Long-range interactions of hydrogen atoms in excited states. III. nS-1S interactions for n >= 3

TL;DR

This paper investigates the long-range van der Waals and Casimir-Polder interactions of excited hydrogen atoms in nS states with ground-state hydrogen, revealing how these interactions scale with quantum number and the importance of quasi-degenerate states.

Contribution

It provides detailed calculations of C_6 coefficients for nS states (n=3 to 12), highlighting their growth and the significance of quasi-degenerate states in long-range interactions.

Findings

C_6 coefficients grow roughly with n^4, reaching over 240,000 for n=12.

Nonretarded van der Waals interactions dominate at short distances, while quasi-degenerate states influence intermediate ranges.

The traditional 1/R^7 Casimir-Polder asymptotics are not applicable to excited-state interactions.

Abstract

The long-range interaction of excited neutral atoms has a number of interesting and surprising properties, such as the prevalence of long-range, oscillatory tails, and the emergence of numerically large can der Waals C_6 coefficients. Furthermore, the energetically quasi-degenerate nP states require special attention and lead to mathematical subtleties. Here, we analyze the interaction of excited hydrogen atoms in nS states (3 <= n <= 12) with ground-state hydrogen atoms, and find that the C_6 coefficients roughly grow with the fourth power of the principal quantum number, and can reach values in excess of 240,000 (in atomic units) for states with n = 12. The nonretarded van der Waals result is relevant to the distance range R << a_0/alpha, where a_0 is the Bohr radius and alpha is the fine-structure constant. The Casimir-Polder range encompasses the interatomic distance range a_0/alpha << R << hbar c/L, where L is the Lamb shift energy. In this range, the contribution of quasi-degenerate excited nP states remains nonretarded and competes with the 1/R^2 and 1/R^4 tails of the pole terms which are generated by lower-lying mP states with 2 <= m <= n-1, due to virtual resonant emission. The dominant pole terms are also analyzed in the Lamb shift range R >> hbar c/L. The familiar 1/R^7 asymptotics from the usual Casimir-Polder theory is found to be completely irrelevant for the analysis of excited-state interactions. The calculations are carried out to high precision using computer algebra in order to handle a large number of terms in intermediate steps of the calculation, for highly excited states.