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Two Level System with thermal population

  1. Nov 10, 2013 #1
    Hello,

    the task is the following:

    Consider a two-level system with thermal population.

    a) Show that the rate equation for the state [itex]N_2[/itex] is the following:

    [itex]\frac{dN_2}{dt}=w(N_1-N_2)-\frac{N_2-N_2^e}{\tau}[/itex]

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

    [itex]N_2^p[/itex] 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 [itex]N_2(t)[/itex] when the system is stimulated by a constant monochromatic signal with spectral density I and frequency [itex]h\nu=(E_2-E_1)[/itex]. For t=0 only thermal population exist.

    My ideas are the following:

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

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

    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: [itex]N_1[/itex] 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.
     
  2. jcsd
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