How do atoms change their phase randomly to produce incoherent light?

From research I found that for one light wave to be coherent, its sources (atoms in a regular light source) need to have the same wavelenght and be in constant phase difference. I can picture why the light has to be monochromatic, because if not, no interference pattern could arise. It is said that most sources are incoherent, exceptions are lasers. Some can be monochromatic but still incoherent, like sodium vapor lamps. Why is that? I figure it might have to do with phase difference between atoms. But how exactly would phase difference vary? What happens to the atoms exactly that causes them to change phase randomly?

How would the resulting wave look like? Wouldn't it still look like a sine wave, because it it the sum of same frequency waves, produced by the numerous atoms? I cannot picture a sine wave that changes phase randomly without being discontinous.

Edit: Would the wave look like this? (There are "sudden" phase changes every 3 units of independant variable)

• Perhaps this answer to a related question could help. Commented Jan 5, 2023 at 16:24

In a laser, the phase of the amplified light is controlled by the light that stimulates it. In a laser oscillator, that's (mostly) the same light, so a "coherent" wave builds up in phase with itself.

But most light sources aren't lasers: the emission is "spontaneous". A useful model is that the spontaneous emission is stimulated by random vacuum fluctuations, leading to random phase fluctuations.

Even in lasers, the phase coherence is limited by vacuum fluctuations propagating upstream from the beam output into the laser. However, remarkably, LIGO famously improves the phase coherence of their lasers by "squeezing" the vacuum in the beam dump, reducing its phase fluctuations!

Some observations:

• Your light can't be perfectly monochromatic if its coherence length is not infinite (a basic result from Fourier analysis). Even having the source not shining for eternity will spread it spectrum slightly (due to the time-frequency uncertainty).

• There need not be discontinuities, you can have an envelope $$E_0(t)$$ such
that $$E(t) = E_0(t) \sin\big(\omega t + \phi(t)\big)$$ and $$E_0$$ is continuous and zero where your phase $$\phi$$ jumps (so in a way, a sequence of wave packets each with a different phase). As long as $$E_0$$ varies on a timescale large compare to the frequency, it will only cause a slight broadening of your spectrum.

You can think of short wave trains arriving one after the other, each with a certain random phase.

Another possibility is that the phase can have some random, slow variation, (but the average of $$\phi$$ over time is some constant).

• The wave train picture is actually pretty close to the truth for something like a gas discharge tube (if you ignore the qunatum aspect of the emitted light): The excited atoms in the tube emit spontaneously (and therfore with random start time), each of these emission events be a short "burst", each with a random phase relative to the others.

• In general the question of coherence is more complex. You have to think about spatial coherence as well as temporal coherence, and the local phase and envelope of your wave result from adding up the incoming light from different points of the source and via different paths.

• If the source was other than a gas discharge tube, an incandescent filament for example, the wave profile would consist on multiple waves packets, one after the another, with phase jumps between them? Being the wave packets the sum of all the present wavelenghts. Commented Jan 5, 2023 at 22:43

It is useful to distinguish between EM radiation and an EM wave.

Radiation occurs when excited subatomic particles in the atom return to a lower state. Photons are emitted in the process. In a thermal radiation, the particles are chaotically on the most different energy levels, the photons of the radiation have the most different wavelengths and directions. In lasers, the emitting electrons usually have only one energy difference at which they radiate. However, the emission still occurs at chaotic times. The emission is largely monochromatic, but without wave character.

An EM wave is created when the surface electrons on an emitter are periodically accelerated synchronously. The rising and falling intensity of the emitted photons stimulate the periodic movement of electrons in a receiver, which we can then measure.