Decoherence and oscillations of supernova neutrinos
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Abstract
Supernova neutrinos have several exceptional features which can lead to interesting physical consequences. At the production point their wave packets have an extremely small size \(\sigma _{x} \sim 10^{11}\) cm; hence the energy uncertainty can be as large as the energy itself, \(\sigma _{E} \sim E\), and the coherence length is short. On the way to the Earth the wave packets of mass eigenstates spread to macroscopic sizes and separate. Inside the Earth the mass eigenstates split into eigenstates in matter and oscillate again. The coherence length in the Earth is comparable with the radius of the Earth. We explore these features and their consequences. (1) We present new estimates of the wave packet size. (2) We consider the decoherence condition for the case of wave packets with spatial spread and show that it is not modified by the spread. (3) We study the coherence of neutrinos propagating in a multilayer medium with density jumps at the borders of layers. In this case coherence can be partially restored due to a “catchup effect”, increasing the coherence length beyond the usual estimate. This catchup effect can occur for supernova neutrinos as they cross the shock wave fronts in the exploding star or the core of the Earth.
1 Introduction
Detecting the neutrino burst from a galactic supernova will be one of the major and outstanding scientific events of this century. It will bring about an enormous amount of new physics information both on the dynamics of the collapse as well as the explosion and on neutrinos themselves. Hence, a deep understanding of the underlying processes and effects is a must.
Supernovae – the strongest known sources of neutrinos – provide a unique environment for the production and the flavor evolution of neutrino states. In addition to the standard resonant flavor conversion in matter [1], the huge neutrino density makes neutrino–neutrino interactions relevant [2, 3, 4] that can lead to various collective effects: synchronized oscillations [3, 5], bipolar oscillations [6, 7], spectral splits or swaps [8, 9, 10, 11], selfinduced parametric resonance [12], etc. (see, e.g., [13, 14, 15] for reviews). Stimulated flavor transitions can occur due to turbulence in a medium [16].
Given the special conditions with very high temperature and density in the neutrinosphere, where neutrinos are produced, a very short time scale for the microscopic production processes is realized. Consequently, the neutrino states are described by very short wave packets in configuration space. Indeed, previous estimates of the wave packet size were \(\sigma _x = 1.8 \cdot 10^{14} \,\mathrm{cm}\) for neutrinos produced in the core of the protoneutron star [17] and \(\sigma _x = 4.2 \cdot 10^{9} \,\mathrm{cm}\) for neutrinos emitted at a radius of \(1000\,\mathrm{km}\) [18] and thus in a region with much lower matter density. These calculations used Coulomb scattering to determine the mean free path of electrons inside the supernova.

shift of the wave packets of the eigenstates by \(\Delta x_\text {shift}\) due to the difference of group velocities and eventually their separation when \(\Delta x_\text {shift} > \sigma _x\), implying a loss of coherence of the neutrino states, and

spread of wave packets of individual eigenstates due to the presence of different energies in a wave packet.
Propagation decoherence has been studied mainly in vacuum [19, 20, 21, 22, 23, 24, 25, 26, 27], but also in matter [18, 28, 29, 30, 31] and in dense neutrino gases [32]. It is characterized by the coherence length \(L_\text {coh}\) – the distance at which \(\Delta x_\text {shift} \sim \sigma _x\). For distances larger than \(L_\text {coh}\), the eigenstates no longer interfere because their wave packets no longer overlap. Thus, the oscillatory pattern disappears and the oscillation probability becomes baselineindependent.
The coherence is also affected by the detection process. In particular, coherence can be restored by an accurate energy measurement [20], which was confirmed via a quantum field theory calculation [33]. In this case the detector must have a coherent observation time larger than the difference of the arrival times of two packets \(\Delta t_\text {det}\) or (equivalently) a sufficiently good energy resolution, \(\Delta E_\text {res} < 1/\Delta t_\text {det}\). Thus, observable effects depend on characteristics of the detector: coherent time of observation and energy resolution.
The wave packets give the complete picture of the evolution of neutrino states in configuration space. However, in order to determine observational results it is enough to treat the problem in the energy representation, which simplifies the considerations substantially. The results of the energymomentum and configurationspace treatments are equivalenct as far as observations are concerned, at least in the stationary source approximation [20, 34] and if no time tagging is performed.
The effects of source and detector are symmetric and can be described by an effective wave packet which includes characteristics of both source and detector. In fact, information about the wave packet of the source is included in the generated energy spectrum and information about the one of the detector in the detector’s energy resolution.
It is believed that propagation decoherence does not affect the dynamics of the flavor evolution. Observable effects are then determined essentially by the energy spectrum at production, flavor evolution without decoherence, and the energy resolution of the detector. As mentioned above, however, the separation of supernova neutrinos is very fast due to their very short wave packet size. This could affect the dynamics of the flavor evolution in the region of collective effects [35], although a tentative answer was negative. In contrast, in [36] it was claimed that decoherence does influence collective oscillations (essentially due to the nonlinear character of the problem) and lead to nontrivial flavor transformations.
Propagation decoherence is a reversible process; no information is lost in a system when wave packets separate. Hence, coherence can be restored by further propagation in matter if the difference of group velocities changes sign or (as mentioned above) in the detector.
The consideration in configuration space may have some advantage in the case of a complicated matter profile. It helps to obtain a clear physics interpretation of the results of integrating over energy. We will consider in detail the evolution of the wave packets all the way from the production point to a detector, focussing on the consequences of the small wave packet size of supernova neutrinos. We will study decoherence and partial restoration of coherence. New interesting effects are realized in oscillations in the matter of the Earth, which are related to an accidental coincidence of the coherence length and the size of the Earth. Some preliminary results have been published in [35].
Apart from separating, the wave packets also spread, since they comprise waves with different energies [37, 38]. The increase of the size of a packet depends on the absolute values of neutrino masses. Effects of the spread on oscillations (which were not explored extensively before) are among the main objectives of this work.
The paper is organized as follows. In Sect. 2 we consider the production of neutrinos in a supernova and present an improved estimate of their wave packet size. We consider propagation decoherence, generalizing the vacuum results to propagation in matter. In Sect. 3 we consider decoherence in the energymomentum representation. We show the equivalence of separation of the packets in configuration space and energy averaging. In Sect. 4 we study flavor evolution and coherence in a multilayer medium where neutrino states split at each border between layers. We describe the “catchup effect” – partial restoration of the coherence between certain components of the split states. In Sect. 5 we discuss the spread of wave packets both in vacuum and in matter. We show that the coherence conditions for wave packets with and without spread coincide. In Sect. 6 we apply our results to supernova neutrino oscillations in the matter of the Earth.
2 Supernova neutrino wave packets
2.1 Size of the wave packet
The characteristics of the neutrino wave packets (WP) produced in a supernova depend on the phase of the explosion, since physical conditions and contributing processes change. We will consider different phases in order.
2.1.1 Neutronization burst
During the earliest stage of a supernova, mainly electron neutrinos are produced by electron capture, \(p~ e^ \rightarrow n ~\nu _e\). We assume the nucleons to be localized well enough that the size of their WP is negligible (see below). Then the time scale for the electron capture process is given by the interval of time during which the electron WP crosses the proton [20]: \(\tau \simeq \sigma _x^e / v^e \simeq \sigma _x^e\) for relativistic electrons. Here \(\sigma _x^e\) is the size of the electron WP in configuration space and \(v_e\) is the electron velocity. During this time a neutrino is emitted coherently. Consequently, its WP has a size \(\sigma _x \simeq \tau \simeq \sigma _x^e\).^{1}
2.1.2 Accretion and cooling phase
For \(\nu _e\), electron capture remains the main production process. Consequently, the WP size is still given by Eq. (3). Compared to the neutronization burst, the physical parameters in the neutrinosphere change in opposite directions: while the matter density \(\rho \) increases, the electron fraction \(Y_e\) decreases by about an order of magnitude [40, 41]. As a result, we expect the \(\nu _e\) WP size to increase moderately.
Electron antineutrinos \(\bar{\nu }_e\) are mainly produced by positron capture on neutrons, \(n ~ e^+ \rightarrow p ~ \bar{\nu }_e\), so their \(\sigma _x\) is equal to the WP size of the positron. The size of the positron WP can be estimated in the same way as that of the electron WP. The most important scattering processes for positrons are scatterings on electrons and on protons, whose cross section is the same as the one for electron (ee and ep) scattering, Eq. (1). As a consequence, Eq. (3) holds for \(\bar{\nu }_e\) as well.
For the nonelectron neutrinos, \(\nu _\mu \) and \(\nu _\tau \), and their antineutrinos, the physics of production is quite different. The number density of these neutrinos is determined at the number sphere. Outside this sphere neutrinos still scatter efficiently on leptons (until the energy sphere) and on nucleons (until the transport sphere) [42]. Hence, the WP size of nonelectron neutrinos is determined by these scattering processes. In general, the size of the WP after a scattering process is determined by the time of overlap of the incoming WP and thus approximately equal to the size of the larger incoming WP [37]. As a consequence, neutrino WP will continue to broaden as long as they scatter with particles with larger WP sizes. In our case these particles are the leptons, since the nucleon WP are estimated to be very small; see Eq. (7). Consequently, we expect \(\sigma _x\) for \(\nu _\mu \) and \(\nu _\tau \) to equal the electron WP size at the energy sphere. This means that once again the result is given by Eq. (3).
2.2 Propagation decoherence
When neutrinos propagate from large to small densities, \(\Delta v_m\) changes sign near the MSW resonance. This can lead to the interesting phenomenon that WP separate above (in density) the resonance, then approach each other below the resonance and overlap at some point, thus restoring coherence [28, 45]. This can be realized for solar and supernova neutrinos.
In the case of quickly changing density (strong adiabaticity violation) the instantaneous eigenstates may become irrelevant for the description of the flavor evolution. So the WP and group velocities introduced for these eigenstates have limited (or no) sense.
2.3 Wave packet separation and coherence loss in a supernova
When propagating from the neutrinosphere to the surface of the star, neutrinos cross regions with changing conditions which affect propagation and coherence of WP. In the central parts, neutrino–neutrino scattering leads to the potential \(V_{\nu \nu }\) [2, 4]. This potential is much smaller than the usual matter potential V in the neutrinosphere, but it can be comparable to V or even bigger at distances of order \(10^2\,\mathrm{km}\) from the center during later phases of the supernova explosion. \(V_{\nu \nu }\) depends on the neutrino flavor state (i.e., on the neutrino wave function), which leads to the socalled collective oscillations.
3 Decoherence and averaging over neutrino energy
So far we have considered WP in configuration space. We have also assumed that the coherence length is determined entirely by the production process. Alternatively, we can consider decoherence in energymomentum space, where the WP width is \(\sigma _E \simeq \sigma _p \simeq 1/\sigma _x\). This consideration makes it easier to take into account the detection process.
3.1 Coherence in energymomentum space
A zero value of \(\Delta x_\text {shift}\), i.e., no shift of the WP, corresponds to a zero derivative \(\partial \phi /\partial E\). This implies a weak dependence of the oscillation phase on energy in a certain interval, consequently no significant effect of averaging, and thus no decoherence. However, higherorder terms in the Taylor expansion of the oscillation phase lead to \(\Delta \phi \ne 0\) even if \(\partial \phi /\partial E\) vanishes. Therefore, the consideration in momentum space implies that the coherence length increases significantly for \(\partial \phi /\partial E = 0\) but does not become infinite, which is different from what we observed using the configurationspace treatment in Sect. 2.2.
3.2 Impact of the detection process
3.3 Equivalence of wave packet separation and energy averaging
In the case of complete overlap, \(\Delta x_\text {shift} \rightarrow 0\), Eq. (35) gives \(E_T \rightarrow \infty \), which is equivalent to \({\partial \phi }/{\partial E} \rightarrow 0\). In this case the effect of the interference term (deviation from the averaged probability) does not depend on energy, and therefore it is also independent of the energy resolution of the detector (as long as \(\Delta E \gg E\), as required by the Taylor expansion in Eq. (28)).
In the opposite case of large shift \(\Delta x_\text {shift}\), the period becomes very small, so one needs to have very good energy resolution since the condition \(\Delta E \ll E_T\) should be satisfied to observe the interference (oscillatory effect) in the oscillation probability. In configuration space that would correspond to a long coherent observation time with \(\Delta t \sim 1/ \Delta E\), and consequently to restoration of coherence in the detector.
Summarizing the two pictures, in both representations (configuration and momentum space) we start from the eigenvalues of the Hamiltonian \(H_{im}\) and their difference \(\Delta H_m\). In configuration space \(\Delta H_m\) determines the difference of group velocities and the relative shift of the WP. Then comparing the shift with the effective size of the packet \(\sigma _{E,\text {tot}}^{1}\) (which includes both the produced WP and the energy resolution of the detector) determines whether coherence is preserved or lost. In momentum space \(\Delta H_m\) determines the oscillation phase and the oscillatory period in energy \(E_T\). Comparison of the latter with the effective width of the packet \(\sigma _{E,\text {tot}}\) determines whether the oscillatory pattern is observable or averaged to a constant oscillation probability.
Equivalence of the configuration and momentumspace considerations is realized when the whole process is taken into account: production, propagation and detection of neutrinos. The phase \(\phi (E, L)\) is the key (integral) characteristic which takes into account all the relevant (for coherence) features of propagation.
The discussion up to this point shows that WP separation and energy averaging produce equivalent effects in the adiabatic case. In fact, this also follows from theorems in [20, 34], according to which it is impossible to distinguish long and short WP; in particular, whether one can observe coherent effects or not is independent of the size of the WP. In the following, we will consider neutrino oscillations in matter with density jumps, aiming to show explicitly that the equivalence holds under such conditions as well.
4 Coherence in multilayer medium
The picture described in the previous section is modified if adiabaticity is broken. In the following we will consider special (maximal) adiabaticity breaking occurring when neutrinos propagate in a multilayer medium that consists of several layers with constant or adiabatically changing density and abrupt density changes between the layers. In other words, there is a steplike change or jump of density at each border between two layers. This happens in a supernova at the front of the shock waves. Later, neutrinos experience density jumps when they enter the Earth and at the boundary between the mantle and the core. Coherence in such a case can be treated in the same way as before, taking into account splittings of the eigenstates at the borders.
4.1 Splitting of eigenstates
Let us consider the jump of the density between the layers k and \(k+1\). Suppose a neutrino propagates in the layer k, crosses the border and then propagates in the layer \(k + 1\). The eigenstate \(\nu _{i m}^{(k)}\) in the layer k does not coincide with any eigenstate in the layer \(k + 1\). Therefore, when crossing the border, \(\nu _{i m}^{(k)}\) will split into two^{6} eigenstates \(\nu _{j m}^{(k+1)}\) of layer \(k + 1\). Correspondingly, at a density jump each WP splits up into a pair of new packets. After the split, the state will oscillate and the packets will shift according to the group velocity difference in the second layer.
Crossing a medium with n layers, \(2^n\) components of the neutrino state are generated. They correspond to parts of the WP of \(\nu _{1m}^{(n)}\) and \(\nu _{2m}^{(n)}\) with different shifts. Thus, compared to the adiabatic case where one deals with only two WP that either overlap or do not, there are additional possibilities in the multilayer medium. It is possible that some but not all WP overlap in the detector, allowing to observe a part of the interference terms in the oscillation probability. This corresponds to an intermediate case between complete coherence (all WP overlap) and complete decoherence (no WP overlap). Notice that the splitting has sense only in the presence of shift and separation of the WP. If the shift is neglected in each layer we can sum up the components which belong to the same eigenstate and the picture is reduced again to the propagation of two WP.
4.2 Two layers of matter and catchup effect
 1.
For \(\psi = \phi _k\) (\(k = 1,2\)), it can be satisfied only if \(\xi _k = \xi _0 = (\cos 2\theta )^{1}\) and thus \(\Delta v_m^{(k)} = 0\), i.e., the WP do not separate in layer k. For this case we had found the stationaryphase condition already at the end of Sect. 3.1.
 2.For \(\psi = \phi _1  \phi _2\), we obtain Eq. (43), as already discussed. Using (18), we can write this equation asAs \(L_1\) and \(L_2\) are positive, this condition can only be satisfied if \(\xi _1, \xi _2 < (\cos 2\theta )^{1}\), or if \(\xi _1, \xi _2 > (\cos 2\theta )^{1}\), that is, if both densities are below or above the critical density.^{9} This means that either in both layers the eigenstates \(\nu _{1m}\) move faster than \(\nu _{2m}\), or in both layers \(\nu _{1m}\) move more slowly than \(\nu _{2m}\). The overlap occurs in the second layer between the WP of \(\nu _{2m}^{(2)}\) and \(\nu _{1m}^{(2)}\) originating from the transitions$$\begin{aligned} L_2 = L_1 \, \frac{1\xi _1\cos 2\theta }{1\xi _2 \cos 2\theta } \, \frac{R(\xi _2)}{R(\xi _1)} . \end{aligned}$$(50)where the arrows indicate the transitions at the density jumps. The corresponding interference term that is not averaged to zero arises from the second and third terms in the oscillation probability (42). According to Eq. (42) the oscillation depth for this mode is \(4 s_2 c_2 s_{10} c_{10} s_{21}^2 = \sin 2\theta _2 \, \sin 2\theta _{10} \, s_{21}^2\).$$\begin{aligned} \nu _1 \rightarrow \nu _{1m}^{(1)} \rightarrow \nu _{2m}^{(2)} \quad \text {and}\quad \nu _1 \rightarrow \nu _{2m}^{(1)} \rightarrow \nu _{1m}^{(2)} , \end{aligned}$$
 3.For \(\psi = \phi _1 + \phi _2\), we obtain from Eq. (49) using Eqs. (44) and (18)This condition can be satisfied if \(\xi _2> (\cos 2\theta )^{1} > \xi _1\), or if \(\xi _1> (\cos 2\theta )^{1} > \xi _2\), that is, if one density is larger than the critical density and the other one is smaller. Now either \(\nu _{1m}^{(1)}\) moves faster than \(\nu _{2m}^{(1)}\) whereas \(\nu _{1m}^{(2)}\) moves more slowly than \(\nu _{2m}^{(2)}\), or vice versa. The overlapping WP originate from the transitions$$\begin{aligned} L_2 = L_1 \, \frac{\Delta v_m^{(1)}}{\Delta v_m^{(2)}} = L_1 \, \frac{1\xi _1\cos 2\theta }{1\xi _2 \cos 2\theta } \, \frac{R(\xi _2)}{R(\xi _1)} . \end{aligned}$$(51)$$\begin{aligned} \nu _1 \rightarrow \nu _{1m}^{(1)} \rightarrow \nu _{1m}^{(2)} \quad \text {and}\quad \nu _1 \rightarrow \nu _{2m}^{(1)} \rightarrow \nu _{2m}^{(2)} . \end{aligned}$$
The case of two layers of matter with adiabatically varying density and density jump between them can be realized in a supernova. The jump is due to the shock wave. The new eigenstates propagate adiabatically and encounter the shock wave in the MSW transition region, where the change of mixing angle is large (the effect of the shock wave outside this region is very small). The first layer is between inner parts of the collective effects and the shock front, and the second one is between the shock front and the surface of the star. The oscillation phases \(\phi _k\) should be computed for adiabatically varying density, \(\phi _k = \int _k \mathrm{d}x \, \Delta H_m^{(k)}\).
The possibility of two layers and two jumps is realized when there are two shock wave fronts (one can move inward); see, e.g., [47]. In this case the first layer is the one between the shock fronts and the second one is above the outer shock. The result is described by Eq. (42) with similar correspondence as in the first case.
4.3 Generalization
 1.If the detector’s energy resolution is good enough to resolve all interference terms,Coherence is completely preserved. As discussed in Sect. 3.2, it does not matter whether this happens because all WP overlap in the detector or because the detector restores coherence.$$\begin{aligned} \overline{P(E)} \simeq P_\text {coh} = \mathcal {A}(E)^2 . \end{aligned}$$(56)
 2.For a bad resolution, the oscillation probability is given by the incoherent sumCoherence is completely lost. All WP are separated (and the detector does not restore coherence), or averaging due to the bad energy resolution removes even interference terms corresponding to overlapping WP.$$\begin{aligned} \overline{P(E)} \simeq P_\text {decoh} = \sum _r a_r^2 . \end{aligned}$$(57)
 3.It is possible that some but not all interference terms in the probability survive averaging, for instance,We will refer to this case as partial survival of coherence. Survival of coherence is possible for each interference term. Therefore, we have to introduce an individual coherence length for each term.$$\begin{aligned} \overline{P(E)} \simeq \sum _r a_r^2 + 2 a_s a_t \cos (\psi _s\psi _t) . \end{aligned}$$(58)
On the other hand, if the term containing \(\cos \phi _k\) survives, this can be due to a vanishing velocity difference in one layer, \(\Delta v_m^{(k)} = 0\), or due to the detector restoring the coherence of consecutive WP that were separated in layer k. Of course, there is also the trivial possibility that one or more layers are so thin that WP do not separate inside them. Another trivial example of partial survival of coherence is realized in the case of threeneutrino mixing in a single layer when at large enough distances the oscillation modes due to \(\Delta m_{31}^2\) and \(\Delta m_{32}^2\) are averaged to zero, whereas the mode due to the small splitting \(\Delta m_{21}^2\) is not.
Note that the catchup effect described here relies on two ingredients, matter effects and strong (maximal) adiabaticity violation. It is thus different from the increase of the coherence length that is possible if \(\Delta v_m\) changes sign during adiabatic propagation [28, 45]. In that case, there is no splitting of the WP into many components, and only two WP arrive at the detector.
5 Spread of the wave packets
5.1 Separation and spread
5.2 Spread and energy redistribution
The coherenceloss condition (74) does not depend on the absolute neutrino mass scale, since \(\sigma _\text {spread}\) cancels. It depends on \(\Delta m^2\) via \(\Delta x_\text {shift}\).
5.3 Spread of the wave packets in matter
In the realistic situation of a supernova this may happen in the MSW region. However, most of the spread occurs in vacuum on the way to the Earth, and the spread inside the supernova can be neglected.
6 Oscillations of supernova neutrinos inside the Earth
6.1 Neutrino states at the surface of the Earth
As we have discussed in Sect. 4, entering the Earth each mass eigenstate splits into eigenstates \(\nu _{im}\) in the matter of the Earth and oscillates. We neglect the presence of the third neutrino \(\nu _{3m} \simeq \nu _3\) here. This state decouples from the rest of the system producing just a small (given by \(\sin ^2 \theta _{13}\)) average oscillation result. So we will consider twoneutrino oscillations driven by the mass splitting \(\Delta m^2_{21}\).
6.2 The coherence condition in the Earth
Using expressions for the length of trajectories in different layers we can find regions of complete decoherence, partial decoherence and complete coherence in the E–\(\cos \eta \) plane, where \(\eta \) is the nadir angle.
In Fig. 3 we show the lengths of different trajectories in the mantle and the core as functions of nadir angle according to Eqs. (88), (91), and (92) together with the coherence lengths for different neutrino energies (the horizontal lines).
Notice that \(\sigma _E \sim 1\,\mathrm{MeV}\) we have found does not depend on neutrino energy, whereas the energy resolution of the detector \(\Delta E_\text {res} \propto \sqrt{E}\). We find that already at \(E > (2\)–\(3) \,\mathrm{MeV}\) the energy resolution of a detector becomes larger than \(\sigma _E\), and therefore determines the oscillation pattern apart from the cases of catchup.
In the following we will describe separately the effects of propagation along the mantle only and along the corecrossing trajectories.
6.3 Oscillations in the mantle of the Earth
6.4 Oscillations of neutrinos crossing the core
Neutrinos that cross the core of the Earth encounter three layers: mantle, core, and mantle. Thus, they pass three jumps of density: at the surface when entering the Earth, when entering the core and when leaving the core. Therefore, each mass state arriving the Earth splits into \(2^3 = 8\) components reaching a detector.
The first term in Eq. (107) with the amplitude \(c_{10} c_{21}^2\) and without phase factor corresponds to the fastest component of the state: in all three layers it corresponds to \(\nu _{1m}^{(i)}\). Notice that this term has the largest amplitude. In contrast, the last term, with the largest phase, \(2 \phi _1 + \phi _2\) and the amplitude \(s_{10} c_{21}^2\) corresponds to the slowest component when the heaviest component \(\nu _{2m}^{(i)}\) propagates in all three layers. The WPs which correspond to these components cannot produce catchup effect with other components.
The interference terms in the probability (107) are proportional to cosines of all possible differences of the phases. This includes 0, \(\phi _1\), \(\phi _2\), \(\phi _1 + \phi _2\), \(2 \phi _1\), \(2 \phi _1 + \phi _2\), and \(\phi _1  \phi _2\), \(2 \phi _1  \phi _2\) and the same combinations with opposite signs.
 1.There are four contributions to the term with phase \((\phi _1\phi _2)\). All of them correspond to the catchup in the case of two layers considered in Sect. 4.2.According to Eq. (109) this term is suppressed by \(s_{21}^3\). Let us consider the condition of coherence restoration, which in the case of small density is given by Eq. (53). In the first approximation the condition is reduced to equality, \(L_M \simeq L_C\). According to Eqs. (91) and (92) this condition is satisfied for the nadir angle \(\eta \) given by
 (i)
Interference of the third and fifth terms of Eq. (107) corresponds to different motion of the WP in the second (core) and third (mantle) layers: the component \(\nu _{1m}^{(1)}\) splits into \(\nu _{1m}^{(2)}\) and \(\nu _{2m}^{(2)}\) crossing the density jump between the mantle and the core. In the core \(\nu _{1m}^{(2)}\) propagates faster than \(\nu _{2m}^{(2)}\). Both components split further when entering the mantle again. Then \(\nu _{1m}^{(3)}\), which originates from \(\nu _{1m}^{(2)}\), moves faster than \(\nu _{2m}^{(3)}\), and therefore its WP can catch up with the packet of \(\nu _{2m}^{(3)}\) originated from \(\nu _{2m}^{(2)}\). Catchup occurs in the third layer.
 (ii)
Interference of sixth and seventh terms in Eq. (107) is similar to (i) with the only difference that the two interfering channels originate from \(\nu _{2m}^{(1)}\) (in the first layer).
 (iii)
In the case of interference of third and second terms in Eq. (107) (in contast to (ii)) the first (mantle) and the second (core) layers are involved. The catchup occurs in the core; in the third layer \(\nu _{1m}^{(3)}\) propagates in both channels, so neither shift nor phase difference are acquired. The consideration is similar to the previous case.
 (iv)
Interference of fourth and sixth terms is similar to case (iii) with the only difference that in the third layer in both channels \(\nu _{2m}^{(3)}\) propagates.
where \(R_C\) and \(R_E\) are the radii of the core and the Earth; \(\eta _c\) is the nadir angle of the trajectory which touches the core, \(\cos \eta _c = 0.836\). The value \(\cos \eta = 0.89\) gives \(L_m = 3760\,\mathrm{km}\).$$\begin{aligned} \cos \eta \simeq \frac{3}{2\sqrt{2}} \sqrt{1  \frac{R_C^2}{R_E^2}} = \frac{3}{\sqrt{2}} \cos \eta _c \simeq 0.89 , \end{aligned}$$(110)  (i)
 2.The term with phase \((2 \phi _1\phi _2)\) in Eq. (109) originates from the interference of the third and sixth terms of Eq. (107). In turn these terms are due to the chains of transitionsThis interference is a genuine 3layer effect: In the first channel of (111) the WP is faster in the first mantle layer, then slower in the core, then again faster in the second mantle layer. In the second channel of (111) inversely: the WP moves first more slowly, then faster, then again more slowly. This corresponds to the change of subscript indices of neutrino states in (111). So, the order of WP in configuration space changes twice (the WP of the first channel arrives first at the core, in the core the second WP overtakes the first one) and catchup occurs in the third layer. The coherence restoration condition for this interference term is \(2L_M \simeq L_C\), which can be realized according to (91) and (92) for$$\begin{aligned} \nu _{1m}^{(1)} \rightarrow \nu _{2m}^{(2)} \rightarrow \nu _{1m}^{(1)} ,\quad \nu _{2m}^{(1)} \rightarrow \nu _{1m}^{(2)} \rightarrow \nu _{2m}^{(1)} . \end{aligned}$$(111)In this case \(L_C = 3100\,\mathrm{km}\). Unfortunately, the amplitude of this interference term is suppressed very strongly by \(s_{21}^4\). Equations (110) and (112) are conditions for complete overlap of WP; partial overlap of WP and partial catchup can be realized in a wider region of nadir angles and neutrino energies.$$\begin{aligned} \cos \eta \simeq \frac{2}{\sqrt{3}} \sqrt{1  \frac{R_c^2}{R_E^2}} = \frac{2}{\sqrt{3}} \cos \eta _c \simeq 0.965 . \end{aligned}$$(112)
 3.
The first term in Eq. (109) does not contain phases, but we keep it since it also shows the catchup effect. The effect, however, does not depend on energy and therefore it does not change by averaging over the energy and is present also in the incoherent case. There are two contribution to this term: interference of the second and fifth terms in Eq. (107), which both have the same phase \(\phi _1\), and interference of the fourth and seventh terms with common phase \(\phi _1 + \phi _2\). In the case of 2–5 interference one WP moves more slowly in the third layer, whereas the other WP moves more slowly in the first layer (they move with the same high speed in the core). So the second WP catches up with the first one in the third layer. The same is the case for 4–7 interference with the only difference that in the core both WP move with the same small speed. Here the catchup does not depend on energy (if we neglect the energy dependence of the depth of interference). The derivatives of the phases are the same for both channels, and therefore there is no averaging over the energy and the catchup is complete. This term is proportional to \(s_{21}^2\) and thus less suppressed than the others. It would acquire a phase if the third layer was different from the first one. However, in this case the coherence condition would be satisfied when \(L_3 \simeq L_1\).
The Earth matter effect and in particular the catchup effect are further suppressed because of the presence of the \(\nu _2\) component in the arriving supernova neutrino flux. For the \(\nu _2 \rightarrow \nu _e\) transition the probability can be obtained from the previous result by the substitutions \(c_{10} \rightarrow  s_{10}\), \(s_{10} \rightarrow c_{10}\). As a result, all the interference terms Eq. (109) change sign, so the observable effect will be proportional to the difference of the \(\nu _1\) and \(\nu _2\) fluxes, \(F(\nu _1)  F(\nu _2)\). In turn, this difference is determined by the dynamics of flavor transitions in the supernova. An additional suppression can be estimated by a factor 0.1 – 0.3.
One last comment: as we have established, the separation and spread of the WP have sizes of tens of meters. Such a state passes through the detector during about \(t_\text {state} \sim 10^{7}\,\text {s}\). So, in principle, present technology allows one to study parts of the packet. The problem is that the time of emission is not known and even the shortest features of the burst are about a few msec, which is much bigger than \(t_\text {state}\).
7 Conclusions
 1.
We have recalculated the size of the WP of supernova neutrinos, finding a uniquely short length of about \(10^{11}\,\mathrm{cm}\), which corresponds to an energy spread \(\sigma _E \sim 1\,\mathrm{MeV}\) – not much smaller than the neutrino energy itself. \(\sigma _E\) does not depend on neutrino energy and is approximately the same for all neutrino species in all phases of the supernova.
 2.
The coherence length is smaller than \(100\,\mathrm{km}\) for the 1–3 mass splitting and of order \(1000\,\mathrm{km}\) for the 1–2 mass splitting. The separation of the WP arriving at the Earth from a supernova in the galactic center can be as large as \(40\,\mathrm{m}\) for the 1–2 mass splitting and \(E = 15\,\mathrm{MeV}\).
 3.
Each wave packet spreads due to the presence of different energy components in it. The spread is proportional to the neutrino mass squared and can reach a macroscopic size. Usually the separation of the packets is bigger that the spread. An exceptional situation is realized for neutrinos oscillating in the Earth. Here oscillations occur due to the interference of components originating from the split of the WP of a mass eigenstate at the surface of the Earth. In this case the spread of the WP is much larger than their relative shift inside the Earth, so they continue to overlap. We have showed, however, that this does not change the coherence condition, which is determined by the original size of the WP without spread.
 4.
We have explored the coherence condition for supernova neutrinos oscillating in the Earth. The coherence length turns out to be comparable with the sizes of the mantle and the core. Thus, for low energies (\(E < 30\,\mathrm{MeV}\)) coherence is completely lost for most nadir angles. For a large range of energies and nadir angles the loss of coherence is partial. Only for high energies and shallow trajectories decoherence can be neglected.
 5.
We have studied oscillations in a multilayer medium characterized by the adiabatic change of density within layers and sudden jumps of density between layers. This has applications for neutrinos crossing the shock waves in a supernova and for neutrinos propagating inside the Earth along corecrossing trajectories.
 6.
A splitting of eigenstates occurs at each density jump, so for twoneutrino mixing \(2^n\) components are formed after n crossings. This multiplication has no meaning if the shift of the WP within each layer can be neglected. In this case the problem is reduced to the twoneutrino problem. However, if the shift and therefore decoherence are substantial, the multiple splitting has physical sense and can lead to new phenomena. In particular, it leads to the new interesting phenomenon of the partial restoration of coherence due to a “catchup effect”. In other words, in a multilayer medium coherence can be partially restored and the coherence length can be increased beyond the usual estimate. In the simplest realization, for two layers, this happens if a component of the WP that travels faster through layer 1 arrives at a detector at the same time as a component that was slower in layer 1 but faster in layer 2. The described effect yields corrections at the percent level or below for supernova neutrinos oscillating in Earth matter, but, in principle, it can be observed if a very highstatistics signal from a close supernova is detected.
 7.
We have studied decoherence in parallel in configuration and momentum space, checking the equivalence between both representations. Although at first sight the catchup effect seems to depend on the size of the WP, this is not the case in the examples we have considered. We have explicitly shown how this sizeindependence is due to the restoration of overlap of WP, confirming the general (abstract) results of [20] and [34]. In a sense the catchup effect is a nontrivial effect of the averaging of oscillation probabilities over energy. We have verified that in all cases we have studied there is an equivalence between configuration and momentum space. That is, we can choose to do all calculations either with WP that separate or by suitably averaging over energy. The observable oscillatory picture is determined by the initial energy spectrum, the energy resolution of the detector, and by the phase acquired between source and detector as a function of energy, unless time tagging is arranged.
Footnotes
 1.
We define the WP width as the position or energy uncertainty of a particle. We employ the “intermediate wave packet” picture for the neutrinos. This approach produces the correct results for oscillation probabilities, as shown by the quantum field theory treatment of neutrino oscillations in vacuum [33, 37, 39]. We consider ultrarelativistic neutrinos throughout.
 2.
The eigenstates of the Hamiltonian should be considered the “true particles”, in analogy to the concept of quasiparticles in condensed matter physics [43].
 3.
We use the variable \(\xi \) instead of \(\eta = 1/\xi \) in [28].
 4.
The WP still spread; see Sect. 5.
 5.
As neutrinos are highly relativistic, \(\mathrm{d}/\mathrm{d}E \simeq \mathrm{d}/\mathrm{d}p\).
 6.
In the case of twoneutrino mixing.
 7.
The oscillation probability and the phases depend not only on E but also on \(L_k\) and \(\rho _k\), but we do not write these dependences explicitly in the following.
 8.
Alternatively, for a fixed baseline a larger \(\Delta E\) is sufficient to observe the interference term.
 9.
Condition (50) can also be satisfied for arbitrary densities in the nonresonance channel, where \(\xi _k<0\).
 10.
To simplify the discussion we assume that there is a sharp transition from coherence to decoherence, i.e., interference terms are either present in \(\overline{P(E)}\) or disappear completely. Thus, we neglect that interference terms are suppressed but nonzero for a partial overlap of WP.
 11.
Up to a sign arising from our definition of \(\Delta v_m\), see Eq. (12).
Notes
Acknowledgments
We would like to thank Evgeny Akhmedov, Yasaman Farzan, Carlo Giunti, Alessandro Mirizzi, Georg Raffelt, Günter Sigl, Irene Tamborra, Ricard Tomas Bayo, and Mariam Tortola Baixauli for helpful discussions. We acknowledge support from the European Union FP7 ITN Invisibles (Marie Curie Actions, PITNGA2011289442) and from the Max Planck fellowship M.FW.A.KERN0001.
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