Relativistic Landau quantization in non-uniform magnetic field and its applications to white dwarfs and quantum information

AUTHORS:

First, we would like to thank the reviewer very much for carefully reading
our manuscript and assessing it with encouraging comments. Here we respond
their comments one by one mentioning changes made in the manuscript. All the
changes are kept in bold-face and Figures 9 and 10 are replaced by new ones.


REVIEWER:

This work addresses the Landau Quantization (LQ) of electrons in non-uniform
magnetic fields and its applications to problems cutting across different
disciplines of physics, for example equation of state (EoS) and mass-radius
(M-R) relationship of white dwarfs (WDs) as well as the quantum speed of
electrons which might be important for processing the speed of quantum information.
The non-uniform field has a simple power law variation as given by B = B_0 ρ^n ẑ
where n=0 represents the uniform magnetic field.

It is a well known fact that the zeroth Landau level is singly degenerate whereas
it is doubly degenerate for all other levels in a uniform magnetic field. Authors
in this manuscript demonstrate that the degeneracy of energy levels as found in
a uniform field, is lifted in non-uniform magnetic fields. They further note that
there is a definite alignment of energy levels of spin-up and spin-down electrons
as the magnitude of a non-uniform magnetic field changes at each point. These are
new findings. It is also observed that the quantum speed of electrons increases as
'n' grows.

The impact of the LQ of electrons in non-uniform magnetic fields is investigated
on the EoS and M-R relationship of magnetized WDs. A specific example of the EoS for
n=-0.3 is used to compute the M-R curve in Figs. 9. If one carefully compares Fig. 9(a)
with Fig. 8(a), it is observed that the EoS of Fig. 8(a) for n=-0.3 is distinctly
stiffer with respect to the case of the uniform field (n=0), but it is not the
case in Fig. 9(a). This needs an explanation.

AUTHORS:

In Figure 8(a), when we showed EoS for a uniform magnetic field, it is for n=0
throughout the star and field B=B_0. However, in Figure 9a, we showed two EoSs
for the same star: one is EoS of central region with constant magnetic field
where B is higher than B_0, according to the chosen profile in the previous
version, and another is EoS with varying magnetic field. See in Eq. (5.2),
how central field and B_0 are related in the central region of the star,
for the model considered in the previous version, with a varying magnetic
field other than central region. Hence this difference is between Figure 8(a)
and Figure 9(a). In the revised version, however, we have replaced these figures
with better ones for clarity with modified physics and changed the text
accordingly. Note the modifications in section 5.1 in the revised version,
part of which prompted due to reviewer's this question.


REVIEWER:

It is observed in Fig. 8(a) that less number of Landau levels is populated
when n=-0.7 resulting in a softer EoS at lower densities and a stiffer EoS
at higher densities. It would be worth demonstrating the effect of the
non-uniform magnetic field for n=-0.7 on the M-R relation in a new plot.

AUTHORS:

For n=-0.7, the Lorentz force may turn out to be too strong for a realistic
central density of the star to destabilize it. Hence, we do not show
an M-R relation for n=-0.7. However, in the revised version, we have modified
the field profile for the M-R relation to remove this confusion. Note,
generally, that the trend of deviation for EoSs in nonuniform magnetic fields
compared to that of the uniform field is same for different n. However, the
deviation is more prominent with decreasing n, hence you noticed it for n=-0.7.


REVIEWER:

In the introduction, authors mention another alternative approach of the
non-uniform magnetic field in Ref.[27] extensively used in the investigations
of structures of neutron stars and WDs. Authors should compare their current
findings on the M-R relation of WDs with their previous results following
the prescription of the magnetic field variation of Ref. [27] inside the WDs.
This would definitely indicate whether new findings related to WDs in this
manuscript are quantitatively significant or not.

AUTHORS:

The aim of this work is not to propose a field profile opposing a density
dependent profile proposed earlier in Ref. 27, which facilitates M-R relations
influenced by magnetic effects (classical and quantum). The main aim is to show
the Landau quantization in nonuniform magnetic fields and how the quantization
effect changes w.r.to the uniform magnetic field case. For that we have
proposed a field profile, satisfying Maxwell's equations, and solved
the Dirac equation (as opposed to the case of uniform field, when the
Dirac equation has to be solved for a quadratic potential). Subsequently,
as one of the immediate applications, we have attempted to explore the
results in (highly) magnetized white dwarfs. Otherwise, the density dependent
field profile, as proposed in Ref. 27, is quite an effective one, at least
as long as the star is spherical and we quite like it. However, the solution
of Dirac equation in varying magnetic field can not be proceeded with a
density dependent magnetic field profile, which merely talks about field
magnitude at various density points. Nevertheless, in principle, as
indicated in the manuscript, by changing parameters, various field profiles
can be obtained from the density dependent profile as well and we may get
equivalent results of the present one as long as its application is to
white dwarfs. Unless the reviewer is very particular about it, we prefer to
undertake this venture in a future work, in order to avoid deviating from
the main flow of the present manuscript, which is Landau quantization in
varying magnetic fields.

Nevertheless, we have revised the manuscript throughout for a better clarity.