Evolution of the eigenvalues and eigenstates of the single-particle reduced density operator during two-particle scattering

TL;DR

The paper analyzes the time-resolved evolution of the entanglement spectrum $\{p_n(t)\}$ of a single-particle reduced density operator during two-particle scattering in $1$D and $2$D using a finite-volume, momentum-space numerical approach. It reveals that, at late times, the top two eigenvalues closely match the central-momentum transmission and reflection probabilities, with corresponding eigenfunctions forming single-peaked transmitted or reflected wavepackets, while smaller eigenvalues correspond to multi-peaked, spatially separated outcomes that peak during collision. In $2$D, the spectrum grows monotonically to its asymptotic values and smaller-eigenvalue eigenstates exhibit increasing angular structure. The work provides a detailed, time-resolved, discrete-ensemble view of decoherence in a dilute environment, offering a scalable framework for understanding sequential two-particle interactions and their impact on the evolution of quantum systems.

Abstract

A particle initially in a pure state but interacting with some environment evolves into a discrete ensemble of pure states, the eigenstates of its reduced density operator, with ensemble probabilities given by the corresponding eigenvalues. In this work, we use numerics to present explicit results for the time-dependence of these eigenvalues and eigenstates for simple scattering experiments in one and two dimensions. This provides a time-resolved picture of the scattering process, showing in detail how an initial state described entirely in terms of continuous parameters evolves into a discrete set of possible outcomes, each with an associated probability and time-evolving wavefunction. We find that for scattering of Gaussian wavepackets in one dimension, the late time spectrum is dominated by two large eigenvalues nearly equal to the transmission and reflection probabilities associated with the central value of momentum. The corresponding eigenstates appear as single-peaked reflected or transmitted wavepackets. The remaining smaller eigenvalues, which increase to a maximum during scattering and then decrease to small values, correspond to reflected or transmitted wavepackets with multiple spatially separated parts. In this case and also for two-dimensional scattering, we find that successively smaller eigenvalues correspond to probability distributions with successively more peaks. These multi-peaked states correspond to outcomes of the scattering experiment where a particle initially in a single wavepacket ends up in a superposition of separated wavepackets after scattering.

Figures (8)

• Figure 1: Probability distributions for eigenstates corresponding to the five largest eigenvalues of the density operator for a single particle shortly after scattering from another particle via a repulsive short-range interaction. The particle was initially in a pure state Gaussian wavepacket travelling to the right. These represent a discrete set of possible outcomes of the scattering experiment.
• Figure 2: Evolution of the eigenvalues of the reduced density operator for a particle scattering symmetrically off another particle via a delta function interaction, for three different values of the interaction strength, parameterized by the transmission probability $T$. Here $t_0$ is the time at which the initial wavepackets would coincide in the absence of interactions.
• Figure 3: Evolution of the $p_i(t) |\psi_i(x,t)|^2$ for the eigenvectors $\psi_i(x,t)$ of the single particle reduced density matrix for a particle scattering symmetrically off another particle via a repulsive delta function interaction. The interaction strength corresponds to transmission probability $T \approx 0.45$. At late times, the two largest eigenvalues are very close to the reflection and transmission probabilities, and the corresponding eigenvectors have probability distributions as expected for reflected and transmitted wavepackets.
• Figure 4: Evolution of the natural logarithm of the largest 20 eigenvalues of the reduced density operator for a particle scattering symmetrically off another particle via a repulsive delta function interaction. Initial wavepackets are taken to be Gaussian. A truncation to Hilbert space dimension 363,609 ($|n_{max}|=301$) was used for this plot.
• Figure 5: Probability distributions $|\psi_i(x,t)|^2$ for the eigenvectors $\psi_i(x,t)$ of the single particle reduced density matrix at $t=3.75 t_0$ corresponding to the six lowest eigenvalues. The eigenstates with successively lower probabilities are observed to have successively more peaks in their probability distribution.