Exciton-Polaron Spectroscopy: Methods and Insights

• Exciton-polaron spectroscopy encompasses methods that probe excitons dressed by carriers or phonons to reveal many-body interactions in materials.
• It employs techniques such as reflectance-contrast, photoluminescence, and multidimensional coherent spectroscopy to resolve quasiparticle energies and dynamics.
• The approach bridges theoretical models with experimental data, enabling tunable studies in 2D semiconductors, perovskites, and organic materials.

Exciton-polaron spectroscopy is the set of spectroscopic methods used to create, resolve, and analyze excitonic quasiparticles that are dressed by a many-body environment. In charge-tunable two-dimensional semiconductors, the relevant object is the exciton Fermi polaron: a charge-neutral quasiparticle formed when a bound electron–hole pair is immersed in a Fermi sea of excess carriers and becomes dressed by virtual electron–hole-pair excitations in that sea. In lattice-coupled materials, especially low-dimensional perovskites and organic semiconductors, the same label is used for excitonic states dressed by phonons or by coupled charge-separated configurations. As a result, exciton-polaron spectroscopy spans reflectance-contrast spectroscopy, photoluminescence, transient absorption, multidimensional coherent spectroscopy, resonant second-harmonic generation, cavity spectroscopy, and pulsed electrically detected magnetic resonance, with each modality emphasizing a different projection of the dressed quasiparticle spectrum, dynamics, or interaction physics (Liu et al., 2020, Biswas et al., 2023, Sidler et al., 2016).

1. Quasiparticle concept and theoretical formalisms

In doped monolayer semiconductors, the minimal starting point is

$H = H_X + H_{\rm FS} + H_{\rm int},$

with

$$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$

and

$$H_{\rm int} = \sum_{q,k,k'} V_{k-k'}\,X_q^\dagger c_{k'+q}^\dagger c_k X_q.$$

In the strong-coupling regime, the exciton polarizes a cloud of Fermi-sea electron–hole pairs. A convenient variational picture is the “Suris tetron,” in which the many-body state is approximated by a superposition of a bare exciton and a single electron–hole-pair dressing channel,

$$|{\rm Polaron},q\rangle \simeq \alpha_0\,X_q^\dagger|{\rm FS}\rangle + \sum_{k,p} \alpha_{k,p}\,X_{q+p-k}^\dagger c_k^\dagger c_p |{\rm FS}\rangle + \cdots,$$

and the resonant pole of the Green’s function defines the exciton-polaron energy $E_p(q)$ (Liu et al., 2020).

Chevy-type and $T$-matrix approaches recur across the field. In a WSe$_2$/twisted-bilayer-graphene heterostructure, the minimal Hamiltonian is written as

$H = H_{\rm exc} + H_{\rm FS} + H_{\rm int},$

with a corresponding Chevy ansatz for a Rydberg exciton dressed by a Fermi sea. In cavity-coupled monolayers, the same dressed-exciton sector is embedded into a photon-coupled Hamiltonian, and in charge-tunable microcavities the attractive and repulsive polaron branches appear as poles of the exciton Green’s function dressed by both the cavity and the electron gas (Arsenault et al., 16 Jun 2025, Sidler et al., 2016).

A complementary interpretation is provided by the analogy to the Tavis–Cummings model. In the regime where the Chevy ansatz is accurate and the Fermi energy is much smaller than the trion binding energy, an exciton-polaron can be understood as a coherent superposition of a bare exciton in an unperturbed Fermi sea and a bright collective excitation of many trions. This formulation directly addresses the relation between trions and exciton-polarons in optical spectra and gives analytic expressions for oscillator-strength transfer and absorption–emission asymmetry (Imamoglu et al., 2020).

In phonon-dressed systems, the elementary Hamiltonian is instead

$$H = H_{\rm exciton} + H_{\rm phonon} + H_{\rm int},$$

with

$$H_{\rm exciton}=\epsilon_0 a^\dagger a,\qquad H_{\rm phonon}=\sum_q \hbar\omega_q b_q^\dagger b_q,\qquad H_{\rm int}=\sum_q g_q(b_q+b_{-q}^\dagger)a^\dagger a.$$

The polaronic distortion energy is

$$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$0

and for a representative mode $$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$1 the Huang–Rhys factor is

$$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$2

In rigid Dion–Jacobson perovskites, the spectral density

$$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$3

peaks between $$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$4–$$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$5, identifying low-frequency Pb–I octahedral modes as the lattice coordinates that drive exciton-polaron formation (Biswas et al., 2023).

This suggests that “exciton-polaron spectroscopy” is best treated as an umbrella category for spectroscopies of excitonic states dressed either by carrier-density fluctuations, by lattice displacements, or by both. The specific observables differ, but the common structure is the extraction of self-energy shifts, residues, linewidth redistribution, branch interconversion, and interaction-induced cross features.

2. Linear optical spectroscopy in charge-tunable two-dimensional semiconductors

A paradigmatic implementation is the optical spectroscopy of the $$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$6–$$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$7 excitonic Rydberg series in ultraclean monolayer MoSe$$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$8 encapsulated in BN and gated by a thin graphite back-electrode. Two complementary probes are used at $$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$9: reflectance-contrast spectroscopy, in which the differential reflectance contrast

$$H_{\rm int} = \sum_{q,k,k'} V_{k-k'}\,X_q^\dagger c_{k'+q}^\dagger c_k X_q.$$0

and its second derivative $$H_{\rm int} = \sum_{q,k,k'} V_{k-k'}\,X_q^\dagger c_{k'+q}^\dagger c_k X_q.$$1 resolve weak excitonic Rydberg resonances, and photoluminescence spectroscopy, in which a $$H_{\rm int} = \sum_{q,k,k'} V_{k-k'}\,X_q^\dagger c_{k'+q}^\dagger c_k X_q.$$2 laser excites the sample and the emitted photons are collected in a backscattering geometry. At zero doping, the A$$H_{\rm int} = \sum_{q,k,k'} V_{k-k'}\,X_q^\dagger c_{k'+q}^\dagger c_k X_q.$$3, A$$H_{\rm int} = \sum_{q,k,k'} V_{k-k'}\,X_q^\dagger c_{k'+q}^\dagger c_k X_q.$$4, and A$$H_{\rm int} = \sum_{q,k,k'} V_{k-k'}\,X_q^\dagger c_{k'+q}^\dagger c_k X_q.$$5 Rydberg states of the A-exciton are visible in both reflectance and PL, and diamagnetic shifts in fields up to $$H_{\rm int} = \sum_{q,k,k'} V_{k-k'}\,X_q^\dagger c_{k'+q}^\dagger c_k X_q.$$6 yield exciton radii $$H_{\rm int} = \sum_{q,k,k'} V_{k-k'}\,X_q^\dagger c_{k'+q}^\dagger c_k X_q.$$7, $$H_{\rm int} = \sum_{q,k,k'} V_{k-k'}\,X_q^\dagger c_{k'+q}^\dagger c_k X_q.$$8, and $$H_{\rm int} = \sum_{q,k,k'} V_{k-k'}\,X_q^\dagger c_{k'+q}^\dagger c_k X_q.$$9 (Liu et al., 2020).

With increasing carrier density, two spectroscopic trends are central. First, excitonic intensity is progressively quenched, with a critical density that scales inversely with the exciton’s $$|{\rm Polaron},q\rangle \simeq \alpha_0\,X_q^\dagger|{\rm FS}\rangle + \sum_{k,p} \alpha_{k,p}\,X_{q+p-k}^\dagger c_k^\dagger c_p |{\rm FS}\rangle + \cdots,$$0-space extent:

$$|{\rm Polaron},q\rangle \simeq \alpha_0\,X_q^\dagger|{\rm FS}\rangle + \sum_{k,p} \alpha_{k,p}\,X_{q+p-k}^\dagger c_k^\dagger c_p |{\rm FS}\rangle + \cdots,$$1

so that $$|{\rm Polaron},q\rangle \simeq \alpha_0\,X_q^\dagger|{\rm FS}\rangle + \sum_{k,p} \alpha_{k,p}\,X_{q+p-k}^\dagger c_k^\dagger c_p |{\rm FS}\rangle + \cdots,$$2. Second, the new absorption and emission features assigned to the exciton-polaron lines shift monotonically to higher energy, with the gate dependence described empirically by

$$|{\rm Polaron},q\rangle \simeq \alpha_0\,X_q^\dagger|{\rm FS}\rangle + \sum_{k,p} \alpha_{k,p}\,X_{q+p-k}^\dagger c_k^\dagger c_p |{\rm FS}\rangle + \cdots,$$3

The steeper slope for higher Rydberg index is attributed to larger spatial extent and stronger many-body dressing, and the measured gate-dependent energy shifts go beyond the trion description but match exciton-polaron theory (Liu et al., 2020).

A distinctive signature in the same system is the absorption–emission gap. Optical absorption creates an exciton-polaron at zero center-of-mass momentum,

$$|{\rm Polaron},q\rangle \simeq \alpha_0\,X_q^\dagger|{\rm FS}\rangle + \sum_{k,p} \alpha_{k,p}\,X_{q+p-k}^\dagger c_k^\dagger c_p |{\rm FS}\rangle + \cdots,$$4

whereas PL favors emission from a finite-momentum roton minimum,

$$|{\rm Polaron},q\rangle \simeq \alpha_0\,X_q^\dagger|{\rm FS}\rangle + \sum_{k,p} \alpha_{k,p}\,X_{q+p-k}^\dagger c_k^\dagger c_p |{\rm FS}\rangle + \cdots,$$5

so that

$$|{\rm Polaron},q\rangle \simeq \alpha_0\,X_q^\dagger|{\rm FS}\rangle + \sum_{k,p} \alpha_{k,p}\,X_{q+p-k}^\dagger c_k^\dagger c_p |{\rm FS}\rangle + \cdots,$$6

Within a $$|{\rm Polaron},q\rangle \simeq \alpha_0\,X_q^\dagger|{\rm FS}\rangle + \sum_{k,p} \alpha_{k,p}\,X_{q+p-k}^\dagger c_k^\dagger c_p |{\rm FS}\rangle + \cdots,$$7-matrix or Chevy-type treatment the dispersion satisfies

$$|{\rm Polaron},q\rangle \simeq \alpha_0\,X_q^\dagger|{\rm FS}\rangle + \sum_{k,p} \alpha_{k,p}\,X_{q+p-k}^\dagger c_k^\dagger c_p |{\rm FS}\rangle + \cdots,$$8

and develops a local minimum at finite momentum with $$|{\rm Polaron},q\rangle \simeq \alpha_0\,X_q^\dagger|{\rm FS}\rangle + \sum_{k,p} \alpha_{k,p}\,X_{q+p-k}^\dagger c_k^\dagger c_p |{\rm FS}\rangle + \cdots,$$9. The measured absorption–emission gap is nearly zero at low doping and rises linearly with $E_p(q)$0, with a slope that increases from $E_p(q)$1 to $E_p(q)$2 polarons; this is interpreted as a direct optical signature of roton-like dispersion in an exciton-polaron system (Liu et al., 2020).

Related linear spectroscopies extend the same logic to other architectures. In dual-gated bilayer WSe$E_p(q)$3, one layer can host neutral excitons while the hole-filled layer hosts attractive polaron states red-shifted by $E_p(q)$4–$E_p(q)$5, with the two layers displaying distinct linewidths and resonant response (Cha et al., 2024). In double-layer semiconductors, a predicted interlayer exciton-polaron formed by coupling an interlayer exciton to breathing flexural phonons is expected to appear as a zero-phonon line shifted by $E_p(q)$6 together with phonon sidebands at $E_p(q)$7, with relative intensities governed by Huang–Rhys factors (Semina et al., 2020).

3. Ultrafast spectroscopy of formation, branching, and relaxation

Transient absorption has become a principal route for time-resolved exciton-polaron spectroscopy because it separates spectral formation from steady-state line assignment. In rigid Dion–Jacobson-type two-dimensional perovskites, the raw TA maps show a ground-state bleaching at the exciton energy, a below-gap photoinduced absorption attributed to band-gap renormalization, and an above-gap photoinduced absorption. After subtracting population kinetics, the residual traces display damped oscillations. Fourier transformation across probe energy yields a coherent peak at $E_p(q)$8 for FPP and $E_p(q)$9 for FPT, while the oscillation amplitude shows a node and the phase exhibits a $T$0 jump at the excitonic resonance. These are hallmarks of a frequency-modulation wavepacket created by impulsive stimulated Raman scattering and indicate that phonons modulate the excitonic transition energy rather than only the transition dipole (Biswas et al., 2023).

The same measurements quantify coupled cooling and polaron stabilization. The hot-carrier channel is fitted by

$T$1

For both ligands, $T$2–$T$3 and is independent of fluence; $T$4 for FPP is essentially fluence-independent, whereas for FPT $T$5 increases from $T$6 to $T$7 as carrier density rises. Together with extracted lattice displacements and Huang–Rhys factors, these data identify stronger exciton–phonon coupling in FPT and a hot-phonon bottleneck at high excitation density (Biswas et al., 2023).

Monolayer Ruddlesden–Popper lead halide perovskites provide a second TA platform with explicit free-exciton/exciton-polaron deconvolution. In $T$8 and $T$9, the TA spectrum contains a sharp negative bleach tracking the free exciton and a red-shifted positive feature near $_2$0 assigned to the exciton-polaron resonance. At $_2$1, the spectrum is decomposed into two Gaussians,

$_2$2

yielding exciton-polaron binding energies of $_2$3–$_2$4 in $_2$5 and $_2$6–$_2$7 in $_2$8. The branching coefficient is greater than unity for all samples, and a quasi-steady dynamic equilibrium between free excitons and exciton-polarons persists for $_2$9–$H = H_{\rm exc} + H_{\rm FS} + H_{\rm int},$0 at $H = H_{\rm exc} + H_{\rm FS} + H_{\rm int},$1 (Mondal et al., 15 Oct 2025).

In a WSe$H = H_{\rm exc} + H_{\rm FS} + H_{\rm int},$2/twisted-bilayer-graphene heterostructure, pump–probe spectroscopy tracks the formation of Rydberg exciton Fermi polarons after pump injection of Rydberg excitons in WSe$H = H_{\rm exc} + H_{\rm FS} + H_{\rm int},$3. For $H = H_{\rm exc} + H_{\rm FS} + H_{\rm int},$4, the $H = H_{\rm exc} + H_{\rm FS} + H_{\rm int},$5 peak redshifts dynamically by up to $H = H_{\rm exc} + H_{\rm FS} + H_{\rm int},$6 for holes and $H = H_{\rm exc} + H_{\rm FS} + H_{\rm int},$7 for electrons at long delay, while the long-time redshift defines a polaron binding energy that follows

$H = H_{\rm exc} + H_{\rm FS} + H_{\rm int},$8

The relaxation rate extracted from the energy shift rises to $H = H_{\rm exc} + H_{\rm FS} + H_{\rm int},$9 for holes and $$H = H_{\rm exciton} + H_{\rm phonon} + H_{\rm int},$$0 for electrons at the highest doping, linking the spectral evolution to Fermi-sea reorganization in the moiré environment (Arsenault et al., 16 Jun 2025).

These ultrafast studies collectively establish two points. First, exciton-polaron spectroscopy is not limited to static branch assignment: it can time-resolve the birth of the dressed state. Second, the controlling bath can be a mobile Fermi sea, localized moiré carriers, or low-frequency lattice modes, and the spectroscopy remains sensitive to the corresponding self-energy pathway.

4. Multidimensional coherent spectroscopy and interaction physics

Two-dimensional coherent spectroscopy generalizes linear spectroscopy by correlating absorption and emission frequencies while retaining mixing-time information. In a microscopic many-body treatment of trion-polaritons and exciton-polaritons in a charge-tunable TMD monolayer inside a planar microcavity, the light–matter coupling is weighted by the excitonic residue:

$$H = H_{\rm exciton} + H_{\rm phonon} + H_{\rm int},$$1

The rephasing spectrum is

$$H = H_{\rm exciton} + H_{\rm phonon} + H_{\rm int},$$2

and the theory predicts diagonal peaks for upper, middle, and lower polariton branches together with three kinds of off-diagonal cross-peaks. Their oscillation periods,

$$H = H_{\rm exciton} + H_{\rm phonon} + H_{\rm int},$$3

encode branch coherences and give direct access to residues and gate-dependent energy shifts (Hu et al., 2022).

Multidimensional coherent spectroscopy in monolayer WS$$H = H_{\rm exciton} + H_{\rm phonon} + H_{\rm int},$$4 resolves exciton-polaron interactions rather than only single-particle branches. In one-quantum rephasing spectra, diagonal peaks at $$H = H_{\rm exciton} + H_{\rm phonon} + H_{\rm int},$$5 and $$H = H_{\rm exciton} + H_{\rm phonon} + H_{\rm int},$$6 correspond to the repulsive and attractive polaron, and cross-peaks reveal AP–RP coupling. The homogeneous linewidths are $$H = H_{\rm exciton} + H_{\rm phonon} + H_{\rm int},$$7 and $$H = H_{\rm exciton} + H_{\rm phonon} + H_{\rm int},$$8, and the cross-peak amplitude implies a repulsive interaction $$H = H_{\rm exciton} + H_{\rm phonon} + H_{\rm int},$$9 arising from phase-space filling when two excitons compete for the same electrons. In cross-circular spectra, a bound bipolaron state appears as peaks shifted by $$H_{\rm exciton}=\epsilon_0 a^\dagger a,\qquad H_{\rm phonon}=\sum_q \hbar\omega_q b_q^\dagger b_q,\qquad H_{\rm int}=\sum_q g_q(b_q+b_{-q}^\dagger)a^\dagger a.$$0 below the AP diagonal, with binding energies $$H_{\rm exciton}=\epsilon_0 a^\dagger a,\qquad H_{\rm phonon}=\sum_q \hbar\omega_q b_q^\dagger b_q,\qquad H_{\rm int}=\sum_q g_q(b_q+b_{-q}^\dagger)a^\dagger a.$$1 and $$H_{\rm exciton}=\epsilon_0 a^\dagger a,\qquad H_{\rm phonon}=\sum_q \hbar\omega_q b_q^\dagger b_q,\qquad H_{\rm int}=\sum_q g_q(b_q+b_{-q}^\dagger)a^\dagger a.$$2 relative to emission into singlet and triplet attractive polarons, respectively (Muir et al., 2022).

Theoretical work on one-dimensional materials sharpens the role of the excited-state-absorption channel. In that setting, ESE and GSB can be described by a one-particle-hole Chevy ansatz, but ESA requires two-impurity eigenstates. In the weak-interaction limit, the ESA contribution cancels the combined ESE and GSB contributions, leading to weak net spectral structure; in the strong-interaction limit, the ESA dip is displaced from the main diagonal and both diagonal and off-diagonal features survive in the total 2DCS signal. The same framework identifies the induced interaction between two attractive polarons through the shift $$H_{\rm exciton}=\epsilon_0 a^\dagger a,\qquad H_{\rm phonon}=\sum_q \hbar\omega_q b_q^\dagger b_q,\qquad H_{\rm int}=\sum_q g_q(b_q+b_{-q}^\dagger)a^\dagger a.$$3 (Wang et al., 2023).

Organic semiconductors add a vibronic variant. In rr-P3HT thin films, ultrafast two-dimensional electronic spectroscopy shows quartets of split vibronic resonances, negative cross-peaks associated with polaron-pair absorption, and waiting-time oscillations with period $$H_{\rm exciton}=\epsilon_0 a^\dagger a,\qquad H_{\rm phonon}=\sum_q \hbar\omega_q b_q^\dagger b_q,\qquad H_{\rm int}=\sum_q g_q(b_q+b_{-q}^\dagger)a^\dagger a.$$4 that persist for up to about $$H_{\rm exciton}=\epsilon_0 a^\dagger a,\qquad H_{\rm phonon}=\sum_q \hbar\omega_q b_q^\dagger b_q,\qquad H_{\rm int}=\sum_q g_q(b_q+b_{-q}^\dagger)a^\dagger a.$$5. The minimal Holstein-type Hamiltonian couples an exciton and a polaron-pair manifold to a dominant underdamped vibration with $$H_{\rm exciton}=\epsilon_0 a^\dagger a,\qquad H_{\rm phonon}=\sum_q \hbar\omega_q b_q^\dagger b_q,\qquad H_{\rm int}=\sum_q g_q(b_q+b_{-q}^\dagger)a^\dagger a.$$6 and coupling $$H_{\rm exciton}=\epsilon_0 a^\dagger a,\qquad H_{\rm phonon}=\sum_q \hbar\omega_q b_q^\dagger b_q,\qquad H_{\rm int}=\sum_q g_q(b_q+b_{-q}^\dagger)a^\dagger a.$$7, and the rapid rise of the polaron-pair cross-peak within one half of a vibrational period is taken as evidence of coherent vibronic formation of hybrid exciton–polaron-pair states (Sio et al., 2016). A related detection geometry, photoinduced-absorption-detected 2D coherent spectroscopy, measures fourth-order populations through metastable polaronic products and gives access to species that are dark in photoluminescence or may not contribute to steady-state photocurrent (Li et al., 2016).

5. Nonlinear, cavity, and spin-sensitive implementations

Resonant second-harmonic generation can act as a layer-selective exciton-polaron spectrometer. In dual-gate 2H-stacked bilayer WSe$$H_{\rm exciton}=\epsilon_0 a^\dagger a,\qquad H_{\rm phonon}=\sum_q \hbar\omega_q b_q^\dagger b_q,\qquad H_{\rm int}=\sum_q g_q(b_q+b_{-q}^\dagger)a^\dagger a.$$8 at $$H_{\rm exciton}=\epsilon_0 a^\dagger a,\qquad H_{\rm phonon}=\sum_q \hbar\omega_q b_q^\dagger b_q,\qquad H_{\rm int}=\sum_q g_q(b_q+b_{-q}^\dagger)a^\dagger a.$$9, selective hole localization in one layer produces exciton-polarons in the hole-filled layer while preserving neutral excitons in the intrinsic layer. Near resonance, the second-order susceptibility is written as

$$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$00

with a strong exciton pole in the intrinsic layer and a weaker, red-shifted attractive-polaron contribution in the doped layer. Because the two layers are no longer identical, destructive cancellation of the bilayer SHG is lifted. Experimentally, the method yields a $$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$01-fold SHG enhancement at minimal electric field, equivalent to conditions near the dielectric-breakdown threshold but using only $$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$02 of the critical breakdown field. The SHG enhancement is maximal around $$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$03–$$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$04 and directly tracks layer-dependent many-body states (Cha et al., 2024).

Microcavity spectroscopy extends the same physics into the strong-coupling regime. In charge-tunable monolayer MoSe$$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$05 in an open microcavity, increasing electron density transfers oscillator strength from the repulsive exciton-polaron resonance to the attractive-polaron manifold, and both branches form polaritons. The coupled system is described by an exciton, a Fermi sea, and a single cavity mode, with Rabi splittings

$$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$06

Simultaneous observation of polariton formation in both attractive and repulsive branches identifies a regime in which the polariton impurity mass is much smaller than that of the electrons, and the polariton splitting gives a direct measure of oscillator-strength transfer through the quasiparticle residue (Sidler et al., 2016).

Spin-sensitive spectroscopy provides a different window. In pulsed electrically detected magnetic resonance, the exciton-polaron complex is modeled as a three-spin system containing a spin-1 triplet exciton and a spin-$$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$07 polaron coupled by exchange and dipolar interactions. The full time-dependent Hamiltonian includes Zeeman, hyperfine, zero-field dipolar, inter-species exchange, inter-species dipolar, and microwave driving terms. In this framework, transition frequencies and resonance positions yield estimates of the inter-species coupling, while multi-pulse experiments such as Hahn echoes, stimulated echoes, and inversion recovery extract $$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$08, $$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$09, and spin-dependent reaction rates. The spectroscopy is therefore sensitive not only to spectral structure but also to the microscopic kinetics of exciton-polaron quenching complexes (Keevers et al., 2015).

6. Interpretive issues, related phases, and research directions

A persistent interpretive issue is the trion-versus-polaron distinction. In low-doping limits, trions are often described as three-body quasiparticles, but several of the cited studies state explicitly that this picture breaks down as carrier density increases and that the optical resonances are more rigorously understood as exciton Fermi polarons. In monolayer MoSe$$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$10, the gate-dependent energy shifts of the Rydberg series go beyond the trion description but match exciton-polaron theory; in the Tavis–Cummings mapping, the trion sector appears as a bright collective excitation hybridized with the bare exciton; and in the microcavity experiment, the continuous oscillator-strength transfer between repulsive and attractive branches is taken as a direct manifestation of Fermi-polaron physics (Liu et al., 2020, Imamoglu et al., 2020, Sidler et al., 2016).

Exciton-polaron spectroscopy also functions as a probe of phases beyond a normal Fermi sea. In a bilayer excitonic insulator, an optically generated intralayer exciton can be screened by excitations out of the condensate to form interlayer polarons. The theory combines diffusion quantum Monte Carlo for the X–IX biexciton binding energy, a BCS description of the excitonic-insulator ground state, and a single interacting-quasiparticle-pair Chevy-like ansatz for the dynamical screening problem. The resulting absorption spectrum contains attractive and repulsive polaron resonances, and at large layer separation a third X–IX$$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$11 peak can emerge. The low-density attractive-polaron energy is tied directly to the X–IX binding energy, so the spectroscopy becomes a probe of condensate structure and interlayer pairing (Amelio et al., 2022).

Prospective directions in the field are already formulated within current experiments and theories. Gate-tunable Rydberg exciton-polarons have been proposed as a platform for engineering roton minima, exploring excitonic analogues of supersolidity or density-wave instabilities, producing polaron-polaritons with modified dispersion, probing higher-order dressings such as penta- and hexa-electron–hole bound complexes, realizing analogues of roton gases, and studying spin- or valley-polarized polaron fluids (Liu et al., 2020). In perovskites, spacer choice and fabrication route are identified as practical levers for tuning exciton–phonon coupling, exciton-polaron population, hot-phonon bottlenecks, branching ratios, and carrier lifetimes, with explicit device implications for LEDs, photodetectors, and solar cells (Mondal et al., 15 Oct 2025, Biswas et al., 2023). In bilayer WSe$$H_X = \sum_q \epsilon_X(q)\,X_q^\dagger X_q,\qquad H_{\rm FS} = \sum_{k,s} \epsilon_{k,s}\,c_{k,s}^\dagger c_{k,s},$$12, resonant SHG is presented not only as a nonlinear control mechanism but also as a spectroscopic tool for probing layer-dependent quantum states (Cha et al., 2024).

Across these platforms, exciton-polaron spectroscopy has evolved from branch identification into a broader many-body methodology. It now resolves quasiparticle energies, residues, dispersion anomalies, ultrafast formation pathways, coherent branch couplings, interaction-induced cross-peaks, nonlinear susceptibilities, and spin-dependent kinetics within a common spectroscopic framework.