I am reading a book introducing basic concept of laser. It is pretty shocking to me that people can generate beam with almost all photons in the same state. In the book, it said that a two-level atoms (with energy level is $\hbar\omega_0$) can absorb a photon from the laser so to excite the atom. The laser with wavevector $k$ and frequency $\omega$. Here is what confusing me. The laser not necessarily have the same frequency as the atom (i.e. $\omega_0 \neq\omega$), so how come the atom absorb a photon which is not of the frequency $\omega_0$? Can it still absorb photon from laser? If so, where is the extra energy ($\hbar\omega - \hbar\omega_0$)gone?
Also, if this is possible, so how does the momentum of the atom changed if it absorb the photon from the laser? I know that if the atom absorb photon of wavevector $k$, then the momentum change on the atom will be $\hbar k$, does it make sense? It seems someone wrong to me because I think the atom can only absorb photon of frequency $\omega_0$ such that the momentum change should be $\hbar k_0$.
In addition, the book introduce a concept of lifetime of the atomic upper state $\tau$. I am trying to understand the force exert on the atom by absorbing one photon. I know that the force is $\Delta p/\Delta t$. Can I say $\tau$ is just how long it takes to make a momentum change if absorb an photon of wavevector $k_0$ so the force the atom experience is $F = \hbar k_0/\tau$ ?