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

We investigate the gauge invariance of the dynamic (ac) Stark shift under “hybrid” gauge transformations from the “length” ($\vec{E}·\vec{r}$) to the “velocity” ($\vec{A}·\vec{p}$) gauge. By a “hybrid” gauge transformation, we understand a transformation in which the scalar and vector potentials are modified, but the wave function remains unaltered. The gauge invariance of the leading term is well known, while we here show that gauge invariance under perturbations holds only if one takes into account an additional correction to the transition current, which persists only in the velocity gauge. We find a general expression for this current, and apply the formalism to radiative and relativistic corrections to the dynamic Stark effect, which is described by the sum of two polarizability matrix elements.

Figure 1

Feynman diagrams for the self-energy radiative correction to the dynamic Stark shift. Interactions with the external laser field are labeled with ${\omega }_{\mathrm{L}}$.

Figure 2

Ratio of the first-order radiative correction of the dynamic polarizability to the unperturbed dynamic polarizability as a function of the laser photon energy $\omega$ (we set $\hslash =1$), divided by the Hartree energy $E_h$. The data are obtained for the ground state of hydrogen. The quantity $\mathrm{\Delta }Q$ is defined in Eq. (34).

Figure 3

Same as Fig. 2, but in a frequency range which covers the intermediate $2P$ state, where the laser frequency can excite the $1S\text{−}2P$ transition resonantly. The plot is included for reference. Of course, near resonance, the second-order perturbation treatment of the atom-laser interaction, which is the basis for Eq. (2), breaks down, and the dressed-state formalism has to be used (see Ref. [13]). The $2P$ resonance is responsible for the first peak in the radiative correction at $\omega =\frac{3}{8}{E}_h$, and the second pole is due to the zero of the unperturbed matrix element $Q$ at $\omega =0.429538E_h$. All calculations are performed in the nonrecoil approximation. The figure illustrates the dramatic increase of the radiative correction as the resonance is approached.