Scaled Variable Spectra

These are calculations of the scaled-variable spectra of hydrogen, helium, and lithium in an electric field. The classical trajectories were calculated at an electric field strength of 1985 V/cm for various unimportant reasons. The spectra were then constructed using the scaling properties of the amplitudes and phases of the semiclassical recurrences to get spectra that represent what you would see if you did a scaled variables scan over the field strengths indicated on the graphs. Notice that I plot the magnitude of the Fourier transform, not the square of the FT which is the usual definition of "recurrence strength".


This is the field strength range used by Courtney for the experiment and Dando for his figure 3. I get the same thing. Good. This is an m=0 spectrum.


This is a lithium spectrum where core scattering is included. This again matches Dando's published spectrum. There are some minor differences, but all in all good agreement. Also m=0.


Getting back to hydrogen. This is a semiclassical |FT| for the estimated field strength range used for the Wesleyan experiment. This doesn't look too different from the Courtney spectrum, but the energy range corresponds to lower principle quantum numbers. This leads to some interesting effects which are _not_ included in this calculation, and modify the spectrum. More on this later.


This is a graph of the magnitude of semiclassical |FT| in the field strength range of the Wesleyan experiment if there is a random energy error or a bandwidth of the laser of a few parts in 10^-6 atomic units. This "cuts off" the longer period orbits and diminishes their recurrence strength compared to the shorter orbits.


Here is what I get when I do the singlet Helium calculation at the field strengths and laser bandwidth of the Wesleyan experiments. The quantum defects are small, the main difference between this and hydrogen are at actions just below 10 where a set of peaks now exists that are not there in hydrogen (see graph above). The m=1 result was created by switching the laser polarization, same classical orbits as m=0, Goetz and Delos' approx.


This is a stacked plot for m=0 where the magnitude of th FT is plotted for the electric field range scanned in the experiment. This is still hydrogen since no scattering has been put in yet. The dashed red lines are the FT's where resonances of the parallel orbit are giving singularities. I have set them to zero in the solid plots as a temporary fix.


Here is similar plot for m=1. The resonances don't matter much here since the polarization puts them in the node of the outgoing wave. The peaks in the first diagonal band grow and diminish as the initial angle of the classical trajectories move into or out of the peak of the outgoing wave's angular distribution.