# The Quantum Eraser from a Weak Values Perspective

Tom Rivlin  Chemical and Biological Physics Department, Weizmann Institute of
Science, 76100 Rehovot, Israel

###### Abstract

The quantum eraser variant of the double-slit experiment, and its ‘delayed
choice’ sub-variant, are considered from the perspective of weak value and
weak measurement theory (which is briefly reintroduced here). The interference
fringes that appear when measuring certain spin states, which can then be
‘erased’ when measuring other spin states, are shown to be anomalous weak
values that depend on particular post-selection choices. By framing the choice
of spin measurement as a weak value of a certain weak measurement, it is then
made clear what physical claims can and cannot be made about what occurs in
the quantum eraser experiment. Specifically, claims about the choice of spin-
state ‘retrocausally’ influencing the choice of slit(s) for the particles to
travel through are discredited, and a simple framework is presented for
understanding how the fringes arise and why they can be ‘erased’.

††: Eur. J. Phys.

  *<EMAIL_ADDRESS>
  * 02 May 2022

Keywords: weak values, weak measurement, quantum eraser, delayed choice
quantum eraser, quantum theory, post-selection, double-slit

##  1 Background

The quantum eraser thought experiment is a simple, measurable variation on the
double-slit experiment that is often called a quantum paradox, with non-
intuitive implications [1, 2, 3, 4, 5, 6, 7, 8, 9]. A naive interpretation of
the ‘delayed choice’ variant of the experiment can even be erroneously thought
of an example of retrocausality [6, 9]. The supposedly paradoxical nature of
the experiment, however, is actually the result of a misinterpretation. The
resolution to the supposed paradox has been discussed by several authors [7,
8, 9], but the resolution to the paradox has not previously been analysed from
a weak measurement and weak value perspective (though see [10, 11, 12, 13, 14]
for discussions of the related issue of weak measurement of momentum in the
two-slit setup). Here it will be shown that the experiment can be thought of
as a type of weak measurement, similar to how other supposed quantum paradoxes
can be examined from a weak measurement viewpoint [15, 16, 17]. This will help
to clarify what can and cannot be claimed about the experiment, and serves as
an informative introduction to a genre of quantum theory problems that can be
clarified with the weak values perspective.

The explanation for the setup (following the explanation of [8]) is as
follows. Suppose one has a double-slit apparatus, such that a particle can
pass through two narrow, finite-width slits $a$ and $b$ with small separation.
In the typical quantum fashion, the particle’s wavefunction can be written as
a superposition of the states where it passes through $a$ and the states where
it passes through $b$, and these states can interfere.

$\ket{\psi_{1}}=\alpha\ket{\psi_{a}}+\beta\ket{\psi_{b}}.$ (1)

If one were to observe the distribution of particles at a particular choice of
$x$ at a screen on the other side of the slits (after a time $t$ has passed
from the particle hitting the slits – the $t$ variable should be considered
implicit wherever an $x$ variable is imposed), one would see particles
distributed according to:

$|\braket{x}{\psi_{1}}|^{2}=|\alpha|^{2}|\psi_{a}(x)|^{2}+|\beta|^{2}|\psi_{b}(x)|^{2}+\alpha^{*}\beta\psi_{a}^{*}(x)\psi_{b}(x)+\beta^{*}\alpha\psi_{b}^{*}(x)\psi_{a}(x).$
(2)

The cross-terms in the final expression give rise to fringes caused by
interference effects, as is well known from conventional quantum mechanics.
For instance, following an example given in [18], if for some wavenumber $k$
and angular velocity $\omega$ at some time $t$ we have

$\displaystyle\psi_{a}(x)$ $\displaystyle=e^{-i\left(kx-\omega
t\right)}/\sqrt{\pi\left(1+x^{2}\right)}\,,$ (3) $\displaystyle\psi_{b}(x)$
$\displaystyle=e^{-i\left(kx+\pi x-\omega
t\right)}/\sqrt{\pi\left(1+x^{2}\right)}\,,$

this leads to the overall intensity distribution at the screen being given
(for $\alpha=\beta=1/\sqrt{2}$) by

$|\braket{x}{\psi_{1}}|^{2}=\frac{2\cos^{2}{\left(\pi
x/2\right)}}{\pi\left(1+x^{2}\right)}.$ (4)

This expression gives a cosine squared distribution modulated by an envelope –
a standard interference fringe pattern. By contrast, if one were to plot
$|\psi_{a}(x)|^{2}$ on its own, the distribution would have a single, distinct
peak.

The ‘quantum eraser’ variant is the situation where one couples each of the
two states associated with the slits to one of two possible states of an
observable, such as spin along the $z$-axis (or photon polarisation). This
would be achieved by configuring the experiment in such a way that only spin-
up particles (say) go through slit $a$ and spin-down through slit $b$. Then by
measuring spin, we measure which slit was traversed (the “which-way”
information). For orthogonal spin-up and down states $\ket{\uparrow}$ and
$\ket{\downarrow}$, the wavefunction then looks like

$\ket{\psi_{2}}=\alpha\ket{\psi_{a}}\ket{\uparrow}+\beta\ket{\psi_{b}}\ket{\downarrow}.$
(5)

Note that this implies that there are two Hilbert spaces, and the overall
wavefunction of the single system $\ket{\psi_{2}}$ is constructed by taking
tensor products between states in the $\\{\ket{\psi_{a}},\ket{\psi_{b}}\\}$
Hilbert space and states in the $\\{\ket{\uparrow},\ket{\downarrow}\\}$
Hilbert space.

When one now observes the distribution of particles at a point $x$ on the
screen, the distribution takes the form

$|\braket{x}{\psi_{2}}|^{2}=|\alpha|^{2}|\psi_{a}(x)|^{2}+|\beta|^{2}|\psi_{b}(x)|^{2},$
(6)

where the orthonormality of the two spin states eliminates the cross-terms in
Eq. 2. Hence the slits are distinguished from the start, the interference
effects are not present, and the final distribution will look like overlapping
single-slit distributions. This is similar to the case where the interference
is destroyed by a measurement that ‘looks at’ the slits by shining photons on
them and transferring momentum [10, 11].

The ‘eraser’ part of the quantum eraser comes when we consider the following:
spin can be measured along any axis, so let us imagine measuring it along one
orthogonal to the $z$-axis, in orthonormal states
$\ket{\pm}=\left(\ket{\uparrow}\pm\ket{\downarrow}\right)/\sqrt{2}$. Because
of the commutation relation between the two sets of spins, measuring it in
this state ‘destroys’ or ‘erases’ the information about the specific up or
down state it was in – the which-way information. We can write
$\ket{\psi_{2}}$ in terms of this new basis (and $\braket{x}{\psi_{2}}$ will
be the same as in Eq. 6):

$\ket{\psi_{2}}=\frac{1}{\sqrt{2}}\left(\alpha\ket{\psi_{a}}+\beta\ket{\psi_{b}}\right)\ket{+}+\frac{1}{\sqrt{2}}\left(\alpha\ket{\psi_{a}}-\beta\ket{\psi_{b}}\right)\ket{-}.$
(7)

The notable effects occur when one considers sub-ensembles of
$\ket{\psi_{2}}$. Suppose one has a detector that distinguishes particles at
the screen as being in the $\ket{+}$ state or the $\ket{-}$ state. One then
obtains either one of

$\displaystyle|\bra{+}\braket{x}{\psi_{2}}|^{2}$
$\displaystyle=\frac{1}{2}\left(|\alpha|^{2}|\psi_{a}(x)|^{2}+|\beta|^{2}|\psi_{b}(x)|^{2}+\alpha^{*}\beta\psi_{a}^{*}(x)\psi_{b}(x)+\beta^{*}\alpha\psi_{b}^{*}(x)\psi_{a}(x)\right),$
(8) $\displaystyle|\bra{-}\braket{x}{\psi_{2}}|^{2}$
$\displaystyle=\frac{1}{2}\left(|\alpha|^{2}|\psi_{a}(x)|^{2}+|\beta|^{2}|\psi_{b}(x)|^{2}-\alpha^{*}\beta\psi_{a}^{*}(x)\psi_{b}(x)-\beta^{*}\alpha\psi_{b}^{*}(x)\psi_{a}(x)\right).$

These expressions, of course, have interference terms. Contrast that with what
happens when we distinguish particles at the screen as being in the
$\ket{\uparrow}$ or $\ket{\downarrow}$ states:

$\displaystyle|\bra{\uparrow}\braket{x}{\psi_{2}}|^{2}$
$\displaystyle=|\alpha|^{2}|\psi_{a}(x)|^{2},$ (9)
$\displaystyle|\bra{\downarrow}\braket{x}{\psi_{2}}|^{2}$
$\displaystyle=|\beta|^{2}|\psi_{b}(x)|^{2}.$

When we distinguish parts of the distribution in this way, the particle is
only ever detected as having been through slit $a$ for the $\ket{\uparrow}$
case or slit $b$ for the $\ket{\downarrow}$ case.

So, depending on which axis we choose to measure the spin along, we either do
or do not see interference fringes. This has been realised experimentally
(with photon polarisations) [3, 4], and, as expected, interference fringes
were seen for one choice of sub-ensembles, but not for the other.

Of course, one is free to choose which axis to measure the spin along after
the particles have traveled through the slit(s) – this is where the ‘delayed
choice’ name comes from. One could think of this as being “retrocausal” – that
the choice one makes in the present as to which axis to measure spin along
causes the particle in the past to either go through two slits or one – to
interfere or not interfere. Different sub-ensembles being revealed by
interactions between the system and the environment is a well-known phenomenon
[19]. Here the claim, however, is that the experimenter’s choice after the
measurement can alter the sub-ensembles.

The claim is untrue for a number of reasons [6, 9], and the rest of this
discussion will focus on how to understand this experiment using weak
measurement theory. But first it must be noted that the interference patterns
one observes by measuring along the $\pm$ axis are not the same as the
patterns one observes by measuring state $\ket{\psi_{1}}$, the state without
any attached spins. The $|\bra{+}\braket{x}{\psi_{2}}|^{2}$ state’s values are
half those of $|\braket{x}{\psi_{1}}|^{2}$, and the structure of
$|\bra{-}\braket{x}{\psi_{2}}|^{2}$ is a modified version of the
$|\braket{x}{\psi_{1}}|^{2}$ pattern due to the minus signs in the cross-
terms. Measuring the spin along the $\pm$ axis still changes the measurement
outcome from what one would observe without measuring any spin, just not as
drastically as measuring along the up-down axis. And, of course, if one were
to add the two $\pm$ or up-down sub-ensembles together, one would exactly
retrieve the original $|\braket{x}{\psi_{2}}|^{2}$ distribution with no
interference.

##  2 Weak Measurement and Weak Values

Weak measurement theory [20, 21, 22, 23] offers an interesting perspective on
this problem, and will be briefly introduced here (based on [22]) in a form
that facilitates its application to the quantum eraser. Consider an attempt to
obtain an eigenvalue of an observable $\hat{A}$ of a quantum system
$\ket{\phi_{i}}$, conditional on the system being post-selected to be in state
$\ket{\phi_{f}}$ [24]. The weak value of the observable for this post-
selection is then said to be

$\braket{\hat{A}}_{w,i,f}=\frac{\bra{\phi_{f}}\hat{A}\ket{\phi_{i}}}{\braket{\phi_{f}}{\phi_{i}}}.$
(10)

One can then sum over all the possible weak values – all the possible post-
selected states $\ket{\phi_{f}}$ – multiplied by the probabilities of
observing each value. This will give the standard $\braket{\hat{A}}$ for the
quantum system $\ket{\phi_{i}}$. This is simple Bayesian logic [24, 25], but
the caveat is that the weak values of $\hat{A}$ may in general be complex,
unlike the strong values which are guaranteed to be real. Even when not
complex, these values may be highly anomalous – falling well outside of the
regular eigenvalue spectrum of the observable.

This is connected to the idea of weak measurement: a type of measurement which
only perturbs the system it measures by a small amount. Consider a detector
for the observable $\hat{A}$ modeled as a ‘pointer’ coupled to the observable
by an interaction Hamiltonian [22]:

$\hat{H}_{\rm{int}}=g\hat{A}\otimes\hat{P}_{d},$ (11)

where $\hat{P}_{d}$ is the operator representing the momentum of the pointer,
and $g$ is a coupling parameter which is assumed to be small. This implies
that interacting with the system being measured via this Hamiltonian gives the
pointer a ‘kick’, and alters its position, but only by a small amount, and
with a large variance in the final position. With certain assumptions (e.g.
different pointer positions correspond to different eigenvalues), this can be
shown to translate into a large uncertainty about the eigenvalue of $\hat{A}$
that is being ‘pointed’ to by the detector. Unlike with regular ‘strong’ (or
‘von Neumann’) measurements, the system being measured is still in a
superposition of multiple eigenvalues of $\hat{A}$ after the interaction with
the pointer.

Expanding to first order the time-evolution operator up to time $t$,
$\exp{\left(-i\hat{H}_{\rm{int}}t/\hbar\right)}$, sandwiched between the pre-
and post-selected states $\ket{\phi_{i}}$ and $\ket{\phi_{f}}$, allows one to
derive the same expression as in Eq. 10 for the ‘expected’ value of $\hat{A}$.
This shows that weak values are the outcomes of weak measurements. In other
words, after the weak measurement, the system is still in a superposition of
multiple eigenstates of the observable, and the ‘probability’ of observing any
one of those eigenvalues – of the system being in that post-selected state –
is given by the weak value associated with it. Strictly speaking, this is not
actually a probability, but something known as a ‘quasiprobability’, since it
may be complex. (Weak values can also be obtained from sequences of strong
measurements [26], meaning weak values and weak measurements do not always
have to be considered in tandem.) It is notable that the form of the weak
value can be derived both from weak measurements and from considering
conditional probabilities.

##  3 Weak Values of the Quantum Eraser

Returning to the quantum eraser experiment, consider performing for the
$\ket{\psi_{2}}$ system a weak measurement of the position with the operator
$\hat{O}_{x}=\ket{x}\bra{x}$, which is a projection onto the $\ket{x}\bra{x}$
subspace and corresponds to a measurement of the density at position $x$. Here
we also implicitly assume that included in this operator is the fact that time
is propagated forward by some amount $t$, such that the waves can travel from
the slits to the screen. Furthermore, strictly speaking the full operator is a
tensor product of $\hat{O}_{x}$ and an identity operator in the Hilbert space
of the spins – the $\hat{O}_{x}$ operator only modifies states in the
$\\{\ket{\psi_{a}},\ket{\psi_{b}}\\}$ Hilbert space.

Consider first the weak value of the operator for the post-selected state
$\ket{\uparrow}$ (again, with an implicit identity operator in the other
Hilbert space). It is given by

$\braket{\hat{O}}_{x,w,\uparrow}=\frac{\braket{\uparrow}{x}\braket{x}{\psi_{2}}}{\braket{\uparrow}{\psi_{2}}}=\frac{\braket{\psi_{2}}{\uparrow}\braket{\uparrow}{x}\braket{x}{\psi_{2}}}{\braket{\psi_{2}}{\uparrow}\braket{\uparrow}{\psi_{2}}}=\frac{|\alpha|^{2}\braket{\psi_{a}}{x}\braket{x}{\psi_{a}}}{|\alpha|^{2}\braket{\psi_{a}}{\psi_{a}}}=|\psi_{a}(x)|^{2}.$
(12)

Likewise, the weak value of the position post-selected for the
$\ket{\downarrow}$ state is given by $|\psi_{b}(x)|^{2}$. In other words,
post-selecting for outcomes of the quantum evolution of $\ket{\psi_{2}}$ such
that spin-up (down) is observed will only show you the parts of the
distribution consistent with going through slit $a$ ($b$), so interference is
eliminated. This is to be expected from the form of Eq. 9. Note that these
weak values are independent of $\alpha$ and $\beta$.

The probability of observing each of the two post-selected states is

$\displaystyle\braket{\psi_{2}}{\uparrow}\braket{\uparrow}{\psi_{2}}=|\alpha|^{2},$
(13)
$\displaystyle\braket{\psi_{2}}{\downarrow}\braket{\downarrow}{\psi_{2}}=|\beta|^{2},$

and so the expected value of the $x$-projection operator acting on
$\ket{\psi_{2}}$ is obtained by summing over the two weak values multiplied by
their probabilities, as expected. The result is the same as Eq. 6.

A similar argument works for post-selecting one of the $\ket{\pm}$ states when
observing position to calculate $\braket{\hat{O}}_{x,w,+}$ and
$\braket{\hat{O}}_{x,w,-}$:

$\frac{\braket{\pm}{x}\braket{x}{\psi_{2}}}{\braket{\pm}{\psi_{2}}}=\left(|\alpha|^{2}|\psi_{a}(x)|^{2}+|\beta|^{2}|\psi_{b}(x)|^{2}\pm\alpha^{*}\beta\psi_{a}^{*}(x)\psi_{b}(x)\pm\beta^{*}\alpha\psi_{b}^{*}(x)\psi_{a}(x)\right),$
(14)

since $\braket{\psi_{a}}{\psi_{b}}=0$,
$\braket{\psi_{a}}{\psi_{a}}=\braket{\psi_{b}}{\psi_{b}}=1$,
$|\alpha|^{2}+|\beta|^{2}=1$.

Hence we can clearly conclude that one would observe interference fringes in
the $x$-distribution when post-selecting for one of the $\ket{\pm}$ states.
With the up and down states, by contrast, no fringes would be present.

The probabilities for the $\ket{\pm}$ states reduce down to

$\braket{\psi_{2}}{\pm}\braket{\pm}{\psi_{2}}=\frac{1}{2},$ (15)

such that the same value as in Eq. 6 is obtained by the appropriate sum over
weak values multiplied by probabilities. Note that these probabilities are
independent of $\alpha$ and $\beta$, and in the simplest case where
$\alpha=\beta=1/\sqrt{2}$, the probabilities of observing the
$\ket{\uparrow}$, $\ket{\downarrow}$, $\ket{+}$, and $\ket{-}$ states are the
same.

Of course, it is entirely possible for the weak values
$\braket{\hat{O}}_{x,w,\uparrow}$, $\braket{\hat{O}}_{x,w,\downarrow}$,
$\braket{\hat{O}}_{x,w,+}$, or $\braket{\hat{O}}_{x,w,-}$ to be larger than
the value of $|\braket{x}{\psi_{2}}|^{2}$ at certain values of $x$. This can
be seen in Figure 1, which plots the actual intensity and weak values for the
$\ket{\pm}$ states for the distributions in Equation 3 (with
$\alpha=\beta=1/\sqrt{2}$). At each value of $x$, one of the two weak values
$\braket{\hat{O}}_{x,w,+}$, $\braket{\hat{O}}_{x,w,-}$ is larger than
$|\braket{x}{\psi_{2}}|^{2}$.

Figure 1: The actual distribution at the screen for $\psi_{2}$ compared with
the $\ket{\pm}$ weak values (the $\uparrow$ and $\downarrow$ weak values
exactly overlap the actual distribution, and the $|\braket{x}{\psi_{1}}|^{2}$
distribution exactly overlaps the $\braket{\hat{O}}_{x,w,+}$ distribution). At
every value of $x$, one of the two weak values is larger than the actual
observed intensity.

By contrast, the weak values $\braket{\hat{O}}_{x,w,\uparrow}$ and
$\braket{\hat{O}}_{x,w,\downarrow}$, if plotted for each $x$, would both
precisely overlap $|\braket{x}{\psi_{2}}|^{2}$. Since the patterns of the two
slits no longer interfere with each other, and since each of $\ket{\uparrow}$
and $\ket{\downarrow}$ are coupled to precisely one of the two slits, post-
selecting for these states does not change the overall distribution.

At any given $x$, there is a greater chance of observing one of the
$\ket{\pm}$ sub-ensembles of the overall distribution than there is of
observing the distribution itself. Such phenomena are commonly seen with weak
values. For instance, at $x=0$, the intensity of the
$\braket{\hat{O}}_{x,w,+}$ distribution is twice that of the regular
distribution. The explanation for this is clear: the probability of the
particle being in the $\braket{\hat{O}}_{x,w,+}$ distribution is exactly half,
and at that point the intensity of the $\braket{\hat{O}}_{x,w,-}$ distribution
is 0. It is clear, however, that for all four weak value distributions
considered here, there will be no negative weak values, and the probabilities
of observing each of the weak values are strictly positive and less than $1$.

##  4 Discussion

The weak measurement being performed is that of the density of the
wavefunction at $x$, via the operator $\hat{O}_{x}$. The assumption is that
because the measurement is weak, only a small amount of information about the
distribution at the screen is obtained from it, and thus this observable still
has a distribution of possible values. One is then free to choose to partition
this distribution into sub-ensembles using different weak values.

Whether one subdivides the ensemble using a pair of up and down spins, or a
pair of plus and minus spins, the overall probability of observing an electron
at the point $x$ does not change. Once the slits are coupled to certain spin
states, as they are here, the interference fringes will disappear, as
expected. The post-selection choice does not ‘retrocausally’ induce or destroy
a fringe pattern in the spatial distribution of the wavefunction. In fact,
nothing physical about the system is altered by the post-selection.

Within a given post-selection choice, it is possible to obtain different
probabilities, and here it is shown that it is possible to reveal a fringe
pattern from the overall data set by post-selecting an appropriate weak value
to measure. These weak values may even be ‘anomalous’ in that the probability
of observing the particle at a position $x$ conditional on it being in (say)
the $\ket{+}$ state may be larger than the total probability of observing it
there. The correct sum over weak values, however, will always produce the
true, physical, fringeless distribution one expects when summing over all
possibilities, regardless of whether those weak values have fringes.

Also note that the operator only changes one of the two Hilbert spaces of the
system, and the post-selection bra only targets the other of the two Hilbert
spaces. Both of these Hilbert spaces are assumed to interact with the Hilbert
space of the measuring device as per the derivation of the weak value.

Notably, in the experimental realization of the quantum eraser [3], the two
single-slit distributions at the screen significantly overlapped. In these
circumstances, the which-way information gained from performing a measurement
of position for a given particle is small, and that is true whether there is a
fringed distribution or not. One can pick out a pattern with fringes by
subdividing the overlapping distributions of particle detection events in an
appropriate way as described by the weak values.

The mystery of the disappearing fringes in the quantum eraser thought
experiment is nothing more than the result of subdividing a distribution into
particular choices of sub-ensembles. As such, the analysis is amenable to an
interpretation based on weak values and weak measurements, as shown here. The
different spin measurements are shown to be different post-selected states of
a weak measurement of the position, and the appropriate choice of post-
selected state is shown to induce fringes in the distribution despite them not
being present in the overall distribution. The physical implications of these
findings are discussed, and it is explained how the supposed ‘retrocausality’
of the delayed-choice quantum eraser is illusory.

Acknowledgements: This work was supported by Grant No. 408/19 of the Israel
Science Foundation. I also wish to thank Prof. Eli Pollak and Dr. Eli Cohen
for many useful discussions in regards to this work.

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# Interaction of supernova neutrinos with stochastic gravitational waves

Maxim Dvornikov<EMAIL_ADDRESS>Pushkov Institute of Terrestrial Magnetism,
Ionosphere and Radiowave Propagation (IZMIRAN), 108840 Moscow, Troitsk, Russia

###### Abstract

We examine the propagation and flavor oscillations of neutrinos under the
influence of gravitational waves (GWs) with an arbitrary polarization. We
rederive the effective Hamiltonian for the system of three neutrino flavors
using the perturbative approach. Then, using this result, we consider the
evolution of neutrino flavors in stochastic GWs with a general energy density
spectrum. The equation for the density matrix is obtained and solved
analytically in the case of three neutrino flavors. As an application, we
study the evolution of the flavor content of a neutrino beam emitted in a
core-collapsing supernova. We obtain the analytical expressions for the
contributions of GWs to the neutrino fluxes and for the damping decrement,
which describes the attenuation of the fluxes to their asymptotic values. We
find that the contribution to the evolution of neutrino fluxes from GWs,
emitted by merging supermassive black holes, dominates over that from black
holes with stellar masses. The implication of the obtained results for the
measurement of astrophysical neutrinos with neutrino telescopes is discussed.

##  I Introduction

Neutrinos provide a unique possibility to explore physics beyond the standard
model with the help of nonaccelerator methods. Such achievements of neutrino
physics became possible after the observation of oscillations of atmospheric
and solar neutrinos [1]. These experimental facts are the straightforward
indications of the nonzero masses and mixing between different neutrino
flavors. External fields, e.g., the neutrino interaction with background
matter [2] and electromagnetic fields [3], are known to modify the process of
neutrino oscillations.

The gravitational interaction, in spite of its weakness, was found in Refs.
[4, 5, 6] to contribute to the neutrino oscillations dynamics. In the majority
of studies, neutrino oscillations in curved spacetime were examined when
particles move in static gravitational backgrounds, e.g., in the vicinity of a
black hole (BH). It is interesting to analyze the propagation and oscillations
of astrophysical neutrinos in time dependent metrics, e.g., induced by a
gravitational wave (GW).

The studies of the fermions evolution in GWs were carried out in Refs. [7, 8].
Neutrino spin oscillations, i.e. when we deal with transitions between active
left polarized and sterile right polarized particles, in background matter
under the influence of GW were discussed in Ref. [9]. Neutrino flavor
oscillations in GWs, as well as in gravitational fields caused by metric
perturbations in the early universe, were considered in Ref. [10].

In this paper, we continue the research in Refs. [11, 12], where the influence
of stochastic GWs on neutrino flavor oscillations was considered. The main
problem of Refs. [11, 12] was the consideration of astrophysical neutrinos
emitted in decays of charged pions. Although such neutrinos form the major
cosmic neutrinos background, their sources are distributed rather uniformly in
the universe. Thus, the integral flux in a terrestrial detector should be
averaged over the propagation distance of such neutrinos. This fact makes it
difficult to separate the contribution of stochastic GWs on the measured
flavor composition. To avoid this difficulty, we decide to examine the effect
of stochastic GWs on supernova (SN) neutrinos. If an explosion of a core-
collapsing SN happens in our galaxy, firstly, it emits a sizable neutrino flux
to be measured even by existing neutrino telescopes [13]. Secondly, SN is
almost a point-like neutrino source. Hence one should not average over the
neutrino propagation distance. In this situation, we expect that the effect of
stochastic GWs is not smeared.

The present work is motivated by the recent direct detection of GWs by the
LIGO-Virgo collaborations [14]. There are active multimessenger searches of
GWs and high energy neutrinos by existing detectors [15, 16] and suggestions
to implement them in future ones [17]. There are also attempts to observe
stochastic GWs [18, 19], with various methods for their detection being
developed [20].

SN neutrinos were reliably detected in 1987 after the SN explosion in the
Large Magellanic Cloud (see, e.g., Ref. [21]). Since then, the experimental
techniques in construction of neutrino telescopes made great achievements.
Now, if a nearby SN in our galaxy explodes, a huge number of events will be
recorded [22]. As mentioned above, a simultaneous detection of GWs and SN
neutrinos may be possible. Besides a direct neutrino signal from a certain SN,
all collapsing stars in the universe emit neutrinos which form the diffuse SN
neutrino background. There are prospects to measure it by existing and future
neutrino telescopes (see, e.g., Ref. [23]).

This work is organized in the following way. In Sec. II, we formulate the
problem of the propagation of flavor neutrinos in a plane GW with an arbitrary
polarization. Then, we derive the equation for the density matrix for flavor
neutrinos if we deal with stochastic GWs. This equation is exactly solved for
the arbitrary energy spectrum of GWs. Then, in Sec. III, we apply our results
for the description of the interaction of SN neutrinos with stochastic GWs. We
find the corrections to neutrino fluxes and the damping decrement in an
explicit form. Finally, we summarize our results in Sec. IV. The effective
Hamiltonian for flavor oscillations under the influence of GW is rederived in
Appendix A.

##  II Evolution of flavor neutrinos in the GWs background

The system of three active flavor neutrinos $\nu_{\lambda}$,
$\lambda=e,\mu,\tau$, with the nonzero mixing, as well as under the influence
of a plane GW with an arbitrary polarization, obeys the following Schrödinger
equation:

$\mathrm{i}\dot{\nu}=(H_{0}+H_{1})\nu,$ (1)

where $\nu^{\mathrm{T}}=(\nu_{e},\nu_{\mu},\nu_{\tau})$,
$H_{0}=UH_{m}^{(\mathrm{vac})}U^{\dagger}$ is the effective Hamiltonian for
vacuum oscillations in the flavor eigenstates basis,
$H_{m}^{(\mathrm{vac})}=\tfrac{1}{2E}\text{diag}\left(0,\Delta
m_{21}^{2},\Delta m_{31}^{2}\right)$ is the vacuum effective Hamiltonian for
the mass eigenstates $\psi_{a}$, $a=1,2,3$, $\Delta
m_{ab}^{2}=m_{a}^{2}-m_{b}^{2}$ is the difference of the squares of masses
$m_{a}$ of mass eigenstates, $E$ is the mean energy of a neutrino beam, and
$U$ is the unitary matrix which relates flavor and mass bases:
$\nu_{\lambda}=U_{\lambda a}\psi_{a}$. To derive $H_{m}^{(\mathrm{vac})}$ in
Eq. (1) we assume that neutrinos are ultrarelativistic and subtract a proper
diagonal term.

The mixing matrix $U$ can be present in the form,

$U=\left(\begin{array}[]{ccc}1&0&0\\\ 0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\end{array}\right)\cdot\left(\begin{array}[]{ccc}c_{13}&0&s_{13}e^{-\mathrm{i}\delta_{\mathrm{CP}}}\\\
0&1&0\\\
-s_{13}e^{\mathrm{i}\delta_{\mathrm{CP}}}&0&c_{13}\end{array}\right)\cdot\left(\begin{array}[]{ccc}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\end{array}\right),$ (2)

where $c_{ab}=\cos\theta_{ab}$, $s_{ab}=\sin\theta_{ab}$, $\theta_{ab}$ are
the corresponding vacuum mixing angles, and $\delta_{\mathrm{CP}}$ is the CP
violating phase. The values of these parameters can be found in Ref. [24].

The Hamiltonian $H_{1}$ in Eq. (1), which describes the neutrino interaction
with GW, has the form $H_{1}=UH_{m}^{(g)}U^{\dagger}$, where

$H_{m}^{(g)}=H_{m}^{(\mathrm{vac})}\left(A_{c}h_{+}+A_{s}h_{\times}\right),$
(3)

is the Hamiltonian in the mass basis, $A_{c}=\tfrac{1}{2}\sin^{2}\vartheta\cos
2\varphi\cos[\omega t(1-\cos\vartheta)]$,
$A_{s}=\tfrac{1}{2}\sin^{2}\vartheta\sin 2\varphi\sin[\omega
t(1-\cos\vartheta)]$, $h_{+,\times}$ are the amplitudes corresponding to
‘plus’ and ‘cross’ polarizations of GW, $\omega$ is the frequency of GW,
$\vartheta$ and $\varphi$ are the spherical angles fixing the neutrino
momentum with respect to the wave vector of GW, which is supposed to propagate
along the $z$-axis. To derive Eq. (3) we assume that [11] $\omega
L|v_{a}-v_{b}|\ll 1$, $a,b=1,2,3$, where $L$ is the distance of the neutrino
beam propagation and $v_{a}$ is the velocity of a mass eigenstate. Analogously
to $H_{m}^{(\mathrm{vac})}$, we subtract the common diagonal term in
$H_{m}^{(g)}$ in Eq. (3).

The Hamiltonian $H_{m}^{(g)}$ for a circularly polarized GW with
$h_{+}=h_{\times}$ was obtained in Ref. [11] based on the exact solution of
the Hamilton-Jacobi equation for a test particle in a plane GW. In the present
work, we provide a more straightforward perturbative derivation of the same
result which is given in Appendix A; cf. Eq. (29). Of course, the expression
for $H_{m}^{(g)}$ coincides with that in Ref. [11] in the limit
$h_{+}=h_{\times}$.

Now we consider the situation when a neutrino interacts with stochastic GWs.
In this case, the angles $\vartheta$ and $\varphi$, as well as the amplitudes
$h_{+,\times}$, are random functions of time. To study the neutrino motion in
such a background, it is more convenient to deal with the density matrix
$\rho$, which obeys the equation, $\mathrm{i}\dot{\rho}=[H_{0}+H_{1},\rho]$.
Following Ref. [25], we introduce the density matrix in the interaction
picture,
$\rho_{\mathrm{int}}=\exp(\mathrm{i}H_{0}t)\rho\exp(-\mathrm{i}H_{0}t)$. It
satisfies the equation,

$\mathrm{i}\dot{\rho}_{\mathrm{int}}=[H_{\mathrm{int}},\rho_{\mathrm{int}}],$
(4)

where $H_{\mathrm{int}}=\exp(\mathrm{i}H_{0}t)H_{1}\exp(-\mathrm{i}H_{0}t)$.
Using the Baker–Campbell–Hausdorff formula and the fact that that both
$H_{m}^{(\mathrm{vac})}$ and $H_{m}^{(g)}$ are diagonal, we get that
$H_{\mathrm{int}}=H_{1}$. This result is valid even before setting $v_{a}\to
1$ in the phase of the wave in Eq. (A) and omitting the common factors
proportional to the unit matrix in both $H_{m}^{(\mathrm{vac})}$ and
$H_{m}^{(g)}$. The initial condition for $\rho_{\mathrm{int}}$ coincides with
that for $\rho$: $\rho_{\mathrm{int}}(0)=\rho(0)\equiv\rho_{0}$.

We assume that stochastic GWs form a Gaussian random process. In this
situation, all odd correlators of angle factors $A_{c,s}$ and the amplitudes
$h_{+,\times}$ are vanishing. Moreover, we take that $h_{+}$ and $h_{\times}$
are independent. After averaging, the formal solution of Eq. (4) can be
present in the form of a series,

$\displaystyle\left\langle\rho_{\mathrm{int}}\right\rangle=$
$\displaystyle\rho_{0}-[H_{0},[H_{0},\rho_{0}]]\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t_{1}}\mathrm{d}t_{2}\big{\langle}\left[A_{c}(t_{1})h_{+}(t_{1})+A_{s}(t_{1})h_{\times}(t_{1})\right]$
$\displaystyle\times\left[A_{c}(t_{2})h_{+}(t_{2})+A_{s}(t_{2})h_{\times}(t_{2})\right]\big{\rangle}+\dotsb$
$\displaystyle=\rho_{0}-[H_{0},[H_{0},\rho_{0}]]\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t_{1}}\mathrm{d}t_{2}\big{(}\left\langle
A_{c}(t_{1})A_{c}(t_{2})\right\rangle\left\langle
h_{+}(t_{1})h_{+}(t_{2})\right\rangle$ $\displaystyle+\left\langle
A_{s}(t_{1})A_{s}(t_{2})\right\rangle\left\langle
h_{\times}(t_{1})h_{\times}(t_{2})\right\rangle\big{)}+\dotsb,$ (5)

where we show only two nonzero terms in order not to encumber the text.

We can see that the series in Eq. (II) decays into two independent ones
corresponding to different polarizations of GW. Each of these series contains
only either $\left\langle h_{+}(t)h_{+}(0)\right\rangle$ or $\left\langle
h_{\times}(t)h_{\times}(0)\right\rangle$. In the following, we account for all
terms in the expansion in Eq. (II). Further analysis of each of the series
corresponding to different GW polarizations is identical to that in Ref. [12].
Therefore we omit the details.

Now, let us consider the averaging of the angle factors. We should mention
that both the amplitudes $h_{+,\times}(t)$ and the angles $\vartheta(t)$ and
$\varphi(t)$ are random functions of time. Indeed, we consider random
distribution of GWs sources. It means that, when a neutrino interacts with a
certain GW, the angle between a neutrino momentum and the wave vector of GW is
randomly distributed from zero to $\pi$. However, unlike the correlators
$\left\langle h_{+,\times}(t_{1})h_{+,\times}(t_{2})\right\rangle$, which are
taken to be arbitrary, we suppose that both
$\langle\vartheta(t_{1})\vartheta(t_{2})\rangle$ and
$\langle\varphi(t_{1})\varphi(t_{2})\rangle$ are proportional to
$\delta(t_{1}-t_{2})$. This supposition is reasonable since it is based on the
assumption of the uniform distribution of the sources of GWs in the universe.
The form of the correlators $\left\langle
h_{+,\times}(t_{1})h_{+,\times}(t_{2})\right\rangle$ depends on physical
processes underlying the GWs production. Thus, it is inexpedient to take that
the amplitudes are $\delta$-correlated.

We can study, e.g., the correlator $\left\langle
A_{c}(t_{1})A_{c}(t_{2})\right\rangle$, which has the form,

$\left\langle
A_{c}(t_{1})A_{c}(t_{2})\right\rangle=\frac{1}{4}\left\langle\sin^{2}\vartheta_{1}\sin^{2}\vartheta_{2}\cos(2\varphi_{1})\cos(2\varphi_{2})\cos\alpha_{1}\cos\alpha_{2}\right\rangle,$
(6)

where $\alpha_{1,2}=\omega t_{1,2}(1-\cos\vartheta_{1,2})$ and the angles
$\vartheta_{1,2}$ and $\varphi_{1,2}$ correspond to $t_{1,2}$. Since the
random variables $\vartheta$ and $\varphi$ are taken to be
$\delta$-correlated, we get that
$\langle\cos\alpha_{1}\cos\alpha_{2}\rangle=1/2$. Now, we should average Eq.
(6) over directions of incoming GWs,

$\left\langle
A_{c}(t_{1})A_{c}(t_{2})\right\rangle=\frac{1}{16\pi^{2}}\int_{0}^{\pi}\mathrm{d}\vartheta\sin^{4}\vartheta\int_{0}^{2\pi}\mathrm{d}\varphi\cos^{2}(2\varphi)=\frac{3}{128}.$
(7)

Analogously we show that $\left\langle
A_{s}(t_{1})A_{s}(t_{2})\right\rangle=\tfrac{3}{128}$.

The obtained correlators of the angular factors in Eq. (7) should be used in
Eq. (II). As we mentioned above, Eq. (II) splits into two independent series.
Accounting for all terms in the expansions and applying the results of Ref.
[12], we get that $\left\langle\rho_{\mathrm{int}}\right\rangle$ obeys the
equation,

$\frac{\mathrm{d}}{\mathrm{d}t}\left\langle\rho_{\mathrm{int}}\right\rangle(t)=-g(t)\frac{3}{64}[H_{0},[H_{0},\left\langle\rho_{\mathrm{int}}\right\rangle(t)]],$
(8)

where

$g(t)=\frac{1}{2}\int_{0}^{t}\mathrm{d}t_{1}\left(\left\langle
h_{+}(t)h_{+}(t_{1})\right\rangle+\left\langle
h_{\times}(t)h_{\times}(t_{1})\right\rangle\right).$ (9)

In case of circularly polarized GWs with $h_{+}=h_{\times}$, we reproduce the
results of Ref. [12] in Eqs. (8) and (9).

To proceed with the analysis of Eq. (8), we define the new matrix
$\rho^{\prime}=U^{\dagger}\left\langle\rho_{\mathrm{int}}\right\rangle U$,
which is the density matrix in the interaction picture for the neutrino mass
eigenstates. After some matrix algebra, we get that $\rho^{\prime}$ satisfies
the equation,

$\dot{\rho}^{\prime}=-\tilde{g}[H_{m}^{(\mathrm{vac})},[H_{m}^{(\mathrm{vac})},\rho^{\prime}]]=-\tilde{g}M,$
(10)

where $\tilde{g}=\tfrac{3}{64}g$. The matrix $M$ in Eq. (10) has the following
entries: $M_{ab}=(E_{a}-E_{b})^{2}\rho^{\prime}_{ab}$. Note that, here, we use
the original form of $H_{m}^{(\mathrm{vac})}$ before the energy decomposition:
$\left(H_{m}^{(\mathrm{vac})}\right)_{ab}=E_{a}\delta_{ab}$.

Thus, Eq. (10) can be integrated straightforwardly

$\rho^{\prime}_{aa}(t)=\rho^{\prime}_{aa}(0)=\text{const},\quad\rho^{\prime}_{ab}(t)=\rho^{\prime}_{ab}(0)g_{ab},\
a\neq b,\quad
g_{ab}(t)=\exp\left[-(E_{a}-E_{b})^{2}\int_{0}^{t}\tilde{g}(t^{\prime})\mathrm{d}t^{\prime}\right],$
(11)

where the initial condition $\rho^{\prime}(0)$ has the form,
$\rho^{\prime}(0)=U^{\dagger}\left\langle\rho_{\mathrm{int}}\right\rangle(0)U=U^{\dagger}\rho(0)U$.
It can be also expressed in the components as
$\rho^{\prime}_{ab}(0)=\sum_{\sigma}U_{\sigma a}^{*}U_{\sigma
b}P_{\sigma}(0)$, where we assume that
$\rho_{\lambda\sigma}(0)=\delta_{\lambda\sigma}P_{\sigma}(0)$. Here the
emission probabilities $P_{\sigma}(0)$ are proportional to the neutrino fluxes
at a source: $P_{\sigma}(0)\propto\left(F_{\nu_{\sigma}}\right)_{\mathrm{S}}$.

Accounting for Eq. (11), we get the expression for the density matrix for
flavor neutrinos, $\rho=UR\rho^{\prime}R^{\dagger}U^{\dagger}$, where
$\left(R\right)_{ab}=\delta_{ab}\exp(-\mathrm{i}E_{a}t)$. The probability to
detect a certain flavor, after the neutrino beam passes the distance $x\approx
t$, reads $P_{\lambda}(x)=\rho_{\lambda\lambda}(t\approx x)$. Using the
components of $U$ in Eq. (2), it can be represented in the form,

$P_{\lambda}^{(g)}(x)=\sum_{\sigma}P_{\sigma}(0)\left[\sum_{a}|U_{\lambda
a}|^{2}|U_{\sigma a}|^{2}+2\text{Re}\sum_{a>b}U_{\lambda a}U_{\lambda
b}^{*}U_{\sigma a}^{*}U_{\sigma
b}\exp\left(-\mathrm{i}\varphi_{ab}x\right)g_{ab}\right],$ (12)

where $\varphi_{ab}=E_{a}-E_{b}=\frac{\Delta m_{ab}^{2}}{2E}$ are the phases
of neutrino vacuum oscillations. Equation (12), which takes into account the
neutrino interaction with stochastic GWs, should be compared with the
analogous probabilities for neutrino vacuum oscillations
$P_{\lambda}^{(\mathrm{vac})}$, which are derived, e.g., in Ref. [26]. The
difference $\Delta
P_{\lambda}=P_{\lambda}^{(g)}-P_{\lambda}^{(\mathrm{vac})}$, which reveals the
effect of GWs on neutrino oscillations, has the form,

$\displaystyle\Delta P_{\lambda}(x)=$ $\displaystyle
2\sum_{\sigma}P_{\sigma}(0)\sum_{a>b}\left\\{\text{Re}\left[U_{\lambda
a}U_{\lambda b}^{*}U_{\sigma a}^{*}U_{\sigma
b}\right]\cos\left(2\pi\frac{x}{L_{ab}}\right)+\text{Im}\left[U_{\lambda
a}U_{\lambda b}^{*}U_{\sigma a}^{*}U_{\sigma
b}\right]\sin\left(2\pi\frac{x}{L_{ab}}\right)\right\\}$
$\displaystyle\times\left\\{1-\exp\left[-\frac{4\pi^{2}}{L_{ab}^{2}}\int_{0}^{x}\tilde{g}(t)\mathrm{d}t\right]\right\\},$
(13)

where $L_{ab}=\tfrac{4\pi E}{|\Delta m_{ab}^{2}|}$ are the neutrino
oscillations lengths in vacuum.

If we study the interaction between stochastic GWs and neutrinos emitted by
randomly distributed sources, we have to average Eq. (II) over the propagation
distance. It gives $\left\langle\Delta P_{\lambda}\right\rangle=0$. Therefore,
the effect of stochastic GWs on oscillations of such neutrinos is washed out.
The fluxes at a source will coincide with these accounting for only vacuum
oscillations. Thus, the claim in Ref. [12], that stochastic GWs result in
small changes of observed fluxes of neutrinos from randomly distributed
sources, is incorrect. The deviation of fluxes, obtained in Ref. [12], is
likely to stem from an inexactitude of numerical simulations. We avoid this
inexactitude in the present work since we rely on the analytical solution of
Eq. (8).

##  III Application to SN neutrinos

In this section, we apply the obtained results for the description of the
propagation of SN neutrinos in the background of stochastic GWs.

A huge amount of energy is carried away by neutrinos from a core-collapsing
SN. The major neutrino luminosity was reported, e.g., in Ref. [27] to take
place during a $\nu_{e}$-burst, which happens because of the direct Urca
process $e^{-}+p\to n+\nu_{e}$ in the neutronazing matter of a protoneutron
star (PNS). This burst occurs at $\sim(3-4)\,\text{ms}$ after the core bounce
and lasts $\lesssim 0.1\,\text{s}$ [27]; see also references therein. The
neutrino luminosity can reach $\sim 10^{53}\,\text{erg}\cdot\text{s}^{-1}$
during the burst, with almost all of emitted neutrinos being of the electron
type [27].

We start this section with the study of the interaction between stochastic GWs
and SN neutrinos emitted in a $\nu_{e}$-burst. In this situation, the fluxes
at a source are
$\left(F_{\nu_{e}}:F_{\nu_{\mu}}:F_{\nu_{\tau}}\right)_{\mathrm{S}}=(1:0:0)$.
At later moments of time, other neutrino flavors are emitted. The fluxes of
different flavors of SN neutrinos become almost equal by
$t\approx(0.05-0.1)\,\text{s}$ after the core bounce. Thus, the initial
neutrino fluxes are not in the ratio $(1:0:0)$. The evolution of SN neutrinos
with such initial condition in the presence of stochastic GWs is also
discussed in this section.

A collapsing star, owing to its relatively small size, can be considered as an
almost point-like neutrino source. Indeed, the size of a neutrinosphere, i.e.
an effective sphere where neutrinos are trapped inside, is
$L_{\text{source}}\lesssim 100\,\text{km}$ at the moment of a $\nu_{e}$-burst.
The energy of SN neutrinos is $E\sim 10\,\text{MeV}$ [22]. The oscillations
lengths are $L_{21}\approx 330\,\text{km}$ and $L_{31}\approx L_{32}\approx
10\,\text{km}$ for this energy and $\Delta m_{ab}^{2}$ from Ref. [24].
Neutrinos are emitted from any point of a neutrinosphere more or less
isotropically. A terrestrial detector can register all SN neutrinos emitted
towards it. Thus, we have to integrate the densities of the fluxes over the
area of a neutrinosphere, $S_{\mathrm{source}}\sim L_{\mathrm{source}}^{2}$,
and divide the result by $S_{\mathrm{source}}$. We can call this procedure as
the averaging over the emission points. Thus, if we carry out this averaging
for $\Delta P_{\lambda}$ in Eq. (II), the (31)- and (32)-contributions are
smeared. The only nonvanishing contribution is from the solar oscillations
channel (21).

As we mentioned above, other neutrino flavors are emitted after a
$\nu_{e}$-burst, changing the ratio $(1:0:0)$ of the initial fluxes. However,
the absolute value of the SN neutrino luminosity becomes smaller. The
neutrinosphere shrinks at these greater times. Nevertheless, its size remains
greater than $10\,\text{km}$. Thus, only the (21)-oscillations channel gives a
nonzero contribution to Eq. (II) even for $t>t_{\mathrm{burst}}$.

We start by considering neutrinos emitted in a $\nu_{e}$-burst. Accounting for
the ratio of the initial fluxes, we get that $\Delta P_{\lambda}$ for such SN
neutrinos takes the form,

$\displaystyle\Delta P_{\lambda}(x)=$ $\displaystyle
2\left[\text{Re}\left[U_{\lambda 2}U_{\lambda
1}^{*}U_{e2}^{*}U_{e1}\right]\cos\left(2\pi\frac{x}{L_{21}}\right)+\text{Im}\left[U_{\lambda
2}U_{\lambda
1}^{*}U_{e2}^{*}U_{e1}\right]\sin\left(2\pi\frac{x}{L_{21}}\right)\right]\left[1-\exp\left(-\Gamma\right)\right],$
$\displaystyle\Gamma=$
$\displaystyle\frac{4\pi^{2}}{L_{21}^{2}}\int_{0}^{x}\tilde{g}(t)\mathrm{d}t.$
(14)

In Eq. (III), we do not set the sine and cosine factors to zero despite $x\gg
L_{21}$. The propagation distance is huge, but it is fixed.

The correlators of the amplitudes of GWs $\left\langle
h_{+,\times}(t)h_{+,\times}(0)\right\rangle$ are related to the spectral
density $S(f)$ of GW by [28]

$\sum_{ij}\left\langle h_{ij}(t)h_{ij}(0)\right\rangle=\left\langle
h_{+}(t)h_{+}(0)\right\rangle+\left\langle
h_{\times}(t)h_{\times}(0)\right\rangle=\int_{0}^{\infty}\mathrm{d}f\cos(2\pi
ft)S(f),$ (15)

where $f$ is the frequency measured in Hz. In Eq. (15), we use Eq. (27). Then,
we define the function
$\Omega(f)=\tfrac{f}{\rho_{c}}\tfrac{\mathrm{d}\rho_{\mathrm{GW}}}{\mathrm{d}f}$,
where $\rho_{\mathrm{GW}}$ is the energy density of a GW and
$\rho_{c}=0.53\times 10^{-5}\,\text{Gev}\cdot\text{cm}^{-3}$ is the closure
energy density of the universe. Using Eq. (15), we get that
$\Omega(f)=\tfrac{\pi f^{3}}{8\rho_{c}G}S(f)$, where $G=6.9\times
10^{-39}\,\text{GeV}^{-2}$ is the Newton’s constant. The function
$\tilde{g}(t)$ has the form,

$\tilde{g}(t)=\frac{3}{128}\int_{0}^{t}\mathrm{d}t_{1}\left(\left\langle
h_{+}(t)h_{+}(t_{1})\right\rangle+\left\langle
h_{\times}(t)h_{\times}(t_{1})\right\rangle\right)=\frac{3G\rho_{c}}{32\pi^{2}}\int_{0}^{\infty}\frac{\mathrm{d}f}{f^{4}}\sin(2\pi
ft)\Omega(f).$ (16)

Now, choosing the source of stochastic GWs, which is fully characterized by
$\Omega(f)$, we can evaluate $\Delta P_{\lambda}$.

We suppose that stochastic GWs are emitted by randomly distributed merging
supermassive BHs. In the case, we can approximate $\Omega(f)$ by [29]

$\Omega(f)=\begin{cases}\Omega_{0},&\text{if}\quad
f_{\mathrm{min}}<f<f_{\mathrm{max}},\\\ 0,&\text{otherwise},\end{cases}$ (17)

where $\Omega_{0}\sim 10^{-9}$, $f_{\mathrm{min}}\sim 10^{-10}\,\text{Hz}$,
and $f_{\mathrm{max}}\sim 10^{-1}\,\text{Hz}$. The main contribution to
$\Gamma$ in Eq. (III) results from $f_{\mathrm{min}}$. Hence we can put
$f_{\mathrm{max}}\to\infty$ since $f_{\mathrm{max}}\gg f_{\mathrm{min}}$.

We suppose that $0<x<L$, where $L\sim 10\,\text{kpc}$ is the maximal
propagation length, which is taken to be comparable with the Galaxy size $\sim
32\,\text{kpc}$. Using Eqs. (III) and (16), we get the parameter $\Gamma$ in
the form,

$\Gamma=\frac{3G\rho_{c}}{8\pi
L_{21}^{2}}\int_{f_{\text{min}}}^{f_{\text{max}}}\frac{\mathrm{d}f}{f^{5}}\sin^{2}\left(\pi
fx\right)\Omega(f)\approx 7\times 10^{9}\times I\left(\tau,324\right),\quad
I(\tau,\omega_{\text{min}})=\int_{\omega_{\text{min}}}^{\infty}\frac{\mathrm{d}\omega}{\omega^{5}}\sin^{2}\left(\omega\tau\right)\Omega(\omega),$
(18)

where $\tau=x/L$ and $\omega_{\text{min}}=\pi Lf_{\text{min}}$ are the
dimensionless parameters. The function $\Gamma(\tau)$ is shown in Fig. 1 for
$\Omega(\omega)$ corresponding to Eq. (17). In Fig. 1, we depict only the
normal mass ordering case since the inverted ordering is almost excluded
experimentally [30].

Figure 1: (a) The parameter $\Gamma$ versus the neutrino beam propagation
length $x=\tau L$; (b)-(d) the corrections the neutrino fluxes $\Delta
F_{\nu_{\lambda}}\propto\Delta P_{\lambda}$ owing to the neutrino interaction
with stochastic GWs. The parameters of the system are [22, 24, 29] $\Delta
m_{21}^{2}=7.5\times 10^{-5}\,\text{eV}^{2}$ and $E=10\,\text{MeV}$
($L_{21}=3.3\times 10^{2}\,\text{km}$), $\theta_{12}=0.6$, $\theta_{23}=0.85$,
$\theta_{13}=0.15$, $\delta_{\mathrm{CP}}=3.77$, $L=10\,\text{kpc}$,
$\Omega_{0}=10^{-9}$, and $f_{\mathrm{min}}=10^{-10}\,\text{Hz}$. The normal
neutrino mass ordering is adopted.

We can see in Fig. 1 that $\Gamma$ tends to a constant value at $\tau\to 1$.
If we study neutrino fluxes at the Earth, we put $x=L$, or $\tau=1$. Then, we
suppose that the distance between a source, SN, and a detector, the Earth, is
great. It corresponds to the limit $\omega_{\text{min}}\gg 1$. Accounting for
Eq. (17), we can rewrite $\Gamma\to\Gamma_{\oplus}$ in the form,

$\Gamma_{\oplus}=\frac{3G\rho_{c}\Omega_{0}}{8\pi
f_{\text{min}}^{4}L_{21}^{2}}\omega_{\text{min}}^{4}\int_{\omega_{\text{min}}}^{\infty}\frac{\mathrm{d}\omega}{\omega^{5}}\sin^{2}\left(\omega\right)\to
8\times
10^{-2}\times\left(\frac{\Omega_{0}}{10^{-9}}\right)\left(\frac{f_{\text{min}}}{10^{-10}\,\text{Hz}}\right)^{-4}.$
(19)

If $\Omega_{0}=10^{-9}$ and $f_{\text{min}}=10^{-10}\,\text{Hz}$,
$\Gamma_{\oplus}=8\times 10^{-2}$ in full agreement with Fig. 1.

Although Eq. (19) is valid for $\Omega(f)$ in Eq. (17), we can see that
$\Gamma_{\oplus}\to 0$ at great $f_{\text{min}}$. This fact explains the
result of Ref. [12] that stochastic GWs, emitted by coalescing BHs with
stellar masses, do not change the fluxes of neutrinos. Indeed, in that case
[29], $f_{\mathrm{min}}\sim 10^{-5}\,\text{Hz}\gg 10^{-10}\,\text{Hz}$. Hence
$\Gamma_{\oplus}\to 0$ and $\Delta P_{\lambda}\to 0$ despite $\Omega$ is
greater for such sources.

If $\left(F_{\nu_{e}}:F_{\mu}:F_{\nu_{\tau}}\right)_{\mathrm{S}}=(1:0:0)$ for
SN neutrinos emitted in a $\nu_{e}$-burst, the probabilities at the Earth for
vacuum oscillations are [26]

$P_{\lambda}^{(\mathrm{vac})}(x)=\sum_{a}|U_{\lambda
a}|^{2}|U_{ea}|^{2}+2\left\\{\text{Re}\left[U_{\lambda 2}U_{\lambda
1}^{*}U_{e2}^{*}U_{e1}\right]\cos\left(2\pi\frac{x}{L_{21}}\right)+\text{Im}\left[U_{\lambda
2}U_{\lambda
1}^{*}U_{e2}^{*}U_{e1}\right]\sin\left(2\pi\frac{x}{L_{21}}\right)\right\\},$
(20)

where we accounted for the fact that the contributions of the (31)- and
(32)-oscillations channels are washed out after the averaging over the
neutrino emission points on the neutrinosphere surface. The values of the
neutrino fluxes $F_{\nu_{\lambda}}^{(\mathrm{vac})}\propto
P_{\lambda}^{(\mathrm{vac})}$ in Eq. (20) are summarized in Table 1. Since the
fluxes $F_{\nu_{\lambda}}^{(\mathrm{vac})}$ are rapidly oscillating on the
distance $L=10\,\text{kpc}$, in Table 1, we present only the mean values and
the amplitudes of oscillations: $(\text{mean value}\pm\text{amplitude})$. We
also give $\left(\Delta F_{\nu_{\lambda}}\right)_{\oplus}$ at $x\lesssim L$,
i.e. the asymptotic values, which correspond to Figs. 1-1. The mean value of
$\left(\Delta F_{\nu_{\lambda}}\right)_{\oplus}$ equals to zero, as explained
above. Thus, we show only the amplitudes of oscillations of $\left(\Delta
F_{\nu_{\lambda}}\right)_{\oplus}$. One can see in Table 1 that the relative
contribution of stochastic GWs to the measured neutrino fluxes is at the level
of $(5-7)\,\%$.

| $F_{\nu_{\lambda}}^{(\mathrm{vac})}$ | $\left(\Delta F_{\nu_{\lambda}}\right)_{\oplus}$
---|---|---
$\nu_{e}$ | $0.5421\pm 0.4144$ | $\pm 0.0321$
$\nu_{\mu}$ | $0.1834\pm 0.1636$ | $\pm 0.0127$
$\nu_{\tau}$ | $0.2745\pm 0.2587$ | $\pm 0.02$
Table 1: Second column: the fluxes $F_{\nu_{\lambda}}^{(\mathrm{vac})}$ based
on Eq. (20) for different neutrino flavors. Third column: $\left(\Delta
F_{\nu_{\lambda}}\right)_{\oplus}$ corresponding to Figs. 1-1 for various
neutrino types. The parameters of neutrinos and GWs are the same as in Fig. 1.

Now we turn to the discussion of neutrinos which are emitted after a
$\nu_{e}$-burst, i.e. at $t>(3-4)\,\text{ms}$ after the core bounce. As we
mentioned above, the ratio of the emission fluxes is not equal to $(1:0:0)$.
At these times, the fluxes of different flavors eventually become almost
equal. It happens at $t\approx 0.05\,\text{s}$ after the core bounce (see,
e.g., the numerical simulation, carried out in Ref. [31]). At
$t>0.1\,\text{s}$, the absolute values of the fluxes start to decrease [31].

Let us study the influence of stochastic GWs on SN neutrinos emitted at
$t_{\text{burst}}<t<0.1\,\text{s}$. We can use the general Eq. (II) taking
that the emission probabilities are time dependent: $P_{\sigma}(0)\to
P_{\sigma}^{(0)}(\Delta t)$, where $\Delta t=t-t_{\text{burst}}$ and
$t_{\text{burst}}=(3-4)\,\text{ms}$ is the $\nu_{e}$-burst time. We can
approximate $P_{\sigma}^{(0)}(\Delta t)$ by the following dependence:

$P_{\nu_{e}}^{(0)}(\Delta t)=\frac{1}{3}\left[1+2\exp(-5K\Delta
t)\right],\quad P_{\nu_{\mu}}^{(0)}(\Delta t)=P_{\nu_{\tau}}^{(0)}(\Delta
t)=\frac{1}{3}\left[1-\exp(-5K\Delta t)\right],$ (21)

where $K=10\,\text{s}^{-1}$ is the fitting factor. The emission probabilities
in Eq. (21) satisfy $\sum_{\sigma}P_{\sigma}^{(0)}=1$ at any $\Delta t$.

The initial fluxes $(F_{\nu_{\lambda}})_{\text{S}}\propto
P_{\nu_{\lambda}}^{(0)}(\Delta t)$ are shown in Fig 2. The value of
$(F_{\nu_{e}})_{\text{S}}$ at $\Delta t=0$ corresponds to the luminosity $\sim
10^{53}\,\text{erg}\cdot\text{s}^{-1}$ in a $\nu_{e}$-burst. One can see that
$(F_{\nu_{\lambda}})_{\text{S}}$ of different flavors become almost equal at
$\Delta t\approx K^{-1}$.

Figure 2: The SN neutrino fluxes for different post bounce times $\Delta t$.
The parameters of the neutrino system and GWs are the same as in Fig. 1;
$K\Delta t=1$ corresponds to $t=0.1\,\text{s}$ after the core bounce. (a) The
fluxes $(F_{\nu_{\lambda}})_{\text{S}}$ at a source. The blue and green lines
overlap since the fluxes of $\nu_{\mu}$ and $\nu_{\tau}$ coincide [see Eq.
(21)]. (b) The maximal values of the deviations of the fluxes $(\Delta
F_{\nu_{\lambda}}^{(\mathrm{max})})_{\oplus}$ in a detector, owing to the
interaction of SN neutrinos with stochastic GWs.

The detection probabilities in Eq. (II) depend on the propagation length $x$.
We consider the maximal values of $\Delta P_{\lambda}$ when neutrinos arrive
to a terrestrial detector. They are

$\displaystyle\Delta P_{\lambda}^{(\mathrm{max})}=$ $\displaystyle
2\left[1-\exp\left(-\Gamma_{\oplus}\right)\right]\sum_{\sigma}P_{\sigma}^{(0)}(\Delta
t)\sum_{a>b}|U_{\lambda a}U_{\lambda b}^{*}U_{\sigma a}^{*}U_{\sigma b}|,$
(22)

where $\Gamma_{\oplus}=8\times 10^{-2}$. Note that $\Delta
P_{\lambda}^{(\mathrm{max})}$ in Eq. (22) depends on $\Delta t$.

The deviations of the maximal fluxes $(\Delta
F_{\nu_{\lambda}}^{(\mathrm{max})})_{\oplus}\propto\Delta
P_{\lambda}^{(\mathrm{max})}$, owing to the interaction with stochastic GWs,
for different post bounce emission times $\Delta t$ are shown in Fig 2. One
can see that $(\Delta F_{\nu_{\lambda}}^{(\mathrm{max})})_{\oplus}$ at $\Delta
t=0$ coincide with the values given in the third column in Table 1. If $\Delta
t\to K^{-1}$, $(F_{\nu_{\lambda}})_{\oplus}\to 0$. Indeed,
$P_{\sigma}^{(0)}=1/3$ for any neutrino flavor at these emission times. Thus
$\sum_{\sigma}U_{\sigma a}^{*}U_{\sigma b}=\delta_{ab}$ and $\Delta
P_{\lambda}^{(\mathrm{max})}=0$, basing on Eq. (22).

##  IV Discussion

In the present work, we have studied the propagation and flavor oscillations
of astrophysical neutrinos interacting with stochastic GWs. We have rederived
more straightforwardly the effective Hamiltonian for such a system in Appendix
A. The analytical expression for the density matrix of flavor neutrinos has
been obtained in Sec. II. Then, in Sec. III, we have applied the obtained
results for the description of SN neutrinos.

The present research has several advances compared to Ref. [12], where the
interaction of astrophysical neutrinos with stochastic GWs was also studied.
Firstly, we have accounted for two independent polarizations of GWs contrary
to the case of a circularly polarized GW in Ref. [12]. Secondly, now we have
found the exact solution of the equation for the density matrix in the general
case of three neutrino flavors. This fact allowed us to correct some
statements about the asymptotic behavior of neutrino fluxes, which were made
basing on numerical simulations in Ref. [12].

In Ref. [12], we studied the interaction between stochastic GWs and
astrophysical neutrinos emitted in decays of charged pions. Sources of such
neutrinos are distributed more or less uniformly in the universe. Thus one had
to average over the neutrino propagation distance to get the fluxes at the
Earth. This fact made it difficult to extract the contribution of stochastic
GWs to neutrino fluxes. In the present work, we have considered neutrinos
emitted in a SN explosion. It enables us not to perform the averaging over the
propagation distance except for (31)- and (32)-oscillations channels. It is
valid since the oscillations length $L_{21}$ is greater than the radius of the
neutrinosphere, i.e. the size of the neutrino source. The distance $L$ between
a possible SN explosion is great, but it is fixed. Therefore we should not
average over $L/L_{21}$ in the probabilities for neutrino flavors.

We have also obtained the analytical expression for the damping decrement
$\Gamma$; cf. Eq. (19). This result allowed us to evaluate the contribution of
other sources of stochastic GWs, e.g., merging BHs with stellar masses, to the
evolution of neutrino fluxes. The straightforward derivation of the effective
Hamiltonian for flavor neutrinos oscillations, presented in Appendix A, can be
applied to different metrics perturbations besides GWs considered here. Using
this result, we can study, e.g., the evolution of neutrinos in perturbations
in the early universe [10, 32].

In the present work, we have also studied the interaction between stochastic
GWs and SN neutrinos emitted in the time interval
$t_{\mathrm{burst}}<t\lesssim 0.1\,\text{s}$. When $t\approx 0.1\,\text{s}$,
the initial fluxes are almost equal,
$\left(F_{\nu_{e}}:F_{\nu_{\mu}}:F_{\nu_{\tau}}\right)_{\mathrm{S}}=(1:1:1)$.
We have found in Sec. III that, in this situation, the contribution of
stochastic GWs to the evolution of SN neutrinos is washed out; cf. Fig. 2. It
means that the major effect is for neutrinos emitted at a $\nu_{e}$-burst,
which corresponds to
$\left(F_{\nu_{e}}:F_{\nu_{\mu}}:F_{\nu_{\tau}}\right)_{\mathrm{S}}=(1:0:0)$.
The neutrino luminosity in a SN explosion remains significant up to
$10\,\text{s}$ after the core bounce, which is the time scale for the neutrino
driven PNS cooling [33]. However, the influence of stochastic GWs on SN
neutrinos emitted at $t>0.1\,\text{s}$ is vanishing since such neutrinos are
emitted with almost equal probabilities. Thus, in Sec. III, it is inexpedient
to extend $\Delta t$ beyond the $0.1\,\text{s}$ interval.

In Table 1, we have summarized the contributions of stochastic GWs to the
fluxes of flavor neutrinos at the Earth. They are at the level of a few
percent. The current neutrino telescopes are able to detect up to several
thousand neutrinos from SN in our Galaxy [13]. Future detectors, like the
Hyper-Kamiokande, can detect about $7.5\times 10^{4}$ such events [34]. Thus,
the interaction with stochastic GWs can results in the change of the SN
neutrinos fluxes by $\sim\pm 350$ events, in case of the Super-Kamiokande, and
by $\sim\pm 3750$ events, for the Hyper-Kamiokande.

###### Acknowledgements.

I am thankful to A. V. Yudin and J. W. F. Valle for the communications, as
well as to V. A. Berezin for the useful discussion. The work is supported by
the government assignment of IZMIRAN.

##  Appendix A Derivation of the effective Hamiltonian

The action $S_{a}(\mathbf{x},t)$ of a neutrino mass eigenstate with the mass
$m_{a}$, moving in the curved spacetime with the metric $g_{\mu\nu}$, obeys
the Hamilton-Jacobi equation,

$g_{\mu\nu}\frac{\partial S_{a}}{\partial x_{\mu}}\frac{\partial
S_{a}}{\partial x_{\nu}}=m_{a}^{2}.$ (23)

The metric in Eq. (23) is supposed to have the small perturbation $h_{\mu\nu}$
of the Minkowski flat metric $\eta_{\mu\nu}=\text{diag}(+1,-1,-1,-1)$:
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$.

We look for the solution of Eq. (23) in the form,
$S_{a}=S_{a}^{(0)}+S_{a}^{(1)}+\dotsc.$ Here, the zero order term
$S_{a}^{(0)}$ obeys the equation,
$\eta_{\mu\nu}\partial^{\mu}S_{a}^{(0)}\partial^{\nu}S_{a}^{(0)}=m_{a}^{2}.$
The finction $S_{a}^{(0)}$ has the form, $S_{a}^{(0)}(x)=p_{a}^{\mu}x_{\mu}$,
where $p_{a}^{\mu}=(E_{a},\mathbf{p})$,
$E_{a}=\sqrt{m_{a}^{2}+\mathbf{p}^{2}}$, and $\mathbf{p}$ is the constant
neutrino momentum.

In the linear approximation in $h_{\mu\nu}$, the first order term
$S_{a}^{(1)}$ obeys the equation,

$2\eta_{\mu\nu}p_{a}^{\mu}\frac{\partial S_{a}^{(1)}}{\partial
x_{\nu}}+h_{\mu\nu}p_{a}^{\mu}p_{a}^{\nu}=0.$ (24)

We can transform the term $\eta_{\mu\nu}p_{a}^{\mu}\partial^{\nu}S_{a}^{(1)}$
in Eq. (24) as

$\eta_{\mu\nu}p_{a}^{\mu}\frac{\partial S_{a}^{(1)}}{\partial
x_{\nu}}=E_{a}\left(\frac{\partial S_{a}^{(1)}}{\partial
t}+\left(\mathbf{v}_{a}\nabla\right)S_{a}^{(1)}\right)=E_{a}\frac{\mathrm{d}S_{a}^{(1)}}{\mathrm{d}t}.$
(25)

where we use $\mathbf{v}_{a}\equiv\mathbf{v}_{a}^{(0)}=\mathbf{p}/E_{a}$ as
the particle velocity. It means that a neutrino is supposed to move along a
straight line. In general situation, one has that
$\mathbf{v}_{a}=\mathbf{v}_{a}^{(0)}+\delta\mathbf{v}_{a}(t)$ in curved
spacetime. However, since $\delta\mathbf{v}_{a}\propto h_{\mu\nu}$, this term
can be neglected in Eq. (25) because $S_{a}^{(1)}\propto h_{\mu\nu}$ already.

Finally, we get that
$\frac{\mathrm{d}S_{a}^{(1)}}{\mathrm{d}t}=-\frac{1}{2E_{a}}h_{\mu\nu}p_{a}^{\mu}p_{a}^{\nu}$.
The contribution to the effective Hamiltonian can be obtained as
$\left(H_{m}\right)_{aa}=\tfrac{\mathrm{d}S_{a}}{\mathrm{d}t}$. One can check
the validity of this expression in the vaccum case. Thus, we obtain that the
neutrino intraction with a gravitational field, induced by a metric
perturbation, contributes to the effective Hamiltonian in the mass basis as

$\left(H_{m}^{(g)}\right)_{ab}=-\frac{\delta_{ab}}{2E_{a}}h_{\mu\nu}p_{a}^{\mu}p_{a}^{\nu}.$
(26)

Note that we have to take $h_{\mu\nu}$ on the particle trajectory,
$h_{\mu\nu}(\mathbf{x},t)=h_{\mu\nu}(\mathbf{x}(t),t)$.

If we choose a plane GW as the metric perturbation, one has that [35]

$h_{ij}=\left(\begin{array}[]{cc}h_{+}\cos\phi_{a}&h_{\times}\sin\phi_{a}\\\
h_{\times}\sin\phi_{a}&-h_{+}\cos\phi_{a}\end{array}\right),\quad i,j=1,2,$
(27)

and $h_{0\mu}=h_{3\mu}=0$. Here $\phi_{a}=\omega t-kz=\omega
t(1-v_{a}\cos\vartheta)$ is the phase of the wave accounting for the neutrino
motion and $h_{+,\times}$ are the amplitudes of different polarizations of GW.
In Eq. (27), we take the Cartesian coordinates $x^{\mu}=(t,\mathbf{x})$. Using
Eqs. (26) and (27), we obtain that

$\displaystyle\left(H_{m}^{(g)}\right)_{ab}=$
$\displaystyle-\frac{p^{2}}{2E_{a}}\delta_{ab}\sin^{2}\vartheta\left[h_{+}\cos
2\varphi\cos\phi_{a}+h_{\times}\sin 2\varphi\sin\phi_{a}\right],$
$\displaystyle\to\frac{m_{a}^{2}}{2E}\frac{\delta_{ab}}{2}\sin^{2}\vartheta\left[h_{+}\cos
2\varphi\cos\phi+h_{\times}\sin 2\varphi\sin\phi\right],$ (28)

where $\phi=\omega t(1-\cos\vartheta)$. To derive Eq. (A) we assume that
$\omega L|v_{a}-v_{b}|\ll 1$ [11] and omit the common factor propotional to
the unit matrix. Finally, one gets that

$H_{m}^{(g)}=H_{m}^{(\mathrm{vac})}\left(A_{c}h_{+}+A_{s}h_{\times}\right),$
(29)

where $A_{c}=\tfrac{1}{2}\sin^{2}\vartheta\cos 2\varphi\cos[\omega
t(1-\cos\vartheta)]$ and $A_{s}=\tfrac{1}{2}\sin^{2}\vartheta\sin
2\varphi\sin[\omega t(1-\cos\vartheta)]$.

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