Resonance wave functions located at the Stark saddle point

We calculate quantum-mechanically exact wave functions of resonances in spectra of the hydrogen atom in crossed external fields and prove the existence of long-lived decaying quantum states localized at the Stark saddle point. A spectrum of ground and excited states reproducing the nodal patterns expected from simple quadratic and cubic expansions of the potential in the vicinity of the saddle point can be identified. The results demonstrate the presence of resonances in the vicinity of the saddle predicted by simple approximations.

Figure 1

Potential of the hydrogen atom in an external electric field [dashed red grids with the Coulomb singularity in (a) and (b)] in comparison with the second-order (a) and third-order (b) power-series expansions around the saddle in the ($\xi$,$y$) plane with $\xi =x-x_s$. Both approximations reproduce the potential in the vicinity of the saddle at the origin correctly; however, both are only valid very close to the saddle point.

Figure 2

Comparison of the resonance energies (real parts of the complex energy eigenvalues) obtained in exact solutions of the full Hamiltonian (1) (red dotted lines) and the second- (solid blue lines) and third-order (solid green squares) approximations. The results are shown for even (a) and odd (b) $z$ parity separately. Position space representations of the exact quantum resonances labeled by capital letters are shown in Figs. 3 to 6. In (a) and (b) the dashed grey lines represent the saddle point energy.

Figure 3

Density plots of the resonance labeled A1 in Fig. 2 in the planes $z=0$, $y=0$, and $x=x_s$. The solid (magenta) lines mark the saddle point. One can clearly see that the probability density of the resonance is restricted to a close vicinity around the saddle point.

Figure 4

Density plots of the resonances labeled A2, A3, A4, and A5 in Fig. 2 in the $z=0$ plane. For decreasing field strengths one can observe that the resonances are decreasingly centered at the saddle point.

Figure 5

Density plots of the resonances B, C1, C2, and D (cf. Fig. 2) in the $z=0$ plane. The resonances can be assigned to second-order approximations with $n_1=0$ and $n_z\ne 0$. The nodal patterns expected for excited states in the $z$ direction can be observed.

Figure 6

Density plots of the resonances E, F, and G (cf. Fig. 2) corresponding to states of the quadratic approximation excited in $n_1$. Complicated nodal patterns are found.