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I am trying to understand better processes that can cause "spin-flipping" to occur for magnetic moments in a bias field along the z-axis, B

_{z}.

The application of a oscillating field in the xy-plane is well known to me. If the oscillating frequency is at the Larmor frequency, [tex]\omega_L = \gamma B[/tex] then the probability of a spin-flip will be given by [tex]sin(\omega_R t)[/tex], where [tex]\omega_R[/tex] is the Rabi frequency and t is the duration time the field is applied for.

I am trying to understand the Landau-Zener transitions. When I try to search google, I just get lots of hits for quantum qubit transitions. If I am interpreting correctly, the LZ transition is almost just a special case of the Rabi transition if the perturbation is just for a short time? The excitation is normally treated as a perturbing field in the x-axis B_x which varies linearly. The spin-flip probability is something like [tex]exp(-2 \pi \Gamma)[/tex], where [tex]\Gamma = \frac{B_x^2}{d\DeltaE/dt}[/tex]. [tex]\DeltaE[/tex] is the size of the energy gap and for the magnetic moment example, it is [tex] 2 \mu B[/tex]. Am I right?

I am confused by this since why does the energy gap change if you're only applying a small perturbation in B

_{x}?

Also, how can one treat a problem where the variations is in the z-axis? Something like a [tex]\delta_{B_Z}[/tex]? Can this induce spin-flips? Can I use the Landau-Zener transition for this since [tex]\Delta_E[/tex] changes? What if this extra [tex]\delta_{B_Z}[/tex] causes the total B

_{Z}to go through zero?