Long-range atom-wall interactions and mixing terms: Metastable hydrogen

Abstract

We investigate the interaction of metastable 2S hydrogen atoms with a perfectly conducting wall, including parity-breaking S-P mixing terms (with full account of retardation). The neighboring 2P 1/2 and 2P 3/2 levels are found to have a profound effect on the transition from the short-range, nonrelativistic regime, to the retarded form of the Casimir-Polder interaction. The corresponding P state admixtures to the metastable 2S state are calculated. We find the long-range asymptotics of the retarded Casimir-Polder potentials and mixing amplitudes, for general excited states, including a fully quantum electrodynamic treatment of the dipole-quadrupole mixing term. The decay width of the metastable 2S state is roughly doubled even at a comparatively large distance of 918 atom units (Bohr radii) from the perfect conductor. The magnitude of the calculated effects is compared to the unexplained Sokolov effect. , research on related matters has found continuously growing interest over the last decades [5][6][7][8]. In the non-retarded regime (close range), the interaction energy scales as 1/Z 3 with the atom-wall distance Z, while for atom-wall distances large in comparison to a typical atomic wavelength, the interaction energy scales as 1/Z 4 (see Chap. 8 of Ref.[9]). The leading term is given by virtual dipole transitions, while multipole corrections have recently been analyzed in Ref. [10]. The symmetry breaking induced by the wall leads to dipole-quadrupole mixing terms, which lead to admixtures to metastable levels [11,12]. While this effect has been analyzed in the non-retarded van-der-Waals regime [11,12], a fully quantum electrodynamic calculation of this effect would be of obvious interest. This fact is emphasized by the curious observation of a long-range, and conceivably super-long-range (micrometer-scale) interaction of metastable hydrogen 2S atoms with a conducting surface (the so-called Sokolov effect, see Refs. [13][14][15][16]). It is not far-fetched to suspect that this effect could be due to a quantum electrodynamically induced tail of the dipole-quadrupole mixing term in the atom-wall interaction. Namely, for the hydrogen 2S atom, the neighboring 2P 1/2 and 2P 3/2 levels are removed only by the Lamb shift and finestructure, respectively, while it is known that virtual states of lower energy can induce long-range tails in atomwall interactions, as well as in the Lamb shift between plates (see Refs. [17][18][19][20][21][22][23][24][25][26]). The large admixtures typically induced in atomic systems when a metastable level couples to nearly degenerate states of opposite parity suggest that a closer investigation of the hydrogen system is warranted. Atomic units with = 4πǫ 0 = 1 and c = 1/α are used throughout this Rapid Communication, where α is the fine-structure constant. The electron charge is explicitly denoted as e unless stated otherwise.Retardation of the atom-wall interaction.-The quan-

Long-range tails in van der Waals interactions of excited-state and ground-state atoms

“…The suppression mainly is due to the smallness of the Lamb shift; analogous observations have recently been made in Ref. [6], where the 2P admixtures to a reference 2S state in hydrogen have been calculated for atom-wall interactions: A parametrically interesting long-range tail has been identified, but it was found to be suppressed due to the smallness of the Lamb shift.…”

“…The 2P 1/2 states are energetically lower than the reference 2S state but displaced only by the Lamb shift L. Their contribution is suppressed, even in the oscillatory terms, due to the E 4 m,A = L 4 prefactor. In the Lamb shift range R ∼ c/L (when R becomes commensurate with the Lamb shift wavelength), the static polarizability of the 2S state has the Lamb shift in the denominator, so that the 1/R 7 -Wick-rotated term of the interaction energy shift is of order 1/( c/L) 7 (L/ c) −1 = [L/( c)] 6 . For R ∼ c/L, it competes with the oscillatory term which is of the same order of magnitude, namely, [L/( c) 6 .…”

“…In the Lamb shift range R ∼ c/L (when R becomes commensurate with the Lamb shift wavelength), the static polarizability of the 2S state has the Lamb shift in the denominator, so that the 1/R 7 -Wick-rotated term of the interaction energy shift is of order 1/( c/L) 7 (L/ c) −1 = [L/( c)] 6 . For R ∼ c/L, it competes with the oscillatory term which is of the same order of magnitude, namely, [L/( c) 6 . In the given distance range, the interaction energy is of order α 24 m e c 2 , where m e is the electron mass, and thus is negligible.…”

“…Recently, the long-range tails of the interaction between an excited-state and a ground-state atom [1][2][3][4][5] as well as those of the interaction between an excited 2S state with a conducting wall [6], have received considerable attention. The question behind the investigation concerns the existence of long-range tails for excited reference states, for which partially conflicting results have been obtained in the past [7][8][9].…”

Long-range interactions of hydrogen atoms in excited states. I.2S−1Sinteractions and Dirac-δperturbations

“…Mathematically speaking, the inequality R ≪ c/L implies R ≪ c/F because F and L are apart by only a single order-of-magnitude [see Eq. (42)]. The regime c/F < R < c/L can only be accessed reliably by a numerical calculation (see Sec.…”