🔬 atomic physics

Coexisting Regular and Chaotic Dynamics in the Dysprosium Feshbach Spectrum

By combining precise Feshbach spectrum calibration with differential magnetic moment measurements for over 80 resonances in $^{162}$ Dy, this study reveals that quantum chaos in strongly dipolar gases is not uniform but coexists with regular dynamics, depending on the molecular-state composition of the resonances.

Original authors: Julie Veschambre, Alexandre Journeaux, Maxime Lecomte, Alice Belmon, Ethan Uzan, Inès de Verdelhan, Patricia Christina Marques Castilho, Jakub Zakrzewski, Jean Dalibard, Raphael Lopes

Published 2026-06-29

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Julie Veschambre, Alexandre Journeaux, Maxime Lecomte, Alice Belmon, Ethan Uzan, Inès de Verdelhan, Patricia Christina Marques Castilho, Jakub Zakrzewski, Jean Dalibard, Raphael Lopes

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

Imagine a giant, chaotic dance floor where thousands of dancers (atoms) are trying to find partners. In most dance halls, the music is simple, and the dancers pair up in predictable, orderly ways. But in this specific experiment, the "music" is a complex, anisotropic magnetic field, and the dancers are Dysprosium atoms. Because these atoms are so magnetic, they interact in a messy, tangled way that usually looks like pure chaos.

The scientists in this paper wanted to understand the rules of this chaotic dance. They didn't just look at where the dancers paired up (the resonance positions); they also measured how the dancers felt the magnetic field (their "magnetic moments").

Here is the breakdown of their discovery using simple analogies:

1. The "Chaotic" Dance Floor

When the scientists looked at the entire dance floor at once, the pattern of where the atoms paired up looked completely random. It was like looking at a static-filled TV screen. Previous studies suggested this was "quantum chaos"—meaning the atoms were mixing so thoroughly that you couldn't predict their behavior, much like a crowd of people bumping into each other in a mosh pit.

2. The New "Flashlight"

The researchers realized that looking at the whole crowd at once was hiding the truth. They decided to use a new tool: a "flashlight" that shone on the magnetic personality of each pair.

Think of the magnetic moment as a "magnetic ID card." Every pair of atoms has a specific ID card that tells you how strongly they react to a magnetic field. The scientists measured these ID cards for over 80 different pairs.

3. The Big Surprise: Order in the Chaos

When they sorted the dancers based on their "magnetic ID cards," the chaos disappeared. They found that the dance floor wasn't uniform; it had two distinct zones:

The "Center" Zone (The Chaotic Crowd):
Pairs with magnetic ID cards near the average value were behaving chaotically. They were bumping into each other, repelling one another, and mixing wildly. This is like the center of a busy party where everyone is interacting with everyone else. The math describing this looked like "random matrix theory"—the gold standard for chaos.
The "Edge" Zone (The Quiet Corner):
Pairs with magnetic ID cards near the lowest values were behaving very differently. They were orderly, predictable, and didn't seem to care about their neighbors. They followed "Poisson statistics," which is a fancy way of saying they were acting like independent, non-interacting individuals. It was like finding a quiet corner of the party where people were standing still, ignoring the noise.

4. Why Does This Happen?

The paper explains this using a "menu" analogy.

The Center: Imagine a restaurant menu with thousands of dishes that all taste roughly the same (similar magnetic moments). Because there are so many options that are similar, the atoms have many different "partners" they can mix with. This creates a dense web of interactions, leading to chaos and strong repulsion.
The Edge: Now imagine the menu has very few dishes that taste like the "lowest" flavor. There are very few options available. Because there are so few compatible partners, the atoms can't mix much. They stay in their own lane, leading to orderly, predictable behavior.

5. The Takeaway

The main point of the paper is that chaos isn't always the whole story. Even in a system that looks completely messy from a distance, there can be hidden pockets of order if you look closely enough.

By using the "magnetic moment" as a sorting tool, the scientists revealed that the Dysprosium Feshbach spectrum is actually a mix of two worlds: a chaotic, highly interactive world in the middle, and a regular, quiet world at the edges. This proves that to truly understand quantum chaos, you can't just look at the positions of the atoms; you have to understand the specific "ingredients" (the molecular states) that make them up.

In short: They took a messy, chaotic puzzle and found that if you sort the pieces by their color (magnetic moment), you see that some pieces fit together in a wild, tangled mess, while others fit together in a neat, orderly line.

Technical Summary: Coexisting Regular and Chaotic Dynamics in the Dysprosium Feshbach Spectrum

Problem Statement
Strongly dipolar quantum gases, particularly those composed of lanthanides like dysprosium (Dy), erbium, and thulium, exhibit dense Feshbach spectra. Previous studies have associated the level statistics of these spectra with quantum chaos arising from couplings among numerous molecular channels. However, a complete microscopic assignment of resonance positions remains challenging, and global statistical analyses often obscure underlying structures. Theoretical frameworks suggest that chaotic spectra are not necessarily statistically homogeneous; regular and chaotic components can coexist, leading to mixed statistics. A critical gap exists in identifying experimentally accessible observables capable of resolving these hidden components, as global level-spacing analyses can mask distinct statistical behaviors within a single spectrum.

Methodology
The authors investigate the Feshbach spectrum of $^{162}\text{Dy}$ by combining precise resonance localization with spectroscopic measurements of the differential magnetic moments ( $\delta\mu$ ) of the associated molecular bound states.

Resonance Localization: Non-degenerate, spin-polarized samples of $^{162}\text{Dy}$ (prepared in the $|-8\rangle$ state at 350 nK) were subjected to atom-loss spectroscopy. By scanning magnetic fields from 0 to 30 G and measuring atom loss due to two- and three-body recombination, the authors identified 128 Feshbach resonances.
Differential Magnetic Moment Measurement: For 83 of these resonances, the authors performed bound-state spectroscopy using a time-modulated, spin-dependent light shift. By measuring the near-threshold dispersion of molecular binding energies $E(B)$ , they extracted the differential magnetic moment $\delta\mu_i = \mu_i^{\text{mol}} - \mu_{\text{en}}$ via local linear fits to the relation $E(B) = \delta\mu_i(B - B_i)$ .
Statistical Analysis:
- Unfolding: To account for the magnetic-field-dependent density of resonances, the spectrum was unfolded using a second-order polynomial fit to the accumulated number of resonances.
- Brody Parameter ( $\beta$ ): The nearest-neighbor spacing distribution $P(s)$ was analyzed using the Brody distribution, where $\beta=0$ corresponds to Poisson statistics (uncorrelated/regular) and $\beta=1$ to Wigner-Dyson statistics (chaotic/level repulsion). The parameter was extracted via maximum-log-likelihood estimation.
- Average Gap Ratio ( $\langle r \rangle$ ): As a complementary, unfolding-free metric, the authors calculated the average ratio of consecutive level spacings to distinguish between Poisson ( $\approx 0.386$ ) and Gaussian Orthogonal Ensemble (GOE, $\approx 0.531$ ) limits.
- Subsets: The analysis was performed on the full set of 128 resonances, a subset of 83 with measured $\delta\mu$ , and specific subsets grouped by their differential magnetic moments (low-moment vs. central-moment).

Key Results

Global Statistics: The full set of 128 resonances yields a Brody parameter $\beta = 0.53(10)$ , and the subset of 83 resonances yields $\beta = 0.53(13)$ . These values indicate an intermediate level of repulsion, consistent with previous studies that treated the spectrum as a single statistical ensemble.
Magnetic-Moment-Resolved Statistics: When the spectrum is resolved according to the differential magnetic moment, distinct statistical behaviors emerge:
- Low-Moment Subset: Resonances associated with states near the lower edge of the $\delta\mu$ distribution (central value $\delta\mu_c = 6.4\,\mu_B$ ) exhibit $\beta = 0.03(16)$ and $\langle r \rangle = 0.34(6)$ . These values are close to the Poisson limit, indicating weak level repulsion and regular dynamics.
- Central-Moment Subset: Resonances near the center of the $\delta\mu$ distribution (central value $\delta\mu_c = 12.4\,\mu_B$ ) display $\beta = 0.69(25)$ and $\langle r \rangle = 0.52(5)$ . These values show enhanced level repulsion, moving toward the Wigner-Dyson limit characteristic of quantum chaos.
Robustness: Sliding-window analyses and bootstrap resampling confirm that this trend is robust against the choice of window size and experimental uncertainties. The transition from regular to chaotic statistics correlates with the position of the resonance within the differential magnetic moment distribution.

Significance and Claims
The paper claims that the differential magnetic moment acts as an experimentally accessible classifier that reveals hidden structure within the chaotic dysprosium Feshbach spectrum. The authors assert that the spectrum is not statistically uniform; rather, it contains coexisting regular and chaotic components that are obscured in a global analysis but become visible when resonances are grouped by the magnetic character of their underlying molecular states.

The authors provide a microscopic interpretation for this phenomenon:

States near the lower edge of the $\delta\mu$ distribution are constrained by the spin-polarized entrance channel (composed of two atoms in stretched spin states). This constraint limits the number of compatible spin configurations available for mixing, resulting in weak level repulsion and Poisson-like statistics.
States near the center of the distribution can arise from many combinations of spin and orbital angular momentum. These states are embedded in a dense network of energetically close molecular states where anisotropic interactions generate strong mixing and enhanced level repulsion, driving the statistics toward the Wigner-Dyson regime.

The work establishes eigenstate-sensitive spectroscopy as a practical route to uncovering statistical structure in chaotic lanthanide Feshbach spectra. It suggests that the coexistence of regular and chaotic components is a generic feature of molecular spectra governed by anisotropic interactions, where the conserved total angular-momentum projection $M$ constrains accessible configurations, and the magnetic moment serves as a classifier for their statistical character.