🔬 mesoscale physics

Velocity gauge formulation of nonlinear optical response in Floquet-driven systems

This paper establishes a velocity gauge theoretical framework for computing nonlinear optical responses in Floquet-driven quantum systems, demonstrating its equivalence to the length gauge while revealing unique phenomena like photon-assisted transitions and Floquet-induced photocurrents with applications in ultrafast spectroscopy and optoelectronics.

Original authors: S. Sajad Dabiri, Reza Asgari

Published 2026-07-28

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: S. Sajad Dabiri, Reza Asgari

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the world of atoms and electrons not as a quiet, still place, but as a dance floor that never stops moving. In the realm of quantum physics, scientists have discovered that if you shake a material with a rhythmic, repeating light pulse—like a strobe light flashing in perfect time—you can force the electrons to behave in completely new ways. This is the world of "Floquet systems," named after a mathematician who figured out how to describe things that repeat over and over. Think of it like a swing set: if you push the swing at just the right moment every time, it goes higher and higher, gaining energy from your rhythm. In these quantum dance floors, the "push" is a laser, and the "swing" is an electron.

Usually, when we study how materials react to light, we look at them sitting still. But what happens when the material itself is being shaken by a powerful, rhythmic laser? This is where things get tricky. Scientists have two main ways to calculate what happens: one looks at the light as a force pushing on the electron's position (like a hand shoving a ball), and the other looks at the light as a change in the electron's speed (like a car accelerating). Both ways are supposed to give the same answer, like two different maps of the same city. However, when the shaking gets intense and the math gets complicated, these two maps sometimes start to look very different, leading to confusion about what is actually happening. This paper dives into that confusion to see if we can find a clearer path.

The Paper's Mission: A New Way to Count the Dances

The authors of this paper, S. Sajad Dabiri and Reza Asgari, decided to build a new theoretical toolkit to solve this puzzle. They focused on a specific method called the "velocity gauge." If you imagine the electron as a car, the "length gauge" calculates the light's effect by looking at how far the car moves, while the "velocity gauge" looks at how fast the car is going and how that speed changes. The authors argue that for systems being shaken by a rhythmic laser (Floquet systems), the velocity gauge is actually the smoother, more reliable map. It keeps the math tidy and avoids certain mathematical glitches that pop up when you try to use the other method.

To test their new toolkit, they applied it to a classic, simple model known as the "Rabi model." You can think of this model as a tiny, two-story house where an electron can only live on the ground floor or the first floor. When you shine a laser on this house, the electron gets excited and jumps between floors. The authors simulated what happens when this house is shaken by a laser and then probed with a second, weaker light.

What They Found: New Moves and Hidden Currents

Their simulations revealed that when you shake this two-story house, the electrons don't just jump up and down; they start doing some very strange, complex dance moves that are impossible in a still house.

First, they found that the electrons can absorb or emit light in "packets" that don't match the laser's rhythm exactly. Instead of just reacting to the main beat, the electrons can jump up or down by adding or subtracting extra "beats" (photons) from the rhythm. This is called "photon-assisted transition." It's like a dancer who, instead of just stepping on the beat, occasionally adds a quick spin or a skip that wasn't in the original music, creating a whole new rhythm.

Second, they discovered that this shaking can create a "rectified current." In a normal, still system, if you wiggle something back and forth, it just wiggles back and forth; it doesn't go anywhere on average. But in their shaken system, the authors found that even if you shine a steady, non-wiggling light (a "DC" field) on the system, it can generate a flow of electricity that oscillates at the laser's frequency. Conversely, if you shine a light that wiggles at twice the laser's speed, the system can produce a steady, one-way flow of electricity. It's as if shaking a bucket of water back and forth could suddenly make the water flow in a single, steady stream.

The Great Comparison: Two Maps, One City

A major part of the paper was comparing their new "velocity gauge" map against the old "length gauge" map. They ran the numbers for both methods and found that, for almost all the interesting, fast-moving parts of the dance, the two maps agreed perfectly. The frequencies of the light the electrons absorbed and the strength of the currents they generated were identical. This is a huge relief for scientists, as it confirms that both methods are fundamentally describing the same physical reality.

However, they did find one tiny, weird glitch. When they looked at the very specific case of zero frequency (a completely steady, non-wiggling light), the velocity gauge map showed a mathematical "spike" or infinity that didn't make physical sense. The length gauge map, on the other hand, stayed calm and sensible in that specific spot. The authors explain that this spike is a "spurious divergence"—a fake error caused by the way the math is set up in the velocity gauge for this specific limit. It's like a GPS that works perfectly for driving on highways but gets confused if you try to park the car in a driveway.

Why This Matters

The authors conclude that their velocity gauge framework is a powerful, robust tool for understanding these shaken quantum systems. They suggest that with modern lasers that can flash incredibly fast (from terahertz to visible light) and very pure materials, scientists could actually see these effects in a lab. They predict that by carefully controlling these laser rhythms, we could engineer materials that switch their electrical properties on and off or create new types of light sources. While the paper is a theoretical simulation and not a report of a physical experiment yet, it lays the groundwork for future experiments that could lead to next-generation electronic devices and faster, smarter ways to control light and matter.

Technical Summary: Velocity Gauge Formulation of Nonlinear Optical Response in Floquet-Driven Systems

Problem Statement
The study of quantum systems under periodic driving (Floquet systems) has revealed novel phenomena such as topological insulators, time crystals, and light-induced superconductivity. A critical challenge in this field is accurately describing the nonlinear optical responses of these driven systems. While the length gauge has been extensively used to describe static nonlinear effects (e.g., shift and injection currents), its application to Floquet systems presents computational complexities, particularly regarding matrix element calculations and basis truncation errors. Conversely, the velocity gauge offers distinct advantages in strong-field physics and momentum-space treatments but lacks a comprehensive formulation for nonlinear optical responses in intrinsically time-periodic systems. The authors aim to bridge this gap by developing a robust velocity gauge framework that complements existing length gauge approaches, ensuring gauge invariance and computational efficiency.

Methodology
The authors develop a perturbation theory for the density matrix of a time-periodic quantum system described by a bare Hamiltonian $H(k, t) = H(k, t+T)$ .

Gauge Choice: The study utilizes the velocity gauge, where the external probe field is incorporated via the Peierls substitution ( $k \to k + A(t)$ ). This keeps the Bloch Hamiltonian diagonal in momentum space, preserving gauge invariance and simplifying analytical and numerical treatments.
Perturbation Expansion: The density matrix $\rho$ is expanded in powers of the vector potential amplitude $\lambda$ . The Liouville-von Neumann equation is solved recursively to derive expressions for the first-order ( $\rho^{[1]}$ ) and second-order ( $\rho^{[2]}$ ) density matrix corrections.
Conductivity Formulation:
- First-Order: The linear optical conductivity $\sigma^{[1]}$ is derived by evaluating the expectation value of the velocity operator. The resulting formula involves sums over Floquet bands and Fourier components of the Hamiltonian derivatives, incorporating photon-assisted transitions where quasi-energies are shifted by integer multiples of the driving frequency ( $\Omega$ ).
- Second-Order: The nonlinear optical conductivity $\sigma^{[2]}$ is derived by considering terms up to second order in the perturbation. The authors establish a correspondence between the time-averaged response of Floquet systems and static systems, where static transitions are replaced by photon-assisted transitions involving specific Fourier indices ( $j_1, j_2$ ) and modified quasi-energies.
Application to the Rabi Model: The framework is applied to a driven two-level system (the Rabi model). The authors derive the Bloch Hamiltonian in both length and velocity gauges, demonstrating their equivalence via a unitary transformation. Numerical calculations are performed for both gauges to compare results.

Key Results

Analytical Framework: The paper provides explicit formulas for first- and second-order optical conductivities in the velocity gauge for Floquet systems. These formulas account for the periodic nature of the driving field, introducing sidebands and photon-assisted transitions characterized by integer multiples of the driving frequency.
Numerical Validation (Rabi Model):
- Agreement: Numerical solutions for the Rabi model in the velocity gauge show full agreement with the length gauge formulation for finite frequencies. Both gauges predict identical peak frequencies and intensities for nonlinear responses.
- Divergence at Zero Frequency: A key finding is a "spurious divergence" in the first-order optical conductivity at the zero-frequency limit ( $\omega \to 0$ ) within the velocity gauge. This divergence is absent in the length gauge formulation. The authors attribute this to the $1/\omega$ dependence in the velocity gauge expression, which requires careful handling of the Drude peak and expansion of terms to cancel unphysical singularities.
- Nonlinear Phenomena: The simulations reveal rich optical phenomena unique to driven systems, including:
  - Photon-assisted transitions: Transitions occurring at frequencies $\omega_{probe} \pm n\Omega$ .
  - Frequency mixing: Generation of currents at sum and difference frequencies (e.g., $\omega_1 + \omega_2 + n\Omega$ ).
  - Emergent Photocurrents: The generation of rectified currents (DC responses) from AC probe fields, a phenomenon not present in static systems under first-order perturbation.
Gauge Invariance: The matrix elements derived in the velocity gauge are shown to be gauge-invariant, meaning they do not depend on the specific phase choice of Bloch wave functions across the Brillouin zone, a significant advantage for numerical stability.

Significance and Claims
The paper claims to establish a comprehensive theoretical bridge between theoretical models and experimental observations in driven quantum systems.

Complementarity: The velocity gauge formulation is presented as a necessary complement to the length gauge, offering distinct advantages for systems with well-defined momentum-space structures (e.g., atomic and solid-state systems) and for analytical tractability.
Experimental Relevance: The authors assert that the predicted phenomena (photon-assisted transitions, frequency mixing, and Floquet-induced photocurrents) are experimentally accessible using current ultrafast laser technologies (THz to visible range) and pump-probe spectroscopy, provided conditions such as low temperatures and high sample purity are met to maintain Floquet coherence.
Future Directions: The framework is claimed to be extendable to multiband systems and strongly correlated electron materials, providing a foundation for the "Floquet engineering" of quantum materials and the development of next-generation optoelectronic devices.

The authors maintain a modest tone regarding the zero-frequency divergence, acknowledging it as a known artifact of the gauge choice that can be resolved through careful expansion, while emphasizing the overall consistency and utility of the velocity gauge for nonlinear optics in Floquet systems.

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