Two Level System with thermal population

[ktolaus]

Hello,

the task is the following:

Consider a two-level system with thermal population.

a) Show that the rate equation for the state $N_2$ is the following:

$\frac{dN_2}{dt}=w(N_1-N_2)-\frac{N_2-N_2^e}{\tau}$

$w=B_{12}\rho({\nu}) \,\,\, N_2=N_2^p+N_2^e\,\,\, N_1=N-N_2$

$N_2^p$ is the portion of the total population caused by pumping.

b) Show, that using such a system it is impossible to build a CW Laser.

c) Calculate the progress of $N_2(t)$ when the system is stimulated by a constant monochromatic signal with spectral density I and frequency $h\nu=(E_2-E_1)$. For t=0 only thermal population exist.

My ideas are the following:

a) $\frac{dN_2}{dt}$ is just the sum of absorption, stimulated emission and spontaneous emission. Spontaneous emission occurs only from the the "pumped portion":
$\frac{dN_2}{dt}=B_{12}\rho({\nu})N_1-B_{21}\rho({\nu})N_2-A_{21}N_2^p$
$\frac{dN_2}{dt}=B_{12}\rho({\nu})N_1-B_{12}\rho({\nu})N_2-A_{21}(N_2-N_2^e)$
using $A_{21}=\frac{1}{\tau}$ leads to:
$\frac{dN_2}{dt}=w(N_1-N_2)-\frac{N_2-N_2^e}{\tau}$

b) I'm not sure, but I think CW is only possible if $\frac{d^2N_2}{dt^2}=0$:
$\frac{d^2N_2}{dt^2}=B_{12}\rho(\frac{dN_1}{dt}-\frac{dN_2}{dt})-\frac{1}{\tau}\frac{dN_2}{dt}+\frac{1}{\tau}\frac{dN_2^e}{dt}$
$N_2^e$ should be independent on time. Using $\frac{dN_1}{dt}=-\frac{dN_2}{dt}$ leads to:
$0=-2B_{12}\rho\frac{dN_2}{dt}-\frac{1}{\tau}\frac{dN_2}{dt}$

This can't be 0 since the left term depends on the frequency but the right term doesn't.

c) I tried several things but none of them were promising. The problem is: $N_1$ depends on the time.
I really would appreciate it if you gave me a hint.

Sorry for my english, but it's not my mother tongue.

 

[member 553083]

Thanks and best regardsTo solve part c), you can use the equation from part a):\frac{dN_2}{dt}=w(N_1-N_2)-\frac{N_2-N_2^e}{\tau}This equation can be written as: \frac{dN_2}{dt}=wN_1 - (w+\frac{1}{\tau})N_2 + \frac{N_2^e}{\tau}Now, we can use the initial conditions given in the question: N_2(0)=N_2^e and \frac{dN_2}{dt}(0)=0We can also assume that N_1 is constant over time. Using these conditions and the equation above, we can solve for N_2 as a function of time: N_2(t)=N_2^e + (N_1-N_2^e)e^{-(w+\frac{1}{\tau})t}This is the solution for the population of state N_2 when stimulated by a constant monochromatic signal.