Exclusion principle for photons: Spin-statistic selection rules for multiphoton transitions in atomic systems

We establish the existence of spin-statistic selection rules (SSSRs) for multiphoton transitions with equal photons in atomic systems. These selection rules are similar to those for systems of many equivalent electrons in atomic theory. The latter are a direct consequence of the Pauli exclusion principle. In this sense, the SSSRs play the role of the exclusion principle for photons: they forbid some particular states for the photon systems. We established several SSSRs for few-photon systems. (i) First rule: two equivalent photons involved in any atomic transition can have only even values of the total angular momentum, $J$. This selection rule is an extension of the Landau-Yang theorem to the photons involved in atomic transitions. (ii) Second rule: three equivalent dipole photons involved in any atomic transition can have only odd values of the total angular momentum, $J=1,3$. (iii) Third rule: four equivalent dipole photons involved in any atomic transition can have only even values of the total angular momentum, $J=0,2,4$. We also suggest a method for a possible experimental test of these SSSRs by means of laser experiments.

Figure 1

Photon frequency distributions for the two-photon transitions ${\left(1s2s\right)}^1{S}_0\to {\left(1s\right)}^{21}{S}_0+2\gamma \left(E1\right)$ (left panel) and ${\left(1s2s\right)}^3{S}_1\to {\left(1s\right)}^{21}{S}_0+2\gamma \left(E1\right)$ (right panel) in He-like U (nuclear charge $Z=92$). The transition rates $\frac{dW}{d\omega }$ in units ${\text{s}}^{-1}$ are plotted vs the photon frequency $\omega$ in units $\omega /{\Delta }_0$ and $\omega /{\Delta }_1$. The values ${\Delta }_0,{\Delta }_1$ correspond to the energy intervals ${\Delta }_0=E\left(2{}^1{S}_0\right)-E\left(1{}^1{S}_0\right)$ and ${\Delta }_1=E\left(2{}^3{S}_1\right)-E\left(1{}^1{S}_0\right)$, respectively. The calculations are performed fully relativistically, with Dirac one-electron wave functions and relativistic expressions for the electromagnetic vector potentials (photon wave functions). The Dirac wave functions for an extended nucleus described by the Fermi distribution were employed. The interelectron interaction was fully neglected; the expected error is about $1/Z$, i.e., about $1\%$ for $Z=92$. The one-electron wave functions for the initial and final two-electron states were coupled to present $1{}^1{S}_0,2{}^1{S}_0$, and $2{}^3{S}_1$ states, respectively. Summation over the full set of two-electron states taken as products of one-electron Dirac states was performed within the B-spline approach. The calculations were carried out in two relativistic “forms,” corresponding to the nonrelativistic “length” and “velocity” forms [18]; the results coincide with 12 digits. For the ions with lower $Z$ values and for the neutral He atom, the accuracy for $W$ values obtained from calculations with the neglect of interelectron interaction becomes poorer, but the difference between the frequency distributions for $2{}^1{S}_0\to 1{}^1{S}_0+2\gamma \left(E1\right)$ and $2{}^3{S}_1\to 1{}^1{S}_0+2\gamma \left(E1\right)$ transitions remains qualitatively the same.

Figure 2

Photon frequency distribution for the ${\left(1s3d\right)}^3{D}_3\to {\left(1s\right)}^{2;1}{S}_0+2\gamma \left(E2\right)$ two-photon transition. Notations, units, etc., are the same as in Fig. 1. ${\Delta }_3$ denotes the energy difference ${\Delta }_3=E\left(3{}^3{D}_3\right)-E\left(1{}^1{S}_0\right)$. Calculation is performed for a He-like U ion ($Z=92$). The total transition probability (integral value) is $W_{2\gamma }\left(3{}^3{D}_3-1{}^1{S}_0\right)=3.3299\times {10}^4$ ${\text{s}}^{-1}$ both in the relativistic “length” and “velocity” forms, respectively.

Figure 3

Three-dimensional plot for frequency distribution of the transition rate $2{}^3{P}_2\to 1{}^1{S}_0+3\gamma \left(E1\right)$ in He-like uranium. On the vertical axis, the transition rate $\frac{dW}{d{\omega }_{1}d{\omega }_2}$ in ${\text{s}}^{-1}$ is plotted; on the horizontal axes, the photon frequencies are plotted in units ${\omega }_1/{\Delta }_2$ and ${\omega }_2/{\Delta }_2$, where ${\Delta }_2$ denotes the energy difference ${\Delta }_2=E\left(2{}^3{P}_2\right)-E\left(1{}^1{S}_0\right)$. The lowest (zero) point is the point with coordinates ${\omega }_1/{\Delta }_2={\omega }_2/{\Delta }_2=1/3$ at the bottom of the “pit” in the frequency distribution for the transition rate that arises due to SSSR-2. The calculations were done neglecting the interelectron interaction.

Figure 4

Three-dimensional plot for frequency distribution of the transition rate $2{}^3{P}_1\to 1{}^1{S}_0+3\gamma \left(E1\right)$ in He-like uranium. All the details are the same as in Fig. 3. ${\Delta }_1=E\left(2{}^3{P}_1\right)-E\left(1{}^1{S}_0\right)$. In this case, there is no SSSR-2 induced “pit” in the frequency distribution for the transition rate. The calculations are performed under the same approximation as for Fig. 3. The total transition probability (integral value) is $W_{3\gamma }\left(2{}^3{P}_1-1{}^1{S}_0\right)=13.825270\times {10}^6$ ${\text{s}}^{-1}$, $W_{3\gamma }\left(2{}^3{P}_1-1{}^1{S}_0\right)=13.825246\times {10}^6$ ${\text{s}}^{-1}$ in the relativistic “length” and “velocity” forms, respectively.