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

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 van der Waals C6 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 C6 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 ≪ a0/α, where a0 is the Bohr radius and α is the fine-structure constant. The Casimir-Polder range encompasses the interatomic distance range a0/α ≪ R ≪ 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 ≫ 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.

Pressure shifts in high-precision hydrogen spectroscopy: II. Impact approximation and Monte-Carlo simulations

“…The M 6 coefficient becomes smaller with the principal quantum number is being increased as recently observed in Ref. [17]. However, the D 6 coefficients increase as the fourth power of the principal quantum number of the excited reference states.…”

Virtual Resonant Emission and Oscillatory Long-Range Tails in van der Waals Interactions of Excited States: QED Treatment and Applications

“…The very-long-range, oscillatory tail of the van der Waals interaction is relevant only for very large interatomic distances. This conclusion holds for nD-1S interactions as well as nS-1S systems [19,23]. The reason for the suppression is that the numerical coefficients which multiply the parametric estimates given in Eq.…”

“…However, this consideration does not take into account the scaling of the terms with the principal quantum number n. While we find that the D 6 coefficients typically grow as n 4 for a given manifold of states (see also Ref. [23]), the energy differences E m,A for adjacent lower-lying states are proportional to 1/(n − 1) 2 − 1/n 2 ∼ 1/n 3 for large n, and the fourth power of the energy difference E m,A enters the prefactor of the 1/R 2 pole term. Hence, it is interesting to compare the parametric estimates to a concrete calculation for excited nD states; this is also important in order to gauge the importance of the nonresonant contributions to the interaction energy which were left out in Ref.…”

Long-Range Interactions for Hydrogen: 6P–1S and 6P–2S Systems

“…The atomic states can be classified according to the quantum number F z ; the z component of the total angular momentum commutes [11] with the total Hamiltonian given in Equation (3). Within the 6P 1/2 -6P 3/2 -1S 1/2 system, the states in the manifold F z = 3 are given as follows,…”

“…This is mainly due to energetic degeneracies connected with the "exchange" of the states among the two atoms. For nS-1S interactions (atomic hydrogen), this problem has recently been investigated in [3]. It was found that the interesting oscillatory 1/R 2 long-range tails [4][5][6][7][8][9][10] are numerically suppressed and become dominant only for excessively large interatomic distances, in a region where the absolute magnitude of the interaction terms is numerically insignificant.…”