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\begin{document}

%\preprint{APS/123-QED}

%TC:ignore

\section{Title}
Measurement of optically induced broken time-reversal symmetry in atomically thin crystals

\section{Author list}
Florentine Friedrich$^{1,\dagger}$, Paul Herrmann$^{1,\dagger}$, Shridhar Sanjay Shanbhag$^2$, Sebastian Klimmer$^{1,3}$, Jan Wilhelm$^{2}$, and {Giancarlo} Soavi$^{1,4,\star}$

\section{Affiliations}
\noindent
$^1$Institute of Solid State Physics, Friedrich Schiller University Jena, Helmholtzweg 5, 07743 Jena, Germany
\newline
$^2$Institute of Theoretical Physics and Regensburg Center for Ultrafast Nanoscopy (RUN), University of Regensburg, Universitätsstrasse 31, 93053 Regensburg, Germany
\newline
$^3$ARC Centre of Excellence for Transformative Meta-Optical Systems, Department of Electronic Materials Engineering, Research School of Physics,
The Australian National University, Canberra ACT 2601, Australia
\newline
$^4$Abbe Center of Photonics, Friedrich Schiller University Jena, Albert-Einstein-Straße 6, 07745 Jena, Germany
\newline
$^{\dagger}$ Both authours contributed equally.
$^{\star}$ giancarlo.soavi@uni-jena.de

%\keywords{nonlinear optics, SHG, valleytronics, 2d materials}
\maketitle


\section{Abstract}

Time-reversal and space-inversion symmetries are fundamental properties of crystals and play a role in underlying phenomena such as magnetism, topology and non-trivial spin textures. Transition metal dichalcogenides (TMDs) represent an excellent tunable model system to explore the interplay between these symmetries as they can be engineered on demand by tuning the number of layers and \textit{via} all-optical bandgap modulation. In this work, we modulate and study time-reversal symmetry in mono- and bilayer TMDs with all-optical methods using third-harmonic Faraday rotation. We excite the samples using elliptically polarized light, achieve spin-selective bandgap modulation and consequent breaking of time-reversal symmetry. The reduced symmetry modifies the nonlinear susceptibility tensor, causing a rotation of the emitted third-harmonic polarization. With this method, we probe broken time-reversal symmetry in both non-centrosymmetric (monolayer) and centrosymmetric (bilayer) \ce{WS_2} crystals. Furthermore, we discuss how the detected third-harmonic rotation angle directly links to the spin-valley locking in monolayer TMDs and to the spin-valley-layer locking in bilayer TMDs. Our results show a powerful approach to study broken time-reversal symmetry in crystals regardless of space-inversion symmetry, and shed light on the spin, valley and layer coupling of atomically thin semiconductors.

%TC:endignore

\section{Main text}

\section{Introduction}
Time-reversal symmetry (TRS), in combination with space-inversion symmetry (SIS), defines the energy-spin properties and the Berry curvature of crystals~\cite{soavi2025signaturetopologypolarchiral}. Thus, the presence or absence of TRS and SIS is paramount to explain and understand a variety of phenomena in condensed matter physics, ranging from magnetism~\cite{Fiebig2023} to spin-valley locking~\cite{PhysRevLett.108.196802,Mak2012}, anomalous transport~\cite{KLITZING1983525,McIver2020}, and topology~\cite{Bao2022}. Nonlinear optics (NLO) is an established and powerful tool to investigate the effect of spatial symmetries and TRS breaking in crystals~\cite{Fiebig2023, Gopalan2011}. The most relevant example is arguably the study of ferroic materials{~\cite{Fiebig2023, Denev2011, Fiebig2016, Kumar2008, Fiebig1994}, where a variety of physical and optical properties is captured by NLO effects described by the magnetic point groups~\cite{Guccione1963}. Among the applications of NLO to the study of time-noninvariant phenomena, second-harmonic nonlinear Kerr/Faraday rotation plays a key role. It has been predicted and experimentally observed that nonlinear Kerr/Faraday rotation offers higher sensitivity compared to its linear counterpart~\cite{PUSTOGOWA1995269,PhysRevLett.74.3692}. The technique has already been applied to bulk~\cite{ZVEZDIN1997444,PhysRevLett.92.047401} and atomically thin crystals~\cite{Wu2023,10.1063/1.372586}. However, most measurements to date were limited to non-centrosymmetric crystals, while the measurement of broken TRS in centrosymmetric crystals remains a challenge. For centrosymmetric crystals, the most common approach to study broken TRS relies on the electric quadrupole response~\cite{Ahn2024}, and very recently another approach has been demonstrated, based on light-matter interactions with engineered topological optical fields to realize room-temperature valleytronics in centrosymmetric systems~\cite{Mrudul2021, Mitra2024, Tyulnev2024}. We propose a different method to study broken TRS in any system, both centrosymmetric and non-centrosymmetric, based on third-harmonic (TH) Faraday rotation. We follow the notation of Ref.~\cite{Matsubara2012} and define \textit{“nonlinear Faraday rotation”} as the angle between the polarization of the fundamental at frequency $\omega$ and the generated $N$-th harmonic at frequency $N\omega$ for experiments in transmission geometry.

To clarify the operating principle of TH Faraday rotation, we refer to the sketch in  Fig.~1a, which depicts the case of a monolayer TMD with broken SIS. When TRS is preserved, the polarization of the TH signal is parallel to the polarization of the input fundamental beam (FB). For broken TRS, \textit{i.e.}, when there is an asymmetry between the $\pm$K points, the TH signal is rotated with respect to the FB, and the degree of broken TRS is proportional to the rotation angle $\theta$. This can be understood in the context of the Curie principle. Any external perturbation reduces the symmetry of a system, allowing only those symmetry operations that are common to both the unperturbed system and the perturbation. Therefore, the new (reduced) symmetry group of the perturbed system can only be a subgroup of the original (unperturbed) crystal class \cite{curie:jpa-00239814}. In our experiments, the perturbation (\textit{i.e.}, circularly polarized light) prohibits mirror operations along any plane that contains the propagation axis of the light beam, as well as C$_2$ rotations perpendicular to the propagation axis of the light beam (see Fig.~1b). A careful analysis of all allowed crystallographic (time-reversal-invariant) and magnetic point groups shows that the removal of these symmetry operations is responsible for the TH Faraday rotation (see Supplementary Information Section S3). With this simple and direct approach, we detect broken TRS in both monolayer (non-centrosymmetric) and bilayer (centrosymmetric) TMDs. We interpret the results macroscopically, based on the differences between the NLO susceptibilities of crystallographic and magnetic point groups, and microscopically from an analytical model based on the semiconductor Bloch equations for the $\boldsymbol{\chi^{(3)}}$ tensor. With both approaches, we show that broken TRS is induced by off-resonant excitation with circularly polarized light, which is responsible for a spin-selective bandgap opening. This spin-selective off-resonant excitation has different effects in monolayer compared to bilayer TMD samples. In monolayer TMDs the broken SIS leads to spin-valley locking. Thus, off-resonant circularly polarized light creates a valley imbalance by the valley-exclusive optical Stark (OS) and Bloch-Siegert (BS) effects~\cite{Sie2017,Kim2014}. In contrast, in bilayer TMDs, where SIS is preserved, the $\pm$K valleys are spin-degenerate, but the same spin couples to opposite layers in the opposite valleys (spin-valley-layer locking)~\cite{Jones2014}. In this case, spin-selective excitation by circularly polarized light does not create a valley imbalance, but it simultaneously lifts the spin degeneracy in both valleys. This corresponds to an engineered band dispersion where TRS is broken while SIS is preserved. Thus, our results not only define a powerful approach to study broken TRS in both centrosymmetric and non-centrosymmetric crystals, but they also provide a unique tool to investigate the fascinating spin, valley and layer coupling of atomically thin semiconductors, which represent a platform for ultrafast valleytronic logic operations~\cite{Gucci2024}. 

%\begin{figure}[]
%    \centering
%    \includegraphics[width=\linewidth]{Figures/FIG1_new_version.pdf}
%    \caption{\textbf{Principle of TH Faraday rotation and certain symmetry operations for a spin-oriented three-fold rotational symmetric system.} \textbf{a}, For a monolayer TMD, broken TRS is equivalent to an asymmetry between the $\pm$K valleys. TRS breaking introduces new $\boldsymbol{\chi^{(3)}}$ tensor elements, leading to a rotation of the TH output by the angle $\theta$. \textcolor{red}{\textbf{b}, The behavior of a three-fold rotational system with spin-oriented states under the vertical mirror and $C_2$ operation. If mirrored vertically, the system changes the spin state from up (straight black arrow pointing upwards) to down (straight red arrow pointing downwards). The same is true for a two-fold rotation operation ($C_2$ operation) in-plane.}}
%    \label{fig:FIG1_NL_Kerr_rotating_sphere}
%\end{figure}

\section{Symmetry of mono- and bilayer TMDs}

TMDs are the ideal platform to investigate the interplay between SIS and TRS. Both properties can be engineered independently by tuning the number of layers (in the 2H phase, an even/odd number of layers possesses/breaks SIS)~\cite{Zhang2014} and \textit{via} interaction with light~\cite{Sie2017,Kim2014}. In addition, layered materials like TMDs display a strong resonant NLO response~\cite{https://doi.org/10.1002/lpor.202100726} despite their atomic thickness, which enables a variety of applications ranging from all-optical~\cite{Klimmer2021} and electrical tunability~\cite{https://doi.org/10.1002/advs.202401840,Soavi2018}, to ultrafast logic operations~\cite{Li2022,Gucci2024}, and nonlinear valleytronics~\cite{https://doi.org/10.1002/smll.202301126}.

Monolayer TMDs belong to the $\bar6m2$ crystallographic point group. They are direct gap semiconductors with optical transitions at the $\pm$K valleys of the Brillouin zone. Since the $\pm$K transitions are mainly defined by the \textit{d}-orbitals of the transition metal atoms~\cite{Liu2015}, we represent the spin states of the lowest energy transitions on the W sites in Fig.~2. When TRS is preserved, the energy and spin at the valleys must follow $E_{\uparrow} (+\bk) = E_{\downarrow} (-\bk)$, and thus optical transitions for both spins are energetically degenerate and equally probable (Fig.~2a, c). The stereographic projection in Fig.~2b shows a summary of the allowed symmetry operations in a monolayer TMD when TRS is preserved (see Supplementary Information section S2 for details). Breaking of TRS in monolayer TMDs is equivalent to lifting the energy degeneracy between the $\pm$K valleys, thereby making the lowest-energy optical transition allowed for only one spin state (Fig.~2d, f). The symmetry of monolayer TMDs reduces to the $\bar6m'2'$ magnetic point group upon TRS breaking (see Fig.~1b, 2e and Supplementary Information sections S1 and S2).

%\begin{figure}
%    \centering
%    \includegraphics[width=\linewidth]{Figures/FIG2.pdf}
%    \caption{\textbf{Probing TRS breaking in mono- and bilayer \ce{WS_2} \textit{via} TH Faraday rotation.} \textbf{a}, Real-space schematic of a \ce{WS_2} monolayer with broken SIS and preserved TRS. \textbf{b}, Stereographic projection of the $D_{3h} \equiv \bar6m2$ crystallographic point group. \textbf{c}, Energy and spin of the $\pm$K valleys of a \ce{WS_2} monolayer with preserved TRS. Linear excitation leads to linear TH parallel to the input. \textbf{d}, Real-space schematic of a \ce{WS_2} monolayer with broken SIS and TRS. Due to the spin-selective gap opening, one spin dominates. \textbf{e}, Stereographic projection of the $\bar6m'2'$ magnetic point group. Red elements represent symmetry operations that are only allowed under antisymmetry. \textbf{f}, Energy and spin of the $\pm$K valleys of a \ce{WS_2} monolayer with broken TRS. Elliptical excitation leads to the opening of a gap at the $+$K valley by $\mathbf{\Delta} E$ and a rotated TH output. \textbf{g}, Real-space schematic of a \ce{WS_2} bilayer with preserved SIS and TRS. \textbf{h}, Stereographic projection of the $D_{3d} \equiv \bar3m$ crystallographic point group. \textbf{i}, Energy and spin of the $\pm$K valleys of a \ce{WS_2} bilayer with preserved TRS. Within one layer, the spin states differ for each valley because of spin-valley-layer locking. Linear excitation leads to linear TH parallel to the input. \textbf{j}, Real-space schematic of a \ce{WS_2} bilayer with preserved SIS, but broken TRS. Due to spin-selective gap opening one of the spins is now dominant. \textbf{k}, Stereographic projection of the $\bar3m'$ magnetic point group. \textbf{l} Energy and spin of the $\pm$K valleys of a \ce{WS_2} bilayer with broken TRS. Elliptical excitation leads to the opening of a gap by $\mathbf{\Delta} E$ in the top and bottom layers and thus to a rotated TH output.} 
%    \label{fig:FIG2_schematics_mono_bi}
%\end{figure}

In contrast, 2H-stacked bilayer TMDs belong to the $\bar3m$ point group when TRS is preserved. Bilayer TMDs are indirect gap semiconductors~\cite{Splendiani2010}, but they still have a direct gap transition at the $\pm $K valleys. Since both SIS and TRS are preserved, the $\pm$K transition are spin-degenerate. In each valley, opposite spins are coupled to opposite layers (see Fig.~2i), leading to the so-called spin-valley-layer locking~\cite{Jones2014}. Breaking of TRS reduces the symmetry of bilayer TMDs to the $\bar3m'$ magnetic point group: vertical mirror planes and the $C_2$ symmetry operations are only allowed in combination with the antisymmetry operation (see the stereographic projection in Fig.~2k and Supplementary Information section S2 for further details). We stress that breaking of TRS in bilayer TMDs is not equivalent to lifting the energy degeneracy between the valleys, but rather to a split between the spin up/down energy levels in both valleys and in opposite layers (see Fig.~2l). This effect is responsible for a large degree of circular polarization in photoluminescence (PL) experiments, observed even at room temperature in \ce{WS_2} and \ce{WSe_2}~\cite{nayak2016robust,doi:10.1021/acs.jpclett.1c01578}.

Based on these symmetry considerations, we will now demonstrate that broken TRS can be easily detected as a rotation in the TH polarization with respect to the polarization of the FB in both mono- and bilayer TMDs.

\section{Analytical model and experimental results}{\label{sec:experimental_results}}
\subsection{TH Faraday rotation in monolayer WS\textsubscript{2}}

First, we study the non-centrosymmetric \ce{WS_2} monolayer. The excitation of TMD monolayers with circularly polarized light is both spin and valley selective~\cite{Herrmann2025}. In contrast, linearly polarized light interacts equally with both valleys. Thus, when the FB is linearly polarized and TRS is preserved, the emitted TH signal is linearly polarized and parallel to the input FB, as depicted in Fig.~2c. The situation changes when an elliptically polarized pulse interacts with the sample, which can be interpreted as a superposition of linearly and circularly polarized light. While the linearly polarized part has no effect on TRS, the circularly polarized component of the FB creates a valley imbalance by breaking TRS \textit{via} the valley-selective optical Stark and Bloch-Siegert effects~\cite{Herrmann2025,Sie2017}, see Fig.~2f. In this new configuration, the symmetry of the monolayer is reduced to $\bar6m'2'$, as discussed above and depicted in Fig.~2e. This symmetry reduction is equivalent to a modification of the $\boldsymbol{\chi^{(3)}}$ tensor. In addition to the time-invariant so-called \textit{intrinsic} $\chi^{(3)}_\mathrm{int}$ elements, corresponding to the unperturbed $\bar6m2$ point group, time-non-invariant $\chi^{(3)}_\mathrm{TRS}$ elements induced by the broken TRS appear. As we show in detail in Supplementary Information S3, the presence of these additional elements leads to a rotation $\theta$ of the TH polarization compared to the unperturbed case, when assuming normal incidence of the electromagnetic field on the sample and within the electric dipole approximation, according to:

\begin{align}
\label{eq:rotation_angle_ellipse}
    \tan 2\theta = 2\,\frac{\text{Re}(\chi^{(3)}_{\mathrm{int}}\chi^{(3)*}_{\mathrm{TRS}})}{\abs{\chi^{(3)}_{\mathrm{TRS}}}^2-\abs{\chi^{(3)}_{\mathrm{int}}}^2}\approx -2
     \,\frac{\text{Im}(\chi^{(3)}_{\mathrm{TRS}})}{\text{Im}(\chi^{(3)}_{\mathrm{int}})}
\end{align}

where $^*$ denotes the complex conjugate. Here, we have assumed $|\chi^{(3)}_{\mathrm{TRS}}|\ll |\chi^{(3)}_{\mathrm{int}}|$ and $\text{Re}(\chi^{(3)}_{\mathrm{int}})=\text{Re}(\chi^{(3)}_{\mathrm{TRS}})=0$ at resonance (see Supplementary Information sections S4 and S5 for details).

To gain a deeper understanding of the microscopic mechanisms of TRS breaking, we employ analytical expressions for $  \chi^{(3)}_{\mathrm{int}}$ and $\chi^{(3)}_{\mathrm{TRS}}$ derived from the semiconductor Bloch equations~\cite{Aversa1995,Seith2024,Herrmann2025}. 
The expressions depend on the band structure and dipole matrix elements, which we obtain from a two-band TMD monolayer Hamiltonian including the optical Stark and Bloch–Siegert shift that break TRS~\cite{Herrmann2025}.
The final analytical expressions for $  \chi^{(3)}_{\mathrm{int}}$ and $\chi^{(3)}_{\mathrm{TRS}} $ are given in the Supplementary Information section S6, and we use them to evaluate the TH rotation $\theta$ from equation~(\ref{eq:rotation_angle_ellipse}) as
\begin{align}
%\tan 2\gamma=-\,\frac{\hbar}{T_2}\,\frac{\sin2\alpha}{(2\Delta-3\hbar\omega)^2+(\hbar/T_2)^2}\,\frac{3\mathcal{E}^2d^2}{8\Delta}\,. \label{blah}\\
\tan 2\theta&=
-\, \frac{2}{1+(2\Delta-3\hbar\omega)^2/(\hbar/T_2)^2}\;\frac{\boldsymbol{\Delta}E}{\hbar/T_2}
\\[0.5em]
&=
-\, \frac{3}{16}\,\sin2\alpha\;\frac{1}{1+(2\Delta-3\hbar\omega)^2/(\hbar/T_2)^2}\;\frac{\mathcal{E}^2d^2}{\hbar\Delta/T_2}\,. \label{e8}
\end{align}

Here, $\Delta$ is the onsite energy ($2\Delta$ is the optical gap), $T_2$ is the dephasing
time, $\omega$ is the frequency of the beam, and $\boldsymbol{\Delta}E$ is
the difference of the optical Stark and Bloch-Siegert shifts between the two
valleys (Fig.~2).
$\boldsymbol{\Delta}E$ is proportional to the square of field strength $\mathcal{E}$ and the absolute value of
the transition dipole moment ($d$) which is identical in the two valleys.


%\begin{figure}[ht!]
%    \centering
%    \includegraphics[width=\linewidth]{Figures/FIG3.pdf}
%    \caption{\textbf{PL and TH rotation measurements for different ellipticities and pulse energies in monolayer \ce{WS_2}.} \textbf{a}, Comparison of the total emitted TH intensity for linear fundamental polarization (black squares, left axis) as a function of FB wavelength and the emitted PL (pink line, right axis). The inset shows an optical microscopy image of the used \ce{WS_2} sample with the mono- and bilayer regions marked by dotted yellow and orange lines, respectively.  \textbf{b}, TH rotation angle as a function of the FB ellipticity angle and for an excitation pulse energy of \SI{263}{\pico\joule} (orange circles) and \SI{395}{\pico\joule} (pink squares). Solid lines are linear fits to the data. The inset shows the elliptical polarization pattern of the emitted TH for linear (violet triangles) and elliptical (blue circles) input polarization. \textbf{c}, Pulse energy dependence of the TH rotation for \SI{-20}{\degree} (blue circles) and \SI{-30}{\degree} (green squares). Solid lines are linear fits to the data with a fixed intercept of \SI{0}{\degree} rotation at \SI{0}{\pico\joule}. \textbf{d}, Wavelength dependence of the TH rotation angle for an input pulse energy of \SI{395}{\pico\joule} for ellipticity angles of \SI{-20}{\degree} (blue circles) and \SI{-30}{\degree} (green squares). Data in \textbf{b}, \textbf{c} and \textbf{d} are presented as mean values $\pm$ regression standard error.}
%    \label{fig:FIG3_experimental_results_monolayer}
%\end{figure}

We can now compare our theoretical model with experimental results for monolayer \ce{WS_2} (Fig.~3). 
The \ce{WS_2} sample (see Methods for details) in the inset of Fig.~3a consists of a mono-/bilayer region (marked in yellow/orange, respectively). Since breaking of TRS is expected to mainly manifest at optical resonances~\cite{Herrmann2025} (see equation~(\ref{e8}): $2\Delta=3\hbar\omega$), we first perform PL and TH wavelength dependence to identify these resonances. For the TH wavelength dependence, the fundamental beam is linearly polarized. We find the $A1s$ excitonic resonance at $\SI{615}{nm}$ (Fig.~3a), where we notice a slight blue-shift of the TH resonance compared to the PL emission. This can be ascribed to a valley-symmetric bandgap opening in the case of excitation with linearly polarized light \cite{klimmer2025ultrafastcoherentbandgapmodulation,trivizas2025tuningcoherentlightgeneration}, which modulates the bandgap without breaking TRS. In all following experiments (see Methods for details), the TH rotation angle is measured by rotating a polarizer in front of the detector to obtain a polarization-dependent pattern for the TH signal, as shown in the inset of Fig.~3b for two exemplary ellipticity angles of \SI{0}{\degree} (linearly polarized) and \SI{-20}{\degree} (elliptically polarized). Further exemplary polarization patterns are provided in Supplementary Information section S7. The polarization-dependent TH patterns are fitted with a $\cos^2$ function, from which we obtain the rotation angle $\theta$ and the error of the numerical fitting. We fix the wavelength of the FB at~\SI{1830}{nm} to work close to the $A1s$ resonance, and perform two different sets of experiments: (1) we fix the input pulse energy and tune the ellipticity of the input FB by rotating a quarter-wave plate in front of the sample (Fig.~3b); (2) we fix the ellipticity and tune the input pulse energy (Fig.~3c). In addition, we scan the wavelength across the $A1s$ resonance with fixed pulse energy and ellipticity (Fig.~3d).

For the first type of experiments (Fig.~3b) we observe a clear rotation of the main axis of the polarization ellipse, which can be explained by the new elements of the NLO susceptibility due to broken TRS, as discussed above. The TH rotation angle scales linearly with the FB ellipticity: a larger circular component of the FB enhances the effect of TRS breaking. We note that circular THG is forbidden by angular momentum conservation~\cite{Bloembergen1970}, and the total TH intensity scales with the $\cos^2$ of the ellipticity. Thus, since  TH intensity decreases drastically for increasing ellipticity of the FB beam, we investigate the TH rotation only up to an ellipticity angle of \SI{\pm30}{\degree}. 

%\begin{figure}[ht!]
%    \centering
%    \includegraphics[width=\linewidth]{Figures/FIG4.pdf}
%    \caption{\textbf{Comparison of analytical model and experimental results for monolayer \ce{WS_2}.} \textbf{a}, Wavelength-dependent TH rotation for fixed ellipticity of \SI{-20}{\degree} and input pulse energy of \SI{395}{\pico\joule} (blue circles in Fig.~\ref{fig:FIG3_experimental_results_monolayer}d). \textbf{b}, Ellipticity-dependent TH rotation for a fixed input pulse energy of \SI{395}{\pico\joule} (pink squares in Fig.~\ref{fig:FIG3_experimental_results_monolayer}b). \textbf{c}, Pulse energy-dependent TH rotation for a fixed ellipticity of \SI{-20}{\degree} (blue circles in Fig.~\ref{fig:FIG3_experimental_results_monolayer}c).
%    For the analytical calculations, we employ $T_2\eqt 28\,\text{fs}$,  $\Delta\eqt$ \SI{1.05}{\electronvolt}  and two values of the dipole element: $d\eqt\SI{3}{\eangstrom}$ (solid lines) and $d\eqt\SI{3.5}{\eangstrom}$ (dashed lines). Data are presented as mean values $\pm$ regression standard error.
%    }
%    \label{fig:FIG4_theory_monolayer}
%\end{figure}

In the second set of experiments, we fix the ellipticity angle to \SI{-30}{\degree} and \SI{-20}{\degree} and scan the input pulse energy in the range from \SI{40}{\pico\joule} to \SI{355}{\pico\joule}. The results are shown in Fig.~3c, where we observe a close to linear dependence of the TH rotation angle with respect to the input pulse energy, as expected for TRS breaking due to OS and BS shifts~\cite{Herrmann2025}. We highlight that in the fitting of Fig.~3c we fixed the intercept to zero, because the TH rotation must be zero for an unperturbed sample. Finally, we measure the wavelength dependence of the TH rotation angle close to the exciton resonance for two different ellipticity angles of \SI{-20}{\degree} and \SI{-30}{\degree} and a fixed input pulse energy of \SI{395}{\pico\joule} on the sample (Fig.~3d). 
We observe the largest TH rotation for excitation at the $A1s$ resonance, while the rotation angle decreases for off-resonant wavelengths. We further note that even though the TH intensity close to resonances can be affected by bandgap modulations that preserve TRS (\textit{e.g.,} excitation with linearly polarized light \cite{klimmer2025ultrafastcoherentbandgapmodulation,trivizas2025tuningcoherentlightgeneration}), the TH Faraday rotation only measures those changes in the bandgap that effectively break TRS.

Next, we quantitatively compare the experimental results to the analytical equation~\eqref{e8} in Fig.~4. According to equation~\eqref{e8}, the TH rotation~$\theta$ depends on the ellipticity angle~$\alpha$, frequency~$\omega$ and field strength~$\mathcal{E}$ of the fundamental beam. Furthermore, $\theta$ also depends on material parameters: optical gap~$2\Delta$, dipole element~$d$, and dephasing time~$T_2$. To determine $2\Delta$ and $T_2$, we focus on the peak of $\theta(\omega)$ in equation~\eqref{e8} at the TH resonance, where $3\omega\eqt 2\Delta$. From the peak position $\approx \SI{615}{\nano\meter}$ in the wavelength scan (Fig.~4a), we extract~$2\Delta\eqt \SI{2.1}{\electronvolt}$ and from the linewidth of the peak, we obtain $T_2\eqt \SI{28}{\femto\second}$. We note that the theoretical model only includes dephasing mechanisms such as electron-electron and electron-phonon scattering, leading to a homogeneous broadening. However, the experiments are sensitive to any homogeneous and inhomogeneous broadening mechanism, including space-local bandgap modulations caused by defects and strain~\cite{vandeGroep2023}. Our extracted value $T_2\eqt28$\,fs effectively includes all of these broadening mechanisms. 

After determining~$\Delta$ and $T_2$, we report the TH rotation $\theta$ from equation~\eqref{e8} for two values of the dipole element, $d\eqt$\SI{3.0}{\eangstrom} and $d\eqt$\SI{3.5}{\eangstrom}. The theoretical and experimental results show good agreement in their wavelength dependence, with peak heights matching within a factor of two (Fig.~4a). For the ellipticity dependence (Fig.~4b), the analytical model gives a nonlinear curve which describes with excellent accuracy the experimental results. The deviation from the linear dependence, which we use for simplicity in Fig.~3b, can be understood from the $\sin 2\alpha$ dependence in equation~\eqref{e8}, assuming small $\theta$ such that $\tan\theta\approx\theta\propto \sin 2\alpha$. Similarly, we find the linear increase of $\theta$ with the fundamental pulse energy both in experiment and in the analytical model (\textit{via} $\mathcal{E}^2$ in equation \eqref{e8}, Fig.~4c). 


\subsection{TH Faraday rotation in centrosymmetric bilayer WS\textsubscript{2}}

We now demonstrate the possibility to probe broken TRS in centrosymmetric bilayer \ce{WS_2}. 
%\begin{figure}[h!]
%    \centering
%    \includegraphics[width=\linewidth]{Figures/FIG5.pdf}
%    \caption{\textbf{PL and TH rotation measurements for different ellipticity angles and pulse energy values in bilayer \ce{WS_2}.} \textbf{a}, Comparison of the total emitted TH intensity (black squares, left axis) and the emitted PL for linear fundamental polarization (pink line, right axis) as a function of the excitation wavelength. \textbf{b}, Rotation of the emitted TH dependent on the ellipticity angle and for an excitation pulse energy of \SI{263}{\pico\joule} (orange circles) and \SI{447}{\pico\joule} (pink squares). Solid lines are linear fits to the data. \textbf{c}, Pulse energy dependence of the TH rotation for \SI{-20}{\degree} (blue circles) and \SI{-30}{\degree} (green squares). Solid lines are linear fits to the data with a fixed intercept of \SI{0}{\degree} rotation at \SI{0}{\pico\joule}. \textbf{d}, Wavelength dependence of the TH rotation for an input pulse energy of \SI{395}{\pico\joule} for ellipticity angles of \SI{-20}{\degree} (blue circles) and \SI{-30}{\degree} (green squares). Data in \textbf{b}, \textbf{c} and \textbf{d} are presented as mean values $\pm$ regression standard error.}
%    \label{fig:FIG5_experimental_results_bilayer}
%\end{figure}
Breaking of TRS in TMD bilayers follows a similar principle as in the monolayer case (Fig.~2g-l): an elliptical input pulse leads to alternating gap opening (in the $+$K valley of the upper layer, in the $-$K valley of the bottom layer) due to spin-valley-layer locking~\cite{Jones2014}, see Fig.~2l. When TRS is broken, the valleys remain energetically degenerate, but they are no longer spin-degenerate. This can be understood by considering the energy-spin relation of a system where SIS is preserved ($E_{\uparrow} (+\bk) = E_{\uparrow} (-\bk) $) and TRS is broken ($E_{\uparrow} (+\bk) \neq E_{\downarrow} (-\bk) $). Looking at the magnetic point groups, bilayer TMDs belong to $\bar3m$/$\bar3m'$ when TRS is preserved/broken. These point groups have the same in-plane components of the $\boldsymbol{\chi^{(3)}}$ susceptibility as the $\bar6m2$/$\bar6m'2'$ of monolayer TMDs (when TRS is preserved/broken). Thus, for in-plane excitation of the sample, we can apply exactly the same considerations done for monolayer TMDs also to the case of bilayer TMDs. 

We perform the same set of experiments discussed in Fig.~3 also for a bilayer \ce{WS_2} crystal (see Fig.~5). 

We start with PL and TH wavelength dependence to determine the $\pm $K direct transitions of the \ce{WS_2} bilayer (Fig.~5a). The PL signal shows two maxima, which we assign to the momentum direct and phonon-assisted momentum indirect transitions~\cite{Echeverry2016,doi:10.1021/acs.jpcc.9b08602} at \SI{635}{\nano\meter} and \SI{705}{\nano\meter}, respectively. Momentum indirect transitions cannot be probed by coherent and parametric harmonic generation, as they require the exchange of momentum with, \textit{e.g.}, defects or phonons. For an emission wavelength of \SI{625}{\nano\meter} (FB at \SI{1875}{\nano\meter}) we observe the maximum TH and assign the difference between PL and TH measurements to the valley and spin symmetric bandgap modulation induced by linearly polarized light \cite{klimmer2025ultrafastcoherentbandgapmodulation,trivizas2025tuningcoherentlightgeneration}. In our TH experiments, we will study only the momentum direct $\pm $K transitions at a FB wavelength of \SI{1875}{\nano\meter}. 

In analogy with the monolayer case, the ellipticity dependence (Fig.~5b) shows a close to linear dependence of the TH rotation angle \textit{versus} the ellipticity of the FB. The TH rotation in the case of bilayer \ce{WS_2} follows the same rules discussed in the previous paragraph. Next, we study the pulse energy dependence of the TH rotation angle (Fig.~5c) for different values of the ellipticity angle (\SI{-20}{\degree} and \SI{-30}{\degree} in blue circles and green squares, respectively). We observe a close to linear dependence, which is again a clear indication that also in bilayer TMDs broken TRS, for our experimental conditions, arises from all-optical bandgap modulation. Finally, we measure the wavelength dependence of the TH rotation angle for two different ellipticity angles of \SI{-20}{\degree} and \SI{-30}{\degree} and input pulse energy of \SI{395}{\pico\joule} (Fig.~5d). We observe the largest TH rotation for excitation at \SI{625}{\nano\meter}, \textit{i.e.}, at resonance with the momentum direct $\pm$K optical transitions. 

All these experimental findings agree with our analytical model for bilayer WS$_2$, which shows that both valleys contribute equally to the TH rotation. The model (see Supplementary Information section S6 for details) gives a TH rotation angle for the bilayer of:

\begin{align}
\tan 2\theta&=
-\, \frac{2}{1+(2\Delta
%-\lambda
-3\hbar\omega)^2/(\hbar/T_2)^2}\;\frac{\deltagapbl}{\hbar/T_2}
\\[0.5em]
&=
-\, \frac{3}{16}\sin2\alpha\;\frac{1}{1+(2\Delta
%-\lambda
-3\hbar\omega)^2/(\hbar/T_2)^2}\;\frac{\mathcal{E}^2d^2}{\hbar \Delta /T_2}\,. 
\label{e13}
\end{align}

This expression is almost identical to equation~\eqref{e8} obtained in the case of monolayer TMDs, with the main difference that the spin-valley exclusive bandgap modulation of monolayer TMDs (\textit{i.e.}, $\Delta E$) is substituted by the valley-symmetric, but spin-layer selective bandgap modulation (\textit{i.e.}, $\deltagapbl$) in the case of bilayer TMDs. Besides, the primary parametric difference between mono- and bilayer samples is the dephasing time $T_2$, which is shorter in the bilayer, an indirect gap semiconductor with additional non-radiative decay channels compared to the monolayer. As a result, the model in equations~\eqref{e8} and~\eqref{e13}  predicts a reduced TH rotation~$\theta_\text{res}$ at the TH resonance, since $\tan{2\theta_\text{res}}\propto T_2$. This prediction aligns with our experimental observations, where the TH rotation in the bilayer is approximately half compared to that of the monolayer. Finally, we note that also the interlayer coupling (which we neglect in our model, being significantly smaller compared to the spin-orbit-coupling in \ce{W} based TMDs \cite{Liu2015}) could play a role in explaining the lower TH rotation of bilayer compared to monolayer samples.

\section{Conclusion}

We have demonstrated an all-optical method to probe broken TRS in both centrosymmetric and non-centrosymmetric systems. The approach is based on the measurement of the TH Faraday rotation, namely a rotation of the TH polarization angle with respect to the polarization of the input FB. We realize and test TH Faraday rotation in two exemplary crystals: monolayer \ce{WS_2} for the non-centrosymmetric case and bilayer \ce{WS_2} for the centrosymmetric case. In both systems, broken TRS has fundamental implications for the understanding of the light-modulated band structure as well as for technological applications. For monolayers, broken TRS induces a valley imbalance, which could be the source of information for valleytronic operations. In bilayer TMDs, broken TRS interacts with the spin-valley-layer locking, that is unique of this type of layered van der Waals structures. For both samples, we show that broken TRS is induced by all-optical and ultrafast energy-spin modulation, and the effect is probed by a sizeable TH rotation angle up to \SI{12.5}{\degree} under our experimental conditions. These results represent an important step towards the development of new all-optical diagnostic techniques to probe broken TRS in any system, regardless of SIS. In addition, this work provides a viable approach for the realization of all-optical and ultrafast valleytronic devices. 
%TC:ignore

\section{Acknowledgements}
S.S.S. thanks Bashab Dey and Artur Slobodeniuk for their helpful discussions. This work was funded by the German Research Foundation (DFG) (CRC 1375 NOA), project number 398816777 (subproject C4); the International Research Training Group (IRTG) 2675 ‘Meta-Active’, project number 437527638 (subproject A4); and the Federal Ministry for Education and Research (BMBF) project number 16KIS1792 SiNNER. J.W. acknowledges the DFG for funding via the Emmy Noether Programme (project number 503985532), CRC 1277 (project number 314695032, subproject A03) and RTG 2905 (project number 502572516).

\section{Author contributions statement}

F.F., P.H. and G.S. conceived the work. F.F. and P.H. performed the measurements and analyzed the data. F.F., P.H. and G.S. interpreted experimental results. S.K. fabricated the sample. S.S.S. and J.W. developed the analytical model. The manuscript was written with contributions from all co-authors.

\section{Competing Interests Statement}

The authors declare no competing interests. 

\section{Figure Captions}
\subsection{Figure 1}
\textbf{Principle of TH Faraday rotation and certain symmetry operations for a spin-oriented three-fold rotational symmetric system.} \textbf{a}, For a monolayer TMD, broken TRS is equivalent to an asymmetry between the $\pm$K valleys. TRS breaking introduces new $\boldsymbol{\chi^{(3)}}$ tensor elements, leading to a rotation of the TH output by the angle $\theta$. \textbf{b}, The behavior of a three-fold rotational system with spin-oriented states under the vertical mirror and $C_2$ operation. If mirrored vertically, the system changes the spin state from up (straight black arrow pointing upwards) to down (straight red arrow pointing downwards). The same is true for a two-fold rotation operation ($C_2$ operation) in-plane.

\subsection{Figure 2}

\textbf{Probing TRS breaking in mono- and bilayer \ce{WS_2} \textit{via} TH Faraday rotation.} \textbf{a}, Real-space schematic of a \ce{WS_2} monolayer with broken SIS and preserved TRS. \textbf{b}, Stereographic projection of the $D_{3h} \equiv \bar6m2$ crystallographic point group. \textbf{c}, Energy and spin of the $\pm$K valleys of a \ce{WS_2} monolayer with preserved TRS. Linear excitation leads to linear TH parallel to the input. \textbf{d}, Real-space schematic of a \ce{WS_2} monolayer with broken SIS and TRS. Due to the spin-selective gap opening, one spin dominates. \textbf{e}, Stereographic projection of the $\bar6m'2'$ magnetic point group. Red elements represent symmetry operations that are only allowed under antisymmetry. \textbf{f}, Energy and spin of the $\pm$K valleys of a \ce{WS_2} monolayer with broken TRS. Elliptical excitation leads to the opening of a gap at the $+$K valley by $\mathbf{\Delta} E$ and a rotated TH output. \textbf{g}, Real-space schematic of a \ce{WS_2} bilayer with preserved SIS and TRS. \textbf{h}, Stereographic projection of the $D_{3d} \equiv \bar3m$ crystallographic point group. \textbf{i}, Energy and spin of the $\pm$K valleys of a \ce{WS_2} bilayer with preserved TRS. Within one layer, the spin states differ for each valley because of spin-valley-layer locking. Linear excitation leads to linear TH parallel to the input. \textbf{j}, Real-space schematic of a \ce{WS_2} bilayer with preserved SIS, but broken TRS. Due to spin-selective gap opening one of the spins is now dominant. \textbf{k}, Stereographic projection of the $\bar3m'$ magnetic point group. \textbf{l} Energy and spin of the $\pm$K valleys of a \ce{WS_2} bilayer with broken TRS. Elliptical excitation leads to the opening of a gap by $\mathbf{\Delta} E$ in the top and bottom layers and thus to a rotated TH output.

\subsection{Figure 3}

\textbf{PL and TH rotation measurements for different ellipticities and pulse energies in monolayer \ce{WS_2}.} \textbf{a}, Comparison of the total emitted TH intensity for linear fundamental polarization (black squares, left axis) as a function of FB wavelength and the emitted PL (pink line, right axis). The inset shows an optical microscopy image of the used \ce{WS_2} sample with the mono- and bilayer regions marked by dotted yellow and orange lines, respectively.  \textbf{b}, TH rotation angle as a function of the FB ellipticity angle and for an excitation pulse energy of \SI{263}{\pico\joule} (orange circles) and \SI{395}{\pico\joule} (pink squares). Solid lines are linear fits to the data. The inset shows the elliptical polarization pattern of the emitted TH for linear (violet triangles) and elliptical (blue circles) input polarization. \textbf{c}, Pulse energy dependence of the TH rotation for \SI{-20}{\degree} (blue circles) and \SI{-30}{\degree} (green squares). Solid lines are linear fits to the data with a fixed intercept of \SI{0}{\degree} rotation at \SI{0}{\pico\joule}. \textbf{d}, Wavelength dependence of the TH rotation angle for an input pulse energy of \SI{395}{\pico\joule} for ellipticity angles of \SI{-20}{\degree} (blue circles) and \SI{-30}{\degree} (green squares). Data in \textbf{b}, \textbf{c} and \textbf{d} are presented as mean values $\pm$ regression standard error.

\subsection{Figure 4}

\textbf{Comparison of analytical model and experimental results for monolayer \ce{WS_2}.} \textbf{a}, Wavelength-dependent TH rotation for fixed ellipticity of \SI{-20}{\degree} and input pulse energy of \SI{395}{\pico\joule} (blue circles in Fig.~3d). \textbf{b}, Ellipticity-dependent TH rotation for a fixed input pulse energy of \SI{395}{\pico\joule} (pink squares in Fig.~3b). \textbf{c}, Pulse energy-dependent TH rotation for a fixed ellipticity of \SI{-20}{\degree} (blue circles in Fig.~3c).
    For the analytical calculations, we employ $T_2\eqt 28\,\text{fs}$,  $\Delta\eqt$ \SI{1.05}{\electronvolt}  and two values of the dipole element: $d\eqt\SI{3}{\eangstrom}$ (solid lines) and $d\eqt\SI{3.5}{\eangstrom}$ (dashed lines). Data are presented as mean values $\pm$ regression standard error.

\subsection{Figure 5}

\textbf{PL and TH rotation measurements for different ellipticity angles and pulse energy values in bilayer \ce{WS_2}.} \textbf{a}, Comparison of the total emitted TH intensity (black squares, left axis) and the emitted PL for linear fundamental polarization (pink line, right axis) as a function of the excitation wavelength. \textbf{b}, Rotation of the emitted TH dependent on the ellipticity angle and for an excitation pulse energy of \SI{263}{\pico\joule} (orange circles) and \SI{447}{\pico\joule} (pink squares). Solid lines are linear fits to the data. \textbf{c}, Pulse energy dependence of the TH rotation for \SI{-20}{\degree} (blue circles) and \SI{-30}{\degree} (green squares). Solid lines are linear fits to the data with a fixed intercept of \SI{0}{\degree} rotation at \SI{0}{\pico\joule}. \textbf{d}, Wavelength dependence of the TH rotation for an input pulse energy of \SI{395}{\pico\joule} for ellipticity angles of \SI{-20}{\degree} (blue circles) and \SI{-30}{\degree} (green squares). Data in \textbf{b}, \textbf{c} and \textbf{d} are presented as mean values $\pm$ regression standard error.

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%TC:endignore

\section{Methods}

\subsection{Sample preparation and characterization}

The mono- and multilayer \ce{WS_2} sample was mechanically exfoliated onto polydimethylsiloxane (PDMS) from a bulk crystal (HQ graphene), and then transferred onto a fused silica substrate. The different thicknesses (mono- and bilayer) were confirmed by PL and contrast measurements. For the PL measurements, we used a \SI{532}{\nano\meter} diode laser (Cobolt 08-DPL \SI{532}{\nano\meter}) as excitation source, which we focus onto the sample with a 50x objective (M Plan Apo 50x, Mitutoyo). The emitted PL was collected in reflection geometry and guided to the spectrometer (Horiba iHR550) after proper filtering with a longpass dichroic mirror (LPD02-532RU-25, Semrock) and a notch filter (FL532-3, Thorlabs).

\subsection{Polarization-resolved THG}

For all the THG experiments, we use a home-built multi-photon microscope in transmission geometry. An optical parametric oscillator (Levante IR fs from APE), pumped by the output of an Yb-doped mode-locked laser (FLINT FL2-12, LIGHT CONVERSION) with a repetition rate of \SI{76}{\mega\hertz} and a pulse length of $\thicksim \SI{100}{\femto\second}$, generates the FB, which is tunable in the range \SI{1300}{\nano\meter} to \SI{2000}{\nano\meter}. To precisely control the pulse energy of the FB, we use a combination of a fixed Glan-Thompson polarizer (GTH10M, Thorlabs) and a wire grid polarizer (WP25M-UB, Thorlabs), which is mounted on a motorized rotation stage (PRM1/MZ8, Thorlabs). Subsequently, the ellipticity is controlled \textit{via} a quarter-wave plate (RSU 2.4.15, B. Halle) mounted on another motorized rotation stage (PRM1/MZ8, Thorlabs). The FB is focused onto the sample by a 40x mirror objective (LMM-40x-UVV) with a focal spot radius of \SI{1.85}{\micro\meter}. The transmitted FB as well as the emitted TH are collimated by a lens (C330TMD, Thorlabs). To obtain polarization-resolved results, we place another wire grid polarizer (WP25M-UB, Thorlabs) after the collimation and before the detector. We block the transmitted FB by two heat absorbing filters (ISK171, Isuzu) as well as a shortpass filter (FESH0700, Thorlabs). Furthermore, bandpass filters for \SI{610}{\nano\meter} (FBH610-10, Thorlabs) and \SI{625}{\nano\meter} (87791, Edmund Optics) can be added. The TH signal is then detected by a silicon avalanche photodiode (APD440A, Thorlabs) and filtered by a lock-in amplifier (MFLI, Zurich Instruments) in combination with an optical chopper (MC2000B, Thorlabs) that modulates the FB at \SI{971}{\hertz}; or by a single photon avalanche diode (SPAD, C22303-050, Hamamatsu) in combination with a counter (HM8123, Rohde \& Schwarz). 

\subsection{Analytical methods}

For the analytical calculation of polarization-resolved THG, we employ an analytical expression of the third-order NLO susceptibility developed in Ref.~\cite{Aversa1995}. Details are given in the Supplementary Information sections S5 and S6.


\section{Data availability}
The data supporting the plots within this article and other findings of this study are available from the corresponding author upon reasonable request.

\section{Methods-only references}

[42] Aversa, C. \& Sipe, J. E. Nonlinear optical susceptibilities of semiconductors: Results with a
length-gauge analysis. \textit{Phys. Rev. B} \textbf{52}, 14636–14645 (1995). URL \hyperlink{https://link.aps.org/
doi/10.1103/PhysRevB.52.14636.}{https://link.aps.org/
doi/10.1103/PhysRevB.52.14636.}

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