I'm trying to solve the driven Two-Level System (TLS or qubit) question using a Fourier transform of the Schrodinger equation (SHE), but I'm getting stuck on solving the equation.
Given Hamiltonian $$H=\left( \begin{array}{cc} -\frac{\omega _0}{2} & \frac{1}{2} V e^{i t \omega _D} \\ \frac{1}{2} V e^{-i t \omega _D} & \frac{\omega _0}{2} \\ \end{array} \right)$$
and plugging into SHE:
$$i\frac{ d }{\text{dt}}\left( \begin{array}{c} c_a(t) \\ c_b(t) \\ \end{array} \right)=H \left( \begin{array}{c} c_a(t) \\ c_b(t) \\ \end{array} \right)$$
I get a set of two coupled 1st Order differential equations which i then Fourier transform and use the derivative rule and the shift rule to get:
\left( \begin{array}{c} -\omega c_a(\omega )=\frac{1}{2} V c_b\left(\omega -\omega _D\right)-\frac{1}{2} \omega _0 c_a(\omega ) \\ -\omega c_b(\omega )=\frac{1}{2} V c_a\left(\omega +\omega _D\right)+\frac{1}{2} \omega _0 c_b(\omega ) \\ \end{array} \right)
If I shift the $c_b$ equation:
$$-\omega c_b\left(\omega -\omega _D\right)=\frac{1}{2} V c_a\left(\omega _D-\omega _D+\omega \right)++\frac{1}{2} \omega _0 c_b\left(\omega -\omega _D\right)$$
and solve for term i can plug back into the first equation: $$c_b\left(\omega -\omega _D\right)=\frac{V c_a(\omega )}{2 \omega +\omega _0}$$
and solve for $c_a$:
$$c_a(\omega )=\frac{V^2 c_a(\omega )}{(2 \omega )^2+\omega _0^2} $$
So it looks like a lorentzian function with a width of the resonance frequency and centered at 0 multiplies $c_a$, but $c_a$ cancels unless it is 0? or some delta function?
This is where I do not understand how to proceed to solve the problem further? Is part of the challenge that I am solving the problem without boundary conditions, just trying to find the steady state?