The PairInteraction Toolkit for Modeling Rydberg Physics in Alkali and Alkaline-Earth-Like Atoms

arXiv:2605.14993 · physics.atom-ph, quant-ph · Submitted 2026-05-14 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "The PairInteraction Toolkit for Modeling Rydberg Physics in Alkali and Alkaline-Earth-Like Atoms".

Mira: The gist The paper presents a unified theoretical framework for modeling Rydberg atoms and their interactions based on multi-channel quantum defect theory (MQDT) and static electromagnetic Green’s tensors,

Kai: First, who's behind it and why it matters.

Title and authors: Mira: The paper summarizes their approach by combining two big tools: they use Multi-Channel Quantum Defect Theory, or MQDT, to describe the Rydberg states of atoms like strontium and ytterbium. Then they layer that on top with Green’s tensor formalism to figure out how those atoms actually interact when you put them in an environment.

Kai: So MQDT handles the structure of a single atom—how its electron configuration dictates its energy levels—and then the Green’s tensor tells us about the forces between two or more of those atoms, including their own self-interaction.

Lev: That MQDT part sounds intense because you have to deal with coupling schemes and angular momentum quantum numbers for every channel, which is a lot to manage when you start talking about many interacting particles on a real chip.

Mira: Exactly. And the summary shows they propose a new way to label those Rydberg states by using averaged angular quantum numbers, which they think makes it much more unambiguous than just using the principal quantum number alone.

Kai: That seems like a solid way to organize the states, which is crucial when you’re trying to design experiments or even simulations because you need consistent labels across different systems.

Lev: It's good that they’re addressing labeling right there, because if your labeling system is messy, the whole simulation becomes unreliable and you can't trust what you measure.

The paper's summary: Kai: Now the paper talks about how their toolkit is better than what came before. They highlight that they’ve designed a new architecture for PairInteraction to make it faster and easier to use when building Hamiltonians for one or two atoms, especially when you introduce external fields or surfaces.

Mira: And they specifically focus on the Green’s tensor part, showing how the self-interaction needs to be included in the coupling scheme because that’s where a lot of physics gets missed if you ignore it.

Lev: I saw something about performance improvements too; they mention speedups of one order of magnitude when constructing Hamiltonians for two interacting atoms and calculating those interaction potentials, which is important for error correction simulations.

Kai: Right, so the hardware side is optimized to run these calculations quickly, using things like Intel MKL and DuckDB for fast matrix element access. It’s not just a theory; it’s implemented in C++ to be efficient.

Mira: But what really stands out are the use cases they present: they study ytterbium in external fields and get excellent agreement with experimental Stark shifts, extracting effective spin-one/two and spin-one model Hamiltonians <ref:2605.14993#pg1>.

Lev: That’s a big deal because those effective Hamiltonians are what you actually need if you want to map out a specific quantum error correction scheme on actual hardware.

The paper's improvements: Kai: So to wrap up, the paper "The PairInteraction Toolkit for Modeling Rydberg Physics in Alkali and Alkaline-Earth-Like Atoms" gives us a complete framework using MQDT and Green’s tensor formalism that lets us model Rydberg interactions in structured environments with good computational performance.

Mira: The main implication is that this toolkit makes it practical to study how these atoms behave near surfaces or in external fields, showing that the couplings due to self-interaction really matter when you're looking at proximity effects.

Lev: For error correction researchers, the fact they can extract those spin-one and spin-one/two Hamiltonians from two-atom potentials is valuable because it means we have a starting point for modeling more complex many-body interactions on actual physical systems <ref:2605.14993#pg1>.

Kai: It’s about bridging the gap between the theoretical description of these atoms and what you can actually cool, trap, and measure in a lab setting.

Mira: They also point out that their method shows clear deviations at small distances when looking at pair interactions near a surface, which tells us we can’t just ignore those self-interaction terms when working close to boundaries.

Lev: So they clearly show that this isn't just for textbook problems; it’s relevant for understanding the physics right up to the boundary where things get complicated.

Kai: That’s what we have today on "The PairInteraction Toolkit for Modeling Rydberg Physics in Alkali and Alkaline-Earth-Like Atoms." We’ve seen how they built a powerful tool to handle these systems.

Mira: It really shows how combining the MQDT structure with the Green's tensor approach gives us a complete, non-perturbative treatment of these couplings.

Lev: We'll keep an eye on how this toolkit helps researchers move from simple two-atom models toward simulating larger, more relevant many-body systems for error correction.

Conclusion: Kai: So we've looked at how they built this toolkit for modeling Rydberg physics in alkali and alkaline-earth-like atoms, and it really shows how you can combine MQDT with Green’s tensor formalism to get a complete picture of what's happening.

Mira: Exactly. The core idea is that you use the multi-channel quantum defect theory to nail down the single atom states, and then you layer on the green's tensor formalism to figure out all those interactions between them in any geometry.

Lev: What I found interesting is how they structured it so that you can build effective spin-one and spin-one/two Hamiltonians from those two-atom potentials. That means we could actually use these calculations to model many-body systems on real hardware, not just simple pairs.

Kai: Right, so the software itself is pretty solid too; they’ve optimized the implementation in C++ with tools like MKL for linear algebra, and they're showing speedups of one order of magnitude when you’re constructing those Hamiltonians.

Mira: That performance is important because it makes these kinds of calculations practical, allowing us to move past just thinking about them theoretically into actually running simulations on a computer.

Lev: And they did get some really good validation, too; they checked the energy levels and dipole matrix elements against experimental data for 174Yb, which confirms that the MQDT models they used are accurate enough for real measurements.

Kai: So it’s a full loop: you use the theory to build a model, you run it on optimized software, and you compare those results to what people have actually measured in labs.

Mira: The implication is that this framework gives us a non-perturbative way to look at these interactions without having to make any simplifying assumptions about how strong the coupling is.

Lev: It also points out that you can’t just ignore the self-interaction terms, especially when you’re studying things near surfaces, because those couplings become relevant at small distances.

Kai: So that means if someone is designing a new architecture for these atoms, they can use this toolkit to see exactly where those surface effects will come into play.

Mira: It’s a very complete description of the physics happening at the atomic level, whether you're looking at two atoms in space or two atoms near a boundary.

Lev: This paper, "The PairInteraction Toolkit for Modeling Rydberg Physics in Alkali and Alkaline-Earth-Like Atoms," gives us a lot of groundwork for how to build those more complex many-body models you need for error correction.

Kai: It sets a clear direction for how we can use these computational tools to design better systems.

Mira: We’ll see what other papers come out next that try to push the boundaries on these kinds of interactions.

Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology, University of Stuttgart · Atom Computing, Inc. · Institut für Theoretische Physik, Universität Tübingen · Institute of Applied Physics, University of Bonn · Physikalisches Institut, Universität Heidelberg · Max Planck Computing and Data Facility

physics.atom-ph, quant-ph

Submitted: 2026-05-14

Updated: 2026-10-08

Comments: 16 pages, 8 figures

Journal ref: Scientific Reports 16, 31484 (2026)

DOI: 10.1038/s41598-026-72921-0

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 86/100

The gist: The gist The paper presents a unified theoretical framework for modeling Rydberg atoms and their interactions based on multi-channel quantum defect theory (MQDT) and static electromagnetic Green’s

Key concepts

Multi-Channel Quantum Defect Theory (MQDT)
MQDT divides space around an atomic core into inner and outer regions. It uses angular momentum quantum numbers to define different 'channels' for the Rydberg electron. This method helps accurately describe how the electron behaves near the core, which is crucial because core-excited states can significantly affect Rydberg energy levels.
Green’s Tensor Formalism
This formalism characterizes the electromagnetic environment using a Green’s tensor, which describes how fields propagate between two points. It decomposes this tensor into free space and scattering components. This approach allows for the calculation of interactions, like dipole-dipole coupling, in arbitrary geometries.
Rydberg State Labeling
A Rydberg state is labeled by an effective principal quantum number ($ u$). This number is derived from the energy difference between the actual atomic level and a reference energy. This allows researchers to simplify complex atomic structures into a single parameter for modeling interactions, making calculations tractable.
Hamiltonian Decomposition
The total system Hamiltonian is broken down into single-atom contributions (including field effects) and two-body interactions. The interaction terms are treated using the Green's tensor formalism. This structured approach ensures that both the atomic structure and the external environment are accounted for when modeling atom pairs.

Terminology

Summary

The gist The paper presents a unified theoretical framework for modeling Rydberg atoms and their interactions based on multi-channel quantum defect theory (MQDT) and static electromagnetic Green’s tensors, which is essential for interpreting experiments and designing new architectures > ref:14

Theoretical Framework

The framework utilizes MQDT to describe Rydberg states of divalent atoms such as strontium and ytterbium, while the Green’s tensor formalism provides a general approach for calculating interactions between two Rydberg atoms in arbitrary geometries > ref:14

MQDT is required because core-excited states can lie close in energy to the relevant Rydberg series and perturb them, particularly in systems with hyperfine-split cores > ref:1

The total Hamiltonian of the system is given by the sum of single-atom contributions and two-body interactions (Hˆ = Hˆ1 ⊗ 1 + 1 ⊗ Hˆ2 + Hˆ12) > ref:2

The single-atom Hamiltonian is written as Hˆα = Hˆ0,α + Hˆfields,α + Hˆsi,α with α = 1, 2, where Hˆ0,α describes the field-free Hamiltonian containing the energies of the unperturbed atomic levels > ref:2

The interaction between atoms as well as the selfinteraction is treated in the Green’s tensor formalism [27] > ref:2

Multi-Channel Quantum Defect Theory (MQDT)

MQDT divides real space around the ionic core into an inner region r ≤ rc and an outer region r > rc, where r denotes the distance from the ionic core > ref:3

A channel is specified by a set of angular-momentum quantum numbers of the Rydberg electron and the ionic core, in a specific coupling scheme, together with the residual core configuration > ref:3

The total angular wave function is obtained by coupling Fc and j to the total angular momentum Ftot, whose projection along the quantization axis yields mFtot > ref:3

The radial Schrödinger equation can be solved analytically for each FJ-coupled channel i, yielding regular Coulomb function fli(νi, r) and irregular Coulomb function gli(νi, r) > ref:4

The total wave function is written as Ψ(r) = 1/r ∑ i (aifli(νi, r) + bigli(νi, r)) ∣ϕi⟩ > ref:5

The channel coefficients are given by Ai = ai/Ni, together with the normalization condition ∑ i Ai2 = 1 > ref:5

To label a Rydberg state, an effective principal quantum number ν is defined as ν = 1/√2(Iref − E) > ref:12

Green’s Tensor Formalism

The electromagnetic environment is fully characterized by the Green’s tensor G(r, r', ω), which describes the propagation of a field of frequency ω between points r and r' > ref:16

The Green’s tensor can generally be decomposed into a free space (or bulk) contribution and a scattering contribution as G(r, r', ω) = G0(r, r', ω) + GR(r, r', ω) > ref:16

In the static limit, the scattering Green’s tensor simplifies considerably and can often be related to electrostatic image-charge solutions > ref:4

The coherent contributions include the dipole-dipole interaction Hˆ12 = -1/2ε0 ∑α≠β Dˆ αS(rα, rβ)Dˆ β > ref:18

The self-interaction term is described by Hˆsi,α = -1/2ε0 Dˆ αS R(rα, rα)Dˆ α > ref:22

Software Implementation and Performance

The framework is implemented in the open-source software PairInteraction v2, which constitutes a complete rewrite of the previous version > ref:51

The software allows for the construction of Hamiltonians for one or two Rydberg atoms, optionally in the presence of external fields and/or surfaces > ref:7

To achieve efficiency, calculations are implemented and parallelized in C++, using Intel MKL [93] for linear algebra operations and DuckDB [94] for fast access to precomputed matrix elements > ref:8

The benchmarks show speedups of one order of magnitude in the construction of Hamiltonians for two interacting Rydberg atoms and in the calculation of interaction potentials > ref:10

Example Applications

In section VI A, calculations are performed for 174Yb, demonstrating that energy levels and dipole matrix elements are computed accurately using MQDT models > ref:6

The software demonstrates excellent agreement with experimentally measured Stark shifts and extracts effective spin-1/2 and spin-1 model Hamiltonians > ref:6

In section VI B, the Green’s tensor formalism is used to study the self-interaction as well as the Rydberg-Rydberg interactions near a perfectly conducting plate > ref:4

The software allows for constructing effective spin-1 models by tuning states into a Förster resonance, which can be achieved by changing the principal quantum number and using a magnetic field > ref:11

The analysis of pair interactions near a surface shows clear deviations at small distances where the self-interaction becomes relevant > ref:8

In conclusion, the framework provides a non-perturbative, complete treatment of interactions within the static Green’s tensor formalism and demonstrates that these couplings cannot generally be neglected for Rydberg atoms near surfaces > ref:12

The software is available at www.pairinteraction.

Improvements for AI systems

  1. Continuous modeling of Rydberg interactions in structured environments: The system can calculate the self-interaction as well as the Rydberg-Rydberg interactions near a perfectly conducting plate, which is explicitly shown to yield clear deviations at small distances, where the self-interaction becomes relevant.

  2. Accurate state characterization for complex atoms: The system can provide a method to label states by identifying the coupling scheme for which the variance of the angular quantum numbers is smallest, suggesting a more physically meaningful labeling than just principal quantum numbers.

  3. Construction of effective Hamiltonians for many-body systems: The software can construct effective spin-1 and spin-1/2 Hamiltonians from two-atom pair potentials, which can then be used to study many-body systems, allowing simulation of complex Rydberg interactions beyond simple two-atom models.

  4. High-fidelity experimental validation: The system enables the comparison of a computed and experimentally measured Stark map of 174Yb, demonstrating that energy levels and dipole matrix elements are computed accurately, which validates the underlying MQDT models used for state description.

Sources

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