Magnetic control of an exciton–polariton condensate in a van der Waals magnet

MainNon-equilibrium Bose–Einstein condensates in solids have provided a versatile platform for exploring macroscopic quantum phenomena1,2, ranging from superfluid propagation3 to nonlinear interactions4. Condensates of hybrid light-matter quasiparticles5,6, called exciton–polaritons, are particularly promising owing to their combination of photonic coherence and excitonic interactions, which makes them a candidate for quantum communication and cryptography applications7,8. The key challenge, however, lies in controlling the coherent condensate emission. So far, most polariton condensates have been realized in conventional semiconductor microcavities where quasiparticle interactions are weak and tunability typically requires a modification of the structure. Quantum materials that mediate interactions of exciton–polaritons with other quasiparticles, such as phonons or magnons, could provide a compelling path forward. While proximitizing superconducting condensates with spin-ordered materials has led to new functionalities9,10,11,12, exciton–polariton condensation in spin-ordered systems remains largely unexplored.Magnetic van der Waals (vdW) crystals13,14,15 have recently emerged as an attractive class of quantum materials supporting profound interactions between excitons, photons, phonons and magnons16,17,18,19,20,21. Particularly the air-stable layered A-type antiferromagnet CrSBr (Fig. 1a) stands out for its tightly bound, quasi-one-dimensional excitons that are polarized along the crystallographic b axis with giant oscillator strength22. The most prominent exciton resonance of bulk CrSBr occurs near an energy of 1.36 eV. At the surface, this resonance is shifted to typically 1.34 eV (ref. 23). Thus far, strong coupling of excitons to vertically confined photon modes in bulk crystals has been explored, where new magneto-optic correlations24,25, hyperbolic effects26, anisotropy27 and, most recently, signatures of bulk polariton condensation28 emerge. Exfoliated flakes of CrSBr form microscopic wires that are strongly elongated in the a axis, effectively realizing a slab waveguide geometry with additional lateral photon confinement.Fig. 1: Exciton–polaritons in a CrSBr cavity wire.Full size imagea, Crystal structure of CrSBr viewed along the a and c axes (blue spheres, Cr; yellow spheres, S; brown spheres, Br). Red arrows indicate the magnetic order in the antiferromagnetic phase. b, Schematic of the CrSBr microcavity structure. The zoom-in indicates the spatial profile of the laterally quantized polariton modes. An exemplary false-colour micrograph of a CrSBr microcavity is shown in the top-right corner. c, Simulated polariton dispersion along the short axis of the microwire obtained by modelling polaritons as particles in an elongated 2D rectangular potential well (Methods). Inset: in the experiment, we resolve this dispersion when the b axis of the microwire is aligned parallel to the entrance slit of the spectrometer. d, Simulated dispersion along the long axis of the microwire, equivalent to the experimental case where the a axis of our microcavity is aligned parallel to the entrance slit of the spectrometer (inset). e,f, Angle-resolved normalized PL of cavity 1 dispersed along the b axis (e) and along the a axis (f) measured using ultrafast photoexcitation (photon energy, 1.771 eV; \(\varPhi\) = 94 µJ cm−2; pulse duration, 240 fs). PL signals in the boxes delimited by broken white lines are enhanced to increase visibility. The white dots in e indicate the ground state (Q0), and the first (Q1) and second (Q2) excited states obtained from a coupled oscillator model (‘Main’). The black dashed line and curve in f show the dispersions of uncoupled exciton and cavity photon modes. The solid black curves display the theoretically fitted lower polariton branches from the coupled oscillator model.Source dataHere, we demonstrate the condensation of exciton–polaritons in such CrSBr microwires. Under ultrafast optical excitation, our microwires show the fingerprint of polariton condensation—that is, a dramatic increase in the photoluminescence (PL) intensity accompanied by linewidth narrowing and a pronounced energy shift. Optical interferometry of the PL emission reveals a clear increase in its spatial and temporal coherence. As a fully spin-polarized electronic manifold is underlying this condensate, its energy can be magnetically changed by up to 10.5 meV. We find that optical pumping of surface exciton states most efficiently drives condensation, pointing to an ultrafast excitation transfer via resonant magnon and phonon pathways. These spin-coupled exciton–polariton condensates open new avenues for designing and controlling waveguided coherent emission by external magnetic fields, magnons and spin fluctuations. They may hence be used to interface semiconductor optics and future photonic quantum technologies with spin-ordered matter.Magnetic microwire polaritonsTo achieve strong coupling between bulk excitons and a single photon mode confined in the vertical direction, few-nm CrSBr crystals are centred between two planar cavity mirrors using polymethyl methacrylate (PMMA) spacers (Methods). In an optical microscope, we identify few-layer CrSBr crystals that are elongated along the crystallographic a axis with typical lengths of tens of micrometres but exhibit a small width of approximately 1 µm in the b direction (see Fig. 1b for an exemplary microcavity). Such structures can act as microwires that spatially confine light like slab waveguides. The tightest confinement is given in the vertical (c axis) direction. The resulting strong coupling with the bulk exciton resonance and the emergent polaritonic effects are thus overall well described by numerical calculations assuming infinite lateral crystal extensions (Extended Data Fig. 1). A narrow width of the flake along the b axis, however, causes additional quantization29 (Fig. 1b, inset), which is less relevant along the a axis owing to the extended wire length in this direction. This situation can be qualitatively captured by modelling the polariton as a particle in an elongated two-dimensional (2D) rectangular potential well (Methods). The model predicts the confinement to cause a sequence of discrete quantized polariton states along the b axis with energy splittings on the order of tens of millielectronvolts (Fig. 1c). As the polaritons are barely restricted along the a axis, the dispersion of the polariton branches remains essentially parabolic along this direction (Fig. 1d).We first characterized the polariton PL of our microwires in the antiferromagnetic phase (lattice temperature, \({T}_{{\rm{L}}}=5\,{\rm{K}}\)), where the spins are aligned antiparallel between adjacent layers (Fig. 1a). Orienting CrSBr wires perpendicular and parallel to the entrance slit of the spectrometer allows us to image the polaritonic band dispersion along the two orthogonal angles \({\theta }_{b}\) and \({\theta }_{a}\) (Fig. 1c,d, insets). Light emitted at an angle \({\theta }_{a,b}\) originates from exciton–polaritons with energy \(E\) and in-plane momentum \({k}_{a,b}=\left(E/\hslash c\right)\sin {\theta }_{a,b}\) (refs. 1,5,30). As an example, we discuss the light emission of a CrSBr cavity with a width of \(W\approx 1.25\,{\rm{\mu }}{\rm{m}}\) and a length of \(L\approx 40\,{\rm{\mu }}{\rm{m}}\) (labelled cavity 1, Extended Data Fig. 1a), after optical excitation (pulse duration, 240 fs; repetition rate, 85 kHz) at a photon energy of \(\hslash {\omega }_{{\rm{pump}}}=1.771\,{\rm{eV}}\).Resolving the PL spectra along the b axis for −25° {\varPhi }_{\mathrm{th}}\) (Fig. 2d). The abrupt increase in the PL intensity from Q1 is accompanied by a substantial decrease in its linewidth, from 19 meV to 8.7 meV, and an energy increase by 5 meV (Fig. 2e,f). These features are unambiguous hallmarks of exciton–polariton condensation in microcavities. Note that the momentum spread of the polariton distribution along the b axis is not expected to narrow upon condensation because Q1 is a spatially confined state.Final-state stimulation inherently synchronizes the phase of all particles contributing to the condensate. The resulting spontaneous macroscopic spatial and temporal coherence should persist as long as the condensate lives. We prove this key aspect by measuring the first-order correlation function of the polariton PL emission of cavity 1 in a Michelson interferometer setup5,32. The PL emission is first split evenly into the two arms of the interferometer, before one of the beams is inverted centro-symmetrically by a retroreflector. The two beams are then combined by a lens and overlapped in the real-space image plane. Temporal overlap is set by an optical delay line. Figure 3a,b shows the spatially overlapped PL images at a delay time of t = 0 fs below and above the threshold, respectively. While signatures of coherence are absent below \({\varPhi }_{\mathrm{th}}\), interference fringes indicate spatially coherent emission for \(\varPhi > {\varPhi }_{\mathrm{th}}\).Fig. 3: Spatial and temporal coherence of the exciton–polariton condensate.Full size imagea,b, Zero-delay spatial interference pattern of the exciton–polariton PL of cavity 1 at \({T}_{{\rm{L}}}=5\,{\rm{K}}\) and a pump fluence of \(0.8{\varPhi }_{\mathrm{th}}\) (a) and \(2.0{\varPhi }_{\mathrm{th}}\) (b), where \({\varPhi }_{\mathrm{th}}\,\)= 30 µJ cm−2. c–e, Spatial interferogram of the exciton–polariton PL emission measured at delay times of t = −330 fs (c), −130 fs (d) and 0 fs (e) at a pump fluence of \(2.0{\varPhi }_{\mathrm{th}}\). f, Interference fringes averaged along the direction perpendicular to the fringes for t = 0 fs and a pump fluence of \(2.0{\varPhi }_{\mathrm{th}}\). g, Visibility of the interference pattern in f as a function of delay time t at a pump fluence of \(2.0{\varPhi }_{\mathrm{th}}\).Source dataSweeping t allows us to measure the temporal coherence of the condensate emission (Fig. 3c–e). We quantify the coherence time by extracting the visibility of the interference fringes, defined as \(\nu =\left({I}_{\max }-{I}_{\min }\right)/\left({I}_{\max }+{I}_{\min }\right)\), as a function of t (Fig. 3f,g). Here, \({I}_{\max }\) (\({I}_{\min }\)) represents the intensity of a local maximum (neighbouring minimum) of the fringes. The data are well described by a Gaussian profile with a full width at half maximum coherence time of 427 ± 7 fs. This value corresponds to a linewidth of ~4 meV, which is compatible with the width observed in Fig. 2e if the finite resolution of the spectrometer is considered (Methods). Most importantly, the measured coherence time exceeds the spontaneous polariton lifetime by one order of magnitude (Methods), unambiguously proving that the exciton–polaritons in CrSBr undergo condensation. The combination of a superlinear intensity scaling, an energy blueshift of the emission peak and enhanced coherence times signifies that the condensation process emerges spontaneously. Similar signatures are also observed in other comparable microcavities, even at elevated temperatures above 20 K and in PL spectra dispersed along the a axis (Extended Data Fig. 7).Characteristic aspects of polariton condensation in CrSBrSeveral aspects about our polariton condensates deserve special mention. First, we observe condensates only under near-resonant excitation of surface excitons, which lie energetically between the bulk exciton resonance and its polaritons (Fig. 2b and Extended Data Fig. 5). Pumping at a photon energy of 1.771 eV, in contrast, yields no condensation despite the much larger excitation densities created by strong B-exciton absorption21,23. The origin of this difference probably lies in the different relaxation pathways following high-energy excitation. Populating polariton states requires a sequence of scattering events, each constrained by energy–momentum matching. The phase space of such relaxation into the narrow condensate region is strongly reduced, and competing channels, such as redistribution of excitation energy into other excitonic states, radiative loss before relaxation or scattering into a broad incoherent reservoir dominate.Another notable observation is that polariton condensates are formed in excited rather than ground states of our microwires. In two of our samples (Fig. 2 and Extended Data Fig. 3), the energy difference between pump photons and the excited-state condensate emission closely matches the energy of an optical magnon mode of CrSBr (~45 meV), which has recently been revealed in neutron scattering36 and far-infrared spectroscopy37. Besides ultrafast excitation of the magnetic order and magnons38, emission of multiple \({{\rm{A}}}_{{\rm{g}}}^{1}\) phonons, which are known to couple strongly with excitons in CrSBr39,40, or a single \({A}_{{\rm{g}}}^{3}\) phonon should also efficiently populate the condensate (Extended Data Fig. 8). To understand the migration of electronic excitation, we calculated the lattice relaxation of the lowest energy exciton state in CrSBr. Density functional theory calculations reveal that electrons localize on the Cr atoms while holes localize on the S atoms. This charge distribution induces a structural distortion that increases the interlayer separation, coupling efficiently to the interlayer breathing \({A}_{{\rm{g}}}^{1}\) phonon mode (Extended Data Fig. 9). This strongly supports efficient excitation energy transfer via scattering with \({A}_{{\rm{g}}}^{1}\) phonons. Together, these results point to a consistent picture in which resonant phonon- and magnon-assisted scattering channels provide an efficient scattering pathway for transferring excitation energy from surface to bulk states. We suggest that wire-to-wire variations in confinement can tune polariton states into the right energy window for supporting resonant enhancement of these processes, thereby enabling the formation of non-equilibrium condensates in excited microwire states.Lastly, the fluence dependence of polariton energies observed in our study is qualitatively different from previous reports. So far, either a decrease in polariton energies24,41 or a competition between opposite shifts of polariton energies28 was noted with increasing excitation fluence. We, however, observe a monotonic increase in polariton energies (Fig. 2f). Circumventing other relaxation pathways, it thus seems that near-resonant excitation at kilohertz repetition rates isolates a contribution to the polariton energy that shifts its states towards higher energies. This is indeed typically ascribed to repulsive Coulomb interactions between excitons1,7,28,41,42 or saturable absorption effects43,44. In CrSBr, however, additional contributions may arise from the coupling of excitons to spin order: a joint theory–experiment study has recently shown that magnon-induced magnetic disorder can sizably reduce the oscillator strength of excitons in CrSBr23,26. It is therefore important to note that when excitons are strongly coupled to photons, a reduction in oscillator strength decreases the Rabi gap and consequently increases the lower polariton energy. This effect is particularly pronounced for lower polaritons with sizable photon fraction, such as those studied in our microwires. Hence, we suggest polariton energy shifts in CrSBr may involve contributions, not only from exciton–exciton repulsion but also from the (unavoidable) photoexcitation of magnons45,46 and the resulting renormalization of the Rabi gap.Magnetic field control of polariton condensatesIn any case, the exciton–polariton modes in CrSBr emerge from a manifold of spin-polarized electronic orbitals. Hence, spin order should provide a natural control knob for polariton condensates in CrSBr. To explore this idea, we mount a new microwire (cavity 5) into a magneto-optical cryostat with tunable out-of-plane magnetic fields (Extended Data Fig. 10). Figure 4a shows the angle-resolved PL emission from this microwire containing laterally quantized polariton states Q0, Q1 and Q2, the dispersionless emission from surface excitons (XS) and a defect emission peak at 1.265 eV labelled XD (ref. 47).Fig. 4: Magnetic control of an exciton–polariton condensate.Full size imagea, Angle-resolved PL spectra along the b axis for B = 0 T, measured at \({T}_{{\rm{L}}}=10\,{\rm{K}}\) under excitation by a CW laser with a photon energy of 1.907 eV. Q0, Q1 and Q2 represent the quantized polaritonic ground state, the first excited state and the second excited state, respectively. XS and XD are the surface exciton state and a defect state. b, Normalized, angle-integrated PL intensity as a function of the magnetic field B oriented along the c axis. The dashed lines serve as guides to the eye. c, Top: integrated PL intensity of state Q1 as a function of pump fluence at B = 0 T (left) and B = 2 T (right). The data are taken at \({T}_{{\rm{L}}}=10\,{\rm{K}}\) under pulsed excitation with a pump photon energy of 1.325 eV. The blue squares indicate the pump fluences used for the corresponding angle-resolved PL spectra in the bottom panels. Bottom: angle-resolved PL spectra along the b axis for B = 0 T (left) and B = 2 T (right) with a pump fluence of 60 µJ cm−2 and 90 µJ cm−2, respectively. Dashed white lines denote the central energy of state Q1, revealing a magnetically induced energy shift of the condensate of up to 10.5 meV. Solid grey lines mark the photon energy below which our long-pass spectral filter suppressing the incident pump light allows PL intensity to be collected.Source dataWe first tracked the magnetic field evolution of these laterally confined polariton modes under continuous-wave (CW) excitation. As the magnetic field increases from 0 T to 2 T, both the exciton–polariton branches and the surface exciton resonance exhibit pronounced redshifts of more than 10 meV, whereas the emission by localized defects remains nearly unaffected. The shape and magnitude of these shifts indicate their origin from a field-driven reconfiguration of the collective spin order rather than spin Zeeman splitting or the magnetic polaron effects observed with condensates in dilute magnetic semiconductors48,49,50. Importantly, the parabolic spectral shift saturates above 2 T, consistent with the known antiferromagnetic-to-ferromagnetic transition in CrSBr under out-of-plane magnetic bias. Each confined mode, Q0, Q1 and Q2, inherits a distinct magnetic field shift directly from their respective exciton fraction and the absolute shift of bulk excitons (~18 meV (refs. 21,24,39); Extended Data Fig. 4d). Compared with bulk excitons, the shift of surface excitons is notably reduced owing to their different magnetic environment causing a weaker dependence on the external field. All in all, our observations obtained under CW excitation demonstrate that magnetic order directly renormalizes the polaritonic eigenstates.We now extend this magnetic control to the condensation regime. Similar to our other microwires, the quantized polariton mode Q1 exhibits the characteristic superlinear increase in PL intensity at threshold both in the antiferromagnetic (B = 0 T) and ferromagnetic (2 T) phases. This confirms that condensation can be achieved in both the antiferromagnetic and the ferromagnetic configuration. Most strikingly, angle-resolved measurements reveal that the condensed state itself undergoes a field-induced energy shift of up to 10.5 meV. Such large tuning of a macroscopically coherent quantum state mediated by the spin order stands in contrast to previous electric-field-based strategies51,52,53. While the latter perturbatively modify the potential experienced by exciton–polaritons, the magnetic field in CrSBr changes magnetic order, symmetry and, ultimately, the nature of the exciton wavefunction from quasi-one-dimensional species to spatially extended three-dimensional wavefunctions as the material transitions from antiferromagnetic to ferromagnetic order22. This demonstrates spin-order-mediated control of an exciton–polariton condensate and, given the exceptionally large spin-induced energy shift, the concept opens an extremely promising interface between macroscopic quantum states and spintronics.ConclusionWe demonstrate polariton condensation in microwires of the vdW antiferromagnet CrSBr. The polaritons are laterally quantized by the finite dimension of the microwire along its short axis. Near-resonant pumping causes polariton condensation in excited quantized states, indicated by a multi-order increase in polariton PL emission, linewidth narrowing, energy shift and spatio-temporal coherence. As such condensates are intimately coupled to the underlying spin order, their energy can be shifted by up to 10.5 meV through reconfiguration of the magnetic order. Photoexcitation of surface states appears to enable particularly efficient population of polariton condensates in CrSBr microwires through phonon and magnon scattering pathways. Our results open exciting possibilities for interfacing the macroscopic quantum mechanical wavefunction of tailor-made condensates with spin order, magnetic control and spintronic concepts for next-generation quantum technologies. Particularly appealing are the prospects of modulating coherent condensate emission on ultrafast timescales by coupling it to coherent, high-frequency magnons. This platform could enable magnetic-memory integration and efficient microwave-to-optical transduction for quantum information purposes, where microwave photons couple to condensate emission via magnons.MethodsFabrication of the microwire cavitiesA 200-nm-thick gold mirror was evaporated on a silicon wafer and covered by a spacer layer of PMMA (nominal thickness, 70 nm). Subsequently, CrSBr was transferred onto the substrate by micro-mechanical exfoliation, capped by another 70-nm PMMA spacer and a semi-transparent, 40-nm-thin gold mirror to complete the microcavity. The schematic layout is shown in Fig. 1b. Using the transfer matrix formalism for this cavity, we obtain a Q factor of 62 for the bare cavity mode. Including the hybridization with the exciton resonance, the lower polariton mode investigated in Fig. 2 is computed to feature a lifetime broadening of 14 meV, corresponding to a lifetime of 47 fs.Optical spectroscopy and microscopy characterizationsFor the optical measurement, the microcavities are pumped by ultrafast laser pulses (repetition rate: 85 kHz, pulse duration: 240 fs) from an optical parametric amplifier (ORPHEUS; Light Conversion) driven by a commercial regenerative Yb:KGW laser amplifier system (PHAROS 10; Light Conversion). The relatively low repetition rate helps us to avoid pulse-to-pulse recovery time artefacts and allows sufficient pulse energies for future nonlinear optical experiments. For optical imaging of the sample, we use a tungsten–halogen light source. To detect the PL, we use a spectrometer (Acton SP2500) coupled with a Peltier-cooled silicon charge-coupled device (CCD) camera (PIXIS 256) with a resolution of 1,024 × 256 pixels. The width of the entrance slit was kept at several hundred micrometres to achieve sufficient statistics despite the low repetition rate. This setting leads to a linewidth broadening of typically ~6 meV. Cosmic-ray events on the CCD camera and slight asymmetries in Fourier images were digitally corrected. For the experiments of Figs. 1–3, the sample is mounted in a closed-cycle cryostat and cooled down by liquid helium to 5 K. To perform magnetic control studies (Fig. 4) we placed the microwire samples in a magnet cryostat with a large-aperture optical access and a variable out-of-plane magnetic field of up to 5 T.Our optical setup allows us to access both the real-space and momentum-space images of the PL from the sample. To obtain the spectrum and real-space image, the emission from the sample is projected onto the imaging plane, that is, the entrance slit of the spectrometer. To resolve the momentum (angle) distribution of the emission, an additional lens is placed into the beam path, such that the Fourier plane of the objective lens is projected on the entrance slit of the spectrometer.The coherence measurements are conducted in a Michelson interferometer, with a retroreflector mounted in one of the two interferometric arms to invert vertically and horizontally the spatial coordinates of the real-space image. The relative temporal delay between the two arms is controlled by an optical delay line. The two interference images are overlapped by a lens on the slit of the spectrometer and imaged by the CCD camera.Coupled oscillator modelThe exciton–polariton dispersion in Fig. 1f is fitted by the coupled oscillator model1:$$\left(\begin{array}{l}{E}_{\mathrm{cav}}\quad\hslash \varOmega /2\\ \hslash \varOmega /2\quad{E}_{\mathrm{exc}}\end{array}\right)\left(\begin{array}{c}\alpha \\ \beta \end{array}\right)=E\left(\begin{array}{c}\alpha \\ \beta \end{array}\right).$$Here \({E}_{{\rm{cav}}}\) and \({E}_{{\rm{exc}}}\) are the energies of the cavity photon and exciton, respectively. \(\hslash \varOmega\) is the vacuum Rabi splitting, related to the exciton–photon coupling strength \(g\) following \(\hslash {\varOmega }=2g\). Moreover, \(\alpha\) and \(\beta\) are the amplitudes of the eigenvectors. The eigenvalues of the energy can be obtained by diagonalizing the matrix: \({E}_{\pm }\left({k}_{\parallel }\right)=\frac{{E}_{\mathrm{cav}}+{E}_{\mathrm{exc}}}{2}\pm \frac{1}{2}\sqrt{{\left[{E}_{\mathrm{cav}}-{E}_{\mathrm{exc}}\right]}^{2}+{\hslash }^{2}{\varOmega }^{2}}\) with \({E}_{+}\) and \({E}_{-}\) being upper and lower polariton branches. Exciton and photon fractions (Extended Data Fig. 4) are determined by the amplitude squared of the Hopfield coefficients \({X}_{{k}_{\parallel }}\) and \({{C}_{{k}_{\parallel}}}\), respectively. They are given by$${\left|{X}_{{k}_{\parallel }}\right|}^{2}=\frac{1}{2}\left(1+\frac{{\Delta }E\left(k\right)}{\sqrt{{\Delta }E{\left({k}_{\parallel }\right)}^{2}+4{g}_{0}^{2}}}\right){\mathrm{and}}\,{\left|{C}_{{k}_{\parallel }}\right|}^{2}=\frac{1}{2}\left(1-\frac{{\Delta }E\left({k}_{\parallel }\right)}{\sqrt{{\Delta }E{\left({k}_{\parallel }\right)}^{2}+4{g}_{0}^{2}}}\right),$$with \(\Delta E={E}_{{\rm{cav}}}-{E}_{{\rm{exc}}}.\) Owing to both the vertical and lateral quantum confinement of the cavity, the cavity photon mode can be further expressed as27,28$${E}_{\mathrm{cav}}\left(j,{k}_{x}\right)=\frac{\hslash c{k}_{z}}{{n}_{c}}\sqrt{1+{\left[\frac{\left(j+1\right){\rm\pi }}{{L}_{y}}\right]}^{2}\frac{1}{{k}_{z}^{2}}+{\left(\frac{{k}_{x}}{{k}_{z}}\right)}^{2}}.$$Here, \({k}_{z}={\rm\pi }/{L}_{z}\) is the wavevector along the vertical direction of the cavity, with \({L}_{z}\) being the effective length of the cavity. \({k}_{x}\) and \({k}_{y}\) are the lateral wavevectors along the extended direction (a direction in our case) and along the confined direction (b direction), respectively. For each quantized mode \(j\left(\mathrm{0,1,2}\ldots \right)\) along the b direction, \({k}_{y}={\rm\pi }\left(j+1\right)/{L}_{y}\) with \({L}_{y}\) being the width of the cavity.‘Particle in a box’ simulationTo obtain the dispersions shown in Fig. 1c,d, we solve the 2D Schrödinger equation using the finite difference method implemented in an open-source code available at https://github.com/LaurentNevou/Q_Schrodinger2D_demo. The CrSBr flake is modelled as a rectangular potential well, whose dimensions are chosen according to the size of the sample. The barrier is set to 10 eV to mimic an infinite potential well. Consequently, the effective mass is the only free parameter of the model. The best agreement with the experimental data is obtained for effective masses \({m}_{a}={m}_{b}=4.8\times {10}^{-5}{m}_{0}\) along the a axis \(({m}_{a})\) and b axis \(\left({m}_{b}\right)\) with the free-electron mass \({m}_{0}\). For each eigenenergy, the obtained wavefunctions are converted to momentum-space using a fast Fourier transform. Finally, a Gaussian broadening in energy is applied for better comparability with the experiment.Computational details of the density functional theory calculationsElectronic structure calculations were performed using the CP2K suite (version 2025.2) with the Quickstep (GPW) method54. To accurately capture the electronic properties of the system, we used the HSE06 screened hybrid functional55, incorporating D3 dispersion corrections with Becke–Johnson damping56 to properly account for the long-range vdW interactions between the CrSBr layers. The TZVP-MOLOPT-PBE basis set was used in combination with Goedecker–Teter–Hutter (GTH) pseudopotentials. To maintain computational efficiency while treating the exchange correlation, the auxiliary density matrix method was used with a corresponding double-zeta basis set optimized for MOLOPT-GTH basis. The plane-wave cutoff of 800 Ry and the self-consistent field convergence was set to 10−8 Ha. Structural optimizations of the excited-state geometries (excitons) were conducted using the maximum overlap method57 to maintain the desired orbital occupation. All calculations were restricted to the Γ-point of the Brillouin zone. The 2D CrSBr layers were modelled using a 3 × 3 supercell of unit cell with in-plane dimensions of 4.74 Å × 3.55 Å with ferromagnetic intralayer coupling and antiferromagnetic interlayer coupling. To minimize artificial periodic interactions along the z axis, a vacuum spacing was included, resulting in total cell heights of 45 Å for the bilayer and 90 Å for the four-layer systems.