Photon emission due to vacuum instability under the action of a quasi-constant electric field

This paper presents a nonperturbative study of photon emission accompanying electron-positron pair creation in a quasi-constant strong electric field, deriving closed formulas for emission probabilities and distributions while establishing the applicability domain of the locally constant field approximation.

The Gist

The Big Picture: Shaking the Empty Room

Imagine the vacuum of space not as an empty void, but as a calm, frozen lake. In the world of quantum physics, this "lake" is actually teeming with potential energy. If you hit it hard enough, you can create ripples that turn into real particles—specifically, pairs of electrons and their antimatter twins, positrons. This is known as the Schwinger effect.

Usually, to make these particles appear, you need an electric field so incredibly strong it's almost impossible to create in a lab. However, this paper asks a specific question: What happens if you shine a light (emit a photon) while you are trying to create these particles?

The authors are studying a scenario where a strong electric field is turned on for a specific amount of time (like a light switch being flipped on and then off) and seeing how it causes the vacuum to "scream" or emit light while it's being torn apart.

The Problem: Too Complicated to Calculate

Calculating exactly how light is emitted during this chaotic process is like trying to predict the exact path of every single drop of water in a tsunami while also tracking a single bubble rising to the surface. The math gets incredibly messy because the electric field is changing, and particles are being created out of nothing.

In the past, scientists used a shortcut called the Locally Constant Field Approximation (LCFA). Think of this like looking at a fast-moving river and pretending, for a split second, that the water is perfectly still. This makes the math easy, but it's only accurate if the river isn't changing too fast.

The Solution: A New Rulebook for the Shortcut

The authors of this paper have refined this "shortcut" method. They wanted to know: When is it safe to pretend the electric field is constant, even though it's actually turning on and off?

They found a specific "safe zone" where this shortcut works perfectly. Here is how they broke it down:

1. The "Formation Zone" Analogy:
   Imagine a photon (a particle of light) being born. It doesn't happen instantly at a single point; it takes a tiny bit of time and space to "form," like a cloud gathering before it rains.

   • The authors discovered that for high-frequency light (very energetic, fast-vibrating photons), this "formation zone" is determined entirely by the strength of the electric field, not by how long the field has been on.
   • Because this zone is so small and happens so quickly, the electric field doesn't have time to change significantly while the photon is being born. This justifies using their "constant field" shortcut.

2. The "Switch" Effect:
   The electric field in their model is like a switch that flips on at time $t_1$ and off at time $t_2$.

   • If the switch stays on for a long time (a "macroscopic" duration), the messy effects of the switch flipping (the "click" sound of the switch) become negligible compared to the main event.
   • They proved that as long as the field stays on long enough, the light emitted behaves as if the field were constant forever.

What They Found: The Shape of the Light

Once they established that their shortcut was valid, they calculated exactly what this emitted light looks like. They found two main characteristics:

• Direction (The "Flat Pancake"):
  The light isn't emitted in all directions like a lightbulb. Instead, it is emitted mostly in a flat plane, perpendicular (at a 90-degree angle) to the direction of the electric field.

  • Analogy: Imagine the electric field is a vertical pole. The light doesn't shoot up or down the pole; it shoots out sideways, like a ring of fire around the pole.

• Polarization (The "Orientation"):
  Light waves vibrate in specific directions. The authors found that the light has a very specific "vibration" pattern.

  • If the electric field is strong, the light vibrates in a direction that is perpendicular to both the direction the light is traveling and the electric field itself. It's like a strict rule that the light must dance in a specific orientation.

The Limits: When the Shortcut Fails

The paper also defines the boundaries of their discovery.

• Too Slow: If the light is very low frequency (like a deep bass rumble), the "formation zone" becomes huge. The electric field might change significantly while the light is forming, so the shortcut doesn't work.
• Too Fast: If the light is too high frequency, it requires more energy than the electric field can provide in the time it is on. There is a maximum "speed" (frequency) limit for the light, determined by how hard the field pushes and how long it pushes.

Summary

In simple terms, this paper is a quality control manual for a physics shortcut.

The authors proved that when a strong electric field is active for a long time, we can safely use a simplified method to calculate how much light is emitted as the vacuum creates matter. They showed that this light comes out in a specific flat shape and with a specific vibration pattern, provided the light is energetic enough. This helps scientists better understand the complex dance between light, matter, and powerful electric fields without getting lost in impossible math.

Technical Summary

Technical Summary: Photon Emission due to Vacuum Instability under a Quasi-Constant Electric Field

Problem Statement
The paper addresses the radiative processes accompanying vacuum instability (the Schwinger effect) in the presence of a strong, quasi-constant electric field of finite duration $T$. While the locally constant field approximation (LCFA) is a standard tool in strong-field Quantum Electrodynamics (QED) for describing pair creation, its application to radiative processes—specifically photon emission from the vacuum accompanied by electron-positron pair creation—requires careful formulation. Previous works by Nikishov described photon emission amplitudes in constant fields, but these descriptions are satisfactory primarily in weak-field regimes where pair creation is negligible. In strong-field regimes, where pair creation is significant, a perturbative approach with respect to radiative interactions for average values (probabilities) is required rather than for amplitudes. The authors aim to develop a nonperturbative formulation of strong-field QED to derive closed formulas for the total probability of photon emission and to rigorously establish the domain of applicability for the LCFA in this specific context.

Methodology
The authors employ a nonperturbative formulation of strong-field QED based on the generalized Furry representation. The external field is modeled as a $T$-constant uniform electric field directed along the $x$-axis, switching on abruptly at $t_1 = -T/2$ and off at $t_2 = T/2$, with amplitude $E$.

1. Effective Perturbation Theory for Average Values: Instead of calculating probability amplitudes directly, the authors construct a perturbation theory for the total probability of photon emission. They utilize a generating functional for mean values, expressing the total probability of emitting a photon with momentum $k$ and polarization $\vartheta$ from the vacuum as a trace over the out-basis or, equivalently, as an expectation value in the in-vacuum state. This allows the S-matrix to be truncated at the first order with respect to the radiative interaction while retaining the nonperturbative effects of the external field on the vacuum state.
2. Bogolubov Transformation: The in- and out-creation and annihilation operators are related via a Bogolubov transformation, characterized by coefficients derived from the solutions of the Dirac equation in the presence of the time-dependent potential. The mean number of created pairs is determined by these coefficients.
3. High-Frequency Approximation: The analysis focuses on the emission of high-frequency photons ($\omega \gg T^{-1}$). In this regime, the authors utilize the saddle-point method to evaluate the integrals involving Weber parabolic cylinder functions (WPCFs) that arise from the Dirac field solutions.
4. Locally Constant Field Approximation (LCFA): The authors investigate the conditions under which the variation of the external field over the "formation interval" of the photon emission can be neglected. They derive the width of this formation interval and compare it to the scale of field variation.

Key Contributions and Results

• Closed Formulas for Total Probability: The authors derive explicit closed formulas for the total probability of one-photon emission from the vacuum accompanied by pair creation in a constant electric field of finite duration. The probability is expressed in terms of integrals involving WPCFs and confluent hypergeometric functions.
• Domain of LCFA Applicability: A central result is the establishment of the frequency domain where the LCFA is valid for photon emission accompanying pair creation. The authors define a high-frequency range $\omega_{min} < \omega < \omega_{max}$:
  • $\omega_{min} \approx \tau_\gamma \omega_{sc}$, where $\omega_{sc} = \Delta t_{st}^{-1}$ and $\tau_\gamma \gg 1$.
  • $\omega_{max} \approx 2(T/\Delta t_{st})\omega_{sc}$.
    Within this range, the width of the formation interval is determined entirely by the electric field strength $E$ and is independent of the photon frequency and particle momenta. Consequently, the field variation over the formation length is negligible, justifying the LCFA.
• Angular and Polarization Distributions:
  • Directionality: In the high-frequency limit, radiation is directed mainly near the plane orthogonal to the electric field axis ($x$-axis).
  • Polarization: The emission exhibits two linear polarizations. For polarization $\vartheta=2$ (in the plane spanned by the propagation direction and the electric field), the probability contains a small term proportional to $\cos \phi$ (where $\phi$ is the angle between the photon momentum and the $x$-axis). For polarization $\vartheta=1$ (orthogonal to the $kX$ plane), the probability is dominant in the strong-field limit ($\pi \lambda_0 \lesssim 1$).
  • Strong Field Limit: In strong fields, the emission is predominantly polarized in the direction orthogonal to both the wave vector and the electric field.
• Frequency Dependence: The probability of high-frequency emission is bounded above by the work done by the electric field ($2eET$). The frequency grows linearly with time during the field's action, and the probability distribution is governed by an exponential factor $e^{-\pi \rho^2}$, where $\rho$ relates to the photon energy and transverse momentum.

Significance and Claims
The paper claims to provide a rigorous advancement of the LCFA method specifically for radiative processes accompanied by the Schwinger effect. By formulating the problem in terms of average values rather than amplitudes, the authors successfully incorporate the nonperturbative effects of strong fields into the calculation of photon emission probabilities.

The results presented are intended to serve as a foundation for further development of the LCFA proposed in previous works (specifically Ref. [6]). The authors note that their findings regarding the formation interval and the validity of the LCFA in high-frequency regimes can be extended to any slowly varying field configuration where the electric field remains uniform and time-dependent. Additionally, the authors suggest that these results may be applicable to the study of emission in 3D Dirac semimetals, where the energy gap acts as a mass term and critical fields are accessible in laboratory conditions. The work does not propose new experimental setups but rather provides the theoretical framework and closed formulas necessary for analyzing such processes in existing or future strong-field scenarios.