Localization with Hopping Disorder in Quasi-periodic Synthetic Momentum Lattice

Using a Bose-Einstein Condensate in a momentum space lattice, researchers experimentally realized a Generalized Aubry-André model with hopping disorder, demonstrating that uncorrelated disorder enhances localization while spatially correlated disorder induces partial delocalization, thereby establishing momentum space lattices as a versatile platform for studying general disordered quantum systems.

The Gist

The Big Idea: A Quantum "Traffic Jam" in a Synthetic World

Imagine you are trying to walk through a crowded hallway.

• In a normal hallway (a perfect crystal): You can walk straight and fast. Everyone is in their own spot, and there are no obstacles.
• In a messy hallway (disorder): There are random piles of trash, people standing in the way, and uneven floors. You get stuck, bump into things, and can't move forward easily. This is called Anderson Localization. In the quantum world, particles (like atoms) behave like waves, and when they hit enough "mess," they stop moving and get stuck in one spot.

For decades, scientists have studied this using quasi-periodic patterns. Think of this as a hallway with a repeating pattern that almost repeats but never quite does (like a wallpaper design that shifts slightly every few feet). This creates a predictable kind of "mess" that traps particles.

The Problem: Real-world messiness isn't predictable. It's random. It's like someone throwing trash on the floor without a pattern. Scientists wanted to study this "true randomness," but their usual tools (real physical lattices) couldn't create it precisely enough.

The Solution: This paper describes a team of scientists who built a virtual hallway using light and atoms. They didn't build a physical floor; they built a "Momentum Space Lattice" (MSL).

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The Experiment: The "Ghost Train" in a Light Tunnel

1. The Setup: A Train on a Track of Light

The scientists used a cloud of ultra-cold Rubidium atoms (a Bose-Einstein Condensate). Think of these atoms as a single, super-cooperative "ghost train" that can be in many places at once.

Instead of putting the atoms on a physical grid of atoms, they used lasers to create a synthetic track.

• The Track: The "sites" on the track aren't physical locations in space; they are different speeds (momentums) the atoms can have.
• The Hopping: The lasers act like a conductor, gently pushing the "train" from one speed to the next. This is called "hopping."

2. The Twist: Adding the "Disorder"

The scientists wanted to see what happens when you add randomness to the hopping.

• The Control: They created a "Generalized Aubry-André" (GAA) model. This is a fancy way of saying they set up a track with a specific, slightly weird rhythm (quasi-periodic).
• The Chaos: They then added hopping disorder. Imagine the conductor (the laser) suddenly gets a bit jittery. Sometimes the push is strong, sometimes weak, and sometimes the timing is off.
  • Scenario A (Random Jitter): The pushes are completely random, like a drunk conductor.
  • Scenario B (Correlated Jitter): The pushes are random, but if the conductor stumbles once, they stumble a few times in a row before recovering. This creates "smooth patches" of chaos.

3. The Observation: Watching the Train Freeze

They watched how the "ghost train" spread out over time.

• If it spreads out: The atoms are delocalized (free to move).
• If it stays put: The atoms are localized (trapped).

The Surprising Findings

The team discovered two very interesting things about how this "jittery conductor" affects the train:

1. Random Jitter Always Traps the Train
When they added completely random disorder (Scenario A), the train got stuck faster and more often, no matter what the underlying rhythm was.

• Analogy: Imagine trying to run through a hallway where the floor tiles are randomly slippery. Even if the hallway was designed to be easy to run in, the random slips make it impossible to get anywhere. The sharp line between "running" and "stuck" became a blurry gray zone where you are just "kinda stuck."

2. Smooth Jitter Actually Helps the Train Move
This was the most surprising part. When they added "correlated" disorder (Scenario B), where the randomness was smooth and connected, something weird happened in the "trapped" zones.

• Analogy: Imagine the hallway has random slippery spots, but they are arranged in long, smooth patches. If you are stuck in a "trap," these smooth patches act like slippery slides. They allow the atoms to slide past the obstacles that would have normally trapped them.
• The Result: In the regions where the atoms were supposed to be stuck, the smooth randomness actually let some of them escape. It created "partial delocalization."

Why This Matters

The "Virtual" Advantage
The coolest part of this paper is how they did it.

• Old Way: Trying to build a physical grid of atoms and randomly mess it up is like trying to build a Lego castle and then randomly knocking blocks out. You can't control exactly which blocks fall or how they fall.
• New Way (MSL): Because they used light to create the "track," they could program the randomness with perfect precision. They could say, "Make the jump between site 5 and 6 exactly 10% weaker, and make the jump between 6 and 7 follow that pattern."

The Takeaway
This experiment proves that we can use light and cold atoms to simulate complex, messy quantum systems with incredible precision. It's like having a flight simulator for quantum physics.

• We can now test theories about how disorder affects the universe.
• We can design materials that control how electricity or heat flows by engineering specific types of "messiness."
• Most importantly, they showed that how the disorder is arranged (random vs. smooth) changes the outcome just as much as how strong the disorder is.

Summary in One Sentence

Scientists built a "virtual track" made of light to show that while random chaos always traps quantum particles, smooth, connected chaos can actually create hidden pathways that let particles escape their traps.

Technical Summary

1. Problem Statement

While Anderson localization (suppression of transport due to disorder) is well-studied in condensed matter physics, experimental realization of true local disorder and off-diagonal (hopping) disorder in quantum simulators remains challenging.

• Current Limitations: Most experiments rely on quasi-periodic potentials (e.g., the Aubry-André or AA model), which offer deterministic disorder but lack the randomness of true local disorder. Conventional real-space optical lattice simulators struggle to implement arbitrary, site-resolved hopping disorder with tunable spatial correlations.
• The Gap: There is a need for a platform that can simulate general disordered quantum systems, specifically investigating how hopping disorder (variations in tunneling amplitudes) interacts with quasi-periodicity and how spatial correlations in this disorder affect localization transitions.

2. Methodology

The authors utilize a Momentum Space Lattice (MSL) realized with an ultracold $^{87}$Rb Bose-Einstein Condensate (BEC) to simulate a Generalized Aubry-André (GAA) chain with added hopping disorder.

• Experimental Platform:

  • System: A BEC of $\approx 4 \times 10^4$ atoms cooled to 50 nK.
  • Lattice Creation: Multiple Bragg diffractions using counter-propagating 1064 nm laser beams couple discrete momentum states ($p = -20\hbar k, \dots, 20\hbar k$), creating a 1D synthetic lattice of 21 sites.
  • Control: Unlike real-space lattices, the MSL allows independent control of hopping amplitudes ($t_i$) and onsite energies ($\mu_i$) for every site by tuning the electric field amplitude ($E_i$) and detuning ($\delta_i$) of specific laser tones generated by an Arbitrary Waveform Generator (AWG).

• Model Hamiltonian:
  The system simulates a 1D GAA chain with hopping disorder:
  $$H = \sum_i t_i (c^\dagger_{i+1}c_i + h.c.) + \lambda \sum_i \frac{\cos(2\pi\beta i + \phi)}{1 - \alpha \cos(2\pi\beta i + \phi)} c^\dagger_i c_i$$

  • $t_i$: Hopping amplitude with added disorder.
  • $\lambda$: Quasi-periodic potential strength.
  • $\alpha$: Deformation parameter (controls self-duality breaking and mobility edges).
  • $\beta = (\sqrt{5}-1)/2$: Incommensurate ratio.

• Disorder Implementation:

  • Uncorrelated Disorder: $t_i$ drawn independently from a uniform distribution $[1-\sigma, 1+\sigma]$.
  • Correlated Disorder: Generated by convolving uncorrelated noise with a Gaussian kernel $g(r) \sim e^{-r^2/2\xi^2}$, where $\xi$ is the tunable correlation length.

• Diagnostics:

  • Inverse Participation Ratio (IPR): Calculated from site-resolved populations to measure spatial localization ($IPR \sim 1/\chi$ for localized states).
  • Return Probability (RP): Measures the probability of the particle remaining at the initial site, quantifying transport suppression.
  • Numerical Simulation: Theoretical results include ideal tight-binding models and a more rigorous simulation of BEC dynamics (incorporating finite momentum width, off-resonant excitations, and pulse shaping effects) to ensure quantitative comparison.

3. Key Contributions

1. Realization of Arbitrary Hopping Disorder: Demonstrated the first experimental implementation of a GAA model with controllable hopping disorder and tunable spatial correlations in a synthetic momentum lattice.
2. Platform Versatility: Established MSL as a precision quantum simulator capable of resolving correlation-dependent dynamical features, surpassing the limitations of real-space lattice simulators regarding disorder configuration.
3. Quantitative Fidelity: Achieved near parameter-free agreement between experimental data and numerical simulations that account for experimental imperfections (e.g., momentum width, off-resonant driving).

4. Key Results

• Effect of Uncorrelated Hopping Disorder:

  • Adding uncorrelated hopping disorder enhances localization across all phases.
  • It smoothes the sharp localization transition characteristic of the clean AA model, turning it into a crossover between weakly and strongly localized regimes.
  • Even weak quasi-periodicity combined with hopping disorder destroys chiral symmetry protection, leading to the localization of all eigenstates.

• Effect of Spatially Correlated Hopping Disorder:

  • Partial Delocalization: In the localized phase, spatial correlations induce partial delocalization of states near strong hopping bonds.
  • Mechanism: Correlated disorder creates smoother landscapes where "high-hopping" regions allow for quasi-extended states, whereas uncorrelated disorder creates short-scale fluctuations that suppress transport.
  • State-Resolved Analysis: Supplemental analysis reveals a strong anticorrelation ($r \approx -0.96$) between local hopping magnitude and IPR in correlated systems, whereas uncorrelated systems show weak correlation ($r \approx -0.3$).

• Experimental vs. Theoretical Agreement:

  • Experimental IPR and RP data match MSL simulations (which include experimental deviations) quantitatively across a wide range of disorder strengths ($\sigma$) and correlation lengths ($\xi$).
  • Deviations from ideal tight-binding models are attributed primarily to the finite momentum width of the BEC and off-resonant excitations, which are successfully modeled.

5. Significance

• Beyond Quasi-periodicity: This work proves that MSL is a viable platform for studying general disordered quantum systems, moving beyond the limitations of quasi-periodic models to explore true random disorder and complex correlation effects.
• Precision Quantum Simulation: The high degree of control and the quantitative agreement with theory demonstrate that MSL can serve as a "precision quantum simulator" for investigating quantum transport, mobility edges, and many-body localization phenomena.
• Future Directions: The platform enables the study of engineered initial states to probe specific localized/delocalized features (e.g., states near strong bonds) and offers a pathway to simulate higher-dimensional disordered systems or interacting many-body systems with controlled disorder.

In summary, the paper provides a robust experimental framework for simulating complex disorder scenarios, revealing that hopping disorder generally enhances localization, while spatial correlations can induce partial delocalization, offering new insights into the interplay between disorder types and quantum transport.