- #1

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## Homework Statement

## Homework Equations

## The Attempt at a Solution

Very confused by this problem. For one thing, it doesn't specify if there is or isn't any light present to drive the stimulated emission/absorbtion. I guess there's no reason to assume that there is no light - but since the question is asking about lifetimes, that would seem more sensible...[plus introducing an unspecified ##\rho(\omega_{0})## seems odd]

Assuming there is no light involved, then this is just a three level system, with three sets of coupled differential equations describing the behaviour - let ##N_{i}## be the number of atoms in state ##i##:

(1) $$ \frac{dN_{A}}{dt} = -(A_{ac}+A_{ab})N_{A} $$

(2) $$ \frac{dN_{B}}{dt} = A_{ab}(N_{A} -N_{B}) $$

(3) $$ \frac{dN_{C}}{dt} = A_{ac}N_{A}+A_{bc}N_{B} $$

[Though,one of them is made redundant by the fact that total particle number must be constant.]

We can easily solve the first equation:

$$ N_{A} = N_{A0}exp\bigg[ - (A_{ac}+A_{ab})t) \bigg] $$

By substituting this into the equation (2), we can then solve for ##N_{B}##:

$$ \frac{dN_{B}}{dt} +N_{B}A_{ab} = A_{ab}N_{A0}exp\bigg[ - (A_{ac}+A_{ab})t) \bigg] $$

So by using the standard method of integrating factors:

$$ N_{B} = Cexp(-A_{ab}t)-\frac{A_{ab}}{A_{ac}}N_{A0}exp(-A_{ac}+A_{ab}t) $$

But how am I meant to define a lifetime for that?, Assuming ##N_{B}(t=0)=0##, i.e:

$$ N_{B} =\frac{A_{ab}}{A_{ac}}N_{A0}exp(-A_{ab}t)\bigg[1-exp(-A_{ac}t)\bigg]$$

Do we take the dominant exponential to define the lifetime?