# Coherence of oscillations in matter and supernova neutrinos

Yago P. Porto-Silva∗ Alexei Yu. Smirnov†
∗ Instituto de Física Gleb Wataghin - UNICAMP, 13083-859, Campinas, São Paulo,
Brazil
∗ † Max-Planck-Institut für Kernphysik, 69117 Heidelberg, Germany
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the propagation coherence for neutrino oscillations in media with
different density profiles. For each profile, we find the dependence of the
coherence length, $L_{coh}$, on neutrino energy and address the issue of
correspondence of results in the distance and energy-momentum representations.
The key new feature in matter is existence of energy ranges with enhanced
coherence around the energies $E_{0}$ of "infinite coherence" at which
$L_{coh}\rightarrow\infty$. In the configuration space, the infinite coherence
corresponds to equality of the (effective) group velocities of the
eigenstates. In constant density medium, there is a unique $E_{0}$, which
coincides with the MSW resonance energy of oscillations of mass states and is
close to the MSW resonance energy of flavor states. In the case of massless
neutrinos or negligible masses in a very dense medium the coherence persists
continuously. In the adiabatic case, the infinite coherence is realized for
periodic density change. Adiabaticity violation changes the shape factors of
the wave packets (WPs) and leads to their spread. In a medium with sharp
density changes (jumps), splitting of the eigenstates occurs at crossing of
each jump. We study the increase of the coherence length in a single jump and
periodic density jumps - castle-wall (CW) profiles. For the CW profile, there
are several $E_{0}$ corresponding to parametric resonances. We outlined
applications of the results for supernova neutrinos. In particular, we show
that coherence between two shock wave fronts leads to observable oscillation
effects, and our analysis suggests that the decoherence can be irrelevant for
flavor transformations in the central parts of collapsing stars.

##  1 Introduction

Propagation coherence is the condition for realization of oscillations - the
interference effects that lead to time - distance periodic variations of
observables [1, 2, 3, 4, 5, 6, 7, 8, 9]. Correspondingly, decoherence means
the disappearance of the interference and oscillatory pattern.

In the configuration space, the decoherence is produced by relative shift, and
eventually, separation of the wave packets (WPs) that correspond to the
eigenstates of propagation. The coherence length was defined as the distance
at which the wave packets are shifted with respect to each other (separated)
by the size of the wave packet $\sigma_{x}$, or equivalently, when the
interference term in the probability is suppressed by a factor $1/e$. In the
energy-momentum space, the decoherence is due to averaging of the oscillation
phase over the energy interval of the energy uncertainty in a setup. It was
checked that in vacuum and uniform matter, the two considerations are
equivalent and lead to the same value of coherence length [10, 11, 12].

The loss of the propagation coherence does not lead to the loss of information
and can be restored under certain conditions. In the configuration space, the
restoration requires a long enough coherence time of detection. In the energy-
momentum space it requires very good energy resolution. Irreversible loss of
coherence occurs if neutrinos are treated as open quantum systems [13].

Mostly the coherence was studied for vacuum oscillation [1, 2, 3, 4, 5, 6, 7,
8, 9]. In matter with constant (or adiabatically changing) density it was
considered in [11, 14, 15, 16, 12]. It was noticed that at certain energy
(close to the MSW resonance energy), the difference of group velocities of the
WPs vanishes in media with constant density and the coherence length becomes
infinite [10, 11, 12]. If density varies with distance monotonously, then at a
specific point, $x_{0}$, near the MSW resonance, the difference of group
velocities changes sign, and the shift that happened at $x<x_{0}$ can be
undone at $x>x_{0}$, so that the packets overlap can be restored [11, 14, 12].
Decoherence affects resonant oscillations of solar and supernova neutrinos
[15] and, in the case of supernova, decoherence happens before the resonance
region. However, the effects of collective oscillations were not taken into
account in [15]. In ref. [16], a formal solution for the neutrino equation of
motion in general profile including non-adiabaticity was found. Based on it,
the equivalence between $x$\- and $E$-spaces was shown, at least for
situations with $\sigma_{E}\ll E$. Limitations of the WP treatment were also
discussed; in particular, as $\sigma_{E}$ increases, negative energy
components of the WPs become relevant. This can be important in situations
with $\sigma_{E}\sim E$.

The coherence in the inner parts of supernova (SN) is of special interest
[17]. The reason is that the wave packets of SN neutrinos are very short,
$\sigma_{E}\sim E$, suggesting that decoherence can occur at smaller scales
than the scales of possible collective oscillation phenomena [12, 18].
Furthermore, effectively, the problem is non-linear [19]. Therefore the
question arises whether the loss of coherence destroys or not the collective
phenomena [20].

In this paper, we consider coherence and decoherence in matter in detail. We
revisit the case of the infinite coherence length in matter of constant
density and give its physics interpretation. We show that in matter dominated
regime, decoherence is determined by vacuum parameters. We comment on the fact
that oscillations of massless neutrinos do not decohere. In the adiabatic case
$L_{coh}\rightarrow\infty$ is possible if a density profile has periodic
modulations. We then consider decoherence in the presence of adiabaticity
violation. That includes decoherence in matter with single density jump which
can be relevant for propagation through the shock wavefront in SN [12], and in
the castle-wall profile [21, 22, 23, 24, 25]. The latter can give some idea
about the decoherence of collective oscillations in the inner parts of
supernova [19]. The castle-wall profile is the simplest example because it
admits analytical treatment and allows us to draw some conclusions about the
relation between parametric resonances and coherence length.

Decoherence in the configuration ($x$-space) is equivalent to averaging of
oscillation phase in energy or $E$-space for stationary situations [26].
Therefore, for each matter profile, we elaborate on the issue of equivalence
of results in the $x$\- and $E$-spaces. In particular, for the castle-wall
profile, we show that the modification of shape factors in $x$-space due to
the density jumps do not alter its Fourier transform and the initial energy
spectrum of the WPs is preserved.

The paper is organized as follows: in section 2 we study the coherence length
in vacuum and matter with constant and adiabatically varying density. We
emphasize the existence of energies with infinite coherence length and give
interpretation of $L_{coh}\rightarrow\infty$ in position and energy spaces. In
section 3, we analyze coherence in the case of maximal adiabaticity violation.
Section 4 presents the study of coherence in the castle-wall profile. Section
5 is devoted to the applications of our results to supernova neutrinos. In
section 6 we present our conclusions.

##  2 Coherence in matter with constant and adiabatically changing density

In what follows for simplicity, we will consider the coherence in two neutrino
system; the generalization to the case of three neutrinos is straightforward.

###  2.1 Coherence of oscillations in matter

In the energy-momentum space, the decoherence is related to averaging of the
oscillation phase $\phi$ or the oscillatory (interference) term in probability
$P_{int}$ over the energy uncertainty, $\sigma_{E}$, of a set up111There is
subtle aspect related to averaging of $\phi$ and $P_{int}$ which we will
discuss later.. $\sigma_{E}$ gives the width of the wave packet (WP), and in
general, it is determined by the energy uncertainty at the production and
detection, thus including a possibility of restoration of the coherence at the
detection: $1/\sigma_{E}=1/\sigma_{E}^{prod}+1/\sigma_{E}^{det}$ [1, 2, 3, 4,
5, 6, 7, 16, 27, 12].

In the case of uniform medium (vacuum, constant density matter) the
eigenstates and eigenvalues of the Hamiltonian of propagation $H_{im}$
($i=1,\,2$) are well defined [28, 10, 11, 14]. The difference of the
eigenvalues, $\Delta H_{m}\equiv H_{2m}-H_{1m}$, determines the oscillation
phase acquired along the distance $L$:

$\phi_{m}=\Delta H_{m}L.$

For a fixed $L$, variation of the phase with change of the neutrino energy $E$
in the interval $E\pm\sigma_{E}$ equals

$\Delta\phi_{m}=\left(2\sigma_{E}\frac{d\Delta
H_{m}}{dE}+\frac{\sigma_{E}^{3}}{3}\frac{d^{3}\Delta
H_{m}}{dE^{3}}+...\right)L.$ (1)

It increases linearly with $L$ and we define the coherence length, $L_{coh}$,
as the length at which the variation of oscillation phase becomes $2\pi$:

$|\Delta\phi_{m}(L_{coh}^{m})|=2\pi.$ (2)

This condition means that averaging of the oscillatory dependence of the
probability, which is a measure of interference, becomes substantial. Using
the first term of the expansion (1) we obtain from the condition (2)

$L^{m}_{coh}=\frac{\pi}{\sigma_{E}}\left|\frac{d\Delta
H_{m}}{dE}\right|^{-1}.$ (3)

At

$\frac{d\Delta H_{m}(E)}{dE}=0$ (4)

the coherence length becomes infinite [11, 14, 12]. The next terms in the
expansions (1) shift the pole in (3) but do not eliminate it. The shift due to
higher order terms is suppressed if $\sigma_{E}/E\ll 1$.

In vacuum

$\Delta H=\frac{\Delta m^{2}}{2E},$

where $\Delta m^{2}\equiv m_{2}^{2}-m_{1}^{2}$ is the mass square difference.
Consequently, Eq. (3) gives

$L_{coh}=\frac{\pi}{\sigma_{E}}\frac{2E^{2}}{\Delta
m^{2}}=l_{\nu}\frac{E}{2\sigma_{E}},$ (5)

with $l_{\nu}=4\pi E/\Delta m^{2}$ being the vacuum oscillation length. Notice
that $L_{coh}\rightarrow\infty$ in the limits $E\rightarrow\infty$ or $\Delta
m^{2}\rightarrow 0$.

In matter with constant density [12], the difference of eigenvalues for a
given matter potential is given by

$\Delta H_{m}=\frac{\Delta m^{2}}{2E}\sqrt{\left(c_{2\theta}-\frac{2EV}{\Delta
m^{2}}\right)^{2}+s^{2}_{2\theta}}\,,$ (6)

where $\theta$ is the vacuum mixing angle and $c_{2\theta}\equiv\cos 2\theta$
and $s_{2\theta}\equiv\sin 2\theta$. The derivative of (6) equals

$\frac{d\Delta H_{m}}{dE}=\frac{\Delta m^{2}}{2E^{2}}\frac{\left(\frac{\Delta
m^{2}}{2E}-V\cos 2\theta\right)}{\Delta H_{m}}.$ (7)

Consequently, according to Eq. (3) the coherence length in matter equals

$L^{m}_{coh}=\frac{L_{coh}}{l_{m}}\frac{2\pi}{\left|\frac{\Delta
m^{2}}{2E}-V\cos 2\theta\right|}=L_{coh}\left(1-c_{2\theta}\frac{2VE}{\Delta
m^{2}}\right)^{-1}\sqrt{\left(c_{2\theta}-\frac{2VE}{\Delta
m^{2}}\right)^{2}+s^{2}_{2\theta}},$

with $L_{coh}$ given in (5) and $l_{m}=2\pi/\Delta H_{m}$ being the
oscillation length in matter.

Let us consider dependence of $L^{m}_{coh}$ on energy. Its salient feature is
existence of the pole at

$E_{0}=\frac{\Delta m^{2}}{2V\cos 2\theta}.$

That is, the infinite coherence length $L^{m}_{coh}\rightarrow\infty$ is
realized at finite energy $E_{0}$, in contrast to the vacuum case. $E_{0}$ is
related to the MSW resonance energy, $E_{R}=\Delta m^{2}\cos 2\theta/2V$, as

$E_{0}=\frac{E_{R}}{\cos^{2}2\theta}.$ (8)

Thus, $E_{0}>E_{R}$ and it can be very close to $E_{R}$ in case of small
mixing. Since the width of the MSW resonance peak is $2\tan 2\theta
E_{R}\approx 2\sin 2\theta E_{R}$, the pole is within the peak. Therefore, at
$E_{0}$, the transition probability is large.

In terms of $E_{0}$ or $x\equiv E/E_{0}$ the derivative (7) can be rewritten
as

$\frac{d\Delta
H_{m}}{dE}=\frac{2\pi}{El_{\nu}}\frac{1-x}{\sqrt{(1-x)^{2}+x^{2}\tan^{2}{2\theta}}}.$

Consequently, the ratio of the coherence lengths in matter and in vacuum (3)
becomes

$\frac{L^{m}_{coh}}{L_{coh}}=\sqrt{1+\frac{\tan^{2}2\theta}{\left(1-\frac{E_{0}}{E}\right)^{2}}}.$
(9)

The ratio Eq. (9) as function of energy is shown in fig. 1. It has the
universal form which depends on the vacuum mixing only. The value
$L^{m}_{coh}/L_{coh}=2$ corresponds to energies at both sides of the peak
$E=E_{0}\left(1\pm\tan 2\theta/\sqrt{3}\right)^{-1}$. Thus, for small $\theta$
the quantity $\Delta E=(2/\sqrt{3})\tan 2\theta E_{0}$ characterizes the width
of the peak.

At the MSW resonance, $E=E_{R}$, the ratio (9) equals

$\frac{L^{m}_{coh}}{L_{coh}}=\frac{1}{\sin 2\theta}.$ (10)

That is, the coherence length Eq. (10) is enhanced in comparison to the vacuum
length. For example, if $\theta=\theta_{13}=8.5^{\circ}$, we have
$L^{m}_{coh}\approx 12L_{coh}$. In the resonance the oscillation length
increases by the same amount: $l_{m}^{R}=l_{\nu}/\sin 2\theta$.

For $E\ll E_{0}$, i.e. in the vacuum dominance case, we have
$L^{m}_{coh}\approx L_{coh}$. Interestingly, for $E\gg E_{0}$, (matter
dominated region)

$L^{m}_{coh}\approx\frac{L_{coh}}{\cos 2\theta}.$ (11)

That is, $L_{coh}^{m}$ is determined by the vacuum parameters, and for small
$\theta$ it approaches the vacuum $L_{coh}$ again. This has important
implications for supernova neutrinos.

The depth of oscillations in matter given by

$\sin^{2}2\theta_{m}=\frac{\sin^{2}2\theta}{\left(\cos
2\theta-\frac{2EV}{\Delta m^{2}}\right)^{2}+\sin^{2}2\theta}$ (12)

becomes at $E=E_{0}$

$\sin^{2}2\theta_{m}=\cos^{2}\theta.$

For $\theta=8.5^{0}$ the depth is close to maximum:
$\sin^{2}2\theta_{m}\approx 0.98$. Therefore, in matter, strong flavor
transitions can be preserved from decoherence over large distances $L$ from
the source to the detector.

Figure 1: Dependence of the ratio $L^{m}_{coh}/L_{coh}$ (9) on energy for
$\theta=\theta_{13}=8.5^{\circ}$. The ratio diverges at $E_{0}$, and $E_{R}$
is the MSW resonance energy.

###  2.2 Coherence in the configuration space

In the configuration space, the loss of propagation coherence is associated
with relative shift and eventually separation of the wave packets of the
eigenstates. The difference of group velocities of the eigenstates is given by

$\Delta v_{m}=-\frac{d\Delta H_{m}}{dE}.$ (13)

Then spatial separation (relative shift) $x_{shift}$ of the packets after
propagating a distance $L$ equals

$x_{shift}(L)=\int_{0}^{L}\Delta v_{m}dx=-\int_{0}^{L}\frac{d\Delta
H_{m}}{dE}dx.$

The coherence length $L^{m}_{coh}$ can be defined as the distance at which the
separation equals the spatial size of the packets $\sigma_{x}$:
$|x_{shift}(L^{m}_{coh})|\approx\sigma_{x}$. At this point, the overlap of the
packets becomes small. In the case of constant density this condition gives

$L^{m}_{coh}=\sigma_{x}\left|\frac{d\Delta H_{m}}{dE}\right|^{-1}.$ (14)

Since $\sigma_{x}=2\pi/(2\sigma_{E})$ (recall that
$\sigma_{E}\approx\sigma_{p}$ is the half-width), the expression (14)
coincides with expression for the length in the energy representation (3).
This shows the equivalence of the results in the configuration and the energy-
momentum spaces.

According to Eq. (13) at $E=E_{0}$ the eigenstates in matter have equal group
velocities and therefore do not separate. As a result, coherence is maintained
during infinite time: $L^{m}_{coh}\rightarrow\infty$. Thus, in the
configuration space the condition for infinite coherence is

$v_{1}=v_{2}.$ (15)

The infinite coherence energy $E_{0}$ coincides with the resonance energy of
oscillations of the mass eigenstates in matter $\nu_{1}\leftrightarrow\nu_{2}$
[14]. This is not accidental. The origin of the energy dependence of phase,
the difference of group velocities and therefore decoherence is the mass
states. The resonance means that at $E_{0}$ the mass states $\nu_{1}$ and
$\nu_{2}$ are maximally mixed in the eigenstates $\nu_{i}^{m}$. That is,
$\nu_{1}^{m}$ and $\nu_{2}^{m}$ both contain equal admixtures of $\nu_{i}$.
Consequently, the group velocities of $\nu_{1}^{m}$ and $\nu_{2}^{m}$ should
be equal. In other words, in a given flavor state the components $\nu_{1}$ and
$\nu_{2}$ oscillate with maximal depth
$\nu_{1}\rightarrow\nu_{2}\rightarrow\nu_{1}\rightarrow\nu_{2}$, which
compensate separation.

The derivative of the oscillation length in matter equals

$\frac{dl_{m}}{dE}=-\frac{2\pi}{(\Delta H_{m})^{2}}\frac{d\Delta H_{m}}{dE}.$

Therefore the condition of infinite coherence (4) coincides with the extremum
of oscillation length. For a given potential $V$, $l_{m}^{max}=2\pi/(V\sin
2\theta)$, which is larger than the length in the MSW resonance:
$l_{m}^{R}=2\pi/(V\tan 2\theta)$, and the difference is substantial for large
vacuum mixing.

Notice that decoherence is related to uncertainty in energy on a certain
interval of energy. Therefore infinite coherence energy should refer to the
average energy in a wave packet.

Notice that the equivalence of results in two representations originates from
the fact that the same quantity $d\Delta H_{m}/dE$ determines the change of
the oscillation phase with energy (in $E-p$ space) on the one hand side, and
the difference of the group velocities (in $x$-space) on the other hand.

###  2.3 Correlation of the phase difference and delay

The separation between the WP of eigenstates, $t\approx\Delta vX$, is
proportional to their phase difference (oscillation phase), $\phi=\Delta
H_{m}X$:

$t=g(E,V)\phi,$ (16)

where according to (7)

$g(E,V)=\frac{\Delta v}{\Delta H_{m}}=\frac{1}{\Delta
H_{m}}{2E^{2}}=\frac{1}{E}\frac{1-\frac{2VEc_{2\theta}}{\Delta
m^{2}}}{\left(c_{2\theta}-\frac{2VE}{\Delta
m^{2}}\right)^{2}+s^{2}_{2\theta}}.$

In terms of infinite coherence energy, $E_{0}$, it can be rewritten as

$g(E,V)=\frac{2Vc_{2\theta}}{\Delta
m^{2}}\frac{\frac{E_{0}}{E}-1}{\left(c_{2\theta}-\frac{E}{E_{0}c_{2\theta}}\right)^{2}+s^{2}_{2\theta}}.$

For $E\ll E_{0}$ (vacuum case)

$g(E,V)=\frac{1}{E}.$

At $E=E_{0}$ (near the MSW resonance): $g(E,V)=0$ \- there is no loss of
coherence, and the oscillation phase, being non-zero, is suppressed by mixing:

$\phi=\frac{\Delta m^{2}X}{2E_{0}}\tan 2\theta=\phi_{vac}(E_{0})\tan 2\theta.$

For $E\gg E_{0}$ (matter dominated case):

$g(E,V)\approx-\frac{c_{2\theta}\Delta
m^{2}}{2VE^{2}}=-\frac{E_{0}c_{2\theta}^{2}}{E^{2}}.$

Here also the relative separation is strongly suppressed, while oscillations
proceed with large phase $\phi\approx VX$.

Notice also that the relative separation changes the sign at $E_{0}$.
Therefore in layers $a$ and $b$ with two different densities one may have
opposite sign of a delay and also $t^{a}=-t^{b}$ (see below). The phases do
not change the sign and are always positive.

###  2.4 Physics of $L^{m}_{coh}\rightarrow\infty$

Here we present an interpretation of the divergence of $L^{m}_{coh}$ at
certain energy $E_{0}$. The transition probability is given by

$P_{\alpha\beta}(\theta_{m},\phi_{m})=\frac{1}{2}\sin^{2}2\theta_{m}\left(1-\cos\phi_{m}\right).$

It should be averaged over the energy interval $2\sigma_{E}$. The change of
$\sin^{2}2\theta_{m}$ with energy in the interval $E_{0}\pm\sigma_{E}$ equals

$\left|2\sigma_{E}\frac{d}{dE}\sin^{2}2\theta_{m}\right|=2\sigma_{E}\frac{Vc_{2\theta}}{\Delta
m^{2}}=\frac{\sigma_{E}}{E_{0}}\ll 1,$

where we have taken into account that at $E_{0}$ the mixing parameters are
$\sin^{2}2\theta_{m}=\cos^{2}\theta$ and $\cos^{2}2\theta_{m}=\sin^{2}\theta$.
Therefore $\sin^{2}2\theta_{m}$ can be put out of the averaging integral at
$E\sim E_{0}$. Furthermore, to simplify consideration, instead of averaging
over $E$ we will average $P_{\alpha\beta}$ immediately over the phase
$\phi_{m}$ in the interval $2\delta\phi_{m}$ determined by

$2\delta\phi_{m}=\frac{d\phi_{m}}{dE}2\sigma_{E}=\frac{d\Delta
H_{m}}{dE}L2\sigma_{E}.$ (17)

The averaging of the oscillatory term yields

$\frac{1}{2\delta\phi_{m}}\int_{\phi_{m}-\delta\phi_{m}}^{\phi_{m}+\delta\phi_{m}}d\phi_{m}^{\prime}P_{\alpha\beta}(\phi_{m}^{\prime})=\frac{1}{2}\sin^{2}2\theta_{m}(E_{0})\left[1-D(\delta\phi_{m})\cos\phi_{m}\right],$

where

$D(\delta\phi_{m})\equiv\frac{\sin\delta\phi_{m}}{\delta\phi_{m}}$

is the decoherence factor which describes suppression of the interference
term.

According to (17) for $E=E_{0}$ the variation of phase is $\delta\phi_{m}=0$,
and consequently, $D(\delta\phi_{m})\rightarrow 1$. The averaging effect is
small in the region around $E_{0}$. Indeed, $D(\delta\phi_{m})$ has a peak
centered at $\delta\phi_{m}=0$. The half-width at nearly half of maximum,
$D(\delta\phi_{m})=0.5$ corresponds to

$|\delta\phi_{m}|\approx 1.89.$ (18)

Consequently, the half-width of the peak in the energy scale, $\Gamma$, is
determined by the condition

$|\delta\phi_{m}(E_{0}+\Gamma)|=2$

(rounding (18) to $2$). Using this value and expressions in (17) and (7), we
find the relative half-width of the peak:

$\frac{\Gamma}{E_{0}}=\frac{\tan 2\theta}{\sqrt{\pi^{2}d^{2}-1}},$ (19)

where

$d\equiv\frac{\sigma_{E}}{E_{0}}\frac{L}{l_{\nu}}.$

Notice that (19) requires $d\pi>1$, on explicitly
$L>l_{\nu}E/(\pi\sigma_{E})\approx L_{coh}$. For $d\pi\rightarrow 1$, the
width $\Gamma\rightarrow\infty$, which means that coherence is well satisfied
for all the energies. Moreover, the peak disappears (half a height does not
exist).

For $\pi d\gg 1$ the Eq. (19) becomes

$\frac{\Gamma}{E_{0}}=\frac{\tan 2\theta}{\pi d}=\tan
2\theta\frac{E_{0}}{\sigma_{E}}\frac{l_{\nu}}{\pi L}.$ (20)

The larger $L$, as well as $\sigma_{E}$, the narrower the peak. Also,
according to (20) the width of the peak increases with energy:

$\frac{\Gamma}{E_{0}}\propto E_{0}^{2}.$

###  2.5 Adiabatic evolution and infinite coherence

In the case of slow adiabatic density change [10, 11, 14], one can introduce
the instantaneous eigenstates and eigenvalues $H_{im}(x)$ and their difference
$\Delta H_{m}(x)$, which are well-defined quantities. The adiabatic
oscillation phase is given by the integral

$\phi_{m}^{ad}(L)=\int_{0}^{L}dx\Delta H_{m}(N_{e}(x)).$ (21)

A change of the phase with the energy in the interval $2\sigma_{E}$ equals

$\Delta\phi_{m}^{ad}(L)=2\sigma_{E}\int_{0}^{L}dx\frac{d}{dE}\Delta H_{m}(x),$

where we permuted the intergration over $x$ and differentiation over $E$. Then
the condition for decoherence is $\Delta\phi_{m}(L^{m}_{coh})=2\pi$, or
explicitly,

$\left|\int_{0}^{L^{m}_{coh}}dx\frac{d}{dE}\Delta
H_{m}(x)\right|=\frac{\pi}{\sigma_{E}},$

and $d\Delta H_{m}(x)/dE$ is given in (7).

In the configuration space the adiabatic evolution means that there are no
transitions between the eigenstates. Therefore a propagation is described by
two wave packets, which do not change shape factors as in the constant density
case. Separation of the WP equals

$\Delta x(L)=\int_{0}^{L}dx\frac{d}{dE}\Delta H_{m}(x),$ (22)

and from (21), (22) we obtain

$\frac{d\phi_{m}^{ad}}{dE}=\Delta x(L),$

which is the basis of equivalence of results in the $x$\- and
$E$-representations. In particular, the infinite coherence condition,
$d\phi_{m}^{ad}/dE=0$, correspons to zero separation.

Let us consider coherence for different density (potential) profiles. For
monotonous change of the potential the infinite $L_{coh}^{m}$ can not be
realized. Indeed, for a given $E$ the pole condition is satisfied for specific
value of the potential $V_{0}(E)=V(E,x_{0})$ in specific point $x_{0}$. In the
point $x=x_{0}+\Delta x$, where $V(x_{0}+\Delta x)$ is outside the
$L_{coh}$-peak, significant enhancement of coherence is absent and
$L_{coh}^{m}<\Delta x$.

Certain increase of the coherence length can be related to the fact that at
$V_{0}$ the derivative $d\Delta H_{m}(x)/dE$ changes the sign which suppresses
the integral over $x$. This corresponds to the change of sign of the
difference of the group velocities at $E_{0}$ ($V_{0}$).

For a layer with a given length $L$ the energy of zero separation (complete
overlap) is realized (22) when

$\int_{0}^{x_{0}}dx\frac{d}{dE}\Delta
H_{m}(x)=-\int_{x_{0}}^{L}dx\frac{d}{dE}\Delta H_{m}(x),$ (23)

and $x_{0}=x_{0}(E)$.

The infinite coherence can be achieved in the potential with periodic
modulations:

$V(x)=\bar{V}(1+h\sin 2\pi x/X).$ (24)

For a large $X$, the adiabaticity condition is satisfied. Zero separation of
WP, $\Delta x=0$, is realized when the difference of group velocities has
different signs in the first and the second parts of the period. So, there is
a continuous "catch-up" effect.

According to (7) the energy $E_{0}$ of infinite coherence is obtained from the
condition

$lim_{L\rightarrow\infty}\int_{0}^{L}dx\frac{\left(\frac{\Delta
m^{2}}{2E}-V\cos 2\theta\right)}{\Delta H_{m}(x)}=0.$ (25)

Around the zero value of nominator, denominator changes much slower and it can
be put out of integral at some average value of the potential. Then the
condition (25) reduces to

$\frac{\Delta m^{2}}{2E}L-\cos 2\theta\int_{0}^{L}dxV(x)=0.$

Performing explicit integration with the potential (24) we obtain

$\frac{\Delta m^{2}}{2E_{0}}\approx\cos
2\theta\bar{V}+\frac{h}{\pi}\frac{X}{L}\sin^{2}\frac{\pi L}{X}.$ (26)

In the limit $L\rightarrow\infty$ the last term on the RHS of (26) can be
neglected and for $E_{0}$ we obtain the same expression (2.1) as in constant
density case with the average potential $\bar{V}$.

In general, if $V$ varies about some average value $\bar{V}$ (even
irregularly), there may be energy where $L_{coh}^{m}\rightarrow\infty$.

###  2.6 Coherence for massless neutrinos

The limit $L_{coh}^{m}\rightarrow\infty$ is realized for oscillations of
massless neutrinos, as originally introduced by Wolfenstein [28]. Here the
oscillation phase is independent of neutrino energy. Recall that the
propagation decoherence is related to the energy dependence of level splitting
$\Delta H_{m}$ and shows up as averaging over energy. Therefore the
decoherence does not exist for oscillations of massless neutrinos driven by
potentials.

##  3 Coherence and adiabaticity violation

Adiabaticity violation means transitions between the eigenstates of
propagation $\nu_{1m}\leftrightarrow\nu_{2m}$ and therefore change of shape of
the WPs. Here we will consider the cases of extreme adiabaticity violation
when $dV/dx\rightarrow\infty$, which corresponds to density jumps at certain
spatial points.

Extreme adiabaticity violation combined with shift and separation of the WPs
leads to splits and catch up of the WPs [12]. We will first illustrate these
effects considering a single density jump.

###  3.1 Coherence in the case of single density jump

Consider a two layers ($a$ and $b$) profile with density jump at a border
between them. The layers have matter potentials $V_{a}$ and $V_{b}$ and
lengths $L_{a}$ and $L_{b}$, so that the total length equals $L\equiv
L_{a}+L_{b}$. Let $\theta_{k}=\theta_{m}(V_{k})$ be the flavor mixing angle in
layer $k$, ($k=a,\,b$).

Suppose in the beginning of layer $a$ the state is

$\nu(x=0,t)=c_{a}^{\prime}f_{1}(t)\nu_{1}^{a}+s_{a}^{\prime}f_{2}(t)\nu_{2}^{a},$
(27)

where we use abbreviations $c_{a}^{\prime}\equiv\cos\theta_{a}$,
$s_{a}^{\prime}\equiv\sin\theta_{a}$ and $f_{i}(t)$ are the shape factors of
the wave packets normalized as

$\int dt|f_{i}(t)|^{2}=1.$

Evolving to the border between the layers $a$ and $b$ the state (27) becomes

$\nu(L_{a},t)=c_{a}^{\prime}f(t-t_{1}^{a})e^{i2\phi_{1}^{a}}\nu_{1}^{a}+s_{a}^{\prime}f(t-t_{2}^{a})e^{i2\phi_{2}^{a}}\nu_{2}^{a}.$
(28)

Here $t_{i}^{a}$ are the times of propagation in the $a$-layer,
$t_{i}^{a}=L_{a}/v_{i}^{a}$, $v_{i}^{a}$ are group velocities, and
$\phi_{i}^{a}$ are phases acquired by the eigenstates taken at the average
energies in the packets. We assume that in the beginning ($t=0$) the shape
factors of $\nu_{1}^{a}$ and $\nu_{2}^{a}$ are equal.

Crossing the border between $a$ and $b$ each eigenstate $\nu_{i}^{a}$ splits
into eigenstates $\nu_{j}^{b}$ of the layer $b$:

$\nu_{1}^{a}=c_{\Delta}\nu_{1}^{b}+s_{\Delta}\nu_{2}^{b},\,\,\,\,\nu_{2}^{a}=c_{\Delta}\nu_{2}^{b}-s_{\Delta}\nu_{1}^{b},$
(29)

where $\Delta\equiv\theta_{b}-\theta_{a}$. Inserting (29) into (28) we obtain
the state in the beginning of layer $b$. Then to the end of the layer $b$ the
state (28) evolves to

$\displaystyle\nu(L,t)=\left[c_{a}^{\prime}c_{\Delta}f(t-t_{1}^{a}-t_{1}^{b})e^{i2(\phi_{1}^{a}+\phi_{1}^{b})}-s_{a}^{\prime}s_{\Delta}f(t-t_{2}^{a}-t_{1}^{b})e^{i2(\phi_{2}^{a}+\phi_{1}^{b})}\right]\nu_{1}^{b}+$
$\displaystyle\left[c_{a}^{\prime}s_{\Delta}f(t-t_{1}^{a}-t_{2}^{b})e^{i2(\phi_{1}^{a}+\phi_{2}^{b})}+s_{a}^{\prime}c_{\Delta}f(t-t_{2}^{a}-t_{2}^{b})e^{i2(\phi_{2}^{a}+\phi_{2}^{b})}\right]\nu_{2}^{b}.$
(30)

We put out the common phase factor $e^{i2(\phi_{1}^{a}+\phi_{1}^{b})}$ and
introduce the phase differences,

$\phi^{a}\equiv\phi_{2}^{a}-\phi_{1}^{a}\,\,\,\,\text{and}\,\,\,\,\phi^{b}\equiv\phi_{2}^{b}-\phi_{1}^{b}.$
(31)

Then we make the time shift as $t^{\prime}=t-t_{1}^{a}-t_{1}^{b}$, which
allows us to express results in terms of the relative time shifts of the
eigenstates 1 and 2 in the layers $a$ and $b$:

$t^{a}\equiv t_{2}^{a}-t_{1}^{a}\,\,\,\,\text{and}\,\,\,\,t^{b}\equiv
t_{2}^{b}-t_{1}^{b}.$ (32)

Using these quantities and projecting (3.1) onto $\nu_{e}$ we obtain the
amplitude of $\nu_{e}\rightarrow\nu_{e}$ transition over the two layer
profile:

$\displaystyle A_{ee}$ $\displaystyle=$ $\displaystyle
c_{\Delta}\left[c_{a}^{\prime}c_{b}^{\prime}f(t^{\prime})e^{-i\phi}+s_{a}^{\prime}s_{b}^{\prime}f(t^{\prime}-t^{a}-t^{b})e^{i\phi}\right]$
(33) $\displaystyle+$ $\displaystyle
s_{\Delta}\left[c_{a}^{\prime}s_{b}^{\prime}f(t^{\prime}-t^{b})e^{-i\phi^{\prime}}-s_{a}^{\prime}c_{b}^{\prime}f(t^{\prime}-t^{a})e^{i\phi^{\prime}}\right].$

In this expression we put out the factor $e^{i(\phi^{a}+\phi^{b})}$ and
introduce

$\phi\equiv\phi^{a}+\phi^{b}\,\,\,\,\text{and}\,\,\,\,\phi^{\prime}\equiv\phi^{a}-\phi^{b}.$
(34)

###  3.2 Specific examples

Let us consider specific cases.

1\. Suppose $|t^{a}|\gg\sigma_{t}$, that is, the separation of the WPs in the
layer $a$ is larger than the coherence time, which means that the WPs are
completely separated and coherence is lost. Suppose also that
$|t^{b}|\ll\sigma_{t}$, i.e. loss of coherence in the layer $b$ is negligible,
$|t^{b}|\approx 0$. Then taking into account that the shape factors
$f(t^{\prime})$ and $f(t-t^{a})$ do not overlap we obtain from (33)

$\displaystyle P_{ee}$ $\displaystyle=$ $\displaystyle\int dt|A_{ee}|^{2}=\int
dt|f(t^{\prime})|^{2}\left|c_{a}^{\prime}c_{b}^{\prime}c_{\Delta}+c_{a}^{\prime}s_{b}^{\prime}s_{\Delta}e^{i2\phi^{b}}\right|^{2}$
(35) $\displaystyle+$ $\displaystyle\int
dt|f(t^{\prime}-t^{a})|^{2}\left|s_{a}^{\prime}s_{b}^{\prime}c_{\Delta}e^{i2\phi^{b}}-s_{a}^{\prime}c_{b}^{\prime}s_{\Delta}\right|^{2}.$

We assume that change of the oscillation phase $\phi^{b}(t)$ along the WP is
negligible, so that the oscillatory factors can be put out of the integral and
take into account that integrations of the moduli squared of the shape factors
give $1$ due to normalization. As a result, we obtain

$P_{ee}=\left|c_{a}^{\prime}c_{b}^{\prime}c_{\Delta}+c_{a}^{\prime}s_{b}^{\prime}s_{\Delta}e^{i2\phi^{b}}\right|^{2}+\left|s_{a}^{\prime}s_{b}^{\prime}c_{\Delta}e^{i2\phi^{b}}-s_{a}^{\prime}c_{b}^{\prime}s_{\Delta}\right|^{2}.$
(36)

Computing the amplitudes in the energy space, we need to use the plane waves
so that $f(t)=1$, and the energy dependent phases $\phi^{a}(E)$ and
$\phi^{b}(E)$. The probability is given by

$P_{ee}=\int dEF(E)|A_{ee}|^{2},$ (37)

where $F(E)$ is the energy spectrum of neutrinos that corresponds to the WP in
the $E$-$p$ space $F(E)=|f(E)|^{2}$ and $f(E)$ is the Fourier transform of
$f(t)$. The amplitude (33) can be written as

$A_{ee}=e^{-i\phi(E)}A_{1}+e^{i\phi^{\prime}(E)}A_{2},$ (38)

with

$A_{1}=c_{a}^{\prime}\left[c_{b}^{\prime}c_{\Delta}+s_{b}^{\prime}s_{\Delta}e^{i2\phi^{b}(E)}\right],\,\,\,\
A_{2}=s_{a}^{\prime}\left[s_{b}^{\prime}c_{\Delta}e^{i2\phi^{b}(E)}-c_{b}^{\prime}s_{\Delta}\right].$
(39)

Here $A_{i}$ are the oscillations amplitudes of the eigenstates of the layer
$a$, $\nu^{a}_{i}$, in the layer $b$, $\nu_{i}^{a}\rightarrow\nu_{e}$.
Inserting (38) into (37) we have

$P_{ee}=\int dEF(E)\left[|A_{1}|^{2}+|A_{2}|^{2}+2|A_{1}A_{2}|\cos
2(\phi^{a}+\chi)\right],$ (40)

where $\chi\equiv Arg(A_{1}^{*}A_{2})$ depends on the phase $\phi^{b}$. To
compare this with consideration in the $x$-space we assume that $\phi^{a}\gg
1$ (which corresponds to large delay in the layer $a$). In contrast, the phase
$\phi^{b}$ is relatively small, $\phi^{b}\ll 1$. In this case we can put all
the terms but those which depend on $\phi^{a}$ in (40) out of the integral
taking them at average value of $E$ in the spectrum. Then due to normalization
$\int dEF(E)=1$ the expression (40) becomes

$P_{ee}=|\bar{A}_{1}|^{2}+|\bar{A}_{2}|^{2}+2|\bar{A}_{1}\bar{A}_{2}|\int
dEF(E)\cos 2(\phi^{a}+\chi).$ (41)

Here $\bar{A}_{i}$ are the amplitudes at the average value of energy. The last
term is suppressed as $1/\phi^{a}$ and negligible for $\phi^{a}\gg 1$. Thus,
the result (41), with $A_{i}$ in (39), coincides with that in the
$x$-representation (36).

In the example considered above, the result does not depend on $L_{a}$, and
$L_{a}\rightarrow\infty$ is possible. In a sense, one can consider this as the
case of infinite coherence since oscillations can be observed at the arbitrary
long distance from the source. Thus, density jump and splitting of eigenstates
induce or restore the interference and therefore oscillations. Here
suppression of propagation coherence is determined by properties of layer $b$.
That is, the problem is reduced to a single layer with constant density.
Depending on initial mixing and $\Delta\theta$, crossing the density jump can
lead to even stronger interference than at the beginning of layer $a$.

The situation described above is realized, e.g., for the solar and supernova
neutrinos oscillating inside the Earth, where $a$ is the vacuum between the
star and the Earth, and $b$ is the Earth. In the high energy part of solar
neutrinos: $c_{a}^{2}\ll s_{a}^{2}$.

2\. Suppose the layer $b$ has density at which velocities of eigenstates are
equal so that coherence is maintained for arbitrary large $L_{b}$ and
$t^{b}=0$. Furthermore, suppose in the layer $a$ the wave packets are
completely separated, $L_{a}\rightarrow\infty$ which means that dependence on
$\phi^{a}$ disappears. Then according to (33),

$\displaystyle A_{ee}$ $\displaystyle=$ $\displaystyle
c_{a}^{\prime}c_{b}^{\prime}c_{\Delta}f(t)e^{i\phi}-s_{a}^{\prime}c_{b}^{\prime}s_{\Delta}f(t-t^{a})e^{i\phi^{\prime}}+$
(42) $\displaystyle
c_{a}^{\prime}s_{b}^{\prime}s_{\Delta}f(t)e^{-i\phi^{\prime}}+s_{a}^{\prime}s_{b}^{\prime}c_{\Delta}f(t-t^{a})e^{i\phi}.$

The interference of the overlapping parts (they have the same argument in $f$)
equals

$t:\,\,2c_{a}^{\prime 2}c_{b}^{\prime}s_{b}^{\prime}s_{\Delta}c_{\Delta}\cos
2\phi^{b}(t),\,\,\,\,\,\,(t-t^{a}):\,\,-2s_{a}^{\prime
2}c_{b}^{\prime}s_{b}^{\prime}s_{\Delta}c_{\Delta}\cos 2\phi^{b}(t-t^{a}).$
(43)

At a detector, both oscillation phases are equal to $\phi^{b}$, and, the sum
of interference terms (after integration over time) is

$\cos 2\theta_{a}\sin 2\theta_{b}s_{\Delta}c_{\Delta}\cos 2\phi^{b}.$ (44)

This can be compared to the depth of interference in the beginning of the
layer $a$: $0.5\sin^{2}2\theta_{a}$.

Regions of parameters exist where the depth increases after propagation in the
two-layer profile.

Apart from the restoration of the interference, there is another phenomenon
which will be important for our further consideration, namely, change of the
shape of the WP at crossing the border between the layers:

  * •

Each packet $\nu_{i}^{a}$ becomes the two-component one with an overall size
extended by the relative delay in the layer $a$, $t^{a}$.

  * •

According to (42) the amplitudes of components of $\nu_{1}^{b}$ equal
$(c_{a}^{\prime}c_{\Delta},\,\,-s_{a}^{\prime}s_{\Delta})$ for the earlier and
the later ones correspondingly. The amplitudes in $\nu_{2}^{b}$ are
$(c_{a}^{\prime}s_{\Delta},\,\,s_{a}^{\prime}c_{\Delta})$.

  * •

The shape depends on the size of density jump. If, e.g., the jump is small,
$s_{\Delta}\sim s_{a}^{\prime}\sim\epsilon\ll 1$, we find from the previous
item the amplitudes for $\nu_{1}^{b}$: $(1,\,\epsilon^{2})$, and for
$\nu_{2}^{b}$: $(\epsilon,\,\epsilon)$. If $s_{\Delta}\approx c_{\Delta}$, the
packets will have similar form with similar amplitudes $\nu_{1}^{b}$:
$(c_{a}^{\prime},\,-s_{a}^{\prime})$ and $\nu_{2}^{b}$:
$(c_{a}^{\prime},\,s_{a}^{\prime})$. If $s_{\Delta}=1$, $c_{\Delta}=\epsilon$
we have $\nu_{1}^{b}$: $(\epsilon,\,\epsilon)$ and $\nu_{2}^{b}$:
$(1,\,\epsilon^{2})$.

3\. Consider the case $t^{a}=-t^{b}$, when separation in the layer $b$
compensates separation in $a$. In spite of this compensation, and in contrast
to the constant density or adiabatic cases, there is no complete overlap at
the end of layer $b$ due to split of the eigenstates. Only WP of $A_{11}$ and
$A_{22}$ components will overlap completely, where subscripts indicate
eigenstates in which a given final component propagated in layer $a$ and $b$
(e.g. $A_{11}$ is the amplitude of WP $\nu_{1}^{a}\rightarrow\nu_{1}^{b}$).
The component $A_{12}$ will shift forward by $t^{a}$ and $A_{21}$ backward by
$t^{a}$ with respect to overlapping components, or vice versa, depending on
the sign of $t^{a}$. If $|t^{a}|>\sigma_{x}$, only $A_{11}$ and $A_{22}$
interfere and the interference term equals

$0.5\sin 2\theta_{a}\sin 2\theta_{b}c_{\Delta}^{2}\cos 2(\phi^{a}+\phi^{b}),$
(45)

which again can be bigger than interference in the beginning of the layer $a$.

4\. In the case $t^{a}\approx 0$, there is no separation and loss of coherence
in the layer $a$. The components $A_{11}$ and $A_{21}$ as well as $A_{22}$ and
$A_{12}$ will overlap, there is no split of eigenstates at the border
$a\rightarrow b$.

There is no unambiguous way of introducing the coherence length in the
presence of density jump in the profile. Recall that the equivalence in the
$E$\- and $x$-representations is established at the level of averaging of the
oscillation phase and separation of the WP. In the case of a single jump,
there are two different oscillation phases and two different delays due to
splitting of eigenstates. The introduction of a single effective phase and
effective delay is non-trivial. As we will see in sect. 4, this can be done
for periodic structures with density jumps.

###  3.3 Wave packets in the $x$\- and $E$-representations

Split of the eigenstates and delays of splitted components lead to substantial
modifications of the shape factor of the total WPs, $\psi_{i}(x,t)$, in the
$x$-space. At the same time, the Fourier transforms of $\psi_{i}(x,t)$
determine the neutrino energy spectrum. Let us show that despite substantial
modifications of $\psi_{i}(x,t)$, the energy spectrum remains unchanged, as it
should be [26].

Suppose $f(E)$ is the Fourier transform of $f(t)$, then in the
$E$-representation the initial state (27) is

$\nu(x=0,E)=f(E)(c_{a}^{\prime}\nu_{1}^{a}+s_{a}^{\prime}\nu_{2}^{a}).$

The energy spectrum of this state equals

$F(E)=|f(E)c_{a}^{\prime}|^{2}+|f(E)s_{a}^{\prime}|^{2}=|f(E)|^{2}.$ (46)

Consider the simplest element of the profile which leads to modification of
the shape factor: the layer $a$ with constant density and density jump at the
end (general case is just repetition of this element). State (3.1), at the
beginning of layer $b$ ($t^{b}_{i}=0$ and $\phi^{b}_{i}=0$) in terms of
$\psi_{i}(x=L_{a},t)$ is

$\nu(L_{a},t)=\psi_{1}(L_{a},t)\nu_{1}^{b}+\psi_{2}(L_{a},t)\nu_{2}^{b},$ (47)

where the total WPs equal

$\displaystyle\psi_{1}(L_{a},t)$ $\displaystyle=$
$\displaystyle[c_{a}^{\prime}c_{\Delta}f(t)-s_{a}^{\prime}s_{\Delta}f(t-t^{a})e^{2i\phi^{a}}],$
$\displaystyle\psi_{2}(L_{a},t)$ $\displaystyle=$
$\displaystyle[c_{a}^{\prime}s_{\Delta}f(t)+s_{a}^{\prime}c_{\Delta}f(t-t^{a})e^{2i\phi^{a}}].$
(48)

Here we made time shift by $t_{1}^{a}$ and put out common phase
$\phi^{a}_{1}$. In the beginning, $\psi_{i}$ are the elementary WPs,
$f(t)\nu_{i}^{a}$, as in (27), but along the propagation their widths increase
and their forms change.

In $E$-space, the state just after crossing the density jump, $\nu(L_{a})$, in
Eq. (47), is:

$\nu(L_{a},E)=\psi_{1}(L_{a},E)\nu_{1}^{b}+\psi_{2}(L_{a},E)\nu_{2}^{b},$ (49)

The Fourier transform of $f(t-t^{a})$ is $f(E)e^{iEt^{a}}$, so that

$\displaystyle\psi_{1}(L_{a},E)$ $\displaystyle=$
$\displaystyle[c_{a}^{\prime}c_{\Delta}-s_{a}^{\prime}s_{\Delta}e^{-iEt^{a}+2i\phi^{a}}]f(E),$
$\displaystyle\psi_{2}(L_{a},E)$ $\displaystyle=$
$\displaystyle[c_{a}^{\prime}s_{\Delta}+s_{a}^{\prime}c_{\Delta}e^{-iEt^{a}+2i\phi^{a}}]f(E).$

Using these expressions, we find the energy spectrum

$F(E)=|\psi_{1}(L_{a},E)|^{2}+|\psi_{2}(L_{a},E)|^{2}=|f(E)|^{2},$

which coincides with (46).

##  4 Coherence in the castle-wall profile

One can introduce the coherence length in the case of a periodic structure
with a sharp density change. The castle-wall (CW) profile is multiple
repetitions of the two layer structure considered sect. 3. This is an
explicitly solvable example of the periodic profile, which can give an idea
about effects in profiles with sine (or cosine) type of dependence on distance
[21, 24, 25].

###  4.1 Parametric oscillations in the energy-momentum space

The oscillation probability after crossing $n$ periods can be obtained from
the probability for a single period computed above. Indeed, the amplitude
$A_{ee}$ (38),(39) can be written as

$A_{ee}=R+iI_{3},$

where the real and imaginary parts of the amplitude equal respectively, see
[24, 25],

$R=c_{a}c_{b}-s_{a}s_{b}\cos\left(2\theta_{a}-2\theta_{b}\right),$ (50)

and

$I_{3}=-(s_{a}c_{b}\cos 2\theta_{a}+s_{b}c_{a}\cos 2\theta_{b}).$

Then the probability is given by

$P_{ee}=R^{2}+I_{3}^{2}.$

Similarly to $A_{ee}$ one can find the amplitudes $A_{e\mu}$, $A_{\mu e}$ and
$A_{\mu\mu}$ which compose the evolution matrix $U_{L}$ over one period. The
matrix allows to reconstruct the Hamiltonian integrated over the period:
$iH_{CW}L=I-U_{L}$. Diagonalization of this Hamiltonian gives the eigenstates,
i.e. mixing and difference of eigenvalues which allow to write the transition
probability after passing $n$ periods of size $L$,
$P=|U(nL)_{\alpha\beta}|^{2}$, as

$P(\nu_{\alpha}\rightarrow\nu_{\beta},nL)=\left(1-\frac{I_{3}^{2}}{1-R^{2}}\right)\sin^{2}\left(n\xi\right).$
(51)

Here $\xi$ is the effective phase acquired over period:

$\xi\equiv\arccos R.$

The factor in front of sine gives the depth of the parametric oscillations.
The depth is maximal at $I_{3}=0$, or explicitly

$s_{a}c_{b}\cos 2\theta_{a}+c_{a}s_{b}\cos 2\theta_{b}=0,$ (52)

which is the condition of the parametric resonance. For arbitrary mixings the
condition is satisfied for values of phases

$\phi^{a}=\frac{\pi}{2}+k\pi\hskip 14.22636pt\text{and}\hskip
14.22636pt\phi^{b}=\frac{\pi}{2}+k^{\prime}\pi.$

In general, $I_{3}=0$ requires specific correlations between phases and
mixings. The transition probability (51) reaches maximum, $P=1$, when

$n\xi=\frac{\pi}{2}+k\pi.$

In what follows for illustration of the results we will use a castle-wall
profile with $n=5$ periods and the following parameters:

$V_{a}=5.3\times 10^{-4}\,\text{eV}^{2},\,\,\,\,\,L_{a}=4\,{\rm
km},\,\,\,V_{b}=2\times 10^{-5}\,\text{eV}^{2},\,\,\,L_{b}=2\,{\rm km}.$ (53)

We take the vacuum oscillation parameters $\theta=8.5^{\circ}$ and $\Delta
m^{2}=2.5\times 10^{-3}$ $\text{eV}^{2}$.

The transition probability of parametric oscillations Eq. (51), as function of
energy, is shown in fig. 2. The probability reaches maximum at $E\sim 2.2$
(MSW resonance), $3.1$ and $7.7$ MeV, where condition (52) is met.

Figure 2: Transition probability of the $\nu_{e}\rightarrow\nu_{\mu}$
parametric oscillations as a function of neutrino energy for castle-wall
profile with $5$ periods. Parameters of the CW profile are given in Eq. (53).
We used $\theta=8.5^{\circ}$ and $\Delta m^{2}=2.5\times 10^{-3}$
$\text{eV}^{2}$.

###  4.2 Coherence length in a castle-wall profile

In the energy representation we can repeat here the same procedure of
determination of the coherence length as in the case of constant density.
According to Eq. (51), the phase of parametric oscillations equals

$\phi_{n}=n\xi=nL\frac{\xi}{L}.$

Therefore the effective difference of eigenvalues equals

$\Delta H_{cw}=\frac{\xi}{L}.$

We introduce $n_{coh}$, the number of periods over which the coherence is
maintained, so that $L_{coh}=n_{coh}L$. Using Eq. (3) for the coherence length
we find

$n_{coh}=\frac{\pi}{\sigma_{E}}\left|\frac{d\xi}{dE}\right|^{-1}.$

According to (3) and (50)

$\frac{d\xi}{dE}=-\frac{1}{\sqrt{1-R^{2}}}\frac{dR}{dE},$ (54)

and consequently,

$n_{coh}(E)=\frac{\pi\sqrt{1-R^{2}}}{2\sigma_{E}\left|\frac{dR}{dE}\right|}.$
(55)

The condition for infinite coherence length, $L_{coh}\propto
n_{coh}\rightarrow\infty$, is

$\left|\frac{d\xi}{dE}\right|=0,\,\,\,\,{\rm
or}\,\,\frac{dR(E)}{dE}=0,\,\,\,(R\neq 1).$ (56)

Since $R(E)$ is an oscillatory function of $E$, and therefore has several
maxima and minima at certain energies $E_{0}^{i}$, the condition (56) is
satisfied at these energies $E=E_{0}^{i}$.

As in the case of constant density, averaging of probability over the energy
intervals $\sigma_{E}$ leads to the appearance of the decoherence factors
$D_{i}(E)$, (2.4), centered at energies $E_{0}^{i}$. Let us find the widths of
the peaks of $D_{i}(E)$ for fixed length of trajectory $nL$. As in constant
density case, instead of averaging over $E$ we perform averaging over the
effective phase in the interval which corresponds to $2\sigma_{E}$:

$\delta\phi_{n}=2\sigma_{E}n\frac{d\xi}{dE}.$ (57)

The decoherence factor

$D(\delta\phi_{n}(E))=\frac{\sin\delta\phi_{n}}{\delta\phi_{n}}$

has a width $\Gamma$ determined by the condition in (2.4). Plugging
$\delta\phi_{n}$ from (57) in the above equation we obtain

$\left|\frac{d\xi(E_{0}^{i}+\Gamma)}{dE}\right|=\frac{1}{\sigma_{E}n}.$ (58)

For narrow peaks, $\Gamma\ll E_{0}^{i}$, we can expand the left hand side of
Eq. (58) around $E_{0}$; and take into account that $d\xi(E_{0})/dE=0$. Then
Eq. (58) reduces to

$\frac{d^{2}\xi(E)}{dE^{2}}\bigg{|}_{E=E_{0}^{i}}\times\Gamma=\frac{1}{\sigma_{E}n}.$
(59)

Differentiating $d\xi/dE$ in (54) over $E$ we have

$\frac{d^{2}\xi(E)}{dE^{2}}\bigg{|}_{E=E_{0}^{i}}=-\frac{1}{\sqrt{1-R^{2}(E_{0}^{i})}}\frac{d^{2}R(E_{0}^{i})}{dE^{2}}.$
(60)

Then insertion of this into (59) gives

$\Gamma\approx\frac{\sqrt{1-R^{2}(E_{0}^{i})}}{\sigma_{E}n\left|\frac{d^{2}R(E_{0}^{i})}{dE^{2}}\right|}.$

In the limits $E_{0}^{i}\ll E_{a},E_{b}$ and $E_{0}^{i}\gg E_{a},E_{b}$, where
$E_{a}$ and $E_{b}$ are the MSW resonance energies, we have
$\theta_{a}\approx\theta_{b}$, and therefore

$R\approx\cos(\phi^{a}+\phi^{b}).$

Since $d\xi/dE\propto d(\phi^{a}+\phi^{b})/dE=0$ at $E=E_{0}^{i}$ the equation
(60) becomes

$\frac{d^{2}\xi}{dE^{2}}\bigg{|}_{E=E_{0}}\approx\frac{d^{2}(\phi^{a}+\phi^{b})}{dE^{2}}\bigg{|}_{E=E_{0}},$

and the width equals

$\Gamma_{i}\approx\frac{1}{\sigma_{E}n\frac{d^{2}(\phi^{a}+\phi^{b})}{dE^{2}}\big{|}_{E=E_{0}^{i}}}.$

The widths $\Gamma^{i}$ are narrower at lower energies due to faster
variations of the half-phases $\phi^{a}$ and $\phi^{b}$. Similarly to the case
of constant matter density, the ranges of weak averaging effect of transition
probability are smaller at lower $E_{0}^{i}$.

###  4.3 Infinite coherence and parametric resonance

The energies of infinite coherence, $E_{0}^{i}$, are correlated to the
energies of parametric resonances, determined by the condition $I_{3}=0$. This
correlation is analogous to the one in Eq. (8) for constant density. To show
this, we consider the expressions for $dR/dE$ and $I_{3}$ in three different
regions of energies. Starting with the explicit expression for $dR/dE$,

$\frac{dR}{dE}=-s_{a}c_{b}\frac{d\phi^{a}}{dE}-c_{a}s_{b}\frac{d\phi^{b}}{dE}-c_{a}s_{b}\cos(2\theta_{a}-2\theta_{b})\frac{d\phi^{a}}{dE}\\\
-s_{a}c_{b}\cos(2\theta_{a}-2\theta_{b})\frac{d\phi^{b}}{dE}+s_{a}s_{b}\sin(2\theta_{a}-2\theta_{b})\frac{d(2\theta_{a}-2\theta_{b})}{dE}.$
(61)

We obtain the following

  1. 1.

Far from the MSW resonances of both layers: $E\ll E_{a},\,E_{b}$, where
$\theta_{a}\approx\theta_{b}\approx\theta$, or $E\gg E_{a},\,E_{b}$ where
$\theta_{a}\approx\theta_{b}\approx\pi/2$. In both cases
$\cos(2\theta_{a}-2\theta_{b})\approx 1$, and consequently,
$R\approx\cos(\phi^{a}+\phi^{b})$. Therefore

$\frac{dR}{dE}=-\sin(\phi^{a}+\phi^{b})\frac{d(\phi^{a}+\phi^{b})}{dE}$

On the other hand

$I_{3}\approx\sin(\phi^{a}+\phi^{b})\cos 2\theta_{a}.$

So that, the condition $I_{3}\rightarrow 0$ requires
$\sin(\phi^{a}+\phi^{b})=0$ and consequently, $\frac{dR}{dE}\rightarrow 0$,
which implies $n_{coh}\rightarrow\infty$.

  2. 2.

In the MSW resonances: when $E$ coincides with one of the resonance energies
e.g. $E\approx E_{a}$, and is far enough from the other one, $E_{b}$, then
$\theta_{a}=\pi/4$, $\theta_{b}=\theta\approx 0$ (small mixing) or $\pi/2$ and
$d\theta_{b}/dE=0$. Because $E_{a}\sim E_{0a}$, we can take in (61)
$d\phi^{a}/dE=0$ which becomes

$\frac{dR}{dE}=(-c_{a}s_{b}\pm s_{a}c_{b}\cos
2\theta_{a})\frac{d\phi^{b}}{dE}\pm 2s_{a}s_{b}\sin
2\theta_{a}\frac{d\theta_{a}}{dE}.$ (62)

Here the upper sign is for $\theta_{b}\approx 0$ and the lower one
$\theta_{b}\approx\pi/2$. We can rewrite (62) as

$\frac{dR}{dE}=\pm I_{3}\frac{d\phi^{b}}{dE}\pm 2s_{a}s_{b}\sin
2\theta_{a}\frac{d\theta_{a}}{dE}.$ (63)

Assuming that $\sigma_{E}d\phi^{b}/dE\sim 1$ and $\sigma_{E}d\theta_{a}/dE\ll
1$, we find that the second term in (63) shifts the zero of $dR/dE$ relative
to the zero of $I_{3}$. Nevertheless, $E_{0}$ is close to $E_{R}$, and the
closer they are, the smaller $d\theta_{a}/dE$ is relative to $d\phi^{b}/dE$.

  3. 3.

Between the MSW resonances: $E_{a}<E<E_{b}$ or $E_{a}>E>E_{b}$. If resonances
are well separated, we have $|\theta_{a}-\theta_{b}|\approx\frac{\pi}{2}$ or
$\cos(2\theta_{a}-2\theta_{b})\approx-1$. Therefore,

$R\approx\cos(\phi^{a}-\phi^{b}),$

and

$\frac{dR}{dE}=-\sin(\phi^{a}-\phi^{b})\frac{d(\phi^{a}-\phi^{b})}{dE}.$

On the other hand,

$I_{3}\approx-\sin(\phi^{a}-\phi^{b})\cos 2\theta_{1}.$

Consequently, $I_{3}\rightarrow 0$ implies $dR/dE\rightarrow 0$, and
$n_{coh}\rightarrow\infty$.

In the left panel of fig. 3 we show $dR/dE$ and $I_{3}$ as functions of
neutrino energy. $I_{3}$ is null at $2.25$, $3.1$ and $7.7$ MeV, while the
zeros of $dR/dE$, $E_{0}^{i}$ are $2.3$, $3.3$ and $7.7$ MeV. These
coincidences show correlation of the parametric resonance and infinite
coherence. The MSW resonance energies are $E_{a}=2.25$ MeV and $E_{b}=62.7$
MeV, outside the range of the plot.

Figure 3: Left panel: $I_{3}$ and $\frac{d\xi}{dE}$ as function of neutrino
energy. Right panel: Dependence of the $n_{coh}$ in (55) on neutrino energy
with $\sigma_{E}/E=0.1$ MeV. The parameters of CW profile are the same as in
fig. 2.

On the right panel of fig. 3 we show dependence of $n_{coh}$ on $E$ for the CW
profile with parameters (53) and $\sigma_{E}/E=0.1$ MeV. $n_{coh}$ diverges at
the energies $E_{0}^{i}$. Notice that the shape of the peak is similar to MSW
case in fig. 1. Thus, at the parametric resonance energies the averaging of
oscillations is weak, and coherence is enhanced.

###  4.4 Coherence in castle-wall profile in the configuration space

In the $x$-representation, a picture of evolution in the CW profile is the
following. Crossing every border between the layers $a$ and $b$ a given
eigenstate of the layer $a$, $\nu_{i}^{a}$, splits into two eigenstates of the
layer $b$: $\nu_{i}^{a}\rightarrow\nu_{1}^{b},\nu_{2}^{b}$. In turn, at the
next border each eigenstate $\nu_{j}^{b}$ splits into eigenstates of $a$
$\nu_{j}^{b}\rightarrow\nu_{i}^{a}$, etc. The amplitudes of the splits are
determined by change of the mixing angles in matter (29).

As in subsection 4.1, 4.2 we consider that a flavor neutrino enters the layer
$a$ of the first CW period and a detector is placed at the border of $b$ layer
of the last period. After crossing $n$ periods of the CW profile, each
initially produced eigenstate $\nu_{i}^{a}$ ($i=1,2$) splits $2^{2n-1}$ times,
and therefore at a detector there are $2^{2n}$ elementary components. The
eigenstate $\nu_{i}^{b}$ ($i=1,2$), that arrives at a detector (last border),
will be a composition of $2^{2n-1}$ such components. They will arrive at
different moments and have different phases and amplitudes. We will assume
that these elementary WPs are short enough and do not spread. Therefore the
oscillation phases (phase differences) are the same along the elementary WPs.
Each elementary WP has its "history" of propagation determined by type of
eigenstate it showed up in each layer of the profile, e.g.
$\nu_{1}^{a}\rightarrow\nu_{2}^{b}\rightarrow\nu_{2}^{a}\rightarrow\nu_{1}^{b}\rightarrow\nu_{1}^{b}\rightarrow...\rightarrow\nu_{2}^{b}$.

The elementary WPs compose the total WP of eigenstates $\psi_{i}$. Splits of
eigenstates and delays lead to spread of total WP after crossing the CW
profile. This spread affects the level of overlap and coherence. If a detector
is sensitive to the flavor
$\nu_{f}=c_{b}^{\prime}\nu^{b}_{1}+s_{b}^{\prime}\nu^{b}_{2}$, the detected
signal is determined by

$\int
dt\left|c_{b}^{\prime}\psi^{b}_{1}(t)+s_{b}^{\prime}\psi^{b}_{2}(t)\right|^{2}.$
(64)

where $\psi_{i}^{b}$ are the total WP of the eigenstates $\nu_{i}^{b}$ in a
layer $b$ at a detector. The total WP, $\psi^{b}_{j}(t)$, can be found summing
up all $2^{2n-1}$ elementary WPs at the detector.

A given elementary WP at the detector can be characterized by $k_{a}$ and
$k_{b}$ \- the numbers of $a$ and $b$ layers it propagates through as the
second eigenstate, i.e. $\nu_{2}^{m}$, $m=a,b$. Correspondingly, the numbers
of layers it crossed as $\nu_{1}^{m}$ are $(n-k_{a})$ and $(n-k_{b})$. Using
the same notation for phases, $\phi_{i}^{a}$ and $\phi_{i}^{b}$ $(i=1,2)$, as
in sect. 3 $(i=1,2)$ we can write the total phase of a given elementary packet
at a detector:

$\phi(k_{a},k_{b})=\phi_{1}^{a}(n-k_{a})+\phi_{2}^{a}k_{a}+\phi_{1}^{b}(n-k_{b})+\phi^{b}_{2}k_{b}.$
(65)

The phase difference between the elementary WPs characterized by $k_{a},k_{b}$
and $k_{a}^{\prime},k_{b}^{\prime}$ equals

$\phi(k_{a},k_{b},k_{a}^{\prime},k_{b}^{\prime})=\phi^{a}(k_{a}-k_{a}^{\prime})+\phi^{b}(k_{b}-k_{b}^{\prime}),$
(66)

where we used definitions in (31).

Similarly one can find the relative time delays of arrival of the packets
characterized by $k_{a},k_{b}$ and $k_{a}^{\prime},k_{b}^{\prime}$ at a
detector using the definitions (32):

$t(k_{a},k_{b},k_{a}^{\prime},k_{b}^{\prime})=t^{a}(k_{a}-k_{a}^{\prime})+t^{b}(k_{b}-k_{b}^{\prime}).$
(67)

As for the case of a single layer (16), there are the relations between the
total delays (67) and total oscillation phases (66):

$t(k_{a}-k_{a}^{\prime},k_{b}-k_{b}^{\prime})=\frac{d\phi(k_{a}-k_{a}^{\prime},k_{b}-k_{b}^{\prime})}{dE},$
(68)

and

$t(k_{a}-k_{a}^{\prime},k_{b}-k_{b}^{\prime})=g(E,V)\phi(k_{a}-k_{a}^{\prime},k_{b}-k_{b}^{\prime}).$
(69)

Suppose, for definiteness, that $t^{a}$ and $t^{b}$ are positive, then maximal
relative delay (separation) $t^{max}$ corresponds to
$|k_{a}-k_{a}^{\prime}|=|k_{b}-k_{b}^{\prime}|=n$:

$t^{max}=n(t^{a}+t^{b}).$

This is the time difference of arrival of the fastest and the slowest extreme
elementary WP.

$t^{max}$ separation determines the spread of whole WPs. The space between the
fastest and the slowest WPs is filled in by non-extreme WPs. The extreme WPs
are coherent if

$n(t^{a}+t^{b})\leq\sigma_{x},$ (70)

which gives the coherence number of periods (length)

$n_{coh}\leq\frac{\sigma_{x}}{t^{a}+t^{b}}.$ (71)

Under this condition, all other (intermediate) WP-components are also coherent
(overlap). The condition (71) is sufficient but not necessary, since the
coherence broken for extreme packets still can hold for intermediate packets.

Parameters $k_{a},k_{b}$ uniquely determine the phase and the delay of a given
component. Number of components with a given $k_{a}$ and $k_{b}$ equals

${}^{n}_{k_{a}}C\times~{}^{n}_{k_{b}}C,$

where ${}^{n}_{p}C$ is number of combinations of $p$ elements from $n$
elements. Apart from $k_{a}$ and $k_{b}$ the amplitudes of individual
components are determined by the product of various mixing parameters: the
initial and final flavor mixing, e.g. $c_{a}^{\prime}c_{b}^{\prime}$ for
$\nu_{e}\rightarrow\nu_{e}$ channel and $s_{\Delta}$, $c_{\Delta}$ determined
by change of mixing at the borders between the layers $\Delta$:

$A(n,r,h)=(-1)^{h}c^{\prime}_{a}c^{\prime}_{b}s_{\Delta}^{r}c_{\Delta}^{2n-1-r}.$

Factor $s_{\Delta}$ originates from each border at which the eigenstate number
changes: $\nu_{1}^{a}\rightarrow\nu_{2}^{b}$, etc., while $c_{\Delta}$ appears
from the borders without change of eigenstate:
$\nu_{i}^{a}\rightarrow\nu_{i}^{b}$. $r$ is the number of borders where
eigenstate changes; it is determined by the number of merging blocks:
sequences of layers without change of eigenstate. $h$ is given by the number
of blocks with even number of sequential layers in which WP propagates as the
second eigenstate.

For large number of periods there are many elementary packets with the same
values of $k_{a},k_{b},r,h$. We call this number the multiplicity,
$M(k_{a},k_{b},r,h)$. Therefore the total amplitude of all WPs with a given
$k_{a},k_{b}$ equals

$A(k_{a},k_{b})=\sum_{r,h}A(n,r,h)M(k_{a},k_{b},r,h).$ (72)

As for the one period case, here the shape of the total WP depends on mixing
and multiplicities.

For $k_{a}=k_{a}^{\prime}$, $k_{b}=k_{b}^{\prime}$ there is no shift but also
the phase difference is zero. For $k_{a}-k_{a}^{\prime}=1$,
$k_{b}-k_{b}^{\prime}=0$, the phase difference is $\phi^{a}$, while for
$k_{a}-k_{a}^{\prime}=0$, $k_{b}-k_{b}^{\prime}=1$ the phase difference is
$\phi^{b}$, etc. In general, one finds the components with phase differences
varying from $0$ to $n(\phi^{a}+\phi^{b})$.

###  4.5 Resummation

We can obtain the compact expression for the total WPs summing up effects
after each crossing of the CW period representing them as the interference of
the two (total) WPs of $\nu_{1}^{a}$ and $\nu_{2}^{a}$.

We consider first the summation in the basis of eigenstates of the layer $a$:
$\nu^{a}=(\nu^{a}_{1},~{}\nu^{a}_{2})$. Let
$\psi^{k-1}=(\psi_{1}^{k-1},~{}\psi_{2}^{k-1})^{T}$ be the vector describing
the state of system after crossing $k-1$ periods of the CW profile.
$\psi_{i}^{k-1}$ is result of resummation of the elementary WP of
$\nu_{i}^{a}$ in the layer $a$ in the beginning of $k$-period. Then after
crossing of the $k$th period, the state becomes

$\psi^{k}=U_{L}^{a}\psi^{k-1},$ (73)

where the evolution matrix over one period

$U_{L}^{a}=U_{\Delta}D^{b}U_{\Delta}^{\dagger}D^{a}.$ (74)

Here $U_{\Delta}$ is the matrix of the mixing change at crossing the border
between $a$ and $b$:

$U_{\Delta}=U^{a\dagger}U^{b}=\left(\begin{array}[]{cc}c_{\Delta}&s_{\Delta}\\\
-s_{\Delta}&c_{\Delta}\end{array}\right),$ (75)

and $U^{a}$, $U^{b}$ are the flavor mixing matrices in the $a$ and $b$ layers.
$D^{a}$ is the evolution matrix in the layer $a$ which can be written as

$D^{a}=\left(\begin{array}[]{cc}S(t^{a}_{1})e^{i2\phi_{1}^{a}}&0\\\
0&S(t^{a}_{2})e^{i2\phi_{2}^{a}}.\end{array}\right)$ (76)

Here $S(t_{i}^{a})$ is the time-shift operator acting on the shape factor of
the WP in such a way that

$S(t_{i}^{a})f_{i}(t)=f_{i}(t-t_{i}^{a}),~{}~{}~{}~{}S(t_{i}^{a})S(t_{j}^{b})=S(t_{j}^{b})S(t_{i}^{a})=S(t_{i}^{a}+t_{j}^{b})~{}~{}~{}(i,j=a,b).$

We can also require that $S(t)^{\dagger}=S(-t)$, so that $S$ is the unitary:
$S(t)^{\dagger}S(t)=I$.

Similarly one can introduce the evolution matrix for the layer $b$. Inserting
$U_{\Delta}$ and $D^{m}_{i}$ ($m=a,b$) from (75) and (76) into (74) we obtain

$U_{L}^{a}=\left(\begin{array}[]{cc}c_{\Delta}^{2}e^{-i\phi}+s_{\Delta}^{2}S(t^{b})e^{-i\phi^{\prime}}&s_{\Delta}c_{\Delta}[S(t^{b}+t^{a})e^{i\phi}-S(t^{a})e^{i\phi^{\prime}}]\\\
s_{\Delta}c_{\Delta}[S(t^{b})e^{-i\phi^{\prime}}-e^{-i\phi}]&s_{\Delta}^{2}S(t^{a})e^{i\phi^{\prime}}+c_{\Delta}^{2}S(t^{b}+t^{a})e^{i\phi}\end{array}\right),$
(77)

where we subtracted the commom phase factor
$\exp\\{i(\phi_{2}^{b}+\phi_{2}^{a}+\phi_{1}^{b}+\phi_{1}^{a})\\}$ and applied
(31) and (34), as well as performed the time shift by $t_{1}^{b}+t_{1}^{a}$
and used (32). Using (77), we can recover results in sect. 3 for one period.

If all $S=1$, we obtain the matrix which leads to the parametric oscillations
as in [21, 23, 24].

The evolution matrix after $n$ periods equals

$U_{nL}^{a}=(U_{L}^{a})^{n}.$ (78)

According to (73) the total eigenstate is

$\psi^{n}(t)=(U^{a}_{L})^{n}f(t)\psi^{0},$ (79)

where $f(t)$ is the initial shape factor and $\psi^{0}$ are the admixtures of
eigenstates $\nu_{i}^{a}$ in the initial moment of time. The components of
$\psi^{n}$ have the form

$\psi^{n}=\sum_{k_{a},k_{b}}^{n}A(k_{a},k_{b})f(t-k_{a}t^{a}-k_{b}t^{b})e^{i(k_{a}\phi^{a}+k_{b}\phi^{b})}\psi^{0}.$
(80)

Notice from (80) that oscillations in energy are induced not only by
interference between the total WPs of $\psi^{n}_{1}$ and $\psi^{n}_{2}$ but
also by interference of components within amplitudes $\psi_{1}^{n}$ and
$\psi_{2}^{n}$ with amplitudes (72).

The amplitude of the $\nu_{e}\rightarrow\nu_{e}$ transition equals

$A_{ee}(t)=\nu_{e}^{T}(U_{L}^{a})^{n}f(t)\nu_{e},~{}~{}~{}~{}~{}~{}~{}~{}\nu_{e}^{T}=(c_{a},~{}s_{a}).$
(81)

This reproduces the amplitude for two layer case of sect. 4.1. Then the
probability can be found as

$P_{ee}=\int dt|\nu_{e}^{T}U_{nL}^{a}f(t)\nu_{e}|^{2}.$ (82)

Various results can be obtained using the general form of amplitude and
probability. One can consider evolution in the flavor basis, which corresponds
to the derivation of the parametric oscillation probability in [21, 23, 24].
Now the evolution matrix equals

$U_{L}^{f}=U^{b}D^{b}U_{\Delta}^{\dagger}D^{a}U^{a\dagger}=(U^{b}D^{b}U^{b\dagger})(U^{a}D^{a}U^{a\dagger}),$
(83)

where in the last step the eigenstates of the layer $b$ are projected onto the
flavor basis. The matrices in the $a$ eigenstate basis (74) and in the flavor
basis (83) are related as

$U_{L}^{a}=U^{a\dagger}U_{L}^{f}U^{a}.$ (84)

The matrices after $n$ layers:

$(U_{L}^{f})^{n}=U^{a}(U_{L}^{a})^{n}U^{a\dagger}.$ (85)

We can determine the effective Hamiltonian over one period using the evolution
matrix over one period: $U_{L}=I-iHL$. Then diagonalizing the Hamiltonian
gives the effective mixing and the eigenvalues, which, in turn, give the depth
of the parametric oscillations and the phase. Clearly, such a Hamiltonian
depends on mixing $\Delta$ and the phases of individual layers
$\phi^{a},\phi^{b}$.

###  4.6 Effective group velocities and infinite coherence in CW profile

In general, the condition of infinite coherence can be formulated as an
equality of the effective group velocities of the wave packets:

$v_{1}^{eff}=v_{2}^{eff}$ (86)

or $\Delta v^{eff}=0$. Determination $v_{i}^{eff}$ depends on properties of
the density profile and turns out to be non-trivial for the CW profile.

Recall that in the case of constant density $v_{i}^{eff}$ are well defined,
see (15) and related discussion. Since the shape factor does not change, the
group velocity is the velocity of any fixed point of shape factor, e.g.,
maximum. The difference of group velocities $\Delta v=\Delta
v(E,V,\theta,\Delta m^{2})$ does not depend on the oscillation phase. The
velocities are constant and do not change in the course of propagation.

If density changes along the neutrino trajectory, $v_{i}^{eff}$ and $\Delta
v^{eff}$ do depend on distance. In the adiabatic periodically varying density
$v_{i}^{eff}$ can be introduced as the velocities averaged over period:

$v_{i}^{eff}=L^{-1}\int_{0}^{L}dx\frac{dH_{i}}{dE}\bigg{|}_{\bar{E}}.$

Then the condition of infinite coherence is

$\Delta v^{eff}=L^{-1}\int_{0}^{L}dx\frac{d\Delta H}{dE}\bigg{|}_{\bar{E}}=0.$

It corresponds to a situation when $\Delta v^{eff}(x)>0$ in one part of the
period and $\Delta v^{eff}(x)<0$ in another part (23).

The situation is much more complicated if adiabaticity is broken as in the
case of a single jump between two layers, discussed in sect. 3. In the second
layer, the WP has two components, and it is non-trivial to identify the point
of the WP to which the group velocity should be ascribed. Furthermore, just
after crossing the jump, certain parts of the WP will have opposite phases and
therefore interfere destructively, so that the interference is the same as
before crossing. This also affects shape factors (47) and (3.3). In this case,
the infinite coherence has no meaning apart from specific cases which are
reduced to infinite coherence in a single layer (sect. 3).

Infinite coherence (infinite number of periods) has meaning in the castle-wall
profile for the (long-range) parametric oscillations. The effective group
velocities should be defined for a single period of CW profile similarly to
the periodic profile with adiabatic density change considered above.
Furthermore, one should take into account (similarly to single jump case) that
(80)

  * •

Shape factor of the total WPs changes at the crossings, due to splitting and
delays of the components;

  * •

Oscillation phase is different in different parts of the same total WP because
of different histories of the elementary components.

As a result, expressions for $v_{i}^{eff}$ and condition for infinite
coherence should depend not only on phases acquired in the layers $a$ and $b$
but also on the mixing in both layers (77).

Indeed we can obtain the condition in the $x$-representation rewriting
condition (56) for $n_{coh}\rightarrow\infty$ in terms of delays $t^{a}$ and
$t^{b}$ we find:

$[s_{a}c_{b}+c_{a}s_{b}\cos 2\Delta]t_{a}+[c_{a}s_{b}+s_{a}c_{b}\cos
2\Delta]t_{b}=\\\ =s_{a}s_{b}\sin 2\Delta\frac{1}{E}\left(\frac{V_{a}\sin
2\theta_{a}}{\Delta H_{a}}-\frac{V_{b}\sin 2\theta_{b}}{\Delta H_{b}}\right),$
(87)

where we used

$\frac{d2\theta_{m}}{dE}=\frac{V_{m}\sin 2\theta_{m}}{E\Delta H_{m}},$

$m=a,b$. From (87), we see that infinite coherence for parametric oscillations
is unrelated to overlaps of the elementary WPs at the end of each period; this
is the major difference from the cases where adiabaticity is conserved.

There is no simple interpretation of infinite coherence for parametric
oscillations in terms of characteristics of elementary WP. In the Table 1, for
three different $E_{0}$, we show the delays of elementary WPs as well as
phases and mixings in each layer of the CW profile with parameters (53). No
simple correlation is observed.

$E_{0}$ (MeV) |  $t_{a}$ ($\text{MeV}^{-1}$) |  $t_{b}$ ($\text{MeV}^{-1}$) |  $\phi^{a}$(rad) |  $\phi^{b}$(rad) |  $\theta_{a}$(rad) |  $\theta_{b}$(rad)
---|---|---|---|---|---|---
$2.3$ |  $-0.98$ |  $-2.4$ |  $1.6$ |  $2.7$ |  $0.83$ |  $0.15$
$3.3$ |  $1.5$ |  $-1.2$ |  $2.1$ |  $1.8$ |  $1.3$ |  $0.156$
$7.7$ |  $0.4$ |  $-0.21$ |  $3.87$ |  $0.73$ |  $1.5$ |  $0.2$
Table 1: Energies of enhanced coherence length, $E_{0}$, delays, phases and
mixing angles in each of the layers $a$ and $b$ of a CW profile with
parameters (53).

##  5 Applications to supernova neutrino evolution

The issues of coherence in matter are of great relevance for oscillations of
supernova neutrinos (SN). The reasons are enormous distances from the
production points to a detector at the Earth, complicated oscillation and
conversion phenomena inside a star, non-trivial profiles of medium potentials.
Here we briefly describe some effects, while a detailed study will be
presented elsewhere [29].

SN neutrinos are produced at densities $\rho\sim 10^{12}$ $\text{g/cm}^{3}$
and have very short wave packets $\sigma_{x}\sim 10^{-11}$ cm [17, 12, 18].
Propagating to the surface of a star they can undergo the collective
oscillations at $R<100$ km [30, 31, 32, 33] and then the resonance flavor
conversion at $R>1000$ km [34, 35, 36, 37, 38, 39]. The collective effects are
supposed to be active at certain phases of a neutrino burst [31].

Let us first assume that collective oscillations are absent due to damping
effects [40] or due to lack of conditions for these oscillations at certain
time intervals. In this case, the standard resonance conversion picture is
realized: in the production region, the mixing is strongly suppressed so that
the flavor states coincide with the eigenstates of propagation in matter. The
coincidence depends on the type of mass hierarchy. Except for special
situations (time periods and locations), the adiabaticity condition is well
satisfied, and therefore the adiabatic transformations occur when neutrinos
travel to the surface: $\nu_{i}^{m}\rightarrow\nu_{i}$ $(i=1,2,3)$. The
adiabaticity can be broken in fronts of shock waves which affects the
described picture.

If the initial fluxes of $\nu_{\mu}$ and $\nu_{\tau}$ are equal, the results
of conversion depend only on the $\nu_{e}-\nu_{e}$ survival probability, $p$,
in the neutrino channel and on the $\bar{\nu}_{e}-\bar{\nu}_{e}$ probability,
$\bar{p}$, in the antineutrino channel. For definiteness we will consider the
$\bar{\nu}_{e}$ channel. In this case, the flux of $\bar{\nu}_{e}$ at the
surface of the star, and consequently, the Earth equals

$F_{\bar{\nu}_{e}}=\bar{p}F_{\bar{\nu}_{e}}^{0}+(1-\bar{p})F_{\bar{\nu}_{x}}^{0},$
(88)

where $F^{0}_{\bar{\nu}_{e}}$ and $F^{0}_{\bar{\nu}_{x}}$ are the initial
fluxes of $\bar{\nu}_{e}$ and $\bar{\nu}_{x}$ (mixture of $\bar{\nu}_{\mu}$
and $\bar{\nu}_{\tau}$). The second term in (88) corresponds to transitions
$\bar{\nu}_{\mu},\bar{\nu}_{\tau}\rightarrow\bar{\nu}_{e}$. Derivation of the
equation (88) assumes that initially produced $\nu_{i}^{m}$ (which coincide
with the flavor states) are incoherent and evolve independently. Therefore
signals produced by $\nu_{i}^{m}$ or states, to which $\nu_{i}^{m}$ evolve,
sum up incoherently.

For $R\ll 1000$ km, the matter density is much bigger than the density of the
H-resonance associated with $\Delta m^{2}_{13}$. In this matter dominated
range, according to Eq. (11) in sect. 2 the coherence length $L_{coh}^{m}$ is
function of the vacuum parameters (see also fig. 1):

$L_{coh}^{m}=\frac{L_{\nu}}{\cos 2\theta}\frac{E}{2\sigma_{E}}=1100~{}{\rm
km}\left(\frac{E}{20{\rm MeV}}\right)\left(\frac{E}{10\sigma_{E}}\right).$

Therefore the WP of eigenstates will separate before reaching the H-resonance
even if they overlapped at the production. Notice that with an increase on
$E$, the length $L_{coh}^{m}$ increases, but the resonance shifts to outer
layers. Still, some partial coherence may exist.

For definiteness we assume the inverted mass hierarchy. Then according to the
level crossing scheme at the production
$\bar{\nu}_{3}^{m}\approx\bar{\nu}_{e}$,
$\bar{\nu}_{1}^{m}\approx\bar{\nu}_{x}$ and
$\bar{\nu}_{2}^{m}=\bar{\nu}_{x}^{\prime}$. Furthermore, antineutrinos cross
only H-resonance. The dynamics is related to
$\bar{\nu}_{3}^{m}-\bar{\nu}_{1}^{m}$ subsystem, while $\bar{\nu}_{2}^{m}$
essentially decouples appearing as a spectator. Therefore the $\bar{\nu}_{e}$
survival probability can be written as

$\bar{p}=|U_{e1}|^{2}P_{31}+s_{13}^{2}(1-P_{31}),$

where $P_{31}\equiv P(\bar{\nu}_{3}^{m}\rightarrow\bar{\nu}_{1}^{m})$ is the
$\bar{\nu}_{3}^{m}\rightarrow\bar{\nu}_{1}^{m}$ transition probability.

Let us consider the effects of the coherence loss and coherence enhancement
under different circumstances.

1\. In the completely adiabatic case,
$\bar{\nu}_{3}^{m}(\approx\bar{\nu}_{e})$ evolves to $\bar{\nu}_{3}$, so that
$P(\bar{\nu}_{3}^{m}\rightarrow\bar{\nu}_{1}^{m})=0$ and
$\bar{p}=|U_{e3}|^{2}=s_{13}^{2}$. In central parts of a star
$\bar{\nu}_{1}^{m}$ has larger group velocity than $\bar{\nu}_{3}^{m}$. Below
the $H$-resonance, inversely, $\bar{\nu}_{3}^{m}$ moves faster and
$\bar{\nu}_{3}$ will arrive at the Earth first.

2\. Adiabaticity is strongly broken in a shock wave front which can be
considered as the instantaneous density jump (see sect. 3). We will assume
that before the jump (layer $a$) and after the jump (layer $b$) neutrinos
propagate adiabatically. Crossing the jump the eigenstates $\bar{\nu}_{1}^{a}$
and $\bar{\nu}_{3}^{a}$ split into eigenstates $\bar{\nu}_{j}^{b}$. To find
$\bar{p}$ we consider evolution of $\bar{\nu}_{3}^{a}$ which becomes
$c_{\Delta}\bar{\nu}_{3}^{b}-s_{\Delta}\bar{\nu}_{1}^{b}$ after crossing the
jump. Then its components $\bar{\nu}_{j}^{b}$ loose coherence and evolve to
$\bar{\nu}_{j}$ at the surface of a star. Consequently, we find

$\bar{p}=c_{\Delta}^{2}s_{13}^{2}+s_{\Delta}^{2}|U_{e1}|^{2}.$

Strong adiabaticity violation effect on $\bar{p}$ is realized if the change of
mixing in matter, $\Delta$, is large. The latter occurs when the jump appears
in the resonance layer, and the size of the jump is larger than the width of
the resonance layer: $\Delta\rho>\rho_{R}\tan 2\theta_{13}=0.3\rho_{R}$. That
is, even a small jump can produce a strong effect.

As we discussed before, the jump regenerates oscillations, which then
disappear, and $\bar{p}$ does not depend on the oscillation phase. Single
shock front (jump) could lead to phase (coherence) effect, if
$\bar{\nu}_{1}^{a}$ and $\bar{\nu}_{3}^{a}$ are produced in a coherent state
(e.g., as a result of collective oscillations) and coherence is supported till
the jump. In this case, the result of evolution depends on the phase
$\phi^{a}$ acquired in the layer $a$, and this phase is observable. Notice
that formula (88) is invalid in this case.

3\. Let us consider the case of two density jumps that appear due to the
presence of two shock wavefronts: the direct and reversed one. This situation
is analogous to the one described in sect. 3. We denote by $a$, $b$ and $g$
the layers between the production point and the inner (reversed) shock,
between two shocks and outside the outer shock correspondingly. The potential
of the layer $b$, $V_{b}$ can be constant or change adiabatically, the
potentials just before inner shock, $V_{a}$, and after second shock (jump),
$V_{g}$, are not equal (in contrast to CW).

The picture of evolution of $\bar{\nu}_{3}^{m}$ state is the following:

  * •

$\bar{\nu}_{3m}$ propagates adiabatically in the layer $a$ and splits at the
border between $a$ and $b$:
$\bar{\nu}_{3}^{a}=c_{\Delta}\bar{\nu}_{3}^{b}-s_{\Delta}\bar{\nu}_{1}^{b}$;

  * •

the latter state is coherent and oscillates in the layer $b$ acquiring the
oscillation phase $\phi^{b}$:
$c_{\Delta}e^{i\phi^{b}}\bar{\nu}_{3}^{b}-s_{\Delta}\bar{\nu}_{1}^{b}$;

  * •

the eigenstates $\bar{\nu}_{3}^{b}$, $\bar{\nu}_{1}^{b}$ split in the second
jump (the border between $b$ and $g$):
$\bar{\nu}_{3}^{b}=c_{\Delta^{\prime}}\bar{\nu}_{3}^{g}-s_{\Delta^{\prime}}\bar{\nu}_{1}^{g}$,
$\bar{\nu}_{1}^{b}=c_{\Delta^{\prime}}\bar{\nu}_{1}^{g}+s_{\Delta^{\prime}}\bar{\nu}_{3}^{g}$,
where $\Delta^{\prime}\equiv\theta_{g}-\theta_{b}$ is the change of the mixing
angle in the second jump;

  * •

in the layer $g$ the eigenstates $\nu_{j}^{g}$ evolve adiabatically to the
mass eigenstates $\bar{\nu}_{j}^{g}\rightarrow\bar{\nu}_{j}$. Furthermore,
$\bar{\nu}_{1}$ and $\bar{\nu}_{3}$ separate and decohere.

According to this picture we obtain the following expression for the
probability $\bar{p}$:

$\bar{p}=|U_{e1}|^{2}\sin^{2}(\Delta+\Delta^{\prime})+s_{13}^{2}\cos^{2}(\Delta+\Delta^{\prime})-(|U_{e1}|^{2}-s_{13}^{2})\sin
2\Delta\sin 2\Delta^{\prime}\sin^{2}\frac{\phi^{b}}{2}.$ (89)

Notice that $\Delta+\Delta^{\prime}=\theta_{g}-\theta_{a}$.

In fig. 4 we plot $\bar{p}$ computed with Eq. (89) for the following values of
parameters: $V_{a}=6.3\times 10^{-12}$ eV, $V_{b}=3.2\times 10^{-11}$ eV,
$V_{g}=3.2\times 10^{-12}$ eV and $L_{b}=1060$ km. If we assume electron
fraction $Y_{e}=0.5$, the corresponding densities are $\rho_{a}=2\times
10^{2}$ $\text{g/cm}^{3}$, $\rho_{b}=10^{3}$ $\text{g/cm}^{3}$,
$\rho_{g}=10^{2}$ $\text{g/cm}^{3}$. Since the distance between the two shock
fronts is much larger than the oscillation length in matter, $l_{m}<100$ km,
one can naively average the oscillatory terms in (89), i.e. take
$\sin^{2}\phi^{b}/2=0.5$ (the red curve in fig. 4). However, according to our
analysis in sect. 2 in layers of constant and slowly varying densities, the
energy range of enhanced coherence exists around $E_{0}$, where oscillation
effects might not be negligible. This is shown in fig. 4 by the black curve
for the energy resolution $\sigma_{E}/E=0.1$. Here $E_{0}=41$ MeV. This
oscillation effect will survive further propagation to the Earth, the effect
encodes information about distance between shock wave fronts and density
profile. Similar effects of enhanced coherence also survives for adiabatic
propagation between the two shocks according to (26). As the distance between
the two jumps increases, one needs better energy resolution,
$\sigma_{E}/E<0.1$, to observe the effect of enhanced coherence.

Figure 4: Dependence of the $\bar{\nu}_{e}$ survival probability in supernova
on neutrino energy (blue line). Inverted mass hierarchy and 1-3 vacuum mixing
were assumed. The probability averaged over the energy interval
$\sigma_{E}=0.1E$ is shown by black line. The red line shows the probability
without oscillatory term.

Let us consider the possible effect of the coherence loss on the collective
oscillations in central parts of a star. These oscillations are induced by
coherent flavor exchange in the $\nu-\nu$ scattering, and high neutrino
densities at $r<100$ km make them relevant.

The critical issue is whether fast loss of coherence due to very short WPss
destroys collective oscillations. Indeed, the latter imply the $\nu-\nu$
interactions after neutrino production, which can be considered as
$\nu$-detection ("observation") process. Loss of coherence between production
and interaction as well as between different interactions can be important.
Furthermore, effectively the problem becomes non-linear, and therefore, it is
not clear if integration over energies that form the WP can be interchanged
with evolution. Such an interchange is in the basis of equivalence of results
in the $x$\- and $E$-representations [26]. There is no clear answer to this
question [17, 12, 41, 18].

Some new insight into the problem can be obtained using description of the
collective oscillations as evolution of the individual neutrinos propagating
in effective external potentials formed by matter and background neutrinos
[19]. In this case, the problem becomes explicitly linear. Since the
background neutrinos oscillate, the effective potentials have non-trivial
oscillatory dependence in time (distance). These variations of the potentials
are non-adiabatic and therefore strong flavor transitions can be interpreted
as parametric oscillations and parametric enhancement effects. Taking into
consideration this picture one can apply the results of sect. 4.

The following comments are in order:

1\. Propagation decoherence is related to the energy dependence of the
Hamiltonian (and consequently the phases). The collective (fast)
transformations are driven mainly by potentials that dominate over the vacuum
term $\Delta m^{2}/2E$ (the source of $E$ dependence). The vacuum term (with
mixing) triggers conversion at the initial short phase. Therefore, apart from
the initial phase, decoherence is simply absent. If the initial phase is short
enough the decoherence is entirely irrelevant.

2\. In the region of fast collective oscillations, $r_{coll}<100$ km, with
strong matter dominance the coherence length is given by vacuum parameters (5)
and turns out to be $L_{coh}\sim 10^{3}$ km. So, $r_{coll}\ll L_{coh}$ and
decoherence can be neglected for $r<r_{coll}$.

3\. For low energies ($E\sim 10$ MeV) and $\sigma_{E}\sim E$ the length
$L_{coh}<10^{2}$ km can be comparable to the scale of collective oscillations.
However, this estimation of $L_{coh}$ is for constant density. As we
established in sect. 4, the parametric oscillations themselves can
substantially enhance coherence at least in certain energy intervals around
$E_{0}$. So, here we deal with coherence sustained by oscillatory dependence
of potentials.

Notice that consideration of coherence for collective oscillations was
problematic because of strong adiabaticity violation and difficulty to define
the eigenstates. Using analogy with the CW case, this can be done by
integrating the Hamiltonian oven one period, thus eliminating fast variations.
For the averaged Hamiltonian, one can introduce the eigenstates and their
effective group velocities. Then the enhanced coherence corresponds to
approximate equality of these group velocities (87).

##  6 Conclusions

The propagation decoherence occurs due to the difference of the group
velocities, which leads to the shift of the WPs relative to each other and
their eventual separation. In matter, due to refraction, both the group and
phase velocities of neutrinos change with respect to the vacuum velocities.
Thus, the refraction affects the propagation coherence (and decoherence).

The consideration of WPs in the $x$-space with average energy and momentum
produces the same results as consideration of plane waves in the $E$-space and
integration of probability (oscillation phase) over the energy spectrum with
width given by the energy uncertainty. In particular, the same value of the
coherence length follows from both approaches.

For different density profiles, we determined the coherence lengths and their
dependence on neutrino energy. The salient feature in matter is the existence
of infinite coherence length, $L_{coh}\rightarrow\infty$, at certain energies
$E_{0}$, and regions of enhanced coherence around $E_{0}$. In the energy
space, $E_{0}$ is given by zero derivatives of the phase on energy. In the
configuration space, it follows from equality of the group velocities of the
eigenstates. The condition can be described as the energy of minimal averaging
of the interference term of the probability.

The fundamental notion here is the effective group velocities which depend on
the density profiles.

1\. For constant density the infinite coherence $E_{0}=E_{R}/\cos 2\theta$
coincides with the MSW resonance energy in oscillations of mass eigenstates
$\nu_{1}\leftrightarrow\nu_{2}$, and for small vacuum mixing, it is close to
the MSW flavor resonance. The width of the region of enhanced coherence (weak
averaging) is inversely proportional to the energy resolution, $\sigma_{E}$,
and grows according to $\propto E_{0}^{2}$. For very large densities (high
energies), the coherence length is determined by vacuum parameters and becomes
close to the coherence length in vacuum: $L_{coh}^{m}=L_{coh}/\cos 2\theta$.

For massless neutrinos and oscillations driven by matter the coherence is
supported infinitely for all the energies.

2\. In varying density profiles, the infinite coherence is realized in
particular situations. For monotonously and adiabatically changing density the
coherence length can increase toward specific energy $E_{0}$, which
corresponds to certain $V_{0}$, such that for $V>V_{0}$ and $V<V_{0}$
separation have opposite signs and compensate each other.

Infinite coherence can be realized for periodic profile with adiabatic density
change. It corresponds to equality of the group velocities of eigenstates
averaged over the period. $E_{0}$ can be found from the latter condition.

3\. In the presence of adiabaticity violation, transitions between eigenstates
of propagation happen. This leads to modification of WP shape, and
consequently, to change of the effective group velocities.

We considered in details the case of maximal adiabaticity breaking - density
jumps in certain points of space (neutrino trajectory). In this case, two new
effects are realized, which are related to the split of the eigenstates at
borders between layers: (i) spread of total WP, (ii) change of the oscillation
phase along these total WPs.

For the example of a single jump between two layers with different constant
densities, there are two components in the first layer and four components in
the second. Correspondingly there are various group velocities and oscillation
phases at a detector, and it is not possible to introduce the coherence length
in the usual way. Still the medium parameters can be selected so that some
level of coherence can be supported for long distances because the split of
the eigenstates regenerates interference and oscillations.

4\. We checked that modifications of the total WP in the $x$-space do not
change its Fourier transform to the $E$-space, and consequently, the energy
spectrum of neutrinos. Therefore derivation of the infinite coherence
conditions for complicated density profiles can be more accessible in the
$E$-representation.

In certain situations, consideration of WP in the $x$-space simplifies
computations of probabilities, especially when coherence is completely lost.
Moreover, it becomes crucial when time tagging is introduced at the production
or detection, breaking the stationarity condition.

5\. For periodic structures with density jumps, such as the castle-wall
profile, the coherence length can be introduced for the parametric
oscillations (and not for small scale coherence of elementary WP). Then the
coherence can be supported for many periods. For parametric oscillations,
there are several $E_{0}$, each associated with a particular parametric
resonance.

Interpretation of the condition for $E_{0}$ in $x$-space in terms of
elementary WPs propagating in the individual layers of the profile is very
non-trivial. We find that the effective group velocities for one period (and
therefore the infinite coherence condition) are rather complicated functions
of mixing angles, phases and delays acquired by the elementary WPs in two
parts of the period.

6\. Using elementary WPs, we reconstructed total WPs at a detector and studied
their spread and shape change in the course of propagation depending on
parameters of the CW profile. The reconstruction of the total WP from the
elementary WP using operators of time shift is presented. Resummation of the
elementary WP at a detector can be performed using the evolution matrix for
one period expressed in terms of operators of time shifts.

7\. We outlined applications of the obtained results to supernova neutrinos.
In particular, we showed that coherence can be supported between two shock
wavefronts, leading to observable oscillation effects at the Earth.

For collective oscillations due to large matter potentials the coherence is
determined by vacuum parameters and coherence length is much larger than the
scale of fast oscillations. Using interpretation of collective oscillations as
the parametric effects we find that coherence can be further enhanced in
certain energy ranges.

##  7 Acknowledgements

YPPS acknowledges support from FAPESP funding Grants No. 2014/19164-453 6, No.
2017/05515-0 and No. 2019/22961-9.

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