Consider a two level system with a Landau-Zener Hamiltonian of the form $$\hat{H}=\begin{pmatrix}v t&\beta\\\beta&-v t\end{pmatrix}.$$
The Landau-Zener formula provides a closed form for the probability amplitude of jumping to the excited state, given by $P={\rm e}^{-2\pi\beta^2/v}$, which is valid in the limit $t_0\rightarrow\infty$, $t_f\rightarrow+\infty$.
However, consider a different problem with the same Hamiltonian, but now the initial and final time are chosen as to satisfy $v t_0=\alpha_0$ and $v t_f=\alpha_f$, where $\alpha_0$ and $\alpha_f$ are fixed. This takes us to $t_0=\frac{\alpha_0}{v}$ and $t_f=\frac{\alpha_f}{v}$. In this case, $v$ can be interpreted as the velocity of the system to undergo the transition of the diagonal terms from $\alpha_0$ to $\alpha_f$. It takes to the system $\Delta t=\frac{\alpha_f-\alpha_0}{v}$ to undergo from $\alpha_0$ to $\alpha_f$.
Now, my interpretation is that increasing values of $v$ lead to a larger violation of the adiabatic theorem, yielding higher probability of excitation. On the other hand, small values of $v$ imply that the evolution is performed more slowly, and this should lead to a decreasing probability of excitation.
However, when I plot the probability of excitation $P$ as a function of $v$ I find that there are some oscillations with a certain frequency, but I don't see where these come from, as to my understanding and according to the previous analysis, $P(v)$ should be a monotonically increasing function, as a higher velocity of the system should lead to higher probability of excitation.
My questions are therefore:
- (a) Where do these oscillations in $P(v)$ come from?
- (b) This oscillations imply that there are situations where $v_1>v_2$ and $P(v_1)<P(v_2)$, i.e. a higher velocity lead to a lower probability of excitation. To me, this doesn't make any sense. Is there any physical/intuitive explanation of this?
