It's not often that an element of mathematics, developed purely out of curiosity, ends up connecting directly to the front line of quantum technology. But that's just what researchers at the NTT Science and Core Technology Laboratory Group have done, developing an interesting link between a theoretical math model and a physical model that describes how light and matter interact. Their discovery could potentially play a role in the development of quantum computers.

For years, the two models lived in separate worlds. Mathematicians explored something called the non-commutative harmonic oscillator (NCHO), an abstract framework for studying complex vibrations and their symmetries. Physicists, meanwhile, focused on the quantum Rabi model (QRM), which explains how a two-level atom, a simplified version of matter, interacts with electromagnetic waves, absorbing and emitting photons. One, a mathematical framework; the other, a physics model of how light interacts with atoms. Until now, no one had been able to prove that the two were mathematically connected.

When Worlds Collide

The NTT team, bringing together experts in pure mathematics and quantum optics, found a direct mathematical match between the NCHO and the two-photon quantum Rabi model. In this version, an atom interacts with two photons at the same time.

They also showed that if you gradually adjust the conditions of the two-photon model, it turns into the more familiar one-photon QRM. That finding creates a clear bridge between the NCHO and both forms of the QRM, something earlier studies had hinted at but never mapped in detail.

Symmetry is the Key

Proving the connection meant finding a way to show that two very different-looking models were actually the same at a deeper level. The team used representation theory, a branch of mathematics that studies symmetries. It's a way of finding hidden similarities between seemingly unrelated systems. Imagine two different puzzles with different pictures, but with pieces cut to exactly the same shapes. Swap the pieces and they still fit.

The researchers rewrote the complex equations for both models in a different mathematical framework that had the same symmetrical structure. This showed that solving one model was effectively the same as solving the other.

Okay... So?

NTT researchers proved that a piece of abstract mathematics lines up exactly with a real-world physics model used to understand how atoms interact with light. That means knowledge from one field can now be used to advance the other.

The QRM is more than just theoretical noodling. It underpins the design of superconducting quantum bits, or qubits, which are the building blocks of some quantum computers. A deeper understanding of the model, and especially its two-photon version, could lead to new methods for controlling quantum systems.

Linking the NCHO to these physical models also opens access to a rich library of mathematical results built up over several decades. Some of that work connects even to number theory, a branch of mathematics with no obvious ties to quantum hardware. Well, until now, that is.

The hope is that these insights might reveal new ways to manipulate light–matter interactions, potentially improving quantum computing performance, inspiring new device designs, or even leading to more efficient error correction methods for qubits.

From Theory to Experiment

NTT's work is still at a theoretical stage, but its potential is very practical. The one-photon QRM has already been confirmed through experiments. The two-photon version, however, has not yet been fully explored in the lab. Now that its mathematical significance is clearer, researchers have a stronger case for doing so.

The plan is to move in stages: first, make theoretical predictions, then design experiments to test them. If the results match, the 2QRM could become a standard tool in quantum optics research and possibly find its way into experimental quantum devices, sensing applications, and precision measurement systems.

Looking Ahead

For NTT, the discovery also shows the value of its Institute for Fundamental Mathematics, launched in 2021 to explore modern mathematics, while tackling complex societal challenges. By bringing mathematicians and physicists together, the Institute is uncovering links that might otherwise go unnoticed.

In the long run, this kind of cross-disciplinary work could be as important to quantum computing as advances in engineering or materials science. Even the most advanced quantum hardware depends on the equations that describe its behavior; sometimes, those equations start life as pure mathematics, with no clear application.

Could the math behind today’s abstract symmetries become the blueprint for tomorrow’s quantum machines, sensors, and imaging systems? NTT’s latest work suggests it might.

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Daniel O'Connor

Daniel O'Connor joined the NTT Group in 1999 when he began work as the Public Relations Manager of NTT Europe. While in London, he liaised with the local press, created the company's intranet site, wrote technical copy for industry magazines and managed exhibition stands from initial design to finished displays.

Later seconded to the headquarters of NTT Communications in Tokyo, he contributed to the company's first-ever winning of global telecoms awards and the digitalisation of internal company information exchange.

Since 2015 Daniel has created content for the Group's Global Leadership Institute, the One NTT Network and is currently working with NTT R&D teams to grow public understanding of the cutting-edge research undertaken by the NTT Group.