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Background theory and equations

Background theory and equations

The discussion here follows roughly the discussions given by de Angelis et al 2011 and Marsh et al 2017. The method in the code most closely follows the former, but notation often matches the latter.

Alpro deals with relativistic ALPs, such that the relevant propagation equation (in the z direction) is that of a reduced Schrodinger-like equation, which is first-order and given by

\[\left( i \frac{d}{d z} + \omega + {\cal M}_0 \right) \, \psi (z) = 0.\]

where \(\omega\) is the beam energy, \(\psi (z) = (A_x, A_y, a)\) is the beam photon-ALP state vector and \({\cal M}_0\) is the mixing matrix, given by

\[\begin{split}{\cal M}_0 = \begin{bmatrix} \Delta_{xx} & \Delta_{xy} & \Delta^{x}_{ a \gamma} \\ \Delta_{yx} & \Delta_{yy} & \Delta^{y}_{a \gamma} \\ \Delta^{x}_{a \gamma} & \Delta^{y}_{\rm a \gamma} & \Delta_{a a} \\ \end{bmatrix}\end{split}\]

where we neglect the Faraday rotation terms \(\Delta_{yx}\) and \(\Delta_{xy}\), and the other terms are given by (here quoted for the \(x\) component)

\[\begin{split} \Delta^{x}_{ a \gamma} = \frac{g_a B_x}{2} \\ \Delta_{a a} = -\frac{m_a^2}{2 \omega} \\ \Delta_{xx} = \Delta_{p} = -\frac{\omega_p^2}{2\omega} \\\end{split}\]

where \(m_a\) and \(g_a\) are the ALP mass and coupling constant, and are the fundamental ALP parameters in the problem. \(\omega_p\) is the normal plasma frequency. The beam equation above resembles a Schrodinger equation with Hamiltonian \(-(\omega + {\cal M}_0)\). We now define the transfer matrix \(U_0(z, z_0)\) which is the solution of the beam equation with initial condition \(U_0=1\). Then, the state vector at a given distance \(z\) is given by

\[\psi (z) = U_0 (z, z_0) \, \psi (z_0)\]

Idealised Case: B Aligned with y axis

To obtain \(U_0\), we first consider the idealised case in which we choose the magnetic field to be aligned with the (we will generalise this result later). In this case the off-diagonal \(\Delta^{x}_{ a \gamma}\) terms disappear and the mixing matrix becomes

\[\begin{split}{\cal M}_0 = \begin{bmatrix} \Delta_{p} & 0 & 0 \\ 0 & \Delta_{p} & \Delta^{y}_{a \gamma} \\ 0 & \Delta^{y}_{\rm a \gamma} & \Delta_{a a} \\ \end{bmatrix}\end{split}\]

This matrix can be diagonalised and it’s Eigenvalues and Eigenvectors found (see e.g. Appendix A of de Angelis et al 2011). The Eigenvalues are

\[\begin{split} \lambda_1 &=& \Delta_{p} \\ \lambda_2 &=& \frac{1}{2} (\Delta_{p} + \Delta_{a a} - \Delta_{\rm osc}) \\ \lambda_3 &=& \frac{1}{2} (\Delta_{p} + \Delta_{a a} + \Delta_{\rm osc}) \\\end{split}\]

where

\[\Delta_{\rm osc} = \left[ (\Delta_{p} - \Delta_{a a})^2 + 4 (\Delta^{y}_{\rm a \gamma})^2 \right]^{1/2}\]

the Eigenvectors are most conveniently expressed in terms of an ALP mixing angle, \(\alpha\), defined as

\[\alpha = \frac{1}{2} \rm{arctan} \left( \frac{2 \Delta^{y}_{\rm a \gamma}} {\Delta_p - \Delta_{a a}} \right)\]

which is a function of \(\omega, B, g, m_a\) and \(\omega_{pl}\). In this case the Eigenvectors \(T_0, T_1, T_2\) are written purely in terms of \(\sin \alpha\) and \(\cos \alpha\) and resemble rotation matrices (they are given in the Eigenvectors subsection at the bottom of this document).

General Case

In the general case, the magnetic field is not aligned with either axis. If we define the angle \(\phi\) as the angle the magentic field \(\vec{B}\) makes with the y axis, then the general mixing matrix is a rotated version of the idealised mixing matrix, i.e.

\[{\cal M}_0 = {\cal R}^T (\phi) {\cal M}^0_0 {\cal R} (\phi)\]

where

\[\begin{split}{\cal R} (\phi) &= \begin{bmatrix} \cos \phi & \sin \phi & 0 \\ \sin \phi & \cos \phi & 0 \\ 0 & 0 & 0 \end{bmatrix}\\\end{split}\]

Similarly, the generalised transfer matrix $U_0$ is given by

\[{\cal U}_0 = {\cal R}^T (\phi) {\cal U}^0_0 {\cal R} (\phi)\]

In this way, the propagation problem can be described completely by just the two angles \(\alpha\) and \(\phi\) and solved using a series of matrix operations.

Code operation

The code follows more or less exactly the above formalism.

Eigenvectors

\[\begin{split}T_{0} &= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\\ T_{1} &= \begin{bmatrix} 0 & 0 & 0 \\ 0 & \sin^2 \alpha & - \sin \alpha \cos \alpha \\ 0 & - \sin \alpha \cos \alpha & \cos^2 \alpha \end{bmatrix}\\ T_{2} &= \begin{bmatrix} 0 & 0 & 0 \\ 0 & \cos^2 \alpha & \sin \alpha \cos \alpha \\ 0 & \sin \alpha \cos \alpha & \sin^2 \alpha \end{bmatrix}\end{split}\]

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