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(Avoided)

The spin level diagrams vs. field of the previous pages did occasionally cross each other. This special condition makes the two level degenerate. If any non secular part of the Hamiltonian could be neglected in the level calculation, far from the crossing, that interaction might resolve the degeneracy, mix the two levels and yield a very different muon behaviour close to the crossing region.

This is indeed what happens in two cases previously observed, namely that of anisotropic muonium and that of the radical with additional nuclear moments, but let us consider first a generic two level system controlled by magnetic field.

The Hamiltonian in the crossing region will look like:

{$$ \frac {\cal H} h = \begin{bmatrix} \nu_a-\gamma_a B & \epsilon \\ \epsilon & \nu_b+\gamma_b B \end{bmatrix}) $$}

with {$\epsilon\ll \nu_a,\nu_b, \gamma_a B, \gamma_b B$}. Far from the crossing to first order only

{$$ \frac {{\cal H}_0} h = \begin{bmatrix} \nu_a-\gamma_a B & 0 \\ 0 & \nu_b+\gamma_b B \end{bmatrix}$$}

matters and the eigenstates are

{$$ |1\rangle=\begin{bmatrix}1\\ 0\end{bmatrix}\qquad|2\rangle=\begin{bmatrix}0\\ 1\end{bmatrix}.$$}

Close to the crossing field {$B =\frac {\nu_a-\nu_b} {\gamma_a+\gamma_b}$} the two eigenstates are nearly degenerate and it is the non secular part of the Hamiltonian,

{$$ \frac {\cal H^\prime} h = \begin{bmatrix} 0 & \epsilon \\ \epsilon & 0\end{bmatrix} $$}

which determines the new eigenstates. At the crossing field:

{$$ |1^\prime\rangle=\frac 1 {\sqrt{2}} \begin{bmatrix}1\\ 1\end{bmatrix}\qquad|2^\prime\rangle=\frac 1 {\sqrt{2}}\begin{bmatrix}\quad1\\- 1\end{bmatrix}.$$}

The plot shows that, actually, the two levels (green and blue curves) repell each other and never cross. On crossing the ALC condition, however, the green curve show that {$|1\rangle$} changes into {$|2\rangle$} and vice-versa. The red curve (notice the different y-scale) is the longitudinal polarization (see below).

Now suppose that this happens in the spin levels of the muon state, and the original eigenstates 1,2 correspond to, e.g., muon spin up and down, respectively, along the field direction, {$\sigma_{z,\mu}|1,2\rangle=\pm|1,2\rangle$}. Then the mixed states, {$1^\prime,2^\prime$} at crossing will not be eigenstates of {$\sigma_{z,\mu}$}, hence a fraction of the polarization will precess and the constant term, typically equal to 1 in high fields, will decrease. A time-averaged measurement will result in a average polarization drop. This is the signature of an Avoided Level Crossing, as it shown by the red curve in the lower part of the figure.

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LevelCrossing

(Avoided)