High-Order Harmonic Generation in Solids

• High-Order Harmonic Generation in solids is a nonlinear process where intense laser fields trigger electron dynamics governed by complex band structures and scattering events.
• The phenomenon combines intraband acceleration and interband recombination, leading to odd harmonic spectra and plateau structures that are experimentally observable.
• Pulse characteristics, material symmetry, and field strength intricately control the HHG spectrum, enabling probing of ultrafast electron and phonon dynamics.

High-order harmonic generation (HHG) in solids is a highly nonlinear optical process in which the interaction of an intense ultrafast laser with a crystalline or amorphous solid generates radiation at integer multiples (harmonics) of the driving frequency, extending into the extreme ultraviolet (XUV) or even soft X-ray spectral range. Unlike in gases, where electron dynamics are well described by quasi-free motion in a Coulomb field, solid-state HHG is governed by the intricate interplay between the periodic band structure, strong-field-induced carrier dynamics, and scattering processes. This process supports the generation of ultrashort pulses, allows access to electron and phonon dynamics, and serves as a platform for interrogating quantum geometry and topology within condensed matter systems.

1. Microscopic Quantum-Mechanical Framework

The microscopic description of HHG in solids begins with a nonequilibrium quantum approach anchored in the Keldysh formalism. The state of the system is described by the single-particle nonequilibrium Green’s function $G_{0}(k; t, t')$ on the Keldysh contour, incorporating initial thermal equilibrium and real-time dynamics. The inclusion of the time-dependent vector potential $A(t)$—applied via the Peierls substitution $k \to k - A(t)$—yields

$$G_{0}(k; t, t') = i [f_{k} - \theta_c(t, t')] \exp\left\{ -i \int_{t'}^{t} d\tau\, \varepsilon(k - A(\tau)) \right\}$$

where $f_{k}$ is the Fermi–Dirac occupation and $\varepsilon(k)$ is the band dispersion. This framework does not employ perturbative expansion in the applied field and naturally incorporates strong-field, ultrafast-driven carrier dynamics.

Elastic (impurity) and inelastic (phonon) scattering are introduced as self-energies, $\Sigma_{\text{imp}}$ and $\Sigma_{\text{ph}}$, that enter the time-dependent Dyson equation along the discretized Keldysh contour. For impurity (elastic) scattering,

$$\Sigma_{\text{imp}}(t, t') = n_i V_i^2 \sum_p G_0(p; t, t')$$

and for electron–phonon (inelastic) scattering in the Migdal/ Holstein limit,

$$\Sigma_{\text{ph}}(t, t') = ig^2 D_0(t, t') \sum_p G_0(p; t, t')$$

with $D_0(t,t')$ the phonon propagator. The full Green’s function $G(k; t, t’)$ is found via numerical solution of the Dyson equation.

The output observable, the time-dependent current,

$$j(t) = -i \sum_k v(k - A(t)) \lim_{t'\to t} G^<(k; t, t')$$

(where $v(k) = \partial \varepsilon(k)/\partial k$), yields the HHG spectrum through its Fourier transform $$I(\omega) \propto |\omega \int dt\, e^{i\omega t} j(t)|^2$$.

2. Role of Pulse Properties, Symmetry, and Material Parameters

The applied laser pulse is characterized by duration, envelope, carrier frequency, amplitude, and phase, with a prototypical form for the vector potential:

$$A(t) = A_{\text{max}} \sin(\omega t + \phi) \exp\left[-\frac{(t - t_0)^2}{2\sigma^2}\right]$$

Multi-cycle (quasi-monochromatic) pulses result in harmonic spectra with peaks strictly at odd multiples of the fundamental frequency ($\omega_a$) in inversion-symmetric materials. This restriction arises from the antisymmetry of the group velocity under inversion ($v(k) = -v(-k)$), which leads to cancellation of even harmonics. Single-cycle pulses, however, have a broad Fourier spectrum, and the position and broadening of the harmonics are determined by the interplay between instantaneous field strength and pulse envelope.

Material-specific properties enter through the detailed band structure $\varepsilon(k)$. The inclusion of tight-binding or ab initio band structures enables realistic modeling for different materials and crystallographic directions (e.g., distinct responses in silicon (100) vs (111)), allowing exploration of how bandwidth, effective mass, and band anisotropy affect the onset of nonlinearity and the HHG cutoff.

3. Nonlinear Regimes and the Emergence of High-order Harmonics

For weak fields, the HHG process is perturbative, generating primarily low-order (3rd, 5th, etc.) harmonics. As the field strength increases, electrons are accelerated up to the Brillouin zone boundary, inducing pronounced Bragg or Bloch oscillations. In this regime, nonlinear interplay yields:

• Enhanced emission of high-order harmonics;
• Temporal localization, allowing the strongest harmonics to approach or even match the intensity of the first harmonic in certain regimes.

The nonlinearity manifests in the appearance of clear high-order plateau structures whose positions and intensities depend on accessible interband transitions, with the cutoff scaling determined by the maximum traversable band gap with the available vector potential amplitude.

4. Robustness to Scattering and Implications for Spectral Features

Elastic and inelastic scattering mechanisms attenuate the total current amplitude and consequently the HHG intensity, but the symmetry-protected structure of the harmonic spectrum remains robust. Enhanced scattering raises the spectral noise floor and may obscure higher-order peaks but does not alter the odd-harmonic selection rule in inversion-symmetric materials. This robustness is a direct consequence of the fact that the leading features of HHG—cutoff energy, odd-harmonic dominance, plateau structure—are determined by the underlying symmetry and nonperturbative electron dynamics, which remain intact unless the symmetry or long-range order is broken.

5. Interplay of Intraband, Interband, and Scattering-induced Contributions

The time-domain formulation via the Keldysh/Dyson formalism automatically accounts for both intraband acceleration (group velocity drift within a band) and interband polarization (electron-hole recombination). The detailed time-evolution naturally models the competition and interference between these processes. Scattering events serve to reduce coherence and redistribute spectral weight, but as long as symmetry and field-driven population transfer dominate, the essential features persist.

The generic mathematical structure is summarized in the set of equations:

• Field-driven Green’s function: $G_0(k; t, t')$;
• Scattering-induced self-energies: $\Sigma_{\text{imp}}$, $\Sigma_{\text{ph}}$;
• Time-evolved Dyson equation for $G(k; t, t')$;
• Current $j(t)$, constructed from $G^<$;
• HHG spectrum via $|\omega \int dt\, e^{i\omega t} j(t)|^2$.

These equations admit direct extensions to include arbitrary field shapes, arbitrary band structures, and tailored inclusion of disorder.

6. Experimental Relevance and Generality

The Keldysh-based approach successfully reproduces several central features observed in HHG experiments on solids, including:

• The spectrum consisting only of odd harmonics in inversion-symmetric materials;
• The emergence and nonlinear scaling of high-order harmonics with increasing field;
• The location of harmonic peaks and their robustness to moderate levels of elastic and inelastic scattering;
• The sensitive dependence of the HHG spectrum on pulse duration, phase, and material orientation.

The theoretical treatment is general and applies to both model and realistic solids, providing a predictive framework for the design and analysis of HHG experiments in crystalline and disordered systems. Its flexibility allows adaptation to novel regimes, such as ultrashort pulses, materials with strong phonon coupling, and the simulation of real materials with full band-structure input.

In conclusion, the nonequilibrium Green’s function and Keldysh-contour approach, as formulated in detail in (Kemper et al., 2012), provides a comprehensive, first-principles-based method to understand and quantitatively predict the microscopic mechanisms of high-order harmonic generation in solids. This framework establishes how field-driven electron dynamics, symmetry, scattering, pulse characteristics, and material parameters jointly determine the efficiency, cutoff, and structure of HHG in the solid state.