Spontaneous vs Stimulated Emission

[center o bass]

Hi! I wanted to check if I have got the correct interpretation of stimulated vs spontaneous emission.

It seems like stimulated emission is defined as a process for which an excited atom/electron spontaneously relaxes down to a lower energy level.

Stimulated emission on the other hand seems to be defined as the emission of a photon from at atom/electron due to the interaction of that atom/electron with a surrounding electromagnetic field.

However, now I'm reading some quantum optics where the electromagnetic field is quantized.


1) Here the author considers initial and final states of the form $|i\rangle = |A, 0\rangle, \quad |f\rangle = |B, 1_{\vec{k}, a}\rangle$ where A and B is the initial and final states of the atom and 0 and $1_{\vec{k}, a}$ are the initial and final number of photons in the electromagnetic field (wave vector $\vec{k}$ and polarization $a$). One calculates the amplitude for this process to happen through the lowest order interaction term in the hamiltonian.

2) The other type of initial and final states which is considered is of the form $|i\rangle = |A, n_{\vec k ,a}\rangle, \quad |f\rangle = |B, n_{\vec{k}, a} + 1\rangle$.

It seems like the author defines 1) to be spontaneous emission. I.e emission of a photon into a vacuum mode, while he defines 2) to be stimulated emission, i.e. emission into a mode which is already populated. However both these processes involve interactions with the electromagnetic field rendering my previous understanding wrong.

So what is stimulated and spontaneous emission? Is spontaneous emission an interaction with an electromagnetic ground state (vacuum)? Or is there no electromagnetic field at all?
If there is an interaction, the process is not really spontaneous is it?

 

[andrien]

Your question needs some editing.You have written stimulated emission in both of the starting lines.Moreover ,you might have seen a summation while quantizing the electromagnetic field which blows up to infinity (energy) when summed over all modes ,which is avoided by just cutting it off.It is this zero point fluctuation of electromagnetic field which gives rise to spontaneous emission.The process is necessarily spontaneous because one can not avoid it.it is only the result of quantization of electromagnetic field,it does not arises while treating the interaction with classical electromagnetic field of atoms.For more details see Gordon baym 'lectures on quantum mechanics'

 

[Jano L.]

So what is stimulated and spontaneous emission?

There are more views on this, the answer depends on which theory you want to base it on. I recommend Einstein's paper from 1917 On the quantum theory of radiation.

Einstein introduces two processes. First,

"According to Hertz, an oscillating Planck
resonator radiates energy in the well-known way, regardless of whether
or not it is excited by an external field."

and describes it by A coefficient.

Then he introduces second process:

"If a Planck resonator is located in a
radiation field, the energy of the resonator is changed through the
work done on the resonator by the electromagnetic field of the
radiation; this work can be positive or negative, depending on the
phases of the resonator and the oscillating field.
"
and describes it by B coefficient.

So, the difference is that in the spontaneous emission, no external action (light wave from distant source) is required.

Is spontaneous emission an interaction with an electromagnetic ground state (vacuum)?

Atoms cannot physically interact with "state". Rather, it is said that the fluctuations of electromagnetic field may be responsible for bringing the atom down. This is one possible explanation of the process, motivated by the fluctuation-dissipation theorem which connects intensity of damping of motion of a small body in a fluid with the fluctuations of the forces the body experiences from the medium.

However, the fluctuations are not the only explanation possible. There are other ways this effect can be understood, in semi-classical theory (Jaynes), and it is possible to explain damping in classical theory too, without any role of fluctuations.

 

[mathman]

Simple qualitative description:

Spontaneous emission: electron drops from an excited state to a lower state (no outside mechanism) - emitting a photon
Stimulated emission (lasers): photon of the same frequency interacts with electron in excited state which drops to lower state - the emitted photon is coherent with the incoming pnoton

 

[andrien]

emitting a photon

I would say creating a photon and not emitting.