Talk:Kicked top

Review of Reviewer A:

Well written article which demonstrates why 'kicked top' can be considred as one of the standard dynamical models used to study quantum signatures of chaos. Of special importance is the discussion of various symmetries of the quantum system which determine which of three universality classes describe statistical fluctuations of the spectra of evolution operators. The paper covers the entire subject, provides links to the relevant literature and is illustrated with nice figures, so I find it appropriate for Scolarpedia.

The only minor improvement which I could suggest concerns a sentence after eq (3), which reads

"The factor (2j+1)^{-1} appears in the second exponent"

  but it should be 'the FIRST exponent' (or the equation should be modified accordingly).

Author Kus:

Thank you for a correct remark. The text of the article was changed accordingly

Review of Reviewer B:

This article about the "Kicked top" is very interesting and important in the context classical and quantum chaos.

I have some suggestions concerning the presentation and pedagogical aspects:

1) In the first section the authors introduce the kicked top in a general form with arbitary polynomials H_0 and H_1. I have the impression that there is somehow a "generic choice" of these polynomials as examples which is used in most of the studies of the Kicked Top. I think the authors should give from the very beginning very clearly this generic choice in a separate equation, i.e. provide in Eq. (2) the prefactors with parameters (and not only "\propto") which are (in my understanding ?) the parameters \alpha and \tau used in the Figures 1-3, or in other words provide the exact classical Hamiltonian corresponding to these figures. If possible this classical Hamiltonian should also correspond to the quantum version given in Eq. (3), eventually providing a translation between "classical" and "quantum" parameters (due to the factor (2j+1)^{-1}).

2) I think in section 1, below the (modified) Eq. (2) one should also provide an explicit expression for the classical map which is obtained from the time dependant Hamiltonian (for the "generic choice", and without an explicit derivation, only the result), i.e. the explicit equations relating J^{(n+1)} to J^{(n)} and which allows to reproduce the figures 1-3 for anybody with reasonable programming skills in the field.

3) I think there is a visibility problem concerning the figures 1-3, especially 2 and 3. They are still quite well visible on the computer screen (provided a reasonable resolution). However, when printed out on paper the dots are barely visible. If possible it would be nice to increase slightly the dot size of these figures and maybe this can even be done by changing a parameter in the corresponding (source) postscript files or the plot program used to produce these figures. This is a more optional suggestion from my part but I think it would be nice if the authors can consider this suggestion.

4) In Section 4 when the authors speak of different "symmetry classes" and the Wigner surmise for P(S) it would be appropriate to add some reference to Random matrix theory, for example the book of Mehta since many readers are not necessarily familiar with these things. Of course a Scholarpedia reference would be ideal but I have seen that for the moment there is only an unfinished version but which seems already visible by an automatic link. I am not sure how to handle this exactly. Maybe a reference to the book of Mehta for now and replace it later with a Scholarpedia reference ?

5) In the last section about the rotator limit it would be nice to provide the translation of the parameters, i.e. to give an equation "K=..." where K is the kicked rotator parameter and "..." is the expression of the kicked top parameters (always for the "generic choice"). There is already an implicit Scholoarpedia reference (by an automatic link) for the kicked rotator which explains the "K"-parameter. Therefore, I suppose it is not necessary to add an explicit additional reference about the kicked rotator.

6) Optionally one might add some explanation about the localization length, i.e. provide an additional equation of the type:

l \propto D/\hbar^2 \approx K^2/(2\hbar^2) if K\gg 1

and then finally "l \propto ..." in terms of the kicked top parameters using the expression "K=..." given previously (according to point 5) and where D is the Diffusion constant (=> refering to kicked rotator article as explanation). This would also give some (simplified) expression of the resulting localization length in terms of the initial kicked top parameters.

Author Kus :

All improvements suggested by the Referee B in points 1), 2), 4), 5), and 6) were included. For the moment we did not find an effective method to change the visibility of the figures.

Editor notes

a) please, update the figs which are not visible

b) it may be useful to quote other physical systems where the kicked top model naturally appears (e.g. see Phys. Rev. Lett. v.74, p.2098 (1995) with resonant tunneling diode)

c) it will be useful to quote works of other groups which used the kicked top to study propeties of quantum chaos (e.g. P.Jacquod et al Phys. Rev. E v.64, p.055203(R) (2001))