Exclusive Scattering Channels from Entanglement Structure in Real-Time Simulations

Abstract

A scattering event in a quantum field theory is a coherent superposition of all processes consistent with its symmetries and kinematics. While real-time simulations have progressed toward resolving individual channels, existing approaches rely on knowledge of the asymptotic particle wavefunctions. This work introduces an experimentally inspired method to isolate scattering channels in Matrix Product State simulations based on the entanglement structure of the late-time wavefunction. Schmidt decompositions at spatial bipartitions of the post-scattering state identify elastic and inelastic contributions, enabling deterministic detection of outgoing particles of specific species. This method may be used in settings beyond scattering and is applied to detect heavy particles produced in a collision in the one-dimensional Ising field theory. Natural extensions to quantum simulations of other systems and higher-order processes are discussed.

Figures (8)

• Figure 1: Channel isolation in the post-inelastic scattering state. Top: the energy density as a function of position and time with the $|11\rangle$ (dark blue), $|12\rangle$ (light blue), and $|21\rangle$ (orange) states distinguished. The dotted line marks the collision time and the dot-dashed lines show the trajectories of the outgoing particles in each state. Bottom: the energy density as a function of position for the final state in the heatmap in the top panel. The superscripts on the individual particles label the states they belong to. The dashed lines show the positions $n_1$ and $n_2$ where bipartitions can be made to isolate $|11\rangle$, $|12\rangle$, and $|21\rangle$.
• Figure 2: Leading-order channel isolation after particle production. Top left: the energy density $E_n$ as a function of position $n$ and time $t$ for the collision of two $|1\rangle$ particles with $k_i=\pm0.36\pi$. Top right: $E_n$ corresponding to $t=120$ in the top left panel. The energy density of the inclusive post-scattering state $|f\rangle$ is shown in black. The color shading shows the contributions of the individual states: $|11\rangle$ (dark blue), $|12\rangle$ (light blue), and $|21\rangle$ (orange). The positions $n_l$ and $n_r$ of the bipartitions used to isolate the states composing the elastic and inelastic channels are marked by the dashed lines. Bottom: $E_n$ of the exclusive states contributing to $|f\rangle$ in the top right panel.
• Figure 3: Entanglement structure of the post-scattering state. The antiflatness ${\cal F}_{AB}$ (blue) and the entanglement entropy $S_{AB}$ (orange) as a function of initial momentum $k_i$. The dashed line marks the threshold momentum $k_\text{thr}$ above which the process $11\to12$ is kinematically allowed.
• Figure 4: Dispersion relations $E(k)$ (left) and the group velocities $v(k)=dE(k)/dk$ (right) for particles $|1\rangle$ (blue) and $|2\rangle$ (orange) calculated in a $L=\infty$ system using the quasiparticle excitation ansatz. The light blue band on the right plot shows $k_i\pm\sigma_k$ used for the scattering simulations in this work.
• Figure 5: Energy densities $E_n$ for the Schmidt components used in the channel isolation process. The left panel shows $|f_{02,2}\rangle$, which is attributed to higher-order processes. The middle panel shows $|f_1\rangle$, which is the combination of $|21\rangle$ and $|f_{1,1}\rangle$ shown on the right. The state $|f_{1,1}\rangle$ is also attributed to higher-order processes.