⚛️ high-energy theory

Entanglement spectrum and Schmidt mode evolution during two-particle scattering

This paper uses numerical simulations to analyze the time evolution of entanglement spectra and Schmidt modes in one- and two-dimensional particle scattering, revealing how an initial pure Gaussian wavepacket dynamically evolves into a discrete ensemble of outcomes where dominant eigenstates correspond to single-peaked transmitted or reflected packets, while smaller eigenvalues represent multi-peaked superpositions.

Original authors: Arsam Najafian, Mark Van Raamsdonk

Published 2026-10-21

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Arsam Najafian, Mark Van Raamsdonk

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet corners of quantum physics, there is a fundamental puzzle about how the smooth, continuous world of a single particle transforms into the distinct, separate outcomes we observe in the real world. Imagine a particle not as a tiny billiard ball, but as a ripple of probability, a wave that can exist in many places at once. When this wave interacts with its surroundings, it does not simply bounce off; it becomes entangled with the environment, a process where the particle and the world around it become inextricably linked. In the standard view of quantum mechanics, this interaction leads to a "collapse," where the many possibilities of the wave suddenly resolve into one single reality. However, the mathematics describing this transition is notoriously difficult. Physicists have long known that if you look closely at a particle interacting with its environment, its state can be described as a mixture of different possibilities, each with its own chance of happening. But seeing exactly how this mixture forms, moment by moment, has remained a theoretical fog. The question is not just whether the particle ends up in one state or another, but how the continuous wave of the beginning breaks apart into a discrete list of specific outcomes, each with a defined probability and a specific shape.

A team of researchers at the University of British Columbia has now peeled back this fog, using powerful computer simulations to watch this transformation happen in real time. They set up a virtual experiment where two particles, initially separated and moving toward each other, collide and scatter. One particle represents the system being studied, while the other acts as the environment. By tracking the mathematical "fingerprint" of the first particle as it interacts with the second, the researchers were able to map out the evolution of its possible states. They found that as the collision occurs, the single, pure state of the particle does not just blur; it splits into a clear, discrete list of possibilities. At the start, the particle is in one definite state. As the scattering happens, this state evolves into a collection of distinct outcomes, each with a specific probability and a unique wave shape. The researchers did not just calculate the odds of the particle going left or right; they visualized the actual shape of the wave for each of these possibilities as they emerged.

The results reveal a striking pattern in how nature organizes these outcomes. In a one-dimensional collision, where particles move along a straight line, the final state is dominated by two main possibilities: the particle either passes through the other or bounces back. The probabilities of these two outcomes match the standard predictions for transmission and reflection. However, the story is richer than a simple coin toss. The researchers discovered that beyond these two dominant outcomes, there is a long tail of much less likely possibilities. These rare outcomes correspond to the particle ending up in a state that is a superposition of multiple separated locations. For instance, a less likely outcome might show the particle as a wave with two distinct peaks, while an even rarer outcome might show it with three peaks. As the probability of an outcome decreases, the number of separate peaks in its wave shape increases. This suggests that the "collapse" into a single location is not an instantaneous switch, but the result of a hierarchy where the most probable outcomes are the simplest, single-peaked waves, and the complex, multi-peaked superpositions are the rare exceptions that fade away.

This behavior holds true even when the experiment is expanded into two dimensions, where a particle can scatter in any direction. In this scenario, the possible outcomes are not limited to just "left" or "right," but could theoretically be any angle. Yet, the simulation shows that the final state still organizes itself into a discrete set of possibilities. The most likely outcome is a single wave packet moving in a specific direction. The next most likely outcomes are waves with two peaks, then three, and so on. The researchers found that the probability of an outcome is directly linked to the complexity of its shape: the more separated peaks a wave has, the less likely it is to occur. This provides a detailed, time-resolved picture of how a continuous interaction gives rise to a discrete set of results. It shows that the universe does not just pick one outcome at random; it sorts them by complexity, with the simplest, most localized states being the most probable.

The study also explored what happens when the particle starts in a more complex state, already existing in a superposition of two separated locations before the collision. In this case, the interaction with the second particle acts like a measurement. The researchers observed that the collision causes the system to evolve into a mixed state with two significant possibilities. In one scenario, the particles do not interact significantly, and the first particle remains in its original, spread-out superposition. In the other scenario, a collision occurs, and the first particle's wave function "collapses" into a single, localized packet moving in a new direction. Remarkably, the second particle, which acted as the probe, carries away the information about which path was taken. If a collision happened, the probe particle's wave function splits into distinct parts, effectively recording the details of the first particle's initial state. This offers a microscopic view of how measurement works: the environment does not just destroy the superposition; it becomes entangled with it, creating a record of the outcome in the shape of the scattered waves.

These findings, derived from numerical simulations of particles with equal mass and short-range repulsive forces, suggest a universal mechanism for how quantum systems settle into definite states. The researchers confirmed their numerical results with analytical calculations for narrow momentum spreads, showing that the dominance of two main outcomes in one dimension is a robust feature of the physics, not just a quirk of the simulation. They also demonstrated that the multi-peaked states correspond to specific, albeit unlikely, outcomes where the particle remains in a superposition of separated locations. The work bridges the gap between the abstract mathematics of quantum entanglement and the concrete reality of scattering experiments. It shows that the transition from a continuous wave to a discrete set of outcomes is a structured process, where the most probable results are the most localized, and the complexity of the wave shape serves as a marker for its rarity. By visualizing the eigenvalues and eigenstates of the system, the researchers have provided a clear, time-resolved map of how a particle navigates the chaotic dance of interaction to emerge with a specific identity and a specific place in the world.

Technical Summary: Entanglement Spectrum and Schmidt Mode Evolution During Two-Particle Scattering

Problem Statement
The paper addresses the evolution of a continuous quantum system (a single particle) initially in a pure state that interacts with an environment (modeled here as a second particle). While the standard description of such scattering involves calculating transmission and reflection probabilities or angular distributions, the authors investigate the finer-grained structure of the system's reduced density operator, $\rho(t)$ . Specifically, they analyze the time-dependent evolution of the eigenvalues (the entanglement spectrum) and the corresponding eigenvectors (Schmidt modes) of $\rho(t)$ . The central question is how an initial pure state, described by continuous parameters, evolves into a discrete ensemble of pure states with associated probabilities, and what the specific spatial structure of these resulting states is.

Methodology
The authors employ a numerical approach to solve the eigenvalue problem for the reduced density operator of a two-particle system interacting via a short-range potential in one and two dimensions.

System Setup: The system consists of two distinguishable particles of equal mass ( $m_1 = m_2$ ) interacting via a potential $V(|\vec{x}_1 - \vec{x}_2|)$ . The initial state is a product of two separated Gaussian wavepackets approaching each other.
Numerical Scheme: To handle the infinite-dimensional Hilbert space, the authors regulate the problem by placing the system in a finite volume box with periodic boundary conditions. They work in a discrete momentum basis, truncating the Hilbert space to a finite range $[-n_{max}, n_{max}]$ for each momentum variable.
Symmetry Exploitation: Due to overall momentum conservation, the Hamiltonian decomposes into block-diagonal matrices corresponding to specific total momenta. This allows for efficient diagonalization of blocks with dimensions up to $10^6$ .
Evolution: The system is evolved in time by expanding the initial state in the energy eigenbasis of the center-of-mass and relative Hamiltonians and applying phase factors $e^{-iEt/\hbar}$ . The reduced density operator for one particle is computed by tracing out the other, followed by diagonalization to extract the time-dependent spectrum $\{p_n(t)\}$ and eigenstates $\{|\psi_n(t)\rangle\}$ .
Validation: The authors verify that results are independent of the box size $L$ and cutoff $n_{max}$ , ensuring the wavepackets remain well-localized away from artificial boundaries during the simulation.

Key Results
The study reveals a detailed, time-resolved picture of how the entanglement spectrum and Schmidt modes evolve during scattering:

Emergence of Discreteness: The initial spectrum, consisting of a single eigenvalue $p_1=1$ (pure state), evolves into a discrete set of non-zero eigenvalues. At late times, the spectrum stabilizes into a discrete ensemble of outcomes.
One-Dimensional Scattering:
- Dominant Modes: For Gaussian wavepackets scattering off a short-range potential, the late-time spectrum is dominated by two large eigenvalues. These values correspond closely to the standard transmission ( $T$ ) and reflection ( $R$ ) probabilities calculated for the central momentum.
- Eigenstate Structure: The eigenstates corresponding to these two dominant eigenvalues are single-peaked wavepackets representing the transmitted and reflected particles, respectively.
- Sub-dominant Modes: Smaller eigenvalues grow to a maximum during the interaction and then decay to negligible values at late times. The eigenstates associated with these smaller eigenvalues exhibit multi-peaked probability distributions. Specifically, the $k$ -th eigenstate (ordered by decreasing probability) displays $k$ spatial peaks, representing a superposition of distinct spatial locations.
- Delta Function Limit: In the limit of a delta-function potential, the sum of the smaller eigenvalues remains small, and the state is effectively a mixture of just the transmitted and reflected components.
Two-Dimensional Scattering:
- Despite the continuous nature of scattering angles in 2D, the late-time state remains a discrete ensemble.
- The eigenvalues evolve monotonically to their asymptotic values.
- Similar to the 1D case, the eigenstate corresponding to the $k$ -th largest eigenvalue possesses $k$ peaks in its angular probability distribution. Higher-order modes correspond to wavepackets that are more spread out in the angular direction.
Superposition State Scattering (Measurement Model):
- The authors model a scenario where a "system" particle is initially in a spatial superposition of two wavepackets, interacting with a "probe" particle.
- The scattering process results in a mixed state for the system particle dominated by two eigenvalues: one corresponding to the unscattered superposition state and one corresponding to a "collapsed" localized state moving in the opposite direction.
- Crucially, the probe particle's outgoing state carries the information distinguishing these outcomes, effectively encoding the initial superposition.

Analytical Confirmation
The paper provides an analytical derivation for the 1D case with narrow momentum spreads. By assuming the scattering amplitudes $t(p)$ and $r(p)$ are constant over the wavepacket width, the late-time wavefunction is shown to be a superposition of a transmitted term and a reflected term where the particles have exchanged momenta. Tracing out the second particle yields a reduced density operator with exactly two non-zero eigenvalues, $|t|^2$ and $|r|^2$ , and corresponding eigenstates that are the time-evolved wavepackets of the individual particles. This confirms the numerical observation that the late-time state is dominated by two modes. A similar toy model using Mehler's formula for 2D scattering reproduces the observation that the $n$ -th eigenstate has $n$ angular peaks.

Significance and Claims
The authors claim that their work provides a more detailed, time-resolved picture of scattering than traditional cross-section calculations.

Discretization of Continuous Systems: The work demonstrates explicitly how a continuous-variable system evolves into a discrete ensemble of outcomes, bridging the gap between continuous scattering theory and the discrete nature of quantum measurement outcomes.
Structure of Decoherence: The results offer a microscopic view of decoherence, suggesting that interactions in a dilute environment can be understood as a sequence of discrete branching events where the density operator's eigenstates evolve into new ensembles.
Novelty: The authors state that while previous works have studied entanglement entropy or purity in scattering, this is the first presentation of the time-resolved entanglement spectrum combined with real-space eigenfunctions for a continuous-variable scattering problem.
Modesty: The paper does not claim to solve the measurement problem or provide a new interpretation of quantum mechanics. Instead, it offers a concrete, calculable model of how a particle's state evolves during interaction, highlighting the emergence of discrete probabilities and specific spatial structures (multi-peaked states) that arise from unitary evolution.

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