Entanglement spectrum and Schmidt mode evolution during two-particle scattering

The gist: This work uses numerics to present explicit results for the time-dependence of eigenvalues and eigenstates for simple scattering experiments in one and two dimensions, revealing how an initial pure state evolves into a discrete ensemble of pure states with associated probabilities and time-evolving wavefunctions.

Basic Setup

The investigation considers a system particle of mass m1 interacting with an environmental particle of mass m2 via a potential that depends only on the separation between the particles (Page 6). The total Hamiltonian is given by H = p21/2m1 + p22/2m2 + V (x1 − x2) (Page 6). A typical initial state considered is a product state where two particles are described by separated Gaussian wavepackets approaching each other, specifically ψ(x1, x2) = ψ1(x1)ψ2(x2) (Page 6). The evolution of the time-dependent density operator for the first particle is computed as ρ1(x1, x'1, t) = Z d x2 Ψ(x'1, x2, t)Ψ(x1, x2, t)∗ (Page 6).

Numerical Approach

Since finding eigenvalues and eigenvectors of the reduced density operator (Equation 1.1) analytically is intractable on an infinite-dimensional Hilbert space, the researchers resort to a numerical scheme (Page 6). This involves regulating the problem by setting it in a finite volume box with side lengths L and periodic boundary conditions (Page 5). The initial state for each particle is defined in terms of momentum eigenstates n⟩ by Nc, Xc, σ⟩ = N X n∈Z e-(N⃗c−⃗n) 2(2σ) squared e(-2πiX⃗c·⃗n/L) ⃗n⟩ (Page 7). The Hamiltonian in the product basis N1, N2⟩ becomes block-diagonal due to overall momentum conservation, allowing for independent diagonalization of smaller blocks (Page 8).

One Dimension Results

In one-dimensional scattering, the results show that starting with the initial eigenvalue spectrum of a pure state evolves to grow more non-zero eigenvalues that eventually approach some limiting values denoted as p out i (Page 5). For the largest two eigenvalues at late times, they are almost exactly equal to the transmission and reflection probabilities associated with the central value of momentum (Page 9). The corresponding eigenstates for these two largest eigenvalues appear as single-peaked reflected or transmitted wavepackets (Page 10). The remaining smaller eigenvalues grow to a maximum during scattering and then decrease to small values, corresponding to reflected or transmitted wavepackets with multiple spatially separated parts (Page 11). For the six lowest eigenvalues at t = 3.75t0, the eigenstates with successively lower probabilities are observed to have successively more peaks in their probability distribution (Page 12).

Two Dimensions Results

In two-dimensional scattering, the eigenvalues evolve monotonically to their asymptotic values when particles have separated (Page 14). The eigenstates corresponding to successively lower probabilities exhibit a phenomenon where the k largest eigenvalue is observed to have k peaks in its probability distribution at late times, which is similar to one-dimensional scattering (Page 15). For the smaller eigenvalues, the eigenvector for the kth eigenvalue is a wavefunction with k maxima in its probability density at late times, and these wavefunctions are more spread out in the angular direction compared to one dimension (Page 16). The evolution of eigenvalues in two dimensions is monotonic, unlike the one-dimensional case (Page 14).

Discussion and Implications

The results provide a detailed picture of basic interactions experienced by a particle in a dilute environment, where the full evolution can be understood as a sequence of discrete branching events (Page 13). The findings show that an interaction between two localized particles can produce an outgoing state that is a superposition of two or more wavepackets, and future studies should investigate whether such a superposition state scatters with another particle to see if this interaction tends to re-localize the particle (Page 16). These results offer a detailed timeresolved perspective on what happens during scattering that goes beyond standard calculations of cross sections and angular distributions (Page 13). The work is related to previous studies characterizing the evolution and/or final entangled state in terms of purity or von Neumann entropy calculated from the spectrum of the density operator (Page 16).

How it works

The core mechanism involves tracking how an initial pure state evolves into a discrete ensemble of pure states, where eigenvalues represent ensemble probabilities and eigenvectors represent corresponding pure states (Page 2). This emergence of discreteness in a physical problem described by continuous parameters is highlighted as a fascinating aspect of the research (Page 4). The evolution is modeled by considering the full system's state via the Schmidt decomposition: Ψ(t)⟩ = X n p n(t)ψ sys n(t)⟩ ⊗ ψ env(t)⟩ (Page 3). This framework allows for a detailed, time-resolved picture of scattering processes (Page 1).

1. The improved AI system can perform time-resolved quantum state prediction during particle scattering by calculating the evolution of eigenvalues and eigenvectors of the single-particle reduced density operator for continuous-variable systems, providing a time-resolved picture of the scattering process.

2. The system can predict discrete measurement outcomes based on eigenvalues and their corresponding eigenstates, as the paper shows that a particle evolves into a discrete ensemble of pure states, the eigenstates of its reduced density operator, with ensemble probabilities given by the corresponding eigenvalues.

3. The AI can model complex decoherence processes by simulating a sequence of discrete branching events where each eigenstate of the density operator after a given scattering event evolves into a new ensemble after the next scattering event, allowing for a detailed picture of what happens to a particle in an environment consisting of a dilute gas of other particles.

4. The system can characterize the structure of late-time quantum states by analyzing eigenvalue behavior, specifically identifying that the largest two eigenvalues at late times are almost exactly equal to the transmission and reflection probabilities that can be calculated analytically for 1D scattering.

5. The AI can analyze superposition states in scattering events by determining that eigenstates with successively lower probabilities are observed to have successively more peaks in their probability distribution, indicating a particle is in a superposition state of k distinct locations, each with an associated uncertainty.

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.