General Two-State Systems and Rabi Oscillations
I've been working on this for quite a while now and can't seem to figure it out. I am concerned with the dynamics of a general two-state system, specifically the case of a particle in a constant magnetic field coupled with a time dependent magnetic field.
Let's start with the simple case, just a particle in a static magnetic field, say $\vec{B} = B_0 \hat{k}$. Then, the Hamiltonian for the system will come from the interaction of the dipole moment with the field, i.e. $$\hat{H} = -\hat{\mu} \cdot \vec{B}$$ We know that the magnetic dipole moment is proportional to the intrinsic spin of the system as $$\hat{\mu} = \gamma \hat{S},$$ where $\gamma$ is the gyromagnetic ratio. Ok, our Hamiltonian is $$\hat{H} = -\gamma B_0 \hat{S}_z = \omega_0 \hat{S}_z,$$ where $\omega_0$ is known as the Larmor frequency. The dynamics of this is very easy, the spin just undergoes a precession about the $z$-axis. If we start in the state, say $$|\psi(0)> = |+x> = \frac{1}{\sqrt{2}}(|+z> + |-z>),$$ then the evolution will be $$|\psi(t)> = e^{-i \omega_0 t/2}\frac{1}{\sqrt{2}}(|+z> + e^{i \omega_0 t}|-z>).$$ We see that the probability of a "flip" in state to $|-x>$ is given by $$|<-x|\psi(t)>|^2 = |\frac{1}{2}(<+z|-<-z|)(|+z> + e^{i \omega_0 t}|-z>)|^2 = \boxed{\sin^2\Big(\frac{\omega_0}{2}t\Big)}.$$ Ok, we see that it flips at times $T_n = \frac{\pi}{\omega_0}(2n-1)$, where $n$ is just a positive integer. This is all good and well. 649
Now we add a time dependent magnetic field of the form $$\vec{B} = B_1(\cos(\omega t) \hat{i} + \sin(\omega t) \hat{j}).$$ This is an oscillating magnetic field on top of the previous magnetic field in the $z$ direction. Our new Hamiltonian becomes (I am dropping the hats on the operators...) $$H = H_0 + H_1 = \omega_0 S_z + \omega_R(\cos(\omega t) S_x + \sin(\omega t) S_y),$$ where $\omega_R$ is known as the Rabi frequency. This is where I am stuck. We now have a Hamiltonian with spins in every direction (time dependent too), so we cannot simply construct the time evolution operator and act on the initial state as we did before. We have a couple of more options to choose from. I tried constructing the matrix representation of this operator in terms of the basis states (which we choose to be $|\pm z>$). The resulting Hamiltonian will be $$H \doteq \frac{\hbar}{2} \begin{pmatrix} \omega_0 & \omega_R e^{-i\omega t} \ \omega_R e^{i \omega t} & -\omega_0 \end{pmatrix}$$ In fact, this is the most general Hamiltonian for a two-state system. But I am still stuck in that I have no clue how to find the eigenvalues and eigenstates of this Hamiltonian. 526