Reply to ''Comment on 'Recurrences without closed orbits' ''

Recently, we presented quantum spectra of alkali-atoms in a static electric field @1#. We restricted the investigation to cases where the electron has at least one unit of angular momentum in the field direction. We found strong features in the recurrence spectra at short actions which do not correspond to real closed orbits in this system. We dubbed these features ‘‘recurrences without closed orbits.’’ In Main’s Comment @2# on this paper, he demonstrates that orbits exist at the correct action to account for the recurrences in our spectra. These orbits come from the bifurcation of the closed uphill and downhill orbits observed in the m 50 photoabsorption spectra of hydrogen in an electric field into real and complex periodic orbits in the m51 and m 52 spectra. The uphill and downhill orbits no longer exist for mfi0, but each is split into a real periodic orbit and a complex orbit in a type of tangent bifurcation. The real periodic orbits formed in this bifurcation, unlike the original orbits, no longer reach the nucleus. Main proposes that these periodic orbits are the origin of the ‘‘recurrences without closed orbits’’ described in Ref. @1#. This is a reasonable inference and we think that a theory of the short action recurrences must account for these orbits. However, there are conceptual questions that need to be addressed before it can be shown whether or not these orbits actually generate the short action recurrences. The role of the complex, ‘‘ghost,’’ periodic orbits found by Main is interesting and may play a role in the explanation of the recurrences in H. These complex orbits are connected to the tunneling of the wave function into the forbidden region and thus to the overlap of the wave function with the initial state. This would be in line with the approximation made in @1# of treating the on-axis orbits as if they did exist even though there were classically forbidden. A theory based on Main’s orbits should also explain the success of the approximation in reproducing the recurrence strengths and the scaling law followed by the ‘‘recurrences without closed orbits’’ we deduced from the approximation. A possible problem with these orbits is that the electron never gets closer than 12.8 a.u. to the nucleus for m51; thus they do not overlap with the 1s initial state which has a size of 1 a.u.