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## 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 E

_{0}and three higher energy levels E

_{1}<E

_{2}<E

_{3}.

Consider the case where the onl radiative decay in this system occurs between levels E

_{1}and E

_{0}

Imagine that the active medium is optically pumped by a monochromatic source with a frequency ω

_{pump}=(E

_{1}-E

_{0})/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 N

_{0}<N

_{1}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?