Asymmetry in emission of photons with left- and right-hand circular polarizations in two-photon decay

Two-photon decay of the $2s$ state in H-like ions is investigated. We report that asymmetry in the emission of photons with left- and right-hand circular polarizations can be observed in this transition if the initial state has a certain polarization. This asymmetry can be used to measure the polarization of ion beams.

Figure 1

Geometry of the two-photon decay. The vectors ${\mathit{k}}_1$ and ${\mathit{k}}_2$ are the momenta of the emitted photons. The vector $\mathit{\xi }$ defines the axis of the quantization of the electron total angular momentum. The left and right pictures show these vectors before and after the inversion operation.

Figure 2

Feynman graphs corresponding to the direct (a) and exchange (b) terms of the two-photon decay. The letters $i$ and $f$ denote the initial and final states of the electron. The wavy lines describe the emitted photons with momenta $k_1, k_2$ and polarizations ${\lambda }_1, {\lambda }_2$.

Figure 3

The differential transition probability for uranium ion as a function of the angle between the momenta of the emitted photons. The left and right columns correspond to $x=0.5$ and $x=0.25$, respectively. The upper and lower rows demonstrate differential transition probability for two different angles $\alpha$: ${45}^{\circ }$ and ${90}^{\circ }$. The black solid curves present the results for initial projection $m_i=-1/2$ and photon polarizations ${\lambda }_1={\lambda }_2=“+”$ (or $m_i=+1/2, {\lambda }_1={\lambda }_2=“-”$). The red dashed curves gives the results for $m_i=+1/2$ and ${\lambda }_1={\lambda }_2=“+”$ (or $m_i=-1/2, {\lambda }_1={\lambda }_2=“-”$).

Figure 4

The parameter of the asymmetry $\mathrm{\Xi }$ [see Eq. (7)] for uranium ion as a function of the angle between the momenta of the emitted photons. The left and right plots correspond to $x=0.5$ and $x=0.25$, respectively.

Figure 5

The parameter of the asymmetry $\mathrm{\Xi }$ as a function of the atomic number of the ion $Z$ for $x=0.25, \theta ={90}^{\circ }$, and $\alpha ={90}^{\circ }$.

Figure 6

The differential transition probability $M_{m_i}^{{\lambda }_1}\left(\alpha \right)$ [see Eq. (8)] for uranium ion as a function of the angle between the momenta of the emitted photons. The black solid curve presents the results for angle $\alpha ={90}^{\circ }$, initial projection $m_i=+1/2, x=0.003$ (${\omega }_1=306$ eV), and photon polarization ${\lambda }_1=“+”$. The red dashed curve shows the same, but for ${\lambda }_1=“-”$.

Figure 7

The differential transition probability [given by Eq. (11)] as a function of $x$ for uranium ion and $\alpha ={90}^{\circ }$. The black solid curve presents the results for initial projection $m_i=-1/2$ and photon polarizations ${\lambda }_1={\lambda }_2=“+”$ (or $m_i=+1/2, {\lambda }_1={\lambda }_2=“-”$). The red dashed curve gives the results for $m_i=+1/2$ and ${\lambda }_1={\lambda }_2=“+”$ (or $m_i=-1/2, {\lambda }_1={\lambda }_2“-”$).