Observation of average topological phase in disordered Rydberg atom array

This study reports the first experimental observation of disorder-induced average symmetry-protected topological (SPT) phases in a disordered Rydberg atom array, confirming the existence of topologically protected edge modes and ground state degeneracy through spatially resolved correlation functions and quench dynamics.

The Gist

Imagine you have a long line of people holding hands, forming a chain. In the world of physics, these people are atoms, and the way they hold hands represents how they interact with each other.

For a long time, scientists believed that for these chains to have special "superpowers" (called topological phases), the people had to stand in a perfect, orderly grid, and the rules of the game had to be exact and unchanging. If the grid got messy or the rules got fuzzy (due to disorder or noise), scientists thought the superpowers would vanish.

This paper is about a surprising discovery: Sometimes, making a mess actually creates the superpowers.

Here is the story of how the researchers at Tsinghua University proved this, explained with some everyday analogies:

1. The Setup: A Perfectly Organized Dance Floor

The scientists used a line of Rydberg atoms (super-excited atoms that act like giant magnets). They used laser beams (optical tweezers) to hold these atoms in place, like a dance floor where everyone has a specific spot.

• The Goal: They wanted to see if the atoms could form a special "topological" state. Think of this state like a secret handshake that only the people at the very ends of the line know. If the chain is in a "topological" state, the ends are special and protected, while the middle is just normal.
• The Problem: In a perfectly straight line, the atoms were in a "boring" state. The ends weren't special.

2. The Twist: Introducing the "Mess"

Usually, if you shake a line of people or push them out of their perfect spots, you expect the system to break. But the researchers decided to intentionally mess things up.

• They randomly shifted the laser beams holding the atoms, so the atoms were no longer in a perfect grid. It was like telling everyone on the dance floor to take a random step left or right.
• The Surprise: Instead of breaking the system, this "structural disorder" triggered a transformation. The atoms suddenly developed that secret handshake at the ends. The "mess" actually induced the topological phase.

3. The "Average" Secret (The Magic of Statistics)

You might ask, "But if every single line is messy, how can there be a perfect rule?"

• The Analogy: Imagine a room full of people. In any single snapshot, the room looks chaotic. But if you take a photo of the room, then flip the photo upside down, and take another photo, and then average them together, a hidden symmetry appears.
• In physics terms, this is called an "Average Symmetry." Even though any single arrangement of atoms is messy and broken, if you look at all possible messy arrangements together, they balance each other out perfectly. It's like a noisy crowd where, if you listen to the average volume, it's perfectly silent. This "average" balance is what protects the special state at the ends.

4. The Proof: How They Knew It Worked

The scientists didn't just guess; they tested it in three clever ways:

• The "Edge" Test (Single Particle): They sent a single "excited" atom into the chain. In a normal, messy chain, the atom would get lost in the middle. But in their "disorder-induced" chain, the atom got stuck at the very ends, like a ball rolling into a valley at the edge of a hill. This proved the ends were special.
• The "Group Hug" Test (Many Particles): They filled the chain with many atoms. They checked how the atoms "hugged" (correlated) each other. In a normal chain, neighbors hugged tightly. In their special messy chain, the atoms hugged their neighbors across the gaps in a specific pattern that only happens in topological states.
• The "Memory" Test (Quench Dynamics): This is the coolest part. They started the atoms in a highly excited, chaotic state (like a shaken soda can) and watched how fast they calmed down.
  • In the middle of the chain, the atoms calmed down quickly (forgot their initial state).
  • But at the edges? The atoms held onto their "memory" (magnetization) for a long time. They were protected. It's like the people at the ends of the line were wearing noise-canceling headphones, ignoring the chaos happening in the middle.

Why Does This Matter?

This discovery changes how we think about order and chaos.

1. Real-World Resilience: Real-world quantum computers are messy and noisy. This paper suggests that we might not need to fix every tiny error to build a quantum computer. We might be able to design the system so that the noise itself creates the protection we need.
2. New Physics: It proves that "Average Symmetry Protected Topological" phases exist. It's a new category of matter that only appears when you embrace the mess rather than fight it.

In a nutshell: The researchers took a line of atoms, messed it up on purpose, and found that the mess created a shield around the edges, protecting them from the chaos. It's a bit like how a chaotic crowd might accidentally form a perfect circle if you look at the whole group from the right perspective.

Technical Summary

1. Problem Statement

Topological phases of matter, such as topological insulators and the Haldane phase, have traditionally been studied in pure quantum states protected by exact symmetries. However, real-world quantum systems (including solid-state materials and noisy intermediate-scale quantum simulators) inherently exist as mixed quantum states due to decoherence and disorder.

• The Gap: While "average symmetry-protected topological (SPT) phases" have been theoretically predicted to exist in mixed states (where the density matrix is invariant under a symmetry even if constituent pure states are not), they have remained experimentally elusive.
• The Challenge: Existing experimental realizations of disorder-induced topology (e.g., Topological Anderson Insulators) have focused on non-interacting particles or single-particle band topology. There is a lack of experimental evidence for disorder-induced, many-body interacting average SPT phases.

2. Methodology

The authors utilized a programmable Rydberg atom array to simulate a 1D hard-core boson model with structural disorder.

• Experimental Platform:

  • System: A chain of 14 $^{87}$Rb atoms arranged in optical tweezers.
  • Encoding: Two Rydberg states, $|s\rangle = |55S_{1/2}\rangle$ (encoded as $|\uparrow\rangle$) and $|p\rangle = |55P_{1/2}\rangle$ (encoded as $|\downarrow\rangle$), coupled by a 22.1 GHz microwave field.
  • Interactions: The system features long-range dipolar exchange interactions (flip-flop hopping) and van der Waals (vdW) interactions.
  • Disorder Implementation: Structural disorder was introduced by randomly displacing the optical tweezers from their regular lattice sites by an amount $\delta x_j \in [-W/2, W/2]$, where $W$ is the disorder strength. This creates a "Rydberg glass."

• Theoretical Model:

  • The system is described by a 1D staggered Su-Schrieffer-Heeger (SSH) model for hard-core bosons with long-range dipolar interactions.
  • Symmetry: While individual disordered configurations break exact inversion symmetry, the ensemble of all possible configurations possesses an average inversion symmetry (statistical symmetry). This is because every configuration $C$ has an equal probability of appearing as its inversion partner $R_C$.

• Experimental Protocols:

  1. Single-Particle Level: Prepared atoms in the $|s\rangle$ state and used microwave spectroscopy to probe for edge states.
  2. Many-Body Level (Half-filling): Prepared Néel-like product states and used adiabatic evolution (ramping down pinning lasers) to reach the many-body ground state.
  3. Dynamics: Performed quantum quenches from highly excited states to observe the decay of edge vs. bulk spin magnetization.

3. Key Contributions

1. First Experimental Observation: This work reports the first direct observation of a disorder-induced many-body interacting average SPT phase.
2. Verification of Average Symmetry: It demonstrates that topological protection can persist in mixed states via average inversion symmetry, distinct from exact symmetry protection.
3. Many-Body Interaction: Unlike previous disorder-induced topological studies limited to non-interacting systems, this experiment explicitly includes strong many-body interactions (hard-core bosons/dipolar exchange), realizing a genuine interacting topological phase.
4. Comprehensive Characterization: The study validates the phase through multiple independent probes:
   • Topological invariants (polarization $P_S$ and $Z_2$ invariant $P_M$).
   • Ground state degeneracy.
   • Spatially resolved correlation functions.
   • Quench dynamics showing topological protection of edge modes.

4. Key Results

A. Single-Particle Regime

• Edge State Emergence: In a regular lattice ($W=0$), no edge states exist. As disorder strength $W$ increases, a topological phase transition occurs at a critical threshold ($W_c \approx 0.15d$).
• Observation: Microwave spectroscopy revealed pronounced peaks in site occupancy at the edges for $W > W_c$, indicating the emergence of zero-energy edge modes.
• Statistical Degeneracy: While individual disordered samples show split edge energies due to broken symmetry, the ensemble average shows statistically degenerate edge states at zero energy, confirming the average symmetry protection.

B. Many-Body Interacting Regime (Half-Filling)

• Ground State Degeneracy: In the regular lattice, the ground state is unique (trivial). In the disordered lattice, the system exhibits a two-fold statistical degeneracy in the half-filling subspace, a hallmark of the SPT phase.
• Correlation Functions:
  • Regular Lattice: Intracell correlations ($C_{intra}$) are stronger than intercell correlations ($C_{inter}$), indicating a trivial dimerized phase.
  • Disordered Lattice: The trend reverses ($C_{inter} \approx -0.42$ vs $C_{intra} \approx -0.13$), signaling a transition to a non-trivial topological phase where the "effective" dimerization is inverted by disorder.
• Topological Invariant: The authors generalized the topological invariant $P_M$ for the ensemble. Numerical simulations confirmed that $P_M$ quantizes to 0.5 in the disordered phase, consistent with the experimental correlation data.

C. Quench Dynamics and Edge Protection

• Experiment: The system was initialized in a Néel state (highly excited) and evolved under the Hamiltonian.
• Result: In the regular lattice, spin magnetization decayed rapidly to zero for both bulk and edge sites (thermalization). In the disordered topological phase, bulk magnetization decayed rapidly, but edge magnetization decayed significantly slower and stabilized at a finite value.
• Significance: This confirms the existence of topologically protected edge modes that are robust against thermalization in the presence of disorder and interactions.

5. Significance

• Fundamental Physics: This work bridges the gap between theoretical predictions of mixed-state topology and experimental reality. It proves that disorder, often considered detrimental, can actually induce topological order in interacting many-body systems via average symmetries.
• Quantum Simulation: It validates the Rydberg atom array as a powerful platform for simulating complex disordered quantum phases, including those relevant to the "Rydberg glass" and amorphous topological matter.
• Future Directions: The platform opens avenues for studying:
  • Higher-dimensional topological phases in amorphous systems.
  • The interplay between many-body localization (MBL) and topology.
  • Floquet topological phases in disordered settings.

In summary, the paper provides comprehensive evidence that structural disorder can drive a transition from a trivial to a topological phase in an interacting quantum system, protected not by exact symmetry, but by average symmetry, fundamentally expanding the classification of topological matter.