Imagine plucking a guitar string. It vibrates, the sound lingers, and then fades away as the energy drains into the air. Now bring this scene down to the scale of an atom. Can an atom vibrate in the same way, gradually losing energy to its surroundings?
For almost a hundred years, physicists suspected the answer was yes, but they couldn’t describe it properly without breaking Heisenberg’s uncertainty principle, one of the central rules of quantum mechanics. This principle makes sure that nature always keeps some secrets.
For instance, when you measure a particle’s position more precisely, its momentum slips further out of reach. Balancing this uncertainty while trying to explain how a quantum system loses energy proved a mathematical nightmare.
This problem has defeated theorists for decades, and solving it could reshape how scientists measure and manipulate matter at the tiniest scales. However, a new study presents the first exact solution to a damped quantum harmonic oscillator, a system that slowly loses energy. It is basically the quantum twin of a guitar string that gradually quiets down.
Arriving at a nearly impossible quantum solution
The story of damped harmonic systems goes back to 1900, when British physicist Horace Lamb built a simple mathematical model of a particle vibrating inside a solid. In this model, the particle’s motion creates waves in the solid.
Those waves push back, and the particle slowly loses energy. That worked beautifully in classical physics. However, when later scientists tried to adapt Lamb’s idea to the quantum world, things fell apart.
“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, co-study author and a quantum physics student at the University of Vermont, said.
The models couldn’t keep the uncertainty principle intact. Describing the damping precisely meant accidentally allowing forbidden precision in position or momentum. The study authors approached the problem differently. Instead of focusing only on the vibrating atom, they included its full relationship with all the other atoms in the material.
This turned the question into what physicists call a many-body problem, where countless interactions need to be accounted for at once. To make sense of it, the team used a powerful mathematical tool known as a multimode Bogoliubov transformation, a technique that rewrites the system’s equations in a way that makes hidden patterns visible and the problem solvable.
What they found is that the atom settles into a special type of quantum state called a multimode squeezed vacuum. In this state, the random quantum noise in one property (say, the atom’s position) can be reduced below the normal limit, but only by allowing more uncertainty in another property (momentum).
This careful trade-off preserved the uncertainty principle while also capturing the way energy leaks out of the system. Thus, for the first time, the quantum version of damping could be described exactly, without breaking quantum rules.
The significance of the damped harmonic oscillator
The solution to the damped quantum harmonic oscillator might sound abstract, but its implications reach far. One direct outcome of this work is the possibility of measuring position at scales finer than the so-called standard quantum limit. That’s the boundary that usually defines how precise a measurement can be in the quantum world.
Crossing it requires clever tricks with squeezed states of matter or light. In fact, according to the study authors, the Nobel Prize–winning detection of gravitational waves in 2017 also relied on such tricks to measure shifts thousands of times smaller than a proton.
The current study’s solution shows that similar tricks could work for atoms in solids, potentially leading to sensors that act like the tiniest rulers imaginable.
However, these results are theoretical for now. The next challenge is connecting math with experiments, and finding real atomic systems where these predictions can be tested.
The study is published in the journal Physical Review Research.