# Quantum Decoherence Effects: a complete treatment

Gabriela Barenboim<EMAIL_ADDRESS>Instituto de Física Corpuscular,
CSIC-Universitat de València, C/Catedrático José Beltrán 2, Paterna 46980,
Spain  Departament de Física Teòrica, Universitat de València, C/ Dr. Moliner
50, Burjassot 46100, Spain  Alberto M. Gago<EMAIL_ADDRESS>Sección Física,
Departamento de Ciencias, Pontificia Universidad Católica del Perú, Apartado
1761, Lima, Perú

###### Abstract

Physical systems in real life are inextricably linked to their surroundings
and never completely separated from them. Truly closed systems do not exist.
The phenomenon of decoherence, which is brought about by the interaction with
the environment, removes the relative phase of quantum states in superposition
and makes them incoherent. In neutrino physics, decoherence, although
extensively studied has only been analyzed thus far, exclusively in terms of
its dissipative characteristics. While it is true that dissipation, or the
exponential suppression, eventually is the main observable effect, the
exchange of energy between the medium and the system, is an important factor
that has been overlooked up until now. In this work, we introduce this term
and analyze its consequences.

††preprint: IFIC 24/13

##  I Introduction

We do not know yet the absolute neutrino masses. We do know however that these
masses are non-zero and non-degenerate. We learned that much because we have
observed neutrino flavor transitions. There, thanks to the fact that the mass
differences are negligible as compared to the energy uncertainty in the
generation and detection processes, the different mass eigenstates cannot be
identified and therefore, the states of the created and detected neutrinos,
which are flavor eigenstates, are expressed as a coherent superposition of
mass eigenstates. The dynamics of the neutrinos is governed by the quantum
Liouville equation or von Neumann equation for the density matrix. As a
result, the evolution is unitary, and coherence—represented by the density
matrix’s off-diagonal terms—is preserved during the propagation.

This coherence, however, can be lost, as for example happens with solar
neutrinos. The decoherence, in this case, is due to wave packet dissipation,
which results from the separation of neutrino mass states over large distances
because of their disparate group velocities. This process is still unitary and
can be dealt with the standard framework.

Likewise, when the neutrinos are weakly connected to the environment and get
entangled with it during their propagation, a different kind of decoherence
can occur. If the environment is added to our Hilbert space, the evolution
becomes unitary once more. But because of the vastness of the environment and
our ignorance of its properties, we are forced to trace over its degrees of
freedom, which gives, in general, rise to non-unitarity. Nevertheless, under
the conditions assumed in our formalism, probability will be conserved even in
this case.

In neutrino physics, this type of decoherence, called quantum decoherence, has
been studied so far, only focusing on its dissipative aspects. The main reason
is that decoherence is thought to occur as a byproduct of thermalization,
where an open system equilibrates by exchanging energy with its environment.
Although it is true that in the long term, dissipation, that is, the
exponential suppression, takes over, exchanging energy with the medium, i.e.
extracting or dumping energy into the system, is an aspect that cannot be
neglected and has been ignored so far.

To take into account this exchanging energy with the medium, here we will add
to the standard Hamiltonian an interaction term between the neutrino system
and the field from the environment, mimicking the well-known atom-field
Hamiltonian interaction that produces the Rabi oscillations in the context of
quantum optics Rabi:1936lvg . Like in quantum optics, this effective
interaction term can potentially inject energy into the neutrino system and
extract energy from it. The exploration of the phenomenology of this
interaction term is well-motivated since it has been largely ignored in the
current literature on quantum decoherence and neutrinos.

Furthermore, we assume that gravity is the source of the environment, the so-
called gravitational decoherence, and therefore is blind to the particle-
antiparticle difference. It is important to note that recently
Oppenheim:2018igd , it has been pointed out that gravity, even being
fundamentally classical, can cause decoherence in a coherent quantum system
that is coupled to the diffusion of the metric and its conjugate momenta. The
latter reinvigorated the importance of studying how quantum decoherence
affects any quantum coherent system, such as the neutrino system.

##  II Open quantum system approach

The evolution of the neutrino system within a pervasive environment is
achieved by solving the Lindblad equation Lindblad:1975ef , which is used for
a non-model-dependent environment:

$\frac{d\rho(t)}{dt}=-i\left[H,\rho(t)\right]+L\left[\rho(t)\right]$ (1)

In this equation, $\rho(t)$ represents the neutrino density matrix over time,
and $L\left[\rho(t)\right]$ is the linear map Benatti:2000ph  that encompasses
non-standard damping effects. This term opens the possibility of evolving from
pure states to mixed states.

$L\left[\rho(t)\right]$ is defined as follows:

$L\left[\rho(t)\right]=-\frac{1}{2}\sum_{j}\left\\{A^{\dagger}_{j}A_{j}\rho(t)+\rho(t)A^{\dagger}_{j}A_{j}\right\\}+\sum_{j}A_{j}\rho(t)A^{\dagger}_{j}$
(2)

Here, $A_{j}$ represents a set of operators with $j=1,2,...,n^{2}-1$ for $n$
neutrino generations. These operators must be hermitian to ensure the Rover
time increase of the Von Neumann entropy.

On the other hand, the usual way to deal with Eq. (1), assuming a two-neutrino
system, is to write the operators $\rho$, $H$ and $A_{j}$ as:

$\rho=\frac{1}{2}\sum{\rho_{\mu}\sigma_{\mu}},\,\,\,\,H=\frac{1}{2}\sum{h_{\mu}\sigma_{\mu}},\,\,\,\,A_{j}=\frac{1}{2}\sum{a^{j}_{\mu}\sigma_{\mu}}$
(3)

where $\mu$ is running from 0 to 3, $\sigma_{0}$ is the identity matrix and
$\sigma_{k}$, are the Pauli matrices which obey the condition
$\left[\sigma_{i},\sigma_{j}\right]=2i\epsilon_{ijk}\sigma_{k}$. Given the
Hermiticity of the $A_{j}$, the matrix $L_{\mu\nu}(\equiv L[\rho(t)])$ is real
and symmetric and it is written as:
$L_{kj}=\frac{1}{2}\sum_{l,m,n}{(\vec{a}_{n}.\vec{a}_{l})\epsilon_{knm}\epsilon_{mlj}}$
with the components $L_{\mu 0}=L_{0\mu}=0$ due to the probability
conservation. Furthermore, since the elements $L_{kj}$ are defined using
scalar products they satisfy the Cauchy-Schwartz inequalities and the complete
positivity condition, which is to assure that the eigenvalues of $\rho(t)$ are
positive at any time.

Considering all the above, the Eq. (1) can be decomposed as:

$\dot{\rho}_{0}=0,\,\,\,\,\dot{\rho}_{k}=\left(H_{kl}+L_{kl}\right)\rho_{l}=M_{kl}\rho_{l}$
(4)

with $H_{kl}=\sum_{j}h_{j}\epsilon_{jlk}$. Thus, the solution of $\rho(t)$
given in its matrix form is:

$\varrho(t)=e^{\textbf{M}t}\varrho(0)$ (5)

where $\varrho$ is a three-dimensional column vector and $\textbf{M}\equiv
M_{kl}$. The $e^{\textbf{M}t}$ can be expressed by:

$e^{\textbf{M}t}=\textbf{D}e^{\textbf{M}_{D}t}\textbf{D}^{-1}\equiv\left[e^{\textbf{M}t}\right]_{il}={\cal{M}}_{il}(t)=\sum_{k}D_{ik}e^{\lambda_{k}}D^{-1}_{kl}$
(6)

with
$\textbf{M}_{D}=\textbf{Diag}\left(\lambda_{1},\lambda_{2},\lambda_{3}\right)$.
Therefore, we can get the neutrino oscillation probability:

$P_{\nu_{\alpha}\rightarrow\nu_{\beta}}=\text{Tr}\left(\rho^{\alpha}(t)\rho^{\beta}\right)=\frac{1}{2}+\frac{1}{2}\sum_{i,l}\rho^{\beta}_{i}(0){\cal{M}}_{il}(t)\,\rho^{\alpha}_{l}(0)=\frac{1}{2}+\frac{1}{2}(\varrho^{\beta}(0))^{T}\varrho^{\alpha}(t)$
(7)

###  II.1 Hamiltonian of the model

In our approach the effective (total) Hamiltonian comprises two pieces:

$\hat{H}_{\text{eff}}=\hat{H}_{\text{osc}}+\hat{H}_{\text{int}},$

the $\hat{H}_{\text{osc}}$ is the standard Hamiltonian for the two-neutrino
oscillation system, which is given by:

$\hat{H}_{\text{osc}}=\left(\begin{array}[]{cc }\Delta&\\\
&-\Delta\end{array}\right)=\Delta\hat{\sigma_{3}}$ (8)

Here, $\Delta=\Delta m^{2}/2E_{\nu}$. The second piece $\hat{H}_{\text{int}}$
entails the interaction between the neutrino system and the environment-field,
and is given by:

$\hat{H}_{\text{int}}=\left(\begin{array}[]{cc }0&e^{i\alpha}\lambda\\\
e^{-i\alpha}\lambda&0\end{array}\right)=\lambda\cos{\alpha}\,\hat{\sigma_{1}}-\lambda\sin{\alpha}\,\hat{\sigma_{2}}=\lambda\left[\hat{\sigma}_{+}e^{i\alpha}+\hat{\sigma}_{-}e^{-i\alpha}\right]$
(9)

This Hamiltonian term resembles a non-quantized version of the atom-field
interaction Hamiltonian used in quantum optics, where $\lambda$ represents the
intensity of the neutrino-environment field interaction, while $\alpha$ is a
complex phase inherited from the environment field. Meanwhile, the terms
$\hat{\sigma}{+}$ and $\hat{\sigma}{-}$ will allow the neutrino conversion and
survival modes, being their role in the atom-field interaction to produce the
Rabi oscillations, a well-known phenomenon in quantum optics alicki . This
term, which violates Lorentz invariance but preserves CPT, was initially noted
in Benatti:2001fa , yet it has not been thoroughly examined or discussed in
the existing literature on quantum decoherence and neutrinos. Its presence
distinguishes our study from others. It indeed has some similarities to the
well known Standard Model effective (SME) coefficients, as we will see later.

Therefore, in this set-up, $\lambda$ is the dimension-full interaction
strength, and $\alpha$ is a real phase. It has to be emphasized that this
interaction does not change sign when going from neutrino to antineutrino,
unlike the so-called non-standard interactions. Moreover, bounds on this
coupling do not exist yet. The known searches for nonstandard interactions of
this type by measuring the Casimir forces or torsion pendula could potentially
bound exclusively a Yukawa-type deviations from Newtonian gravity for the
element $\lambda_{ee}$ if $SU(2)_{L}$ symmetry holds. However, similar bounds
on non-diagonal elements do not exist.

###  II.2 Neutrino Oscillation Probability

####  II.2.1 Without Quantum Decoherence Effects

Although, according to our hypothesis, the non-diagonal Hamiltonian term in
the $\hat{H}_{\text{eff}}$ is a contribution from the environment, in this
calculation, we will determine the neutrino oscillation probability without
incorporating the dissipative terms. This oscillation probability (without
decoherence effects) has a very rich phenomenology on its own and, indeed,
constitutes the primary focus of this study.

For performing the aforementioned calculation we need to diagonalize the
$\hat{H}_{\text{eff}}$, which turns out to be:

$\hat{H}_{\text{Diag}}=\frac{1}{2E_{\nu}}\left(\begin{array}[]{cc }\Delta
m_{\text{eff}}^{2}&\\\ &-\Delta
m_{\text{eff}}^{2}\end{array}\right)=\frac{\Delta
m_{\text{eff}}^{2}}{2E_{\nu}}\,\hat{\sigma_{3}}$ (10)

with a new effective square mass difference:

$\Delta_{\text{eff}}=\frac{\Delta
m_{\text{eff}}^{2}}{2E_{\nu}}=\sqrt{\displaystyle{\Delta^{2}+\lambda^{2}}}\equiv\Delta
m_{\text{eff}}^{2}=\sqrt{\displaystyle{(\Delta
m^{2})^{2}+4\,\lambda^{2}\,E_{\nu}^{2}}}$ (11)

which in the case $\lambda\ll\Delta$ implies111It is important to keep in mind
that $\Delta\sim 10^{-21}\mbox{GeV}$ for the atmospheric mass difference and
energies of a few GeV

$\frac{\Delta
m_{\text{eff}}^{2}}{2E_{\nu}}=\Delta\left(1+\frac{\lambda^{2}}{2\Delta^{2}}\right)=\frac{\Delta
m^{2}}{2E_{\nu}}+\left(\frac{\lambda^{2}}{\Delta m^{2}}\right)E_{\nu}$ (12)

In this limit, a clear parallel can be drawn between the
$\left(\frac{\lambda^{2}}{\Delta m^{2}}\right)$ term and the SME
$c$-coefficients Kostelecky:2003fs , which appear alongside a kinematic phase
scaling linearly with $E_{\nu}$. Even upon further expansion of $\frac{\Delta
m_{\text{eff}}^{2}}{2E_{\nu}}$, the correspondence between our model and the
$c$-coefficients associated with (odd) higher powers of
$E_{\nu}^{2n-1}(n=2,..)$ remains valid, with the $c$-coefficients being
proportional to $\left(\frac{\lambda^{2n}}{(\Delta m^{2})^{2n-1}}\right)$.
From this perspective, our model offers a framework to establish a direct
relationship between the $c$-coefficients of higher-order terms in the SME and
the higher-order contributions in the series expansion of $\frac{\Delta
m_{\text{eff}}^{2}}{2E_{\nu}}$.

The another ingredient for the diagonalization (and the calculation of the
neutrino oscillation probability) is the mixing matrix, which connects the
neutrino flavor states with the effective vacuum mass eigenstates. This is
given by:

$U(\beta+\theta,\alpha)=\left(\begin{array}[]{cc}e^{i(\pi-\alpha)}\cos(\beta+\theta)&\sin(\beta+\theta)\\\
e^{-i\alpha}\sin(\beta+\theta)&\cos(\beta+\theta)\\\ \end{array}\right)$ (13)

Where:

$\tan^{2}(\beta+\theta)=\dfrac{\lambda\sin
2\theta+\sqrt{\displaystyle{\Delta^{2}+\lambda^{2}}}-\Delta\cos
2\theta}{-\lambda\sin
2\theta+\sqrt{\displaystyle{\Delta^{2}+\lambda^{2}}}+\Delta\cos 2\theta}$ (14)

here, $\theta$ is the vacuum mixing angle. This formula exhibits the resonant
behavior we see in the case of non-standard interactions, when

$\lambda=\Delta\cot 2\theta$ (15)

However in our case, the resonance will be present in both channels. The
mixing angle in a medium, then will increase if

$\lambda\sin 2\theta>\Delta\cos 2\theta$

or

$2\lambda E>\Delta m^{2}\cot 2\theta$

and will decrease in the opposite case.

In the small interaction regime,

$\displaystyle\tan^{2}(\beta+\theta)$ $\displaystyle=$
$\displaystyle\tan^{2}\theta+\frac{\lambda\,\tan\theta\,\sec^{2}\theta}{\Delta}$
$\displaystyle\tan^{2}(\beta+\theta)$ $\displaystyle=$
$\displaystyle\tan^{2}\theta\left(1+\frac{2\lambda E}{\sin\theta\,\Delta
m^{2}}\right)$ (16)

Given that both the mass difference and the mixing angle are similarly
affected for neutrinos and antineutrinos in our gravitationally pervasive
environment, their propagation in such a medium does not induce a spurious,
medium-induced CP violation. Therefore, not only CPT is conserved but also
CP222For genuine CPT violation in the mass terms see Barenboim:2002tz ,
consequently, we can anticipate the following CPT conserving relations:

$P(\nu_{\alpha}\longrightarrow\nu_{\alpha})=P(\overline{\nu}_{\alpha}\longrightarrow\overline{\nu}_{\alpha})$
$P(\nu_{\alpha}\longrightarrow\nu_{\beta})=P(\overline{\nu}_{\beta}\longrightarrow\overline{\nu}_{\alpha})$

and also the ones that preserves CP (in the two generation case)

$P(\nu_{\alpha}\longrightarrow\nu_{\beta})=P(\overline{\nu}_{\alpha}\longrightarrow\overline{\nu}_{\beta})$

In the three generation case, (genuine) CP violation will appear, and this
equation will be no longer valid. Medium effects will induce fake CP violation
as well. The inclusion of (non-diagonal) decoherence effects from the
environment can reveal the presence of Majorana CP phases in the neutrino
oscillation formula Benatti:1997xt ; Carrasco ; Capolupo .

Now, with all the ingredients in hand, we can give the expression of the
neutrino oscillation probability 333The neutrino oscillation probability can
also be calculated using the density matrix formalism (Eq. (7)). The
$\rho_{e}\left(0\right)$ and $\rho_{\mu}\left(0\right)$ are given in the
Appendix A:

$P_{\nu_{\alpha}\rightarrow\nu_{\beta}}=\sin^{2}2(\beta+\theta)\sin^{2}\left(\frac{\sqrt{\Delta^{2}+\lambda^{2}}}{2}L\right)$
(17)

for $\alpha\neq\beta$. As expected, the formula above is similar, at some
level, to the vacuum Rabi Oscillations. Particularly, the
$\sqrt{\Delta^{2}+\lambda^{2}}$ plays the same role as the Rabi frequency,
setting the pace for energy exchange between the neutrino system and the
environment-field with $\Delta$ be analogous to the detuning. While the
oscillation amplitude is given by 444It is interesting to note that similar
amplitude can be obtained from Reference Barger  by setting
$2\theta_{b}=\pi/2+2\theta_{m}$ and $\eta=0$, given in the reference’s
notation, where $\theta_{m}=\theta$ and $\delta b=\lambda$. :

$\sin^{2}2(\beta+\theta)=\dfrac{\left(\Delta\sin 2\theta+\lambda\cos
2\theta\right)^{2}}{\Delta^{2}+\lambda^{2}}$ (18)

the oscillation amplitude clearly confirms the resonant condition
$\lambda=\Delta\cot 2\theta$ quoted before. It is straightforward to note that
the oscillation amplitude also satisfies an anti-resonant condition at
$\lambda=-\Delta\tan{2\theta}$.

Now, considering the oscillation amplitude and the Eq. (17) in

Figure 1: Amplitude of the oscillation as a function of $\lambda/\Delta$ for
different choices of the vacuum mixing angle

the small interaction regime (i.e. expanding the probability up to the first
order in $\lambda/\Delta$) we get the following expression for the two-
generation probability for the model:

$\displaystyle P_{\nu_{\alpha}\rightarrow\nu_{\beta}}$ $\displaystyle\approx$
$\displaystyle\sin^{2}2\theta\left(1+\left(\frac{4E_{\nu}\lambda}{\Delta
m^{2}}\right)\cot{2\theta}\right)\sin^{2}\left(\frac{\Delta}{2}L\right)$ (19)
$\displaystyle\approx$
$\displaystyle\left(\sin^{2}2\theta+\left(\frac{2E_{\nu}\lambda}{\Delta
m^{2}}\right)\sin{4\theta}\right)\sin^{2}\left(\frac{\Delta}{2}L\right)$ (20)

from the above expression, we can see that the amplitude of the oscillation
can be amplified or reduced, depending on the mixing angle. Specially
interesting is the case $\theta\approx\pi/4$, where the suppression is
remarkable, as we will see later. Notice that this modulation of the mixing
angle, which scales linearly with the energy, and unlike to matter effect does
not depend on the density of the medium, offers a unique opportunity to
search/bound these kinds of effects. It should be noted that if the SME CPT
preserving but Lorentz violating terms are the first-order contribution of a
full decoherence model, its modification of the kinematic phase should be
accompanied by this higher-order modification of the mixing angle.

Considering the full expression of $\sin^{2}2(\beta+\theta)$ as given in Eq.
(18), we illustrate its behavior in Figure(1) for three different values of
$\theta$: $\pi/4,\pi/8$ and $\pi/16$. For $\theta=\pi/4$, the peak for
$\sin^{2}2(\beta+\theta)$ is achieved when $\lambda=0$ as ruled by the
resonant behavior given in Eq. (15), while, for the rest of the values of
$\lambda/\Delta$, it is suppressed regardless of whether $\lambda$ is positive
or negative This suppression is due to the attenuation factor
$(1+(\lambda/\Delta)^{2})^{-1}=(1+(2E_{\nu}\lambda/\Delta m^{2})^{2})^{-1}$,
which diminishes as $|\lambda/\Delta|$ increases (or the increase of $E_{\nu}$
for a fixed $\lambda$), as we previously observed in Eq. (20). In the scenario
where $\theta=\pi/8$, the peak, as expected from Eq. (15), is at
$\lambda=\Delta$, while the valley is, in general, at
$\lambda=-\Delta\tan{2\theta}$ which in this case is at $\lambda=-\Delta$. On
the other hand, the suppression is governed by the factor
$(1+2(\lambda/\Delta)/(1+(\lambda/\Delta)^{2}))$. For $|\lambda/\Delta|<1$,
this factor approximately takes the form $(1+2(\lambda/\Delta))$. This
explains why within the range $[-1,1]$, the $\sin^{2}2(\beta+\theta)$
increases (decreases) for positive (negative) $\lambda$. Outside this range,
the amplitude of $\sin^{2}2(\beta+\theta)$ diminishes as the attenuation
factor converges to $(1+2/(1+(\lambda/\Delta)^{2}))$. For $\theta=\pi/16$,
although we observe a behavior akin to that of $\theta=\pi/8$, it is different
the interval in which $\sin^{2}2(\beta+\theta)$ either increases or decreases
due to the distinct weighting of $\lambda$ and $\Delta$ in the complete
expression of $\sin^{2}2(\beta+\theta)$. The peak is reached at
$\lambda=2.4142\,\Delta$ while the valley is at $\lambda=-0.4142\,\Delta$.

For completeness, the probability transitions are shown in Figure(2) where the
effects of the mixing angle distortions due to the interaction with the
environment are moderated by the oscillation driven by the kinematic phase,
which in turn is also modified as shown before. There it can be clearly seen
that the amplitude of the oscillation, is the one depicted in Figure(1), while
the position of the maximum of the oscillation is dictated by the modulation
of the effective mass difference.

Figure 2: Transition probability as a function of $\lambda/\Delta$ for
different choices of the vacuum mixing angle for $\Delta m^{2}L/(4E)\approx 1$

####  II.2.2 With Quantum Decoherence Effects

Given the non-diagonal nature of our Hamiltonian, $\hat{H}_{\text{eff}}$, we
find ourselves in a scenario akin to the one shown in Carpio:2017nui . Thus,
the most straightforward approach is where the decoherence matrix remains
unaltered after rotating the non-diagonal basis to the diagonal one. The
latter property, as highlighted in Carpio:2017nui , allows us to directly
engage with $\hat{H}_{\text{eff}}$ and to compute the neutrino oscillation
probability, including the quantum decoherence effects, following the standard
methodology Gago:2002na ; Barenboim:2004wu . The decoherence matrix that
satisfies this criterion is
$\textbf{L}=-\textbf{Diag}\left(\gamma,a,\gamma\right)$ Carpio:2017nui .

Using the Eq. (7) we can write the neutrino oscillation probability (the
corresponding ${\cal{M}}_{il}$ are listed on the Appendix B):

$\displaystyle P_{\nu_{\alpha}\rightarrow\nu_{\beta}}(t)=\frac{1}{2}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\left\\{\left[\left(\rho_{1}^{\alpha}\rho_{1}^{\beta}+\rho_{2}^{\alpha}\rho_{2}^{\beta}\right)\cosh{\left(\frac{\Omega
t}{2}\right)}\right.\right.$ (21) $\displaystyle+$
$\displaystyle\left.\left(\frac{2\Delta_{\text{eff}}\left(\rho_{1}^{\alpha}\rho_{2}^{\beta}-\rho_{2}^{\alpha}\rho_{1}^{\beta}\right)+\left(-\gamma+a\right)\left(\rho_{1}^{\alpha}\rho_{1}^{\beta}-\rho_{2}^{\alpha}\rho_{2}^{\beta}\right)}{\Omega}\right)\sinh{\left(\frac{\Omega
t}{2}\right)}\right]e^{-\left(\gamma+a\right)t}$ $\displaystyle+$
$\displaystyle\rho_{3}^{\alpha}\rho_{3}^{\beta}e^{-2\gamma t}\Bigg{\\}}\,,$

where $\Omega=\sqrt{-4\Delta_{\text{eff}}^{2}+\left(\gamma-a\right)^{2}}$.
Considering $\rho_{1}^{\alpha}=2\,\mathfrak{Re}(U_{\alpha 1}^{\ast}U_{\alpha
2})$, $\rho_{2}^{\alpha}=-2\,\mathfrak{Im}(U_{\alpha 1}^{\ast}U_{\alpha 2})$,
and $\rho_{3}^{\alpha}=|U_{\alpha 1}|^{2}-|U_{\beta 2}|^{2}$ we have:

$\displaystyle\rho_{1}^{\alpha}\rho_{1}^{\beta}+\rho_{2}^{\alpha}\rho_{2}^{\beta}$
$\displaystyle=$ $\displaystyle 4\,\mathfrak{Re}\left(U^{*}_{\alpha
1}U_{\alpha 2}U_{\beta 1}U^{*}_{\beta 2}\right)$
$\displaystyle\rho_{1}^{\alpha}\rho_{2}^{\beta}-\rho_{2}^{\alpha}\rho_{1}^{\beta}$
$\displaystyle=$ $\displaystyle-4\,\mathfrak{Im}\left(U^{*}_{\alpha
1}U_{\alpha 2}U_{\beta 1}U^{*}_{\beta 2}\right)$
$\displaystyle\rho_{3}^{\alpha}\rho_{3}^{\beta}$ $\displaystyle=$
$\displaystyle-1+2\sum_{l=1,2}|U_{\alpha l}|^{2}|U_{\beta
l}|^{2}=-1+2\delta_{\alpha\beta}-4\,\mathfrak{Re}\left(U^{*}_{\alpha
1}U_{\alpha 2}U_{\beta 1}U^{*}_{\beta 2}\right)$ (22)

With the relations above we can re-write Eq. (21) as follows:

$\displaystyle P_{\nu_{\alpha}\rightarrow\nu_{\beta}}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{1}{2}\left\\{\left[4\,\mathfrak{Re}\left(U^{*}_{\alpha
1}U_{\alpha 2}U_{\beta 1}U^{*}_{\beta 2}\right)\cosh{\left(\frac{\Omega
t}{2}\right)}\right.\right.$ (23) $\displaystyle+$
$\displaystyle\left.\left(\frac{-8\,\Delta_{\text{eff}}\mathfrak{Im}\left(U^{*}_{\alpha
1}U_{\alpha 2}U_{\beta 1}U^{*}_{\beta
2}\right)+4\,\left(-\gamma+a\right)\mathfrak{Re}\left(U^{*}_{\alpha
1}U_{\alpha 2}U^{*}_{\beta 1}U_{\beta
2}\right)}{\Omega}\right)\sinh{\left(\frac{\Omega
t}{2}\right)}\right]e^{-\left(\gamma+a\right)t}$ $\displaystyle+$
$\displaystyle\left(-1+2\delta_{\alpha\beta}-4\,\mathfrak{Re}\left(U^{*}_{\alpha
1}U_{\alpha 2}U_{\beta 1}U^{*}_{\beta 2}\right)\right)e^{-2\gamma
t}\Bigg{\\}}\,,$

Replacing the mixing matrix elements from Eq. (13) we got the following
expression:

$\displaystyle P_{\nu_{\alpha}\rightarrow\nu_{\beta}}(t)=\frac{1}{2}$
$\displaystyle-$ $\displaystyle\frac{1}{2}e^{-2\gamma
t}+\left(\delta_{\alpha\beta}+\frac{1}{2}\sin^{2}2(\beta+\theta)\right)\,e^{-2\gamma
t}$ (24) $\displaystyle-$
$\displaystyle\frac{1}{2}\sin^{2}2(\beta+\theta)\cosh\left(\frac{\Omega
t}{2}\right)e^{-\left(\gamma+a\right)t}$ $\displaystyle+$
$\displaystyle\frac{1}{2}\left(\gamma-a\right)\sin^{2}2(\beta+\theta)\cos
2\alpha\left(\frac{\sinh{\left(\frac{\Omega
t}{2}\right)}}{\Omega}\right)e^{-\left(\gamma+a\right)t}$

substituting $|\Omega|\rightarrow 2\Delta_{\text{eff}}$, $\gamma=a=0$ in the
latter equation the Eq. (17) can be easily recovered.

To stress how the new (complete) decoherence scenario differs from what has
been done so far, we can analyze a very simplified scenario. Lets concentrate
on the transition probability only, i.e. $\alpha\neq\beta$ and assume
$\gamma=a$. In this case, the last term vanishes, $\Omega\rightarrow
2i\Delta_{\text{eff}}$ and the probability takes the following form

$\displaystyle P_{\nu_{\alpha}\rightarrow\nu_{\beta}}(t)=\frac{1}{2}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\left(1-2\sin^{2}2(\beta+\theta)\sin^{2}\left(\frac{\Delta
m_{\text{eff}}^{2}L}{4E_{\nu}}\right)\right)\,e^{-2\gamma t}$
$\displaystyle=\frac{1}{2}$ $\displaystyle-$
$\displaystyle\frac{1}{2}\left(1-\dfrac{2\left(\Delta\sin 2\theta+\lambda\cos
2\theta\right)^{2}}{\Delta^{2}+\lambda^{2}}\sin^{2}\left(\frac{\sqrt{\Delta^{2}+\lambda^{2}}}{2}L\right)\right)\,e^{-2\gamma
t}$ (25)

where it is pretty transparent that the intensity of the neutrino-environment
field interaction parametrized by $\lambda=-\Delta\tan{2\theta}$ is
independent of the $\gamma$-effect, being that the frequency of the energy
exchange is controlled by $\sqrt{\Delta^{2}+\lambda^{2}}/2$. Figure(3)
represents the transition probabilities in three different scenarios where the
new term is present, with (solid) and without (dashed) $\gamma$-term, for a
maximal mixing angle in vacuum $\theta=\pi/4$, where $\sin(\Delta L/2)$ has
been set to one for $L\simeq 800$ km. No matter effects have been included.
For the sake of comparison, the no decoherence scenario is also shown. As
expected, and given that $\theta=\pi/4$, we can see that as long as $\lambda$
increases, the oscillation amplitude decreases in relation to the standard
case. Similarly, the shift of the oscillation frequency (energy exchange
frequency) is larger as $\lambda$ grows. Figure(4) describes the same
scenarios for a mixing angle of $\theta=\pi/8$. For this case, the effect is
the opposite for the oscillation amplitude, as it increases when $\lambda$
increases. The behavior of the shift in oscillation frequency is similar to
the previous case; i.e., the shift is larger for larger $\lambda$.

Figure 3: Transition probabilities, for a maximal mixing angle in vacuum, are
presented in three different scenarios,
$\lambda/\Delta=\left\\{1/4,1/2,1\right\\}$, with (solid) and without (dashed)
exponential suppression. The oscillation has been set to reach its maximum for
$L\simeq 800$Km. The standard scenario is also shown for comparison. In the
scenarios, including exponential suppression,$\gamma=8.2\cdot
10^{-24}{\mbox{GeV}}=\left(24000{\mbox{km}}\right)^{-1}$ has been used. Figure
4: Same plot as in Figure(3) but for $\theta=\pi/8$

##  III Conclusions

In this work, we go beyond other neutrino decoherence studies performed so
far, which have only considered the dissipative effects of the environment. We
introduce the possibility of interaction between the neutrino system and an
environmental field. This Hamiltonian term, mimicking the atom-field
Hamiltonian used in quantum optics, represents an energy exchange between the
neutrino system and the environmental field—losing or gaining energy, akin to
Rabi oscillations. This unexplored interaction opens up a very interesting
phenomenology.

One of our most interesting findings is that this interaction can induce
resonant behavior, maximizing oscillation amplitude. Conversely, the
oscillation amplitude can be suppressed. The amplification or suppression of
the oscillation amplitude, resulting from the oversight of this interaction,
could be a potential source of distortion in the measurement of oscillation
parameters for future neutrino accelerator experiments, such as those
referenced in Arguelles:2022tki . Specially interesting are the prospects at
DUNE and T2HK.

Besides, these effects can be probed in the light of the actual high-energy
atmospheric neutrino data, or we can further assess their impact in the high-
energy astrophysical neutrino observations. The reference to these test
scenarios is based on the fact that the resonant condition is achieved when
$2E_{\nu}\lambda=\Delta m^{2}\cot{2\theta}$, implying that, with increasing
neutrino energy, we can be sensitive to smaller values of $\lambda$.

On the other hand, the interference phase $\Delta
m_{\text{eff}}^{2}/4E_{\nu}=\sqrt{\Delta^{2}+\lambda^{2}}/2$, when considering
a small $\lambda$, could be hard to probe at ${\mathcal{O}}(1\,\text{GeV})$
energies. This is due to the shift at the first order scaling as
$\propto\lambda^{2}E_{\nu}$. Therefore, utilizing the high-energy atmospheric
data set for this purpose would be advantageous. It is evident that there is
still much to be done, such as expanding our study to the three-neutrino
generation framework, which includes matter effects. This expansion also
involves, as mentioned earlier, contrasting the neutrino-environment field
interaction in its three-neutrino generation version with current data and
considering future projections.

All in all, there isn’t a single quantum system that is completely closed or
isolated. Rather, coherent dynamics—as characterized by a Schrödinger
equation—usually is only valid for very short timescales before the coupling
between the open system and its surroundings takes over, resulting in
coherence loss and the beginning of more classical behavior. Therefore,
specially in the case of neutrinos, which travel long distances and come
almost unaltered from the past, a complete understanding of the phenomenology
associated to the interaction with the environment not restricted only to
decoherence is mandatory.

## Acknowledgments

This work is supported by the Spanish grants CIPROM/2021/054 (Generalitat
Valenciana) and PID2020-113775GB-I00 (AEI/10.13039/501100011033), by the
European ITN project HIDDeN (H2020-MSCA-ITN-2019/860881-HIDDeN), Marie
Skłodowska-Curie Staff Exchange grant ASYMMETRY (HORIZON-
MSCA-2021-SE-01/101086085-ASYMMETRY) and by Dirección de Fomento de la
Investigación at Pontificia Universidad Católica del Perú, through Grants No.
DFI-2021-0758 and CONCYTEC through Grant No.60-2015-FONDECYT. A.M. Gago wants
to express his gratitude to the Departament de Física Teòrica, Universitat de
València for their warm hospitality during the development of this work. He
also wants to thank F. de Zela and E. Massoni for the valuable discussions and
suggestions.

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## Appendix A: $\rho_{e}\left(0\right)$ and $\rho_{\mu}\left(0\right)$

$\rho_{e}\left(0\right)=\left(\begin{array}[]{cc}\cos^{2}(\beta+\theta)&-e^{i\alpha}\cos(\beta+\theta)\sin(\beta+\theta)\\\
e^{-i\alpha}\cos(\beta+\theta)\sin(\beta+\theta)&\sin^{2}(\beta+\theta)\\\
\end{array}\right)$ (A-1)
$\rho_{\mu}\left(0\right)=\left(\begin{array}[]{cc}\sin^{2}(\beta+\theta)&e^{i\alpha}\cos(\beta+\theta)\sin(\beta+\theta)\\\
e^{-i\alpha}\cos(\beta+\theta)\sin(\beta+\theta)&\cos^{2}(\beta+\theta)\\\
\end{array}\right)$ (A-2)

## Appendix B: ${\cal{M}}_{il}$ elements

$\displaystyle{\cal{M}}_{11}(t)$ $\displaystyle=$
$\displaystyle\left(\cosh{\left(\frac{\Omega
t}{2}\right)}+\left(-\gamma+a\right)\frac{\sinh{\left(\frac{\Omega
t}{2}\right)}}{\Omega}\right)e^{-\left(\gamma+a\right)t}$
$\displaystyle{\cal{M}}_{12}(t)$ $\displaystyle=$ $\displaystyle
2\Delta_{\text{eff}}\,\frac{\sinh{\left(\frac{\Omega t}{2}\right)}}{\Omega}$
$\displaystyle{\cal{M}}_{21}(t)$ $\displaystyle=$
$\displaystyle-2\Delta_{\text{eff}}\,\frac{\sinh{\left(\frac{\Omega
t}{2}\right)}}{\Omega}$ $\displaystyle{\cal{M}}_{22}(t)$ $\displaystyle=$
$\displaystyle\left(\cosh{\left(\frac{\Omega
t}{2}\right)}-\left(-\gamma+a\right)\frac{\sinh{\left(\frac{\Omega
t}{2}\right)}}{\Omega}\right)e^{-\left(\gamma+a\right)t}$ (B-1)

where ${\cal{M}}_{13}(t)={\cal{M}}_{31}(t)=0$,
${\cal{M}}_{23}(t)={\cal{M}}_{32}(t)=0$ and
$\Omega=\sqrt{-4\Delta_{\text{eff}}^{2}+\left(\gamma-a\right)^{2}}$


$v_{4,1}$     $\mathbf{7.27^*}$     8.0482     2.4939     7.5606     1.8723     5.0499     12.1542

$v_{4,2|1}$     $\mathbf{9.98^*}$     9.9089     3.3581     9.082     2.6316     5.7849     16.4214

$v_{5,1}$     $\mathbf{5.41^*}$     5.8013     1.5465     5.4467     1.1149     3.8427     8.8646

$v_{5,3|1}$     $\mathbf{5.26^*}$     6.0324     1.6404     5.6626     1.3035     4.0945     8.8652

$v_{5,4|1,3}$     $\mathbf{7.47^*}$     8.14     2.2743     7.8704     1.7779     5.0633     13.045

$v_{6,5}$     $\mathbf{5.27^*}$     8.34     2.4175     7.6211     3.1237     5.2655     12.8108

$v_{7,1}$     $\mathbf{9.75^*}$     10.0117     3.058     9.1846     2.4324     6.2343     15.6522

$v_{7,2|1}$     $\mathbf{8.17^*}$     8.7747     2.8075     8.2079     2.0742     5.1824     13.6875

$v_{7,5|1,2}$     $\mathbf{6.35^*}$     8.0564     2.6115     7.6028     2.271     4.7868     12.7598

$v_{8,1}$     $\mathbf{9.98^*}$     10.0132     3.3093     9.4096     2.5236     6.014     16.2632

$v_{8,3|1}$     $\mathbf{6.79^*}$     7.6349     2.6396     7.0186     1.9867     4.4484     12.7467

$v_{9,1}$     $\mathbf{6.86^*}$     7.1971     1.9426     6.7725     1.5081     4.8446     10.8707

$v_{9,4|1}$     $\mathbf{6.65^*}$     7.7157     2.4767     7.2372     1.9109     4.5809     12.0884

$v_{9,7|1,4}$     $\mathbf{5.52^*}$     8.6611     2.6835     8.2426     3.2541     4.9682     13.7083

The true values, the mean, mean absolute error (MAE) of the mean, the median, the MAE of the median, the 5th quantile ($q_{0.05}$), and the 95th quantile ($q_{0.95}$) of the posterior median estimates of $v_{xy|z}$ and $\bar\varphi_{xy|z}$ out of the 200 replications for simulation (S2).

§.§ Parameters in the stock copula

This section gives the sequential estimation results for simulation (S2). <ref>, <ref>, <ref> and <ref> are four graphs summarizing the parameter estimation for $a_{xy|z}$, $b_{xy|z}$, $v_{xy|}z$, and $\bar\varphi_{xy|z}$. These graphs are similar to the graphs in <ref>, <ref>, <ref> and <ref> presented in Section <ref>, and the observations regarding the estimation results are similar to those in Section <ref>.

A graph summarizing the parameter estimates for $a_{xy|z}$ by level in simulation (S2). The copulas are in ascending order based on the true values of $a_{xy|z}$ in each level for a better visualization. The green curves indicate true values, the blue curves indicate mean estimates, the orange curves indicate the median estimates, and the pair of dashed curves represent the 90% credible intervals of the sequential estimates out of the 200 replications.

A graph summarizing the parameter estimates for $b_{xy|z}$ by level in simulation (S2). The copulas are in ascending order based on the true values of $b_{xy|z}$ in each level for a better visualization. The green curves indicate true values, the blue curves indicate mean estimates, the orange curves indicate the median estimates, and the pair of dashed curves represent the 90% credible intervals of the sequential estimates out of the 200 replications.

A graph summarizing the parameter estimates for $v_{xy|z}$ by level in simulation (S2). The copulas are in ascending order based on the true values of $v_{xy|z}$ in each level for a better visualization. The green curves indicate true values, the blue curves indicate mean estimates, the orange curves indicate the median estimates, and the pair of dashed curves represent the 90% credible intervals of the sequential estimates out of the 200 replications.

A graph summarizing the parameter estimates for $\bar\varphi_{xy|z}$ by level in simulation (S2). The copulas are in ascending order based on the true values of $\bar\varphi_{xy|z}$ in each level for a better visualization. The green curves indicate true values, the blue curves indicate mean estimates, the orange curves indicate the median estimates, and the pair of dashed curves represent the 90% credible intervals of the sequential estimates out of the 200 replications.

§ COMPARING THE SEQUENTIAL ESTIMATOR AND THE POSTERIOR MEAN/MEDIAN ESTIMATORS

In our empirical study, we conduct a moving-window study with a total of 173 windows, in each of which we include 92 stocks. Estimating stock copulas using MCMC sampling is infeasible in this case. For $m=10$, $p=92$ and $T=750$, for each stock, it took 6 hours to run 20,000 iterations in the MCMC sampling. The sequential estimation, however, only takes 30 seconds.

In this section, we compare the sequential estimates to the posterior mean/median estimates. We conduct 20 replications with different data sets generated from the GC-GARCH model with the same underlying network in <ref> as in simulation (S2). The true parameters are also fixed across replications. We calculate the MAEs of the estimates of all 100 stocks, by level. <ref> shows the time series plots of the average MAE, within the same level, of the sequential estimates and the posterior mean/mode/median estimates of (a) $\bar\varphi_{xy|z}$, (b) $a_{xy|z}$, (c) $b_{xy|z}$, and (d) $v_{xy|z}$. The x-axes are the levels.

* In (a), we observe that the MAEs of the sequential estimates of $\bar\varphi_{xy|z}$ (dashed blue line) are similar to the MAEs of the mean and median MCMC estimates (respectively the solid red and dotted green lines).

* In (b), the MAEs for $a_{xy|z}$ using sequential estimation are smaller than the MAEs of the MCMC estimates in most of the levels.

* In (c), we observe that the sequential estimates of $b_{xy|z}$ have slightly larger MAEs for the first four levels, and the sequential estimates actually work better for the subsequent levels.

* In (d), the MAEs of the sequential estimates and the MCMC median estimates of $v_{xy|z}$ are similar.

Summarizing the above four points, we observe that sequential estimates work as well as the MCMC estimates.

The average MAEs for $\bar\varphi_{xy|z}$.

The average MAEs for $a_{xy|z}$.

The average MAEs for $b_{xy|z}$.

The average MAEs for $v_{xy|z}$.

The time series plots of the average MAEs of the sequential estimates, the posterior mean and posterior median estimates of (a) $\overline\varphi_{xy|z}$, (b) $a_{xy|z}$, (c) $b_{xy|z}$, and (d) $v_{xy|z}$, plotted by level.

§ LIST OF STOCKS

Name                Stock symbol     Name                Stock symbol
CKH HOLDINGS        0001.HK          CSPC PHARMA         1093.HK

CLP HOLDINGS        0002.HK          CHINA RES LAND      1109.HK

HK & CHINA GAS     0003.HK          CK ASSET            1113.HK

HSBC HOLDINGS       0005.HK          YANKUANG ENERGY     1171.HK

POWER ASSETS        0006.HK          SINO BIOPHARM       1177.HK

HANG SENG BANK      0011.HK          CHINA RAIL CONS     1186.HK

HENDERSON LAND      0012.HK          BYD COMPANY         1211.HK

SHK PPT             0016.HK          ABC                 1288.HK

NEW WORLD DEV       0017.HK          AIA                 1299.HK

GALAXY ENT          0027.HK          PICC GROUP          1339.HK

MTR CORPORATION     0066.HK          ICBC                1398.HK

HANG LUNG PPT       0101.HK          PSBC                1658.HK

TSINGTAO BREW       0168.HK          CRRC                1766.HK

GEELY AUTO          0175.HK          GF SEC              1776.HK

ALI HEALTH          0241.HK          CHINA COMM CONS     1800.HK

CITIC               0267.HK          PRADA               1913.HK

WH GROUP            0288.HK          COSCO SHIP HOLD     1919.HK

CHINA RES BEER      0291.HK          SANDS CHINA LTD     1928.HK

TINGYI              0322.HK          CHOW TAI FOOK       1929.HK

SINOPEC CORP        0386.HK          MINSHENG BANK       1988.HK

HKEX                0388.HK          COUNTRY GARDEN      2007.HK

CHINA RAILWAY       0390.HK          ANTA SPORTS         2020.HK

TECHTRONIC IND      0669.HK          CHINA VANKE         2202.HK

CHINA OVERSEAS      0688.HK          SHENZHOU INTL       2313.HK

TENCENT             0700.HK          PING AN             2318.HK

CHINA TELECOM       0728.HK          MENGNIU DAIRY       2319.HK

AIR CHINA           0753.HK          PICC P&C           2328.HK

CHINA UNICOM        0762.HK          LI NING             2331.HK

LINK REIT           0823.HK          GREATWALL MOTOR     2333.HK

PETROCHINA          0857.HK          PRU                 2378.HK

XINYI GLASS         0868.HK          SUNNY OPTICAL       2382.HK

ZHONGSHENG HLDG     0881.HK          BOC HONG KONG       2388.HK

CNOOC               0883.HK          CPIC                2601.HK

HUANENG POWER       0902.HK          CHINA LIFE          2628.HK

CONCH CEMENT        0914.HK          ENN ENERGY          2688.HK

CHINA LONGYUAN      0916.HK          STANCHART           2888.HK

CCB                 0939.HK          ZIJIN MINING        2899.HK

CHINA MOBILE        0941.HK          BANKCOMM            3328.HK

LONGFOR GROUP       0960.HK          CICC                3908.HK

XINYI SOLAR         0968.HK          CM BANK             3968.HK

SMIC                0981.HK          BANK OF CHINA       3988.HK

CITIC BANK          0998.HK          CMOC                3993.HK

CKI HOLDINGS        1038.HK          CITIC SEC           6030.HK

HENGAN INT'L        1044.HK          CMSC                6099.HK

CHINA SOUTH AIR     1055.HK          CEB BANK            6818.HK

CHINA SHENHUA       1088.HK          HTSC                6886.HK

The $p=92$ stocks included in the portfolio.


# Toward Human-Like Social Robot Navigation: A Large-Scale, Multi-Modal,
Social Human Navigation Dataset

Duc M. Nguyen, Mohammad Nazeri, Amirreza Payandeh, Aniket Datar, and Xuesu
Xiao All authors are with the Department of Computer Science, George Mason
University {mnguy21, mnazerir, apayande, adatar<EMAIL_ADDRESS>
###### Abstract

Humans are well-adept at navigating public spaces shared with others, where
current autonomous mobile robots still struggle: while safely and efficiently
reaching their goals, humans communicate their intentions and conform to
unwritten social norms on a daily basis; conversely, robots become clumsy in
those daily social scenarios, getting stuck in dense crowds, surprising nearby
pedestrians, or even causing collisions. While recent research on robot
learning has shown promises in data-driven social robot navigation, good-
quality training data is still difficult to acquire through either trial and
error or expert demonstrations. In this work, we propose to utilize the body
of rich, widely available, social human navigation data in many natural human-
inhabited public spaces for robots to learn similar, human-like, socially
compliant navigation behaviors. To be specific, we design an open-source
egocentric data collection sensor suite wearable by walking humans to provide
multi-modal robot perception data; we collect a large-scale ($\sim$50 km, 10
hours, 150 trials, 7 humans) dataset in a variety of public spaces which
contain numerous natural social navigation interactions; we analyze our
dataset, demonstrate its usability, and point out future research directions
and use cases.111Website: https://cs.gmu.edu/~xiao/Research/MuSoHu/

##  I INTRODUCTION

Social navigation is the capability of an autonomous agent to navigate in a
way such that it not only moves toward its goal but also takes other agents’
objective into consideration. Most humans are proficient at such a task,
smoothly navigating many public spaces shared with others on a daily basis:
humans form lanes or groups among crowds, use gaze, head movement, and body
posture to communicate navigation intentions, wait in line to enter congested
areas, or give way to others who are in a rush. With an increasing amount of
autonomous mobile robots being deployed in public spaces [1, 2, 3, 4], those
robots are also expected to navigate among humans in a similar, human-like,
socially compliant manner.

Figure 1: Data collection in natural human-inhabited public spaces with the
open-source sensor suite including 3D LiDAR, stereo and depth camera, IMU,
microphone array, and 360° camera.

However, the autonomous navigation performance of these mobile robots is still
far from satisfactory. Despite extensive robotics effort to create efficient
and collision-free autonomous navigation systems, we still witness the “frozen
robot” problem in dense crowds and robots moving against upcoming foot traffic
or cutting too close to moving humans. Unfortunately, due to such
deficiencies, there is increasing fear about the public adoption and even
safety of humans around these robots [5, 6]. The current lack of safe and
socially compliant navigation systems still presents a major hurdle preventing
service robots being widely adopted.

One avenue toward socially compliant robot navigation is using machine
learning for robots to learn the variety of unwritten social norms, for which
traditional cost functions are hard to design. For example, Reinforcement
Learning (RL) [7] uses trial-and-error experiences while Imitation Learning
(IL) [8] requires expert demonstrations. However, both of these learning
paradigms require an extensive amount of training data, which is difficult to
acquire: RL in the real world is extremely expensive due to the limited
availability of robots, while RL in simulation requires a good model of social
navigation interactions of humans, which are what roboticists are trying to
create in the first place; IL requires demonstration datasets collected on
robot platforms, mostly through expensive human teleoperation at scale [9].

Considering the goal of creating socially compliant robot navigation and the
availability of many humans that excel at such a task, this work leverages the
easily accessible social human navigation data in public spaces for mobile
robots to learn from. To be specific, we first present an open source, first-
person-view, social human navigation data collection sensor suite that can be
worn on the head of a walking human and provide easy access to a large body of
readily available, high-quality, natural social navigation data in the wild
for robot learning, as shown in Fig. 1. Our design includes a set of different
robotic sensors: a 3D Light Detection and Ranging (LiDAR) sensor, stereo and
depth camera, Inertia Measurement Unit (IMU), microphone array, and 360°
camera. We open-source our design and software so the sensor suite can be
easily replicated and used to collect social human navigation data in
different places. Second, with the new data collection suite, we introduce our
Multi-modal Social Human Navigation dataset (MuSoHu): a large-scale,
egocentric, multi-modal, and context-aware dataset of human demonstrations of
social navigation. MuSoHu contains approximately 10 hours, 150 trajectories,
50 kilometers of socially compliant navigation demonstrations collected by 7
human demonstrators that comprise multi-modal data streams from different
sensors, in both indoor and outdoor environments—within the George Mason
University campus and the Washington DC metropolitan area. We also provide
annotations of interesting social interaction events and of the navigation
contexts (i.e., “casual”, “neutral”, and “rush”) for each of the trials.
Third, we present analysis in terms of human and robot social navigation and
point out future research directions and anticipated use cases of our dataset.

##  II RELATED WORK

In this section, we review related work in social robot navigation and
learning from human datasets.

###  II-A Social Robot Navigation

To organically integrate service robots into the fabric of our society, these
robots must be capable of moving in human-inhabited spaces in a socially
compliant manner. One difficulty in creating such socially compliant
navigation systems is to hand-craft appropriate rules or cost functions to
cope with unwritten social norms in public spaces [10]. Therefore, researchers
have sought help from machine learning and aimed at _learning_ socially
compliant navigation behaviors in a data-driven manner [11].

RL has shown success in learning a variety of behaviors from simulated trial-
and-error experiences [12, 13, 14]. However, the high fidelity of simulated
social interactions required by RL for social navigation poses its own
challenges and requires a good understanding and then analytical
representation of the unwritten social norms to create such simulated
interactions, which is the difficulty in social navigation in the first place.
Additionally, the reward function in RL needs to be carefully-designed but can
still be brittle [15].

To address such issues, IL [16] utilizes expert demonstrations to learn
socially compliant navigation behaviors [17, 18]. Kretzschmar et al. [19] has
proposed Inverse Reinforcement Learning (IRL) to learn the reward function
from demonstrations for social navigation policies. Behavior Cloning [16] has
treated the social navigation problem as supervised learning and regressed to
an end-to-end motion policy that maps from perception to actions. However, to
facilitate IL, a large corpus of socially compliant navigation demonstration
data is essential. For example, the Socially Compliant Navigation Dataset
(scand) [9] is a recent effort to provide social robot navigation behaviors
demonstrated by human teleoperation.

###  II-B Learning from Human Datasets

scand [9] is a recent dataset that aims at tackling the challenges of socially
compliant robot navigation. scand includes socially compliant, human
teleoperated robot navigation demonstrations in indoor and outdoor
environments on The University of Texas at Austin campus. Using scand,
researchers have shown that IL policies can be trained end-to-end for
socially-aware global and local planners for robot navigation. However, scand
requires a significant amount of cost and effort to set up and deploy the
robot platforms in the wild and to collect large-scale human-teleoperated
robot navigation demonstrations to cover the plethora of interesting social
interactions in public spaces. Furthermore, how people react differently to a
teleoperated mobile robot followed by a human operator is also unclear.

Considering the difficulty in acquiring large-scale real-world data,
researchers have also looked into utilizing recorded videos of human
activities in the wild. For example Ego4D [20] is an egocentric video dataset,
which offers daily-life activity video of different scenarios (house-hold,
outdoor, workplace, leisure, etc.) captured by different humans wearing
cameras from different locations worldwide. Ego4D offers a solution to the
scalability of datasets by introducing a standard and wearable design so many
people can collect data in real-world, daily settings from different parts of
the world. However, Ego4D is not specifically designed for robotics (hence the
lack of common robot sensors and perception like LiDAR, depth camera, IMU, and
odometry), so it is difficult for mobile robots to directly learn socially
compliant navigation behaviors from the raw video feed in Ego4D.

Inspired by the pros and cons of both scand and Ego4D, we introduce a wearable
data collection sensor suite specifically designed to provide data to enable
social robot navigation. It allows us to collect social human navigation data
from the perspective of a suite of multi-modal robotic sensors in our daily
life with a small setup overhead (i.e., with a wearable helmet). We provide a
large-scale social human navigation dataset, which can be easily extended in
the future by robotics researchers all around the world, show that human-like
social robot navigation behaviors can be learned through such a dataset, and
point out future research directions and anticipated use cases of our dataset.

##  III SENSOR SUITE

We design and make publicly available a data collection device, which is
wearable by a human walking in public spaces and provides multi-modal
perceptual streams that are commonly available on mobile robot
platforms.222https://github.com/RobotiXX/MuSoHu-data-collection We also
process the raw data to extract human navigation behaviors, i.e., the paths
and actions taken by the human demonstrator to navigate through social spaces.

To be specific, our data collection sensor suite is equipped with a 3D LiDAR,
a stereo and depth camera with built-in IMU, a microphone array to provide
ambient sound, and a 360° camera that offers spherical view of the
environment. All the sensors are mounted to a helmet via open-sourced hardware
to capture egocentric data of the demonstrator during social navigation. To
stream and store real-time social human navigation data, all sensors are
connected to a laptop carried by the demonstrator with wired connections (Fig.
1 middle).

###  III-A 3D LiDAR

As most mobile robots use LiDARs as a reliable sensor to acquire accurate and
robust geometric information about the environment, we include a 3D LiDAR to
capture such information around the human demonstrator. Considering the
different heights of the mounting locations (on robot vs. on our helmet), we
use a 3D LiDAR to collect 3D point clouds, which can be converted to 2D scans
at different heights if necessary. We choose a Velodyne Puck VLP-16 for our
sensor suite, which has a range of 100 meters and generates up to 600,000
points/second, across a 360° horizontal and 30° vertical field of view. The 3D
LiDAR is mounted on the top of the helmet to record spatial measurements of
the surrounding.

###  III-B Stereo and Depth Camera

RGB cameras provide visual and semantic information of the environment. In
addition to the geometric information provided by the LiDAR, semantics also
plays a vital role in social navigation interactions. For example, humans use
gesture, gaze, and body posture to explicitly or implicitly convey
navigational intentions and facilitate interactions. Those behaviors can be
used to understand the intentions of other people sharing the same space but
are difficult to capture with 3D LiDAR alone. For our sensor suite, we choose
Stereolabs ZED 2, a stereo camera with depth sensing and a built-in IMU (see
below for more details), considering its compact form factor and efficient
power consumption (in contrast to other RGB-D cameras that require a separate
power supply, ZED 2 can be efficiently powered by the same USB cable for data
transmission). The camera is positioned in the front of the helmet, with the
optical axis pointing forward. The wide 120° field of view captures
interesting social interactions happening in front of and from the sides of
the human demonstrator.

###  III-C IMU

Many mobile robots are also equipped with IMUs to measure linear accelerations
and rotational speeds. Therefore, we also utilize the built-in IMU from the
ZED 2 camera and record their raw measurements. It is worth to note that due
to the difference between walking humans and wheeled or tracked robots that
drive, the IMU readings collected in our dataset may be significantly
different than those from such types of mobile robots, especially the
acceleration along the vertical axis. We posit that to leverage the IMU data
in MuSoHu, special techniques such as transfer learning [21] may be necessary.

###  III-D Odometry / Actions

Similar to scand, we collect visual-inertia odometry provided by the ZED 2
camera. Such positional odometry provides learning data of navigation path and
can be utilized to learn robot global planners. Different than scand, in which
the robot navigation actions can be directly recorded as teleoperation
commands, our data collection hardware does not have access to such actions,
i.e., how the human demonstrator walks. Therefore, we extract linear and
angular velocities from the positional odometry using the difference between
two consecutive odometry frames.

###  III-E 360° Camera

In addition to the forward facing stereo and depth camera, we also collect
360° RGB video to provide better situational awareness of the surrounding. The
360° vision also includes all possible sensory information that can be
provided by active pan-tilt cameras onboard many mobile robot platforms. We
use a Kodak Pixpro Orbit360 4K VR Camera, mounted on top of the 3D LiDAR, to
collect 360° images. The camera has a very compact form factor with two lenses
integrated in one camera body to provide spherical 360° view and does not
introduce extra weight compared with other bulky 360° cameras. Note that due
to software limitations the camera’s webcam mode does not allow both lenses to
stream live video to the laptop, so we save the spherical 360° view from both
lenses to an SD card in the camera.

###  III-F Microphone Array

Although not commonly used for navigation tasks, microphones are available on
many mobile robot platforms, e.g., for verbal communications. Furthermore,
recent research has started to investigate using sound for navigation [22].
Considering the extra information provided by this different perception
modality, we also include a microphone array, a Seeed Studio ReSpeaker Mic
Array v2.0, to collect ambient sound during social human navigation.

##  IV DATASET

Figure 2: Three example data frames in Old Town Alexandria, VA, Springfield
Towncenter, VA, and National Mall, Washington DC. 360° view (top), 3D LiDAR
point cloud (bottom left), and depth image (bottom right) are shown for each
data frame.

The sensor suite described in Sec. III is designed to be easily replicable by
any research group and to collect data worldwide. But we collect an initial
Multi-modal Social Human navigation dataset (MuSoHu) on the George Mason
University campus and in the Washington DC metropolitan area (Fig.
2).333https://dataverse.orc.gmu.edu/dataset.xhtml?persistentId=doi:10.13021/orc2020/HZI4LJ

###  IV-A Data Collection Procedure

To collect multi-modal, socially compliant, human-level navigation
demonstrations to learn future robot navigation, seven human demonstrators
wear the sensor suite helmet and navigate to predefined goals in public spaces
in a socially compliant manner. We choose navigation scenarios with frequent
social interactions in various indoor and outdoor environments at different
time periods (e.g., after class or during weekends). The sensor suite’s
superior portability (i.e., only a helmet and a laptop) also allows us to
record portions of MuSoHu in other settings in the Washington DC Metropolitan
Area, including Fairfax, Arlington, and Springfield in Virginia and the
National Mall in DC. Notably, for a trajectory at a certain location at the
same time period, in many cases, we record three trials to capture three
navigation contexts, i.e., _casual_ , _neutral_ , and _rush_ , in which
walking speed and safety distance from others may vary, in order to encourage
different social navigation interactions based on different contexts. We
intend such context awareness in MuSoHu to be useful for future studies on
context-aware social navigation, e.g., social compliance when facing someone
who is about to be late for a class is different than that when facing someone
who is taking a casual afternoon stroll in the park.

For each trajectory, all sensor data are collected using the Robot Operating
System (ros) Bag functionality, except the 360° camera, which does not allow
data streaming of both built-in cameras to provide spherical 360° view to ros.
Therefore, we store the 360° video on an SD card and provide synchronization
using a movie clapboard.

###  IV-B Dataset Analyses

####  IV-B1 Labeled Annotations of Social Interactions

MuSoHu includes a list of textual tags for each trajectory that describe the
different social interactions that occur along the path. We expand beyond the
tags from scand and the full list of 17 predefined labels can be found in
Table I (with five new tags in bold font).

TABLE I: Descriptions of Label Tags Contained in MuSoHu. Tag |  Description |  # Tags
---|---|---
Against Traffic |  Navigating against oncoming traffic |  124
With Traffic |  Navigating with oncoming traffic |  90
Street Crossing |  Crossing across a street |  51
Overtaking |  Overtaking a person or groups of people |  62
Sidewalk |  Navigating on a sidewalk |  60
Passing Conversational Groups |  Navigating past a group of 2 or more people that are talking amongst themselves |  52
Blind Corner |  Navigating past a corner where the human cannot see the other side |  46
Narrow Doorway |  Navigating through a doorway where the human opens or waits for others to open the door |  23
Crossing Stationary Queue |  Walking across a line of people |  24
Stairs |  Walking up and/or down stairs |  17
Vehicle Interaction |  Navigating around a vehicle |  13
Navigating Through Large Crowds |  Navigating among large unstructured crowds |  19
Elevator Ride |  Navigating to, waiting inside, and exiting an elevator |  7
Escalator Ride |  Navigating to and riding an escalator |  2
Waiting in Line |  Waiting in Line to enter congested areas |  2
Time: Day |  Navigation during day time |  65
Time: Night |  Navigation during night time |  32

####  IV-B2 Example Data Frames

In Fig. 3, we show the corresponding linear and angular velocities (filtered
by Savitzky-Golay filter to smooth out high frequency noises caused by walking
gait) and navigation path taken by the human demonstrator in the three
scenarios shown in Fig. 2. In the first scenario, the demonstrator navigates
around a right corner and avoids an upcoming family; in the second scenario,
the demonstrator makes a 90° right-hand turn, while avoiding people in an
indoor food court; in the third scenario, the demonstrator dodges (right-left-
right) a dense crowd during a right-hand turn. Both linear and angular
velocities and navigation path provide learning signals for mobile robots.

Figure 3: Linear (Blue) and Angular (Red) Velocities and Navigation Path
(Green) Taken by the Human Demonstrator.

####  IV-B3 Proof-of-Concept Usage

We use a small subset of MuSoHu data (ten navigation trials) to train a
Behavior Cloning policy that maps from raw LiDAR input to linear and angular
velocity (Fig. 3). The learned policy is deployed on two physical robots, an
AgileX Hunter SE (an Ackermann steering wheeled vehicle) and a Unitree Go1 (a
quadruped robot), both of which exhibit collision avoidance behavior learned
from MuSoHu (Fig. 4).

Figure 4: Learned Obstacle Avoidance Behavior from MuSoHu.

##  V ANTICIPATED USE CASES

MuSoHu’s large body of socially compliant human navigation data with multi-
modal robotic perception collected in natural public spaces in the wild
provide opportunities for many future research directions.

###  V-A Learning Social Robot Navigation

The primary purpose of MuSoHu is to provide a large corpus of training data
for mobile robots to learn socially compliant navigation behaviors. As we
demonstrate in our preliminary experiments, robot navigation behaviors similar
to the human behaviors in MuSoHu can be learned end-to-end using Behavior
Cloning. Other imitation learning methods, such as IRL, can utilize MuSoHu to
learn a socially compliant cost function for downstream navigation planners
[19].

The replicability of our sensor suite makes collecting social human navigation
data very easy. We intend the sensor suite to be replicated by different
research groups to collect data in different countries worldwide. An even
larger and also more diverse corpus of data opens up orthogonal research
directions that are not currently possible. For example, new social robot
navigation systems can be developed that are culturally dependent, i.e., the
way a mobile robot moves can be fit into different culture contexts. For
example, imagine a contact-tolerant culture where pedestrians are comfortable
with walking very closely to each other vs. a contact-averse culture where
people prefer to keep distance.

###  V-B Imitation Learning with Various Constraints

One potential challenge, in other words, opportunity for future research, is
how to address the difference in human and robot navigation. Human navigation
is based on legged locomotion, while most mobile robots are wheeled or
tracked. Different motion morphologies caused by such an embodiment mismatch
[23] may require extra care to be taken during learning. Transfer learning
techniques [21] may provide one promising avenue to leverage the full
potential of MuSoHu. In addition to the different motion morphologies, despite
our choice of sensor modalities to align with robot perception, viewpoint
mismatch still exists: to avoid occluding the 3D LiDAR view, it is mounted on
top of the helmet, which is higher than most robots’ LiDAR position; all
perceptual data are subject to the effect of walking gait cycles, e.g., the
cyclic motion along the vertical axis, which does not exist for most mobile
robots. Therefore, imitation learning from mismatched observation techniques
[7] need to be investigated to address the perceptual mismatch between MuSoHu
and mobile robots.

###  V-C Studying Social Human and Robot Navigation

One question being debated frequently is _should roboticists build robots to
navigate in public spaces in the same way as humans?_ Our MuSoHu dataset,
along with its future extensions in different countries worldwide, and scand
provide a way to investigate related problems. Assuming the navigation
behaviors in MuSoHu and scand are the optimal way of human and robot social
navigation in public spaces respectively, we can analyze both datasets to see
whether the human and robot behaviors are the same, similar, or completely
different. Another way is to build social robot navigation systems with the
data in MuSoHu and scand and evaluate the learned social navigation behaviors
with standardized protocols and metrics [24] to see wheter there is any
difference between the two and if yes which way is preferred by people that
interact with the robots.

###  V-D Real-to-Sim Transfer for Social Navigation

Creating high-fidelity social navigation simulation environments has been a
focus of social robot navigation researchers [25, 26]. A realistic simulator
that can induce real-world human-robot social interactions that conform with
the underlying unwritten social norms will facilitate social robot navigation
research on multiple fronts, such as reinforcement learning based on simulated
trial and error, large scale validation and evaluation of new social
navigation systems before their real-world deployment, and objective and
repeatable benchmark and comparison among multiple social navigation systems.
However, existing social navigation simulators rely on simplified human-robot
interaction models, e.g., the Social Force Model [27] or ORCA [28]. Such a
sim-to-real gap [29] may cause problems when the navigation systems learned,
evaluated, or compared in simulation are deployed in the real world.

The MuSohu dataset provides another alternative and promising avenue toward
shrinking such a sim-to-real gap through real-to-sim transfer to improve
social navigation simulation. The data collected in the wild in MuSoHu enable
researchers to synthesize natural, real-world, human-robot social navigation
interactions in simulation. Approaches can be developed to learn such
interaction models from the natural interactions in MuSoHu, which can be used
to control simulated agents, robots or humans, in a high-fidelity simulator.

###  V-E Investigating Robot Morphology for Social Navigation

Human-human social navigation interactions embody a large set of interaction
modalities, which are frequently present in MuSoHu. For example, in addition
to avoiding other humans as moving obstacles, humans use gaze, head movement,
and body posture to communicate navigation intentions in crowded spaces; they
use body or natural language to express their navigation mindset or context
(e.g., they are in a rush and apologize for being less polite). Most current
mobile robots, however, do not possess such capabilities to communicate their
navigation intentions and contexts in an efficient manner. Analyzing the
human-human social navigation interaction modalities in MuSoHu will shed light
on what other robot morphology may be useful to facilitate efficient social
navigation, such as adding a robot head with gaze [30], turn signals [31], or
gait features (for legged robots) [32] to disambiguate navigation intentions,
or adding voice [33] to signal the urgency of the robot’s navigation task.

##  VI CONCLUSIONS

We present a large-scale, multi-modal, social human navigation dataset,
MuSoHu, to allow robots to learn human-like, socially compliant navigation
behaviors. Our open-sourced hardware and software design allows our portable
sensor suite to be easily replicated and used to collect social human
navigation data in a variety of public spaces worldwide. Such an easy access
to a variety of natural social navigation interactions in human-inhabited
public spaces in the wild is shown in our preliminary experiments to be useful
to learn social robot navigation. We point out future anticipated use cases
and research directions of MuSoHu to develop socially compliant robot
navigation.

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