Strong-field ionization and gauge dependence of nonlocal potentials

Nonlocal potential models have been used in place of the Coulomb potential in the Schrödinger equation as an efficient means of exploring high-field laser-atom interaction in previous works. Although these models have found use in modeling phenomena including photoionization and ejected electron momentum spectra, they are known to break electromagnetic gauge invariance. This paper examines if there is a preferred gauge for the linear field response and photoionization characteristics of nonlocal atomic binding potentials in the length and velocity gauges. It is found that the length gauge is preferable for a wide range of parameters.

Figure 1

Curves relating $V_0$ and $\sigma$ for constant values of $E_0$ that satisfy Eq. (19). Shown here for the first five hydrogen states, the Gaussian nonlocal potential supports a (single) bound state of arbitrary energy.

Figure 2

The normalized configuration space wave function $\psi \left(\right|\mathbf{r}\left|\right)$ is given by the Fourier transform of Eq. (16) (shown here for $E_0=0.5$). The variable $\sigma$ is used as a fitting parameter.

Figure 3

The static polarizability $\alpha (\omega =0)$ as calculated from Eqs. (35) and (36). In the limit $\sigma \to 0$, the nonlocal potential is equivalent to a Gaussian potential, and the polarizability is gauge independent. For positive values of $\sigma$, the length-gauge system is more easily polarized by a (dc) applied electric field.

Figure 4

The dynamic polarizability in the length and velocity gauges, with a single photon resonance at $\omega =E_0$. The solid lines represent the $\alpha \left(\omega \right)$ given by Eqs. (35) and (36) ($\omega \ge E_0\to \omega +i\delta$), and the crosses represent the simulated low-field response via the total dipole [Eq. (25)].

Figure 5

The ionization rates predicted by the PMPB model and the nonlocal length and velocity gauge formulations as a function of laser frequency and intensity ($100\times 100$ data points, interpolated). The laser parameters here span the multiphoton (high frequency, low intensity) and tunnel (low frequency, high intensity) ionization regimes. The length gauge and PMPB ionization rates agree well over the parameter space shown; the velocity gauge ionization rate generally underestimates in the multiphoton regime and overestimates in the tunnel regime. Slices along constant intensity and frequency are shown in Figs. 6 and 7 for direct comparison.

Figure 6

The ionization rate as a function of intensity for $I_0=2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ [6]; the values of $\sigma$ for the length and velocity gauge potentials were calibrated at this intensity, at 800 nm (visible here as the crossing point for all rates). The velocity gauge overestimates the rate towards the tunnel regime and underestimates it in the multiphoton regime, while the PMPB and length gauge rates predict similar rates. The rates are also plotted for $I_0=2\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ [6].

Figure 7

The PMPB photoionization rate and nonlocal (length gauge) rate also show agreement as a function of intensity for 800-nm light; the velocity gauge ionization rate does not (7). At 400 nm (7), the velocity gauge ionization rate has the same power dependence as the length gauge and PMPB rates, but strongly underestimates the magnitude for the chosen fitting parameter ($\sigma =4.785$).