Gauge-invariant relativistic strong-field approximation

The problem of the choice of gauge in the relativistic strong-field approximation (SFA) is analyzed. The main motivation is to obtain a relativistic ionization amplitude in the SFA which in the nonrelativistic limit coincides with the conventional well-accepted results of the SFA in the length gauge. A gauge-invariant formulation of the SFA is derived which is applicable both in the nonrelativistic as well as in the relativistic regime of laser-induced strong-field ionization phenomena. The gauge invariance is achieved by means of employment of an eigenstate of the physical energy operator for the initial atomic state. As an example, a comparison of predictions of the gauge-invariant theory with conventional SFA results in the radiation gauge is given for above-threshold ionization in the relativistic regime.

Figure 1

The region of parameters $\xi$ and Z where different ionization regimes of a hydrogenlike ion take place. The thin curve at the bottom corresponds the condition of $\gamma =1$. The tunneling ionization (TI) regime takes place for the parameters of the shaded region [see Eqs. (38, 39)], and the over-the-barrier ionization (OBI) for the parameters above the thick curve.

Figure 2

Photoelectron spectrum in a linearly polarized laser field with angular frequency $\omega =0.05\mathrm{a.u.}$ and electric field strength $E=3.4\mathrm{a.u.}$ via ${\log }_{10}\left({\mid M_{\mathbf{p}}\mid }^2\right)$ in Eq. (36), with final electron momentum $\mathbf{p}=(p_x,0,p_z)$ and emission angle $\theta =\arccos (p_x∕p)=0.18$. The model potential is adapted to ${\mathrm{B}}^{4+}$ with ionization potential $I_p=12.5\mathrm{a.u.}$ The electron energy is scaled in multiples of the ponderomotive energy $U_p=⟨A^2∕2⟩$. The photoelectron spectrum in gray is evaluated via the standard SFA in the velocity gauge, the one in black via the gauge-invariant SFA.