🔬 mesoscale physics

Long-lasting Topological Entanglement in a Monitored Rashba Nanowire

This paper demonstrates that the topological value of disconnected entanglement entropy in a monitored Rashba nanowire persists for a time linear in system size despite boundary dissipation, a phenomenon driven by the interplay between non-conserved particle number, topological manifold degeneracy, and the eventual poisoning of ballistic quasiparticles.

Original authors: Emanuele Guida, Giulia Salatino, Gianluca Passarelli, Angelo Russomanno, Procolo Lucignano

Published 2026-06-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Emanuele Guida, Giulia Salatino, Gianluca Passarelli, Angelo Russomanno, Procolo Lucignano

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

The Big Picture: A Quantum Wire Under a Microscope

Imagine a very thin, special wire made of quantum materials. This wire has a secret "topological" superpower: it holds a special kind of connection (entanglement) between its two ends, even though they are far apart. Scientists call this a "Majorana mode." It's like having two magic coins at opposite ends of a long table that always land on the same side, no matter how far apart they are.

Usually, if you poke or disturb these magic coins, the connection breaks instantly. But this paper asks a fascinating question: What happens if we constantly "watch" this wire and let it lose a few particles here and there?

The researchers found something surprising: Even though the wire is being watched and losing particles, that special connection between the ends doesn't break immediately. Instead, it survives for a surprisingly long time—specifically, a time that gets longer the longer the wire is.

The Cast of Characters

The Wire (Rashba Nanowire): Think of this as a quantum highway. Unlike a normal highway where cars (particles) are conserved, this highway allows cars to appear and disappear.
The Magic Coins (Majorana Modes): These are the special states living at the very ends of the wire. They are the source of the "topological" connection.
The Watcher (Monitoring): Imagine a security camera constantly taking snapshots of the wire. In the quantum world, "watching" isn't passive; it actually changes the system. Every time the camera "sees" a particle leave, it's like a quantum jump.
The Poisoning (Quasiparticles): Sometimes, the watcher sees a particle leave, and this creates a "glitch" or a "poisoned" particle that starts running down the wire.

The Discovery: Why It Lasts Longer

In a previous study on a different type of wire (called the SSH chain), if the watcher saw a particle leave at the very end, the magic connection broke instantly. It was like if one of your magic coins was knocked off the table, the connection vanished immediately.

But this Rashba wire is different. Here's why:

Analogy 1: The Switching Doors

In this wire, the two ends are connected to a special "switching room." Because the wire doesn't strictly conserve the number of particles, when a particle is lost at one end, the system doesn't just break; it switches.

Imagine the two magic coins are in a room with two doors: Door A and Door B.
When a particle is lost, the system doesn't destroy the coins. Instead, it flips a switch, moving the coins from Door A to Door B (or vice versa).
The "connection" (the topological value) is still there; it just changed its state. The system is still in a "topological" mode, just a different version of it.

Analogy 2: The Messenger and the Long Hallway

So, if the connection doesn't break immediately, when does it finally break?

When the particle is lost at the end, it also creates a "messenger" (a finite-energy quasiparticle).
This messenger is like a runner starting a sprint from one end of the hallway.
The runner has to travel all the way across the wire to reach the other end to mess up the connection.
The Key Finding: The time it takes for the runner to cross the hallway depends on how long the hallway is.
- If the wire is short, the runner arrives quickly, and the connection breaks fast.
- If the wire is very long, the runner takes a long time to get there.
Therefore, the "magic connection" survives for a time that is directly proportional to the length of the wire. In a very long wire, the connection lasts a very long time.

The "Poisoning" Effect

The paper calls this process "quasiparticle poisoning." Usually, scientists worry that losing a particle poisons the whole system instantly. This paper shows that in this specific setup, the poison doesn't kill the system instantly. It just starts a timer. The poison creates a runner, and the system stays healthy until that runner reaches the other side.

Summary of Results

The Setup: They simulated a quantum wire being constantly watched (monitored) while losing particles.
The Result: The special "topological" connection between the ends stayed strong for a long time.
The Rule: The longer the wire, the longer the connection lasts. The lifetime of the connection grows linearly with the size of the system.
The Reason: The system can switch between different "safe" states when particles are lost, rather than breaking. The connection only breaks when the "messengers" created by the loss have enough time to travel across the entire wire.

Why This Matters (According to the Paper)

This is important because it shows that topological quantum states are more robust than we thought. Even if you are constantly watching them and they are losing particles, they don't just vanish instantly. They have a "grace period" that depends on the size of the system. This helps us understand how these special quantum states behave in real-world, imperfect conditions where they are constantly interacting with their environment.

Technical Summary: Long-lasting Topological Entanglement in a Monitored Rashba Nanowire

Problem Statement
This work investigates the stability of symmetry-protected topological phases under continuous monitoring, specifically focusing on the fate of the disconnected entanglement entropy (DEE) in a Rashba nanowire hosting Majorana zero modes. While previous studies on the monitored Su-Schrieffer-Heeger (SSH) chain established that boundary dissipation rapidly destroys topological edge entanglement, the behavior of superconducting systems (which lack particle number conservation) under similar conditions remained unclear. The central problem is to determine how local quantum jumps, acting as a model for quasiparticle poisoning, affect the non-local entanglement structure of a topological superconductor and whether the topological value of the DEE persists in the presence of such dissipation.

Methodology
The authors employ a quantum-trajectory approach to unravel the Lindblad master equation governing the open quantum system.

Model: The system is a one-dimensional Rashba nanowire with proximity-induced $s$ -wave superconductivity and an external magnetic field, described by a mean-field Hamiltonian that maps to an effective 1D spinless $p$ -wave superconductor (Kitaev chain) in the low-energy limit.
Dissipation: The environment is modeled via local, spin-dependent particle-loss jump operators ( $L_{i\sigma} = \sqrt{\gamma_\sigma} c_{i\sigma}$ ). The study considers various spatial profiles for these jumps: homogeneous dissipation, bulk-only, and edge-only configurations.
Dynamics: The evolution is simulated using the stochastic Schrödinger equation, where the state remains Gaussian along individual quantum trajectories due to the quadratic nature of the Hamiltonian and linear jump operators. This allows for efficient numerical evolution of the correlation matrix rather than the full density matrix.
Observable: The primary metric is the Disconnected Entanglement Entropy (DEE), defined as $S_D = S(A) + S(B) - S(A \cup B) - S(A \cap B)$ , where $A$ and $B$ are specific disconnected subsystems. The DEE is calculated for individual trajectories and then ensemble-averaged.
Analytical Benchmark: To interpret the numerical results, the authors analyze a simplified, fully dimerized Kitaev chain, deriving the effect of a local jump on the topological ground state analytically.

Key Contributions and Results

Linear Scaling of Topological Lifetime: The primary finding is that the topological value of the DEE ( $\log 2$ ) persists for a time $t_c$ that scales linearly with the system size $N$ . This holds true regardless of whether dissipation acts in the bulk, at the boundaries, or uniformly across the chain. This contrasts sharply with the SSH model, where boundary dissipation leads to a system-size-independent lifetime.
Robustness Against Disorder: The linear scaling of the DEE plateau duration is robust against weak static disorder in the chemical potential.
Mechanism of Persistence: The authors attribute this extended lifetime to two specific features of the superconducting model:
- Absence of Particle Conservation: Unlike the SSH model, the Rashba nanowire does not conserve particle number.
- Ground State Degeneracy: The topological phase possesses a two-fold degenerate ground state manifold characterized by different fermion parities.
- Parity Switching vs. Excitation Propagation: Local quantum jumps at the boundaries do not immediately destroy the topological mode. Instead, they induce transitions (parity switches) between the two degenerate ground states, effectively creating and annihilating the Majorana mode. The DEE remains quantized during these switches.
Role of Quasiparticle Poisoning: The eventual decay of the DEE is not caused by the immediate loss of the edge mode but by the generation of finite-energy quasiparticle excitations near the boundary. These excitations propagate ballistically through the chain. The DEE is destroyed only when these propagating quasiparticles reach the opposite boundary (or the entanglement cuts), disrupting the long-range correlations. This propagation time is proportional to the system size ( $L/v$ ), explaining the linear scaling of the lifetime.
Statistical Analysis of Jumps: Analysis of the jump-induced variations ( $\Delta S_D$ ) reveals that individual quantum jumps do not instantaneously alter the DEE value. The histogram of $\Delta S_D$ is sharply peaked at zero, confirming that the non-local entanglement is preserved locally immediately after a jump event.

Significance and Claims
The paper claims that the DEE serves as a direct probe of how local poisoning events are converted into a delayed decay of non-local topological entanglement. The key significance lies in demonstrating that for superconducting topological phases, the topological entanglement structure is inherently more robust against boundary monitoring than in particle-conserving systems like the SSH chain. The authors argue that the "poisoning" of the topological state is a dynamical process governed by the ballistic spreading of finite-energy quasiparticles, rather than an instantaneous collapse of the edge mode. Consequently, in the thermodynamic limit, the topological entanglement lifetime diverges, suggesting that topological order in monitored superconducting systems can survive for arbitrarily long times despite local dissipation. The work extends the trajectory-resolved approach from the SSH model to Majorana systems, providing a unified understanding of how ground-state degeneracy and non-conservation laws influence topological stability under continuous observation.