$\phi(\kappa_{\theta}(\gamma))\geq-C_{2}\geq c{\rm
d}_{X}(\gamma,\operatorname{\mathrm{id}})-C.$

If $T\geq T_{0}$, fix a partition

$0=t_{0}<t_{1}<\dots<t_{m}=T$

with the following properties:

  1. (1)

$t_{n+1}-t_{n}\geq T_{0}$ for $n=0,\dots,m-1$,

  2. (2)

$\gamma_{n}:=f(t_{n})\in\Gamma$ for $n=0,\dots,m$ and

  3. (3)

if $0=s_{0}<s_{1}<\dots<s_{m^{\prime}}=T$ is another partition with the first
two properties, then $m^{\prime}\leq m$.

Then by Equation (16),

$\phi(\kappa_{\theta}(\gamma))=\sum_{n=0}^{m-1}\Big{(}\phi(\kappa_{\theta}(\gamma_{n+1}))-\phi(\kappa_{\theta}(\gamma_{n}))\Big{)}\geq\sum_{n=0}^{m-1}\Big{(}\phi(\kappa_{\theta}(\gamma_{n}^{-1}\gamma_{n+1}))-C_{0}\Big{)}.$

Thus to complete the proof it suffices to verify that

$\displaystyle\phi(\kappa_{\theta}(\gamma_{n}^{-1}\gamma_{n+1}))\geq
c(t_{n+1}-t_{n})+C_{0}$ (18)

for each $n=0,\dots,m-1$. Indeed, if this were the case, then

$\phi(\kappa_{\theta}(\gamma))\geq c\sum_{n=0}^{m-1}(t_{n+1}-t_{n})=cT\geq
c{\rm d}_{X}(\gamma,\operatorname{\mathrm{id}})-C.$

We will now prove Equation (18). Fix $n\in\\{0,\dots,m-1\\}$. If
$t_{n+1}-t_{n}<(B+2)T_{0}$, then Equation (17) implies that

$\phi(\kappa_{\theta}(\gamma_{n}^{-1}\gamma_{n+1}))\geq 1+C_{0}\geq
c(t_{n+1}-t_{n})+C_{0}.$

If $t_{n+1}-t_{n}>(B+2)T_{0}$, then by the maximality of the partition and the
fact that $B+2>4$, there exist $P\in\mathcal{P}$, $\eta\in\Gamma$ and
$a,b\in[t_{n},t_{n+1}]$ such that:

  1. (1)

$a\in[t_{n},t_{n}+T_{0})$, $b\in(t_{n+1}-T_{0},t_{n+1}]$ (hence $a<b$),

  2. (2)

$f(a),f(b)\in\eta P$, and

  3. (3)

$f|_{[a,b]}$ is contained in the combinatorial horoball associated to $\eta
P$.

Then by applying Equation (16) to the geodesic $t\mapsto\gamma_{n}^{-1}f(t)$,

$\displaystyle\phi(\kappa_{\theta}(\gamma_{n}^{-1}\gamma_{n+1}))$
$\displaystyle\geq\phi(\kappa_{\theta}(\gamma_{n}^{-1}f(a)))+\phi(\kappa_{\theta}(f(a)^{-1}f(b)))+\phi(\kappa_{\theta}(f(b)^{-1}\gamma_{n+1}))-2C_{0}$
$\displaystyle\geq\phi(\kappa_{\theta}(f(a)^{-1}f(b)))-2C_{0}-2C_{2},$

where the last equality holds by the definition of $C$. Since
$f(a)^{-1}f(b)\in P$ and

${\rm d}_{X}(f(a)^{-1}f(b),\operatorname{\mathrm{id}})={\rm
d}_{X}(f(a),f(b))>BT_{0},$

by Lemma 10.2, we then have

$\displaystyle\phi(\kappa_{\theta}(\gamma_{n}^{-1}\gamma_{n+1}))$
$\displaystyle\geq c_{1}{\rm
d}_{X}(f(a)^{-1}f(b),\operatorname{\mathrm{id}})-2C_{0}-C_{1}-2C_{2}$
$\displaystyle=\frac{c_{1}}{2}{\rm d}_{X}(f(a),f(b))+C_{0}+c_{1}T_{0}$
$\displaystyle\geq c(t_{n}-t_{n-1})+C_{0}.$

This completes the proof. ∎

##  Appendix A Proof of Proposition 2.3

In this appendix, we prove Proposition 2.3. We start with an observation about
the linear case.

######  Lemma A.1.

If $\mathsf{U}\subset\operatorname{\mathsf{SL}}(d,\operatorname{\mathbb{R}})$
is a connected unipotent group with Lie algebra $\mathfrak{u}$, then there
exist $C_{0}>1$ and a positive polynomial
$P:\mathfrak{u}\rightarrow\operatorname{\mathbb{R}}$ such that

$\frac{1}{C_{0}}P(Y)^{1/2}\leq\sigma_{1}(e^{Y})\leq C_{0}P(Y)^{1/2}$

for all $Y\in\mathfrak{u}$.

###### Proof.

Define $P:\mathfrak{u}\rightarrow\operatorname{\mathbb{R}}$ by

$P(Y)=\sum_{1\leq i,j\leq d}\left[e^{Y}\right]_{i,j}^{2}.$

Since $\mathfrak{u}$ is nilpotent, see [7, Section 4.8], $P$ is a polynomial.
Observe that the Euclidean norm $\left\|\cdot\right\|_{\rm
Euc}:\mathrm{End}(\operatorname{\mathbb{R}}^{d})\to\operatorname{\mathbb{R}}$
and the first singular value
$\sigma_{1}:\mathrm{End}(\operatorname{\mathbb{R}}^{d})\to\operatorname{\mathbb{R}}$
are both norms on the vector space
$\mathrm{End}(\operatorname{\mathbb{R}}^{d})$, so there exists $C_{0}>1$ such
that

$\frac{1}{C_{0}}\leq\frac{\sigma_{1}(X)}{\left\|X\right\|_{\rm Euc}}\leq
C_{0}$

for all $X\in\mathrm{End}(\operatorname{\mathbb{R}}^{d})$. Since
$P(Y)^{1/2}=\left\|e^{Y}\right\|_{\rm Euc}$ for all $Y\in\mathfrak{u}$, the
lemma follows. ∎

###### Proof of Proposition 2.3.

Proof of (1). Since $Z(\mathsf{G})$ is trivial, $\operatorname{\mathsf{U}}$ is
isomorphic to ${\rm Ad}(\operatorname{\mathsf{U}})$. Since each element of
${\rm Ad}(\operatorname{\mathsf{U}})$ is unipotent, the matrix logarithm

$\log(A)=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{(A-\operatorname{\mathrm{id}})^{n}}{n}$

is well defined on ${\rm Ad}(\operatorname{\mathsf{U}})$. So the exponential
map of $\mathsf{SL}(\mathfrak{g})$ induces a diffeomorphism ${\rm
ad}(\mathfrak{u})\rightarrow{\rm Ad}(\mathsf{U})$, which implies that the
exponential map of $\mathsf{G}$ induces a diffeomorphism
$\mathfrak{u}\rightarrow\mathsf{U}$.

Proof of (2). See for instance [37, Prop. 10.14].

Proof of (3). See for instance [42, Prop. 3.4.2].

Proof of (4). Let $d:=\dim\mathfrak{g}$ and fix a linear identification
$\mathfrak{g}=\operatorname{\mathbb{R}}^{d}$. This induces identifications
$\mathsf{GL}(\mathfrak{g})=\mathsf{GL}(d,\operatorname{\mathbb{R}})$ and
$\mathfrak{sl}(\mathfrak{g})=\mathfrak{sl}(d,\operatorname{\mathbb{R}})$.
Using the root space decomposition, we can pick our identification so that
${\rm ad}(\mathfrak{a})$ is a subgroup of the diagonal matrices in
$\mathsf{GL}(\mathfrak{g})=\mathsf{GL}(d,\operatorname{\mathbb{R}})$

Since $Z(\mathsf{G})$ is trivial, ${\rm
ad}:\mathfrak{g}\rightarrow\mathfrak{sl}(\mathfrak{g})$ is injective, so the
map

$\sigma_{1}\circ{\rm ad}:{\mathfrak{g}}\to\operatorname{\mathbb{R}}$

is a norm on $\mathfrak{g}$. Hence there exists $C_{1}>1$ such that

$\left\|X\right\|\leq C_{1}\sigma_{1}({\rm ad}(X))$

for all $X\in\mathfrak{g}$. Then, since ${\rm ad}(\mathfrak{a})$ is a subgroup
of the diagonal matrices,

$\left\|Y\right\|\leq C_{1}\sigma_{1}({\rm ad}(Y))=C_{1}\log\sigma_{1}({\rm
Ad}(e^{Y}))$

for all $Y\in\mathfrak{a}$. Hence, by the $\mathsf{KAK}$-decomposition,

$\displaystyle\left\|\kappa(g)\right\|\leq C_{1}\log\sigma_{1}({\rm
Ad}(e^{\kappa(g)}))\leq C_{1}C_{2}^{2}\log\sigma_{1}({\rm Ad}(g))$

for all $g\in\mathsf{G}$, where

$C_{2}:=\max_{k\in\operatorname{\mathsf{K}}}\,\sigma_{1}({\rm Ad}(k)).$

So by Lemma A.1 and part (3),

$\displaystyle\left\|\kappa(e^{Y})\right\|$ $\displaystyle\leq
C_{1}C_{2}^{2}\log\sigma_{1}(e^{{\rm ad}(Y)})\leq C_{1}C_{2}^{2}\log
C_{0}+\frac{C_{1}C_{2}^{2}}{2}\log P({\rm ad}(Y))$

for all $Y\in\mathfrak{u}$. Thus, to prove (4), it suffices to show that there
is some $A,a>0$ such that

$\displaystyle P({\rm ad}(Y))\leq A(1+\left\|Y\right\|)^{a}$ (19)

for all $Y\in{\mathfrak{u}}$.

Again, since $Z(\mathsf{G})$ is trivial, the map
$\left\|\cdot\right\|^{\prime}:\mathfrak{g}\to\operatorname{\mathbb{R}}$ given
by

$\left\|Y\right\|^{\prime}:=\max_{1\leq i,j\leq d}\left|[{\rm
ad}(Y)]_{i,j}\right|.$

is a norm, and so is bilipschitz to $\left\|\cdot\right\|$. At the same time,
observe that there is a polynomial function $Q$ of one variable with positive
coefficients such that

$P({\rm ad}(Y))\leq Q(\left\|Y\right\|^{\prime})$

for all $Y\in\mathfrak{u}$. Observe that there is some $B,b>0$ such that

$Q(\left\|Y\right\|^{\prime})\leq B(1+\left\|Y\right\|^{\prime})^{b}$

for all $Y\in\mathfrak{u}$, so Equation (19) holds. So (4) follows.

Proof of (5). For
$\psi\in\sum_{\alpha\in\Delta}\operatorname{\mathbb{Z}}_{\geq
0}\cdot\omega_{\alpha}$ let
$\chi_{\psi}:=\psi+\sum_{\alpha\in\Delta}\omega_{\alpha}$. By Proposition 4.6,
for each such $\psi$ there exist
$d_{\psi},N_{\psi}\in\operatorname{\mathbb{N}}$ and a irreducible
representation
$\Phi_{\psi}:\mathsf{G}\rightarrow\operatorname{\mathsf{SL}}(d_{\psi},\operatorname{\mathbb{R}})$
such that

$e^{\chi_{\psi}(\kappa(g))}=\sigma_{1}(\Phi_{\psi}(g))^{1/N_{\psi}}$

for all $g\in\mathsf{G}$. By part (3),
$\Phi_{\psi}(\mathsf{U})\subset\operatorname{\mathsf{SL}}(d_{\psi},\operatorname{\mathbb{R}})$
is unipotent with Lie algebra ${\rm d}\Phi_{\psi}(\mathfrak{u})$. So by Lemma
A.1 there exist $A_{\psi}>1$ and a positive polynomial
$P_{\psi}:\mathfrak{u}\rightarrow\operatorname{\mathbb{R}}$ such that

$\frac{1}{A_{\psi}}P_{\psi}(Y)^{1/(2N_{\psi})}\leq
e^{\chi_{\psi}(\kappa(e^{Y}))}\leq A_{\psi}P_{\psi}(Y)^{1/(2N_{\psi})}$ (20)

for all $Y\in\mathfrak{u}$. Since
$\omega_{\alpha}=\chi_{\omega_{\alpha}}-\chi_{0}$, (5) follows from Equation
(20) with

$R_{\alpha}(Y):=\frac{P_{\omega_{\alpha}}(Y)^{N_{0}}}{P_{0}(Y)^{N_{\omega_{\alpha}}}},$

$M_{\alpha}:=2N_{0}N_{\omega_{\alpha}}$ and
$C_{\alpha}:=A_{\omega_{\alpha}}A_{0}$. ∎

##  Appendix B Proof of Theorem 4.4

In this appendix we prove Theorem 4.4. As mentioned before, Theorem 4.4 was
established in [41] in the special case when
$\mathsf{G}=\operatorname{\mathsf{SL}}(d,\operatorname{\mathbb{R}})$. In the
following argument we use Proposition 4.6 to reduce to this special case.

######  Theorem B.1.

Assume $Z(\mathsf{G})$ is trivial and $\operatorname{\mathsf{P}}_{\theta}$
contains no simple factors of $\mathsf{G}$. Suppose $\Gamma\subset\mathsf{G}$
is a $\operatorname{\mathsf{P}}_{\theta}$-Anosov subgroup relative to
$\mathcal{P}$.

  1. (1)

If $X$ is a Groves–Manning cusp space for $(\Gamma,\mathcal{P})$ and
$M:=\mathsf{G}/\operatorname{\mathsf{K}}$ is a Riemannian symmetric space
associated to $\mathsf{G}$, then there exist $c>1$, $C>0$ such that

$\frac{1}{c}{\rm
d}_{M}(\gamma\operatorname{\mathsf{K}},\operatorname{\mathsf{K}})-C\leq{\rm
d}_{X}(\gamma,\operatorname{\mathrm{id}})\leq c{\rm
d}_{M}(\gamma\operatorname{\mathsf{K}},\operatorname{\mathsf{K}})+C$

for all $\gamma\in\Gamma$.

  2. (2)

If $P\in\mathcal{P}$, then $P$ is a cocompact lattice in a closed Lie group
$\mathsf{H}$ with finitely many components. Moreover,

    1. (a)

$\mathsf{H}=\mathsf{L}\ltimes\mathsf{U}$ where $\mathsf{L}$ is compact and
$\mathsf{U}$ is the unipotent radical of $\mathsf{H}$.

    2. (b)

$\mathsf{H}^{0}=\mathsf{L}^{0}\times\mathsf{U}$ and $\mathsf{L}^{0}$ is
Abelian.

Fix $\mathsf{G}$, $\operatorname{\mathsf{P}}_{\theta}$, $\Gamma$, and
$\mathcal{P}$ satisfy the assumptions of Theorem 4.4. Let
$\chi:=\sum_{\alpha\in\theta}\omega_{\alpha}$, and let
$N\in\operatorname{\mathbb{N}}$,
$\Phi:\mathsf{G}\rightarrow\operatorname{\mathsf{SL}}(d,\operatorname{\mathbb{R}})$
and
$\xi:\operatorname{\mathcal{F}}_{\theta}\rightarrow\operatorname{\mathcal{F}}_{1,d-1}(\operatorname{\mathbb{R}}^{d})$
satisfy Proposition 4.6 for $\chi$. Let $\mathsf{G}_{\star}$ denote the
Zariski closure of $\Phi(\mathsf{G})$ in
$\operatorname{\mathsf{SL}}(d,\operatorname{\mathbb{R}})$.

######  Lemma B.2.

$\Phi$ is injective, $\mathsf{G}_{\star}$ is semisimple and
$\Phi(\mathsf{G})=\mathsf{G}_{\star}^{0}$.

###### Proof.

Since $\mathsf{G}$ is semisimple, $\ker\Phi$ is either discrete or contains a
simple factor of $\mathsf{G}$. Since
$\xi:\operatorname{\mathcal{F}}_{\theta}\rightarrow\operatorname{\mathcal{F}}_{1,d-1}(\operatorname{\mathbb{R}}^{d})$
is a $\Phi$-equivariant embedding, $\ker\Phi$ must act trivially on
$\operatorname{\mathcal{F}}_{\theta}$. So
$\ker\Phi\subset\operatorname{\mathsf{P}}_{\theta}$. By assumption
$\operatorname{\mathsf{P}}_{\theta}$ contains no simple factors of
$\mathsf{G}$, so $\ker\Phi$ is discrete. Since $\mathsf{G}$ is connected, for
every $g\in\mathsf{G}$, there is a continuous path in $\mathsf{G}$ connecting
$\operatorname{\mathrm{id}}$ and $g$. Since $\ker\Phi$ is normal, this implies
that for any $h\in\ker(\Phi)$, there is a continuous path in $\ker(\Phi)$
between $h$ and $ghg^{-1}$. The discreteness of $\ker(\Phi)$ then implies that
$ghg^{-1}=h$. Since both $g$ and $h$ are arbitrary, we see that $\ker\Phi$ is
contained in the center of $\mathsf{G}$, and so $\ker\Phi$ is trivial.

By construction
$\Phi(\mathsf{G})\subset\operatorname{\mathsf{SL}}(d,\operatorname{\mathbb{R}})$
is irreducible and contains a proximal element and hence $\mathsf{G}_{\star}$
is a semisimple Lie group by [11, Lem. 2.19].

Since ${\rm d}\Phi(\mathfrak{g})$ is the Lie algebra of $\Phi(\mathsf{G})$,

${\rm Ad}(h){\rm d}\Phi(\mathfrak{g})={\rm d}\Phi(\mathfrak{g})$

for all $h\in\Phi(\mathsf{G})$. So

${\rm Ad}(h){\rm d}\Phi(\mathfrak{g})={\rm d}\Phi(\mathfrak{g})$

for all $h\in\mathsf{G}_{\star}$. Since $\mathsf{G}$ is connected,
$\Phi(\mathsf{G})$ is a connected normal subgroup of $\mathsf{G}_{\star}^{0}$,
and thus is an almost direct product of simple factors of
$\mathsf{G}_{\star}^{0}$.

Suppose for contradiction that there is a simple factor
$\mathsf{H}\subset\mathsf{G}^{0}_{\star}$ that does not lie in
$\Phi(\mathsf{G})$. Since $\Phi$ is irreducible and $\mathsf{H}$ commutes with
$\Phi(\mathsf{G})$, we may apply Schur’s lemma to deduce that
$\mathsf{H}\cong\operatorname{\mathbb{R}}$, which is impossible since
$\mathsf{G}_{\star}^{0}$ is semisimple. Thus,
$\Phi(\mathsf{G})=\mathsf{G}_{\star}^{0}$. ∎

Recall that ${\rm d}_{M}$ satisfies ${\rm
d}_{M}(g\operatorname{\mathsf{K}},\operatorname{\mathsf{K}})=\left\|\kappa(g)\right\|$
for all $g\in\mathsf{G}$. So to prove part (1) it suffices to prove the
following.

######  Lemma B.3.

If $X$ is a Groves–Manning cusp space for $(\Gamma,\mathcal{P})$, then there
exist $c>1$, $C>0$ such that

$\frac{1}{c}\left\|\kappa(\gamma)\right\|-C\leq{\rm
d}_{X}(\gamma,\operatorname{\mathrm{id}})\leq
c\left\|\kappa(\gamma)\right\|+C$

for all $\gamma\in\Gamma$.

###### Proof.

Let
$N:=\operatorname{\mathsf{SL}}(d,\operatorname{\mathbb{R}})/\mathsf{SO}(d)$ be
the symmetric space associated to
$\operatorname{\mathsf{SL}}(d,\operatorname{\mathbb{R}})$ and let
$x_{0}:=\mathsf{SO}(d)\in N$. Since $\Phi(\Gamma)$ is
$\operatorname{\mathsf{P}}_{1,d-1}$-Anosov relative to $\mathcal{P}$ and
$\ker\Phi$ is trivial, by [41, Thm. 1.7], there exist $c_{1}>1$, $C_{1}>0$
such that

$\frac{1}{c_{1}}{\rm d}_{N}(\Phi(\gamma)x_{0},x_{0})-C_{1}\leq{\rm
d}_{X}(\gamma,\operatorname{\mathrm{id}})\leq c_{1}{\rm
d}_{N}(\Phi(\gamma)x_{0},x_{0})+C_{1}$

for all $\gamma\in\Gamma$. Then there exist $c_{2}>1$ such that

$\frac{1}{c_{2}}\log\sigma_{1}(\Phi(\gamma))-C_{1}\leq{\rm
d}_{X}(\gamma,\operatorname{\mathrm{id}})\leq
c_{2}\log\sigma_{1}(\Phi(\gamma))+C_{1}$ (21)

for all $\gamma\in\Gamma$.

Lemma B.2 implies that ${\rm d}\Phi({\mathfrak{a}})$ is a Cartan subspace of
the Lie algebra of $\mathsf{G}_{\star}$. By [32, Thm. 7], ${\rm
d}\Phi({\mathfrak{a}})$ is conjugate to a subspace of the symmetric matrices
in $\mathfrak{sl}(d,\operatorname{\mathbb{R}})$, which in turn implies that
${\rm d}\Phi({\mathfrak{a}})$ is conjugate to a subspace of the diagonal
matrices. So there exists $c_{3}>1$ such that

$\frac{1}{c_{3}}e^{\sigma_{1}({\rm d}\Phi(X))}\leq\sigma_{1}(e^{{\rm
d}\Phi(X)})\leq c_{3}e^{\sigma_{1}({\rm d}\Phi(X))}$

for all $X\in\mathfrak{a}$. Since $\Phi$ is injective, so is ${\rm d}\Phi$.
Hence there exists $c_{4}>1$ such that

$\frac{1}{c_{4}}\sigma_{1}({\rm d}\Phi(X))\leq\left\|X\right\|\leq
c_{4}\sigma_{1}({\rm d}\Phi(X))$

for all $X\in\mathfrak{g}$. Finally, since $\operatorname{\mathsf{K}}$ is
compact,

$D:=\max_{k\in\operatorname{\mathsf{K}}}\sigma_{1}(\Phi(k))$

is finite.

Now if $g\in\mathsf{G}$, then by the $\mathsf{KAK}$-decomposition,

$\displaystyle\log\sigma_{1}(\Phi(g))$ $\displaystyle\leq 2\log
D+\log\sigma_{1}\left(e^{{\rm d}\Phi(\kappa(g))}\right)$ $\displaystyle\leq
2\log D+\log c_{3}+\sigma_{1}({\rm d}\Phi(\kappa(g)))$ $\displaystyle\leq
2\log D+\log c_{3}+c_{4}\left\|\kappa(g)\right\|$

and likewise

$\log\sigma_{1}(\Phi(g))\geq-2\log D-\log
c_{3}+\frac{1}{c_{4}}\left\|\kappa(g)\right\|.$

Combining these estimates with Equation (21) completes the proof. ∎

######  Lemma B.4.

If $P\in\mathcal{P}$, then $P$ is a cocompact lattice in a closed Lie subgroup
$\mathsf{H}\subset\mathsf{G}$ with finitely many components. Moreover,

  1. (1)

$\mathsf{H}=\mathsf{L}\ltimes\mathsf{U}$ where $\mathsf{L}$ is compact and
$\mathsf{U}$ is the unipotent radical of $\mathsf{H}$.

  2. (2)

$\mathsf{H}^{0}=\mathsf{L}^{0}\times\mathsf{U}$ and $\mathsf{L}^{0}$ is
Abelian.

###### Proof.

Let $\mathsf{H}_{\star}$ denote the Zariski closure of $\Phi(P)$ in
$\operatorname{\mathsf{SL}}(d,\operatorname{\mathbb{R}})$. Since
$\Phi(\Gamma)$ is $\operatorname{\mathsf{P}}_{1,d-1}$-Anosov relative to
$\mathcal{P}$, [41, Prop. 4.2 and Thm. 8.1] imply that $\Phi(P)$ is a
cocompact lattice in $\mathsf{H}_{\star}$. Moreover

  1. (1)

$\mathsf{H}_{\star}=\mathsf{L}_{\star}\ltimes\mathsf{U}_{\star}$ where
$\mathsf{L}_{\star}$ is compact and $\mathsf{U}_{\star}$ is the unipotent
radical of $\mathsf{H}_{\star}$.

  2. (2)

$\mathsf{H}^{0}_{\star}=\mathsf{L}^{0}_{\star}\times\mathsf{U}_{\star}$ and
$\mathsf{L}^{0}_{\star}$ is Abelian.

Then $\mathsf{U}:=\Phi^{-1}(\mathsf{U}_{\star})$,
$\mathsf{L}:=\Phi^{-1}(\mathsf{L}_{\star}\cap\mathsf{G}_{\star}^{0})$ and
$\mathsf{H}:=\Phi^{-1}(\mathsf{H}_{\star}\cap\mathsf{G}_{\star}^{0})$ satisfy
the lemma. ∎

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# Information break of time symmetry in the macroscopic limit

Miroslav Holeček<EMAIL_ADDRESS>New Technologies Research Center,
University of West Bohemia, Plzeň 301 00, Czech Republic

###### Abstract

The evident contrast between the time symmetry of fundamental microscopic laws
and the time asymmetry of macroscopic processes is a challenging physical
problem. The observation of unitary evolution of a general physical system by
an idealized observer whose recording and processing information is time
neutral is studied in a full information context. In the macroscopic limit, a
contradiction between observations and predictions done at different times
appears. It implies that the observer cannot read records performed at times
when the entropy of the system is larger.

††preprint: APS/123-QED

The closed macroscopic system evolves so that its entropy cannot decrease. It
has a convincing statistical explanation given by L. Boltzmann in the late
19th century Boltzmann1896 ; Lebowitz1993 ; GLTZ2019 ; Price2010 . The problem
is, however, that the time symmetry of laws of microscopic physics guarantees
that the same statistical analysis used in the opposite time direction leads
to the conclusion that the entropy cannot be lower in the past too
Lebowitz1993 ; Callender2021 . It can be illustrated on a simple example (see
Fig. 1).

Consider two observers detecting _independently_ the macroscopic state of a
gas in box at two different times. At $t_{a}$, Alice detects the gas at the
left part occupying the volume $V/2$, Bob finds it as filling the whole volume
$V$ at $t_{b}$. Bob does not know the past of the gas but using statistical
analysis he deduces that the probability that its actual microstate $x_{b}$ is
a final microstate of a process $V^{\prime}\to V$, where $V^{\prime}$ is
smaller then $V$ (e.g. $V^{\prime}=V/2$), is in order of $10^{-10^{20}}$, i.e.
it is practically impossible.

Figure 1: Immediately after the removing of the partition the system is closed
and the unitary microscopic process, $x_{a}\to x_{b}$, realizes. At $t_{a}$,
Alice predicts the macroscopic change $V/2\to V$, Bob concludes by the same
statistical arguments used at $t_{b}$ that only the process $V\to V$ was
possible.

When studying this situation in a broader context we see that, before $t_{a}$,
there must exist some operations guaranteeing the initial conditions of the
gas at $t_{a}$ Penrose2005 . Such considerations lead into the so called _past
hypothesis_ Albert2000  that the past of the system tends to states with an
extremely low entropy Lebowitz1993 ; Earman2006 . Using this hypothesis Bob
_knows_ that the past states have lower entropy and cannot derive the previous
(incorrect) result.

Namely any statistical analysis depends on attainable _information_. The past
hypothesis brings general information that prevents us from analyzing the past
situation in the same way as the future one. Hence the observer must know
beforehand in which time direction the past is. We humans know it though it is
not easy to explain why Szelag2004 ; Wackermann2005 ; WittWass2009 ;
MlodBrun2013 ; CollellFauquet2015 . In the objective physical description of
reality, however, the situation depicted at Fig. 1 should be described without
human observers – Alice and Bob could be robots (computers endowed with some
sensors). What does then mean the conflict of Alice’s observation at $t_{a}$
and Bob’s statistical prediction concerning $t_{a}$?

In this Letter, we analyze such situations from a pure information viewpoint
Jaynes1957 ; MarNorVed ; SagUed2012 ; DefJar2013 ; BarSei2014b ; ParHorSag2015
. The observer here is a robot strictly determined by doing only these
operations: it performs a measurement at time $t$, gains coarse-grained
information ${\cal I}(t)$ about the state of a system, processes it anyway,
and records the result. The question is at _which_ time this record can be
read. The answer is simple: if the observed system is macroscopic and it is
isolated between times $t$ and $t^{\prime}$ (its microstate evolves unitary)
the record can be read at $t^{\prime}$ _only_ if its entropy fulfills
$S(t^{\prime})\geq S(t)$. If $S(t^{\prime})>S(t)$ the _time symmetry is
broken_ : records written at $t$ can be read at $t^{\prime}$ while records
written at $t^{\prime}$ cannot be read at $t$.

It nicely corresponds to the fact that we remember the past but not the
future. Moreover, the results may have a relation to the quantum-mechanical
phenomenon studied in Ref. Maccone2009 : the decrease of entropy during the
evolution of a system quantum-mechanically entangled with its observer is
accompanied by erasure of records about the observed process (e.g. in
observer’s brain).

The contribution is organized as follows. First, the concepts like
”observation” and ”record” are defined by using the special model of the
information gathering and utilizing system (IGUS) Hartle2005 . Information
gathered by the IGUS defines the (information) entropy Jaynes1957  that allows
to derive an information form of the fluctuation theorem Crooks2000 ; Wang2002
; Sev2008 ; SeifHafeziJarzynski2021 , Eq. (3). In the macroscopic limit, it
implies the break of time symmetry in possibility of reading records of the
IGUS. It determines the time arrow in the direction of increasing entropy. The
problem of Loschmidt’s time reverse Loschmidt1876  (as a spin-echo experiment
Echo2016 ) is then discussed in the light of the gained result.

_Observation and records_. Consider the evolution of a physical system ${\bf
S}$ that is observed by an observer ${\cal O}$. The observer may be a human
who passively detects the varying current situation, an experimenter or a
robot connected with an experimental device measuring given actual parameters,
etc. Whichever the case, ${\cal O}$ gains immediate information ${\cal I}(t)$
about ${\bf S}$ at a time $t$ (a set of coarse-grained data concerning the
observed state of ${\bf S}$).

Information ${\cal I}(t)$ is transformed into a complex set of records that
become various physical representations of this information Landauer1999  and
can serve as a memory Wolpert1992 ; Holecek2019 : changes in human brain
cells, photos, experimental data saved at hard disks, etc. In Ref. Hartle2005
, the concept of information gathering and utilizing systems (IGUS) is
introduced to represent a simplified version of ${\cal O}$. The IGUS has $n+1$
registers. The actual information ${\cal I}(t)$ is stored in the register
$P_{0}$ so that the existing content of $P_{0}$ is relocated into $P_{1}$,
$P_{1}\to P_{2},\ldots,P_{n-1}\to P_{n}$, and the content of $P_{n}$ is
erased.

The ordered set of registers $P_{i}$ determines the time orientation: the
content of $P_{i}$ is the past with respect to that of $P_{i-1}$. We assume,
however, that the observer has no predetermined time direction (is time
neutral). That is why we introduce a time neutral IGUS that has only the
register $P_{0}$. It stores the actual information ${\cal I}(t)$. $P_{0}$
cannot serve as a memory: if information from another time, ${\cal
I}(t^{\prime})$, is detected it is automatically stored in $P_{0}$ and ${\cal
I}(t)$ is erased.

The memory of IGUS is a single storage ${M}$ in which it can record its
knowledge about ${\bf S}$. The record exists till new information is recorded
into ${M}$. The knowledge of the IGUS about ${\bf S}$ is ${\cal I}(t)$ (with
$t$ being the time of the last observation) or any information transformed
from ${\cal I}(t)$. For example, it can recalculate data included in ${\cal
I}(t)$ by the use of existing physical laws and gain information ${\cal
J}(t|t^{\prime})$ about possibly observed state of ${\bf S}$ at another time
$t^{\prime}$ (see Fig. 2). We call ${\cal J}(t|t^{\prime})$ the _prediction_
though there is no order of times $t$ and $t^{\prime}$; obviously ${\cal
J}(t|t)={\cal I}(t)$.

Figure 2: Two records in the IGUS ”brain”: information about the actual
situation (${\cal I}(t)$) is in register $P_{0}$. This record is erased
whenever a new observation of ${\bf S}$ is done. The second register, ${M}$,
is a simple memory of IGUS. It can save information ${\cal I}(t_{0})$ from an
arbitrary time $t_{0}$ or any transformation (processing) of this information.
A typical transformed information is a prediction of possible state of ${\bf
S}$ at another time.

The state of IGUS is thus defined by the content of $P_{0}$ and ${M}$, i.e.
${\cal I}(t)$ and ${\cal I}_{tr}(t_{0})$, respectively, where ${\cal
I}_{tr}(t_{0})$ is a transformed information ${\cal I}(t_{0})$ (e.g. ${\cal
I}_{tr}(t_{0})={\cal I}(t_{0})$). In the case when ${\cal I}_{tr}(t_{0})={\cal
J}(t_{0}|t)$ the IGUS records two information about ${\bf S}(t)$. The both
information are gained from correct physical observations, possible processed
by the use of valid physical laws. It implies that ${\cal I}(t)$ and ${\cal
J}(t_{0}|t)$ cannot be in contradiction.

It is worth emphasizing that the prediction is usually based on incomplete
data (it is done only on a coarse-grained knowledge of the system at a given
time moment). Hence its accuracy can be relatively low and it may have
sometime only a probabilistic character. Nevertheless, even so we can imagine
predictions that are clearly inconsistent with the observed data. Excluding
such situations (i.e. demand that the IGUS cannot be at a state $[P_{0},M]$ in
which $P_{0}$ contradicts $M$) is thus crucial for connecting a robotic IGUS
with a physical reality. Notice that it is also a leading principle in our
research of physical world – if an inconsistency is found out we must either
correct the used laws or look for a mistake in our evaluation of experiments
or observations.

_Entropy_. From an information theoretic perspective, entropy is associated
with observer’s ignorance (lack of information) as to the microstate of the
system Jaynes1957 ; Crooks1999 . In other words, the observer has some
information about the system, ${\cal I}_{S}$, and entropy is information that
is necessary to ”add” to observer’s knowledge to determine the actual
microstate, $x$. The entropy thus can be written as $S(x|{\cal I}_{S})$. The
entropy is a _state_ quantity Callen1985 ; LiebYngvason2013  what means that
information ${\cal I}_{S}$ must depend only on actual data gained at a
concrete time. Hence ${\cal I}_{S}$ is information gathered by our special
IGUS, i.e. ${\cal I}_{S}={\cal I}$.

The concrete form of ${\cal I}$ can be very diverse: it can be the actual
value of the system microstate, $x$, the value of a thermodynamic quantity
$\alpha$ (e.g. $\alpha=V$), the quantum projector
$X=\sum_{1}^{n}|\psi_{i}\rangle\langle\psi_{i}|$ describing an incomplete
knowledge about the system in a mixed state StrasWint2021 , or parameters of
any coarse-graining description BaronTVEMWG2007 . Whatever the case, we can
identify ${\cal I}$ with a subset $\Gamma({\cal I})\subset\Gamma_{\bf S}$,
where $\Gamma_{\bf S}$ is the state space of the studied system ${\bf S}$ and
the actual microstate $x\in\Gamma({\cal I})$. Using the Shannon
characterization of information Shannon1948 ; LombardiHolikVanni2016  we can
identify the entropy of the microstate of the system Crooks1999 ,

$S(x|{\cal I})=-k_{B}\ln p(x|{\cal I}),$ (1)

where $p(x|{\cal I})$ is a probability that the system is at the microscopic
state $x$ if we know that $x\in\Gamma({\cal I})$, and $k_{B}$ is the Boltzmann
constant. Eq. (1) implies that if ${\cal I}_{1}$ is more precise then ${\cal
I}_{2}$, i.e. $\Gamma({\cal I}_{1})\subset\Gamma({\cal I}_{2})$, then
$S(x|{\cal I}_{1})\leq S(x|{\cal I}_{2})$ (with the use of conditional
probabilities).

_Time evolution_. Imagine an experimental setup arranged in a closed lab with
two observers, Alice and Bob. Alice comes into the lab at $t_{a}$, gains
information ${\cal I}_{a}={\cal I}(t_{a})$ about a system ${\bf S}$ and leaves
the lab. Bob does the same at $t_{b}$ when he gains information ${\cal
I}_{b}={\cal I}(t_{b})$ (see Fig. 1). There is a single memory cell ${M}$
outside the lab. After leaving the lab, each observer reads ${M}$ and compare
its content with own observation (if it is relevant). Then she/he makes a
transformation of her/his observation (${\cal I}\to{\cal I}_{tr}$) and records
it in ${M}$. The observers are independent and can communicate only via the
record in ${M}$. Hence Alice and Bob can be formally identified with one IGUS
with a single register $P_{0}$ including either ”${\cal I}_{a}$ at $t_{a}$” or
”${\cal I}_{b}$ at $t_{b}$”.

The system ${\bf S}$ is isolated in between the times when Alice and Bob
perform their observations and its evolution is unitary: the microstate at
time $t_{a}$, $x_{a}$, is transformed to the microstate at $t_{b}$, $x_{b}$,
via a one-to-one mapping, ${\cal F}$, defined on $\Gamma_{\bf S}$, i.e.
$x_{b}={\cal F}(x_{a})$. Denote $G_{a}$ the maximal subset of $\Gamma({\cal
I}_{a})$ so that $G_{b}\equiv{\cal F}(G_{a})\subset\Gamma({\cal I}_{b})$. If
$x_{a}\in G_{a}$ then the conditional probabilities $\pi=p/P_{a\to b}$ on
$G_{a}$ and $\overline{\pi}=p/P_{b\to a}$ on $G_{b}$ fulfill

$\pi(x_{a})=\overline{\pi}({\cal F}(x_{a})),$ (2)

whereas $P_{a\to b}=\sum_{x\in G_{a}}p(x|{\cal I}_{a})$ is the probability
that information ${\cal I}_{a}$ gained by Alice at $t_{a}$ implies that Bob
gains information ${\cal I}_{b}$ at $t_{b}$, i.e. ${\cal J}(t_{a}|t_{b})={\cal
I}_{b}$ is valid with the probability $P_{a\to b}$. Similarly, $P_{b\to
a}=\sum_{x^{\prime}\in G_{a}}p(x^{\prime}|{\cal I}_{b})$ is the probability
that information ${\cal I}_{b}$ gained by Bob at $t_{b}$ implies that Alice
gains information ${\cal I}_{a}$ at $t_{a}$, i.e. ${\cal J}(t_{b}|t_{a})={\cal
I}_{a}$ with the probability $P_{b\to a}$. Eqs. (1,2) relates the
probabilities $P_{a\to b}$ and $P_{b\to a}$,

$\frac{P_{b\to a}}{P_{a\to b}}=e^{-k_{B}^{-1}\Delta S},$ (3)

where $\Delta S=S(x_{b}|{\cal I}_{b})-S(x_{a}|{\cal I}_{a})\equiv
S(t_{b})-S(t_{a})$.

In the context of IGUS it means, for example, that if the register $P_{0}$
includes ”${\cal I}_{a}$ at $t_{a}$” it can be written ”${\cal I}_{b}$ at
$t_{b}$ with the probability $P_{a\to b}$” in the register $M$. If the system
is microscopic or mesoscopic $k_{B}/|\Delta S|$ is not negligible and the
prediction has only stochastic character.

_Macroscopic limit_. The existence of macroscopic limit means that it is
possible to formulate physics in the limit $k_{B}\to 0$ so that $\Delta S$
remains nonzero. Hence $|\Delta S|/k_{B}\to\infty$ which simulates the
description of macroscopic processes when $|\Delta S|/k_{B}\sim 10^{20}$. In
the macroscopic limit, Eq. (3) implies that $P_{a\to b}=0$ whenever $\Delta
S<0$. Hence Alice must predict that observed information at $t_{b}$ must be
connected with entropy $S(t_{b})\geq S(t_{a})$ and Bob must predict that
observed information at $t_{a}$ must be connected with entropy $S(t_{a})\geq
S(t_{b})$. If $\Delta S\neq 0$ one prediction must be wrong.

It is exactly the situation depicted at Fig. 1: Alice observes (the IGUS has
in $P_{0}$) ”$V(t_{a})=V/2$”. Bob must conclude that the gas cannot occupy the
volume less then $V$ at any time, i.e. he records ”$V(t_{a})=V$” in ${M}$. If
Alice can read Bob’s record (i.e. if there exists the state of IGUS
$[P_{0},{M}]=[V(t_{a})=V/2,V(t_{a})=V]$) we get the situation when the
contents of $P_{0}$ and ${M}$ are in contradiction. On the other hand, if Bob
reads Alice’s prediction he does not indicate any controversy since the record
in ${M}$ is ”$V(t_{b})=V^{\prime}$, $V^{\prime}\geq V/2$” and Bob knows (the
IGUS has in $P_{0}$) that ”$V(t_{b})=V$”.

The only conclusion is that the IGUS cannot be in the state
$[V(t_{a})=V/2,V(t_{a})=V]$, i.e. Alice cannot read Bob’s prediction. It
breaks the symmetry between times $t_{a}$ and $t_{b}$: the record written at
$t_{a}$ can be read at $t_{b}$ but the record written at $t_{b}$ cannot be
read at $t_{a}$. This symmetry break has the general validity. Since
information can be transferred (can _flow_) into a different time only via
some records Maccone2009 ; Barbour1999  this conclusion means that information
concerning macroscopic observations of an isolated system can flow from $t$ to
$t^{\prime}$ only if $S(t^{\prime})\geq S(t)$.

This symmetry break is in agreement with our everyday observation. Information
gained at $t_{0}$ cannot flow to any time $t<t_{0}$: we cannot influence the
past. On the other hand, sending information to future (via performed records)
is a typical human activity: writing, planning, etc. The message written
morning can be read the same day afternoon. The opposite order is impossible.

It implies why we can remember only the past. Namely any memory is a record in
our brain. The record done at $t$ can be read at $t_{0}$ only if $t<t_{0}$. If
we imagine that Bob is the older Alice (i.e. Alice at $t_{b}$) then Alice can
record the actual situation observed at $t_{a}$ into ${M}$ (some cells in her
brain), i.e. ”$V(t_{a})=V/2$” in ${M}$. At $t_{b}$, Alice gains information
${\cal I}(t_{b})=V$ (has it in $P_{0}$). Since $t_{b}>t_{a}$ she can read
${M}$, i.e. she _remembers_ the situation at $t_{a}$: she knows that the gas
occupied the volume $V/2$ at $t_{a}$.

It seems, however, as a contradiction with Bob’s prediction done at $t_{b}$:
${\cal J}(t_{b}|t_{a})=V$. This prediction, however, is derived (transformed)
only from ${\cal I}(t_{b})$ corresponding to the entropy $S(x|{\cal I})$ at
$t_{b}$. But Alice (as older Bob) has information about ${\bf S}$ at $t_{b}$
that is not ${\cal I}(t_{b})$ but ${\cal I}_{\star}=({\cal I}(t_{b}),{\cal
I}(t_{a}))$. Information ${\cal I}_{\star}$, however, does not define entropy.
Eq. (3) then cannot be used (the knowledge of Alice is ${\cal I}_{\star}$ that
implies that $x_{a}\in G_{a}$ and probabilities $\pi$, $\overline{\pi}$ and
Eq. (2) must be used instead).

_Time reverse_. Notice that the direction of information flow is given only by
the difference of entropy – no aspects of microscopic dynamics play here a
role. A nice illustration is an idea of a hypothetical twin of our universe
whose microscopic state is a time reversed state of our universe. The
molecules in the box at Fig. 1 thus have opposite signs of velocites in this
universe and the dynamic operator ${\cal F}$ works as follows: ${\cal
F}(x_{b}^{R})=x_{a}^{R}$, where $x^{R}$ is the reverse of the microstate
MaesNetocny2003 . But time does not flow from $t_{b}$ to $t_{a}$ since
information can flow _only_ from $t_{a}$ to $t_{b}$ ($S(t_{a})<S(t_{b})$ is
valid in the twin universe too). The passage of time in such a twin universe
must be the same as that in ours, i.e. in the agreement with the second law of
thermodynamics Carroll2008 .

The existence of time reversed states in _our_ universe then seems to be
paradoxical. Imagine that there exists a possibility of realizing the
Loschmidt thought experiment, i.e. performing a sudden change of the sign of
velocities of all molecules, $x\to x^{R}$, by an external action with the
system at the moment when the gas occupies the whole box after the unitary
evolution with the entropy increase, i.e. at time $t_{b}$ at Fig. 1. The next
unitary evolution brings the system into the state $x_{a}^{R}$, i.e. to the
macrostate when the gas is in the left half of the box (see Fig. 3). The
passage of time (as information flow) then must have the opposite direction in
the stage depicted within the dashed box at Fig. 3. How can it be?

Figure 3: The time reversion, $R$, of the microstate at the moment when the
gas fills the whole box can bring the gas again to the left part of the box.
The process must be strictly unitary: any external perturbation changes a very
special initial condition of the second half of the process,
$x^{R}\in(G_{b})^{R}$, and no backward time evolution can happen. The standard
thermodynamic process – the spontaneous gas expansion – is not sensitive on
the perturbation: nearly any perturbed microstate tends to
$\Gamma(\alpha_{b})$.

To answer the question we must ask _who_ can observe the problematic process,
$V\to V/2$. It is important realizing that the initial conditions of _this_
process has to be prepared by the standard thermodynamic process $V/2\to V$
and the subsequent time reversion. Namely the microstate before the reversion
has to be within an extremely small, precisely given set $G_{b}$. It implies
that the preparing process has to be strictly unitary and no external
perturbation can happen.

After the reversion, the microstate of the system belongs into the small,
special set $(G_{b})^{R}$. The microstates from this set represent the gas
occupying the whole volume at the macroscopic scale. But how this macrostate
can be observed? It is necessary to break the isolation and interact with the
system. This interaction – whatever small – perturbs the initial condition,
the microstate leaves the set $(G_{b})^{R}$ and the system does not follow the
process $V\to V/2$ (and $V\to V$ is expected instead).

Hence the whole Loschmidt process must be realized in an absolute isolation
that is not realistic ECHO . If so, the observer can still observe only the
beginning and the end of the process, i.e. $V/2\to V/2$. It would be a strange
situation but it were not in contradiction with the second law of
thermodynamics since $\Delta S=0$ during this process.

_Concluding discussion_. The main idea of Boltzmann’s statistical program is
that the macrostate with a higher entropy is overwhelmingly larger then that
with a lower one. Hence the microstate wanders into this huge set in
overwhelmingly many cases Penrose2005 . There are two problematic points: (i)
a certain dependence on the concrete form of microscopic dynamics Earman2006 ,
(ii) problematic conclusions obtained in the opposite time direction.

The content of Eq. (3) is similar: the probability that ${\cal F}(x)$ or
${\cal F}^{-1}(x)$ with $x$ being a random microstate from a huge subset
$\Gamma$ of the state space ”hits” a very small target is extremely low. The
interpretation is, however, different: $P_{a\to b}$ or $P_{b\to a}$ are
probabilities of the rightness of _predictions_ of an observer who has
information _only_ from the time when $x\in\Gamma$. The result is valid for
each unitary dynamics and the inconsistency in one time direction has a deep
physical meaning in the macroscopic limit. Namely if a consistent physical
description of the macroscopic world exists the macroscopic limit cannot give
contradictory physical results.

The assumption that _records_ of predictions concerning times with lower
entropy _cannot be read_ at those times gives a consistent picture without
contradictions. The crucial (and most subtle) point of this consideration is
that Eq. (3) is valid if it concerns observers who have information _only_
from a single time moment. Pictures like Fig. 1 are deceptive: they present
the situation as if it were known at both times. But information of this kind
cannot be connected with entropy.

Entropy – as a state quantity – must be related to information concerning a
concrete time moment. The question is if entropy at $t$ can be defined also
from a prediction ${\cal J}(t_{0}|t)$ that is also information concerning $t$.
Let us define $S^{\prime}(t)\equiv S(x(t)|{\cal J}(t_{0}|t))$. Information
${\cal J}(t_{0}|t)$ is, however, only transformation of information ${\cal
I}(t_{0})$ which implies that it cannot be more precise then ${\cal
I}(t_{0})$, i.e. $S^{\prime}(t)\geq S(t_{0})$. If $S(t)<S(t_{0})$, i.e. we do
the prediction towards the past, it cannot be in accord with the observation
at $t$ since $S^{\prime}(t)$ is _always_ different from $S(t)$
($S(t)<S^{\prime}(t)$). Towards the future, $S^{\prime}(t)$ can equal $S(t)$
since $S(t)\geq S(t_{0})$.

This consideration outlines again the important role of predictions (i.e.
processed information gained from external impulses at different times). It is
worth stressing that information about a concrete event that happens at a
given time is usualy ’read’ by humans via various physical stimuli (visual,
auditory, etc.) that are recorded at different times Szelag2004 . The
permanent ’recalculation’ of records (in the sense of ${\cal I}\to{\cal
I}_{tr}$ defined above) thus appears as important element of our perception of
time.

Concerning the whole concept we may ask for the role of probabilities since
the observer may not have any idea about probabilities of possible
microstates. The answer is that Eq. (3) is used here only in the macroscopic
limit in which a concrete probability distribution does not play a role
(entropy $S(x|{\cal I})$ can be defined without using the concept of
probabilities, the crucial result that $P_{a\to b}$ or $P_{b\to a}$ tends to
zero is then equivalent to ${\cal I}\to\infty$ Holecek2021 ).

There are many other questions connected with the derived break of time
symmetry, especially the relation to the perceived time passage or the concept
of time on a relativistic spacetime block Barbour1999 ; Rovelli2004 ; BLOCK .
It would be also appealing to find an interconnection with the past hypothesis
(that can be interpreted as new additional information) including its
cosmological context Price2010 ; Earman2006 ; Carroll2008 ; Wald2006 . We also
do not explain here the mechanism why the record ’vanishes’ at times when the
entropy is lower. It might have an interesting relation to quantum-mechanical
phenomenon analyzed in Ref. Maccone2009 . Our approach may also contribute to
the question of emergence of the macroscopic world ParHorSag2015 . The
information description of the macroscopic limit can be instructive here.

_Acknowledgment_. The author is indebted to Ján Minár for his help and
constructive discussions, and to Phill Strasberg for inspirational and
critical comments concerning the foundations of thermodynamics and statistical
physics. The work is supported by the New Technologies Research Center of the
West Bohemia University in Pilsen.

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# A fiber detector to monitor ortho-Ps formation and decay

B. Rienäcker  Corresponding author<EMAIL_ADDRESS>Physics Department,
CERN, 1211 Geneva 23, Switzerland  R. S. Brusa  Department of Physics,
University of Trento, via Sommarive 14, 38123 Povo, Trento, Italy  TIFPA/INFN
Trento, via Sommarive 14, 38123 Povo, Trento, Italy  R. Caravita  TIFPA/INFN
Trento, via Sommarive 14, 38123 Povo, Trento, Italy  S. Mariazzi  TIFPA/INFN
Trento, via Sommarive 14, 38123 Povo, Trento, Italy  L. Penasa  Department of
Physics, University of Trento, via Sommarive 14, 38123 Povo, Trento, Italy
TIFPA/INFN Trento, via Sommarive 14, 38123 Povo, Trento, Italy  F. Pino
Department of Physics and Astronomy ”Galileo Galilei”, University of Padova,
Via Marzolo, 8, 35131 Padova, Italy  O. A. Ranum  Gravitation Astroparticle
Physics Amsterdam (GRAPPA), University of Amsterdam, Science Park 904, 1098 XH
Amsterdam, The Netherlands  G. Nebbia  INFN Padova, via Marzolo 8, 35131
Padova, Italy

(August 28, 2024)

###### Abstract

We describe a novel method to use a scintillating fiber detector similar to
the Fast Annihilation Cryogenic Tracking (FACT) used at the antimatter
experiment AE$\bar{\hbox{g}}$IS to monitor the presence of ortho-positronium.
A single scintillating fiber was coupled to a photomultiplier tube and
irradiated by flashes of about $6\text{\times}{10}^{6}\text{\,}$
$511\text{\,}\mathrm{keV}$ $\gamma$-rays produced by
$\approx$$10\text{\,}\mathrm{ns}$ long positron pulses. The results were used
to demonstrate the ability to track the creation and annihilation of ortho-
positronium atoms over time in cryogenic and highly magnetic environments by
using the FACT detector as a “digital calorimeter”.

###### pacs:

36.10.Dr, 78.70.Bj, 29.40.Mc

##  I Introduction

Positronium (Ps), the bound state of an electron and a positron, is a purely
leptonic matter-antimatter system with many applications in fundamental
research [1, 2, 3, 4, 5, 6, 7]. In particular, positronium can be used to
induce the charge exchange reaction with cold trapped antiprotons in order to
efficiently create antihydrogen. This was recently achieved by
AE$\bar{\hbox{g}}$IS (Antimatter Experiment: Gravity, Interferometry,
Spectroscopy) [8] and is also planned to be used by GBAR (Gravitational
Behaviour of Antihydrogen at Rest) [9]. Both experiments are located at the
Antiproton Decelerator (AD) facility at CERN. The AE$\bar{\hbox{g}}$IS
collaboration produces ground state positronium by implanting positron bunches
into a nanochanneled silicon target [10, 11]. A substantial fraction of the
long-lived positronium atoms in a triplet state (ortho-positronium with total
spin $S=1$) subsequently cools by inelastic collisions with the channel walls
and escapes from the nanoporous target at almost environmental temperature
into vacuum. Such cold Ps is then excited to Rydberg states by laser
irradiation [12] before reaching the trapped antiproton plasma and forming
antihydrogen. Such experiments require ultra-high vacuum, a homogeneous
magnetic field in the order of few Tesla and cryogenic temperatures, making it
difficult to set up an efficient detector to monitor and track positronium
creation and decay.

The commonly used technique for Ps detection, namely Single Shot Positron
Annihilaton Lifetime Spectroscopy (SSPALS) with scintillation detectors [13],
is stringent limited in narrow-spaced environments at liquid helium
temperatures with high magnetic fields and small coverable solid angles such
as is occurring at AE$\bar{\hbox{g}}$IS [14, 15]. Therefore, positronium
formation is monitored in a destructive way by laser-induced photoionization
and by imaging the ionized positrons, which are axially confined by a
$1\text{\,}\mathrm{T}$ magnetic field, with a microchannel plate and a
phosphor screen [16, 17]. Antihydrogen formation and annihilation, on the
other hand, can be monitored by means of the Fast Annihilation Cryogenic
Tracking (FACT) scintillating fiber detector surrounding the production region
[18, 19]. The FACT detector has been designed to detect pions produced by the
annihilation of antihydrogen atoms. Nevertheless, the scintillating fibers
could be used as well to non-destructively monitor the production and the
excitation of ortho-positronium by the detection of $3\gamma$ annihilation
following the in-flight decay of o-Ps in vacuum. In order to determine such a
possibility, we first ran Monte Carlo simulations of the response of such a
fiber detector to gamma radiation. We then performed experimental tests on one
of the scintillating fibers as is installed in the FACT detector and studied
its response to positron bursts from the AE$\bar{\hbox{g}}$IS positron system.

##  II A ”digital calorimeter” for o-Ps: Principle and simulations

The FACT detector of AE$\bar{\hbox{g}}$IS is made of 794 scintillating fibers
distributed in four layers with cylindrical symmetry and a length of
$200\text{\,}\mathrm{mm}$. The fiber type is _Kuraray SCSF-78M_ , multi-
cladded with $1\text{\,}\mathrm{mm}$ in diameter. There are two layers of
scintillating fibers at radial distances of $70\text{\,}\mathrm{mm}$ and
$98\text{\,}\mathrm{mm}$ from the central beam axis. Each layer consists again
of two layers, but one is shifted against the other in order to avoid blind
spots in the detection area. Per single layer, the fiber centers are
horizontally separated by $1.2\text{\,}\mathrm{mm}$ and the radial distance of
the shifted layer is increased by $0.8\text{\,}\mathrm{mm}$ (see Fig. 1).

Figure 1: Schematic of the FACT cross section. In total 794 scintillating
fibers in four layers in cylindrical geometry (inner diameter is
$70\text{\,}\mathrm{mm}$, outer diameter is $98\text{\,}\mathrm{mm}$) are used
to detect annihilation products. The Ps converter is positioned in the center,
but radially displaced by $20\text{\,}\mathrm{mm}$ to be aligned with
AE$\bar{\hbox{g}}$IS positron transfer line.

Each scintillating fiber is mechanically and optically coupled to a clear
fiber, which transports scintillation light to a Multi-Pixel Photon Counter
(MPPC, type _Hamamatsu S10362-11-100C_), which produces a small electrical
signal at its output. A hit is registered only when its signal overcomes a
given threshold. The MPPC output is connected to a fast monolithic amplifier
(_Mini-Circuits MAR-6+_), whose signal is fed into a discriminator which then
returns the time over threshold (ToT), digitized by a FPGA acquisition board
[18, 20]. The positron/Ps converter is positioned in the center of FACT, but
the radial distance from the central axis is approximately
$20\text{\,}\mathrm{mm}$ so that it is aligned with the positron transfer
line. When positrons hit the converter, ortho-positronium is created which
flies for several millimeters before it annihilates into predominately three
gamma quanta with energies ranging between 0 and $511\text{\,}\mathrm{keV}$.
This continuous o-Ps annihilation spectrum is described by the so-called Ore-
Powell formula [21], which was used for the following Monte Carlo simulation.

With the 2014 version of the PENELOPE code [22] we simulated the response of
an array of scintillating fibers in the same geometry as FACT to a discrete
Ore-Powell energy spectrum of $\gamma$-radiation. Fig. 2 shows the simulated
detector response to a delta pulse of $2\text{\times}{10}^{6}\text{\,}$ gamma
quanta originating from the target region for three different energy
thresholds. The vertical axis represents the number of gamma quanta leaving a
signal above the threshold in each fiber. Exploiting the symmetry of the
entire setup, only 1/8 of all available fibers needed to be simulated for this
proof-of-principle. They were numbered from 0 to 99, with the central fiber of
the entire array being placed at the zero position of the abscissa in the
simulation. The three different curves show the number of gamma quanta leaving
more than $50\text{\,}\mathrm{k}\mathrm{eV}\mathrm{e}\mathrm{e}$ (black
squares), $100\text{\,}\mathrm{k}\mathrm{eV}\mathrm{e}\mathrm{e}$ (white
squares) or $200\text{\,}\mathrm{k}\mathrm{eV}\mathrm{e}\mathrm{e}$ (white
circles) of energy in each fiber. The reduction of the number of counts
reflects the effect of the solid angle, which is decreasing by approximately
$70\text{\,}\mathrm{\char 37\relax}$ when going from the central fiber (#0) to
the outermost fiber (#99). This basic response signal is valid for one layer
of fibers with constant radius.

Figure 2: Simulated response of an array of scintillating fibers to a pulse of
$2\text{\times}{10}^{6}\text{\,}$ o-Ps annihilation gamma quanta with applied
energy thresholds of 50, 100 and 200 keVee, respectively.

Considering its solid angle
$\Omega\approx$0.09\text{\,}\mathrm{s}\mathrm{r}$$, the central fiber would be
irradiated by $1.4\text{\times}{10}^{4}\text{\,}$ photons originating from the
position of the Ps converter. However, the simulation result with an energy
threshold of $50\text{\,}\mathrm{k}\mathrm{eV}\mathrm{e}\mathrm{e}$ as plotted
in Fig. 2 shows that only 115 gamma quanta leave a sizeable signal inside the
fiber. This is mainly due to the intrinsic efficiency of the fiber yielding
the ability to ”switch it on”, but of course also partly due to the applied
threshold. In order to validate the simulation, we estimated this intrinsic
efficiency of one fiber at a given energy threshold by taking into account the
solid angle of the central fiber, and compared this to measurements existing
in literature for similar systems. The intrinsic efficiency of one fiber at an
energy threshold of $150\text{\,}\mathrm{k}\mathrm{eV}\mathrm{e}\mathrm{e}$ is
$0.3\text{\,}\mathrm{\char 37\relax}$ as yielded by the PENELOPE simulation.
This compares well with the efficiency of $0.23\text{\,}\mathrm{\char
37\relax}$ measured in Ref. [23] on a $1\text{\,}\mathrm{mm}$ thick plastic
scintillator for gamma radiation of the same energy regime and detection
threshold.

In order to be able to simulate the detector response to a decaying cloud of
Ps atoms, we first have to consider a typical ortho-positronium SSPALS
spectrum as is shown in Fig. 3. This spectrum was obtained by implanting a
pulse of $3\text{\times}{10}^{6}\text{\,}$ positrons with a time spread of
about $10\text{\,}\mathrm{ns}$ full-width-half-maximum (FWHM) and with a
kinetic energy of roughly $3\text{\,}\mathrm{keV}$ into a positron/Ps
converter target. The annihilation signal was recorded with a PbWO4
scintillation detector positioned about $4\text{\,}\mathrm{cm}$ away from the
Ps converter target inside a dedicated test chamber. The used positron system
and the Ps converter are described in detail elsewhere [24, 11]. As shown in
Fig. 3, the positrons hit the Ps converter target at time $t=0$, leading to an
almost instantaneous peak of positron annihilation into
$511\text{\,}\mathrm{keV}$ gamma quanta. In the absence of o-Ps formation, the
signal quickly approaches the noise level. Conversely, in the presence of o-Ps
being emitted into vacuum, the signal at times greater than
$50\text{\,}\mathrm{ns}$ is proportional to the number of gamma quanta emitted
by the decaying cloud of o-Ps as a function of time with the characteristic
ground state vacuum lifetime of $142\text{\,}\mathrm{ns}$. Due to the
exponential nature of the decay, one can calculate the number of annihilation
products occurring in any time interval of a chosen width as a function of the
initial number of ortho-Ps atoms.

Figure 3: Examples for SSPALS spectra measured with a PbWO4 scintillation
detector for an aluminum target (no Ps formation, grey curve) and for a
positron/Ps converter (Ps formation, black curve). The SSPALS spectrum in
presence of the converter shows an exponentially decreasing Ps tail with the
typical ground state vacuum lifetime of about $142\text{\,}\mathrm{ns}$. The
best exponential fit is reported as the dashed line. The slight deviation from
the theoretical Ps lifetime value comes from systematic effects such as the
non-linear detector response function at high signal amplitudes.

We then take into account the $200\text{\,}\mathrm{MHz}$ clock as used in the
FACT acquisition system, which sets a lower limit for the sampling time of
$5\text{\,}\mathrm{ns}$ [18]. We measured the average recovery time
$\tau_{\text{MPPC}}$ of one Hamamatsu MPPC and its amplifier with an
oscilloscope, amounting to less than $5\text{\,}\mathrm{ns}$ following the
detection of a single photon. We thus chose a time integration interval of
$20\text{\,}\mathrm{ns}$ for the simulation of the o-Ps decay, which surely is
a long enough recovery time for the readout electronics after detecting single
photons. In case too many photons arrived at the fiber within this chosen time
window, one would adjust the threshold on the corresponding MPPC until the
recovery time is again within an acceptable range. If we now count the number
of fibers firing above that threshold inside the chosen time interval and
track this activity over time, while taking into account the position of each
fiber and thus the resulting solid angles, we will finally reproduce the
exponentially decaying number of o-Ps atoms. In other words, the response to
the o-Ps decay hinges on the statistical nature of fibers being switched _on_
or _off_ under the exponentially decreasing bombardment with annihilation
gamma quanta. Note that the choice of an optimal threshold for each MPPC is
crucial, as too high values cut away the important long tail of o-Ps decay,
while a too low threshold would saturate the MPPCs for a too long time — up to
several $100\text{\,}\mathrm{ns}$ — due to the intense prompt positron
$2\gamma$-annihilation pulse. If there was no formation of Ps in the target,
all implanted positrons would rapidly annihilate into $2\gamma$ quanta,
producing a spectrum that shows a large number of fibers firing at $t=0$, and
then rapidly approaching noise level within the first
50-$100\text{\,}\mathrm{ns}$.

This noise level in turn can be estimated using the dark noise level of the
real FACT detector at low energy thresholds, which amounts to
$100\text{\,}\mathrm{kHz}$ per channel [18]. This means, each of the 794
fibers fires $0.002$ times within a time interval of $20\text{\,}\mathrm{ns}$.
The way this will affect the counting when taking into account all fibers is
described by a Poisson distribution, with an expectation value
$\lambda=794\cdot 0.002=1.6$ counts within a $20\text{\,}\mathrm{ns}$ time
interval as was used for the simulation. Thus if the recorded counts exceed
the noise level several tens of nanoseconds after positron implantation, it
must be due to o-Ps annihilation.

In summary, the method proposed here consists of counting the number of fibers
arranged in a cylindrical array that are firing within a fixed time interval,
e.g. $20\text{\,}\mathrm{ns}$ as explained above, and tracking such a number
as a function of the time. Taking into account the geometry of the detector
array, one can find the exponentially decreasing number of gamma quanta
originating from o-Ps annihilation in flight and even infer the approximate
amount of originally formed positronium atoms.

##  III Experiments with a single fiber

We coupled one Kuraray scintillating fiber to a fast photomultiplier tube (PMT
_Hamamatsu R1450_). The characteristics of such a fiber-PMT assembly are shown
in Table 1 on the left side. This should be compared to the characteristics of
the MPPC on the right side as used in the FACT detector of
AE$\bar{\hbox{g}}$IS. Both systems are very similar, therefore the results we
obtain with our fiber-PMT assembly are indicative for the combination of
fiber-MPPC as used in FACT.

| Fiber-PMT | Fiber-MPPC
---|---|---
Spectral response | $300-$600\text{\,}\mathrm{nm}$$ | $320-$900\text{\,}\mathrm{nm}$$
$\lambda$ of max. response | $420\text{\,}\mathrm{nm}$ | $440\text{\,}\mathrm{nm}$
Nominal voltage | $-1500\text{\,}\mathrm{V}$ | $70\text{\,}\mathrm{V}$
Gain | $1.7\text{\times}{10}^{6}\text{\,}$ | $2.4\text{\times}{10}^{6}\text{\,}$
Rise time | $1.8\text{\,}\mathrm{ns}$ | $1.2\text{\,}\mathrm{ns}$
Decay time | $6\text{\,}\mathrm{ns}$ |  $<$$5\text{\,}\mathrm{ns}$
Table 1: Characteristics of systems where a PMT and a MPPC are coupled to a
scintillating fiber. Both systems are very similar, thus results obtained with
fiber-PMT can directly be compared to fiber-MPPC as is used for the FACT
detector.

The fiber was shielded from ambient light by a black plastic coverage. One of
the polished ends was coupled to the PMT via a black plastic holder by an air-
gap without use of optical grease or bonding. This fiber-PMT assembly was
tested by using a 133Ba source whose emission spectrum shows a dominant peak
at $356\text{\,}\mathrm{keV}$ energy [25]. The X-ray peaks at around
$31\text{\,}\mathrm{keV}$ energy were strongly suppressed, since the source
was sealed in a brass casing of about $5\text{\,}\mathrm{mm}$ thickness. As a
consequence, the dominant peak of the Ba-spectrum can be used as a proxy for
the mean energy of the continuous annihilation spectrum of o-Ps. By recording
the number of counts with a Tektronix TDS5054B oscilloscope, its trigger
threshold being set to $5\text{\,}\mathrm{mV}$, we confirmed the ability of a
$1\text{\,}\mathrm{mm}$ thick Kuraray scintillating fiber to detect gamma rays
in the energy range between $300-$$400\text{\,}\mathrm{keV}$. When the Ba
source was located about $10\text{\,}\mathrm{mm}$ away from the fiber, the
background count rate of $150\text{\,}\mathrm{Hz}$ increased to several
$\text{\,}\mathrm{kHz}$.

In order to experimentally test the response of a Kuraray fiber to a burst of
positrons, the aforementioned positron test chamber of AE$\bar{\hbox{g}}$IS
(sketched in Fig. 4) was used. Pulses with about
$3\text{\times}{10}^{6}\text{\,}$ positrons were produced using the
AE$\bar{\hbox{g}}$IS positron system, which is located at the Antiproton
Decelerator (AD) facility at CERN. The loose end of the fiber was rolled up
into three loops of about $5\text{\,}\mathrm{cm}$ in diameter for a total
length of about $47\text{\,}\mathrm{cm}$. The 3-loop end of the fiber has been
inserted in the detector “pit” of the positron test chamber of
AE$\bar{\hbox{g}}$IS with the loop axis pointing towards an aluminium target.
The positron pulses were steered onto the target, which led on average to a
peak amplitude of about $100\text{\,}\mathrm{mV}$ on the used oscilloscope.
This resembled the average response of our fiber-PMT assembly to the full
$2\gamma$ annihilation signal consisting of about
$6\text{\times}{10}^{6}\text{\,}$ gamma quanta with $511\text{\,}\mathrm{keV}$
energy. We assumed all positrons to annihilate on the surface of the aluminum
target into two gamma quanta with no production of Ps or positron back-
reflection. The background signal was acquired by switching off the magnets in
the transfer line so that no positrons were transferred from the source region
to the test chamber.

Figure 4: Test chamber, where the Kuraray scintillating fiber is tested with a
pulse of $3\text{\times}{10}^{6}\text{\,}$ positrons.

The 3-loop fiber was positioned in the detector pit at distances of 4, 14,
$18\text{\,}\mathrm{cm}$ from the aluminium target, and between 50 and 100
measurements were taken for each position recording the signal amplitudes. The
resulting histograms are shown in Fig. 5. One can see that as distances from
the target increase the distributions shift towards smaller signal amplitudes
and have a smaller width. Moreover, the output of the fiber at greater
distances was increasingly often not distinguishable from the noise level,
which then added to the histogram as a zero signal. This was due to the
decreasing solid angle covered by the 3-loop fiber in each position. The solid
angles relative to the tested distances are
$0.161\text{\,}\mathrm{s}\mathrm{r}$, $0.013\text{\,}\mathrm{s}\mathrm{r}$ and
$0.008\text{\,}\mathrm{s}\mathrm{r}$, respectively.

Figure 5: Number of occurrences as a function of pulse amplitudes for
different distances, i.e. solid angles. Error bars are left out, as this is a
qualitative indication only. Solid lines are meant as a guide to the eye.

We can now derive the probability for a single fiber to register a signal
above a certain energy threshold by calculating for each distance the ratio
between the number of counts higher than a given threshold and the sum of all
occurrences. In Fig. 6 we report the percent-probability that a fiber
activates, i.e. that a photon burst with constant intensity deposits enough
energy into the fiber that it exceeds the threshold. The corresponding
thresholds are indicated in the abscissa.

Figure 6: Probability of detecting a constant flash of
$6\text{\times}{10}^{6}\text{\,}$ photons from different distances as a
function of the detection threshold. Equivalently, this can be interpreted as
the probability to detect bursts of photons with varying intensities, but from
a constant arbitrarily chosen solid angle (here
$0.09\text{\,}\mathrm{s}\mathrm{r}$). Again the error bars were left out for
this qualitative result.

##  IV Simulated response of the ”digital calorimeter”

The variation of the distance and therefore of the solid angle can be thought
to be equivalent to a change in the number of annihilating positrons. In other
words, the measurements at different distances simulate the responses of one
particular fiber with a fixed solid angle to flashes of gamma radiation with
_different_ intensities — exactly as it occurs during the exponential decay of
o-Ps in free flight. Referring to the central fiber of a detector with the
geometry of FACT, i.e. with a solid angle of
$0.09\text{\,}\mathrm{s}\mathrm{r}$, one can directly use these sets of
measurements at different distances in order to obtain the probability of this
fiber to respond to nearly instantaneous flashes of gamma quanta with varying
intensities. The curves in Fig. 6 can therefore also be interpreted as the
percent-probability that a fiber covering a solid angle of
$0.09\text{\,}\mathrm{s}\mathrm{r}$ activates due to a flash of
$1.1\text{\times}{10}^{7}\text{\,}$ (circles),
$8.8\text{\times}{10}^{5}\text{\,}$ (rectangles) or
$5.3\text{\times}{10}^{5}\text{\,}$ (triangles) annihilation photons,
respectively.

We can now construct the response of the central fiber to a decaying cloud of
o-Ps atoms at different times. The positronium cloud is generated by a pulse
of $N_{0}$ positrons implanted at time $t=0$. It then exponentially decays
with its typical vacuum annihilation lifetime of $142\text{\,}\mathrm{ns}$.
The initial o-Ps conversion ratio of the positron/Ps converter targets at
AE$\bar{\hbox{g}}$IS is 0.3 [11]. Then taking only the fraction surviving
magnetic quenching inside the present $1\text{\,}\mathrm{T}$ field
($66\text{\,}\mathrm{\char 37\relax}$), one finally finds a conversion
efficiency of $20\text{\,}\mathrm{\char 37\relax}$. From this, we obtained the
number of gamma quanta emitted within intervals of $20\text{\,}\mathrm{ns}$ at
any given time. As we have shown above, a fiber has a chance to get activated
with an certain intrinsic efficiency that depends on the set energy threshold
when flashes of gamma rays irradiate it. Combining this effect with the
simulated geometrical responses of the entire fiber array as was given for
three thresholds in Fig. 2, we obtained the statistical number of
scintillating fibers being activated at the same time, $T_{firing}$.

The outcome is given in Fig. 7, which shows the number of fibers firing during
the production and the decay of o-Ps for two different intensities of the
initial positrons pulse at $t=0$. With the threshold set to
$25\text{\,}\mathrm{mV}$ and a positron pulse containing
$2\text{\times}{10}^{8}\text{\,}$ particles, the number of active fibers
saturates for about $100\text{\,}\mathrm{ns}$ and then it decreases. When
using only half the positrons, we observe no saturation but the decreasing
tail. The threshold of $25\text{\,}\mathrm{mV}$ has been chosen for this
specific example and was used to verify the correct behaviour of the
simulation. In reality, it depends on the intrinsic response of the single
fiber-amplifier assembly, the number of implanted positrons and the aspired
geometry and thus needs to be adjusted when using another system as is
presented here. Notably, by regarding the convolution with the detector
geometry, the resulting decrease of active fibers resembles the decay of o-Ps
in free flight such as is usually tracked by the aforementioned SSPALS method
(black curve in Fig. 3). An estimate of the errors has been done for these
particular tests by taking into account the statistical fluctuation on the
number of gamma rays that leave a sizeable amount of energy, i.e. above the
chosen $25\text{\,}\mathrm{mV}$ threshold, in the central fiber. This yielded
an error ranging from $3-$$4\text{\,}\mathrm{\char 37\relax}$ at short times
where almost all the fibers are hit and respond, to about
$10-$$12\text{\,}\mathrm{\char 37\relax}$ at the longest times where a reduced
number of fibers respond.

Figure 7: Simulated response of a fiber array detector such as FACT to
$1\text{\times}{10}^{8}\text{\,}$ positrons (circles) and
$2\text{\times}{10}^{8}\text{\,}$ positron (squares) as a function of time
after positron implantation and conversion to Ps. As threshold
$25\text{\,}\mathrm{mV}$ was used for this test.

##  V CONCLUSIONS

A novel method to use an array of scintillating fibers coupled to an
amplification and detection system in order to track ortho-Ps formation and
decay in cryogenic and magnetic environments over time has been tested. A
single fiber coupled to a fast PMT was irradiated by flashes of about
$6\text{\times}{10}^{6}\text{\,}$ gamma quanta produced by a burst of
positrons from the AE$\bar{\hbox{g}}$IS positron system. The response of the
fiber obtained at different distances from the radiation source demonstrated
the possibility to setup a fiber array in the geometry of the FACT detector of
AE$\bar{\hbox{g}}$IS and use it to monitor the annihilation signal of decaying
o-Ps in free flight. By introducing the precise geometry and efficiencies of a
cylindrical fiber array as is used in FACT into a Monte Carlo simulation, we
provided a proof-of-principle on the capability of this scintillating fiber
detector to monitor the o-Ps formation and decay with a time resolution of at
least $20\text{\,}\mathrm{ns}$. Hence, this method could provide a non-
destructive measurement of o-Ps annihilation in ground state with a dynamic
range of about 9.5 bits inside a narrow-spaced, cryogenic, evacuated and
magnetic environment. An on-leading development of this method consists of
tracking the decay of laser-excited Rydberg-Ps atoms over time, which have a
lifetime of the order of microseconds.

## ACKNOWLEDGMENTS

The authors wish to thank the AE$\bar{\hbox{g}}$IS collaboration at CERN for
permitting the use of the positron apparatus. This work was supported by:
Istituto Nazionale di Fisica Nucleare (INFN Italy); the CERN Fellowship
program and the CERN Doctoral student program; European’s Union Horizon 2020
research and innovation program under the Marie Sklodowska-Curie grant
agreements No. 754496, FELLINI.

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