# Pertubation and density matrix

1. Aug 2, 2009

### KFC

Hi there, I am reading a text by Robert W. Boyd "Nonlinear optics", in page 228, he used pertubation theory on two-level system and let the steady-state solution of the dynamics equation of density matrix as

$$w = w_0 + w_1 e^{-i\delta t} + w_{-1}e^{i\delta t}$$

where $$w=\rho_{bb} - \rho_{aa}$$ is the inversion of population between level b (excited) and level a (ground), $$\Omega+\delta$$ is the frequency of the probe field, $$\Omega$$ is the frequency of the pump field. I have few questions

1) the author said the solution shown above is steady-state solution, but why it is time-dependent?

2) we know that, $$w$$ must be real, so $$w_1=w_{-1}^*$$, but if for $$w_1$$ only, is there any physical significance? Why we have to consider the first-order solution like that? What contribution of $$w_1$$ and $$w_{-1}$$ made?

2. Aug 3, 2009

### weejee

1) Steady-state solutions need not be independent of time. (Consider the case of a forced harmonic oscillator)

3. Aug 3, 2009

### KFC

Yes, in that case, it is time dependent. But this is so confusing. In wiki, about steady state (http://en.wikipedia.org/wiki/Steady_state), it puts : "A system in a steady state has numerous properties that are unchanging in time. ..." So if the properties are unchanging in time, why it can be time related?

Moreover, for the problem I mentioned above, it is about several first-order ODEs. It obtins the steady soultions by solving the equations when the time derivative of the variables equal to ZERO. If the time derivative of the variables equal ZERO, the only possibiliy is the solution is time-independent or constant, isn't it?