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Report of the Editorial Board Member -- *******/Zhang
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The authors consider the non-Markovian dynamics in a system with
countably many, dense energy levels. They analyze the derivation of
Fermi's Golden Rule: they replace the generally non-equidistant energy
spectrum of the system under consideration with an equidistant one and
obtain an equidistant sampling of the sinc^2 function in Eqs. (4) and
(5) for the expression of the transition probability. Then using a
trick with the Fourier transforms, the authors found a simple
expression for the first order transition probability Eqs. (4) and
(11) which tells us the following: (i) the transition probability is a
piece-wise linear function of time, the kinks are located at
m*2*pi/delta, where m is an integer, delta is the distance between two
adjacent energy levels closest to the final state of the system (ii)
the slope of the different sections of the transition probability
depends sensitively on the exact location of the energy of the final
state relative to the neighboring states.

The authors worked out two examples to illustrate their result: they
applied their formula to the tight binding model in a finite chain,
where the potential of one of the sites is modulated periodically. In
this system the validity conditions of their approximate formula can
be easily fulfilled, PROVIDED that the final state of the transition
is far from the edge of the energy spectrum. The authors chose the
initial states close to the bottom of the energy spectrum, whereas the
modulation couples states nearly resonantly around the middle of the
energy spectrum. There the spacing of the energy levels varies slowly.
The authors compared the prediction of their approximate formula with
the numerical solution of the Schrodinger equation and found good
agreement for low amplitude periodic modulation. They also show how
the approximate result gets further and further from the numerical
solution for longer time AND/OR for larger modulation amplitude.

The non-Markovian dynamics in various physical systems have been
extensively studied recently. Such systems are for example (1) the
spontaneous emission of two level atoms interacting with the
electromagnetic reservoir in a photonic crystal, where the photonic
band edge is tuned close to the atomic transition. Here the decay time
depends strongly on the position of the excited state energy relative
to the band edge. (2) spread of a local excitation in a tight binding
model. Some good examples to such a study can be found in the papers:

paper I: E. Rufeil-Fiori, H.M.Pastawski: "Survival probability of a
local excitation in a non-Markovian environment: Survival collapse,
Zeno and anti-Zeno effects" Physica B 404 (2009) 2812–2815

paper II: E. Rufeil-Fiori, H.M.Pastawski: "Non-Markovian decay beyond
the Fermi Golden Rule: Survival collapse of the polarization in spin
chains" Chemical Physics Letters 420 (2006) 35–41

Here the system dynamics is calculated exactly using the Green's
function method. The probability of staying in the initial state
exhibits "quantum interference" features which cannot be explained
with Fermi's Golden Rule, see Fig 2 in paper I, and Fig 3 and 4 in
paper II.

Returning back to the manuscript under consideration, the method
developed by the authors is interesting, although it seems that it
does not reveal as many features of the system dynamics as the
seemingly more elaborate Green's function method. I have written a
simple MATLAB program to plot the "exact" form of the time dependence
of the first order transition probability in Eq. (2), i.e. using the
system's original discrete energy level distribution. For ki=41 I
retrieved quite well the author's curve. However, for ki=141 the first
and second kinks could be well identified, but they were rather
rounded and the second section of the curve was already quite wavy. In
this latter case the final states with energy approx. Ei+omega lay
close to the edge of the spectrum, hence Ek and delta_k varied a lot
between adjacent states. This result agrees with the authors
requirement.

From these results it follows that a quantitative estimate about the
accuracy and limitation of the approximate solution would be necessary
to apply the method in practice. In particular, for how long the
predicted probability is a good approximation after the first kink?
Concerning accuracy, there are two factors: the validity of the
author's equal energy spacing simplification and the validity of the
first order perturbation theory in a long time evolution.

I agree with the referee that deviation from the Fermi's Golden Rule
have been discussed by many authors. However, I have not encountered so
far the approach of the authors.

In summary, I find that the analysis presented in the manuscript is
incomplete. A comparison with a more elaborate analytical solution is
missing. An estimate of the accuracy of the long time prediction would
also be necessary. Hence I do not support the publication of the
manuscript in the present form in the Physical Review A.

Dr. Zsolt Kis
Editorial Board Member
Physical Review A