Nonperturbative effects in second harmonic generation

Abstract: Second-harmonic generation (SHG) is a quintessential probe of inversion symmetry breaking in condensed matter. While perturbative $χ^{(2)}$ processes are well-documented, the nonperturbative regime under intense driving remains largely unexplored. In this Letter, we develop a nonperturbative Floquet-Keldysh theory to describe SHG in two-band systems. Our analysis reveals the emergence of two distinct types of nonperturbative saturation: a transition from the conventional $E^2$ scaling to a linear $E$ dependence, and a stronger saturation regime where the SHG response becomes independent of the field amplitude. These behaviors are analytically shown to be governed by one-photon and two-photon resonance processes, respectively. By applying our formalism to a tight-binding model of monolayer GeS, we demonstrate that these specific scaling behaviors are observable in realistic materials and are fully consistent with large-scale numerical Floquet-matrix calculations.

• The paper reveals distinct SHG saturation regimes, with a crossover from quadratic to linear scaling via one-photon resonances and a field-independent plateau via two-photon resonances.
• It employs a nonperturbative Floquet-Keldysh approach to derive analytic expressions that are confirmed by large-scale numerical simulations using a monolayer GeS model.
• The findings provide practical insights for nonlinear optical devices by identifying experimentally accessible saturation thresholds in strong-field regimes.

Nonperturbative Effects in Second Harmonic Generation: Floquet-Keldysh Analysis

Overview

The paper "Nonperturbative effects in second harmonic generation" (2604.05710) presents a detailed theoretical investigation of SHG in two-band systems under strong optical driving. Leveraging a nonperturbative Floquet-Keldysh formalism, the authors uncover and classify two distinct saturation behaviors in the SHG response as a function of driving field amplitude: a crossover from quadratic to linear scaling governed by one-photon resonances, and a saturation to a field-independent response controlled by two-photon resonances. These analytic predictions are rigorously validated using large-scale numerical Floquet calculations for monolayer GeS, highlighting both the formal and practical significance of strong-field nonlinear optics.

Nonperturbative Floquet-Keldysh Formalism

The authors develop their framework starting from a two-band system subjected to monochromatic light, expressed through a time-dependent Bloch Hamiltonian $H(t) = H_0(\bm{k} + e\bm{A}(t)/\hbar)$ with the vector potential $\bm{A}(t)$ encoding the electromagnetic drive. By expanding $H(t)$ in powers of $\bm{A}(t)$, up to second order, Floquet components $H_n$ for $n = 0, \pm 1, \pm 2$ are identified, capturing both paramagnetic and diamagnetic coupling.

The resulting static Floquet Hamiltonian constructed in the extended photon-number space enables representation of all photon-dressed states and their couplings. The system's coupling to a fermionic bath is incorporated via a dissipation rate $\Gamma$ within the Keldysh formalism, yielding closed-form expressions for the lesser Green's function and the time-resolved current.

This formalism ensures a unified treatment from the perturbative to the strongly nonperturbative regime, capturing all relevant resonance channels between dressed bands for arbitrary field strengths.

Analytical Saturation Mechanisms

By isolating relevant photon-number subspaces through the RWA, the authors provide explicit analytic forms for the SHG current under both one-photon and two-photon resonant conditions.

• One-photon resonance: Involves coupling between the dressed valence band ($m=1$) and conduction band ($m=0$), with the field-dependence entering via parametric coupling. The analytical form predicts a crossover from the conventional perturbative $J \propto E^2$ scaling to a linear $\bm{A}(t)$0 regime as the Rabi frequency surpasses the dissipation rate.
• Two-photon resonance: Involves direct coupling ($\bm{A}(t)$1) via a diamagnetic term. Here, the SHG current transitions from perturbative quadratic scaling to a saturated field-independent value when the nonlinear coupling term dominates, marking a more pronounced nonperturbative effect.

Both processes are controlled by field-dependent denominators: the transition to the nonperturbative regime occurs when $\bm{A}(t)$2 (one-photon) or $\bm{A}(t)$3 (two-photon) terms become comparable to or larger than $\bm{A}(t)$4.

Application to Monolayer GeS and Numerical Validation

To substantiate the theoretical analysis, the framework is applied to a realistic tight-binding model for monolayer GeS—a prototypical gapped Dirac system with strong inversion symmetry breaking and large nonlinear responses.

Two methods are compared:

• Method 1: Analytical integration using truncated $\bm{A}(t)$5 Floquet Hamiltonians for the dominant photon processes.
• Method 2: Full numerical solution of the Floquet-Keldysh equations using a $\bm{A}(t)$6 matrix to include up to high-order multiphoton effects.

Key findings are:

• For sub-gap excitation ($\bm{A}(t)$7 eV), one-photon transitions are inaccessible. The observable SHG response transitions from $\bm{A}(t)$8 scaling to a constant, field-independent plateau at high intensities, quantitatively matching both analytic and numerical treatments.
• For above-gap excitation ($\bm{A}(t)$9 eV), the response transitions from $H(t)$0 scaling to a robust linear dependence ($H(t)$1) at large fields, again with analytical and numerical agreement.

The saturation thresholds—set by the interplay of Rabi frequency and dissipative relaxation—are within experimentally accessible field ranges ($H(t)$2–$H(t)$3 V/m), making experimental observation of these effects viable with current ultrafast laser technologies.

Theoretical and Practical Implications

The analysis establishes that SHG, in the nonperturbative regime, is governed by the underlying photon-dressed Floquet eigenstructure. The qualitative change in response—linear or saturated—serves as a direct, experimentally observable signature of the dominant resonance pathway.

Practically, the identification of constant saturation via two-photon processes suggests fundamental upper bounds for nonlinear device operation and points toward new opportunities for saturable absorbers or nonlinear switches based on quantum geometric design. The validity of low-dimensional analytical truncations even at extreme fields underscores the efficiency of reduced models for predicting nonlinear optical phenomena in realistic materials.

Theoretically, the results generalize the established understanding of nonperturbative shift current phenomena to wider classes of second-order effects. The work suggests that the full nonlinear optical response under strong driving can be classified according to controllable Floquet resonance mechanisms, with geometric and material selectivity.

Conclusion

The presented Floquet-Keldysh theory systematically addresses the nonperturbative regime of SHG in two-band systems, specifically elucidating two distinct saturation mechanisms: a linear regime controlled by one-photon resonance and a stronger field-independent plateau governed by two-photon resonance. The framework accurately captures the nonlinear response in realistic materials such as monolayer GeS, as validated by comprehensive numerical simulations. These insights extend the theoretical foundation for strong-field nonlinear optics and motivate targeted experimental investigations of nonperturbative effects in advanced quantum materials.