Why is two level laser impossible even with non-optical pump

[misko]

I think I understand why two level laser is not possible if we use optical pumping.

However, I don't understand why we can't create laser with two energy levels that are pumped by non optical pump? Why we can't create population inversion that way? For example, if we could somehow pump electrons from level 1 to level 2 with some collision processes then we could create population inversion on the level 2, right? Then electrons from level 2 would be able to go back to level 1 with stimulated emission. Why is this not possible?

 

[DrDu]

I think the first MASER, the ammonia maser is based on a two-level system. The molecules in the higher energetic state are separated passing them through a quadrupole field, so that afterwards you have inversion of the occupation.

 

[misko]

Then why is the statement "2 level laser is not possible" is repeated over and over again?

 

[DrDu]

Folklore

 

[misko]

So if professor on the exam asks me "why can't we create lasers with only 2 energy levels?" and if I answer "oh actually we can, I know that every book on this topic says we can't but don't worry about that it's just folklore", what grade do you think I will get?

Anyone else want to contribute to the discussion?
I still don't understand it.

 

[blue_leaf77]

Optically pumped two level system cannot give rise to population inversion because the the stimulated emission and absorption rate between both levels are equal. Denoting the upper population as $N_2$ and the lower one $N_1$, the rate equation for $N_2$ is
$$
\frac{dN_2}{dt} = B_{12}I(\omega)N_1 - B_{21}I(\omega)N_2 - A_{21}N_2
$$
where $B_{12}I(\omega)$ the transition probability corresponding to the absorption from the lower to upper states (note that it is proportional to the intensity of the light at the resonant frequency between the two levels because we have assumed optical pumping), $W_{21}N_2I(\omega)$ the transition probability corresponding to the stimulated emission from the upper to lower states, and $A_{21}$ spontaneous emission probability. Now, we have $B_{12}N_1 = B_{21}$ and let's seek out the expression of ratio of the populations at the steady state
$$
\frac{dN_2}{dt} = 0 = B_{12}I(\omega)N_1 - B_{21}I(\omega)N_2 - A_{21}N_2
$$
which can be rearranged to give
$$
\frac{N_2}{N_1} = \frac{B_{12}I(\omega)}{B_{21}I(\omega) + A_{21}} < 1
$$
Thus, optical pumping cannot yield population inversion in a two level atom.
I acknowledge that this part of laser physics has also cast an enigmatic problem in my mind because the above steps was derived under optical pumping condition, this suggests that other pumping methods such as electron collision pumping may be able to lead to population inversion in two level systems. Other sources like http://spie.org/publications/optipedia-pages/press-content/fg12/fg12_p03_optical_pumping and http://physics.stackexchange.com/questions/72080/lasing-in-a-2-level-system always specified optical pumping to seemingly safe their argument in case that later two level population inversion can be achieved via non-optical pumping. However, I just found in page 292 in https://books.google.de/books?id=x3VB2iwSaxsC&pg=PA290&lpg=PA290&dq=population+inversion+with+two+level+system&source=bl&ots=21GKcwzCb9&sig=rBn8cKwKWz221pJEFTupKnW_FuI&hl=en&sa=X&ved=0ahUKEwin8tnv597KAhUMFiwKHQcMDfwQ6AEIUzAI#v=onepage&q=population inversion with two level system&f=false, that at least for electron collision pumping two level inversion is not possible either due to the Boltzmann distribution between the two populations,
$$
\frac{N_2}{N_1} = e^{-\Delta E/(kT_e)}
$$
which cannot even reach unity unless the electron temperature $T_e$ becomes infinity.

 

[DrClaude]

Thus, optical pumping cannot yield population inversion in a two level atom.

There are some special exceptions. Here is a quote from W. Demtröder, Atoms, Molecules and Photons:

Under special conditions it is also possible to achieve inversion for a short time in a two-level system, if the pumping time is short compared to all relaxation times of the system and even shorter than the Rabi oscillation time T_R = π · h/(M_ik · E(ν_ik )), where M_ik is the matrix element for the transition i → k and E is the electric field vector of the pump wave. These conditions, however, apply only to very few real systems that are specially designed.

 

[blue_leaf77]

There are some special exceptions. Here is a quote from W. Demtröder, Atoms, Molecules and Photons:

I have heard that excimer laser effectively operates between only two levels, I think this laser falls into the category described by Demtroeder.