Multiple-Phonon Addition Processes

• Multiple-phonon addition processes are quantum phenomena where two or more phonons interact via absorption, emission, or merging, fundamentally reshaping energy spectra.
• They enable advanced applications such as quantum state engineering, non-linear photoluminescence, and damping control in systems like superfluids and optomechanical cavities.
• These processes are rigorously analyzed using diagrammatic techniques and chain-fraction summation methods to resolve resonant phonon satellites and inter-level couplings.

Multiple-phonon addition processes constitute a class of physical phenomena in which two or more phonons are absorbed, emitted, or merged either by quasiparticles or within collective excitations of solids and quantum gases. Such processes are central to understanding the renormalization of energy spectra, phonon-mediated emission/absorption, non-linear photoluminescence, quantum state engineering, and damping in superfluids. The underlying mechanisms involve intricate interplay between quantum many-body interactions, resonance conditions, and system-specific couplings, and are described by diagrammatic techniques, polaron physics, and perturbative or non-perturbative methods.

1. Diagrammatic Summation and the Multiplicative Chain-Fraction Method

The theoretical analysis of multiple-phonon addition processes in quasiparticles interacting with phonons is fruitfully addressed via the Feynman–Pines diagram technique. For multi-level localized quasiparticles coupled to dispersionless optical (polarization) phonons at $T=0$, the system is governed by a Fröhlich-type Hamiltonian,

$$\hat H = \sum_{\mu=1}^\tau E_\mu a_\mu^\dagger a_\mu + \sum_{\bf q} \Omega b_{\bf q}^\dagger b_{\bf q} + \sum_{\mu,\mu'=1}^\tau \sum_{\bf q} \varphi_{\mu\mu'} a_\mu^\dagger a_{\mu'} (b_{\bf q} + b_{-{\bf q}}^\dagger)$$

The Green's function $G_{\mu\mu'}(\omega)$ expands via a mass operator (MO), $M_{\mu\nu}(\omega)$, which collects all phonon-mediated self-energies. High-order expansions produce both "multiplicative" (all internal indices coincide) and "non-multiplicative" diagrams. The multiplicative diagrams encapsulate all virtual processes where the same level accumulates sequential phonons, enabling their exact resummation into a branched continued fraction: $$m_{\mu\nu}^{[m]}(\xi) = \sum_{\lambda=1}^\tau \frac{\alpha_{\mu\lambda}\alpha_{\lambda\nu}}{\xi_\lambda-1 - m_{\lambda\lambda}^{[m]}(\xi)}$$ This continued-fraction captures the infinite hierarchy of multi-phonon additions, allowing closed-form, numerically efficient computation of phonon-induced spectrum restructuring. Non-multiplicative contributions are subleading and iteratively suppressed under successive chain-fraction summations (Tkach et al., 2019, Tkach et al., 2018).

2. Renormalized Energy Spectra and Phonon Satellites

In the two-level case, the MO chain-fraction construction yields a renormalized spectral structure consisting of the ground-state polaron shift and an infinite set of "phonon satellites"—each satellite corresponding to a quasiparticle dressed by $l$ virtual phonons. The coupled Dyson equations for the dressed Green's functions,

$g_\mu(\xi) = \frac{1}{\xi_\mu - m_\mu(\xi)}$

lead to renormalized eigenenergies located at $\xi = l$ and $\xi = \delta + l$, i.e., at

$$E_1 + l\Omega, \quad E_2 + l\Omega, \qquad l = 0, 1, 2, \dots$$

These satellites are grouped into bands whose widths scale approximately as $\Delta e_{\mu, l} \propto \alpha_{\mu\mu}$ and reflect the strength of the intra-level electron–phonon coupling. Interlevel phonon coupling ($\alpha_{12}$) induces anti-crossing complexes and hybridized ladders, particularly when the detuning $\delta$ is near integer multiples of $\Omega$, manifesting strong multi-phonon resonance and multi-level anticrossings (Tkach et al., 2019, Tkach et al., 2018).

3. Multi-Phonon Addition in Optomechanical and Cavity QED Systems

Quantum engineered platforms such as cavity QED and cavity optomechanical systems facilitate controlled n-phonon addition processes. A typical realization involves a two-level emitter or optical cavity mode strongly coupled to a mechanical (phonon) mode, with coherent driving. The system Hamiltonian in the rotating frame,

$$H = \Delta \sigma^\dagger \sigma + \omega_m b^\dagger b + g \sigma^\dagger \sigma (b+b^\dagger) + \Omega (\sigma + \sigma^\dagger)$$

exhibits anti-Stokes and Stokes resonances when the laser detuning satisfies $\Delta = n\omega_m$. In this regime, the system supports resonant transitions between $|g,0\rangle$ and $|e,n\rangle$ (or their cavity analogs), mediated by an effective n-phonon coupling

$$g_n \sim \Omega \left(\frac{g}{\omega_m}\right)^n \frac{1}{\sqrt{n!}}$$

These resonances produce super-Rabi oscillations between the zero- and n-phonon states, and, under inclusion of dissipation, give rise to highly pure n-phonon bundle emission (addition), with emission rates and purities tunable by coupling strengths, detuning, and loss rates (Bin et al., 2019, Macovei et al., 2022).

4. Resonant Multi-Phonon Absorption and Anti-Stokes Photoluminescence

In materials with strong electron-phonon coupling and soft lattice modes, efficient absorption (addition) of multiple phonons underlies phenomena such as anti-Stokes photoluminescence (ASPL). In CsPbBr$_3$ nanocrystals, the carrier–phonon interaction forms polaron states characterized by a binding energy $E_\text{pol} = g^2/(\hbar\omega_0)$. When incident photons are tuned below the gap by $M\hbar\omega_0$, resonant $M$-phonon absorption transfers an electron to the conduction band plus $M$ phonons. The rate for such a process follows

$$W_M = (2\pi/\hbar) |\mu E|^2 \rho(E_c) \; e^{-S} \frac{S^M}{M!} [n_B(T)]^M \delta(\Delta - M\hbar\omega_0)$$

where $S = (g/\hbar\omega_0)^2$ is the Huang–Rhys factor, $n_B(T)$ the phonon occupation, and $\delta$ imposes multi-phonon resonance. ASPL quantum efficiencies near unity are achieved when the Franck–Condon overlap $S^M e^{-S}$ is maximized ($M \approx S$), a regime attainable via material engineering (Zhang et al., 2023).

5. Multi-Phonon Processes in Superfluids: Universal Damping Mechanisms

In pair-condensed Fermi gases (BEC–BCS crossover), phonon addition manifests through resonant $2\rightarrow1$ ("Beliaev-Landau") and $2\rightarrow2$ ("Landau-Khalatnikov") scattering. The effective Hamiltonian incorporates up to four-phonon interactions. The dominant low-temperature damping channel depends crucially on the sign of the phonon dispersion curvature $\gamma$:

• For $\gamma>0$ (upward concavity), energy conservation allows $2\leftrightarrow1$ processes, which yield damping rates scaling as $q^5$ at $T=0$ with universal prefactors and significant $q^7$ corrections beyond mean-field theory.
• For $\gamma<0$ (downward concavity), only $2\leftrightarrow2$ processes are resonant at small $q$, with $T^7$ scaling at low temperatures. Rescaling momenta and temperature-independent factors leads to universal formulas for these damping processes that interpolate between BEC and BCS regimes (Kurkjian et al., 2016).

6. Experimental Manifestations and Parameter Regimes

Realization and observation of multi-phonon addition processes require:

• Strong electron–phonon or photon–phonon coupling,
• Well-resolved resonance conditions (e.g., $\omega_m \gg \kappa, \gamma$ for optomechanical systems),
• Cryogenic operation to suppress thermal phonon population,
• Tunable detuning $\Delta$ to achieve multi-phonon resonance,
• Engineered materials with large Huang–Rhys factors and soft LO phonon modes.

Practical platforms include semiconductor quantum dots in nanomechanical resonators, superconducting qubits with surface-acoustic-wave coupling, optomechanical cavities, and nanocrystal samples exhibiting strong polaronic effects (Bin et al., 2019, Zhang et al., 2023, Macovei et al., 2022).

7. Physical Interpretation and Outlook

Multiple-phonon addition reflects the quantum nonlinearity of the interacting system—sequential or simultaneous absorption/emission of phonons fundamentally alters excitation spectra, emission properties, damping dynamics, and device functionality. Chain-fraction summation techniques rigorously capture the full ladder of virtual and real multi-phonon processes and clarify the spectral transitions between non-resonant and resonant coupling regimes. These processes underpin quantum-state engineering, quantum information transport (phonon guns), high-efficiency up-conversion, and non-linear damping in supersolids and ultra-cold gases. Advances hinge on material optimization for electron-phonon coupling and resonance engineering, with ongoing work extending beyond the polaronic and diagrammatic paradigms to time-dependent and strongly non-equilibrium scenarios (Tkach et al., 2019, Zhang et al., 2023, Tkach et al., 2018).