Jul 9, 2026 -- Underpinning much of modern technology, from smartphones to scanning tunneling microscopes to particle colliders, there is Fermi’s Golden Rule.

Named for 20th century Italian-American physicist Enrico Fermi (but actually discovered by British physicist Paul Dirac), the rule is a formula that connects what can be measured in an experiment — such as how fast atoms “jump” between energy states — to the microscopic properties of a quantum mechanical system. The formula is taught in every undergraduate quantum physics class.

Yet scientists sometimes misapply it. They either misjudge the conditions under which the formula works, or they miss the “window” for its use.

A “user manual” for Fermi’s Golden Rule would be a boon to researchers, says Yale physicist Nir Navon — and now he and his lab partners have provided one.

“We put one of the most famous formulas in all of quantum mechanics to the test, and found where it works and where it fails, including ways that many physicists weren’t fully aware of,” said Navon, an associate professor of physics in Yale’s Faculty of Arts and Sciences and senior author of a new study in the journal Nature Physics. “We’re telling everyone who uses it to take a breath first and check their ingredients.”

It was by chance that Navon and his team even came across their blueprint for Fermi’s Golden Rule.

As part of earlier research with quasiparticles — emergent quantum objects that many scientists use as proxies to understand interacting quantum systems — Navon and his colleagues used precise radio frequency control to change the spin state of atoms cooled to nanokelvin temperatures, billionths of a degree above absolute zero.

Their control over the atoms was so precise they could see exactly when the Golden Rule governing the rate of energy transitions was in play.

“We realized we could see the full picture — before, during, and after the Golden Rule applies,” Navon said. “It was really beautiful. We opened a door we weren’t expecting.”

Based on their observations, the researchers highlighted two areas where scientists tend to misapply the formula. First, the perturbation that drives the quantum transition must be weak. Second, the formula is only valid within a specific window of observation time.

Neither condition is as simple as it sounds. What counts as “weak” is not always obvious — it depends on the intrinsic energy scales of the system being probed, which are not always known in advance. The time window is equally subtle; it opens only after the system has had a chance to respond to the perturbation and closes before the initial state has been modified too much.

And, crucially, the width of the window is itself determined by the very properties of the system under study.

For well-understood systems — an isolated atom in an electromagnetic field, for instance — physicists can work out both conditions with precision. But for assemblies of strongly interacting particles — like the ultracold quantum gases Navon’s team works with — the calculation is intractable, even with supercomputers.

The result is a circular problem: the Golden Rule is needed to interpret the experimental data, but knowing where the Golden Rule applies requires information that only those experiments can provide.

Navon noted that having more guidance for using the formula will become even more important as the pace of research into quantum mechanics continues to accelerate. Fields as varied as chemistry, materials science, semiconductor engineering, and quantum computing all make use of the Golden Rule.

“In a sense, we are raising awareness,” Navon said. “While this formula is golden, its assumptions should never be taken for granted.”

The study’s experimental effort was led by Jianyi Chen, a student in physics in Yale’s Graduate School of Arts and Sciences, with significant contributions from Songtao Huang, another physics student in GSAS. Co-authors are YGSAS students Alan Tsidilkovski and Gabriel Assumpção, former Yale students Yunpeng Ji and Grant Schumacher, and Alexander Schuckert of the University of Maryland.

Funding for the research came from the National Science Foundation, the David and Lucile Packard Foundation, and the Alfred P. Sloan Foundation.