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Population Inversion & Pumping

If we have a large number of atoms (say No) in thermal equilibrium in different energy states obeys the Maxewell Boltzmann Statistics. 
If we assume that at temperature T Kelvin, the population in Energy state E1 & E2 is N1 & N2. Then according to Boltzmann law of distribution :

The negative exponent of this equation indicates that N1 > N2 at equilibrium (because E2 > E1 ).
It means that the number of atom in higher energy states (N2) is less than number of atoms in lower energy states(N1) as shown in fig.
Under such circumstances, the probability of stimulated emission is much less than probability of spontaneous emission ,but In spontaneous emission, the photons emitted by various atoms are in random phase while in stimulated emission they are in the phase fullfiling the requirements of LASER.
The basic requirements of LASER,  we must have :

i) The higher energy state should have a longer mean life time, i.e. it should be a meta stable state.
ii) The number of atoms in higher energy state E2 must be greater than E1, i.e.
           E1 > E2     but    N2 > N1
The situation in which the number of atoms in higher energy state is greater than that in lower energy stateis called population inversion.


The process of achieving population inversion is called pumping.
There are various types of pumping process but the most common is optical pumping used in Ruby LASER.
In optical pumping, the population inversion is about by three level scheme. To understand the working, Consider a material whose atoms when raised from ground state even to short lived excited state E3( which has mean life of 10^-8 s) donot fall back to state even because transition from E3 to E1 is forbidden by selection rule, but transition from E3 to an intermediate meta stable state E2 (which has a longer life of 10^-3 s)  is allowed.
When the atom of substance absorb a photon of frequency :

It goes to E3 state by stimulated absorption process.Some of the atom decay spontaneously to metastable state E2 ,where they live much longer time of 10^-3s(which is 10^5 times of life of E3.
This leads to a situation that state E2 has more atoms than state E1 which means that a population inversion has been achieved.
After achieving a population inversion, if the atoms in meta stwble state E2 are exposed to photons of frequency (v)
       In this process,  a single photon produces a very large number of identical photons, so that radiation beam is now highly amplified. 
Thus an intense coherent beam of light in direction of incident photons emerges from the substance.

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