Can You Produce a Laser with a 4-Level System?

[schmeling]

Homework Statement

We will be considering an active medium that consists of atoms with four energy levels. These energy levels are the ground energy level E₀ and three higher energy levels E₁<E₂<E₃.
Consider the case where the onl radiative decay in this system occurs between levels E₁ and E₀
Imagine that the active medium is optically pumped by a monochromatic source with a frequency ω_pump=(E₁-E₀)/ħ. Can a laser be produced in this scenario? Explain your answer.

Homework Equations

The Attempt at a Solution

If we assume that the spacing between the higher energy levels is larger than the spacing between the ground and first excited states, we conclude that this is just an attempt at creating a two level laser which cannot work. It is impossible to obtain population inversion in this setting, the maximum that can be achieved is an equal number of electrons in the ground and first excited states. However if we assume that the spacing of the energy levels is the same, then when the first excited state starts being populated, it is possible that another transition will take place promoting the electrons into higher energy levels.
The difference between a three level laser and this set up is in the intermediate step of the electron having to pass through the first excited state.
My attempt was to write out the rate of change of the number of electrons in each level in a steady state. They all need to equal zero. Then by rearanging and solving the inequality N₀<N₁ we would get the condition for population inversion. So far I am getting rubbish results.
For anyone familiar with this problem, should I continue attempting this via the steady state method or is there some sort of underlying flaw in my reasonsing here?

 

[mfb]

If we assume that the spacing between the higher energy levels is larger than the spacing between the ground and first excited states, we conclude that this is just an attempt at creating a two level laser which cannot work.

Do you need that assumption? There are no radiative decays between the other levels, and even if there would, I don't see why you would need it. The argument you give stays valid.

 

[schmeling]

Thanks for your reply mfb. My reasoning behind the assumption is the following:

1. There are more than two energy levels in the atom.
2. If they are evenly spaced then the monochromatic pump can also cause transitions between the first excited and higher states (once first excited gets populated).
3. I am exploring the possibility of population inversion in this setting.

Of course, if it is impossible to get population inversion this way then we can safely get rid of the assumption but I am yet to be convinced that population inversion cannot be achieved here.

I can imagine a situation where N₀=N₁=N₂=N₃
Now an electron in N₂ decays to N₁ non radiatively (independent of pump) and we have a very small population inversion. If an electron in N₃ is very short lived then it decays down to N₂ preventing the pump causing the N₁ electron to come back up and so the pump causes stimulated emission with N₁>N₀.

 

[mfb]

If they are evenly spaced

That won't happen for realistic setups. And even then, there is no way to get state inversion (apart from statistical fluctuations, those don't count).