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# Proximity control of interlayer exciton-phonon hybridization in van der Waals heterostructures

## Abstract

Van der Waals stacking has provided unprecedented flexibility in shaping many-body interactions by controlling electronic quantum confinement and orbital overlap. Theory has predicted that also electron-phonon coupling critically influences the quantum ground state of low-dimensional systems. Here we introduce proximity-controlled strong-coupling between Coulomb correlations and lattice dynamics in neighbouring van der Waals materials, creating new electrically neutral hybrid eigenmodes. Specifically, we explore how the internal orbital 1s-2p transition of Coulomb-bound electron-hole pairs in monolayer tungsten diselenide resonantly hybridizes with lattice vibrations of a polar capping layer of gypsum, giving rise to exciton-phonon mixed eigenmodes, called excitonic Lyman polarons. Tuning orbital exciton resonances across the vibrational resonances, we observe distinct anticrossing and polarons with adjustable exciton and phonon compositions. Such proximity-induced hybridization can be further controlled by quantum designing the spatial wavefunction overlap of excitons and phonons, providing a promising new strategy to engineer novel ground states of two-dimensional systems.

## Introduction

Heterostructures of atomically thin materials provide a unique laboratory to explore novel quantum states of matter1,2,3,4,5,6,7,8,9,10,11. By van der Waals stacking, band structures and electronic correlations have been tailored, shaping moiré excitons1,2,3,4,5, Mott insulating1,8,9,10, superconducting7,10, and (anti-)ferromagnetic states6,7,8. The emergent phase transitions have been widely considered within the framework of strong electron–electron correlations12,13,14. Yet, theoretical studies have emphasized the role of electron–phonon coupling in atomically thin two-dimensional (2D) heterostructures, which can give rise to a quantum many-body ground state featuring Fröhlich polarons, charge-density waves, and Cooper pairs15,16,17,18. Unlike in bulk media, electronic and lattice dynamics of different materials can be combined by proximity. In particular, coupling between charge carriers and phonons at atomically sharp interfaces of 2D heterostructures are widely considered a main driving force of quantum states not possible in the bulk, such as high-Tc superconductivity in FeSe monolayer (ML)/SrTiO3 heterostructures11, enhanced charge-density wave order in NbSe2 ML/hBN heterostructures19 and anomalous Raman modes at the interface of WSe2/hBN heterostructures20. However, disentangling competing effects of many-body electron–electron and electron–phonon coupling embedded at the atomic interface of 2D heterostructures is extremely challenging and calls for techniques that are simultaneously sensitive to the dynamics of lattice and electronic degrees of freedom.

Here, we use 2D WSe2/gypsum (CaSO4·2H2O) heterostructures as model systems to demonstrate proximity-induced hybridization between phonons and electrically neutral excitons up to the strong-coupling regime. We tune a Coulomb-mediated quantization energy—the internal 1s–2p Lyman transition of excitons in WSe2—in resonance with polar phonon modes in a gypsum cover layer (Fig. 1a) to create new hybrid excitations called Lyman polarons, which we directly resolve with phase-locked few-cycle mid-infrared (MIR) probe pulses. Engineering the spatial shape of the exciton wavefunction at the atomic scale allows us to manipulate the remarkably strong exciton–phonon coupling and to induce a crosstalk between energetically remote electronic and phononic modes.

## Results

### Rydberg spectroscopy of Lyman polarons

We fabricated three classes of heterostructures—a WSe2 ML, a 3R-stacked WSe2 bilayer (BL), and a WSe2/WS2 (tungsten disulfide) heterobilayer (see Supplementary Note 1)—by mechanical exfoliation and all-dry viscoelastic stamping (see “Methods”). All samples were covered with a mechanically exfoliated gypsum layer and transferred onto diamond substrates. Figure 1b shows an exemplary optical micrograph of the WSe2 ML/gypsum heterostructure, where strong photoluminescence can be observed from the WSe2 ML, attesting to the radiative recombination of 1s A excitons (Fig. 1c). The MIR transmission spectrum of gypsum (Fig. 1d) features two absorption peaks caused by the vibrational $$\nu_4$$ and $$\nu_3$$ modes of the $${\mathrm{SO}}_4$$ tetrahedral groups at 78 and 138 meV, respectively (see “Methods”). These modes are spectrally close to the internal resonance between the orbital 1s and 2p states of excitons in WSe221,22 and are, thus, ideal for exploring the polaron physics that arises from the proximity-induced exciton–phonon coupling at the van der Waals interface. If the coupling strength exceeds the linewidth of both modes one may even expect exciton–phonon hybridization as the excitonic Lyman transition is resonantly dressed by the spatially nearby phonon field (Fig. 1a). In this proximity-induced strong-coupling scenario, Lyman polarons would emerge as new eigenstates of mixed electronic and structural character.

In the experiment, we interrogate the actual spectrum of low-energy elementary excitations by a phase-locked MIR pulse. The transmitted waveform is electro-optically sampled at a variable delay time, tpp, after resonant creation of 1s A excitons in the K valleys of WSe2 by a 100 fs near-infrared pump pulse (see “Methods”). A Fourier transform combined with a Fresnel analysis directly reveals the full dielectric response of the nonequilibrium system (see “Methods”). The pump-induced change of the real part of the optical conductivity, $$\Delta \sigma _1$$, and of the dielectric function, $$\Delta \varepsilon _1$$, describe the absorptive and inductive responses, respectively. The dielectric response of a photoexcited WSe2 ML covered with hBN at tpp = 0 ps (Fig. 2, gray spheres) is dominated by a maximum in $$\Delta \sigma _1$$ (Fig. 2a) and a corresponding zero crossing in $$\Delta \varepsilon _1$$ at a photon energy of 143 meV (Fig. 2b). This resonance matches with the established internal 1s–2p Lyman transition in hBN-covered WSe2 MLs21 and lies well below the E1u phonon mode in hBN (~172 meV, see “Methods”).

In marked contrast, $$\Delta \sigma _1$$ features a distinct mode splitting for the WSe2/gypsum heterostructure (Fig. 2a, red spheres). The two peaks and corresponding dispersive sections in $$\Delta \varepsilon _1$$ (Fig. 2b, red spheres) are separated by ~35 meV and straddle the internal 1s–2p Lyman resonance of the WSe2/hBN heterostructure. Interestingly, each peak is much narrower than the bare 1s–2p transition in the WSe2/hBN heterostructure. Since the background dielectric constants (neglecting phonons) of gypsum and hBN are similar, the bare 1s–2p Lyman resonance in a gypsum-covered WSe2 ML is expected to appear at an energy close to 143 meV, which gives rise to only a small detuning (ΔE ≈ 5 meV) to the vibrational $$\nu_3$$ mode in gypsum (138 meV). The prominent splitting of $$\Delta \sigma _1$$ of ~35 meV in the WSe2/gypsum heterostructure clearly exceeds the detuning energy and, thus, implies that the two new resonances are indeed Lyman polarons caused by strong-coupling.

### Interlayer exciton–phonon hybridization

Hybridization between the intra-excitonic resonance and a lattice phonon across the van der Waals interface should lead to a measurable anticrossing signature. To test this hypothesis, we perform similar experiments on the WSe2 BL/gypsum heterostructure, where the intra-excitonic transition can be tuned through the phonon resonance. Strong interlayer orbital hybridization in the WSe2 BL shifts the conduction band minimum from the K points to the Λ points, leading to the formation of K–Λ excitons ($${\mathrm{X}}^{{\mathrm{K}} - {{\Lambda }}}$$) with wavefunctions delocalized over the top and bottom layer22,23. Such interlayer orbital hybridization, which is also commonly observed in other 2D transition metal dichalcogenide (TMD) heterostructures1,2,3,4,5,8,9,24, renders the internal 1s–2p Lyman transition more susceptible to many-body Coulomb renormalization than in a single ML. This offers a unique opportunity to tune the intra-excitonic resonance from 87 to 69 meV by merely increasing the excitation fluence from 5 to 36 µJ cm−2 (see Supplementary Note 2).

Figure 3a displays the MIR response of the WSe2 BL/gypsum heterostructure at tpp = 3 ps and various excitation densities. Strikingly, we observe a distinct anticrossing near the 1s–2p Lyman transition of K–Λ excitons in the WSe2 BL and the $$\nu_4$$ mode of gypsum upon increasing the excitation density. This is unequivocal evidence of hybridization of exciton and phonon modes across the atomic interface. In addition, the absorption for all excitation densities exhibits a discernible shoulder at a photon energy of ~115 meV (Fig. 3a, red arrow), which is very close to the 1s–2p resonance of K–K excitons ($${\mathrm{X}}^{{\mathrm{K}} - {\mathrm{K}}}$$)22. Such a transition is indeed expected to occur at short delay times tpp < 1 ps, when the bound electron–hole pairs are prepared in the K valleys through direct interband excitation. However, the subtle interplay between 2D confinement and interlayer orbital overlap in the BL gives rise to a complex energy landscape1,2,3,4,5,8,9,24, where the lowest-energy exciton state is given by K–Λ species. Thus, sub-picosecond thermalization of the electron to Λ valleys via intervalley scattering22,23,25 should render the 1s–2p transition of K–K excitons weak. Yet, we clearly observe its spectral signature during the entire lifetime (see Supplementary Note 3). In addition, a new absorption band appears above the $$\nu_3$$ resonance of gypsum at an energy of ~150 meV (Fig. 3a, blue arrow). Its spectral position is nearly independent of the excitation density. We will show next that these surprising observations hallmark interlayer exciton–phonon hybridization involving as many as two phonon and two exciton resonances across the atomic interface, at once.

The dominant anticrossing feature in Fig. 3a occurs at an energy close to the $$\nu_4$$ mode of gypsum (78 meV) and the 1s–2p resonance of K–Λ excitons in WSe2 (69–87 meV, depending on the excitation fluence), while additional optical transitions emerge at energies close to the $$\nu_3$$ mode (138 meV) and the 1s–2p resonance of K–K excitons (115 meV). Therefore, we consider how the 1s–2p transition of K-Λ and K–K excitons (see Supplementary Note 4) hybridize with $$\nu_3$$ and $$\nu_4$$ phonons in gypsum. The electron–phonon interaction is commonly described by the Fröhlich Hamiltonian, which is linear in the phonon creation and annihilation operators and couples only states differing by one optical phonon26. The energetically lowest excited states of the uncoupled system, in which only one of the two exciton species or one of the two phonons is excited, can be denoted as $$\left| {2p^{{\mathrm{K}} - \Lambda },\,0\nu_4,\,1s^{{\mathrm{K}} - {\mathrm{K}}},\,0\nu_3} \right\rangle$$, $$\left| {1s^{{\mathrm{K}} - {{\Lambda }}},\,1\nu_4,\,1s^{{\mathrm{K}} - {\mathrm{K}}},0\nu_3} \right\rangle$$, $$\left| {1s^{{\mathrm{K}} - {{\Lambda }}},\,0\nu_4,\,2p^{{\mathrm{K}} - {\mathrm{K}}},\,0\nu_3} \right\rangle$$, $$\left| {1s^{{\mathrm{K}} - {{\Lambda }}},\,0\nu_4,\,1s^{{\mathrm{K}} - {\mathrm{K}}},\,1\nu_3} \right\rangle$$. The coupling between different states is illustrated in Fig. 3c. Using these basis vectors, we derive an effective Hamiltonian

$$H_{{\mathrm{eff}}} = \left( \begin{array}{*{20}{c}} {E_{1s - 2p}^{{\mathrm{K}} - {{\Lambda }}}} & {V_1} & 0 & {V_2} \\ {V_1} & {E_{{\mathrm{ph}}}^{\nu_4}} & {V_3} & 0 \\ 0 & {V_3} & {E_{1s - 2p}^{{\mathrm{K}} - {\mathrm{K}}}} & {V_4} \\ {V_2} & 0 & {V_4} & {E_{{\mathrm{ph}}}^{\nu_3}} \end{array} \right).$$
(1)

Here, $$E_{1s - 2p}^{{\mathrm{K}} - {{\Lambda }}}$$ $$( {E_{1s - 2p}^{{\mathrm{K}} - {\mathrm{K}}}})$$ and $$E_{{\mathrm{ph}}}^{\nu_4}$$ $$( {E_{{\mathrm{ph}}}^{\nu_3}} )$$ denote the 1s–2p resonance energy of K–Λ (K–K) excitons and the energy of the $$\nu_4$$ ($$\nu_3$$) mode, respectively, whereas $$V_1$$, $$V_2$$, $$V_3$$, and $$V_4$$ describe the exciton–phonon coupling constants (Fig. 3c, red arrows). At exciton densities for which $$E_{1s - 2p}^{{\mathrm{K}} - {{\Lambda }}}$$ is tuned through $$E_{{\mathrm{ph}}}^{\nu_4}$$, the Hamiltonian shows that the resonant exciton–phonon hybridization leads to an avoided crossing. Quantitative comparison between experiment and theory can be achieved by directly solving the effective Hamiltonian and yields four new hybrid states ($$\left| {{{\Psi }}_n} \right\rangle$$, n = 1, 2, 3, 4) that consist of a superposition of the basis modes.

Figure 3b displays a 2D map of the simulated optical conductivity of the new polaron eigenstates $$\left| {{{\Psi }}_n} \right\rangle$$ as a function of the probe energy ($$\hbar \omega$$) and the position of $$E_{1s - 2p}^{{\mathrm{K}} - {{\Lambda }}}$$. Since the excitons in WSe2 are largely thermalized as K–Λ species on a sub-picosecond scale, the oscillator strength of the resulting polarons observed thereafter depends on their projection $$P_n = \left\langle {{{\Psi }}_n|2p^{{\mathrm{K}} - {{\Lambda }}},0\nu_4,1s^{{\mathrm{K}} - {\mathrm{K}}},\,0\nu_3} \right\rangle$$ onto the bare zero-phonon K–Λ exciton (see Supplementary Note 3). To validate our model, we fit the simulated optical conductivity to the experimental data (Fig. 3a). Again, we set $$E_{{\mathrm{ph}}}^{\nu_4}$$ = 78 meV and $$E_{{\mathrm{ph}}}^{\nu_3}$$ = 138 meV (see Fig. 1d), and $$E_{1s - 2p}^{{\mathrm{K}} - {\mathrm{K}}}$$ = 115 ± 5 meV (ref. 22), while $$E_{1s - 2p}^{{\mathrm{K}} - {{\Lambda }}}$$ and the oscillator strength of the 1s–2p transition of the K–Λ exciton serve as fit parameters. For oscillator strengths similar to published values in ref. 22, the numerical adaption yields excellent agreement between theory and experiment and reproduces all optical transitions (Fig. 3a) and the prominent anticrossing (Fig. 3d).

The coupling constants retrieved from fitting the model to the experimental data amount to $$V_1 \approx V_3 =$$ 20 ± 2 meV and $$V_2 \approx V_4 =$$ 31 ± 2 meV (see Supplementary Note 3), even exceeding values reported in quantum dots27. This result is remarkable given that in our experiments strong-coupling is only achieved by proximity across the van der Waals interface. The ratio $$\frac{{V_2}}{{V_1}} \sim \frac{{V_4}}{{V_3}} \sim \sqrt 2$$ qualitatively reflects the relative dipole moments of the $$\nu_4$$ and $$\nu_3$$ modes (see Supplementary Note 3). Our analysis also allows us to assign the high-frequency features in Fig. 3a to $$\left| {{{\Psi }}_3} \right\rangle$$ and $$\left| {{{\Psi }}_4} \right\rangle$$. Even when the K–K excitons are weakly populated at tpp = 3 ps and the $$\nu_3$$ phonon resonance is far-detuned from the 1s–2p resonance of K–Λ excitons, the strong-coupling scenario allows for these Lyman polarons to emerge. In addition, by increasing the excitation density, many-body Coulomb correlations shift the bare 1s–2p resonance of K–Λ excitons and thereby modify the Lyman composition of $$\left| {{{\Psi }}_n} \right\rangle$$, as shown in Fig. 3e. For example, for $$E_{1s - 2p}^{{\mathrm{K}} - {{\Lambda }}}$$ = 86.5 meV, $$\left| {{{\Psi }}_1} \right\rangle$$ consists of 40% (6%) 1s–2p Lyman transition of the K–Λ (K–K) exciton and 10% (44%) $$\nu_3$$ ($$\nu_4$$) phonon (see Supplementary Note 3).

### Shaping the interlayer exciton–phonon coupling strength

The interlayer exciton–phonon hybridization can be custom-tailored by engineering the spatial overlap of exciton and phonon wavefunctions on the atomic scale. To demonstrate this possibility, we create spatially well-defined intra- ($${\mathrm{X}}^{{\mathrm{intra}}}$$) and interlayer exciton ($${\mathrm{X}}^{{\mathrm{inter}}}$$) phases by interfacing the WSe2 ML with a WS2 ML in a WSe2/WS2/gypsum heterostructure. Unlike in the WSe2 BL/gypsum heterostructure, the intralayer excitons in WSe2 are now spatially separated from gypsum by the WS2 ML (Fig. 4a). Ultrafast charge separation at the interface between WSe2 and WS2 depletes the Lyman resonance of $${\mathrm{X}}^{{\mathrm{intra}}}$$, while the transition of $${\mathrm{X}}^{{\mathrm{inter}}}$$ emerges on the sub-picosecond timescale21. Figure 4b shows $$\Delta \sigma _1$$ of the WSe2/WS2/gypsum heterostructure at tpp = 1 and 10 ps. The MIR response arising from exciton–phonon coupling is qualitatively similar to that observed in the WSe2 BL/gypsum heterostructure. This is partly because both the inter- ($$E_{1s - 2p}^{{\mathrm{inter}}}$$ = 69 meV) and intralayer ($$E_{1s - 2p}^{{\mathrm{intra}}}$$ = 114 meV) 1s–2p resonance in the WSe2/WS2 heterostructure are similar to $$E_{1s - 2p}^{{\mathrm{K}} - {{\Lambda }}}$$ (69–87 meV, depending on the excitation fluence) and $$E_{1s - 2p}^{{\mathrm{K}} - {\mathrm{K}}}$$ (115 meV) in the WSe2 BL, respectively. However, a direct comparison between $$\Delta \sigma _1$$ of both systems reveals a strong enhancement of the oscillator strength of the $$\left| {{{\Psi }}_4} \right\rangle$$ mode and larger splitting between $$\left| {{{\Psi }}_1} \right\rangle$$and $$\left| {{{\Psi }}_2} \right\rangle$$ in the WSe2/WS2/gypsum heterostructure (Fig. 4c). By fitting the experimental data with our coupling model, we found that the large oscillator strength of $$\left| {{{\Psi }}_4} \right\rangle$$ arises from enhanced phonon coupling to $${\mathrm{X}}^{{\mathrm{inter}}}$$ of V1 = 22 ± 2 meV and V2 = 36 ± 2 meV, which may be related with the dipolar nature of interlayer excitons. Meanwhile, the interlayer exciton–phonon coupling strength, which results directly from $${\mathrm{X}}^{{\mathrm{intra}}}$$ confined in the WSe2 layer, amounts to V3 = 16 ± 2 meV and V4 = 25 ± 2 meV (see Supplementary Note 5). The reduction of coupling strength, V3 and V4, by at least 20% compared to the WSe2 BL/gypsum case can be attributed to the atomically small spatial separation of the WSe2 ML from gypsum, which illustrates how exciton–phonon interaction could be fine-tuned in search for new phases of matter.

## Discussion

Our results reveal that even charge-neutral quasiparticles can interact with phonons across a van der Waals interface in the strong-coupling limit. Controlling excitonic wavefunctions at the atomic length scale can modify the coupling strength. We expect important implications for the study of polaron physics with charged and neutral excitations in a wide range of atomically thin strongly correlated electronic systems. In particular, polarons are known to play a crucial role in the formation of charge-density waves in Mott insulators and Cooper pairs in superconductors16,19. Moreover, excitons in TMD heterostructures embody important properties arising from the valley degree of freedom and can be engineered from topologically protected edge states of moiré superlattices1,2,3,4,13,14,28,29,30. In the future, it might, thus, even become possible to transfer fascinating aspects of chirality and nontrivial topology to polaron transport.

## Methods

### Sample preparation

All heterostructure compounds were exfoliated mechanically from a bulk single crystal using the viscoelastic transfer method31. We used gypsum and hBN as dielectric cover layers. The vibrational $$\nu_4$$ and $$\nu_3$$ modes of the $${\mathrm{SO}}_4$$ tetrahedral groups32 in gypsum (Fig. 1d) are close to the internal 1s–2p transition of excitons in WSe2. In contrast, the prominent E1u mode in hBN at 172 meV33 is far-detuned from the internal 1s–2p exciton transition in the WSe2 layer. The exfoliated gypsum, hBN, and TMD layers were inspected under an optical microscope and subsequently stacked on top of each other on a diamond substrate with a micro-positioning stage. To remove any adsorbates, the samples were annealed at a temperature of 150 °C and a pressure of 1 × 10−5 mbar for 5 h. The twist angle of the WSe2 BL was ensured by the tear and stack method: Starting from an extremely large exfoliated monolayer, only half of it is transferred onto the substrate. Consequently, transferring the remaining part of the ML onto the diamond substrate yields a perfectly aligned WSe2 BL.

### Ultrafast pump-probe spectroscopy

Supplementary Figure 6a depicts a schematic of the experimental setup. A home-built Ti:sapphire laser amplifier with a repetition rate of 400 kHz delivers ultrashort 12-fs NIR pulses. The output of the beam is divided into three branches. A first part of the laser output is filtered by a bandpass filter with a center wavelength closed to the interband 1s A exciton transition in the WSe2 layer, and a bandwidth of 9 nm, resulting in 100-fs pulses. Another part of the laser pulse generates single-cycle MIR probe pulses via optical rectification in a GaSe or an LGS crystal (NOX1). The probe pulse propagates through the sample after a variable delay time tpp. The electric field waveform of the MIR transient and any changes induced by the nonequilibrium polarization of the sample are fully resolved by electro-optic sampling utilizing a second nonlinear crystal (NOX2) and subsequent analysis of the field-induced polarization rotation of the gate pulse. Supplementary Figure 6b shows a typical MIR probe transient as a function of the electro-optic sampling time teos. The MIR probe pulse is centered at a frequency of 32 THz with a full-width at half-maximum of 18 THz (Supplementary Fig. 6c, black curve) and a spectral phase that is nearly flat (Supplementary Fig. 6c, blue curve). Using serial lock-in detection, we simultaneously record the pump-induced change ΔE(teos) and a reference Eref(teos) of the MIR electric field as function of teos.

### Extracting the dielectric response function

To extract the pump-induced change of the dielectric function of our samples with ultrafast NIR pump-MIR probe spectroscopy, we use serial lock-in detection. Hereby, a first lock-in amplifier records the electro-optic signal of our MIR probe field. Due to the modulation of the optical pump, the transmitted MIR probe field varies by the pump-induced change $${{\Delta }}E( {t_{{\mathrm{eos}}},t_{{\mathrm{pp}}}} )$$. This quantity is read out in a second lock-in amplifier at the modulation frequency of the pump. Simultaneously, the electro-optic signal is averaged in an analog low-pass to obtain a reference signal $$E_{{\mathrm{ref}}}( {t_{{\mathrm{eos}}}} ) = \frac{1}{2}( E_{{\mathrm{ex}}}( {t_{{\mathrm{eos}}},t_{{\mathrm{pp}}}} ) + E_{{\mathrm{eq}}}( {t_{{\mathrm{eos}}}} ))$$, where $$E_{{\mathrm{eq}}}\left( {t_{{\mathrm{eos}}}} \right)$$ is the signal after transmission through the sample in thermal equilibrium and $$E_{{\mathrm{ex}}}( {t_{{\mathrm{eos}}},t_{{\mathrm{pp}}}} ) = E_{{\mathrm{eq}}}( {t_{{\mathrm{eos}}}} ) + {{\Delta }}E( {t_{{\mathrm{eos}}},t_{{\mathrm{pp}}}} )$$ is the signal after transmission through the excited sample at $$t_{{\mathrm{pp}}}$$. From these quantities $$E_{{\mathrm{eq}}}\left( {t_{{\mathrm{eos}}}} \right)$$ and $$E_{{\mathrm{ex}}}\left( {t_{{\mathrm{eos}}}} \right)$$ are directly extracted. Subsequently, a Fourier transform for a fixed $$t_{{\mathrm{pp}}}$$ yields $$E_{{\mathrm{eq}}}\left( \omega \right)$$ and $$E_{{\mathrm{ex}}}( {\omega ,t_{{\mathrm{pp}}}} )$$, which in turn provides us with the complex-valued field transfer coefficient of our layered structure

$$T_{{\mathrm{pr}}}\left( {\omega ,t_{{\mathrm{pp}}}} \right) = T_{{\mathrm{pi}}}\left( {\omega ,t_{{\mathrm{pp}}}} \right) T_{{\mathrm{eq}}}\left( \omega \right) = \frac{{E_{{\mathrm{ex}}}\left( {\omega ,t_{{\mathrm{pp}}}} \right)}}{{E_{{\mathrm{eq}}}\left( \omega \right)}} T_{{\mathrm{eq}}}\left( \omega \right),$$
(2)

where $$T_{{\mathrm{eq}}}\left( \omega \right)$$ is the equilibrium field transmission coefficient and $$T_{{\mathrm{pi}}}( {\omega ,t_{{\mathrm{pp}}}} )$$ denotes the pump-induced change thereof. These quantities are completely defined by the equilibrium dielectric function $$\varepsilon \left( \omega \right)$$ and its pump-induced change $${{\Delta }}\varepsilon \left( {\omega ,t_{{\mathrm{pp}}}} \right)$$. By using the established optical transfer-matrix formalism34, we express the experimentally measured $$T_{{\mathrm{pr}}}( {\omega ,t_{{\mathrm{pp}}}} )$$ with the dielectric function. Finally, we insert the known equilibrium dielectric function and numerically invert the optical transfer-matrix formalism to extract the coveted quantity $${{\Delta }}\varepsilon ( {\omega ,t_{{\mathrm{pp}}}} )$$ discussed in the main text. Owing to the extremely thin sample thickness, challenges associated with Fabry–Perot resonances are unproblematic here and the inversion algorithm is especially stable and quantitatively reliable.

## Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request to give guidance to the interested party.

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## Acknowledgements

We thank Martin Furthmeier for technical assistance, and Philipp Steinleitner, Philipp Nagler, Alexander Graf, and Anna Girnghuber for preliminary studies and discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through Project ID 314695032—SFB 1277 (subproject A05) and project HU 1598/8. The Marburg group acknowledges funding from the European Union’s Horizon 2020 research and innovation program under grant agreement no. 881603 (Graphene Flagship) and the Deutsche Forschungsgemeinschaft (DFG) through SFB 1083 (subproject B9).

## Funding

Open Access funding enabled and organized by Projekt DEAL.

## Author information

Authors

### Contributions

The study was conceived by P.M., C.-K.Y., and R.H. and supervised by C.-K.Y., E.M., and R.H. P.M., C.-K.Y., M.L., and R.H. carried out the experiments, P.M., M.L., and I.H. prepared the heterostructures, and P.M., C.-K.Y., G.B., and E.M. carried out the theoretical modelling. All authors analyzed the data, discussed the results, and contributed to the writing of the manuscript.

### Corresponding authors

Correspondence to Chaw-Keong Yong or Rupert Huber.

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### Competing interests

The authors declare no competing interests.

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Merkl, P., Yong, CK., Liebich, M. et al. Proximity control of interlayer exciton-phonon hybridization in van der Waals heterostructures. Nat Commun 12, 1719 (2021). https://doi.org/10.1038/s41467-021-21780-6

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• DOI: https://doi.org/10.1038/s41467-021-21780-6