Dynamics of edge modes in monitored Su-Schrieffer-Heeger Models

This paper demonstrates that while dissipation generally disrupts edge mode dynamics in the monitored Su-Schrieffer-Heeger model, selectively protecting the chain's edges allows for the recovery of unitary-like features, underscoring the critical influence of spatial dissipation patterns on these quantum systems.

The Gist

The Big Picture: A Noisy Quantum Chain

Imagine a long line of people holding hands, forming a chain. In the world of quantum physics, this is called the Su-Schrieffer-Heeger (SSH) model. Under perfect conditions, this chain has a special "secret handshake" at its two ends (the edges). These ends are connected in a spooky, invisible way called entanglement, even though they are far apart. This is a "topological" feature, meaning it's a robust property of the whole system, like a knot that can't be untied just by pulling on the rope.

However, in the real world, nothing is perfect. The chain is constantly being poked, prodded, and watched by the environment. This is called dissipation or noise. Usually, when you watch a quantum system too closely or let it interact with the environment, that special "secret handshake" at the ends gets destroyed, and the chain loses its special properties.

The Experiment: Watching the Chain in Real-Time

The authors of this paper wanted to see what happens to these edge connections when the chain is being "monitored." Instead of just looking at the average result of many experiments (which hides the details), they looked at individual quantum trajectories.

Think of it like this:

• The Average View: If you take a blurry photo of a crowd, you just see a gray mass.
• The Trajectory View: If you put on special glasses and watch one specific person in the crowd move step-by-step, you see exactly how they react to every bump and shove.

In this study, the "bumps" are called quantum jumps. These are random events where the environment interacts with the chain. The researchers tracked how the "secret handshake" (measured by a tool called Disconnected Entanglement Entropy, or DEE) changed after every single jump.

The Key Discovery: Location Matters More Than Type

The researchers tested two main scenarios regarding where the "noise" (dissipation) hits the chain:

1. The "Uniform Noise" Scenario: Imagine the entire chain is being poked randomly from head to toe.

   • Result: The special connection at the ends breaks very quickly. The "secret handshake" is lost.

2. The "Protected Edges" Scenario: Imagine the noise only hits the middle of the chain, leaving the two ends completely untouched and safe.

   • Result: Surprisingly, the "secret handshake" at the ends survives! Even though the middle of the chain is chaotic and noisy, the ends remain connected for a very long time.

The Analogy: Think of the chain as a long, fragile bridge. If you shake the whole bridge, it collapses. But if you only shake the middle section and leave the two anchor points (the edges) perfectly still, the connection between the anchors remains strong. The paper found that where the noise hits is more important than what kind of noise it is.

The "First Jump" Surprise

The researchers also looked at the very first time the environment poked the chain. They found a fascinating difference depending on where that first poke happened:

• If the first poke hits an edge: The "secret handshake" is destroyed instantly and completely. It's like cutting the rope at the anchor point; the connection is gone in a split second.
• If the first poke hits the middle: The connection survives. The chaos in the middle doesn't immediately ruin the special bond at the ends.

They also found that the type of noise (whether it preserves certain symmetries or breaks them) didn't matter as much as the location. Whether the noise was "symmetry-preserving" or "symmetry-breaking," if it hit the edge, the connection broke. If it stayed in the middle, the connection held.

The Role of the "Push" (Quench)

The study also looked at what happens if you suddenly change the rules of the chain (a "quantum quench") while it's being noisy.

• If the chain is noisy everywhere, changing the rules doesn't save the connection; it still breaks.
• However, if the edges are protected from noise, the connection stays strong for a long time, regardless of whether the rules changed or not.

The Bottom Line

The main takeaway is that spatial protection is key. You don't need to stop all the noise in the universe to keep a quantum system's special edge properties alive. You just need to shield the edges.

If you can keep the "ends" of your quantum chain safe from the environment's random jolts, the special topological connection will survive, even if the rest of the chain is a mess. This suggests that for future quantum technologies, we might not need perfect isolation for the whole system—just for the critical parts at the edges.

Technical Summary

Technical Summary: Dynamics of Edge Modes in Monitored Su-Schrieffer-Heeger Models

Problem and Motivation
The study of topological phases in open quantum systems governed by Lindblad dynamics presents significant challenges, particularly regarding the definition of suitable topological markers for non-equilibrium and dissipative states. While the "tenfold way" classification has been extended to open systems, it often relies on the steady-state properties of the density matrix, which may obscure the dynamical behavior of topological edge modes. Furthermore, topological markers are typically nonlinear functions of the quantum state; thus, averaging over quantum trajectories (which yields the density matrix) is not equivalent to studying the topology embedded within individual trajectories. This paper investigates the dynamics of topological edge modes in the monitored Su-Schrieffer-Heeger (SSH) model, focusing on how dissipation and quantum jumps affect the stability of edge states. The authors aim to determine whether the spatial arrangement of dissipation or its symmetry classification (within the dissipative tenfold way) plays a more critical role in the survival of topological features.

Methodology
The authors analyze a one-dimensional free-fermion SSH chain with open boundary conditions, initialized in its topological ground state. The system evolves under a combination of unitary Hamiltonian dynamics and non-unitary dissipation described by the Lindblad equation. The study employs the "quantum jump" unraveling approach, where the system's evolution is modeled as a stochastic Schrödinger equation: smooth non-Hermitian evolution interrupted by discrete quantum jumps.

Two distinct types of dissipative dynamics are considered, classified according to Ref. [6]:

1. Symmetry-Preserving Dissipation (SPD): Jump operators act on individual sites (injection on sublattice A, extraction on sublattice B), preserving the generalized symmetries of the BDI class.
2. Symmetry-Breaking Dissipation (SBD): Jump operators act on pairs of sites, breaking the generalized symmetries.

Crucially, the authors investigate both uniform dissipation (acting on the entire chain) and non-uniform dissipation (acting only on the central bulk, leaving the edges untouched).

To diagnose the topological properties, the authors utilize two tools:

• Two-point correlation functions: To track the spreading and decay of edge mode correlations.
• Disconnected Entanglement Entropy (DEE): A nonlinear measure defined as $S_D = S_A + S_B - S_{A \cup B} - S_{A \cap B}$. Unlike the von Neumann entropy, DEE is robust for detecting symmetry-protected topological phases and quantifies the entanglement between topological edge states. Since the state along a single trajectory remains pure, DEE can be calculated directly. The study analyzes both the ensemble average of the DEE and the statistical distribution of its variations ($\Delta S_D$) induced by individual quantum jumps.

Key Results

1. Spatial Localization of Dissipation is Paramount:
   The most significant finding is that the spatial profile of dissipation dictates the stability of edge modes more than the symmetry class of the Lindbladian.

   • Uniform Dissipation ($\alpha=1$): When dissipation acts on the entire chain (including boundaries), the topological edge modes are rapidly destroyed. The DEE deviates from its quantized value (2) after a finite time that is independent of the system size.
   • Bulk-Only Dissipation ($\alpha=0.8$): When dissipation is confined to the central bulk, leaving the edges untouched, the topological edge modes remain stable. In this regime, the time scale ($t_c$) for the DEE to deviate from its initial value scales linearly with the system size ($L$). This recovers the behavior observed in unitary (closed) systems, demonstrating that edge modes can be protected from decoherence if the boundary is shielded from dissipation.

2. Role of the Coherent Hamiltonian:
   The authors examine the interplay between the coherent part of the Lindbladian and dissipation.

   • In the "unquenched" case (Hamiltonian matches the initial state), dissipation at the boundary leads to a rapid decay of DEE.
   • In the "quenched-to-trivial" case (Hamiltonian is trivial, initial state is topological), the coherent dynamics generates entanglement that competes with the destructive effect of quantum jumps. This competition can delay the decay of the DEE, creating a transient "plateau" even when the Hamiltonian is trivial. However, this protection is only effective if the edges are shielded from dissipation; if the boundary is affected, the DEE eventually decays regardless of the coherent contribution.

3. Statistical Analysis of Quantum Jumps:
   By analyzing the distribution of DEE variations ($\Delta S_D$) conditioned on the location of quantum jumps, the authors reveal the microscopic mechanism of topological destruction:

   • SPD Dynamics: A single quantum jump occurring at a boundary site causes an abrupt, quantized drop in DEE ($\Delta S_D = -2$), corresponding to the immediate destruction of the long-range entangled Bell pair formed by the edge modes.
   • SBD Dynamics: Due to the non-local nature of the jump operators, the destruction of entanglement is more gradual. The first jump at the boundary results in a smaller, non-quantized reduction ($\Delta S_D \approx -0.38$), with the complete destruction ($\Delta S_D = -2$) occurring only after subsequent jumps.
   • Bulk Jumps: When dissipation is restricted to the bulk ($\alpha=0.8$), jumps do not produce the characteristic peaks associated with edge destruction, confirming that edge modes are robust against bulk noise.

Significance and Claims
The paper claims that for monitored quadratic open systems, the spatial localization of dissipation is the primary factor determining the dynamical stability of topological edge modes, superseding the relevance of the symmetry classification (SPD vs. SBD) within the dissipative tenfold way.

The study demonstrates that:

• Topological edge modes in the SSH model can be effectively protected from decoherence by confining dissipation to the bulk, allowing the system to retain unitary-like scaling properties (linear scaling of stability time with system size).
• The "tenfold way" classification for quadratic Lindbladians, while useful for single-particle spectra, is insufficient for predicting the dynamical evolution of entanglement properties and edge mode stability.
• Monitoring quantum trajectories provides access to nonlinear topological markers (like DEE) that reveal phenomena, such as the discrete destruction of edge entanglement by boundary jumps, which are hidden in the ensemble-averaged density matrix.

The authors conclude that understanding the spatial patterns of dissipation is crucial for engineering robust topological phases in open quantum systems, and that the interplay between coherent dynamics and dissipation can be tuned to enhance the stability of topological signatures.