A Reduced Action Integral for Photon-Photon Interactions in Vacuum

TL;DR

The paper develops a reduced action integral framework to model photon-photon interactions in vacuum arising from the Euler--Heisenberg Lagrangian. By representing each light pulse with trial functions and applying a variational principle to the reduced action, it derives equations of motion for observable pulse parameters such as centroid, spot size, phase, and polarization. Three representative examples—phase modulation, birefringence, and frequency mixing—demonstrate the method's ability to predict centroid deflections, polarization dynamics, and new-frequency generation without full-field simulations. This approach offers a fast, parameter-centric tool to guide experimental geometries and optimize configurations for detecting quantum vacuum nonlinearities. The framework is poised for extensions to more complex pulse geometries and for integration with gradient-based optimization to design experiments probing photon-photon scattering.

Abstract

Electromagnetic waves propagating through vacuum can polarize virtual electron-positron pairs; this polarization, in turn, nonlinearly modifies their propagation. A semi-classical nonlinear wave equation describing the propagation is derived from the Euler--Heisenberg Lagrangian density, which captures vacuum polarization effects up to the one-loop level. Here, we present a reduced-action-integral approach that enables rapid modeling of nonlinear phenomena arising from the Euler--Heisenberg Lagrangian. Application of the variational principle to the reduced action provides equations of motion for familiar light-pulse parameters, such as spot size, phase, polarization, and phase-front curvature, without requiring full-field simulations. Three examples demonstrate the utility of the approach: phase modulation, birefringence, and frequency mixing.

Figures (5)

• Figure 1: (a) The third-order nonlinear response $(\bm{\mathrm{\mathcal{P}}}, \bm{\mathrm{\mathcal{M}}})$ of a polarized virtual-pair medium couples multiple input pulses (unprimed). Several processes can modify the outbound pulses (primed): (b) Phase modulation induced by one or more pulses ($E_2$, $E_3$) can deflect another pulse ($E_1$). The blue arrow chain depicts the resulting centroid trajectory of $E_1$ as it deviates from its initial heading. (c) The relative polarization of $E_1$ and $E_2$ affects the strength of their coupling, resulting in a birefringence that modifies the polarization of $E_1$. For instance, an initially linearly polarized field $E_1$ can become elliptically polarized, as illustrated by the blue ellipses. (d) Three spectrally distinct pulses ($E_1$, $E_2$, and $E_3$) mix to generate a fourth pulse ($E_4$) at a new frequency ($\omega_4$).
• Figure 2: The reduced-action-integral approach: (a) Construct an action integral for the nonlinear, paraxial evolution of all spectrally distinct or spatiotemporally disjoint fields. (b) Substitute physically motivated trial functions for the fields, parameterized by familiar light-pulse quantities, such as the spot size, power (amplitude), and phase. (c) Integrate over the coordinates orthogonal to the propagation direction to obtain a reduced action integral. (d) Apply the principle of least action to derive equations for the evolution of the pulse quantities.
• Figure 3: Deflection angle $\Theta_\mathrm{c}^{(1)}$ of a probe pulse $E_1$, resulting from its collision with two pump pulses, $E_2$ and $E_3$, as a function of $z$ and the probe moving-frame coordinate, $t-z/c$. All pulses have the same frequency $\hbar \omega =1.24$ eV, same minimum spot size $\tilde{w}_1=\tilde{w}_2 =\tilde{w}_3= 2$$\mu$m, and share a focal plane at $z = 0$. Each pump has a peak power of $25$ PW and a duration of $20$ fs. The pump pulses are transversely offset by $x_{\mathrm{c},2}=x_{\mathrm{c},3}=1.4$$\mu$m and are delayed by $\mp 70$ fs for $E_2$ and $E_3$, respectively, relative to the arrival time of the central time slice of the probe pulse at focus ($t-z/c = 0$, $z=0$). The temporal extent of the pump pulses are shown by the pairs of red and violet lines. The line brightness decreases with increasing distance from the focal plane, reflecting the intensity of the pumps.
• Figure 4: The relative phase difference, $\Delta \theta^{(1)}_1$, between two polarization components of an x-ray probe pulse, $E_{11}$ and $E_{12},$ due to its interaction with a counter-propagating optical pump pulse, $E_2$, as a function of $z$ and the moving-frame coordinate of the probe $t-z/c$. The x-ray probe has a central frequency $\hbar\omega_1 = 25$ keV and a minimum spot size $\tilde{w}_1 = 1$$\mu$m. The optical pump has a central frequency $\hbar\omega_2 = 1.24$ eV, a minimum spot size $\tilde{w}_2 = 2$$\mu$m, a peak power of $25$ PW, and a duration of $20$ fs. The pulses share a focal point in the plane $z=0$. The two red lines mark the front and back edges of the pump. The decreasing line brightness with increasing distance from the focal plane conveys the local intensity of the pump.
• Figure 5: Generation of a third-harmonic signal pulse, $E_4$ ($\hbar \omega_4 = 4.65$ eV), from the interaction of two second-harmonic pump pulses, $E_1$ and $E_2$ ($\hbar \omega_{1}=\hbar \omega_{2} = 3.1$ eV), and a fundamental pump pulse, $E_3$ ($\hbar \omega_3 = 1.55$ eV). The pumps have unequal spot sizes $\tilde{w}_1=2$$\mu$m, $\tilde{w}_2=3$$\mu$m, and $\tilde{w}_3=5$$\mu$m, equal durations $\tau_1=\tau_2=\tau_3=20$ fs, and peak powers $4\tilde{P}_1=4\tilde{P}_2=\tilde{P}_3 = 25$ PW. (a) Power of the generated signal, $P_4$, along the optical axis, $z_4$, as a function of its moving-frame coordinate $t-z_4/c$. (b) Spectral density of the signal at a $z_4$ location beyond the overlap volume, where its generation has ceased; integration gives a total of $295$ third-harmonic photons. (c) At the location of maximum power growth [center star in (a) at $t-z_4/c = 0$ fs and $z_4= 0$$\mu$m], the different spot sizes of the pumps results in a signal with an elliptical transverse profile (outlined in violet) and an off-axis centroid. (d) After the signal generation ceases [right star in (a) at $t-z_4/c = 0$ fs and $z_4 = 5$$\mu$m], the centroid moves to the origin.