Energy-resolved transport of ultracold atoms across the Anderson transition: theory and experiment

This paper presents a quantitative theoretical framework based on the self-consistent theory of localization that accurately describes the energy-resolved transport of ultracold atoms across the three-dimensional Anderson transition, successfully benchmarking against numerical simulations and explaining experimental density profiles by distinguishing the contributions of Bose-condensed and thermal atoms.

The Gist

The Big Picture: The "Lost in the Crowd" Experiment

Imagine you are at a massive, chaotic music festival (the disordered potential). The crowd is packed tight, and people are moving in every direction. If you try to walk from one side to the other, you will constantly bump into people, get pushed around, and your path will be a mess. This is what happens to particles in a "disordered" material.

In physics, there is a famous phenomenon called Anderson Localization. It's like a magical rule where, if the crowd is chaotic enough, you don't just get slowed down—you get stuck. You can't move at all, no matter how hard you try. The particles get "frozen" in place because their waves cancel each other out.

For a long time, scientists have wanted to study exactly how this happens. But there was a problem: in previous experiments, the "people" (atoms) were all moving at different speeds. Some were running, some were walking, some were standing still. It was like trying to study traffic flow when everyone is driving at random speeds; you couldn't tell if a car was stuck because of the traffic or just because it was a slow driver.

The Breakthrough: Tuning the "Radio"

This paper describes a new experiment and a new way of thinking about the data.

1. The Experiment (The Tuning Fork):
The researchers used a special trick involving radio waves (like tuning a radio station). They had a cloud of ultra-cold atoms (a Bose-Einstein Condensate, which is like a super-organized group of atoms acting as a single wave).

• The Old Way: They would dump all the atoms into the chaotic crowd at once.
• The New Way: They used a radio pulse to "select" only the atoms with a very specific speed (energy). It's like using a bouncer at a club who only lets in people wearing a specific color shirt.
• The Result: They could now watch a group of atoms that were all moving at the exact same speed as they tried to navigate the chaotic crowd. By changing the "shirt color" (the energy), they could see what happened when the atoms were fast (diffusive), slow (localized), or right on the edge of getting stuck (critical).

2. The Theory (The Weather Forecast):
The experimental data was great, but it was hard to predict exactly what would happen using standard math. The math was too heavy for computers to handle for such a large system.

The authors developed a new theoretical toolbox (based on something called the "Self-Consistent Theory").

• The Analogy: Imagine trying to predict the weather. You don't need to track every single raindrop. Instead, you look at the average wind speed and humidity to predict if it will rain.
• The Innovation: This new theory does the same thing for atoms. It doesn't track every single collision. Instead, it calculates the "average probability" of an atom moving from point A to point B, while accounting for the fact that the atoms are waves that can interfere with themselves.
• The Secret Sauce: The theory realized that the experiment wasn't perfect. Even with the radio filter, there were still a few "slow" atoms mixed in with the "fast" ones (thermal atoms). The theory had to account for this mix. It's like realizing your weather forecast was wrong because you forgot to account for a sudden cold front. Once they added this "thermal mix" into the math, the theory matched the experiment perfectly.

The Three Regimes (The Three Scenarios)

By tuning the energy of the atoms, they observed three distinct behaviors, which the theory successfully predicted:

1. The Diffusive Regime (The Rush Hour):

   • What happens: The atoms have high energy. They move through the crowd, bumping into people but eventually making it across. They spread out like a drop of ink in water.
   • The Theory: Predicts a smooth, bell-shaped curve of where the atoms end up.

2. The Localized Regime (The Traffic Jam):

   • What happens: The atoms have low energy. The crowd is so chaotic that the waves cancel out. The atoms get stuck in one spot and don't move.
   • The Theory: Predicts that the atoms will stay in a tight, exponential cluster, never spreading out.

3. The Critical Regime (The Edge of Chaos):

   • What happens: This is the "Goldilocks" zone. The energy is just right—on the edge between moving and getting stuck. The atoms spread out, but in a weird, slow way that doesn't fit normal rules.
   • The Theory: This is the hardest part to predict. The theory correctly identified this strange, slow spreading pattern (called "anomalous diffusion").

Why This Matters

Think of this paper as the instruction manual for a very complex machine.

• Before: Scientists could build the machine (the experiment) and see it work, but they didn't have a perfect manual to explain why it worked or to predict exactly how it would behave in new situations.
• Now: They have written a manual (the theory) that is accurate, easy to use, and explains the data perfectly, even accounting for the "imperfections" (the thermal atoms) in the real world.

This is a big deal because it gives scientists a powerful tool to study how waves move through messy environments. This isn't just about atoms; the same math applies to light moving through fog, sound moving through a forest, or even electrons moving through new types of computer chips.

In short: They figured out how to tune the speed of atoms to see exactly when they get stuck in a mess, and they wrote a new math book that perfectly explains the rules of the game.

Technical Summary

1. Problem Statement

The paper addresses the challenge of quantitatively describing the dynamics of matter waves in three-dimensional (3D) disordered quantum systems, specifically across the Anderson transition (the transition from diffusive to localized transport).

• Context: While Anderson localization has been observed in 1D, 2D, and 3D systems, previous experimental setups suffered from broad atomic energy distributions. This prevented the direct observation of the transition and the study of the critical regime near the mobility edge ($E_c$).
• Specific Challenge: A recent experiment (Ref. [40]) successfully achieved energy-resolved transport using a radio-frequency (rf) transfer scheme to prepare ultracold atoms with narrow energy distributions in a 3D speckle potential. However, a fully quantitative theoretical framework linking these microscopic experimental conditions to the observed spatial density profiles was missing.
• Goal: To develop a theoretical model capable of accurately predicting the time-evolution of atomic density profiles across the mobility edge, accounting for the specific spectral and spatial properties of the experimentally prepared states, including the presence of thermal atoms.

2. Methodology

The authors developed a tailored implementation of the Self-Consistent Theory (SCT) of localization and benchmarked it against ab initio numerical simulations and experimental data.

A. Theoretical Framework (SCT)

• Initial State Modeling: The authors modeled the rf-transfer process where atoms are moved from a disorder-free state $|1\rangle$ to a disorder-sensitive state $|2\rangle$. The initial wave function $|\psi_0\rangle$ is defined as a filtered version of the condensate wave function: $|\psi_0\rangle = f(\hat{H} - E_f)|\phi\rangle$.
• Energy Distribution: The theory incorporates the specific energy distribution $D(E; E_f)$ of the atoms. Crucially, this distribution is a weighted sum of a Bose-Einstein Condensate (BEC) component and a thermal component (arising from the $\sim25\%$ thermal fraction in the initial cloud).
• Disorder-Averaged Density: The authors derived an expression for the disorder-averaged density $n(r,t)$ by summing over all energies weighted by $D(E; E_f)$ and the disorder-averaged propagator $P_E(r, r', t)$.
• Propagator Calculation: Using the SCT, the propagator is calculated via a generalized diffusion coefficient $D(\omega)$, which is renormalized by quantum interference (Cooper diagrams). The authors solved the self-consistent equation for $D(\omega)$ in 3D, introducing a UV cutoff to handle divergences. This yields distinct asymptotic behaviors for the diffusion coefficient depending on whether the energy is above, at, or below the mobility edge.
• Predictions: The theory predicts:
  • Diffusive ($E > E_c$): Gaussian spreading ($\langle r^2 \rangle \propto t$).
  • Critical ($E = E_c$): Anomalous spreading ($\langle r^2 \rangle \propto t^{2/3}$) with Airy function profiles.
  • Localized ($E < E_c$): Exponential profiles with time-independent width ($\langle r^2 \rangle \approx \text{const}$).

B. Numerical Simulations

• Setup: Ab initio simulations were performed on a 3D grid ($100 \mu m^3$) using a blue-detuned speckle potential with parameters matching the experiment ($V_R/h = 416$ Hz).
• Technique: To avoid the prohibitive cost of diagonalizing the 3D Hamiltonian, the authors used Chebyshev polynomial expansions to apply the energy filter operator and the time-evolution operator.
• Benchmarking: The simulations served as a ground truth to validate the SCT approximations under conditions of strong disorder and large spatial scales.

C. Experimental Comparison

• Data: The theory was compared against experimental data from Ref. [40], measuring 1D density profiles $n_{1d}(z,t)$ at various times (up to 5s) and target energies ($E_f$) spanning the mobility edge.
• Fitting: The SCT parameters ($\alpha_0, \beta$) were fitted to the experimental central density evolution, while the mobility edge $E_c$ was fixed based on prior experimental determination.

3. Key Contributions

1. Tailored SCT Implementation: The authors extended the SCT to 3D and adapted it to handle non-ideal experimental conditions, specifically the finite width of the energy filter and the mixed BEC-thermal nature of the initial atomic cloud.
2. Quantitative Agreement: The framework successfully reproduces experimental density profiles across all regimes (diffusive, critical, localized) with high precision, a feat difficult to achieve with standard large-scale numerical simulations due to computational cost.
3. Role of Thermal Atoms: The study highlights that the thermal fraction of the atomic cloud is critical for interpreting experimental data. While the BEC component dominates, the thermal tail significantly affects the density profile tails, especially near the mobility edge and in the localized regime. Ignoring this leads to discrepancies between theory and experiment.
4. Validation of Scaling Laws: The work confirms the theoretical scaling laws for wave-packet spreading ($t$, $t^{2/3}$, and constant) across the mobility edge using real experimental data.

4. Results

• Benchmarking: The SCT predictions matched the ab initio numerical simulations almost perfectly for times up to 7 seconds and energies near the mobility edge. Deviations in the diffusive regime tails were attributed to finite-size effects in the numerical box, which the theory correctly accounted for by using periodic boundary conditions.
• Experimental Fit:
  • Diffusive Regime ($E_f > E_c$): The theory accurately predicted the broad, Gaussian-like expansion of the cloud.
  • Localized Regime ($E_f < E_c$): The theory reproduced the time-independent, exponentially decaying spatial profiles.
  • Critical Regime ($E_f \approx E_c$): The model captured the intermediate, slow-expansion dynamics characteristic of the critical point.
• Thermal Contribution: Comparisons showed that using a pure BEC model ($f_c=1$) failed to reproduce the experimental tails in the localized regime. Including the 25% thermal fraction ($f_c=0.75$) yielded excellent agreement, confirming that the thermal component acts as a "broad pedestal" in the energy distribution that influences the spatial profile.
• Parameters: The fitted SCT parameters ($\alpha_0 \approx 4.29 \, s^{-1/3}$, $\beta \approx 2.97 \, \mu m \cdot s^{-1/3}$) were consistent with theoretical expectations, though slightly smaller than ideal simulation values, likely due to residual interactions or weak confinement in the experiment.

5. Significance

• Theoretical Toolbox: The paper provides a versatile and computationally efficient theoretical toolbox for describing 3D disordered quantum systems. It bridges the gap between microscopic theory and macroscopic experimental observables where ab initio simulations are often too expensive.
• Experimental Insight: It validates the rf-transfer scheme as a robust method for exploring the Anderson transition and demonstrates that precise characterization of the initial state (including thermal fractions) is essential for quantitative analysis.
• Future Applications: The framework is readily extensible to other platforms (e.g., fermionic systems) and more complex scenarios, such as including weak interactions or transverse confinement. It opens the door to studying critical exponents, multifractality, and percolation transitions in random potentials using cold atoms.

In summary, this work establishes a rigorous quantitative link between theory and experiment for 3D Anderson localization, resolving previous discrepancies by accounting for the specific spectral and thermal properties of the prepared atomic states.