# Spin-orbit interaction with large spin
in the semi-classical regime

Didier Robert111 Laboratoire de Mathématiques Jean Leray, Université de
Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France
Email<EMAIL_ADDRESS>
###### Abstract

We consider the time dependent Schrödinger equation with a coupling spin-orbit
in the semi-classical regime $\hbar\searrow 0$ and large spin number
$\sf{s}\rightarrow+\infty$ such that $\hbar^{\delta}\sf{s}=c$ where $c>0$ and
$\delta>0$ are constant. The initial state $\Psi(0)$ is a product of an
orbital coherent state in $L^{2}(\mathbb{R}^{d})$ and a spin coherent state in
a spin irreducible representation space ${\mathcal{H}}_{2\sf{s}+1}$. For
$\delta<1$, at the leading order in $\hbar$, the time evolution $\Psi(t)$ of
$\Psi(0)$ is well approximated by the product of an orbital and a spin
coherent state. Nevertheless for $1/2<\delta<1$ the quantum orbital leaves the
classical orbital. For $\delta=1$ we prove that this last claim is no more
true when the interaction depends on the orbital variables. For the Dicke
model, we prove that the orbital partial trace of the projector on $\Psi(t)$
is a mixed state in $L^{2}(\mathbb{R})$ for small $t>0$.

##  1 Introduction

We consider here the Schrödinger equation for a system of particles with large
spin number when the spin and the position variables are coupled:

$\displaystyle i\hbar\partial_{t}\Psi(t)$ $\displaystyle=\hat{H}(t)\Psi(t),$
(1.1) $\displaystyle\Psi(0)$ $\displaystyle=\varphi_{z_{0}}\otimes\psi_{\bf
n_{0}},\;\;{\rm in}\;\;L^{2}(\mathbb{R}^{d})\otimes{\mathcal{H}}_{2\sf{s}+1}$

where $\varphi_{z_{0}},\psi_{\bf n_{0}}$ are respectively Schrödinger, spin
coherent states and

$\hat{H}(t)=\hat{H}_{0}(t)+\hbar\widehat{\mathfrak{C}}(t)\cdot{\bf S}.$ (1.2)

$\hat{H}_{0}(t)$ has a scalar $\hbar$-Weyl symbol as well as
$\hat{\mathfrak{C}}_{j}(t)$ and ${\bf S}=(S_{1},S_{2},S_{3})$ are spin
matrices representation in an irreducible space ${\mathcal{H}}_{2\sf{s}+1}$ of
dimension $2\sf{s}+1$, where $\sf{s}$ is the total spin number.
A particular case is the Pauli equation for the electron ($\sf{s}=1/2$).

$\hat{H}=\frac{1}{2}\left(\hbar D-A\right)^{2}+\hat{V}+\hbar B\cdot\sigma,$

where $D=i^{-1}\nabla_{x}$, $x\in\mathbb{R}^{3}$, $A=(A_{1},A_{2},A_{3})$ is a
magnetic potential, $V$ an electric potential, $B=(B_{1},B_{1},B_{1})$ the
magnetic field in $\mathbb{R}^{3}$;
$\sigma=(\sigma_{1},\sigma_{2},\sigma_{3})$ are the Pauli matrices:

$\sigma_{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\;\sigma_{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\;\sigma_{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$

For $\sf{s}=1/2$ we have $S_{k}=\frac{\sigma_{k}}{2}$.
More generally the Weyl symbol of the interaction is $\hbar H_{1}(t,X)$ where
$\displaystyle{H_{1}(t,X)=\sum_{1\leq k\leq
3}\mathfrak{C}_{k}(t,X)S_{k}}=\mathfrak{C}(t,X)\cdot{\bf S}$.
If the spin $\sf{s}$ is fixed, in the semi-classical regime, $\hbar\searrow
0$, $H_{1}(t,X)$ is the subprincipal symbol of $\hat{H}(t)$. Hence we can get
a semiclassical approximation at any order for $\Psi(t)$ using generalized
coherent states as it will be recalled later. For details see [3], Chapter 14
and [4] for matrix principal symbol $H_{0}(t)$.
The spin matrices $S_{k}$ are realized as hermitian matrices in the Hermitian
space ${\mathcal{H}}_{2\sf{s}+1}$ such that
$S_{k}=d{\mathcal{D}}^{(\sf{s})}(\sigma_{k}/2)$ are defined by the derivative
at $\mathbb{I}$ of the irreducible representation ${\mathcal{D}}^{(\sf{s})}$
in ${\mathcal{H}}_{2\sf{s}+1}$. In particular from the Lie algebra
$\mathfrak{su}(2)$ we have the commutation relations

$[S_{k},S_{\ell}]=i\epsilon_{k,\ell,m}S_{m}.$ (1.3)

$\mathfrak{C}_{k}(t,X)$ are real scalar symbols in $X\in\mathbb{R}^{2d}$. So
the full Weyl symbol of $\hat{H}(t)$ is the matrix
$H(t,X)=H_{0}(t,X)\mathbb{I}+\hbar\mathfrak{C}(t,X)\cdot{\bf S}$, with the
spin operator ${\bf S}=(S_{1},S_{2},S_{3})$.
For simplicity we shall assume in all this paper that the Weyl symbols
$H_{0}(t)$ and $\mathfrak{C}(t,X)$ are subquadratic [3](p.409). It follows
that the quantum Hamiltonian $\hat{H}(t)$ generates a propagator
$\mathcal{U}(t,t_{0})$ in the Hilbert space
$L^{2}(\mathbb{R}^{d},{\mathcal{H}}_{2\sf{s}+1})$. (see Proposition 123 of [3]
easily extended to systems).

It is known (see [3] ( Ch.1 and Chap.7) that the Schrödinger coherent states
are labelled by the phase space $T^{*}(\mathbb{R}^{d})$ and the spin (or
atomic) coherent states are labelled by the sphere $\mathbb{S}^{2}$. So it is
natural to study the semi-classical limit of the propagator with the phase
space $T^{*}(\mathbb{R}^{d})\times{\mathbb{S}}^{2}$. In particular we can
reformulate the propagation for coherent states $\varphi_{z}\otimes\psi_{\bf
n}$ labelled by $(z,{\bf n})\in\mathbb{R}^{2d}\times\mathbb{S}^{2}$ where
$\varphi_{z}$ is a Schrödinger coherent state and $\psi^{(\sf{s})}_{\bf
n}=\psi_{\bf n}$ is a spin coherent state.
Let us recall our notations.

  * •

Heisenberg translations in $L^{2}(\mathbb{R}^{d})$:

$\hat{T}(z)=\exp\left(\frac{i}{\hbar}p\cdot\hat{x}-q\cdot\hat{p}\right)$

where $z=(q,p)\in\mathbb{R}^{d}\times\mathbb{R}^{d}$, $\hat{x}$ is
mutiplication by $x$ and $\hat{p}=\frac{\hbar}{i}\nabla_{x}$.

  * •

Schrödinger coherent states:

$\varphi_{z}=\hat{T}(z)\varphi_{0},\;\;\varphi_{0}(x)=(\pi\hbar)^{-d/4}\exp\left(\frac{|x|^{2}}{2\hbar}\right)$

  * •

$SU(2)$ and the sphere $\mathbb{S}^{2}$: let

$\mathbf{n}=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta),\;0\leq\theta<\pi,\;0\leq\varphi<2\pi;$

To any $\mathbf{n}\in\mathbb{S}^{2}$ we associate the transformation in
$SU(2)$,

$g=g_{\mathbf{n}}=\exp\left(i\frac{\theta}{2}((\sin\varphi)\sigma_{1}-(\cos\varphi)\sigma_{2})\right)$
(1.4)

  * •

irreducible representations and spin coherent states: let $\sf{s}$ be an half
integer and ${\mathcal{D}}^{\sf{s}}$ the irreducible representation in the
Hilbert space ${\mathcal{H}}_{2\sf{s}+1}$ of dimension $2\sf{s}+1$. To
${\mathbf{n}}\in\mathbb{S}^{2}$ is associated the spin coherent states 222this
construction follows from the computation of the isotropy subgroups of the
$SU(2)$ action: we get roughly $SU(2)/{\rm ISO}\approx\mathbb{S}^{2}$ (see [3]
for details).

$\psi_{\mathbf{n}}={\mathcal{D}}^{\sf{s}}(g_{\mathbf{n}})\psi_{0},$

where $\psi_{0}$ is a unit eigenvector of $S_{3}$ in
${\mathcal{H}}_{2\sf{s}+1}$ with minimal eigenvalue $-\sf{s}$.

Let us consider uncorrelated initial states $\Psi_{z_{0},{\bf
n}_{0}}=\varphi_{z_{0}}\otimes\psi^{(\sf{s})}_{{\bf n}_{0}}$, where
$\varphi_{z_{0}}$ is a Schrödinger (orbital) coherent state and
$\psi^{(\sf{s})}_{{\bf n}_{0}}$ a spin coherent state. When the spin number
$\sf{s}$ is fixed we have the following result proved in [2] and revisited in
[3].

######  Theorem 1.1.

[2] For the initial state $\Psi_{z_{0},{\bf
n}_{0}}=\varphi^{\Gamma_{0}}_{z_{0}}\otimes\psi^{(\sf{s})}_{{\bf n}_{0}}$ we
have

$\mathcal{U}(t,t_{0})\Psi_{z_{0},{\bf n}_{0}}={\rm
e}^{\frac{i}{\hbar}S(t,t_{0})+i\sf{s}\alpha(t)}\varphi_{z(t)}^{\Gamma(t)}\otimes\psi^{(\sf{s})}_{{\bf
n}(t)}+O(\sqrt{\hbar})$ (1.5)

where $z_{0}\mapsto z(t)$ is the classical flow for the Hamiltonian for
$H_{0}(t)$, $S(t,t_{0})$ is the classical action, the covariance matrix
$\Gamma(t)$ is computed from the dynamics generated by the linearized flow of
$H_{0}(t)$ and $\alpha(t)$ is a real phase computed from the spin motion ${\bf
n}(t)$ which satisfies the Landau-Lifshitz [7] equation

$\partial_{t}{\bf n}(t)=\mathfrak{C}(t,z(t))\wedge{\bf n}(t).$ (1.6)

######  Remark 1.2.

In Theorem 1.1 the motion of the spin depends on the motion along the orbit
but the orbit motion is independent of the spin.

######  Remark 1.3.

Theorem 1.1 is a particular case of propagation of coherent states for systems
with a subprincipal term of order $\hbar$ and a principal term $H_{0}(t)$
(here scalar) without crossing eigenvalues. Moreover one can get a complete
asymptotic expansion in power of $\sqrt{\hbar}$ mod $O(\hbar^{\infty})$.
For smooth crossings some non-adiabatic results are proved in [4].

The first new result in this paper is a generalization of Theorem 1.1 for
$\sf{s}\rightarrow+\infty$ and $\hbar\searrow 0$ such that
$\sf{s}\hbar^{\delta}=c$ where $0<\delta<1$ and $c>0$ are constants. Let us
denote $\kappa:=\hbar\sf{s}$ which is here a small positive parameter for
$\delta<1$.

######  Theorem 1.4.

Let us assume that $0<\delta<1$. For the initial state $\Psi_{z_{0},{\bf
n}_{0}}=\varphi_{z_{0}}\otimes\psi^{(\sf{s})}_{{\bf n}_{0}}$, the solution of
(1.1) for the Hamiltonian (1.2) satisfies:

$\mathcal{U}(t,t_{0})\Psi_{z_{0},{\bf n}_{0}}={\rm
e}^{\frac{i}{\hbar}S(t)+i\sf{s}\alpha(t)}\varphi_{z(t)}^{\Gamma(t)}\otimes\psi^{(\sf{s})}_{{\bf
n}(t)}+O(\sqrt{\hbar}+\kappa)),$ (1.7)

where the dynamics of the coherent states satisfies the following system of
equations:

$\displaystyle\dot{q}$
$\displaystyle=\partial_{p}H_{0}(t,q,p)+\kappa\partial_{p}\mathfrak{C}(t,q,p)\cdot{\bf
n}(t)$ (1.8) $\displaystyle\dot{p}$
$\displaystyle=-\partial_{q}H_{0}(t,q,p)-\kappa\partial_{q}\mathfrak{C}(t,q,p)\cdot{\bf
n}(t)$ $\displaystyle\dot{\bf n}(t)$
$\displaystyle=\mathfrak{C}(t,q,p)\wedge{\bf n}(t).$

Moreover $S(t)$ is the action along the trajectory $z(t)$ and $\alpha(t)$ is
the action along the trajectory ${\mathbf{n}}(t)$ of the classical spin, in
the time interval $[t_{0},t]$.

Let be $(z^{\kappa}_{t},{\bf n}^{\kappa}_{t})$ the flow satisfying (1.8) with
$z^{\kappa}_{0}=z_{0}$, ${\bf n}^{\kappa}_{0}={\bf n}_{0}$.

######  Corollary 1.5.

Let be $1/2>\varepsilon>0$ small. Then we have

  * •

if $\sf{s}\leq c\hbar^{-1/2+\varepsilon}$ then the Theorem 1.4 is valid by
taking $\kappa=0$ in (1.8), with the error term $O(\hbar^{\varepsilon})$. So
in this case the quantum orbital follows the classical trajectory for
$H_{0}(t)$ at the leading order in $\hbar$.

  * •

$If\sf{s}\approx c\hbar^{-1/2-\varepsilon}$ then the quantum orbital motion
depends on the spin motion: in general we cannot take $\kappa=0$ in (1.8). If
$\nabla_{X}\mathfrak{C}(t_{0},z_{0})\neq 0$ the orbital Gaussian
$\varphi_{z(t)}^{\Gamma(t)}$ follows the new trajectory $z^{\kappa}_{t}$ and
not $z^{\kappa}_{t}$. Moreover there exist $\tau>0$, $c>0$, such that

$|z^{\kappa}_{t}-z^{0}_{t}|\geq c|t-t_{0}|\hbar^{1/2-\varepsilon},\;{\rm
for}\;\;|t-t_{0}|\leq\tau.$ (1.9)

(see Lemma A.1)

######  Remark 1.6.

Notice that the coherent state $\varphi_{z}^{\Gamma}(x)$ is localized in any
disc $\\{|X-z|\leq c\hbar^{1/2-\varepsilon}\\}$, $c>0,\varepsilon>0$, in the
phase space $\mathbb{R}^{2d}$. More precisely its Wigner function
${\mathcal{W}}_{\varphi_{z}^{\Gamma}}(X)$ satisfies for some $\mu>0$, $C>0$,

$|{\mathcal{W}}_{\varphi_{z}^{\Gamma}}(X)|\leq(\pi\hbar)^{-d}{\rm
e}^{-\frac{\mu}{\hbar}|X-z|^{2}}.$

And its Husimi function ${\mathcal{H}}_{\varphi_{z}^{\Gamma}}(X)$ satisfies,
for some $C^{\prime}>0$ and $\mu^{\prime}>0$,

$0\leq{\mathcal{H}}_{\varphi_{z}^{\Gamma}}(X)\leq C^{\prime}\hbar^{-d}{\rm
e}^{-\frac{\mu^{\prime}}{\hbar}|X-z|^{2}},$

(for more properties of the Husimi functions see [3], section 2.5).
Hence from (1.9) we get that ${\mathcal{H}}_{{\varphi_{z(t)}^{\Gamma(t)}}}(X)$
is $O(\hbar^{\infty})$ in a neighborhood of size $\hbar^{1/2-\varepsilon/2}$
of $z^{0}_{t}$ for $0<t_{1}<t<t+\tau$.

Let us consider now the case $\kappa>0$ fixed and $\hbar\searrow 0$. We shall
see that Theorem 1.4 cannot be satisfied in general. In particular for the
Dicke model considered by Hepp-Lieb (for $\hbar=1$) [5] (See also [11] for a
recent review).

$\hat{H}_{\rm Dic}=\hbar\omega_{c}{\bf a}^{\dagger}{\bf
a}+\omega_{3}\hat{S}_{3}+\frac{\lambda}{\sqrt{N}}({\bf a}^{\dagger}+{\bf
a})\hat{S}_{1},$

In quantum optics this model describes the interaction between light and
matter where the light (photons) is a field operator. Here we consider a toy
model with an quantum harmonic oscillator for photons and $N$ atoms of matter
being in two spin levels.
$\omega_{c}>0$, $\omega_{3}\geq 0$, $N=2\sf{s}+1$ is the dimension of the spin
states space, $\lambda>0$ a coupling constant, ${\bf
a}=\frac{1}{\sqrt{2}\hbar}(x+\hbar\partial_{x})$, ${\bf
a}^{\dagger}=\frac{1}{\sqrt{2}\hbar}(x-\hbar\partial_{x})$ are the usual
creation and annihilation operators (satisfying $[{\bf a},{\bf
a}^{\dagger}]=1$) and $\hat{S}_{j}=\hbar S_{j}$. Here the Fock space is simply
$L^{2}(\mathbb{R})$.

A more explicite expression for $\hat{H}_{\rm Dic}$ considered here is the
following

$\hat{H}_{\rm
Dic}=\frac{\omega_{c}}{2}(-\hbar^{2}\partial_{x}^{2}+x^{2})+\omega_{3}\hat{S}_{3}+\frac{2\lambda}{\sqrt{N\hbar}}x\hat{S}_{1}.$
(1.10)

With this elementary model we get a contradiction with Theorem 4.7 of [2]. In
the regime $\hbar\approx\sf{s}^{-1}$, $\sf{s}\nearrow\infty$ the orbital
trajectory blows up into a mixed state for $t>0$ (”decoherence of the orbital
state”).

######  Remark 1.7.

Why to consider large spin quantum systems?
Notice that for $N$ atoms of spin 1/2 the spin Hilbert space should be of
dimension $2^{N}$ corresponding to the tensor product
$\otimes^{N}{\mathcal{D}}^{1/2}$ where ${\mathcal{D}}^{1/2}$ is the
representation of degree 2 of $SU(2)$ for the spin 1/2. Using the Clebsch-
Gordon formula we see that the representation $\otimes^{N}{\mathcal{D}}^{1/2}$
contains the irreducible representation ${\mathcal{D}}^{N/2}$ of $SU(2)$ of
maximal degree $N+1$ corresponding to an effective spin $\sf{s}=\frac{N}{2}$.
So that in this settting the large spin limit is also the large number of
atoms limit (thermodynamic limit).

######  Theorem 1.8.

Let be $0<\kappa=\hbar\sf{s}$, $\Psi(0)=\varphi_{z_{0}}(x)\psi_{{\bf n}_{0}}$
and $\Psi(t)={\rm e}^{-\frac{it}{\hbar}\hat{H}_{\rm Dic}}\Psi(0)$ with
$\Psi(0)=\varphi_{z_{0}}\otimes\psi_{\bf n_{0}}$.
Then there exists $c_{0}>0$ such that for any $0<\mu<1/2$ and $\hbar\searrow
0$, we have

${\rm tr}[{\rm tr}_{{\mathcal{H}}_{2s+1}}\Pi_{\Psi(t)}]^{2}\leq(1+c_{0}\kappa
t^{2})^{-1/2}+O(\hbar^{\mu}),$ (1.11)

where $\Pi_{\Psi}$ is the projection on the pure state $\Psi\in
L^{2}(\mathbb{R})\otimes{\mathcal{H}}_{2s+1}$, ${\rm
tr}_{{\mathcal{H}}_{2s+1}}\Pi_{\Psi}$ is the partial trace of $\Pi_{\Psi}$ in
${\mathcal{H}}_{2s+1}$ and the last trace in (6.7) is computed for operators
in the Hilbert space $L^{2}(\mathbb{R},\mathbb{C})$.

######  Remark 1.9.

The previous result shows that for a spin $\sf{s}$ of order $\hbar^{-1}$ the
orbital evolution is transformed into a mixed state as the spin interaction is
switched on and $\hbar$ is small enough.
For a density matrix $\hat{\rho}$, ${\mathcal{P}}[\hat{\rho}]:={\rm
tr}[\hat{\rho}^{2}]$ is called the purity of the density matrix $\hat{\rho}$.
Here it is applied for $\hat{\rho}_{\sf{s}}(t):={\rm
tr}_{{\mathcal{H}}_{2s+1}}\Pi_{\Psi(t)}$. This is related with the von-Neumann
entropy which is defined as ${\rm S}_{vN}[\hat{\rho}]=-{\rm
tr}[\hat{\rho}\log\hat{\rho}]$. So we have easily ${\rm
S}_{vN}[\hat{\rho}]\geq 1-{\mathcal{P}}[\hat{\rho}]$. For a pure state ${\rm
S}_{vN}[\hat{\rho}]=0$ and ${\mathcal{P}}[\hat{\rho}]=1$.
Then from (6.7) we get that the density matrix ${\rm
tr}_{{\mathcal{H}}_{2s+1}}\Pi_{\Psi(t)}$ has a positive von Neumann entropy
for $t>0$ and $\hbar>0$ small enough.
In the analysis of the Dicke model one consider that the orbital part is an
open sub-system of the closed total system (orbital+spin). For other open
systems and time evolution of coherent states (like the ”Schrödinger cat”) we
refer to [3], Chapter 13 for more details.

##  2 Preliminaries

###  2.1 Reduction to the interaction propagator

It is convenient to annihilate the scalar part $H_{0}(X)$ by considering the
interaction representation for the propagator $\mathcal{U}(t,t_{0})$ of the
Hamiltonian $\hat{H}(t)$.
Hence we have
$\mathcal{U}(t,t_{0})=\mathcal{U}_{0}(t-t_{0}){\mathcal{V}}(t,t_{0})$, where
$\mathcal{U}_{0}(t-t_{0})={\rm e}^{-i\frac{t-t_{0}}{\hbar}\hat{H}_{0}}$ and
the propagator ${\mathcal{V}}(t,t_{0})$ must satisfy

$i\hbar\partial_{t}{\mathcal{V}}(t,t_{0})=(\widehat{\mathfrak{C}}_{I}(t)\cdot{\bf
S}){\mathcal{V}}(t,t_{0}).$

For $1\leq k\leq 3$ the Weyl symbols ${\mathfrak{C}}_{I,k}(t,X)$ is computed
by the Egorov Theorem. In particular its principal term is given by

${\mathfrak{C}}_{I,k}(t,X)={\mathfrak{C}}_{k}(t,\Phi_{0}^{t-t_{0}}(X)).$

So for simplicity, in what follows we shall assume that $H_{0}\equiv 0$ and
consider the simpler interaction spin-orbit Hamiltonian $\hat{H}_{\rm
int}(t)=\hbar\widehat{\mathfrak{C}}(t)\cdot{\bf S}$.
For the Dicke model $H_{0}$ is the harmonic oscillator so we have:

$\Phi_{0}^{t}(x,\xi)=(\cos(\omega_{c}t)x(0)+\sin\omega_{c}t\xi(0),-\sin(\omega_{c}t)x(0)+\cos(\omega_{c}t)\xi(0),$

hence $H_{\rm
int}(t,x,\xi)=(\cos(\omega_{c}t)x+\sin(\omega_{c}t)\xi)S_{1}+\hbar\omega_{3}S_{3}$.

###  2.2 A realization of spin-$\sf{s}$ representation

Because our aim is to perform explicit computations for the spin side we
choose to work with a concrete representation (see for example [3], chap.7).
We assume here that ${\mathcal{V}}^{(\sf{s})}={\mathcal{H}}_{2\sf{s}+1}$, the
complex linear space of homogenous polynomials of degree $2\sf{s}$ in two
variables $(z_{1},z_{2})\in\mathbb{C}^{2}$. ${\mathcal{H}}_{2\sf{s}+1}$ is an
Hermitian space for the scalar product defined such that the monomials (named
Dicke states):

${\mathscr{D}}^{(\sf{s})}_{m}=\frac{z_{1}^{\sf{s}+m}z_{2}^{\sf{s}-m}}{\sqrt{(\sf{s}+m)!(\sf{s}-m)!}},$

where $-\sf{s}\leq m\leq\sf{s}$ and $m$ is an integer or half an integer
according $\sf{s}$ is.
In ${\mathcal{H}}_{2\sf{s}}$ the spin operators are realized as

$\displaystyle S_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(z_{1}\partial_{z_{1}}-z_{2}\partial_{z_{2}}\right),\;S_{+}=z_{1}\partial_{z_{2}},\;S_{-}=z_{2}\partial_{z_{1}}$
$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\frac{S_{+}+S_{-}}{2},\;S_{2}=\frac{S_{+}-S_{-}}{2i},$ (2.1)

with the commutation relations of the Lie algebra $su(2)$ of the group
$SU(2)$:

$[S_{3},S_{\pm}]=\pm S_{\pm}.$

Recall that $g\in SU(2)$ if $g=\begin{pmatrix}\alpha&-\bar{\beta}\\\
\beta&\alpha\end{pmatrix}$, $\alpha,\beta\in\mathbb{C}$,
$|\alpha|^{2}+|\beta|^{2}=1$.
Spin coherent states are defined by the action of $SU(2)$ on the Dicke state
of minimal weight: ${\mathscr{D}}^{(\sf{s})}_{-\sf{s}}$. As known, to get a
good parametrization of coherent states we choose a family of transformations
in $SU(2)$ indices by the sphere $\mathbb{S}^{2}$. This is obtained by
computing the isotropy subgroup ${\rm ISO}$ of the $SU(2)$ action: $g\in{\rm
ISO}$ if and only if $g=\begin{pmatrix}\alpha&0\\\
0&\bar{\alpha}\end{pmatrix}$, $\alpha={\rm e}^{i\psi},\psi\in\mathbb{R}$. Then
we get that the space of orbits $SU(2)\backslash{\rm ISO}$ can be identified
with the sphere $\mathbb{S}^{2}$ (for details see for example [3]).

Let us denote

$\mathbf{n}=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta),\;0\leq\theta<\pi,\;0\leq\varphi<2\pi;$

To any $\mathbf{n}\in\mathbb{S}^{2}$ we associate the transformation in
$SU(2)$,

$g=g_{\mathbf{n}}=\exp\left(i\frac{\theta}{2}(\sin\varphi\sigma_{1}-\cos\varphi\sigma_{2})\right)$
(2.2)

where $\sigma_{1},\ \sigma_{2}$ are the Pauli matrices which satisfy the
commutation relations

$[\sigma_{k},\sigma_{l}]=2i\epsilon_{k,\ell,m}\sigma_{m}$ (2.3)

For $\sf{s}=\frac{1}{2}$ we have $S_{k}=\frac{\sigma_{k}}{2}$. So if
$g\mapsto{\mathcal{D}}^{(\sf{s})}g$ is the representation of $SU(2)$ in
${\mathcal{H}}_{2\sf{s}+1}$ we have

${\mathcal{D}}^{(\sf{s})}g_{\mathbf{n}}=\exp\left(\frac{\theta}{2}({\rm
e}^{i\varphi}S_{-}-{\rm e}^{-i\varphi}S_{+})\right)$ (2.4)

######  Definition 2.1.

The coherent states of $SU(2)$ are defined in the representation space
${\mathcal{H}}_{2\sf{s}+1}$ as follows:

$|\mathbf{n}\rangle={\mathcal{D}}^{\sf{s}}(g_{\mathbf{n}}){\mathscr{D}}^{(\sf{s})}_{-\sf{s}}:=\psi_{\mathbf{n}}^{(\sf{s})}.$
(2.5)

It is some time convenient to consider a complex parametrization of the sphere
$\mathbb{S}^{2}$ using the stereographic projection ${\mathbf{n}}\mapsto\eta$,
on the complex plane and an identification of the coherent states
$|\mathbf{n}\rangle$ with the state $|\eta\rangle:=\psi_{\eta}$ defined as
follows:

$|\eta\rangle=(1+|\eta|^{2})^{-j}\exp(\eta S_{+})|\sf{s},-\sf{s}\rangle,$

More precisely we denote $|\eta\rangle=|{\mathbf{n}}\rangle$ with the
following correspondence:

$\mathbf{n}=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta),\quad\eta=-\tan\frac{\theta}{2}e^{-i\varphi}$

The geometrical interpretation is that $-\bar{\eta}$ is the stereographic
projection of $\bf n$.
Recall the following expression of $g_{\mathbf{n}}$

$g_{\mathbf{n}}=\begin{pmatrix}\cos\frac{\theta}{2}&-\sin\frac{\theta}{2}{\rm
e}^{-i\varphi}\\\ \sin\frac{\theta}{2}{\rm
e}^{i\varphi}&\cos\frac{\theta}{2}\end{pmatrix}$ (2.6)

For further investigations we shall need two results:
1) compute the derivative of $t\mapsto T(g_{{\mathbf{n}}_{t}})$ for a $C^{1}$
path on $\mathbb{S}^{2}$.
2) compute the adjoint action of $T(g_{\mathbf{n}})$ on $S_{k}$, $1\leq k\leq
3$.
From (2.4) we get

$\displaystyle\partial_{\varphi}T(g_{\mathbf{n}})$ $\displaystyle=$
$\displaystyle\frac{i}{2}T(g_{\mathbf{n}})\left(\sin\theta({\rm
e}^{-i\varphi}S_{+}+{\rm e}^{i\varphi}S_{-})+(1-\cos\theta)S_{3}\right)$ (2.7)
$\displaystyle i\partial_{\theta}T(g_{\mathbf{n}})$ $\displaystyle=$
$\displaystyle\frac{i}{2}T(g_{\mathbf{n}})({\rm e}^{i\varphi}S_{-}-{\rm
e}^{-i\varphi}S_{+})$ (2.8)

######  Lemma 2.2.

Let be a $C^{1}$ path on $\mathbb{S}^{2}$: $t\mapsto(\theta_{t},\varphi_{t})$.
Then we have

$\displaystyle\partial_{t}T(g_{\mathbf{n}})=\frac{i}{2}T(g_{\mathbf{n}})\left(\sin\theta({\rm
e}^{-i\varphi}S_{+}+{\rm
e}^{i\varphi}S_{-})+(1-\cos\theta)S_{3}\right)\dot{\varphi}_{t}$
$\displaystyle+\frac{1}{2}T(g_{\mathbf{n}})\left({\rm e}^{i\varphi}S_{-}-{\rm
e}^{-i\varphi}S_{+}\right)\dot{\theta}_{t}.$ (2.9)

To compute $\dot{\theta}$, $\dot{\varphi}$ we use the Riemann model for
$\mathbb{S}^{2}$. Let be $-\bar{\eta}\in\mathbb{C}$ the stereographic
projection of ${\mathbf{n}}$ on the equatorial plane, from the south pole
${\mathbf{n}}_{so}$. The coordinates of ${\mathbf{n}}$ are given by

$n_{1}=-\frac{\eta+\bar{\eta}}{1+|\eta|^{2}},\;\;n_{2}=\frac{\eta-\bar{\eta}}{i(1+|\eta|^{2})},\;\;n_{3}=\frac{1-|\eta|^{2}}{1+|\eta|^{2}}.$

So we have

$\eta=-\tan\left(\frac{\theta}{2}\right)\,{\rm
e}^{-i\varphi},\;\cos\theta=\frac{1-|\eta|^{2}}{1+|\eta|^{2}},\;\sin\theta\,{\rm
e}^{i\varphi}=-\frac{2\,\bar{\eta}}{1+|\eta|^{2}}.$ (2.10)

Then we get easily

$\displaystyle\dot{\theta}=\frac{\eta\dot{\bar{\eta}}+\bar{\eta}\dot{\eta}}{|\eta|(1+|\eta|^{2})},\;\;\dot{\varphi}=\frac{i}{2}\left(\frac{\dot{\eta}}{\eta}-\frac{\dot{\bar{\eta}}}{\bar{\eta}}\right).$
(2.11)

######  Lemma 2.3.

$-i\partial_{t}T(g_{{\mathbf{n}}_{t}})=iT(g_{{\mathbf{n}}_{t}})(A\dot{\theta}+B\dot{\varphi})$

with

$(A\dot{\theta}+B\dot{\varphi})=-\frac{1}{1+|\eta|^{2}}(\dot{\eta}S_{+}-\dot{\bar{\eta}}S_{-})+\frac{|\eta|^{2}}{1+|\eta|^{2}}\left(\frac{\dot{\eta}}{\eta}-\frac{\dot{\bar{\eta}}}{\bar{\eta}}\right)S_{3}$

Proof. Standard computations. $\square$

Let us denote
$S_{k}(\mathbf{n})={\mathcal{D}}^{\sf{s}}(g_{\mathbf{n}})^{*}S_{k}{\mathcal{D}}^{\sf{s}}(g_{\mathbf{n}})$,
$k=+,-,3$ or $k=1,2,3$.
In the following Lemma similar results are stated in [6], section 2.6, and
proved by a different method.

######  Lemma 2.4.

If $(\theta,\varphi)$ are the coordinates of $\mathbf{n}$ on $\mathbb{S}^{2}$,
we have,

$\displaystyle S_{3}(\theta,\varphi)$ $\displaystyle=$
$\displaystyle\cos\theta.S_{3}-\sin\theta\left(\frac{{\rm
e}^{i\varphi}S_{-}+{\rm e}^{-i\varphi}S_{+}}{2}\right)$ (2.12) $\displaystyle
S_{+}(\theta,\varphi)$ $\displaystyle=$ $\displaystyle{\rm
e}^{i\varphi}\sin\theta.S_{3}+\frac{\cos\theta+1}{2}S_{+}+\frac{\cos\theta-1}{2}{\rm
e}^{2i\varphi}S_{-}$ (2.13) $\displaystyle S_{-}(\theta,\varphi)$
$\displaystyle=$ $\displaystyle{\rm
e}^{-i\varphi}\sin\theta.S_{3}+\frac{\cos\theta+1}{2}S_{-}+\frac{\cos\theta-1}{2}{\rm
e}^{-2i\varphi}S_{+}$ (2.14) $\displaystyle
S_{1}(\theta,\varphi)=\left(\frac{\cos\theta+1}{2}+\frac{\cos\theta-1}{2}\cos
2\varphi\right)S_{1}+\frac{\cos\theta-1}{2}\sin 2\varphi\,S_{2}$
$\displaystyle+\cos\varphi\sin\theta\;S_{3}$ $\displaystyle
S_{2}(\theta,\varphi)=\frac{\cos\theta-1}{2}\sin 2\varphi
S_{1}+\left(\frac{\cos\theta+1}{2}-\frac{\cos\theta-1}{2}\cos
2\varphi\right)S_{2}$ $\displaystyle+\sin\theta\sin\varphi S_{3}.$ (2.16)

Proof. We write ${\mathcal{D}}^{\sf{s}}(g_{\mathbf{n}})={\rm
e}^{\frac{\theta}{2}L}$ where $L={\rm e}^{i\varphi}S_{-}-{\rm
e}^{-i\varphi}S_{+}$. Let be $S$ one of the spin operator, we have

$\partial_{\theta}S(\theta,\varphi)=\frac{1}{2}{\rm
e}^{-\frac{\theta}{2}L}[S,L]{\rm e}^{\frac{\theta}{2}L}$

and the commutation relations

$[L,S_{3}]={\rm e}^{i\varphi}S_{+}+{\rm e}^{-i\varphi}S_{+},\;[L,S_{-}]=-2{\rm
e}^{-i\varphi}S_{3},\;[L,S_{+}]=-2{\rm e}^{i\varphi}S_{3}.$

So we find

$\displaystyle\partial_{\theta}S_{3}(\theta,\varphi)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}({\rm e}^{i\varphi}S_{-}(\theta,\varphi)+{\rm
e}^{-i\varphi}S_{+}(\theta,\varphi))$ (2.17)
$\displaystyle\partial_{\theta}S_{+}(\theta,\varphi)$ $\displaystyle=$
$\displaystyle{\rm e}^{i\varphi}S_{3}(\theta,\varphi)$ (2.18)
$\displaystyle\partial_{\theta}S_{-}(\theta,\varphi)$ $\displaystyle=$
$\displaystyle{\rm e}^{-i\varphi}S_{3}(\theta,\varphi).$ (2.19)

hence

$\partial_{\theta}^{2}S_{3}(\theta,\varphi)=-S_{3}(\theta,\varphi).$ (2.20)

Solving the differential equation (2.20) we get (2.12). $\square$

Now we come to the spin-coherent states. Le be ${\mathbf{n}}_{0}$ the north
pole on $\mathbb{S}^{2}$ corresponding to the Dicke state
${\mathscr{D}}^{(\sf{s})}_{-\sf{s}}:=\psi_{{\mathbf{n}}_{0}}$. The following
Lemma is basic for our next computations.

######  Lemma 2.5.

For any $\mathbf{n}\in\mathbb{S}^{2}$ we have

${\mathbf{S}}\psi_{\mathbf{n}}=-\sf{s}{\mathbf{n}}\psi_{\mathbf{n}}+\sqrt{\frac{\sf{s}}{2}}\,{\mathbf{v}}({\mathbf{n}})\psi_{1,{\mathbf{n}}}$

where
$\psi_{1,{\mathbf{n}}}={\mathcal{D}}^{(\sf{s})}(g_{\mathbf{n}}){\mathscr{D}}^{(\sf{s})}_{1-\sf{s}}$
and ${\mathbf{v}}({\mathbf{n}})=(v_{1},v_{2},v_{3})\in\mathbb{C}^{3}$ is
defined as

$\displaystyle v_{1}(\mathbf{n})$ $\displaystyle=$
$\displaystyle\frac{\cos\theta+1}{2}+\frac{\cos\theta-1}{2}{\rm
e}^{-2i\varphi}$ (2.21) $\displaystyle v_{2}(\mathbf{n})$ $\displaystyle=$
$\displaystyle\frac{\cos\theta+1}{2i}-\frac{\cos\theta-1}{2i}{\rm
e}^{-2i\varphi}$ (2.22) $\displaystyle v_{3}(\mathbf{n})$ $\displaystyle=$
$\displaystyle-\sin\theta{\rm e}^{-i\varphi}$ (2.23)

In particular we have ${\mathbf{n}}\cdot{\mathbf{v}}(\mathbf{n})=0$.

Proof. The formulas are proved from elementary computations using Lemma 2.4
and that $S_{3}\psi_{{\mathbf{n}}_{0}}=-\sf{s}\psi_{{\mathbf{n}}_{0}}$,
$S_{+}\psi_{{\mathbf{n}}_{0}}=\sqrt{2\sf{s}}\,\psi_{1,{\mathbf{n}}_{0}}$,
$S_{-}\psi_{{\mathbf{n}}_{0}}=0$. In particular we recover here a well known
property of the spin coherent states:

${\mathbf{n}}\cdot{\mathbf{S}}\psi_{\mathbf{n}}=-\sf{s}\psi_{\mathbf{n}}.$

$\square$

######  Remark 2.6.

In Lemma 2.5 the formulas have only two terms, a leading term of order
$\sf{s}$ and a second term of order $\sqrt{\sf{s}}$. In Lemma 4.5 of [2] the
authors claims that the expansion is an asymptotic power serie in
$\sf{s}^{-1}$. This contradicts our computations.

It is also convenient to write down the evolution of the spin matrices on the
Riemann sphere, denoting $S_{k}(\eta)=S_{k}(\mathbf{n})$ where
$\eta\in\mathbb{C}$ is the complex coordinate of
${\mathbf{n}}\in\mathbb{S}^{2}$.

######  Lemma 2.7.

$\displaystyle S_{3}(\eta)$ $\displaystyle=$
$\displaystyle\frac{1-|\eta|^{2}}{1+|\eta|^{2}}S_{3}+\frac{\eta
S_{+}+\bar{\eta}S_{-}}{1+|\eta|^{2}},$ $\displaystyle S_{+}(\eta)$
$\displaystyle=$
$\displaystyle-2\frac{\bar{\eta}}{1+|\eta|^{2}}S_{3}+\frac{1}{1+|\eta|^{2}}S_{+}-\frac{\eta^{2}}{1+|\eta|^{2}}S_{-}$
$\displaystyle S_{-}(\eta)$ $\displaystyle=$
$\displaystyle-2\frac{\eta}{1+|\eta|^{2}}S_{3}+\frac{1}{1+|\eta|^{2}}S_{-}-\frac{{\bar{\eta}}^{2}}{1+|\eta|^{2}}S_{+}.$

###  2.3 The classical spin space

Let us recall that the sphere $\mathbb{S}^{2}$ has a natural symplectic form:
$\sigma_{\bf n}(u,v)=(u\wedge v)\cdot{\bf n}=\det(u,v,\bf n)$, where ${\bf
n}\in\mathbb{S}^{2}$, $u,v\in T_{\bf n}(\mathbb{S}^{2})$.
Let be $H:\mathbb{R}^{3}\rightarrow\mathbb{R}$ a smooth function. Its
resriction to $\mathbb{S}^{2}$ defined an Hamiltonian vector field $X_{H}$ on
$\mathbb{S}^{2}$ satisfying $dH(Y)=\sigma_{\bf n}(Y,X_{H}),\;\forall Y\in
T_{\bf n}(\mathbb{S}^{2})$. So we get the Hamilton equation (named in this
context the Landau equation):

$\dot{\bf n}=\nabla H\wedge\bf n.$

In complex coordinates (2.10) the covariant symbol
$H_{c}(t,\eta,\bar{\eta})=\langle\psi_{\eta},\hat{H}(t)\psi_{\eta}\rangle$ of
the Hamiltonian $\hat{H}(t)=\hbar{\mathfrak{C}}(t)\cdot{\bf S}$, becomes

$H_{c}(t,\eta,\bar{\eta})=(1+|\eta|^{2})^{-1}(\mathfrak{C}_{3}(t)(1-|\eta|^{2})-({\mathfrak{C}}_{-}(t)-\bar{\eta}+{\mathfrak{C}}_{-}(t)+\eta))$

where $\mathfrak{C}_{\pm}=\mathfrak{C}_{1}\pm i\mathfrak{C}_{2}$. Recall that
the symplectic form on the Riemann sphere $\hat{\mathbb{C}}$ is
$\sigma_{c}=2i(1+|\eta|^{2})^{-2}d\eta\wedge d\bar{\eta}$. So the Hamilton
equation in $\hat{\mathbb{C}}$ becomes

$\dot{\eta}=\frac{(1+|\eta|^{2})^{2}}{2i}\partial_{\bar{\eta}}H_{c}(t,\eta,\bar{\eta}).$
(2.24)

This is the Landau-Lifschitz equation $\eqref{eq:LL}$ in complex coordinates.
Following [10] the symplectic two form satisfies
$d{\mathcal{\theta}}_{c}=\sigma_{c}$ where the one-form is
$\mathbf{\theta}_{c}=i^{-1}\left(\partial_{\bar{\eta}}Kd\bar{\eta}-\partial_{\eta}Kd\eta\right)$
and $K(\eta,\bar{\eta})=2\log(1+\eta\bar{\eta})$ is the Kähler potential for
$\hat{\mathbb{C}}$.
In particular the action $\Gamma$ for $H_{c}$ is the one form in
$\hat{\mathbb{C}}_{\eta}\times\mathbb{R}_{t}$ satisfying
$d\Gamma_{c}=\mathbf{\theta}_{c}-H_{c}dt$. So, along an Hamiltonian path in
time interval $[0,T]$, the action is given by (see Appendix):

$\gamma(T)=\int_{0}^{T}\left(\frac{\Im(\eta_{t}\dot{\bar{\eta}}_{t})}{2(1+|\eta_{t}|^{2})}-H_{c}(\eta_{t},\bar{\eta}_{t})\right)dt$
(2.25)

In the same Appendix it is proved that $\alpha(t)=\gamma(t)$ where $\alpha(t)$
is the phase given in Theorem 1.4.

###  2.4 The Schrödinger coherent states

We shall use some well known formulas concerning the Heisenberg translations
operators $\hat{T}(z)$ and coherent states.

######  Lemma 2.8.

Le be $t\mapsto z_{t}=(q_{t},p_{t})$ a $C^{1}$ path in the phase space
$\mathbb{R}^{2d}$. Then we have

$i\hbar\partial_{t}\hat{T}(z_{t})=\hat{T}(z_{t})\left(\frac{1}{2}\sigma(z_{t},\dot{z}_{t})+\dot{q}_{t}\cdot\hbar\nabla_{x}-\dot{p}_{t}\cdot
x\right).$ (2.26)

######  Lemma 2.9.

[3, chap.2] Assume that $A$ a sub-polynomial symbol. Then for every $N\geq 1$,
we have

$\hat{A}\varphi_{z}=\sum_{|\gamma|\leq
N}\hbar^{\frac{|\gamma|}{2}}\frac{\partial^{\gamma}A(z)}{\gamma!}\Psi_{\gamma,z}+{\mathcal{O}}(\hbar^{(N+1)/2}),$
(2.27)

the estimate of the remainder is uniform in $L^{2}(\mathbb{R}^{n})$ for $z$ in
every bounded set of the phase space and

$\Psi_{\gamma,z}=\hat{T}(z)\Lambda_{\hbar}{\rm Op}^{w}_{1}(z^{\gamma})g.$
(2.28)

where $g(x)=\pi^{-n/4}{\rm e}^{-|x|^{2}/2}$, ${\rm Op}^{w}_{1}(z^{\gamma})$ is
the 1-Weyl quantization of the monomial :
$(x,\xi)^{\gamma}=x^{\gamma^{\prime}}\xi^{\gamma^{\prime\prime}}$,
$\gamma=(\gamma^{\prime},\gamma^{\prime\prime})\in\mathbb{N}^{2d}$. In
particular ${\rm Op}^{w}_{1}(z^{\gamma})g=P_{\gamma}g$ where $P_{\gamma}$ is a
polynomial of the same parity as $|\gamma|$.

##  3 The Schrödinger equation for the spin-orbit interaction

Recall our reduced Hamiltonian
$\hat{H}(t)=\hbar\widehat{\mathfrak{C}}(t)\cdot{\bf S}$ and the Schrödinger
equation

$(i\hbar\partial_{t}-\hat{H}(t))\Psi(t)=0,\;\Psi(0)=\varphi_{z_{0}}\otimes\psi_{{\mathbf{n}}_{0}}.$
(3.1)

Let us consider the following ansatz

$\Psi_{\rm app}(t)={\rm
e}^{\frac{i}{\hbar}\gamma(t)}\varphi_{z(t)}^{\Gamma(t)}\otimes\psi^{(\sf{s})}_{{\bf
n}(t)}$

such that for $\hbar^{\delta}\sf{s}=\kappa>0$ and some $\mu>0$ we have:

$(i\hbar\partial_{t}-\hat{H}(t))\Psi_{\rm app}(t)=O(\hbar^{1+\mu})\;\;{\rm
for}\;\;\hbar\rightarrow 0$ (3.2)

hence from Duhamel formula we should have

$\|\Psi(t)-\Psi_{\rm app}(t)\|=O(\hbar^{\mu}).$

If $\mathfrak{C}(t)$ depends only on time $t$ then the ansatz $\eqref{ans}$
gives an exact solution for some $\gamma(t)$, ${\mathbf{n}}_{t}$ [2].

######  Proposition 3.1.

Let us assume that $\hat{H}(t)=\hbar{\mathfrak{C}}(t)\cdot{\bf S}$, where
${\mathfrak{C}}(t)$ depends only on time. Then the ansatz (3.2) is exact for
any $\sf{s}$. More explicitly we have

$\Psi_{\rm app}(t)={\rm
e}^{i\sf{s}\alpha(t)}\varphi_{z_{0}}\otimes\psi^{(\sf{s})}_{{\bf n}(t)}$ (3.3)

The classical spin motion $\bf{n(t)}$ follows the Landau-Lifshitz equation

$\dot{\mathbf{n}}_{t}={\mathfrak{C}}(t)\wedge{\mathbf{n}}_{t}.$ (3.4)

and the phase $\alpha(t)$ is the classical spin action computed in (2.25).

Proof. This is well known (see a proof in Appendix B).

##  4 The regime $\sf{s}=c\hbar^{-\delta}$, $0<\delta<1$

In this section we give a proof of Theorem 1.4. We have two small parameters
$\hbar$ and $\kappa:=\sf{s}\hbar=c\hbar^{1-\delta}$.
In this regime we shall prove that the formula (1.5) is still valid with the
error estimate $O(\sqrt{\hbar}+\kappa)$.
Let us consider the ansatz

$\Psi^{\sharp}(t)={\rm
e}^{\frac{i}{\hbar}S(t,t_{0})+i\sf{s}\alpha(t)}\varphi_{z(t)}^{\Gamma(t)}\otimes\psi^{(\sf{s})}_{{\bf
n}(t)}.$ (4.1)

As for the scalar Schrödinger equation (see [3], Chap. 4) we shall compute a
path $t\mapsto(z(t),{\mathbf{n}}(t))$, a covariance matrix $\Gamma(t)$ and
phases $S(t,t_{0}),\alpha(t,t_{0})$ such that (1.5) is satisfied.
It is enough to prove the result for the interaction propagator
${\mathcal{V}}(t,t_{0})$ such that the orbital Hamiltonian $H_{0}(t)$ is
absent. The full result is obtained by applying the scalar propagator of the
orbital motion.
It is enough to prove, for the norm in
$L^{2}(\mathbb{R}^{d},{\mathcal{H}}_{2\sf{s}+1})$, we have

$i\hbar\partial_{t}\Psi^{\sharp}(t)=\hat{H}(t)\Psi^{\sharp}(t)+O(\hbar(\sqrt{\hbar}+\kappa)).$

By standard computations we get asymptotic expansions in $\hbar$ for each term
$i\hbar\partial_{t}\Psi^{\sharp}(t)$ and $\hat{H}(t)\Psi^{\sharp}(t)$ mod an
error $O(\hbar(\sqrt{\hbar}+\kappa))$.
Recall the notations

$\displaystyle\varphi^{\Gamma}_{z}$ $\displaystyle=$
$\displaystyle\hat{T}(z)\Lambda_{\hbar}g^{\Gamma}_{0},\;g_{0}(x)=(\pi)^{-d/4}c_{\Gamma}{\rm
e}^{i<x,\Gamma x>},\Lambda_{\hbar}f(x)=\hbar^{-d/4}f(x/\sqrt{\hbar})\;\;{\rm
and}$ $\displaystyle\psi_{\bf n}$ $\displaystyle=$ $\displaystyle
T(g_{\mathbf{n}})\psi_{{\bf n}_{0}}.$ (4.2)

Using (2.26), (2.2), Lemma 2.3, Lemma 2.9, and Taylor formula around $z_{t}$
at the order 3, the Schrödinger equation for the ansatz (3.2) is transformed
as follows.

$\displaystyle\left(\frac{1}{2}\sigma(z_{t},\dot{z}_{t})-\partial_{t}S(t)-\sf{s}\hbar\dot{\alpha}(t)+\sqrt{\hbar}(\dot{q}_{t}\cdot\nabla_{x}-\dot{p}_{t}\cdot
x)\right)g_{0}\otimes\psi_{{\mathbf{n}}_{0}}+$ (4.3) $\displaystyle
g_{0}\otimes\hbar(A\dot{\theta}+B\dot{\varphi})\psi_{{\mathbf{n}}_{0}}$
$\displaystyle=$ $\displaystyle\sum_{1\leq k\leq
3}\left(\mathfrak{C}_{k}(t,z_{t})+\sqrt{\hbar}\nabla_{X}\mathfrak{C}_{k}(t,z_{t}){\rm
Op}^{w}_{1}(X)\right)g_{0}\otimes\hbar
S_{k}({\mathbf{n}}_{t})\psi_{{\mathbf{n}}_{0}}+O(\hbar(\sqrt{\hbar}+\kappa)).$

But we have

$(\dot{q}_{t}\cdot\nabla_{x}-\dot{p}_{t}\cdot x)g_{0}=(i\dot{q}-\dot{p})\cdot
xg_{0},\;\;{\rm and}\;\;{\rm Op}^{w}_{1}(a\cdot X)g_{0}=(\alpha+i\beta)\cdot
xg_{0}$

where $a\cdot X=\alpha\cdot x+\beta\cdot\xi$. From Lemma 2.3 we have

$A\dot{\theta}+B\dot{\varphi})=-\sqrt{2\sf{s}}\frac{\dot{\eta}_{t}}{1+|\eta_{t}|^{2}}\psi_{1,{\mathbf{n}}_{0}}+\sf{s}\frac{|\eta_{t}|^{2}}{1+|\eta_{t}|^{2}}\left(\frac{\dot{\eta}_{t}}{\eta_{t}}-\frac{\dot{\bar{\eta_{t}}}}{\bar{\eta}_{t}}\right)\psi_{{\mathbf{n}}_{0}}.$
(4.4)

From Lemma 2.7 we have also

$\displaystyle S_{3}(\eta)\psi_{{\mathbf{n}}_{0}}$ $\displaystyle=$
$\displaystyle-\sf{s}\frac{1-|\eta|^{2}}{1+|\eta|^{2}}\psi_{{\mathbf{n}}_{0}}+\sqrt{2\sf{s}}\frac{\eta}{1+|\eta|^{2}}\psi_{1,{\mathbf{n}}_{0}},$
$\displaystyle S_{+}(\eta)\psi_{{\mathbf{n}}_{0}}$ $\displaystyle=$
$\displaystyle
2\sf{s}\frac{\bar{\eta}}{1+|\eta|^{2}}\psi_{{\mathbf{n}}_{0}}+\sqrt{2\sf{s}}\frac{1}{1+|\eta|^{2}}\psi_{1,{\mathbf{n}}_{0}}$
$\displaystyle S_{-}(\eta)\psi_{{\mathbf{n}}_{0}}$ $\displaystyle=$
$\displaystyle
2\sf{s}\frac{\eta}{1+|\eta|^{2}}\psi_{{\mathbf{n}}_{0}}-\sqrt{2\sf{s}}\frac{{\bar{\eta}}^{2}}{1+|\eta|^{2}}\psi_{1,{\mathbf{n}}_{0}}$

Now we get the equations to compute the ansatz by identifying the coefficients
of $(\hbar,\kappa)$ in (4.3). Easy computations give the following results.

  * •

Projection on $g_{0}\otimes\psi_{{\mathbf{n}}_{0}}$: the coefficient of
$\hbar^{0}$ gives the classical action $S(t,t_{0})$ and the coefficient of
$\kappa$ gives the spin action $\alpha(t)$. In particular. we get

$\dot{\alpha}(t)=\frac{\Im(\eta_{t}\dot{\bar{\eta}}_{t})}{2(1+|\eta_{t}|^{2})}-H_{c}(\eta_{t},\bar{\eta}_{t}),$
(4.5)

where
$H_{c}(\eta,\bar{\eta})=\langle\psi_{\eta},{\mathfrak{C}}(t)\cdot{\mathbf{S}}\psi_{\eta}\rangle$.

  * •

Projection on $xg_{0}\otimes\psi_{{\mathbf{n}}_{0}}$ determines the Hamilton
equation for the orbit $z_{t}$.

  * •

projection on $xg_{0}\otimes\psi_{1,{\mathbf{n}}_{0}}$ determines the Landau-
Lifshitz equation for ${\mathbf{n}}(t)$.

Notice that for the interaction dynamics the covariant matrix $\Gamma$ is
constant, it depends only on the orbital dynamics for $H_{0}(t)$ not in the
interaction with the spin, contrary to the orbital motion when
$\kappa\gg\sqrt{\hbar}$.
Using the same method as in the scalar case considered in [3], Chapter 4 we
can complete the proof of Theorem 1.4.

##  5 The regime $\hbar\sf{s}=\kappa$=constant

The ansatz 3.2 is very natural when considering the classical analogue of
(3.1). We assume $\hbar\sf{s}=\kappa>0$ and we keep $\hbar$ as our semi-
classical parameter. For simplicity assume that $\kappa=\frac{1}{2}$.
The symbol of $\hat{H}(t)=\widehat{\mathfrak{C}}(t)\cdot\hat{\bf S}$, is
$H(t,q,p;{\bf n})=-\kappa\mathfrak{C}(t,q,p)\cdot{\bf n}$, defined on
$T^{*}(\mathbb{R})\times\mathbb{S}^{2}$, (recall that the covariant symbol of
${\bf S}$ is $-s{\bf n}$, see [3], prop. 90) we get the classical system of
equations

$\displaystyle\dot{q}$ $\displaystyle=$
$\displaystyle-1/2\partial_{p}\mathfrak{C}(t,q,p)\cdot{\bf n},$
$\displaystyle\dot{p}$ $\displaystyle=$ $\displaystyle
1/2\partial_{q}\mathfrak{C}(t,q,p)\cdot{\bf n},$ $\displaystyle\dot{\bf n}$
$\displaystyle=$ $\displaystyle\mathfrak{C}(t,q,p)\wedge{\bf n}.$ (5.1)

When $\mathfrak{C}(t)$ depends only on time there is no orbit interaction with
the spin and using section 2.2 we can compute the phase $\alpha(t)$ such that
(3.2) is the exact solution of (3.1). $\square$
But in the following computations we shall see that (3.2) is not possible if
$\nabla_{X}{\mathfrak{C}}(t,X)\neq 0$.
For simplicity we assume here that for $1\leq k\leq 3$, $\mathfrak{C}_{k}(t)$
is a linear form on $\mathbb{R}^{2d}$, $\mathfrak{C}_{k}(t,X)=a_{k}(t)\cdot
X=\alpha_{k}(t)\cdot x+\beta_{k}(t)\cdot\xi$, $X=(x,\xi)$.
Let us revisit the computations of Section 4 in the particular case. Denote
${\mathfrak{C}}_{\pm}={\mathfrak{C}}_{1}\pm i{\mathfrak{C}}_{2}$.
Let us denote $(0)_{L}$ and $(0)_{R}$ the coefficient of $\hbar^{0}$ on left
and right side of (4.3). In the same way we introduce $(1/2)_{L,R}$ and
$(1)_{L,R}$. for the coefficients of $\hbar^{1/2}$ and $\hbar^{1}$. Just
compute to get

$\displaystyle(0)_{L}=\frac{1}{2}\sigma(z_{t},\dot{z}_{t})-\dot{\gamma}(t)-\frac{1}{2}\frac{|\eta_{t}|^{2}}{1+|\eta_{t}|^{2}}\left(\frac{\dot{\eta}_{t}}{\eta}-\frac{\dot{\bar{\eta_{t}}}}{\bar{\eta}_{t}}\right)$
(5.2)
$\displaystyle(0)_{R}=-\frac{1-|\eta_{t}|^{2}}{1+|\eta_{t}|^{2}}{\mathfrak{C}}_{3}(t,z_{t})-\frac{{\bar{\eta}}^{2}}{1+|\eta|^{2}}{\mathfrak{C}}_{-}(t,z_{t})-\frac{{\bar{\eta}}^{2}}{1+|\eta|^{2}}{\mathfrak{C}}_{+}(t,z_{t})$

The equation $(0)_{L}=(0)_{R}$ determine $\gamma(t)$ when $z_{t}$, $\eta_{t}$
are known.
In the linear case consider here $\nabla_{X}\mathfrak{C}(t)$ is independent on
$X$. So we have:

$\displaystyle(1/2)_{L}$ $\displaystyle=$
$\displaystyle(-i\dot{q}-\dot{p})xg_{0}\otimes\psi_{{\mathbf{n}}_{0}}-\frac{\dot{\eta}_{t}}{1+|\eta_{t}|^{2}}g_{0}\otimes\psi_{1,{\mathbf{n}}_{0}}$
$\displaystyle(1/2)_{R}$ $\displaystyle=$
$\displaystyle-\left(\frac{1-|\eta_{t}|^{2}}{1+|\eta_{t}|^{2}}(\alpha_{3}(t)+i\beta_{3}(t))+\frac{\eta_{t}}{1+|\eta_{t}|^{2}}(\alpha_{+}(t)+i\beta_{+}(t))\right.$
(5.3) $\displaystyle+$
$\displaystyle\left.\frac{\bar{\eta}_{t}}{1+|\eta_{t}|^{2}}(\alpha_{-}(t)+i\beta_{-}(t))\right)\cdot
xg_{0}\otimes\psi_{0}$
$\displaystyle+\left({\mathfrak{C}}_{3}(t)\frac{\eta_{t}}{1+|\eta_{t}|^{2}}+{\mathfrak{C}}_{-}(t)\frac{1}{1+|\eta_{t}|^{2}}\right.$
$\displaystyle-\left.{\mathfrak{C}}_{+}(t)\frac{{\bar{\eta}}^{2}_{t}}{1+|\eta_{t}|^{2}}\right)g_{0}\otimes\psi_{1,{\mathbf{n}}_{0}}$

We denote ${\mathfrak{C}}(t):={\mathfrak{C}}(t,z_{t})$. From the equation
$(1/2)_{L}=(1/2)_{R}$, using that the states $xg_{0}\otimes\psi_{0}$ and
$g_{0}\otimes\psi_{1,{\mathbf{n}}_{0}}$ are orthogonal in
${\mathcal{H}}_{2\sf{s}}$, we obtain a system of coupled equations which
determines a trajectory $(z_{t},\eta_{t})$ in
$\mathbb{R}^{2d}\times\mathbb{S}^{2}$. In particular we get again for
${\mathbf{n}}_{t}$ the Landau equation (3.4) for ${\mathfrak{C}}(t,z_{t})$ but
here the time dependent equation for $z_{t}$ depends on ${\mathbf{n}}_{t}$.
Now let us consider $(1)_{L,R}$. First we have $(1)_{L}=0$. Let us compute
$(1)_{R}$ which is a term supported by the the mod
$xg_{0}\otimes\psi_{1,{\mathbf{n}}_{0}}$.

$\displaystyle(1)_{R}$ $\displaystyle=$
$\displaystyle-\left(\frac{\eta_{t}}{1+|\eta_{t}|^{2}}(\alpha_{3}(t)+i\beta_{3}(t))+\frac{1}{1+|\eta_{t}|^{2}}(\alpha_{+}(t)+i\beta_{+}(t))\right.$
$\displaystyle+$
$\displaystyle\left.\frac{1}{1+|\eta_{t}|^{2}}(\alpha_{-}(t)+i\beta_{-}(t))\right)\cdot
xg_{0}\otimes\psi_{1,{\mathbf{n}}_{0}}.$

Then we get that $(1)_{R}\neq 0$ if $\nabla_{X}\mathfrak{C}(t)\neq 0$.

######  Remark 5.1.

The section 4.2 of [2] use in a fondamental way their Lemma 4.5 concerning the
action of the spin operators on spin coherent states. But the conclusion of
this Lemma is false as we have shown in (4). In particular the $\sqrt{\sf{s}}$
term in our Lemma 2.5 is at the origin of the wrong ansatz (3.2) in the regime
$\hbar\sf{s}=\kappa$.

##  6 More on the Dicke model

###  6.1 Preliminary computations and reductions

A simpler form of the interaction Hamiltonian for the Dicke model is

$\hat{H}_{\rm Dint}(t)=((\cos t)x+(\sin t)\hbar D_{x})\hbar S_{1}.$ (6.1)

More generally consider the Hamiltonian
$\hat{H}(t)=(\alpha(t)x+\beta(t)D_{x}))A$, where $A$ is a bounded Hermitian
operator in the Hilbert space ${\mathcal{H}}$. Let be ${\mathcal{U}}(t)$ the
propagator for the Schrödinger equation with initial data at $t=0$.

$i\partial_{t}\Psi(t)=\hat{H}(t)\Psi(t)$

Then we have the following lemma related with Campbell-Hausdorff formula.

######  Lemma 6.1.

There exist two scalar functions $c(t)$ and $\tilde{c}(t)$ such that

${\mathcal{U}}(t)={\rm e}^{-\frac{i}{2}c(t)A^{2}}{\rm e}^{-ib(t)D_{x}A}{\rm
e}^{-ia(t)xA}$ (6.2)

where $c(t)=\int_{0}^{t}\alpha(\tau)b(\tau)d\tau$,
$a(t)=\int_{0}^{t}\alpha(\tau)d\tau$, $b(t)=\int_{0}^{t}\beta(\tau)d\tau$, and

${\mathcal{U}}(t)={\rm e}^{-\frac{i}{2}\tilde{c}(t)A^{2}}{\rm
e}^{-i(b(t)D_{x}+a(t)x)A}$ (6.3)

where $\tilde{c}(t)=c(t)-a(t)b(t)$.

###### Proof.

Let us first remark that (6.3) is a direct consequence of the Campbell-
Hausdorff formula. So it is enough to prove (6.2). Let $V(t)={\rm
e}^{-ib(t)D_{x}A}{\rm e}^{-ia(t)xA}$. By a direct computation and a
commutation we get that that

$i\partial_{t}V(t)=(\dot{a}(t)xA+\dot{b}(t)D_{x})V(t)+a(t)b(t)A^{2}.$

So we get (6.2) with $\dot{c}(t)=\alpha(t)b(t)$. Now from the Campbell-
Hausdorff formula, we get (6.3) with $\tilde{c}(t)=c(t)-a(t)b(t)$. ∎

$\square$.

Let us assume that $\Psi(0,x)=\varphi_{z_{0}}(x)\psi_{n_{0}}$, the product of
a Schrödinger coherent state and a spin coherent state. Let us assume first
that $\beta=0$. Then the time evolution is given by
$\Psi(t,x)=\varphi_{z_{0}}(x){\rm e}^{-ia(t)xS_{1}}\psi_{\bf n_{0}}$. We have
seen (Proposition 3.1) that

${\rm e}^{-ia(t)xS_{1}}\psi_{\bf n_{0}}={\rm
e}^{i{\sf{s}}\gamma(t,x)}\psi_{{\bf n}(t,x)}.$

The classical spin ${\bf n}(t,x)$ is given here by

$\displaystyle n_{1}(t,x)$ $\displaystyle=$ $\displaystyle
n_{0,1}=\cos\theta_{0},$ $\displaystyle n_{2}(t,x)$ $\displaystyle=$
$\displaystyle\sin\theta_{0}\cos(\theta(t,x)),$ $\displaystyle n_{3}(t,x)$
$\displaystyle=$ $\displaystyle\sin\theta_{0}\sin(\theta(t,x)),$

where $\theta(t,x)=\theta_{0}-a(t)x$. The phase $\alpha(t,x)$ is given by

$\dot{\alpha}(t,x)=\frac{\alpha(t)x}{1+|\eta|^{2}}\left(\frac{n_{1}(t,x)}{1+n_{3}(t,x)}+|\eta|^{2}\frac{n_{2}(t,x)}{1+n_{3}(t,x)}\right)$
(6.4)

with $\gamma(t_{0},x)=0$ and $|\eta|^{2}=\frac{1-n_{3}^{2}}{(1+n_{3})^{2}}$.
So we have the formula

$\Psi(t,x)=\varphi_{z_{0}}(x){\rm e}^{i{\sf{s}}\gamma(t,x)}\psi_{{\bf
n}(t,x)}.$

If $\sf{s}$ is frozen ($\hbar$-independent ) we can use the Taylor formula and
we get

$\|\Psi(t)-\varphi_{z_{0}}{\rm e}^{i{\sf{s}}\gamma(t,q_{0})}\psi_{{\bf
n}(t,q_{0})}\|_{L^{2}(\mathbb{R},{\mathcal{H}}_{\sf{s}})}=O(\sqrt{\hbar})$

which is a compatible with the ansatz (3.2) for $\mu=1/2$.

For $\sf{s}\hbar=\kappa>0$ we also consider Taylor expansion.

$\gamma(t,x)=\gamma(t,q_{0})+\partial_{x}\gamma(t,q_{0})(x-q_{0})+\partial_{x}^{2}\gamma(t,q_{0})\frac{(x-q_{0})^{2}}{2}+O(|x-q_{0}|^{3})$

Hence

$\varphi_{z_{0}}(x){\rm e}^{i{\sf{s}}\gamma(t,x)}=\varphi_{z_{0}}(x){\rm
e}^{i{\sf{s}}(\gamma(t,q_{0})+\partial_{x}\gamma(t,q_{0})(x-q_{0})+\partial_{x}^{2}\gamma(t,q_{0})\frac{(x-q_{0})^{2}}{2})}+O(\kappa\sqrt{\hbar})$

in this formula we see that the spin motion add a momentum to the orbit motion
(the linear term in $(x-q_{0})$ and a quadratic contribution to the Gaussian.
But the Taylor argument to eliminate the $x$-dependence does not work for the
spin part $\psi_{{\bf n}(t,x)}$.
The reason is the following. From explicit formulas [3], we know that

$|\langle\psi_{\bf n},\psi_{\bf m}\rangle|={\rm e}^{\sf{s}\log(1-\frac{|{\bf
n}-{\bf m}|^{2}}{4})},$ (6.5)

hence we have

$\|\psi_{\bf n}-\psi_{\bf m}\|^{2}\approx 2(1-{\rm e}^{-s\frac{|\bf n-\bf
m|^{2}}{4}})$

With the localization by $\varphi_{z_{0}}$ for $x$ in a neighborhood of
$q_{0}$ we have $|\bf n(t,x)-\bf n(t,q_{0})|$ of order $\hbar$. Assume that
$0<\theta_{0}<\pi/2$ and $f=1$. Then there exist $c_{1}>,c_{2}>0$ such that
for $t>0$ we have

$\|\psi_{\bf n(t,x)}-\psi_{\bf n(t,q_{0})}\|^{2}\geq c_{1}(1-{\rm
e}^{-c_{2}\sf{s}t^{2}|x-q_{0}|^{2}}).$

Using that $\sf{s}=\frac{\kappa}{\hbar}$, we get with $c_{3}>0$, for $t_{0}>0$
small enough,

$\|\varphi_{z_{0}}(x)(\psi_{\bf n(t,q_{0})}-\psi_{{\bf
n}(t,x)})\|^{2}_{L^{2}(\mathbb{R}_{x},{\mathcal{H}}_{2\sf{s}+1})}\geq
c_{3}\kappa t^{2},\;\forall t\in]0,t_{0}].$

This shows that the classical spin ${\bf n}(t,x)$ has to depend not only on
the orbit but also on the position $x$ on the orbit which is not compatible
with the ansatz (3.2).

Another and more accurate way to understand the last computations is related
with an entanglement-decoherence phenomenon for the time evolution of the
initial state $\Psi(0)=\varphi_{z_{0}}\otimes\psi_{\bf n_{0}}$, when the
interaction with a large spin system is switched on.
We shall see that for $t\in[0,t_{0}],\kappa>0$ and
$\sf{s}=\frac{\kappa}{\hbar}$, then $\Psi(t,x)$ is an entangled state which
means that it is not possible to have a decomposition like
$\Psi(t)=\varphi(t)\otimes\psi(t)$ with $\varphi(t)\in L^{2}(\mathbb{R})$ and
$\psi(t)\in{\mathcal{H}}_{2\sf{s}+1}$.
We use the partial traces in the Hilbert space
$L^{2}(\mathbb{R},\mathbb{C})\otimes{\mathcal{H}}_{2\sf{s}+1}=L^{2}(\mathbb{R},{\mathcal{H}}_{2\sf{s}+1})$.
Recall that in a tensor product of Hilbert spaces
${\mathcal{H}}={\mathcal{H}}_{1}\otimes{\mathcal{H}}_{2}$, for a trace class
operator $A$ in ${\mathcal{H}}$, the partial trace of $A$ on
${\mathcal{H}}_{2}$ is the unique trace class operator in ${\mathcal{H}}_{1}$,
denoted ${\rm tr}_{{\mathcal{H}}_{2}}(A)$, such that the following Fubini
identity is satisfied for any bounded operator $B$ on ${\mathcal{H}}_{1}$,

${\rm tr}_{{\mathcal{H}}_{1}}({\rm tr}_{{\mathcal{H}}_{2}}(A)B)={\rm
tr}_{\mathcal{H}}(A(B\otimes{\mathbb{I}}_{{\mathcal{H}}_{2}})).$

We shall use the following invariance property: if $U_{k}$ are invertible
operators in ${\mathcal{H}}_{k}$, and $U=U_{1}\otimes U_{2}$, then we have

${\rm tr}_{{\mathcal{H}}_{2}}(U^{-1}AU)=U_{1}^{-1}{\rm
tr}_{{\mathcal{H}}_{2}}(A)U_{1}.$

Notice that if ${{\mathcal{H}}_{2}}=\mathbb{C}$ then we have
${\mathcal{H}}_{1}={\mathcal{H}}_{1}\otimes\mathbb{C}$ and ${\rm
tr}_{\mathbb{C}}(A)=A$.
Let
$\Psi=\psi_{1}\otimes\psi_{2}\in{\mathcal{H}}_{1}\otimes{\mathcal{H}}_{2}$.
and denote by $\Pi_{\Psi}$ the orthogonal projector on $\Psi$. Then we have
${\rm tr}_{{\mathcal{H}}_{2}}\Pi_{\Psi}=\Pi_{\psi_{1}}$.
Physical interpretation: if $A$ is a density matrix in ${\mathcal{H}}$ (non
negative operator with trace 1) then ${\rm tr}_{{\mathcal{H}}_{2}}(A)$ is also
a density matrix in ${\mathcal{H}}_{1}$ which represents the state of the sub-
system ${\mathcal{H}}_{1}$.
Suppose that the total system satisfies a Schrödinger equation with an initial
state $\Psi_{0}=\psi_{1}\otimes\psi_{2}$ (pure state in ${\mathcal{H}}$). For
time $t>0$ the density matrix $\Pi_{\Psi(t)}$ of the total system is pure but
the density matrix ${\rm tr}_{{\mathcal{H}}_{2}}(\Pi_{\Psi(t)})$ of the
subsystem in ${\mathcal{H}}_{1}$ is not necessary a pure state because it is
not isolated (decoherence ). When the rank of ${\rm
tr}_{{\mathcal{H}}_{2}}(\Pi_{\Psi(t)})$ is $\geq 2$ the sub-system (1) can
occupy at least two orthogonal pure states with probabilities in $]0,1[$, like
for the Schrödinger cat.

Let be $\Psi\in
L^{2}(\mathbb{R},\mathbb{C})\otimes{\mathcal{H}}_{2\sf{s}+1})\simeq
L^{2}(\mathbb{R},{\mathcal{H}}_{2\sf{s}+1})$. A simple computation gives

${\rm
tr}_{{\mathcal{H}}_{2\sf{s}+1}}(\Pi_{\Psi})f(x)=\int_{\mathbb{R}}f(y)\langle\Psi(y),\Psi(x)\rangle_{{\mathcal{H}}_{2\sf{s}+1}}dy,\;\;\forall
f\in L^{2}(\mathbb{R}).$

In the same way we have also

${\rm
tr}_{L^{2}(\mathbb{R})}u=\int_{\mathbb{R}}\langle\Psi(x),u\rangle_{{\mathcal{H}}_{2\sf{s}+1}}\Psi(x)dx,\;\;\forall
u\in{{\mathcal{H}}_{2\sf{s}+1}}.$

The orbital density matrix $\rho_{\rm O}(t):={\rm
tr}_{{\mathcal{H}}_{2\sf{s}+1}}(\Pi_{\Psi(t)})$ has the integral kernel
$K(t,x,y)={\rm e}^{i\sf{s}(\gamma(t,x)-\gamma(t,y)}\tilde{K}(t,x,y)$ where

$\tilde{K}(t,x,y)=\overline{\varphi_{z_{t}}(x)}\varphi_{z_{t}}(y)\langle\psi_{{\bf
n}(t,x)},\psi_{{\bf n}(t,y)}\rangle_{{\mathcal{H}}_{2\sf{s}+1}}.$ (6.6)

So we have

${\rm tr}(\rho_{O}(t)^{2})=\int_{\mathbb{R}^{2}}|\tilde{K}(t,x,y)|^{2}dxdy.$

Recall that $\hat{H}(t)=\alpha(t)xS_{1}$.

######  Lemma 6.2.

There exists $c_{0}>0$ such that for any $0<\mu<1/2$ we have

$\int_{\mathbb{R}^{2}}|\tilde{K}(t,x,y)|^{2}dxdy\leq(1+c_{0}\kappa
t^{2})^{-1/2}+O(\hbar^{\mu})$ (6.7)

Notice that $\hat{K}(t)$ is a non negative operator of trace 1. So let be
$\lambda_{j}(t)$ the eigenvalues of $\rho_{O}(t)$ (in decreasing order with
multiplicities). So if $t>0$, $\kappa>0$ and $\hbar$ small enough, we get from
the Lemma that

$\sum_{j}\lambda_{j}^{2}(t)<\sum_{j}\lambda_{j}(t)=1$

hence $\hat{K}(t)$ has at least two eigenvalues $<1$ when the spin interaction
is switched on (recall that $\hat{K}(0)$ is the projector on a pure state).

Proof of Lemma 6.2.
We use (6.5) to compute the Schwartz kernel (6.6) and Lemma 6.3
So we get for any $\mu>0$,

$\int_{\mathbb{R}^{2}}|\tilde{K}(t,x,y)|^{2}dxdy\leq(\pi\hbar)^{-1}\int_{\\{u^{2}+v^{2}\leq
r_{0}\\}}{\rm e}^{-\frac{1}{\hbar}(u^{2}+v^{2}+c_{1}\kappa
t^{2}(u-v)^{2})}dudv$

Then a direct computation gives the estimate:

$\int_{\mathbb{R}^{2}}|\tilde{K}(t,x,y)|^{2}dxdy\leq(1+2c_{1}\kappa
t^{2})^{-1/2}+O(\hbar^{\mu})$

$\square$

Let us now consider the full interaction Hamiltonian for the Dicke model:
$\hat{H}_{\rm Dint}(t)=((\cos t)x+(\sin t)\hbar D_{x})\hbar S_{1}$. By a
symplectic transform this Hamiltonian is conjugate to the previous one. So
Lemma 6.2 is also true in this case.

###  6.2 Proof of Theorem 1.8

Strategy:
1) Reduce to the interaction picture with the propagator
${\mathcal{V}}(t,t_{0})$:
Denote $\Psi_{I}(t)={\mathcal{V}}(t,t_{0})\Psi_{0}$. Then we have
$\Pi_{\Psi(t)}={\mathcal{U}}_{0}(t-t_{0})\Pi_{\Psi_{I}(t)}{\mathcal{U}}_{0}(t_{0}-t)$.
Hence we have

${\rm tr}_{{\mathcal{H}}_{2s+1}}\Pi_{\Psi(t)}={\mathcal{U}}_{0}(t-t_{0}){\rm
tr}_{{\mathcal{H}}_{2s+1}}\Pi_{\Psi_{I}(t)}{\mathcal{U}}_{0}(t_{0}-t)$

But ${\mathcal{U}}_{0}(t-t_{0})$ is a unitary operator in
$L^{2}(\mathbb{R},\mathbb{C})$ so it is enough to prove the result for
$\Psi_{I}(t)$.
2) Proof of the result if $\omega_{3}=0$
Let us consider the time dependent Hamiltonian
$\hat{H}(t)=(\alpha(t)x+\beta(t)D_{x}))S_{1}$. Using (6.3), to compute partial
trace on ${\mathcal{H}}_{2\sf{s}+1}$ it is enough the consider the propagator:

$W(t):={\rm e}^{-i(b(t)\hbar D_{x}+a(t)x)S_{1}}$

Finally by a symplectic rotation $R(t)$ we get a metaplectic transformation
$\hat{R}(t)$ in $L^{2}(\mathbb{R},\mathbb{C})$ such that

$\hat{R}(t)W(t)\hat{R}^{*}(t)={\rm
e}^{\tilde{\alpha}(t)xS_{1}}:={\tilde{W}}(t).$

We have already proved above the decoherence for the evolution of the initial
state $\Psi(0,x)=\varphi_{z_{0}}(x)\psi_{n_{0}}$ by ${\tilde{W}}(t)$. So the
proof for $\omega_{3}=0$ is achieved.
3) The case $\omega_{3}\neq 0$.
Now the interaction is computed from the decoupled Hamiltonian
$\hat{K}_{0}=\hat{H}_{0}+\hbar\omega_{3}S_{3}$. So we have
$\mathcal{U}(t)=\mathcal{U}_{0}(t){\mathcal{V}}(t)$, where
$\mathcal{U}_{0}(t)={\rm e}^{-i\frac{t}{\hbar}\hat{K}_{0}}$ and the propagator
${\mathcal{V}}(t)$ must satisfy

$i\hbar\partial_{t}{\mathcal{V}}(t)=\hat{K}_{I}(t){\mathcal{V}}(t),\;\;{\mathcal{V}}(0)=\mathbb{I},$

where

$\hat{K}_{I}(t)=\left(\alpha(t)x+\beta(t)\hbar
D_{x}\right)\left(\cos(\omega_{3}t)\hat{S}_{1}+\sin(\omega_{3}t)\hat{S}_{2}\right).$

Like in the case $\omega_{3}=0$, to compute the partial trace it is enough to
consider the case $\beta(t)=0$. Hence we can conclude, like for
$\omega_{3}=0$, using Proposition 3.1, (6.5) and the following Lemma

######  Lemma 6.3.

Let us consider the Landau equation depending on the parameter
$x\in\mathbb{R}$,

$\partial_{t}{\bf n}(t)=\mathfrak{C}(t,x)\wedge{\bf n}(t).$

Assume that $\mathfrak{C}_{3}\equiv 0$,
$\nabla_{x}\mathfrak{C}_{1}(t,x_{0})\neq 0$ and $n_{3}(0)\neq 0$. Then there
exists $c_{0}>0$, $t_{0}>0$, $r_{0}>0$ such that

$|{\bf n}(t,x)-{\bf n}(t,y)|\geq c_{0}t|x-y|,\;\;{\rm
if}\;\;|x_{0}-y|+|x_{0}-x|\leq r_{0},0\leq t\leq t_{0}.$ (6.8)

###### Proof.

From the Landau-Lifshitz equation we get
$|\partial_{s}\partial_{x}n_{2}(s,u)|\geq c_{1}>0$ from $s$ and $u-x_{0}$
small enough. Then (6.8) follows. ∎

######  Remark 6.4.

It seems possible that these results coud be extended to more general spin-
orbit interaction like $\widehat{\mathfrak{C}}_{1}S_{1}$ such that
$\nabla_{X}{\mathfrak{C}}_{1}(X)\neq 0$ (principal type condition) by
constructing a unitary Fourier-integral operator $U$ such that
$U\widehat{\mathfrak{C}}_{1}U^{*}\approx\hat{x}$.

##  Appendix A Classical perturbations of Hamiltonians

Our aim here is to analyze the consequence on the trajectories of the
perturbation of order $\kappa$ in (1.8) for the critical regime
$\sf{s}\approx\hbar^{-1/2}$. More generally let us consider times dependent
smooth vector fields in $\mathbb{R}^{m}$, $F(t,X),G(t,X)$ and the differential
equations

$\dot{Y}(t)=F(t,Y(t)),\;\dot{X}(t)=F(t,X(t))+\kappa
G(t,X(t)),\;X(0=Y(0)=X_{0}.$

######  Lemma A.1.

There exists $t_{0}>0,\kappa_{0}>0$ such that for we have

$\|X(t)-Y(t)\|=\kappa t\|G(0,X_{0})\|+O(\kappa t^{2}),\;{\rm
for}\;\;0<t<t_{0},0<\kappa<\kappa_{0}.$

In particular if $G(0,X_{0})\neq 0$ we have, for $c_{2}>0$,

$\|X(t)-Y(t)\|\geq c_{2}\kappa t,\;\;{\rm for}\;\;0<t<t_{0}.$

###### Proof.

The following argument is standard to compare solutions of differential
equations.
In a first step, we get for some $c_{1}>0$, $t_{0}>0$, that we have

$\|X(t)-Y(t)\|\leq c_{1}\kappa,\;\;{\rm for}\;\;0<t<t_{0}.$

Then we use the variation equation with the linearized equation around $Y(t)$
to get

$X(t)-Y(t)=\kappa\int_{0}^{t}R(s,t)G(s,Y(s))ds+O(\kappa^{2}t^{2}).$

So we get the Lemma with $t_{0}>0$ small enough. ∎

##  Appendix B Proof of Proposition 3.1

Here we can assume $\hbar=1$.
We are using complex coordinates on the sphere for the Hamiltonian (Section
2.3):

$H_{c}(\eta,\bar{\eta})=(1+|\eta|^{2})^{-1}(\mathfrak{C}_{3}(1-|\eta|^{2})-(\mathfrak{C}_{-}\bar{\eta}+\mathfrak{C}_{+}\eta))$

Recall that $\eta=\frac{in_{2}-n_{1}}{1+n_{3}}$ for $\bf
n=(n_{1},n_{2},n_{3})$ ($\eta(0,0,-1)=\infty$)
Let us compute the time derivative of the ansatz (3.3), like in Section 5. The
classical dynamics of the spin is given by (2.24).
By a straightforward computation we get the spin phase $\alpha$:

$\alpha(t)=\int_{0}^{t}\left(\frac{\Im(\dot{\eta}\bar{\eta})}{2(1+|\eta|^{2})}-H_{c}(\eta,\bar{\eta})\right)d\tau=\gamma(t).$

## References

  * [1] F.T Arecchi and E. Courtens and R. Gilmore and H. Thomas. Atomic Coherent States in Quantum Optics Physical Review A, Vol.6, No.6, p.2211-2237, December 1972.
  * [2] J. Bolte and R. Glaser. Semi-classical propagation of coherent states with spin-orbit interaction Annales Henri Poincaré 6, p. 625-656, 2004.
  * [3] M. Combescure and D. Robert. Coherent states and applications in mathematical physics. Theoretical and Mathematical Physics. Springer, 2nd edition 2021.
  * [4] C. Fermanian-Kammerer, C. Lasser and D. Robert. Propagation of wave packets for systems presenting codimension one crossings. Comm. Math. Phys., 385(3), p. 1685-1739 (2021).
  * [5] K. Hepp and E. Lieb. On the superradiant phase transition for molecules in a quantized radiation field: the Dicke maser model. Annals of Physics. 76 (2):p.360-404 (1973).
  * [6] J. Fröhlich, A. Knowles and E. Lenzmann. Semi-Classical Dynamics in Quantum Spin Systems. Letters in Mathematical Physics, 82(2), p. 275-296 (2007).
  * [7] L. Landau and E. Lifshitz. On the theory of dispersion of magnetic permeability in ferromagnetic bodies . Phys. Zeitsch. der Sow., 8:153–169, (1935).
  * [8] Y. Kosmann-Schwarzbach. Groupes et symétries Éditions de l’École Polytechnique (2006)
  * [9] E. Lieb. A proof of an entropy conjecture of Wehrl. Communications in Mathematical Physics, 62 p. 1-13, (1978).
  * [10] E. Onofri A note on coherent state representations of Lie groups Journal of Mathematical Physics. Vol. 16, No. 5 p. 1087-1089. (1975)
  * [11] M. Roses and G. Dalla Torre Dicke model PLOS, https://doi.org/10.1371/journal.pone.0235197 Published: September 4, ’2020).


./images/

# Designing Trustful Cooperation Ecosystems is Key to the New Space
Exploration Era

Renan Lima Baima  0000-0002-6281-8153 University of LuxembourgFINATRAX
Research Group, SnT29, Avenue J.F KennedyKirchbergL-1855Luxembourg
<EMAIL_ADDRESS>,  Chovet Loïck  0000-0003-4025-9095 University of
LuxembourgSpaceR Research Group, SnT29, Avenue J.F
KennedyKirchbergL-1855Luxembourg<EMAIL_ADDRESS>,  Johannes Sedlmeir
0000-0003-2631-8749 University of LuxembourgFINATRAX Research Group, SnT29,
Avenue J.F KennedyKirchbergL-1855Luxembourg<EMAIL_ADDRESS>,  Miguel
Angel Olivares Mendez  0000-0001-8824-3231 University of LuxembourgSpaceR
Research Group, SnT29, Avenue J.F KennedyKirchbergL-1855Luxembourg
<EMAIL_ADDRESS>and  Gilbert Fridgen  0000-0001-7037-4807
University of LuxembourgFINATRAX Research Group, SnT29, Avenue J.F
KennedyKirchbergL-1855Luxembourg<EMAIL_ADDRESS>
(2024; 4 September 2023; 22 November 2023; 21 December 2023)

###### Abstract.

In the emerging space economy, autonomous robotic missions with specialized
goals such as mapping and mining are gaining traction, with agencies and
enterprises increasingly investing resources. MRS research has provided many
approaches to establish control and communication layers to facilitate
collaboration from a _technical_ perspective, such as granting more autonomy
to heterogeneous robotic groups through auction-based interactions in mesh
networks. However, stakeholders’ competing _economic_ interests often prevent
them from cooperating within a proprietary ecosystem. Related work suggests
that distributed ledger technology (DLT) might serve as a mechanism for
enterprises to coordinate workflows and trade services to explore space
resources through a transparent, reliable, non-proprietary digital platform.
We challenge this perspective by pointing to the core technical weaknesses of
blockchains, in particular, increased energy consumption, low throughput, and
full transparency through redundancy. Our objective is to advance the
discussion in a direction where the benefits of DLT from an economic
perspective are weighted against the drawbacks from a technical perspective.
We finally present a possible DLT-driven heterogeneous multirobot systems
(MRS) for map exploration to study the opportunities for economic
collaboration and competitiveness.

Autonomous Agent, Blockchain, Coordination, Coopetition, Multi-Agent Systems,
Multi-Robot Systems, Space Economy

††copyright: rightsretained††journalyear: 2024††doi:
10.1145/3639476.3639760††conference: New Ideas and Emerging Results ; April
14–20, 2024; Lisbon, Portugal††booktitle: New Ideas and Emerging Results
(ICSE-NIER’24), April 14–20, 2024, Lisbon, Portugal††price: 15.00††isbn:
979-8-4007-0500-7/24/04††ccs: Computing methodologies Distributed computing
methodologies††ccs: Applied computing Aerospace††ccs: Applied computing
Astronomy††ccs: Applied computing Information integration and
interoperability††ccs: Applied computing Cross-organizational business
processes††ccs: Computer systems organization Peer-to-peer architectures††ccs:
Computer systems organization Self-organizing autonomic computing††ccs:
Computer systems organization Robotic autonomy††ccs: Computer systems
organization External interfaces for robotics

DLT

    distributed ledger technology
ISRU

    in situ resource utilization
MRS

    multirobot systems
ESA

    european space agency

##  1\. Introduction

The development of the new space economy was bootstrapped thanks to policies
and agreements by governments and international organizations (Al-Akhar,
2019). The industry’s transition from a centralized model to a decentralized
free market system (Orlova et al., 2020) has increased the number of companies
and startups in the new space economy, with 80% of the revenue generated in
the last 16 years coming from private investments (Conn, 2021). Major space
agencies, including NASA (NASA, 2016), have outlined goals for in situ
resource utilization (ISRU), such as destination reconnaissance, mapping, and
resource acquisition. These plans aim to establish a long-term human presence
on celestial bodies (e.g., Moon and Mars) through robotic lunar missions that
utilize ISRU technology (Crawford, 2015) to perform tasks from mining to
manufacturing solar panels.

MRS are groups of robots that collaborate to complete specific tasks or
support each other based on individual capabilities. They can comprise both
homogeneous and heterogeneous groups of robots and find applications in
diverse fields such as precision farming, search and rescue, and space
exploration (Pretto et al., 2021). The main advantage of using MRS is that a
team of geographically distributed robots can outperform single robots
(Deshpande and Luntz, 2003). However, its effectiveness hinges on robust,
data-integrated coordination mechanisms that facilitate the exchange of norms,
maintain team identity, and promote group confidence (Strobel et al., 2018).
To avoid the issue of information asymmetry in coordination mechanisms and the
resulting “market for lemons” dilemma (Akerlof, 1970), it is crucial to
minimize the knowledge gap between sellers and buyers. While financial
incentives and sanctions can encourage cooperation and aid in meeting legal
and regulatory requirements, managing resources and coordinating interactions
among multiple organizations and countries can be challenging. In
_coopetitive_ MRS (Ghobadi and D’Ambra, 2011), various operators’ economic
incentives often compete, making it difficult to achieve common objectives and
ensure participation. With over 60 countries involved in space activities
(Borowitz, 2019), many actors may be reluctant to join a centralized platform
due to concerns about monopolistic or oligopolistic ownership, conflicts of
interest, and treaty violations (Leon, 2018).

As organizations increasingly collaborate to achieve shared objectives
(Harrison and St. John, 1996), deploying cross-border and cross-organization
MRS arguably becomes increasingly critical for future missions. However, the
efficiency of ISRU in such heterogeneous MRS missions depends on effective
communication and coordination among robots (Martinez Rocamora et al., 2023),
such as the location and availability of critical resources, e.g., water and
iron. The challenge of aligning the economic interests of competing
stakeholders in a shared, non-proprietary digital space is expected to shape
the new era of space exploration (Orlova et al., 2020). DLT has been proposed
to facilitate collaboration among these entities, enabling autonomous,
decentralized, peer-to-peer decision-making in MRS for space missions (De
Filippi and Leiter, 2021; Ibrahim et al., 2021). In this regard, DLT has shown
promise in previous research (Khan et al., 2019; De Filippi and Leiter, 2021).
In the context of space missions, it could allow robots to autonomously
negotiate resource usage, such as shelters, without a central authority.
However, it is essential to note that DLT is not without technical
limitations. Although it offers similar automation capabilities to centralized
platforms through smart contracts, it is generally less efficient due to the
inherent replication of transaction processing and storage (Butijn et al.,
2020; Sedlmeir et al., 2020). Consequently, deploying DLT in MRS and space
exploration missions is particularly challenging (Triantafyllou et al., 2021),
given the limited resources in such a remote environment. Additionally, the
inherent transparency of blockchains complicates data access management, as
information is either visible to all participants or none (Sedlmeir et al.,
2022).

This New Ideas and Emerging Results paper intentionally steers from presenting
an exhaustive list of applications and challenges to showcase practical
application development. It emphasizes exploring new ideas and the suitable
needs of using DLT for non-proprietary market-based coordination of MRSs in
space. Our approach is not about pioneering a specific, feasible application
in space exploration and multirobotic coordination. We revisited the market-
based MRS space mapping case presented by Dias et al. (2006), which we have
already begun implementing to evaluate the next steps. Our primary
contribution is investigating the economic benefits and assessing potential
solutions’ technical challenges in space exploration requirements (Lorenz,
2020; Zlot et al., 2002; Garcıa et al., 2018). By leveraging DLT for
distributed open coordination, we aim to address the global cost-efficiency
problem in market-based MRS coordination (Quinton et al., 2023), advocating
for creating a trustworthy cross-organizational platform. Section 2 introduces
the foundations of DLT, including smart contracts and tokens, and reviews
existing works on DLT in MRS and space. Section 3 details the technical
challenges of using DLT in space MRS. Finally, Section 4 presents a use case
for distributed auction-based mapping exploration, focusing on the ESA-ESRIC
Space Resource Challenge (Link et al., 2021). We conclude in Section 5.

##  2\. Related Work

We draw upon systematic literature reviews on market-based approaches in MRS
(Dias et al., 2006; Quinton et al., 2023), research on space MRS (Gao and
Chien, 2017), and the application of DLT in robotics and the space industry
(Aditya et al., 2021; Ibrahim et al., 2021).

DLT gained prominence with its first application, the cryptocurrency Bitcoin
(Nakamoto, 2008). As a subset of DLT, blockchain technology is characterized
by the replicated synchronized transaction processing and a decentralized
consensus algorithm with different levels of participation, permissionless vs.
permissioned, and access, public vs. private (Butijn et al., 2020). DLT offers
novel opportunities for digital interaction in a non-proprietary digital
infrastructure, such as creating trust within consortiums and enabling new
trading streams via Fungible and Non-Fungible Tokens (NFTs). NFTs can
represent, among many other possibilities, ownership of an non-interchangeable
asset, such as collectibles or art pieces (Hartwich et al., 2023). Besides the
simple transfer of ownership, DLT also allows implementing programming logic
through uploading code known as smart contracts and invoking this code’s
exposed methods through transactions. In essence, smart contracts allow DLT to
provide the same functionality as a centralized platform, with two exceptions:
first, non-deterministic methods are not supported, and second, the smart
contract code, as well as all its inputs, intermediary results, and outputs,
when triggered through a transaction are available to all DLT nodes owing to
the replicated execution and storage of transactions (Kannengiesser et al.,
2022).

Market-based approaches have been suggested for various applications in MRS
(Quinton et al., 2023) and, specifically, space mapping (Dias et al., 2006).
These approaches often leverage auction systems as suitable tools to
efficiently decide which resources to use in a competitive space environment
(Zlot et al., 2002). Previous research suggests that robots can offer
dependable services to each other by incorporating DLT (Lonshakov et al.,
2018), thereby opening up possibilities for space exploration, such as selling
maps, facilitating decentralized decision-making, and enabling autonomous
behavior. Integrating DLT is also said to leverage agent cooperation and
communication aimed at detection, learning, and autonomous penalization
(Pittaras, 2022), improving heterogeneous robots’ capabilities allowing them
to perform complex tasks collaboratively.

Applications of DLT in space have also been suggested for recording orbital
positions, mining licenses, and managing space traffic (Consensys Space,
2018). Another suggested application of DLT in space is to support the
efficient and effective coordination of MRS missions, such as in-orbit
operations (Cordes, 2018), satellite formations, surface, and planetary
exploration. By enabling robots to communicate and make transactions and
decisions autonomously, DLT is said to support the independent operation of
individual robots and enable them to adapt to dynamic environments. Examples
of single-entity deployments of DLT that consider economic transactions
include the AIRA (Lonshakov et al., 2018) project, which introduces the
concept of “Robonomics,” where heterogeneous robots and humans can contract
and offer services and receive payment through smart contracts.

Despite advancements, earlier suggestions have focused on centralized single-
entity settings. As such, they are not considering the main reason for using
DLT in a multi-participatory and competitive space environment and the novel
challenges that may arise from it. In other words, all the platform
functionalities that related work relies on could also be provided as a
service through a dedicated provider that creates the needed digital
infrastructure for one or multiple platforms in space (Orlova et al., 2020) or
uses a decentralized peer-to-peer communication network to create such a
platform. Moreover, existing works still need to adequately address the need
for flexible information representation and autonomous robot task learning in
dynamic environments (Baima and Colombini, 2021). A better understanding of
the economic opportunities and technical challenges is indispensable to assess
and implement DLT in this context.

##  3\. Research Gap and Open Challenges

Our discussion is grounded in existing research that tackles the challenges of
DLT from various angles, such as resource consumption, performance, and
transparency (Sedlmeir et al., 2022, 2020). While Aditya et al. (2021) have
explored the general challenges that DLT faces in robotics, we focus on five
specific open problems particularly relevant to DLT in space (Ibrahim et al.,
2021) and market-based MRS (Dias et al., 2006; Quinton et al., 2023).

The first challenge is achieving high levels of autonomy in resource-limited
environments, a recurring issue in space exploration (Triantafyllou et al.,
2021; Lorenz, 2020). While ontologies and semantic web have been proposed as a
solution (Memduhoglu and Basaraner, 2018), they often face difficulties
dealing with data heterogeneity and system stability in resource-limited
settings. Furthermore, establishing a fault-proof infrastructure and a
trustless environment is crucial for autonomous coordination (Quinton et al.,
2023; Orlova et al., 2020). Given the constraints on storage and computational
capacity in space robots due to the need for temperature and radiation
hardening (Gao and Chien, 2017), thorough modeling and testing of DLT in MRS
are essential. This includes evaluating public vs. private networks,
permissionless vs. permissioned topologies, and possible ISRU-specific
consensus mechanisms, such as consensus-based auction and bundle algorithms
(Quinton et al., 2023). The most advanced existing solutions for heterogenous
MRS, such as (Schuster et al., 2020) and (Dettmann et al., 2022), assume a
fully cooperative system. Evaluating a coopetitive system remains an open
problem.

The second challenge involves the universal cost/reward function for MRS in
market-based coordination (Quinton et al., 2023). Despite various proposals to
improve bid valuation, such as distance-based metrics or performance indexes,
the computational complexity of combinatorial auctions still needs to improve
(Dias et al., 2006). Consequently, researchers have explored alternatives,
such as clustering from single-item auctions to consensus-based bundle
algorithms. However, current solutions often rely on proprietary frameworks,
needing more transparency and trustless requirements for a digital platform in
the emerging space economy (Orlova et al., 2020). To optimize ISRU global
efficiency, companies must consider outsourcing mapping data to other MRS
companies willing to sell instead of individually mapping it.

Thirdly, public narratives about blockchain technology are vital for its
acceptance in governing outer space (De Filippi and Leiter, 2021), and
misconceptions about high energy consumption have been a barrier (Sedlmeir et
al., 2020). The replicated processing of transactions storage using blockchain
still increases the total resource consumption in terms of bandwidth,
computing power, and electricity by a factor of $N$ in a blockchain network of
$N$ nodes compared to a centralized platform, without taking necessary backups
into account (Sedlmeir et al., 2020). Nevertheless, design decisions such as
not using proof of work as a consensus mechanism and approaches like sharding,
roll-ups, and off-chain payment channels can somewhat mitigate these
inefficiencies (Strobel et al., 2018; Sedlmeir et al., 2020; Lonshakov et al.,
2018; Hartwich et al., 2023). Moreover, further investigation is needed to
understand how to maintain trust and collaboration in a _coopetitive_ economy
(Ghobadi and D’Ambra, 2011).

Fourth, for a comprehensive evaluation of opportunities and challenges, DLT
research should also investigate alternatives, such as solutions that propose
a cloud-based system that could be used in spacecraft networks to address data
complexity and heterogeneity (Baldor et al., 2013). These alternatives require
further investigation, especially concerning latency sensitivity in DLT
solutions for satellite communication (Ibrahim et al., 2021). Primarily,
latency characteristics should be investigated, as DLT is known to be highly
latency sensitive despite the safety guarantees even in non-synchronous
environments (Guggenberger et al., 2022; Gervais et al., 2016).

Lastly, recent works have demonstrated the potential of using artificial
intelligence (AI) agents to autonomously learn incremental affordances (Baima
and Colombini, 2021) and achieve Pareto efficiency and improve equality and
productivity in competitive and dynamic market simulations (Zheng et al.,
2020). While these results promise to promote universal shared efficient
knowledge, there needs to be more comprehensive coverage and insights on
autonomous economic interactions in the existing literature on machine
economies in general (Hartwich et al., 2023) and DLT in heterogeneous space
MRS in specific. DLT has been suggested to manage heterogeneous space systems
and efficiently support decentralized behavior (Ferrer et al., 2018). The
increasing interest from the scientific community and industry participants
(Weinzierl and Sarang, 2021) also supports the need for further research on
cost-effective MRS. Traditional optimization techniques have also been used to
achieve decentralized, shared economies without DLT (Pilgerstorfer and
Pournaras, 2017); further performance comparisons are needed to understand the
approximation heuristics of decentralized combinatorial optimization systems.

##  4\. Research Plan

By outlining the methodology, objectives, and critical components of our
research plan, grounded in a hypothetical scenario developed as part of the
ESA-ESRIC space resource challenge (Link et al., 2021), we aim to explore the
integration of blockchain technology extending to the proposed market-based
MRS for space exploration (Dias et al., 2006).

Figure 1. Scelestial mapping scenario, where each zone color represents a
company’s collaboration

In the given scenario, depicted in Figure 1, six main actors are involved: the
Service Orderer (SO), an Earth-based organization responsible for mission
directives, and five Service Providers (SP) – entities that execute the tasks
in-situ – as roles elaborated in Table 1. While the decentralized MRS
coordination concept dates back to the 1980s (Miller, 1985; Aubert et al.,
1986), recent surveys lacking related approaches (Quinton et al., 2023; Aditya
et al., 2021) indicate that the full potential of DLT in this domain still
needs to be explored. To fill the research gaps, we prioritize in our plan the
following essential requirements:

  1. i.

Network: Enable a mesh network that lets clients create job postings and
robots to communicate and coordinate trustfully via market-based tasks (Dias
et al., 2006; Quinton et al., 2023).

  2. ii.

Data Sharing Transparency: Ensure transparency in data sharing to foster a
_coopetitive_ environment with reduced information asymmetry (Ghobadi and
D’Ambra, 2011; Lorenz, 2020).

  3. iii.

Robot Agnostic System: Develop a system compatible with any robotic platform
and scalable to growing demands (Dias et al., 2006).

  4. iv.

Data Loss Resistance: Implement safeguards to ensure data integrity and
accessibility, even in the face of local system failures or network
disruptions (Orlova et al., 2020; NASA, 2016; Lorenz, 2020).

Table 1. List of service providers.

SP | Color | Robotic Fleet | Main Focus
---|---|---|---
A | Pink |  | Communications satellites,
---
Multiple antennas
| Moon-earth
---
communications
B | Blue | Medium size offline fleet of robots | Mapping
C | Green | Few robots w/ embedded sensors | Resource analysis
D | Yellow | Large fleet of small robots |  | Fast mapping with
---
less precision
E | None | A single robot | Mapping

In the scenario depicted in Figure 1, each polygon, structured using the
Goldberg polyhedron, represents a sub-map owned and traded by network members
(Goo et al., 2019). This approach reduces the coordination complexity and
allows for a fixed data size for each map segment, optimizing the data storage
on the blockchain. However, because different sections of the map may
represent distinct territories to be explored, ownership of the gathered
information is transferred to the client rather than the explorer. When a
robot creates data, it is desirable to represent it as an NFT to prove the
authenticity of the knowledge produced and shared on the network. Such would,
for instance, prevent unauthorized marketing as the original explorer, yet
excessive information about the map would render its value because other DLT
nodes could retrieve it for free.

Our initial implementation involves a decentralized system of three rovers
designed for resource identification, context analysis, and environmental
mapping. These rovers have components comparable to space-graded hardware and
can run blockchain nodes. The system uses a mesh network for communication and
centralized data processing. However, as the MRS for exploration design
decision, the ultimate aim is to transition to a fully decentralized system
supported by our successful simulations under similar Lunar network
conditions. While our research plan offers a comprehensive approach to
integrating DLT with MRS, as we argued in Section 2, a critical aspect of
future research will be to focus on optimizing the throughput and resource
consumption of blockchain nodes. The initial data organization strategy
significantly enhanced data transmission efficiency, particularly regarding
critical bandwidth usage. Yet, it is essential to investigate the specific
conditions under which DLT is most effective compared to authenticated and
accountable bilateral communication based on digital signatures.

##  5\. Conclusion

This paper has presented a comprehensive research plan for integrating DLT
with market-based heterogeneous MRS in space exploration. We have critically
examined the existing literature to identify significant research gaps and
articulated the necessity for interdisciplinary endeavors encompassing legal,
regulatory, engineering, organizational, and economic aspects. Our proposed
research agenda, while ambitious, aims to develop an MRS architecture
incorporating DLT and enabling autonomous economic decision-making among
robotic agents. We have also highlighted the unique technical challenges that
arise when deploying DLT-based systems in the harsh conditions of outer space.
While the primary focus is on space applications, the principles, and
architectures discussed have broader implications, potentially revolutionizing
economic routes that could be adapted for terrestrial applications.

It is important to note that while this paper emphasizes practical application
and feasibility, the challenges and use cases presented are not exhaustive.
Our objective is to stimulate further research and discussion in this area,
and we invite other scholars to extend, critique, or build upon our proposed
framework. By laying the groundwork for future research, we hope to contribute
to developing more efficient, transparent, and autonomous systems for space
exploration. Designing such a system integrating DLT with MRS can
significantly advance the field of software engineering, especially for
robotics and automation software, offering new avenues for innovation and
collaboration.

###### Acknowledgements.

This research was funded in part by the Luxembourg National Research Fund
(FNR) in the FiReSpARX (ref. 14783405) and PABLO (ref. 16326754) projects, and
by PayPal, PEARL grant reference 13342933. For the purpose of open access, and
in fulfillment of the obligations arising from the grant agreement, the author
has applied a Creative Commons Attribution 4.0 International (CC BY 4.0)
license to any Author Accepted Manuscript version arising from this
submission.

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  *[DLT]: distributed ledger technology
  *[MRS]: multirobot systems
  *[ISRU]: in situ resource utilization
  *[MRSs]: multirobot systems