrSBr microcavity structure used in
the simulations. b,c Simulated reflectance and measured PL spectra of two
CrSBr cavity samples. Crystal thickness $d_{sim}$ is extracted from the
simulations, while $d_{exp}$ is determined by atomic force microscopy. $E_{X}$
indicates the energy of excitons used in the simulation.

Excellent agreement is obtained for all samples using
$\Delta_{X}=$908\text{\,}\mathrm{meV}\text{/}$$, with a typical deviation of
<5% between the crystal thickness extracted by simulations and those
determined by atomic force microscopy. Few meV shifts of the exciton
transition energy are observed from sample to sample that we attribute to
strain effects caused by the different thermal expansion of the materials
surrounding the CrSBr crystals.

In CrSBr bulk crystals on a $\textrm{SiO}_{2}/\textrm{Si}$ substrate an
additional resonance is apparent at $\sim$1.38\text{\,}\mathrm{eV}\text{/}$$,
similar to the recently observed $X^{*}$ transition Klein  _et al._ (2022a).
While it is barely visible in PL, the appearance in the differential
reflectivity measurements suggests finite oscillator strength and band related
transitions. It cannot be explained by upper polariton modes of the 1s exciton
as shown in Fig. S2b. Experimental reflectance spectra are well described by
adding an additional $X^{*}$ resonance to Eq. (S1) with an energy splitting of
the 1s exciton and the $X^{*}$ of $\Delta
E\sim$14\text{\,}\mathrm{meV}\text{/}$$. Compared to the 1s exciton resonance,
the $X^{*}$ has a significantly weaker oscillator strength of
$\Delta_{X^{*}}\sim$55\text{\,}\mathrm{meV}\text{/}$$. As a result, the
induced refractive index change has a negligible effect on the polariton
states below 1.37 eV that are analyzed in the main manuscript. The origin of
$X^{*}$ is still unresolved. It may involve momentum indirect transitions due
to the strong extension of the wavefunction along the $\Gamma-X$ direction or
transitions between the split conduction bands and the valence band Klein  _et
al._ (2022a).

The simulated polariton dispersion shown in Fig. 1D of the main text is
determined by calculating reflectance spectra as a function of the elevation
angle $\phi$, thereby changing the in-plane component of the incidence wave
vector according to $k_{\parallel}=k_{0}\cos{\phi}$. Given the strong
anisotropy of CrSBr, care has to be taken regarding the crystal orientation.
If the CrSBr crystal is oriented with the $b$–axis parallel to the slit of the
spectrometer ($0^{\circ}$) the exciton polarization axis $\bm{P_{b}}$
coincides with the momentum direction $k_{b}$ imaged by the spectrometer (c.f.
Fig. S4a), in which case the measured modes are TE polarized. By rotating the
CrSBr cavity crystal by $45^{\circ}$ (azimuth) with respect to the plane-of-
incidence (c.f. Fig. S4b), the polariton modes get mixed TE and TM
polarization, which results in a stronger polariton dispersion given that
$\varepsilon_{c}<\varepsilon_{b}$, as shown in Fig. S4c.

Figure S4: Role of crystal orientation in the measured polariton dispersion a
Upper panel: A CrSBr cavity crystal oriented with $b$–axis parallel to the
slit of the spectrometer ($0^{\circ}$). Hence, the exciton polarization axis
$\bm{P_{b}}$ coincides with the momentum direction $k_{b}$ imaged by the
spectrometer. Lower panel: The same CrSBr cavity crystal rotated by
$45^{\circ}$. A wave-plate rotates the exciton polarization axis $\bm{P_{b}}$
parallel to the slit, but the rotation of the $b$–axis with respect to the
spectrometer changes the momentum direction imaged by the spectrometer. b
Rotation of the sample images a stronger polariton dispersion, as demonstrated
by the simulated reflectance map for $0^{\circ}$ and $45^{\circ}$
configurations.

####  S2.1.2 Quantum model of strong light-matter coupling in bulk crystals

While a semiclassical theory is well suited to calculate optical spectra and
polariton modes, a quantum mechanical approach is required to characterize the
composition of the underlying polariton states. A quantum treatment of the
exciton-photon interaction is based on a microsopic theory of the coupling
between the excitons and the quantized electromagnetic modes. The
approximation of keeping only quadratic terms yields a Hamiltonian which can
be diagonalized exactly and the mixed exciton-photon modes are the polariton
modes of the system.

We consider light polarized in b-direction and assume that the CrSBr crystal
of thickness L is embedded in a microcavity with perfectly reflecting walls.
The whole system keeps the translational invariance along the plane orthogonal
to the z–direction. The dispersion of the electromagnetic modes is given by
$\omega^{c}_{\mathbf{q},q_{z}}=v\sqrt{\mathbf{q}^{2}+q_{z}^{2}}$ with a
z-dependence according to $\cos(q_{z}z)$. Here, $\mathbf{q}$ is the in-plane
wave vector, $q_{z}=\pi n/L$ with integer values $n$ and
$v=c/\sqrt{\varepsilon_{b}}$.

The interaction between exciton and photon states can be calculated by
representing the vector potential $\mathbf{A}(\mathbf{r})$ in second-
quantization Savona  _et al._ (1994):

$\displaystyle\mathbf{A}(\mathrm{r})=\sum_{\mathbf{q},q_{z}}\sqrt{\frac{2\pi\hbar
v}{\Omega(\mathbf{q}^{2}+q^{2}_{z})^{\frac{1}{2}}}}[b_{\mathbf{q},q_{z}}e^{i\mathbf{q}\cdot\rho}-b^{\dagger}_{\mathbf{q},q_{z}}e^{-i\mathbf{q}\cdot\rho}]\cos(q_{z}z)\mathbf{e}_{\mathbf{q},q_{z}}$
(S3)

where $\Omega$ is the normalization volume of the photon eigenmodes and
$\mathbf{e}_{\mathbf{q},q_{z}}$ are the unit vectors of the transversal photon
polarization.

Under weak excitation conditions and using the Coulomb gauge the exciton-
photon interaction can be written in resonant approximation as Savona  _et
al._ (1994)

$\displaystyle
H_{\text{rad}}=-i\sum_{\mathbf{q},k_{z},q_{z}}C_{\mathbf{q},q_{z},k_{z}}(b^{\dagger}_{\mathbf{q},q_{z}}X_{\mathbf{q},k_{z}}-h.c.)\,.$
(S4)

In this expression, we define $X_{\mathbf{q},k_{z}}$,
$X^{\dagger}_{\mathbf{q},k_{z}}$ as the annihilation and creation operator for
an exciton with given in-plane wave vector $\mathbf{q}$ and subband index
$k_{z}$. The exciton operator obeys Bose commutation relations. The
interaction strength is given by

$\displaystyle
C_{\mathbf{q},q_{z},k_{z}}=\frac{\omega^{X}_{\mathbf{q},q_{z}}}{c}\sqrt{\frac{2\pi\hbar
v}{L}}(q^{2}+q^{2}_{z})^{\frac{1}{2}}(\mathbf{e}_{\mathbf{q},q_{z}}\cdot\mathbf{d}^{cv}_{\mathbf{q},q_{z}})\psi_{X}(\mathbf{r}=0)I_{k_{z},q_{z}}$
(S5)

where $\omega^{X}_{\mathbf{q},q_{z}}$ is the exciton dispersion and

$\displaystyle
I_{k_{z},q_{z}}=\int^{L/2}_{-L/2}dz\rho_{k_{z}}^{cv}(z)e^{-iq_{z}z}$ (S6)

describes the overlap between the confinement function of electrons and holes
in z-direction $\rho_{k_{z}}^{cv}(z)$ and the electromagnetic mode. In the
bulk crystal limit we can assume that
$I_{k_{z},q_{z}}\approx\delta_{k_{z},q_{z}}$.

A simplified expression for the exciton-photon coupling strength can be
obtained assuming a weak $q$-dependence of the exciton dispersion and dipole
matrix elements for relevant photon momenta:

$\displaystyle C_{\mathbf{q},q_{z}}\approx
g_{0}\sqrt{\frac{\omega_{X}}{\omega^{c}_{\mathbf{q},q_{z}}}}$ (S7)

with $g_{0}=\sqrt{\frac{\hbar\omega_{X}\Delta}{2\varepsilon_{b}}}$ and
$\Delta$ the exciton oscillator strength. The bilinear exciton-photon
Hamiltonian

$\displaystyle
H=\sum_{\mathbf{q},q_{z}}\hbar\omega_{X}X^{\dagger}_{\mathbf{q},q_{z}}X_{\mathbf{q},q_{z}}+\sum_{\mathbf{q},q_{z}}\hbar\omega^{c}_{\mathbf{q},q_{z}}+H_{rad}$
(S8)

can be diagonalized by introducing polariton operators

$\displaystyle
p_{\mathbf{q},q_{z}}=u(\mathbf{q},q_{z})X_{\mathbf{q},q_{z}}+v(\mathbf{q},q_{z})b_{\mathbf{q},q_{z}}$
(S9)

as a linear combination of exciton and photon operators Hopfield (1958). The
idea is to demand the polariton operators to obey Bose commutation relations
and the coefficients $u(\mathbf{q},q_{z})$ and $v(\mathbf{q},q_{z})$ are
chosen so that the exciton-photon Hamiltonian becomes diagonal. As a result
one obtains the polariton spectrum

$\displaystyle\Omega^{UP/LP}_{\mathbf{q},q_{z}}=\frac{1}{2}(\hbar\omega_{X}+\hbar\omega^{c}_{\mathbf{q},q_{z}})\pm\frac{1}{2}\sqrt{(\hbar\omega_{X}-\hbar\omega^{c}_{\mathbf{q},q_{z}})^{2}+4C_{\mathbf{q},q_{z}}^{2}}$
(S10)

showing upper (LP) and lower (LP) polariton branches. At the crossing point
between exciton and cavity dispersion
($\hbar\omega_{X}=\hbar\omega^{c}_{\mathbf{q},q_{z}}$), the polariton
splitting is $2g_{0}$, which can be viewed as the vacuum-field Rabi splitting
referred to in the main manuscript. Based on the material parameters of CrSBr
obtained using the semiclassical approach, we can estimate a Rabi energy of
$g_{0}=~{}$120\text{\,}\mathrm{meV}\text{/}$$. The exciton and photon fraction
can be derived from the coefficients in Eq.S9 and noting that
$|u_{\mathbf{q},q_{z}}|^{2}+|v_{\mathbf{q},q_{z}}|^{2}=1$.

In general, the dispersion relation obtained by a semiclassical and quantum
mechanical approach can be seen to be equivalent provided that the complex
quantum-mechanical energies and the poles of the transmission coefficient
coincide Savona  _et al._ (1995), which is a challenge for complex cavity
designs. Thus, we rely on the experimentally measured polariton dispersions
and use (S7) and (S10) to estimate $\omega^{c}_{\mathbf{q},q_{z}}$ for each
polariton mode (cf. Fig. S5).

Figure S5: Exciton-photon hybridization in CrSBr cavities. Color-coded exciton
fraction $\bm{X}$ plotted for the different polariton branches of our 580 nm-
thick cavity sample.

###  S2.2 Enhancement of exciton-polariton states in crystals with external
cavity mirrors

Figure S6: Cavity-enhanced polariton emission. a Simulated reflectance of a
580 nm crystal enclosed by a bottom dielectric Bragg mirror comprising of
8-pairs of alternating $\textrm{SiO}_{2}$/SiN layers and a 35 nm-thin Au
mirror. b Reflectance line-cut at normal incidence ($0^{\circ}$) shows
multiple sharp polariton modes with cavity quality factors around 300. c
Respectively color-coded electric field profiles extracted for the energies
indicated in a and b. Electric fields at the energy of polariton branches are
strongly enhanced by the external cavity mirrors, while states at other
energies are suppressed. Compared to those, the polariton field strength is
enhanced by a factor of 300. d Simulated reflectance of a 580 nm-thin CrSBr
flake on top of a $\textrm{SiO}_{2}/\textrm{Si}$ substrate with
$\textrm{SiO}_{2}$ thickness of 285 nm. e Reflectance line-cut at normal
incidence ($0^{\circ}$) shows multiple polariton modes with cavity quality
factors around 30. f Electric field profiles extracted for the energies
indicated in d and e. Compared to the external cavity, the enhancement of
polariton states over uncoupled states only amounts to a factor of 10.

###  S2.3 Emission and reflectance of crystals with and without external
cavity mirrors

Figure S7: Low-temperature PL emission and differential reflectance contrast
dR/dE. a,c In CrSBr bulk crystals enclosed by external cavity mirrors, optical
states observed in reflectance and PL perfectly match the theoretically
predicted exciton-polariton dispersion. b,d In bare CrSBr bulk crystals on
$\textrm{SiO}_{2}/\textrm{Si}$ substrates, reflectance measurements match the
predicted exciton-polariton dispersion, but other, unidentified states are
observed in the low-temperature PL response at energies below the exciton-
polaritons, highlighting the benefit of adding external cavity mirrors.

###  S2.4 Polarization-dependent PL emission and reflectance of the 580 nm
cavity sample

Figure S8: Polariton emission and reflectance of the 580 nm cavity sample. a
Angle-resolved polariton PL emission recorded without polarization optics in
the detection path. b Angle-resolved differential reflectance analyzed for
polarization $P$ along the in-plane $b$–axis shows exciton-polariton modes. c
Angle-resolved differential reflectance analyzed for polarization $P$ along
the in-plane $a$–axis shows a purely photonic cavity mode. Data recorded at
1.6 K.

##  S3 Theoretical analysis of the effects of coherent and incoherent magnons

Here, we examine the influence of the magnetic order and its fluctuations on
exciton and exciton-polariton energies in CrSBr. Specifically, our analysis
considers effects on the exciton energy produced by external magnetic fields
and optically induced coherent magnons in the presence of spin canting at low
temperatures. Moreover, we evaluate the role of non-zero temperatures both in
a model based on incoherent magnons, as well as by including temperature
effects in a description of sub-lattice magnetizations. We thus derive an
analytic model for the temperature- and field-dependence of the exciton energy
in CrSBr.

###  S3.1 Coupling of magnetic and electronic structure at zero temperature

We first analyze the relation between the electronic and magnetic structure of
CrSBr at zero temperature in the presence of an out-of-plane (spin-canting)
external magnetic field, analogous to the discussion presented in Ref. Wilson
_et al._ (2021). This will serve as a framework for our further analysis
detailed below.

####  S3.1.1 Magnetic ground state with applied magnetic field

We consider the antiferromagnet to be described by the magnetic free energy
density:

$\displaystyle F$
$\displaystyle=J\boldsymbol{M}_{A}\cdot\boldsymbol{M}_{B}+K_{h}\left(M_{Ac}^{2}+M_{Bc}^{2}\right)-K_{e}\left(M_{Ab}^{2}+M_{Bb}^{2}\right)-\mu_{0}H_{0}\left(M_{Ac}+M_{Bc}\right),$
(S11)

where $\boldsymbol{M}_{A,B}$ are the magnetizations, assumed spatially
uniform, of sub-lattices A and B. In Eq. (S11) above, $J$ ($>0$) parameterizes
the antiferromagnetic exchange between the two sub-lattices, $K_{h}$ ($>0$)
accounts for the hard-axis anisotropy in the out-of-plane $c$–direction,
$K_{e}$ ($>0$) captures the easy-axis anisotropy along the $b$–axis, and the
final term on the right represents Zeeman energy due to externally applied
field $\boldsymbol{H}_{\mathrm{ext}}=H_{0}\hat{\boldsymbol{c}}$.

Figure S9: Schematic depiction of the two sub-lattice magnetizations in
equilibrium. We assume an easy axis along $\hat{\boldsymbol{b}}$ and a hard
axis along $\hat{\boldsymbol{c}}$. An external magnetic field is applied along
$\hat{\boldsymbol{c}}$.

Parameterizing the equilibrium configuration as depicted in Fig. S9 via the
angle $\alpha$, the free energy density simplifies to:

$\displaystyle F$ $\displaystyle=-JM_{0}^{2}\cos
2\alpha+2KM_{0}^{2}\sin^{2}\alpha-2\mu_{0}H_{0}M_{0}\sin\alpha-2K_{e}M_{0}^{2},$
(S12)

where $M_{0}$ is the magnitude of sub-lattice magnetizations
$\boldsymbol{M}_{A,B}$, and we define $K\equiv K_{e}+K_{h}$. The equilibrium
configuration is determined by minimizing the free energy:

$\displaystyle\frac{\partial F}{\partial\alpha}$ $\displaystyle=0,$ (S13)
$\displaystyle\implies\sin\alpha$
$\displaystyle=\frac{\mu_{0}H_{0}}{2M_{0}\left(J+K\right)},\quad\frac{\mu_{0}H_{0}}{2M_{0}\left(J+K\right)}<1$
$\displaystyle\alpha$
$\displaystyle=\pi/2,\quad\frac{\mu_{0}H_{0}}{2M_{0}\left(J+K\right)}\geq 1.$
(S14)

####  S3.1.2 Exciton energy shift in a spin-canting external field

Considering the role of interlayer tunneling and its spin dependence, the
exciton energy shift may be expressed as Wilson  _et al._ (2021):

$\displaystyle\Delta E$
$\displaystyle=\Delta_{B}\cos^{2}\left(\frac{\theta}{2}\right)=\Delta_{B}\cos^{2}\left(\frac{\pi}{2}-\alpha\right),$
(S15) $\displaystyle=\Delta_{B}\sin^{2}\left(\alpha\right),$ (S16)

where $\theta$ is the angle between $\boldsymbol{M}_{A}$ and
$\boldsymbol{M}_{B}$ (Fig. S9), $\Delta_{B}$ is the maximum field-induced
shift of the exciton energy, and $\alpha$ is given by Eq. (S14). Thus, we see
that the exciton energy shift varies quadratically with the applied magnetic
field until a ferromagnetic configuration is realized, after which the exciton
energy remains constant (cf. bell-like curve in Fig. 2A of the main
manuscript).

####  S3.1.3 Exciton energy shift in the presence of coherent magnons

A recent pump-probe experiment demonstrated the dynamic aspects of this
coupling between the magnetic and electronic structure in CrSBr using coherent
magnons induced by ultrashort optical excitation Bae  _et al._ (2022). In a
canted configuration, where $\theta_{0}$ is the equilibrium angle between the
two sub-lattice magnetizations governed by a static external field, coherent
magnons produce a time-dependence of the angle $\theta$

$\displaystyle\theta(t)=\theta_{0}+\Delta\theta_{0}\exp(-\kappa t)\cos(\omega
t)\,,$ (S17)

where $\kappa$ is the magnon decay rate and $\omega$ is their frequency.

If we assume that $\Delta\theta_{0}\ll\theta_{0}$, we can express the effect
of coherent magnons on the exciton energy by

$\displaystyle\Delta E$ $\displaystyle=\Delta
E(\theta=\theta_{0})+\evaluated{\derivative{\theta}\Delta
E}_{\theta=\theta_{0}}(\Delta\theta_{0}\exp(-\kappa t)\cos(\omega t))$ (S18)
$\displaystyle\derivative{\theta}\Delta E$
$\displaystyle=\Delta_{B}\cos\left(\frac{\theta}{2}\right)\left(-\sin\left(\frac{\theta}{2}\right)\right)$
$\displaystyle=-\frac{\Delta_{B}}{2}\sin(\theta)$
$\displaystyle\evaluated{\derivative{\theta}\Delta E}_{\theta=\theta_{0}}$
$\displaystyle=-\frac{\Delta_{B}}{2}\sin(\theta_{0})$ (S19)

By inserting Eq. S19 into Eq. S18, we thus obtain an expression for the energy
of excitons in the presence of spin canting and coherent magnons,

$\displaystyle\Delta
E(t)=\Delta_{B}\cos^{2}\left(\frac{\theta_{0}}{2}\right)-\frac{\Delta_{B}}{2}\,\sin(\theta_{0})\,\Delta\theta_{0}\,\exp(-\kappa
t)\,\cos(\omega t)\,.$ (S20)

This expression is used to numerically simulate the response of exciton-
polaritons to coherent magnons in Fig. 3B of the main manuscript (cf. also
Section S4.2).

###  S3.2 Coupling of magnetic and electronic structure at low temperature

To address the effect of finite temperatures, we consider the same magnet and
anisotropies as in the previous section [Eq. (S11)], but first derive a
description in the absence of any applied magnetic field, i.e., $H_{0}=0$.
Thus, at zero temperature, the ground state corresponds to antiparallel sub-
lattice magnetizations oriented along the $b$–axis. However, when one
considers non-zero temperatures, thermal fluctuations cause the magnetizations
to fluctuate about their equilibrium positions. At temperatures much smaller
than the Néel temperature, these fluctuations are synonymous with the presence
of thermally generated incoherent magnons in the system. An instantaneous
configuration of the magnet is depicted in Fig. S10, where $\alpha$, $\beta$,
and $\theta$ become random variables. We now examine the impact of such
fluctuations, representing the presence of incoherent magnons in the system,
on exciton energies.

####  S3.2.1 Exciton energy shift in the presence of magnetic fluctuations

Figure S10: Schematic depiction of the two sub-lattice magnetizations at a
moment of time. No external magnetic field is applied now and thus, the
equilibrium configuration corresponds to $\alpha=\beta=0$. At a given instant
of time, $\alpha$ and $\beta$ have random values due to thermal fluctuations.

Following the previous section, we may write the instantaneous exciton energy
assuming that the electrons adapt instantly to the relatively slow magnetic
fluctuations:

$\displaystyle\Delta E$
$\displaystyle=\Delta_{B}\cos^{2}\left(\frac{\theta}{2}\right)=\Delta_{B}\cos^{2}\left(\frac{\pi}{2}-\frac{\alpha+\beta}{2}\right),$
(S21) $\displaystyle=\Delta_{B}\sin^{2}\left(\frac{\alpha+\beta}{2}\right),$
(S22)
$\displaystyle=\Delta_{B}\left[\sin\left(\frac{\alpha}{2}\right)\cos\left(\frac{\beta}{2}\right)+\sin\left(\frac{\beta}{2}\right)\cos\left(\frac{\alpha}{2}\right)\right]^{2},$
(S23)
$\displaystyle=\Delta_{B}\left[\sin^{2}\left(\frac{\alpha}{2}\right)\cos^{2}\left(\frac{\beta}{2}\right)+\sin^{2}\left(\frac{\beta}{2}\right)\cos^{2}\left(\frac{\alpha}{2}\right)+\frac{1}{2}\sin\alpha\sin\beta\right].$
(S24)

The experimentally recorded shift in the exciton energy is obtained by
averaging, denoted by the operator $\left\langle{\cdot}\right\rangle$, over
the thermal fluctuations:

$\displaystyle\left\langle{\Delta E}\right\rangle$
$\displaystyle=\Delta_{B}\left[\left\langle{\sin^{2}\left(\frac{\alpha}{2}\right)}\right\rangle\left\langle{\cos^{2}\left(\frac{\beta}{2}\right)}\right\rangle+\left\langle{\sin^{2}\left(\frac{\beta}{2}\right)}\right\rangle\left\langle{\cos^{2}\left(\frac{\alpha}{2}\right)}\right\rangle\right],$
(S25)

where we have assumed that the thermal fluctuations in sub-lattices A and B
behave independently, and
$\left\langle{\sin\alpha}\right\rangle=\left\langle{\sin\beta}\right\rangle=0$.
The latter assumption is well-justified since the average value of $\alpha$
and $\beta$ is 0. The former assumption essentially disregards the
correlations that are generated by the antiferromagnetic exchange. This
assumption is justified at temperatures larger than a few Kelvins since the
antiferromagnetic exchange in the material under investigation has been found
to be small (less than 1 T) Cham  _et al._ (2022), and can be neglected when
we consider thermal fluctuations at finite temperatures.

####  S3.2.2 Exciton energy shift in the presence of incoherent magnons

It is much more convenient and powerful to describe these incoherent
fluctuations in terms of magnons, rather than the classical Landau-Lifshitz
description that we have followed thus far. We now bridge the two pictures and
relate the exciton energy with the incoherent magnon density. To this end, we
note that the average projection of the sub-lattice magnetizations along the
equilibrium $b$–direction, $M_{Ab}$ and $M_{Bb}$, are reduced by the existence
of magnons:

$\displaystyle M_{Ab}$ $\displaystyle=M_{0}\cos\alpha,$ (S26) $\displaystyle
M_{Ab}^{2}$
$\displaystyle=M_{0}^{2}\cos^{2}\alpha=M_{0}^{2}(1-\sin^{2}\alpha),$ (S27)
$\displaystyle\left\langle{M_{Ab}}\right\rangle$ $\displaystyle\approx
M_{0}-\frac{M_{0}}{2}\left\langle{\sin^{2}\alpha}\right\rangle.$ (S28)

Equating this reduction in the magnetization to what is caused by magnons, we
obtain:

$\displaystyle\frac{M_{0}}{2}\left\langle{\sin^{2}\alpha}\right\rangle$
$\displaystyle=n_{A}\hbar\gamma,$ (S29)
$\displaystyle\implies\left\langle{\sin^{2}\alpha}\right\rangle$
$\displaystyle=\frac{2n_{A}\hbar\gamma}{M_{0}},$ (S30)

where we have assumed the sub-lattice A magnons to bear spin $\hbar$, $\gamma$
($>0$) is the sub-lattice gyromagnetic ratio, and $n_{A}$ is the density of
magnons with spin opposite to the A sub-lattice equilibrium spin. Similarly,
we obtain:

$\displaystyle\left\langle{\sin^{2}\beta}\right\rangle$
$\displaystyle=\frac{2n_{B}\hbar\gamma}{M_{0}},$ (S31)

for the density of magnons with spin opposite to the B sub-lattice equilibrium
spin.

Substituting these in Eq. (S25), we obtain our final result:

$\displaystyle\left\langle{\Delta E}\right\rangle$
$\displaystyle=\Delta_{B}\left[\left\langle{\sin^{2}\left(\frac{\alpha}{2}\right)}\right\rangle\left\langle{\cos^{2}\left(\frac{\beta}{2}\right)}\right\rangle+\left\langle{\sin^{2}\left(\frac{\beta}{2}\right)}\right\rangle\left\langle{\cos^{2}\left(\frac{\alpha}{2}\right)}\right\rangle\right],$
(S32)
$\displaystyle=\Delta_{B}\left[\left\langle{\sin^{2}\left(\frac{\alpha}{2}\right)}\right\rangle+\left\langle{\sin^{2}\left(\frac{\beta}{2}\right)}\right\rangle-2\left\langle{\sin^{2}\left(\frac{\alpha}{2}\right)}\right\rangle\left\langle{\sin^{2}\left(\frac{\beta}{2}\right)}\right\rangle\right],$
(S33)
$\displaystyle\approx\Delta_{B}\left[\left\langle{\sin^{2}\left(\frac{\alpha}{2}\right)}\right\rangle+\left\langle{\sin^{2}\left(\frac{\beta}{2}\right)}\right\rangle\right],$
(S34)
$\displaystyle=\Delta_{B}\frac{\left(n_{A}+n_{B}\right)\hbar\gamma}{2M_{0}},$
(S35)

where we have employed
$\sin\left(\alpha/2\right)\approx\left(\sin\alpha\right)/2$ and similar for
$\beta$, valid when $\alpha,\beta\ll 1$.

According to Eq. (S35), the shift of the exciton energy should vary linearly
with the magnon temperature, since $n_{A,B}\sim k_{B}T$. However, this is not
the complete story because our analysis assumes $M_{0}$ and other parameters
entering the free energy to be temperature independent. When one takes this
into account, for example by considering that $M_{0}$ is decreasing as the
temperature increases Cham  _et al._ (2022), the energy shift above bears a
quadratic contribution in addition to the linear one. We address this regime
by accounting for the reduction of the sublattice magnetizations below.

####  S3.2.3 Exciton energy shift in the presence of incoherent magnons and a
saturation magnetic field

In the previous section, we have considered zero external magnetic field. If
we consider a large applied field, the ground state of the system is in a FM
configuration, i.e., the two sub-lattice magnetizations align along the
external magnetic field. We now consider the exciton energy shift under such a
configuration.

Figure S11: Schematic depiction of the two sub-lattice magnetizations at a
moment of time. A large external magnetic field is assumed such that the
equilibrium configuration corresponds to
$\boldsymbol{M}_{A}=\boldsymbol{M}_{B}=M_{0}\hat{\boldsymbol{b}}$ [Eq. (S14)].
$\theta_{A}$ and $\theta_{B}$, with $\theta=\theta_{A}+\theta_{B}$, now become
the random variables with zero mean value associated with the thermal
fluctuations.

Depicting the instantaneous configuration in Fig. S11, the exciton energy
shift is evaluated as:

$\displaystyle\Delta E$
$\displaystyle=\Delta_{B}\cos^{2}\left(\frac{\theta}{2}\right)=\Delta_{B}\cos^{2}\left(\frac{\theta_{A}+\theta_{B}}{2}\right),$
(S36)
$\displaystyle=\Delta_{B}\left[\cos\left(\frac{\theta_{A}}{2}\right)\cos\left(\frac{\theta_{B}}{2}\right)-\sin\left(\frac{\theta_{A}}{2}\right)\sin\left(\frac{\theta_{B}}{2}\right)\right]^{2},$
(S37)
$\displaystyle=\Delta_{B}\left[\cos^{2}\left(\frac{\theta_{A}}{2}\right)\cos^{2}\left(\frac{\theta_{B}}{2}\right)+\sin^{2}\left(\frac{\theta_{A}}{2}\right)\sin^{2}\left(\frac{\theta_{B}}{2}\right)-\frac{1}{2}\sin\theta_{A}\sin\theta_{B}\right].$
(S38)

A shift equivalent to that recorded in our time-integrated optical experiments
presented in Fig. 3 of the main manuscript is obtained by averaging over
thermal fluctuations:

$\displaystyle\left\langle{\Delta E}\right\rangle$
$\displaystyle=\Delta_{B}\left[\left\langle{\cos^{2}\left(\frac{\theta_{A}}{2}\right)}\right\rangle\left\langle{\cos^{2}\left(\frac{\theta_{B}}{2}\right)}\right\rangle+\left\langle{\sin^{2}\left(\frac{\theta_{A}}{2}\right)}\right\rangle\left\langle{\sin^{2}\left(\frac{\theta_{B}}{2}\right)}\right\rangle\right],$
(S39)
$\displaystyle=\Delta_{B}\left[1-\left\langle{\sin^{2}\left(\frac{\theta_{A}}{2}\right)}\right\rangle-\left\langle{\sin^{2}\left(\frac{\theta_{B}}{2}\right)}\right\rangle+2\left\langle{\sin^{2}\left(\frac{\theta_{A}}{2}\right)}\right\rangle\left\langle{\sin^{2}\left(\frac{\theta_{B}}{2}\right)}\right\rangle\right],$
$\displaystyle\approx\Delta_{B}\left[1-\left\langle{\sin^{2}\left(\frac{\theta_{A}}{2}\right)}\right\rangle-\left\langle{\sin^{2}\left(\frac{\theta_{B}}{2}\right)}\right\rangle\right].$
(S40)

Employing the analysis similar to that in the previous section, we may express
this in terms of the magnon densities as follows

$\displaystyle\left\langle{\Delta E}\right\rangle$
$\displaystyle=\Delta_{B}\left(1-\frac{\left(n_{A}+n_{B}\right)\hbar\gamma}{2M_{0}}\right).$
(S41)

Hence, we note that the exciton energy shift due to a non-zero population of
incoherent magnons in this case is in the opposite direction as compared to
the previous case Eq. (S35). This makes sense as in the current configuration,
the energy shift is already at maximum ($\Delta E=\Delta_{B}$) at zero
temperature and the thermal fluctuations can only reduce the effect.

##### Discussion

The analysis above has employed the hierarchy of energy scales (temperature
much larger than antiferromagnetic exchange and anisotropies) present in the
system and provided a simplified way to estimate the effects. A more rigorous
analysis of the magneto-electric effect due to incoherent magnons, especially
at low temperatures, should consider the fact that antiferromagnetic magnons
do not always bear spin 1, when accounting for anisotropies that are important
at low energies Kamra  _et al._ (2017, 2020). In fact, for the dominant easy-
plane anisotropy in the material under investigation, the low-energy magnons
bear a much smaller spin than 1. Such effects have been disregarded in our
analysis and are expected to weaken the linear-in-temperature exciton energy
shift at low temperatures.

####  S3.2.4 Exciton energy shift induced by changes in sub-lattice
magnetizations for a broader temperature range

So far, we have used the expression for the exciton energy shift

$\displaystyle\Delta E=\Delta_{B}\cos^{2}\left(\frac{\theta}{2}\right)\,,$
(S42)

which holds only as long as temperature-induced changes in the sub-lattice
magnetizations are small. To account for the temperature dependence of the
sublattice magnetization over a broad temperature range up to the ordering
temperature, we express the exciton energy shift in terms of the
magnetization:

$\displaystyle\Delta E^{\prime}$
$\displaystyle=k\,\boldsymbol{M}_{A}\cdot\boldsymbol{M}_{B}$
$\displaystyle=k\,{M_{S}}^{2}\cos\theta\,,$ (S43)

where $M_{S}\equiv M_{S}(T)$ is the temperature-dependent magnetization of
each sub-lattice and $k$ is a constant.

Since they are based on the same underlying phenomena, the two different
expressions for the exciton energy shift, $\Delta E$ and $\Delta E^{\prime}$,
should be consistent.

$\displaystyle\Delta E^{\prime}$ $\displaystyle=k\,{M_{S}}^{2}\cos\theta$
$\displaystyle=k\,{M_{S}}^{2}\left(2\cos^{2}\frac{\theta}{2}-1\right)$
$\displaystyle=2k\,{M_{S}}^{2}\cos^{2}\frac{\theta}{2}-k\,{M_{S}}^{2}\,$ (S44)

Using $\Delta_{B}=2k\,{M_{S}}^{2}$, we obtain

$\displaystyle\Delta E^{\prime}$
$\displaystyle=\Delta_{B}\cos^{2}\left(\frac{\theta}{2}\right)-k\,{M_{S}}^{2}$
$\displaystyle\Delta E^{\prime}$ $\displaystyle=\Delta E-k\,{M_{S}}^{2}\,,$

and thus $\Delta E$ and $\Delta E^{\prime}$ are fully consistent, but as we
show in the following, $\Delta E^{\prime}$ is much more convenient to describe
the temperature dependence of exciton energies in CrSBr. To account for the
second-order phase transition, we assume a temperature-dependence of the sub-
lattice magnetization

$\displaystyle\frac{M_{S}}{M_{0}}=\left(1-\frac{T}{T_{corr}}\right)^{\beta}\,$
(S45)

where $M_{0}$ is the magnetization at $T=0$, and $T_{corr}$ is the temperature
up to which short-range correlations are observed (see Section S4.2). As a
result, the temperature- and field-dependence of the exciton energy in CrSBr
is given by

$\displaystyle\Delta E^{\prime}$
$\displaystyle=k\,{M_{0}}^{2}\cos(\theta)\left(1-\frac{T}{T_{corr}}\right)^{2\beta}\,,$
(S46)

and explicit expressions for the temperature-dependence of the exciton energy
for AFM ($\theta=\pi$) and FM ($\theta=0$) configurations are

$\displaystyle\Delta E^{\prime}$
$\displaystyle=-\,\frac{\Delta_{B}}{2}\left(1-\frac{T}{T_{corr}}\right)^{2\beta}$
$\displaystyle\text{for AFM}\,,$ (S47) $\displaystyle\Delta E^{\prime}$
$\displaystyle=+\,\frac{\Delta_{B}}{2}\left(1-\frac{T}{T_{corr}}\right)^{2\beta}$
$\displaystyle\text{for FM}\,,$ (S48)

For $T\ll T_{corr}$, we thus recover the linear temperature-dependence
expected from the above analysis of incoherent magnons in the low temperature
limit:

$\displaystyle\Delta
E^{\prime}=\frac{\Delta_{B}}{2}\,\cos\theta_{0}\,\left(1-\frac{2\beta
T}{T_{corr}}\right)\,.$ (S49)

In this analysis, $\Delta_{B}=-17.5$ meV, as determined by numerical
simulation of reflectance spectra.

##  S4 Experimental and numerical analysis of polariton-magnon coupling

###  S4.1 Measurement of the coupling between polaritons and coherent magnon

Figure S12: Polariton-magnon coupling dynamics probed by resonant transient
reflectance. a Schematic illustrating the transient reflectivity measurement
employing a pump pulse with above-gap energy and a resonantly tuned probe
pulse with a time delay $\Delta t$. b Corresponding polariton dispersion
measured at 1.6K by polariton PL emission showing two closely-spaced polariton
branches with exciton fractions $\bm{X}\approx 0.9$. Red-shaded area indicates
the spectrum of the probe pulse. c Pump-probe signal for different magnetic
fields ($\bm{B_{ext}}\parallel c$). d Fourier transform of the pump-probe
dynamics obtained by scanning the magnetic-field. Inset: Fourier spectrum
obtained at $\bm{B_{ext}}=0.9$ T showing a $\sim$1 GHz frequency splitting
attributed in a recent report to the coupling between acoustic phonons and
magnons Bae  _et al._ (2022).

###  S4.2 Numerical analysis of polariton-magnon coupling: Coherent magnons

Our experimental results presented in Fig. 2B of the main manuscript allow us
to model the time-dependent response of exciton-polaritons to a density of
coherent magnons. We modify Eq. S20 to include the dependence of magnetic
shifts on the exciton fraction $\bm{X}$,

$\displaystyle E_{pol}(t,\bm{X})$ $\displaystyle=\bm{X}\Delta E(t)$ (S50)
$\displaystyle=\bm{X}\,\Delta_{B}\cos^{2}\left(\frac{\theta}{2}\right)-\bm{X}\,\frac{\Delta_{B}}{2}\,\sin(\theta_{0})\,\Delta\theta_{0}\,\exp(-\kappa
t)\,\cos(\omega t)\,.$ (S51)

For the simulation of the oscillatory part of the transient differential
reflectance $R(t)-R_{osc}(t)$ of polaritons shown in Fig. 3B of the main
manuscript, we reduce Eq. S51 to $\Delta
E_{pol}(t,\bm{X})=\bm{X}\,\Delta_{\theta_{0}}\,\exp(-\kappa t)\,\cos(\omega
t)$ and choose $\Delta_{\theta_{0}}=2$ meV, $\kappa=0.5$ ns and $\omega=2\pi
f_{mag}$ with $f_{mag}=12.5$ GHz (at $\bm{B_{ext}}=1.2$ T, cf. Fig. S12) to
match our experiments. To isolate the effect of $\bm{X}$ on the oscillatory
signal induced by coherent magnons, we omit the detuning-dependent spectral
width of polariton branches in our samples and simply describe their
reflectance by a normalized Lorentzian function with a constant width of 4.5
meV and a peak energy given by $\Delta E_{pol}(t,\bm{X})$.

###  S4.3 Measurement of the coupling between polaritons and incoherent magnon

The response of exciton-polaritons to incoherent magnons predicted by our
analytic theory in Section S3 is demonstrated by the temperature- and field-
dependent PL emission of all polariton branches in our 580 nm-thick cavity
sample. Based on the results plotted in Fig. S13 we make three main
observations: i ) As a function of temperature, most exciton-polariton
branches shift towards lower energies for $B=0$ and towards higher energies
for $\bm{B}\geq B_{sat}$. ii ) Polariton branches with small detuning rapidly
disappear from the spectrum, while branches with large detuning are observed
at much higher temperatures. iii ) The temperature- and field-dependence of
polaritons is significantly altered by detuning $\delta_{pol}$ and the exciton
fraction $\bm{X}$.

Figure S13: Energies of different polariton branches in the 580 nm-thin cavity
sample obtained by temperature- and field-resolved PL measurements. PL peak
energies extracted from polariton branches with different detuning
$\delta_{pol}$ for $B$=0 and $\bm{B}\geq B_{sat}$.

###  S4.4 Numerical analysis of polariton-magnon coupling: Incoherent magnons

To better understand the physical mechanisms behind these observations, we
simulate the polariton temperature dependence in transfer-matrix calculations,
thereby considering the influence of different parameters on the dielectric
function, $\epsilon(E,T,\bm{B_{ext}})$, which is the foundation for our
numerical analysis:

$\epsilon(E,T,\bm{B_{ext}})=\epsilon_{b}^{\infty}(T)+\frac{f_{X}(T)}{E_{X}(T,\bm{B_{ext}})^{2}-E^{2}-i\gamma(T)E}\,,$
(S52)

where, $E$ is the energy, $T$ the sample temperature, $\bm{B_{ext}}$ is the
external magnetic field, and
$f_{X}(T)=2\Delta_{X}\sqrt{{E_{X}(T)}^{2}-(\Gamma(T)/2)^{2}}$ denotes an
effective oscillator strength that includes the exciton energy $E_{X}$, the
intrinsic exciton oscillator strength $\Delta_{X}$, and the broadening of the
exciton line-width $\Gamma(T)$.

First, we account for the different contributions to the temperature- and
field-dependence of excitons in CrSBr,

$E_{X}(T,\bm{B_{ext}})=E_{X}(0)+\Delta E_{X}^{mag}(\bm{B_{ext}},T)+\Delta
E_{X}^{ph}(T)\,$ (S53)

including changes in the exciton energy due to incoherent magnons, $\Delta
E_{X}^{mag}(\bm{B_{ext}},T)$, and phonons, $\Delta E_{X}^{ph}(T)$. Our
numerical description of the effects of incoherent magnons is based on the
results of our analytic model presented in Eqs. S47 and S48. We explicitly use
$T_{corr}=$ 180 K, instead of $T_{N}$, to account for the fact that magnetic
correlations are clearly visible in our experiments up to temperatures
exceeding $T_{N}$. Good agreement between our numerical model and the
experimental data is obtained for $\beta\approx 0.4$. The corresponding
decrease and increase of the exciton energy for AFM and FM configurations,
induced by the steadily increasing density of incoherent magnons, are plotted
in Fig. S14a.

Besides incoherent magnons, phonons couple to excitons, inducing further
temperature-dependent changes in the exciton energy that can be described by
O’Donnell and Chen (1991):

$\Delta
E_{X}^{ph}(T)=S\langle\hbar\omega\rangle(\coth(\langle\hbar\omega\rangle)/2k_{B}T-1)\,,$
(S54)

where $S=0.5$, $\langle\hbar\omega\rangle$ is the average phonon energy
(extracted from Fig. S14d, see discussion below), and $\hbar$ is Planck’s
constant. The sum of the terms in Eq. S53 is plotted in Fig. S14c for AFM
($B_{ext}=0$) and FM ($\bm{B_{ext}}\geq B_{sat}$) magnetic configurations.

Figure S14: Contributions to the temperature dependence of polaritons. a
Coupling of excitons to incoherent magnons for both AFM (${B_{ext}}=0$) and FM
($\bm{B_{ext}}>B_{sat}$). b Coupling of excitons to phonons. c Summed
contributions of magnons and phonons. d Temperature-dependence of the
dielectric constant $\epsilon_{a}$ measured via the temperature-dependence of
the optical mode along the intermediate $a$–direction (cf. also Fig. S3C) and
fitted by the Einstein function (see text). e Temperature-induced broadening
of the excitons line-width. f Phenomenological temperature-dependence of the
dielectric constant $\epsilon_{b}(\infty)$ in the magnetic easy axis direction
derived from comparison with the experimental results shown in Figs. S9 & 10\.

While the dielectric tensor element $\epsilon_{a}^{\infty}$ does not
explicitly affect the temperature dependence of our exciton-polaritons, we can
use it to experimentally access other relevant parameters. By measuring the
temperature dependence of the uncoupled optical mode in the $a$–direction in
our 580 nm cavity sample, we obtain $\epsilon_{a}^{\infty}(T)$, which we fit
to the Einstein relation Dubrovin and Pisarev (2020) to extract the average
optical phonon energy, $\langle\hbar\omega\rangle=$17.5 meV, used in Eq. S54,
directly from our measurements. This value is in reasonable agreement with the
$A_{g}^{1}$ phonon mode observe in Raman experiments Klein  _et al._ (2022a).
Experimentally, we can further conclude that $\epsilon_{a}^{\infty}(T)$ is
neither significantly affected by an external magnetic field, nor by magnetic
phase transitions in CrSBr, indicating that the coupling of the magnetic order
to the dielectric tensor is negligible in the $a$–direction.

We use the extracted phonon energy to calculate the temperature-dependent
broadening of the exciton line-width $\Gamma(T)$ Jin  _et al._ (2020) (see
Fig. S14e), which is in very good agreement with line-widths observed in PL
measurements of excitons in bi- and few-layer samples Wilson  _et al._ (2021);
Klein  _et al._ (2022b).

Figure S15: Optical resonances in the 580 nm-thin cavity sample obtained by
temperature-dependent reflectivity measurements. a Polariton branches obtained
by analyzing the reflectivity signal along the crystal $b$-direction. b
Uncoupled optical mode analyzed along the crystal $a$-direction.

To capture the role of the polariton photon component in our simulations, we
account for changes in the refractive index that determine the energy of
photons in our cavities by constructing the temperature-dependence of the
dielectric tensor element $\epsilon_{b}^{\infty}(T)$ based on the two main
contributions shown in Fig. S14f. In the first step, we include the effects of
phonons on the dielectric tensor element using the Einstein relation (dashed
line) and the average phonon energy $\langle\hbar\omega\rangle=$17.5 meV
extracted from our measurements. In the second step, we assume that the
temperature-induced decrease of the magnetization $\bm{M}(T)$ along the
magnetic easy axis ($b$–axis) reduces $\epsilon_{b}^{\infty}$. Like the field-
induced splitting, we postulate that this effect occurs up to temperatures
$T\approx T_{corr}$.

In summary, considering the different contributions to
$\epsilon(E,T,\bm{B_{ext}})$ shown in Fig. S14 allows us to numerically
simulate the temperature-dependence of all polaritons branches in our 580 nm
cavity sample. Figure S15 demonstrates the overall good agreement between the
simulated energies and those determined by optical reflectance experiments. In
particular, our simulations reproduce the three main conclusions deducted from
our experiments: i ) Decrease (increase) of exciton-polariton energies under
increasing temperatures (shown here only for $B=0$). ii ) Rapid disappearance
of the polariton branches with small detuning and high exciton fraction. iii )
The dependence of the thermal response of exciton-polaritons on detuning and
exciton fraction $\bm{X}$.

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