In a study published July 7th, 2025, in the journal Physical Review Research, University of Vermont researchers discovered a precise solution to a model that acts as a “damped quantum harmonic oscillator,” a guitar-string sort of motion at the atomic scale.
UVM professor Dennis Clougherty (right) and his student Nam Dinh wondered if there are systems in the atomic scale that behave like the vibrating motion of a guitar string in the Newtonian world. They found that the answer is yes, and solved a 90-year-old problem in quantum physics. Image Credit: Darkforxelixir/Shutterstock.com
A plucked guitar string can vibrate for several seconds before becoming quiet. A playground swing that has been empty of its passengers will eventually come to rest. These are referred to as “damped harmonic oscillators” by physicists, and they are well understood using Newton's equations of motion.
However, things are peculiar in the tiny world of atoms, which follow the strange rules of quantum physics. Dennis Clougherty, a professor at UVM, and his pupil Nam Dinh pondered whether there exist atomic systems that exhibit Newtonian behavior similar to a guitar string's vibrating motion.
If so, can we construct a quantum theory of the damped harmonic oscillator?
Dennis Clougherty, Professor, University of Vermont
It turns out that theorists have been attempting, with varying degrees of success, to use quantum physics to describe these damped harmonic systems for around 90 years.
“The difficulty involves preserving Heisenberg’s uncertainty principle, a foundational tenet of quantum physics,” explained Clougherty, who has been a Physics Professor at the University of Vermont since 1992.
The renowned Heisenberg uncertainty principle demonstrates that there is a fundamental limit to the accuracy with which the position and momentum of a particle can be determined simultaneously, in contrast to the human-scale world of, for instance, arcing rockets or bouncing balls. The more precisely one attribute is measured at the atomic level, the less precisely the other can be determined.
Lamb Chopped
The UVM researchers analyzed a model created by British physicist Horace Lamb in 1900, long before Werner Heisenberg was born and quantum physics had evolved. Lamb wanted to explain how a vibrating particle in a solid might lose energy to it.
Using Newton’s equations of motion, Lamb demonstrated that elastic waves generated by the particle's motion reflect on the particle itself, causing it to dampen and vibrate with less energy over time.
In classical physics, it is known that when objects vibrate or oscillate, they lose energy due to friction, air resistance, and so on. But this is not so obvious in the quantum regime.
Nam Dinh, PhD Student, University of Vermont
Clougherty and Dinh (who received a BS in physics from UVM in 2024, a master's degree in 2025, and are currently pursuing a PhD in mathematics) reconstructed Lamb's model for the quantum world and discovered a solution with funding from the National Science Foundation and NASA.
Clougherty added, “To preserve the uncertainty principle, it is necessary to include in detail the interaction of the atom with all the other atoms in the solid. It is a so-called many-body problem.”
Tiny Tools?
They resolved the issue “through a multimode Bogoliubov transformation, which diagonalizes the Hamiltonian of the system and allows for the determination of its properties,” they explain. This resulted in a state referred to as a “multimode squeezed vacuum.”
To summarize, UVM researchers were able to mathematically reformulate Lamb’s system, allowing the oscillating behavior of an atom to be fully defined in exact terms.
Accurately pinpointing the position of one atom might result in the world's smallest tape measure: new ways for measuring quantum distances and other ultra-precision sensing technologies. These possible uses stem from an essential result of the UVM scientists' recent work: it anticipates how the uncertainty in the location of the atom evolves when it interacts with the other atoms in the solid.
“By reducing this uncertainty, one can measure position to an accuracy below the standard quantum limit,” Clougherty noted.
In physics, there are certain absolute limits, such as the speed of light, and Heisenberg’s uncertainty principle forbids exact measurement of a particle. However, certain quantum methods can minimize this uncertainty beyond normal limits, such as calculating the particle's behavior in a particular “squeezed vacuum” state, which lowers quantum randomness noise in one variable (location) while increasing it in another (momentum).
This type of mathematical maneuver was responsible for the development of the first practical gravitational wave detectors, which can monitor changes in distance a thousand times smaller than the nucleus of an atom and were given the Nobel Prize in 2017. Who knows what the UVM theorists’ finding of a new quantum solution to Lamb's century-old model might reveal?
Journal Reference:
Clougherty, D. P. and Dinh, N. H. (2025). Quantum Lamb model. Physical Review Research. doi.org/10.1103/9fxx-2x6n.