Strong coupling of virtual negative states in the Kapitza-Dirac effect

TL;DR

This work investigates how negative-energy states influence the two-photon Kapitza-Dirac diffraction of electrons in a standing light wave. By formulating the Dirac equation with a standing-wave vector potential and solving it numerically, alongside perturbative analyses and classical-ponderomotive treatments, the authors demonstrate that coupling to negative-energy intermediate states can dominantly contribute to the diffraction amplitude, especially at small transverse momenta $p_3\ll mc$, and that this quantum behavior has a consistent relativistic classical counterpart through gamma-corrected ponderomotive dynamics. The results connect the diffraction dynamics to virtual electron-positron pair processes in a fully quantized theory and highlight the limitations of non-relativistic treatments in capturing these effects, offering a coherent relativistic framework with potential experimental relevance. The findings deepen the interpretation of Kapitza-Dirac scattering in relativistic regimes and suggest further exploration of negative-state contributions and their connections to fundamental QED processes like Compton scattering and Feynman-Stückelberg interpretations.

Abstract

Negative states are an intrinsic property of relativistic quantum theory and related to anti-particles in the context of the Dirac sea concept. We show that negative states can dominantly contribute to the diffraction amplitude in the quantum dynamics of the two-photon Kapitza-Dirac effect. We draw our conclusion by investigating solutions from time-dependent perturbation theory, where the perturbative solutions are in match with numeric solutions of the relativistic quantum system and also with the numeric and analytic solutions from the relativistic equations of motion of a classical point-like electron in an external standing wave light field. While our numeric solutions assume a strong laser field, the analytic solutions indicate that negative state coupling remains dominant for arbitrary low field amplitudes, where in the single-photon case (Compton scattering) negative state coupling can be mathematically associated with the interaction of a virtual electron-positron pair in the context of a quantized theory in old-fashioned perturbation theory.

Figures (5)

• Figure 1: Occupation probabilities as a function of the full interaction time $T$ for the dynamics of the simulated the two-photon Kapitza-Dirac effect. (a) Displayed are the absolute value squares of the only non-vanishing expansion coefficients $c^{+\uparrow}$ and $c^{-\uparrow}$ for a simulation of the equation of motion \ref{['eq:inserted_dirac_equation']} with initial condition \ref{['eq:FSME_initial_conditions']} and parameters $k_L=0.02\,mc/\hbar$, $\Delta T=5 \times 2 \pi/\omega$ and $eA_0 = 0.01\,mc^2$. The observed Rabi period of 1600 laser cycles can be deduced analytically, as done in Eq. \ref{['eq:rabi_period']} by a calculation using time dependent perturbation theory. The gray dotted line marks the simulation time at which the diffraction probability as a function of the transverse momentum $p_3$ is displayed in Fig. \ref{['fig:Simulation_and_Predicted']}. Panels (b) and (c) show the quantities $|\tilde{c}_0^{+,\uparrow}|^2$ and $|\tilde{c}_2^{+,\uparrow}|^2$, respectively, for which the simulation in panel (a) is redone with the matrix entries $V^{\pm,s;\mp,s^\prime}_{n,n^\prime}$ set to zero.
• Figure 2: Diffraction probability \ref{['eq:c_plus_diffraction_probablity']} of the two-photon Kapitza-Dirac effect as a function of the transverse electron momentum $p_3$. The parameters and initial condition \ref{['eq:FSME_initial_conditions']} are as for Fig. \ref{['fig:time_evolution']} with $T=150 \times 2\pi/\omega$, where the first data point of the numeric solution at $p_3=0$ corresponds to the gray dot at the gray dotted time mark in Fig. \ref{['fig:time_evolution']}(a). The results of the numeric solutions of the differential equation \ref{['eq:inserted_dirac_equation']} are marked by red boxes and the analytic short-time solution from time-dependent perturbation theory (perturbative solution) of Eq. \ref{['eq:coherent_sum_expression']} is displayed by a solid black line. The green dotted line marks the square of the electron's classical momentum in laser propagation direction (axis on the right) for the case of the relativistic description of the classical equations of motion \ref{['eq:relativistic_motion_equation']}. This classical solution is performed with a vector potential amplitude of $eA = 10^{-3}\,mc^2$, with all other parameters being consistent with the values used for the quantum calculations. The cyan dashed line marks the low momentum approximation \ref{['eq:dirac_perturbative_diffraction_probability_approximation']} of the perturbative relativistic quantum solution, where we mention that the relativistic classical solution \ref{['eq:relativistic_ponderomotive_force']} exhibits the same scaling with respect to the transverse momentum $p_3$ as the analytic quantum solution.
• Figure 3: Coupling paths in perturbation theory which are corresponding to the four intermediate states in Eq. \ref{['eq:intermediate_quantized_states']} in the context of Compton scattering ahrens_2020_two_photon_bragg_scattering. The interaction starts with the single electron $e_0^-$, single photon $\gamma_+$ state $\ket{e^-_0,\gamma_+}$ and also ends with s single electron $e_2^-$, single photon $\gamma_-$ state $\ket{e^-_2,\gamma_-}$, where the electron acquires two photon momenta $0\rightarrow 2$ and the photon momentum is reversed $+\rightarrow -$. The quantum paths in the negative energy continuum (corresponding to $F_a^-$) are associated with the state configuration of an electron plus an electron-positron pair $\ket{e^-_2 e^+_1 e^-_0}$, $\ket{e^-_2 e^+_1 e^-_0,\gamma_+\gamma_-}$ in a fully quantized theory.
• Figure 4: Diffraction probabilities for different coupling paths of the two-photon Kapitza-Dirac effect as a function of the transverse momentum $p_3$. Displayed is the perturbatively solved diffraction probability $|c_2^+(T)|^2$ from Eq. \ref{['eq:coherent_sum_expression']} and the spin resolved positive and negative coupling strengths $|c_2^{+,s}(T)|^2_{\gamma'}$ from Eq. \ref{['eq:individual_coupling']}. The probability $|c_2^+(T)|^2$ is the same as the black line in Fig. \ref{['fig:Simulation_and_Predicted']}, where its shape appears differently due to the logarithmic scaling in this figure. We observe that the coupling path with a negative intermediate electron state $|c_2^{+,\uparrow}(T)|^2_{-}$ is dominating the diffraction probability for $p_3<mc$ .
• Figure 5: Interaction geometry of Compton scattering, in a frame of reference in which the electron is reversing its momentum $\boldsymbol{p}$. (a) In this reference frame the photon is back-scattering by $180^\circ$ for a situation of the Kapitza-Dirac effect with zero transverse momentum $p_3=0$. (b) In contrast, the photon scatters off by some angle $\vartheta$ for situations with large transverse momenta $p_3 \gg m c$ in the Kapitza-Dirac effect.