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Frame rotation

 

Consider CRE-filtered beam. The photons of the beam are characterized by its momentum, second quantization axis, Lambda-variable and chi-variable. Introduce laboratory right-hand coordinate system $ \{X, Y, Z\}$ with Y-axis alond the photon second quantization axis and Z-axis along the photon 3-momentum.

The CRE-filtered beam passes through conerefringent element, denoted as CRE′. It has its own coordinate system $ \{X', Y', Z'\}$ and Lambda-parameter $ \Lambda'$ . $ \{X', Y', Z'\}$ is rotated relative $ \{X, Y, Z\}$ about the Z-axis by an angle $ \omega$ .

The passage of CRE-filtered beam through CRE′ can be considered as a passage through one Stern-Gerlach filter. The passage does not change the photon 3-momentum.

Fine splitting neglected

If the fine Poggendorff splitting is neglected, the photon state before the CRE′ filter can be denoted by $ \ket{in}=\ket{\Lambda, \chi}$. CRE′ filter splits the $ \ket{in}$ -beam in two beams, which are in basis states of CRE′. The second quantization axes of both beams is either along (up) or opposite (down) to the Y′- axis. The photons of one of the beams are always in up-state, denote it by $ \ket{ u }=\ket{\Lambda_{u}, \chi_{u}}$ . The photons from the other one can be in a down or in up state depending on the variable $ \sigma=sign(\Lambda' - \Lambda)$ . They are in a down-state if $ \sigma = -1$ and in up-state if $ \sigma = +1$ ; denote this state by $ \ket{\sigma}=\ket{\Lambda_{\sigma}, \chi_{\sigma}}$ .

Transition amplitudes

We are interested for the transition amplitudes from $ \ket{in}$ -state to the final $ \ket{u}$ - and $ \ket{\sigma}$ -states. The experiment demonstrates that:

$\displaystyle \Braket{u \vert in} = cos {\displaystyle \frac{\omega}{2} }$

$\displaystyle \Braket{\sigma \vert in } = sin {\displaystyle \frac{\omega}{2} }$

Photon variables

We are interested for the values of the Lambda- and chi-variables of the photon final states. The experiment demonstrates that:

$\displaystyle \Lambda_{u} = \Lambda + \Lambda'$

$\displaystyle \Lambda_{\sigma} = \vert \Lambda - \Lambda' \vert$

$\displaystyle \chi_{u} \equiv \chi + {\displaystyle \frac{\omega}{2} }\ \ \ (mod \pi)$

$\displaystyle \chi_{\sigma} \equiv \sigma \chi_{u}\ \ \ (mod \pi)$

 

 

May 24, 2006
Author: T Kirilov