The quantum world is like a kitchen where particles suddenly appear out of "nothing." But you need a chef — a classical field acting as an external source. Glauber's theorem, long known in optics, has now been generalized: it shows exactly how the field "cooks up" particles, be they photons or gravitons.

The authors took a Hamiltonian (the energy recipe of the system) with terms up to second order in creation and annihilation operators. It turned out that the time evolution operator can be decomposed into three simple actions: displacement, squeezing, and rotation. It's as if a complex particle dance boiled down to three basic moves. The equations for these moves are no longer operator equations but ordinary numerical ones, easy to solve on a computer or even by hand.

This approach is universal. In experiments searching for axions (dark matter candidates), a microwave cavity acts as a classical background, converting axions into photons. This is the linear regime — a simple displacement. But in black hole mergers, gravitons are born via quadratic interactions — here you need both squeezing and rotation. Thermal initial states, important for real cavities, also fit into the scheme. As a result, you can honestly calculate not just the average particle number but also finer statistics — for instance, how "noisy" the photons are in a detector.