Signatures of the classical transition state in atomic quantum spectra

We perform quantum-mechanically exact calculations of resonances in the spectrum of the hydrogen atom in crossed external fields and establish a close connection between the classical transition state in phase space and features in the quantum spectrum. By varying the external field strengths, structures are revealed which are surprisingly similar to the quantized energy levels of the classical electron motion in the vicinity of the saddle point obtained with an approximation of the potential. The results give clear evidence for signatures of the transition state in quantum spectra.

Figure 1

(Color online) Potential of the hydrogen atom in an external electric field in the $\left(\xi ,y\right)$ plane [cf. Eq. (2)] with $\xi =x-x_s$ denoted by the red grid with the Coulomb singularity. Reactive trajectories must pass the Stark saddle point at $\left(\xi =0,y=0\right)$ in whose vicinity the transition state is located. The blue grid without singularity describes a quadratic approximation around the saddle point [cf. Eq. (3)].

Figure 2

(Color online) (a) Comparison of the quantized energies of the electron motion near the transition state (solid blue lines) with the exact quantum energies (red points) on the line $\gamma =0.008\alpha$, $f=0.0003\alpha$ for $0<\alpha <1$ as defined in Eqs. (14a, 14b, 14c) in the $\left(\gamma ,f\right)$ space for even $z$ parity. Only exact quantum energies with $\left|\text{Im}\left(E\right)\right|<0.0006$ are shown. The exact resonances which trace the approximate energies [Eq. (8)] are shown by a larger line strength. The lowest (dashed green) line represents the saddle-point energy. The black frame marks the region magnified in Fig. 3. (b) Same situation for states with odd $z$ parity.

Figure 3

(Color online) Same situation as depicted in Fig. 2. The region around the avoided level crossings near the state with $n_z=2$ and $n_1=0$ is magnified.

Figure 4

(Color online) Same comparison of the quantized energies of the electron motion near the transition state (solid blue lines) with exact quantum energies (red points) as in Fig. 2 but on the line $\gamma =0.01{\alpha }^{\prime }$, $f=0.0001+0.0002{\alpha }^{\prime }$ for $0<{\alpha }^{\prime }<1$ defined in Eqs. (15a, 15b, 15c). Only exact quantum energies with $\left|\text{Im}\left(E\right)\right|<0.0002$ are shown. The saddle-point energy is marked by the lowest (dashed green) line.