Bound Excited States of Fröhlich Polarons

• The paper identifies that bound excited states in Fröhlich polarons represent internal excitations arising from impurity vibrations and relaxation modes of the polaronic cloud.
• The analysis employs variational approaches, Bogoliubov transformations, and numerical simulations to quantify energy shifts, spectral weights, and threshold phenomena.
• The study highlights how dimensionality and coupling strength shape spectroscopic signatures, providing actionable insights for experimental verification in polarizable media.

Bound excited states of Fröhlich polarons are discrete energy levels above the polaron ground state, separated from the essential spectrum by a finite gap, and associated with internal vibrations or quantum fluctuations of the polaronic dressing cloud. These states represent internal polaron excitations rather than trivial phonon excitations and are central to the understanding of optical spectra, non-linear response, and transient dynamics in polarizable media. Their existence and properties depend sensitively on the dimensionality, coupling regime, and the details of the electron–phonon interaction.

1. Fröhlich Model, Polaron Ground State, and Self-Trapped Potential

The canonical Fröhlich Hamiltonian in one and higher dimensions describes a mobile charge carrier (electron or impurity) interacting with a continuum of optical phonons (or more generally Bogoliubov modes in a Bose–Einstein condensate). The dimensionless coupling constant $\alpha$ quantifies the strength of the interaction. In the strong coupling regime, $\alpha \gg 1$, the Landau–Pekar approach becomes accurate: the carrier induces a self-consistent polarization field, leading to an effective trapping potential (Casteels et al., 2010, Taylor et al., 3 Jun 2025). The ground state is localized and can be well approximated by a variational product of the lowest harmonic oscillator wavefunction for the polaron and a coherent state of phonons (or Bogoliubov excitations): $$|\Psi_0\rangle = |\psi_{\text{HO}}^{(0)}\rangle \otimes U|0\rangle,$$ where $$U = \exp\left[\sum_k(f_k a_k - f_k^* a_k^\dagger)\right]$$ is a canonical displacement operator determined variationally.

2. Mechanisms and Characterization of Bound Excited States

Bound excited states arise from two principal mechanisms: (a) internal excitations of the impurity within the self-induced potential (e.g., harmonic oscillator excited states), and (b) quantized fluctuations (“relaxation modes”) of the coupled electron-phonon field around the Pekar minimizer. In both cases, the phonon (or Bogoliubov) field dynamically relaxes to accommodate the polaron’s new electronic configuration.

Landau–Pekar/Strong Coupling Limit

In the strong coupling limit, the impurity potential is deep, and the low-lying spectrum forms an approximate ladder of “Relaxed Excited States” (RES), each corresponding to an impurity harmonic oscillator eigenstate with quantum number $n$ and a redressed phonon cloud (Casteels et al., 2010). The energy functional for the $n$th state is minimized with respect to the width parameter $\lambda$: $$E_n(\lambda) = \frac{1}{2\lambda^2(3/2+n)} - \frac{\alpha \mu}{2\pi^2\lambda} \int d^3q \frac{\rho^2(q)}{2\lambda^2 + q^2},$$ with optimal $\lambda_n$ determined for each $n$.

Quantum Fluctuations and Bogoliubov Theory

Beyond the pure variational (mean-field) picture, quantized vibrational modes around the Pekar state are captured by expanding the action to quadratic order, yielding a set of harmonic oscillators with frequencies $\omega_j$ set by the Hessian of the Pekar functional (Taylor et al., 3 Jun 2025, Mitrouskas et al., 2022). The excitation energies are

$$E_j / (\hbar\omega_{LO}) \simeq - \frac{\alpha^2}{3} - \frac{1}{2} \sum_i(1 - \omega_i) + \omega_j,$$

where $j = 0$ is the ground state, $j > 0$ are bound excited states, and the spectrum is discrete and converges below the continuum at integer multiples of $\hbar\omega_{LO}$.

In the continuum limit of strong coupling, the number of such bound states diverges (Taylor et al., 3 Jun 2025, Mitrouskas et al., 2022). Each corresponds to a quantized fluctuation mode of the polaronic cloud, and their spatial structure is set by the eigenfunctions of the Hessian operator.

3. Dimensionality, Onset, and Threshold Phenomena

The occurrence and properties of bound excited states depend strongly on model dimensionality and coupling:

• In one-dimensional systems, an arbitrarily large number of bound excited states exist for strong coupling; the onset for the first bound excited state is observed at a coupling strength $\alpha_c \approx 1.73$ (Taylor et al., 3 Jun 2025).
• In higher dimensions and different polaronic environments (e.g., BEC impurities), there exists a minimal coupling, depending on quantum number $n$, below which no self-localized excited bound state is supported (Casteels et al., 2010).
• In weak coupling (small $\alpha$), it is rigorously established that there are no bound excited eigenvalues below the essential spectrum, i.e., only the ground state is bound (Seiringer, 2022).

Threshold phenomena are signaled by the critical coupling or binding strength needed for a discrete excitation to detach from the phonon continuum. The actual numerical threshold depends on details such as phonon dispersion, impurity mass, and cutoff procedures.

4. Spectroscopic Identifiers: Transition Energies and Spectral Weights

Transition energies between the ground state and RES or quantum fluctuation modes serve as spectroscopic fingerprints. The difference in minimized energies,

$$\Delta E = E_{\text{RES}}(\lambda_{\text{RES}}) - E_{\text{ground}}(\lambda_{\text{ground}}),$$

corresponds to optical or Bragg transition frequencies measurable in experiment (Casteels et al., 2010). In the one-dimensional model, the spectral weights $Z_j^{(P)} = |\langle \text{vac}|j\rangle|^2$ of the excited states can be substantial (e.g., $Z_1^{(0)} \approx 0.2$ near the threshold), making these states observable in linear spectroscopy (Taylor et al., 3 Jun 2025).

The excitation spectrum as a function of $\alpha$ reveals that with increasing coupling:

• Both the ground and excited state energies are renormalized downward.
• The spectral weight of the ground state decreases and that of excited states increases non-trivially.
• The dressing cloud's internal structure broadens in excited states, as shown by the momentum-resolved phonon occupation.

5. Numerical Methods and Theoretical Tools

Accurate characterization of bound excited states, especially at intermediate coupling, demands non-perturbative numerical techniques. The Full Configuration Interaction Quantum Monte Carlo (FCIQMC) approach, applied to discretized models, allows stochastic projection onto excited eigenstates. Crucially, annihilation between positive and negative walkers counteracts the sign problem induced by nodal structures in excited states, stabilizing numerical results as walker number increases (Taylor et al., 3 Jun 2025).

In strong coupling, semiclassical expansion about the Pekar solution, diagonalization of quadratic fluctuation Hamiltonians, and Bogoliubov transformations yield the analytic bound state structure (Mitrouskas et al., 2022). The approach extends to finite momentum $P$ and permits the paper of bound states via fiber Hamiltonians.

6. Experimental Relevance and Physical Implications

Bound excited states of Fröhlich polarons manifest as discrete lines or features in absorption, photoluminescence, or Bragg spectra—offering direct evidence of strong electron–phonon coupling and internal polaron structure. The transition frequencies’ dependence on the coupling parameter, impurity–host mass ratio, and other tunable variables allows for detailed comparison with theoretical predictions and potentially for precision measurements of polaronic properties (Casteels et al., 2010, Taylor et al., 3 Jun 2025).

Their existence is tightly linked to polaron self-trapping and has implications for transport, optical response, and quantum coherence in polar materials, polaronic semiconductors, and ultracold atomic gases. The diverging number of bound states and their scaling with $\alpha$ underscore the qualitative change in spectral structure as the system transitions from weak to strong coupling regimes.

7. Summary Table: Characteristic Features

Regime	Bound Excited States	Threshold for Occurrence	Spectral Weight
Weak coupling ($\alpha < \alpha_c$)	Absent	No discrete states apart from ground	All spectral weight in ground state
Intermediate/Strong coupling ($\alpha > \alpha_c$)	Present: infinite tower (in 1D, strong-coupling, etc.)	$\alpha > \alpha_c \sim 1.73$ (1D), state-dependent in 3D	Substantial in first excited state ($Z \sim 0.2$), redistributes as $\alpha$ increases

These results demonstrate the richness and complexity of the internal excitation spectrum of Fröhlich polarons and provide a quantitative roadmap for both theoretical investigation and experimental detection of bound excited states in diverse polaronic systems.