Nonperturbative effects in second harmonic generation

This paper develops a nonperturbative Floquet-Keldysh theory for second harmonic generation in two-band systems, revealing distinct saturation regimes governed by one- and two-photon resonances and validating these predictions through numerical simulations on monolayer GeS.

The Gist

The Big Picture: Shining a Light on Matter

Imagine you have a special crystal (like a piece of glass or a thin sheet of atoms) and you shine a laser beam at it. In the world of light, this is usually a boring, predictable event. If you shine a red laser (frequency $\omega$), the crystal might glow back with a slightly different color, but usually, it just reflects the red light.

However, some special crystals have a "magic trick" called Second Harmonic Generation (SHG). When you shine red light on them, they don't just reflect red; they actually double the energy of the light and spit out blue light (frequency $2\omega$).

For decades, scientists understood this trick using a simple rule: The brighter the red light, the brighter the blue light. Specifically, if you double the brightness of the red laser, the blue light gets four times brighter. This is the "weak light" rule, and it works perfectly for normal flashlights or dim lasers.

The Problem: What Happens with a Super-Bright Laser?

The authors of this paper asked: "What happens if we turn the laser up to maximum power?"

In the past, nobody knew what to expect because standard physics breaks down when the light is incredibly intense. It's like trying to predict how a car behaves by only driving it at 10 mph, then suddenly asking what happens if you drive it at 200 mph. The old rules might not apply.

The researchers wanted to explore this "extreme light" territory to see if the crystal's magic trick changes.

The Discovery: Two New Ways the Light "Gives Up"

Using a complex mathematical toolkit (called Floquet-Keldysh theory, which is like a super-advanced way to track how atoms dance to the rhythm of light), the authors discovered that when the laser gets too strong, the crystal stops following the old rules. Instead of getting brighter and brighter, the blue light output hits a "ceiling" or a "speed limit."

They found two distinct types of ceilings, depending on how the light interacts with the atoms:

1. The "One-Step" Climb (Linear Saturation)

• The Analogy: Imagine a person trying to climb a ladder. In the weak light world, if you push them harder (more light), they climb twice as fast. But in the strong light world, they hit a "one-step" limit. No matter how hard you push, they can only climb one rung at a time.
• The Result: The blue light stops growing quadratically (4x, 9x, 16x) and starts growing linearly (2x, 3x, 4x). It's still getting brighter, but much slower than before. This happens when the light energy matches a specific "one-step" jump between electron energy levels.

2. The "Two-Step" Wall (Constant Saturation)

• The Analogy: Now imagine a person trying to jump over a very high wall. If you push them gently, they can't jump. If you push them harder, they jump higher. But if you push them too hard, they hit a ceiling so hard that they just stop going up entirely. They are stuck at the same height, no matter how much force you apply.
• The Result: The blue light stops growing completely. It hits a flat plateau. Even if you double the laser power, the blue light output stays exactly the same. This happens when the light energy matches a "two-step" jump.

The Test Case: The "GeS" Crystal

To prove this wasn't just math on a page, the authors tested their theory on a real material called Monolayer GeS (a single layer of Germanium and Sulfur atoms).

• The Experiment: They simulated shining different colored lasers at this crystal.
• The Outcome:
  • When they used a laser that required a "two-step" jump, the blue light hit a flat wall (Constant Saturation).
  • When they used a laser that required a "one-step" jump, the blue light grew slowly (Linear Saturation).
• The Verification: They compared their new math formulas against massive computer simulations (like running a super-detailed video game of the atoms). The results matched perfectly, proving their theory is correct.

Why Does This Matter?

1. New Physics: It shows that nature has hidden "speed limits" for how fast materials can react to light. We thought light could make things brighter forever; now we know there are hard stops.
2. Better Tech: This could help engineers design better optical switches, ultra-fast computers, or sensors. If you know exactly when a material will stop responding to light, you can use that to create precise controls.
3. A New Diagnostic Tool: By watching how the light saturates (does it go flat? does it go linear?), scientists can now "listen" to the atoms to figure out exactly how they are structured and how they are moving.

Summary

Think of the crystal as a musical instrument.

• Weak Light: If you pluck the string gently, the volume goes up smoothly.
• Strong Light (Old Theory): We thought if you plucked it harder, it would get much louder.
• Strong Light (New Discovery): The authors found that if you pluck it too hard, the string hits a physical limit. Sometimes it just gets a little louder (Linear), and sometimes it stops getting louder entirely (Constant), regardless of how hard you pull.

This paper maps out exactly where those limits are and how to find them in real materials.

Technical Summary

1. Problem Statement

Second-harmonic generation (SHG) is a fundamental nonlinear optical process used extensively to probe inversion symmetry breaking in condensed matter. While the behavior of SHG in the weak-field regime is well-described by perturbation theory (where the response scales as $I_{2\omega} \propto I_{\omega}^2$, or $J \propto E^2$), the behavior under intense laser fields remains largely unexplored.

• The Gap: Standard $\chi^{(2)}$ descriptions fail when light-matter coupling becomes strong. Recent studies on shift currents (a related $\chi^{(2)}$ process) have shown deviations from perturbative scaling (transitioning to linear $J \propto E$), suggesting that SHG may also exhibit complex nonperturbative saturation regimes.
• The Goal: To develop a theoretical framework capable of describing SHG in the nonperturbative regime and to identify the specific scaling behaviors and physical mechanisms governing these effects.

2. Methodology

The authors employ a nonperturbative Floquet-Keldysh formalism to analyze a two-band system under periodic driving by monochromatic light.

• Hamiltonian Formulation:

  • The system is modeled using the Peierls substitution, $H(t) = H_0(\mathbf{k} + e\mathbf{A}(t)/\hbar)$, where $\mathbf{A}(t)$ is the vector potential of the driving field.
  • The time-dependent Hamiltonian is expanded up to the second order in the vector potential to extract Fourier components ($H_n$).
  • A static Floquet Hamiltonian ($H_F$) is constructed in an extended photon-dressed state space.

• Current Calculation:

  • The time-dependent current is derived using the lesser Green's function ($G^<$) within the Keldysh formalism.
  • The SHG current component at frequency $2\Omega$ is extracted by isolating terms where the photon index difference $n - m = 2$.

• Analytical Approximation (Rotating-Wave Approximation - RWA):
  The authors identify two distinct resonance channels and truncate the infinite Floquet matrix into effective $2 \times 2$ Hamiltonians for each:

  1. One-Photon Resonance: Truncates to the subspace of the valence band (dressed with 1 photon) and the conduction band (0 photons). Coupling is mediated by $H_{\pm 1}$.
  2. Two-Photon Resonance: Truncates to the subspace of the valence band (dressed with 2 photons) and the conduction band (0 photons). Coupling is mediated by the diamagnetic term $H_{\pm 2}$.

• Validation:
  The analytical results derived from the truncated models are compared against large-scale numerical calculations using a full $22 \times 22$ Floquet matrix (Method 2) to ensure accuracy beyond the RWA.

3. Key Contributions

The paper makes three primary theoretical contributions:

1. Derivation of Nonperturbative SHG Currents: The authors derive analytical expressions for the SHG current ($J_{1ph}$ and $J_{2ph}$) that include field-dependent saturation factors in the denominator, valid beyond the perturbative limit.
2. Identification of Two Distinct Saturation Regimes:
   • Regime A (One-Photon Dominated): A transition from the conventional quadratic scaling ($J \propto E^2$) to a linear dependence ($J \propto E$).
   • Regime B (Two-Photon Dominated): A stronger saturation mechanism where the SHG response becomes independent of the field amplitude ($J \propto \text{constant}$), resulting in a plateau.
3. Physical Mechanism Clarification: The authors demonstrate that these scaling behaviors are governed by the competition between the Rabi frequency ($\Omega_R \sim eE$) and the dissipation rate ($\Gamma$). The saturation arises from intensity-induced broadening (power broadening) compensating for the driving strength.

4. Results

The theory was applied to monolayer GeS, a gapped Dirac material with broken inversion symmetry ($C_{2v}$ point group), known for its giant shift-current response.

• Sub-Gap Driving ($\hbar\Omega = 1.5$ eV):

  • One-photon transitions are energetically forbidden.
  • The system is dominated by the two-photon resonance.
  • Result: The SHG current initially follows $E^2$ scaling but saturates to a constant value at high intensities. This confirms the "constant saturation" regime.

• Above-Gap Driving ($\hbar\Omega = 5.0$ eV):

  • One-photon resonance is allowed and dominant.
  • Result: The SHG current transitions from $E^2$ to a clear linear dependence ($J \propto E$) as intensity increases.

• Agreement:

  • The analytical predictions (Method 1) showed excellent agreement with the full numerical Floquet calculations (Method 2) across both regimes, even at extreme field strengths ($10^7$–$10^8$ V/m).
  • This suggests that for SHG, higher-order photon-dressing processes primarily renormalize the effective bandgap (dynamic Stark shift) rather than opening entirely new resonance channels, provided the driving frequency avoids high-order multiphoton resonances with deeper bands.

5. Significance

• Fundamental Physics: The work extends the understanding of nonlinear optics from the perturbative $\chi^{(2)}$ regime into the nonperturbative territory. It reveals that strong light-matter coupling qualitatively alters the nature of nonlinear responses, creating distinct "fingerprints" (linear vs. constant scaling) that identify the dominant excitation pathway.
• Experimental Feasibility: The predicted saturation thresholds for GeS ($\sim 10^7$–$10^8$ V/m) are accessible using modern mid-infrared or terahertz ultrafast lasers without causing optical damage, making experimental verification feasible.
• Diagnostic Tool: The field dependence of SHG can serve as a direct probe of the underlying Floquet resonance structure in driven crystals. Observing a linear scaling indicates one-photon dominance, while a plateau indicates two-photon dominance.
• Material Control: The findings suggest that intense light fields can be used to dynamically tune and suppress second-order optical responses in topological and noncentrosymmetric materials, offering new avenues for controlling optical nonlinearities.