Functional form of the imaginary part of the atomic polarizability

Abstract

The dynamic atomic polarizability describes the response of the atom to incoming electromagnetic radiation. The functional form of the imaginary part of the polarizability for small driving frequencies ω has been a matter of long-standing discussion, with both a linear dependence and an ω 3 dependence being presented as candidate formulas. The imaginary part of the polarizability enters the expressions of a number of fundamental physical processes which involve the thermal dissipation of energy, such as blackbody friction, and non-contact friction. Here, we solve the longstanding problem by calculating the imaginary part of the polarizability in both the length (" d · E") as well as the velocity-gauge (" p · A") form of the dipole interaction, verify the gauge invariance, and find general expressions applicable to atomic theory; the ω 3 form is obtained in both gauges. The seagull term in the velocity gauge is found to be crucial in establishing gauge invariance.

Dynamic dipole polarizabilities of heteronuclear alkali dimers: optical response, trapping and control of ultracold molecules

“…However, Eq. (14) is not similar to imaginary parts obtained in second quantization [44,47,51,52] (which do not solve the controversy either). The main reason is the following.…”

“…(8)- (11)), it is odd with respect to ω [43,44]. Moreover, Im(α − 1 (ω)) scales as ω and not ω 3 for vanishing frequencies [47], especially because the width γ 2 is ω-independent.…”

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

“…A different prescription, which puts the poles into the lower half of the complex plane, has recently been used in Ref. [20]. In this latter study, one considers the relative permittivity ǫ r (ω) of a dilute gas and its relation to the dynamic dipole polarizability α(ω) of the gas atoms,…”

Relativistic and radiative corrections to the dynamic Stark shift: Gauge invariance and transition currents in the velocity gauge

“…One should add that the derivation here is related to the one recently presented in Appendix A of [8] in the context of the gauge invariance of radiative corrections to the two-photon decay width, and to Ref. [9,10] for the gauge invariance of the imaginary part of the atomic polarizability.…”

“…[11]], and carry out the gauge transformation of the wave function at t = ±∞, where it amounts to the identity transformation (because the perturbing fields vanish). All processes which allow for such a description, have been found to be gauge invariant under "hybrid" transformations [5,[7][8][9][10].…”