Extreme plasmons

Abstract: Nanosciences largely rely on plasmons which are quasiparticles constituted by collective oscillations of quantum electron gas composed of conduction band electrons that occupy discrete quantum states. Our work has introduced non-perturbative plasmons with oscillation amplitudes that approach the extreme limit set by breakdown in characteristic coherence. In contrast, conventional plasmons are small-amplitude oscillations. Controlled excitation of extreme plasmons modeled in our work unleashes unprecedented Petavolts per meter fields. In this work, an analytical model of this new class of plasmons is developed based on quantum kinetic framework. A controllable extreme plasmon, the surface "crunch-in" plasmon, is modeled here using a modified independent electron approximation which takes into account the quantum oscillation frequency. Key characteristics of such realizable extreme plasmons that unlock unparalleled possibilities, are obtained.

• The paper introduces a non-perturbative quantum kinetic model that describes extreme plasmons through large-amplitude electron oscillations in quantum gases.
• It employs the Wigner phase-space formalism and a modified independent electron approximation to capture quantum and relativistic effects.
• The study presents the surface crunch-in plasmon mode for controllable PV/m field generation, with promising implications for particle acceleration and nanophotonics.

Non-Perturbative Quantum Kinetic Modeling of Extreme Plasmons

Introduction

The paper "Extreme plasmons" (2404.02087) presents a comprehensive theoretical framework for a new class of plasmons—termed "extreme plasmons"—characterized by non-perturbative, large-amplitude collective oscillations of conduction electrons in quantum electron gases. Unlike conventional plasmons, which are well-described by small-amplitude, linearized models, extreme plasmons approach the coherence limit set by the breakdown of phase ordering, enabling electromagnetic fields in the petavolt-per-meter (PV/m) regime. The work develops an analytical quantum kinetic model, introduces the surface "crunch-in" plasmon as a controllable realization, and provides explicit conditions for excitation and field estimation, with implications for both fundamental nanoscience and high-field applications.

Quantum Kinetic Framework and Model Development

Quantum Degeneracy and Correlation

The analysis begins by establishing the quantum nature of conduction electron gases in metals and semiconductors, characterized by high degeneracy ($\chi \gtrsim 1$) and strong correlation ($\Gamma \gtrsim 1$) at room temperature for densities $n_0 \sim 10^{18-24}\,\mathrm{cm}^{-3}$. The quantum state is fundamentally distinct from classical plasmas, with discrete energy and momentum states governed by Fermi-Dirac statistics and the Pauli exclusion principle. This underpins the collisionless, collective behavior necessary for sustaining plasmons.

Wigner Phase-Space Formalism

To capture the non-perturbative, large-amplitude dynamics, the paper employs the Wigner quantum phase-space formalism. The Wigner function $f_W(\mathbf{r},\mathbf{p},t)$ provides a quasi-probability distribution over position and momentum, enabling the treatment of quantum systems in a manner analogous to classical kinetic theory but without violating the uncertainty principle. The evolution is governed by the Wigner equation, coupled to Poisson's equation for the self-consistent electrostatic potential.

Modified Independent Electron Approximation

For the large-amplitude regime, the many-body problem is reduced via a modified independent electron approximation, justified by the dominance of kinetic and lattice potential energies over electron-electron interactions at high excitation. This allows the dynamics to be modeled by single-particle equations of motion, with quantum corrections incorporated through a material- and excitation-dependent quantum factor $\mathscr{F}_Q(\mathbf{k},\mathbf{p})$.

Breakdown of Perturbative Approaches

The work rigorously demonstrates that conventional fluid and linearized quantum models (e.g., those of Pines and Bohm) are inapplicable in the extreme regime, where the oscillation amplitude $\delta$ approaches the plasmonic wavelength $\lambda_Q$. Here, the collective wavevector $k$ and electron momenta $p$ become large, and the system is no longer amenable to perturbative expansion in small parameters such as $\alpha$. The quantum factor $\mathscr{F}_Q$ encapsulates the non-perturbative enhancement of plasmon frequency and field strength.

Surface Crunch-In Plasmon: Controlled Excitation and Kinetic Model

Experimental Realizability and Constraints

The paper introduces the surface "crunch-in" plasmon as a practical mechanism for exciting extreme plasmons. This mode is excited by a high-density, ultra-relativistic charged particle beam propagating inside a hollow conductive tube. The design ensures:

• Collisionless interaction (beam propagates in vacuum core)
• Elimination of pre-pulse energy (intrinsic to accelerator-based beams)
• Maximized energy exchange (azimuthal enclosure by quantum electron gas)
• Tunability via nanofabrication (control over $n_0$, $r_t$, $\Delta w$)
• Continual focusing (enabled by the electrostatic nature of the crunch-in mode)

Kinetic Equation and Existence Conditions

The radial dynamics of conduction electrons are modeled in cylindrical coordinates, with the equation of motion for the radial displacement $r(z,t)$ given by:

$$\gamma_e m_e \frac{\partial^2 r(z,t)}{\partial t^2} = F_{\rm coll} + eE_b$$

where $F_{\rm coll}$ is the net collective force (including lattice and electron compression), $E_b$ is the beam field, and $\gamma_e$ is the relativistic factor. The model incorporates the Heaviside function to account for the transition between wall and core regions. The existence and excitation conditions for the crunch-in plasmon are derived analytically, relating beam and material parameters to the required amplitude and spatial confinement.

Field Estimation

The model yields explicit expressions for the peak radial and longitudinal electric fields:

• Radial field:

$$E_r \sim \mathscr{F}_Q \Theta \frac{Q_b}{r_t \sigma_z}$$

• Longitudinal field:

$$E_z \sim \frac{\mathscr{F}_Q^2}{\kappa} \sqrt{n_0 r_t} \frac{\sqrt{Q_b}}{\sigma_z}$$

For representative parameters ($n_0 = 2 \times 10^{22}\,\mathrm{cm}^{-3}$, $r_t = 100\,\mathrm{nm}$, $Q_b = 315\,\mathrm{pC}$, $\sigma_z = 400\,\mathrm{nm}$), the model predicts $E_r \sim 9\,\mathrm{TV/m}$ and $E_z \sim 5.7\,\mathrm{TV/m}$ (modulo $\mathscr{F}_Q$ and geometric factors), in agreement with prior simulation results. For lower-density semiconductors, fields in the $\sim 10\,\mathrm{GV/m}$ range are predicted.

Relativistic Quantum Effects and Coherence Limit

The analysis highlights two key quantum-relativistic effects:

• Relativistic quantum tunneling: At high excitation, conduction electrons can tunnel across the material-vacuum interface on attosecond timescales, enabling radial oscillations that cross the tube boundary.
• Relativistically induced ballistic transport: The mean free path of conduction electrons increases by orders of magnitude as their velocity approaches $c$, further suppressing collisional dissipation and enabling sustained coherence.

The coherence limit is set by the onset of trajectory overlap and phase decoherence, beyond which the collective plasmonic mode breaks down. The model provides a quantum-corrected expression for the maximum sustainable field before this limit is reached.

Implications and Future Directions

Theoretical Implications

The work establishes that extreme plasmons constitute a fundamentally new regime of collective quantum dynamics, inaccessible to classical or perturbative quantum models. The non-perturbative quantum kinetic approach, with explicit inclusion of material- and excitation-dependent corrections, is essential for accurate modeling. The results challenge the adequacy of fluid and linearized theories for high-field nanoplasmonics and motivate further development of quantum kinetic simulation tools.

Practical Applications

The ability to controllably excite PV/m fields at nanometric scales opens avenues for:

• Compact, high-gradient particle acceleration
• Nano-wiggler-based gamma-ray sources
• Exploration of strong-field QED phenomena (e.g., vacuum polarization near the Schwinger limit)
• Ultrafast, high-field nanophotonics and optoelectronics

The surface crunch-in plasmon provides a concrete pathway for experimental realization, leveraging advances in nanofabrication and accelerator technology.

Experimental Outlook

The paper outlines ongoing and planned experimental efforts to validate the model, including the use of tunable semiconductor structures and high-brightness particle beams. Precise determination of the quantum factor $\mathscr{F}_Q$ and geometric parameters ($\Theta$, $\kappa$) will require dedicated measurements and possibly the development of new diagnostics for ultrafast, high-field phenomena at the nanoscale.

Conclusion

This work presents a rigorous, non-perturbative quantum kinetic theory for extreme plasmons, introducing the surface crunch-in plasmon as a controllable, experimentally accessible realization. The model provides explicit existence and excitation conditions, field estimates, and highlights the necessity of quantum kinetic approaches for accurate description. The predicted PV/m fields and associated phenomena have significant implications for both fundamental science and advanced technological applications, motivating further theoretical, computational, and experimental investigation.