\section{Introduction}

Several different methods to construct minimal polynomials of Salem numbers have been investigated in the literature (see e.g.\ \cite{BDGPS,boyd,grossmcmullen,smyth}). Various authors associate Salem numbers to Coxeter polynomials and use this relation in order to construct Salem numbers (cf.\ for instance \cite{cannwagr,floydplotnick,grohirmcm,lakatos,mckrowsmy}). In this paper we follow the very explicit approach of Gross et al.~\cite{grohirmcm} and provide precise information on the decomposition of Coxeter polynomials of certain star-like trees into irreducible factors, thereby giving estimates on the degree of the occurring Salem factor.

To be more precise, let $r, a_0,\ldots, a_r \in \NN$ such that $a_0\ge 2,\ldots, a_r \ge 2$. We consider the star-like tree
 $T=T (a_0,\ldots, a_r)$ with $r+1$ arms of  $a_0-1,\ldots, a_r-1$ edges, respectively. According to \cite[Lemma 5]{mckrowsmy}
the Coxeter polynomial of $T (a_0,\ldots, a_r)$ is given by
$$
R_{T (a_0,\ldots, a_r)} (z) =\prod_{i=0}^{r} \Bigl( \frac{z^{a_i}-1}{z-1}\Bigr)\Bigl(z+1- z \sum_{i=0}^{r}  \frac{z^{a_i-1}-1}{z^{a_i}-1}\Bigr).
$$
Note that $R_T$ can be written as
\begin{equation}\label{eq:decomp}
R_T(z)=C(z)S(z),
\end{equation}
where $C$ is a product of cyclotomic polynomials and $S$ is the minimal polynomial of a Salem number or of a quadratic Pisot number. Indeed, by the results of \cite{pena},  the zeros of $R_T$ are either real and positive or  have modulus 1. The decomposition \eqref{eq:decomp} now follows from \cite[Corollaries~7 and~9, together with the remark after the latter]{mckrowsmy}, as these results imply that $R_T$ has exactly one irrational real positive root of modulus greater than $1$.

For Coxeter polynomials corresponding to star-like trees with three arms we are able to say much more about the factors of the decomposition \eqref{eq:decomp}. In particular, we shall prove the following result.

\begin{theorem}\label{thm1}
Let  $a_0,  a_1, a_2 \in \ZZ$  such that $a_2 > a_1 >a_0>1$ and
$(a_0, a_1, a_2)\ne (2,3,t)$ for all $t\in \set{4, 5, 6}$. Further, let $T:=T (a_0, a_1, a_2)$ be the star-like tree with three arms of  $a_0-1, a_1-1, a_2-1$ edges, and let $\lambda$ be its largest eigenvalue. Then $\tau >1$ defined by
$$
\sqrt {\tau}+ 1/\sqrt {\tau}=  \lambda
$$
is a Salem or a quadratic Pisot root of the Coxeter polynomial $R_{T}$ of  $T$.
If $S$ is the minimal polynomial of $\tau$  then we can write
\begin{equation}\label{eq:clear}
R_T(x)=S(x)C(x),
\end{equation}
where $C$ is a product of cyclotomic polynomials of orders bounded by $420 (a_2-a_1+a_0-1)$ whose roots have multiplicity bounded by an effectively computable constant $m(a_0,a_2-a_1)$. Thus
\begin{equation}\label{SalemLowerBound}
\deg S \ge \deg R_T - m(a_0,a_2-a_1)\sum_{k\le 420 (a_2-a_1+a_0-1)}\varphi(k),
\end{equation}
where $\varphi$ denotes Euler's $\varphi$-function.
\end{theorem}

\begin{remark}[Periodicity properties of cyclotomic factors]\label{rem:period}
Gross et al.~\cite{grohirmcm} study certain Coxeter polynomials and prove periodicity properties of their cyclotomic factors. Contrary to their case, our Coxeter polynomials $R_T$ do not have the same strong separability properties (cf.\ Lemma~\ref{lem:mult}). For this reason, we could not exhibit analogous results for $C(x$), however, we obtain weaker periodicity properties in the following way.

In the setting of Theorem~\ref{thm1} assume that $a_0$ as well as $a_2-a_1$ are constant. For convenience set $S_{a_1}=R_{T(a_0,a_1,a_2)}$ and let $\zeta_k$ be a root of unity of order $k$. It follows from \eqref{expzdarst} below (see \eqref{pzdef} for the definition of $P$) that $S_{a_1}(\zeta_k)=0$ if and only if  $S_{a_1+k}(\zeta_k)=0$, i.e., the
fact that the $k$-th cyclotomic polynomial divides $S_{a_1}$ depends only on the residue class of $a_1 \pmod k$.
Therefore, setting $K:=\lcm\{1,2,\ldots,  420 (a_2-a_1+a_0-1)\}$,
the set of all cyclotomic polynomials dividing $S_{a_1}$ is determined by the residue class of $a_1 \pmod K$.

If we determine the set $\{k \,:\, k \le 420 (a_2-a_1+a_0-1),\, S_{a_1}(\zeta_k) = 0 \}$ for all $a_1 \le K$ we thus know exactly which cyclotomic factor divides which of the polynomials $S_{a_1}$ for $a_1\in\mathbb{N}$. Obviously, this knowledge would allow to improve the bound \eqref{SalemLowerBound}.
\end{remark}

\begin{remark}[Degrees of the Salem numbers]\label{rem:degree}
Theorem~\ref{thm1} enables us to exhibit Salem numbers of arbitrarily large degree.
 Indeed, if $a_0$ and the difference $a_2-a_1$ are kept small and  $a_1\to \infty$ then \eqref{SalemLowerBound} assures that $\deg S \to \infty$. We also mention here that Gross and McMullen \cite[Theorem~1.6]{grossmcmullen} showed that for any odd integer
$n\ge 3$ there exist infinitely many {\it unramified} Salem numbers of degree $2 n$;
recall that a Salem polynomial $f$ is said to be unramified if it satisfies $|f(-1)|=|f(1)|=1$.
The construction pursued in this work substantially differs from ours: it is proved that every unramified Salem polynomial arises from
an  automorphism of an indefinite lattice.
\end{remark}

If  two of the arms of the star-like tree under consideration get longer and longer, the associated Salem numbers converge to the $m$-bonacci number $\varphi_m$, where $m$ is the (fixed) length of the third arm. This is made precise in the following theorem.

\begin{theorem}\label{th:m}
Let $a_1 > a_0 \ge 2$ and $\eta \ge 1$ be given and set $a_2 = a_1+\eta$.  Then, for $a_1\to\infty$, the Salem root $\tau(a_0, a_1, a_2)$ of the  Coxeter polynomial associated with $T (a_0, a_1, a_2)$ converges to $\varphi_{a_0}$, where the degree of $\tau(a_0, a_1, a_2)$ is bounded from below by \eqref{SalemLowerBound}.
\end{theorem}

Besides that, we are able to give the following result which is valid for more general star-like trees.

\begin{theorem}\label{th:general}
Let $r\ge 1$, $a_r>  \cdots > a_1 > a_0 \ge 2$, and choose $k\in\{1,\ldots,r-1\}$.  Then, for fixed $a_0,\ldots,a_k$ and $a_{k+1},\ldots, a_r \to \infty$, the Salem root $\tau(a_0, \ldots , a_r)$ of the Coxeter polynomial associated with $T (a_0, \ldots, a_r)$ converges to the dominant Pisot root of
\begin{equation}\label{eq:qz}
Q(z)=(z+1-r+k)\prod_{i=0}^k(z^{a_i}-1)-z\sum_{i=0}^k(z^{a_i-1}-1)
\prod_{\begin{subarray}{c} j=0 \\ j\not=i \end{subarray}}^k (z^{a_j}-1).
\end{equation}
\end{theorem}

\section{Salem numbers generated by Coxeter polynomials of star-like trees}

For convenience, we introduce the polynomial
\begin{equation}\label{pzdef}
\begin{array}{rrl}
\displaystyle P(z)&:=&\displaystyle (z-1)^{r +1} R_{T (a_0,\ldots, a_r)} (z) \\[3pt]
&=&\displaystyle \Bigl( \prod_{i=0}^{r} (z^{a_i}-1)\Bigr) (z+1)- z \sum_{i=0}^{r}  \Bigl( (z^{a_i-1}-1) \prod_{j=0, j\ne i}^{r} (z^{a_j}-1)\Bigr).
\end{array}
\end{equation}
Of course, like $R_T$,  the polynomial $P$ can be decomposed as a product of a Salem (or quadratic Pisot) factor times a factor containing only cyclotomic polynomials.

Now, we concentrate on star-like trees with three arms, i.e., we assume that $r=2$.

\begin{lemma} \label{pblock}
Let $a_2 > a_1 > a_0$. Then for $T(a_0,a_1,a_2)$ the polynomial $P(z)$ reads
\begin{equation}\label{expzdarst}
  P(z)= z^{a_1 + a_2} Q(z) + z^{a_1 + 1} R(z)+ S(z)
\end{equation}
with
\begin{eqnarray*}
 Q(z) &=& z^{a_0+1} - 2 z^{a_0}+ 1, \\
  R(z) &=& z^{a_2-a_1+a_0-1} - z^{a_2-a_1} + z^{a_0-1} -1,\\
  S(z) &=& -z^{a_0 +1} + 2z -1.
  \end{eqnarray*}
Moreover,
$$
\max\{ \deg(Q), \deg(R), \deg(S) \} = a_2-a_1+a_0-1, \quad \deg (P)= a_0 + a_1 +a_2 + 1,
$$
and the (naive) height of $P$ equals $2$.
\end{lemma}
\begin{proof}
This can easily be verified by direct computation.
\end{proof}

\begin{lemma}\label{lem:mult}
Let $a_2 > a_1 > a_0$ and let $P$ as in \eqref{pzdef} be associated to $T(a_0,a_1,a_2)$. Then there exists an effectively computable constant $m=m(a_0,a_2-a_1)$ which bounds the multiplicity of every root $z$ of $P$ with $|z|=1$.
\end{lemma}

\begin{proof}
Observe, that $1$ is a root of $Q$, $R$, and $S$. Thus, for technical reasons, we work with $\tilde P(z)=P(z)/(z-1)$ and, defining $\tilde Q(z)$, $\tilde R(z)$, and $\tilde S(z)$ analogously, we write
\[
\tilde P(z)= z^{a_1 + a_2} \tilde Q(z) + z^{a_1 + 1} \tilde R(z)+ \tilde S(z).
\]
Our first goal is to bound the $n$-th derivatives $|\tilde P^{(n)}(z)|$ with $|z|=1 $ away from zero. To this end we define the quantities
\begin{align*}
\eta(a_0)&:=\min\{|\tilde Q(z)| \,:\, |z|=1 \} > 0, \\
E_n=E_n(a_0)&:=\max\{|\tilde Q^{(k)}(z)| \,:\, 1\le k\le n,\, |z|=1 \},  \\
F_0=F_0(a_0,a_2-a_1)&:=\max\{|\tilde R(z)| \,:\,  |z|=1 \}, \\
F_n=F_n(a_0,a_2-a_1,n)&:=\max\{|\tilde R^{(k)}(z)| \,:\, 1\le k\le n,\, |z|=1 \},  \\
G_n=G(a_0,n)&:=\max\{|\tilde S^{(n)}(z)| \,:\, |z|=1 \}.
\end{align*}
For $n\ge 1$ one easily computes that (note that $(x)_{n}=x(x-1)\cdots(x-n+1)$ denotes the Pochhammer symbol)
\begin{align*}
\tilde P^{(n)}(z) =& (a_1+a_2)_{(n)}\tilde Q(z)z^{a_1+a_2-n} +
(a_1+1)_{(n)}\tilde R(z)z^{a_1+1-n} \\
&+ \sum_{k=0}^{n-1}{n\choose k}(a_1+a_2)_{(k)}\tilde Q^{(n-k)}(z)z^{a_1+a_2-k} \\
&+ \sum_{k=0}^{n-1}{n\choose k}(a_1+1)_{(k)}\tilde R^{(n-k)}(z)z^{a_1+1-k}
+ \tilde S^{(n)}(z).
\end{align*}
Now for  $|z|=1$ we estimate
\begin{align}
|\tilde P^{(n)}(z)| \ge & (a_1+a_2)_{(n)}\eta(a_0) - 2^{-n+1}(a_1+a_2)_{(n)}F_0 \nonumber \\
&- 2^{n-1}(a_1+a_2)_{(n-1)}E_n - 2^{n-1}(a_1+1)_{(n-1)}F_n - G_n\label{eq:betrag}\\
\ge& (a_1+a_2)_{(n)} \Big(\eta(a_0) - 2^{-n+1}F_0 - \frac{2^{n-1}(E_n+F_n)}{a_1+a_2-n+1} - \frac{G_n}{(a_1+a_2)_{(n)}}\Big). \nonumber
\end{align}
Now we fix $a_0$ and the difference $a_2-a_1$. Then we choose $n_0=n_0(a_0,a_2-a_1)$ such that
$$
\eta(a_0) - 2^{-n_0+1}F_0 >0.
$$
In view of \eqref{eq:betrag} there exists a constant $c=c(a_0,a_2-a_1)$ such that for all $a_1,a_2$ with $a_1+a_2 > c$ (with our fixed difference) we have $|\tilde P^{(n_0)}(z)|  > 0$ for all $z$ with $|z|=1$. If, on the other hand, $a_1+a_2 \le c$, then we have $\deg \tilde P \le c+a_0$. Therefore, in any case, the multiplicity of a root of $\tilde P$ on the unit circle is bounded by $\max(n_0,c+a_0)$ and the result follows by taking $m=\max(n_0,c+a_0)+1$.
\end{proof}

The following lemma is a simple special case of Mann's theorem.

\begin{lemma}\label{abcz}
Let   $a, b, c, p, q  \in \ZZ$ such that  $(p,q)\ne (0,0)$ and $a,b,c$ nonzero.
If $\zeta$ is  a  root of unity such that
$$a \zeta^p + b \zeta^q  +c= 0$$
then  the order of $\zeta$ divides $\; 6 \, \gcd (p, q).$
\end{lemma}

\begin{proof}
This is a special case of \cite[Theorem 1]{mann}.
\end{proof}

For subsequent use we recall some notation and facts (used in a similar context in \cite{grohirmcm}).
A  {\em divisor} on the complex plane is a finite sum
$$  D=\sum_{j  \in J}  a_j \cdot z_j$$
where $a_j \in \ZZ\setminus \set{0} $ and $${\rm supp \,} (D):= \set{z_j \in  \CC\;:\; j \in J}$$
is the support of $D$; $D$ is said to be  {\em effective} if all its coefficients are positive.

\medskip

The set of all divisors on $\CC$ forms the abelian group ${\rm Div \,} (\CC)$, and the natural
 evaluation map $\sigma: {\rm Div \,} (\CC) \to \CC $ is given by
$$
\sigma ( D) =\sum_{j  \in J}  a_j z_j\,.
$$

\medskip

A  {\em polar rational polygon}  (prp) is an effective divisor  $D=\sum_{j  \in J}  a_j \cdot z_j$ such that each $z_j$ is a root of unity and $ \sigma ( D) =0\,.$ In this case the order ${\rm o } (D) $ is the cardinality of the subgroup of
$ \CC \setminus \set{0} $ generated by the roots of unity $\set{z_j / z_k \;:\; j,k\in J}$.
The prp $D$ is called  {\em primitive} if there do not exist non-zero prp's $D'$ and $D''$ such that $D=D' + D''$.
In particular, the coefficients of $D', D''$ are positive, thus each prp can be expressed as a sum of primitive prp's.

\bigskip

Every polynomial  $f \in \ZZ[X] \setminus \set{0}$ can be uniquely written in the form
\begin{equation}\label{dcf}
  f=\sum_{j \in J} \varepsilon_j a_j X^j
\end{equation}
with $J \subseteq \set{0, \ldots, \deg  (f)}$, $\varepsilon_j=\pm 1$ and $a_j > 0$. We call
$$\ell (f):= \Card (J)$$
the length of $f$. For $\zeta \in \CC$ with $f(\zeta)=0$ we define the effective divisor of $f$ (w.r.t. $\zeta$) by
$$D f (\zeta):= \sum_{j \in J}  a_j (\varepsilon_j \zeta^j) \,.$$

\bigskip

\begin{proposition}\label{farc}
Let $a_2 > a_1 > a_0$ and let $P$ as in \eqref{pzdef} be associated to $T(a_0,a_1,a_2)$. If   $\zeta$ is  a  root of unity such that  $P(\zeta) = 0$ then
the order of  $\zeta$ satisfies
$$
\mathrm{ord}(\zeta) \le 420 (a_2-a_1+a_0-1).
$$
\end{proposition}

\begin{proof}
We follow the proof of \cite[Theorem 2.1]{grohirmcm} and write the polynomials $Q, R, S$
in the form
$$Q (X) =\sum Q_j (X), \qquad R (X) =\sum R_j (X), \qquad S (X) =\sum S_j (X)$$
with finite sums of integer polynomials such that
$$D P (\zeta)= \sum_{j}  D P_j (\zeta)= \sum_{j} \bigl( \zeta^{a_1 + a_2} D Q_j(\zeta) +  \zeta^{a_1 + 1} D R_j(\zeta)+  DS_j(\zeta)\bigr)$$
is a decomposition of the divisor  $ D P (\zeta)$ into primitive polar rational polygons $ D P_j (\zeta)$, thus for every $j$ the sum
$$\zeta^{a_1 + a_2} Q_j(\zeta) + \zeta^{a_1 + 1} R_j(\zeta)+ S_j(\zeta)$$
is the evaluation of the primitive prp $ D P_j (\zeta)$.
Observe that in view of Lemma \ref{pblock} the (naive) height of the polynomials $Q_j, R_j, S_j$ does not exceed  $2$ since the coefficients cannot increase when performing the decomposition of a prp into primitive prp's.

\medskip

Case 1: \qquad $\max \set{\ell (Q_j), \ell (R_j), \ell (S_j)} > 1$ for some $j$

\medskip

 Let us first assume $\ell (Q_j)> 1$ for some $j$. The ratio of any two roots of unity occurring in $D Q_j (\zeta)$
 can be written in the form $\pm \zeta^e$ with $1 \le e \le \deg (Q_j) \le \deg (Q)$. Therefore we have
 $$\frac{\mathrm{ord}(\zeta)}{2 \deg (Q) } \le {\rm o \,} (D P_j (\zeta)) .$$
 By Mann's Theorem \cite{mann}, $ {\rm o \,} (D P_j (\zeta))$ is bounded by the product of primes
 at most equal to
 $$\ell (P_j)\le \ell (Q_j) +\ell (R_j) + \ell (S_j)\le \ell (Q) +\ell (R) + \ell (S)\le 3+4+3=10\,.$$
 The product of the respective primes is at most $2\cdot 3 \cdot 5\cdot 7=210$. Therefore by Lemma \ref{pblock} we find
 $$\frac{\mathrm{ord}(\zeta)}{2 (a_0+1)}=\frac{\mathrm{ord}(\zeta)}{2 \deg (Q) } \le 210,$$
 which yields
 $$\mathrm{ord}(\zeta) \le 420 \; (a_0+1).$$
 Analogously, the other two cases yield
 $$\mathrm{ord}(\zeta) \le 420 \; (a_2-a_1+a_0-1) \qquad \text{ or } \qquad\mathrm{ord}(\zeta)\le 420 \; (a_0+1),$$
 and we conclude
\begin{equation}\label{resge1}
\mathrm{ord}(\zeta) \le 420 \; \max \set{a_0+1, \, a_2-a_1+a_0-1 } =420 \; (a_2-a_1+a_0-1).
\end{equation}

\medskip

Case 2:  \qquad $\max \set{\ell (Q_j), \ell (R_j), \ell (S_j)} \le 1$ for  all $j$

\medskip

In this case, $D P_j (\zeta)$ is either of the form
\begin{equation}\label{Pdouble}
D P_j (\zeta) = c_{j1}\zeta^{b_{j1}} + c_{j2}\zeta^{b_{j2}}
\end{equation}
or of the form
\begin{equation}\label{Ptriple}
D P_j (\zeta) = c_{j1}\zeta^{b_{j1}} + c_{j2}\zeta^{b_{j2}}  + c_{j3}\zeta^{b_{j3}} ,
\end{equation}
where $c_{ji}\in \{-2,\ldots, 2\}$ by Lemma~\ref{lem:mult}. We distinguish two subcases.

\medskip

Case 2.1: There exists $j$ such that $D P_j (\zeta)$ is of the form \eqref{Ptriple}.

\medskip

 In this situation $D P_j (\zeta)$ can be written more explicitly as
\begin{equation}\label{Pt1}
D P_j (\zeta) = c_{j1}\zeta^{a_1+a_2+\eta_1} + c_{j2}\zeta^{a_1+\eta_2}  + c_{j3}\zeta^{\eta_{3}}
\end{equation}
or
\begin{equation}\label{Pt2}
D P_j (\zeta) = c_{j1}\zeta^{a_1+a_2+\eta_1} + c_{j2}\zeta^{a_2+\eta_2}  + c_{j3}\zeta^{\eta_{3}},
\end{equation}
where $\eta_1\in\{0,a_0,a_0+1\}$, $\eta_2\in\{1,a_0\}$, and $\eta_3\in\{0,1,a_0+1\}$. If $D P_j (\zeta)$ is as in \eqref{Pt1} then $P_j (\zeta)=0$ implies that
\[
c_{j1}\zeta^{a_1+a_2+\eta_1-\eta_3} + c_{j2}\zeta^{a_1+\eta_2-\eta_3}  + c_{j3} =0.
\]
Now, using Lemma~\ref{abcz} we gain
\[
\mathrm{ord}(\zeta) \mid 6 \mathrm{gcd}(a_1+a_2+\eta_1-\eta_3, a_1+\eta_2-\eta_3)
\]
which yields
\[
\mathrm{ord}(\zeta) \mid 6(a_2-a_1+\eta_1-2\eta_2+\eta_3),
\]
hence,
\begin{equation}\label{tripleresult}
\mathrm{ord}(\zeta) \le 6(2 a_0 + a_2 - a_1).
\end{equation}
If $D P_j (\zeta)$  is as in \eqref{Pt2}, by analogous arguments we again obtain \eqref{tripleresult}.

\medskip

Case 2.2:  For all $j$ the divisor $D P_j (\zeta)$ is of the form \eqref{Pdouble}.

\medskip

In this case we have to form pairs of the 10 summands of $D P(\zeta)$ to obtain the divisors $D P_j (\zeta)$. As $\ell(R)=\ell(S)=4$ there must exist $j_1,j_2$ such that $\ell(R_{j_1})=0$ and $\ell(S_{j_2})=0$.  In what follows, $c_{ij}\in\{-2,\ldots,2\}$, and  $\eta_1,\eta_1'\in\{0,a_0,a_0+1\}$, $\eta_2,\eta_2'\in\{1,a_0\}$, and $\eta_3,\eta_3'\in\{0,1,a_0+1\}$.
Then $D P_{j_1} (\zeta)$ is of the form
\begin{equation}\label{pairj1}
D P_{j_1} (\zeta) = c_{j_11}\zeta^{a_1+a_2+\eta_1} + c_{j_13}\zeta^{\eta_{3}}
\end{equation}
which yields
\begin{equation*}\label{pairj1eq}
c_{j_11}\zeta^{a_1+a_2+\eta_1-\eta_3} + c_{j_13} = 0,
\end{equation*}
and, hence,
\begin{equation}\label{pairj1ord}
\mathrm{ord}(\zeta)\mid 2(a_1+a_2+\eta_1-\eta_3).
\end{equation}
For $D P_{j_2} (\zeta)$ we have two possibilities.  Either we have
\begin{equation}\label{pairj21}
D P_{j_2} (\zeta) = c_{j_21}\zeta^{a_1+a_2+\eta'_1} + c_{j_22}\zeta^{a_1+\eta'_2} .
\end{equation}
This yields
\begin{equation*}\label{pairj21eq}
c_{j_21}\zeta^{a_2+\eta'_1-\eta_2'} + c_{j_22} = 0,
\end{equation*}
and, hence,
\begin{equation}\label{pairj21ord}
\mathrm{ord}(\zeta)\mid 2(a_2+\eta_1'-\eta_2').
\end{equation}
The second alternative for $D P_{j_2} (\zeta)$ reads
\begin{equation}\label{pairj22}
D P_{j_2} (\zeta) = c_{j_21}\zeta^{a_1+a_2+\eta'_1} + c_{j_22}\zeta^{a_2+\eta'_2}.
\end{equation}
This yields
\begin{equation*}\label{pairj22eq}
c_{j_21}\zeta^{a_1+\eta'_1-\eta_2'} + c_{j_22} = 0,
\end{equation*}
and, hence,
\begin{equation}\label{pairj22ord}
\mathrm{ord}(\zeta)\mid 2(a_1+\eta_1'-\eta_2').
\end{equation}
If $P_{j_2}$ is of the form \eqref{pairj21}, then \eqref{pairj1ord} and \eqref{pairj21ord} yield
that
\[
\mathrm{ord}(\zeta)\mid 2 \mathrm{gcd}(a_1+a_2+\eta_1-\eta_3, a_2+\eta_1'-\eta_2'),
\]
hence, $\mathrm{ord}(\zeta)\mid 2(a_1-a_2+\eta_1-2\eta_1'+2\eta_2'-\eta_3)$ and therefore
\begin{equation}\label{endres2}
\mathrm{ord}(\zeta)\le 2(a_2-a_1+3a_0+1).
\end{equation}
If $P_{j_2}$ is of the form \eqref{pairj22}, then \eqref{pairj1ord} and \eqref{pairj22ord} yield
\[
\mathrm{ord}(\zeta)\mid 2 \mathrm{gcd}(a_1+a_2+\eta_1-\eta_3, a_1+\eta_1'-\eta_2'),
\]
hence, \eqref{endres2} follows again.

Summing up, the proposition is proved by combining  \eqref{resge1}, \eqref{tripleresult}, and \eqref{endres2}.

 \end{proof}

Combining results from \cite{mckrowsmy} with our previous considerations we can now prove the main theorem of this paper.

\begin{proof}[Proof of Theorem~\ref{thm1}]
The fact that $\tau$ is either a Salem number or a quadratic Pisot number  as well as the decomposition of $R_T$ given in \eqref{eq:clear} follows immediately from~\eqref{eq:decomp}. The bound on the orders of the roots of the cyclotomic polynomials is a consequence of Proposition~\ref{farc}. Together with Lemma~\ref{lem:mult} this proposition yields the estimate $\eqref{SalemLowerBound}$ on the degree of $S$, where the explicitly computable constant $m(a_0,a_2-a_1)$ is the one stated in Lemma~\ref{lem:mult}.
 \end{proof}

\section{Convergence properties of Salem numbers generated by star-like trees}

In this section we prove Theorems~\ref{th:m} and~\ref{th:general}.
In the following proof of Theorem~\ref{th:m} we denote by $M_m(x)=x^m-x^{m-1}-\dots-x-1$ the minimal polynomial of the $m$-bonacci number $\varphi_m$.

\begin{proof}[Proof of Theorem~\ref{th:m}]
Note that $Q(x)=(x-1)M_{a_0}(x)$ holds. The theorem is proved if for each $\varepsilon > 0$ the polynomial $P(x)$ has a root $\zeta$ in the open ball $B_\varepsilon(\varphi_{a_0})$ for all sufficiently large $a_1$. We prove this by using Rouch\'e's Theorem. Let $\varepsilon > 0$ be sufficiently small and set $C_\varepsilon : =\partial B_\varepsilon(\varphi_{a_0})$. Then $\delta := \min\{|M_{a_0}(x)|\,:\, x \in C_\varepsilon \} >0$. Thus, on $C_\varepsilon$ we have the following estimations.
\begin{align}
\label{est}
|x^{a_1+1}R(x)+S(x)| &< (\varphi_{a_0}+\varepsilon)^{a_1+1}4(\varphi_{a_0}+\varepsilon)^{\eta+a_0-1} + 4(\varphi_{a_0}+\varepsilon)^{a_0+1},
\\
|x^{a_1+a_2}Q(x)| &> (\varphi_{a_0}-\varepsilon)^{2a_1+\eta}(\varphi_{a_0}-1-\varepsilon)\delta.
\nonumber
\end{align}
Combining these two inequalities yields
\begin{equation}\label{P_est}
|P(x)| > (\varphi_{a_0}-\varepsilon)^{2a_1+\eta}(\varphi_{a_0}-1-\varepsilon)\delta -
4(\varphi_{a_0}+\varepsilon)^{a_1+a_0+\eta} - 4(\varphi_{a_0}+\varepsilon)^{a_0+1}.
\end{equation}
Since \eqref{est} and \eqref{P_est} imply that for sufficiently large $a_1$ we have
\[
|P(x)| > |x^{a_1+1}R(x)+S(x)| \qquad (x\in C_\varepsilon),
\]
Rouch\'e's Theorem yields that $P(x)$ and $x^{a_1+a_2}(x-1)M_{a_0}(x)$ have the same number of roots in $B_\varepsilon(\varphi_{a_0})$. Thus our assertion is proved.
\end{proof}

To prove Theorem~\ref{th:general} we need the following auxiliary lemma.

\begin{lemma}\label{lem:aux}
Let $r\ge 1$, $a_r>  \cdots > a_1 > a_0 \ge 2$, and choose $k\in\{1,\ldots,r-1\}$. If $P$ is as in \eqref{pzdef} and $Q$ as in \eqref{eq:qz} then, for fixed $a_0,\ldots,a_k$ and $a_{k+1},\ldots, a_r$ sufficiently large, we have
\[
\#\{\xi \in \mathbb{C}\;:\, P(\xi)=0,\, |\xi| > 1\} = \#\{\xi \in \mathbb{C}\;:\, Q(\xi)=0,\, |\xi| > 1\} = 1.
\]
\end{lemma}

\begin{proof}
First observe that
\begin{equation}\label{eq:PQesti}
P(z)= z^{a_{k+1} + \cdots + a_r}Q(z) + O(z^{a_{k+2} + \cdots + a_r+\eta})
\end{equation}
for some fixed constant $\eta \in \mathbb{N}$.Since by ~\eqref{eq:decomp} the polynomial $P$ has exactly one (Salem) root outside the unit disk, it is sufficient to prove the first equality in the statement of the lemma.

We first show that $Q$ has at least one root $\xi$ with $|\xi| > 1$. This is certainly true for $k < r-2$ as in this case we have $|Q(0)|>1$. For $k\in \{r-2,r-1\}$ we see that
\[
Q^{(\ell)}(1)=0,\;\hbox{for} \;  0\le \ell < a_0+\cdots+a_k - 1, \quad Q^{(a_0+\cdots+a_k - 1)}(1)<0.
\]
As the leading coefficient of $Q$ is positive, this implies that $Q(\xi)=0$ for some $\xi >1$.

Now we show that $Q$ has at most one root $\xi$ with $|\xi| > 1$. Assume on the contrary that there exist two distinct roots $\xi_1,\xi_2\in\mathbb{C}$ of $Q$ outside the closed unit circle. Applying Rouch\'e's Theorem to $P(z)$ and $z^{a_{k+1}+\cdots+a_r}Q(z)$ shows by using \eqref{eq:PQesti} that also $P$ has two zeroes outside the closed unit circle which contradicts the fact that $P$ is a product of a Salem polynomial and cyclotomic polynomials, see~\eqref{eq:decomp}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{th:general}]
From Lemma~\ref{lem:aux} and \eqref{eq:qz} we derive that
\[
Q(z)=C(z)T(z)z^s,
\]
where $s\in\{0,1\}$ and $T$ is a Pisot or a Salem polynomial. To show the theorem we have to prove that $T$ is a Pisot polynomial. It suffices to show that $C(z)T(z)$ is not self-reciprocal.

We distinguish three cases. If $k < r-2$ then $|C(0)T(0)|>1$, hence, as this polynomial has leading coefficient $1$ it cannot be self-reciprocal.

Denote the $\ell$-th coefficient of the polynomial $f$ by $[z^\ell]f(z)$. If $k=r-2$ we have $[z]C(z)T(z)=2(-1)^{r-1}$ and $[z^{a_0+\cdots+a_{r-2}}]C(z)T(z)=-r$. As $k>0$ we have $r>2$ and again the polynomial cannot be self-reciprocal.

Finally for $k=r-1$ we have $[z]C(z)T(z)=0$ and $[z^{a_0+\cdots+a_{r-1}-1}]C(z)T(z)=-r$ which again excludes self-reciprocity of $C(z)T(z)$.
\end{proof}

\comment{
\rmus{Finding cyclotomic parts in polynomials:   \cite[Section 2]{beukerssmyth}}

\rmus{Modifying and applying techniques developped in  \cite[Section~7]{grossmcmullen}?}
}

\section{Concluding remarks}

In this note we have studied Coxeter polynomials of star-like trees with special emphasis on star-like trees with three arms. It would be nice to extend Theorem~\ref{th:m} to star-like trees $T(a_0,\ldots, a_r)$ with four and more arms in order to get lower estimates on the degrees of the Salem polynomials involved in Theorem~\ref{th:general}. In fact, the estimate on the maximal multiplicity of the irreducible factors of $R_T$ contained in Lemma~\ref{lem:mult} can be carried over to star-like trees with larger values of $r$. Concerning Proposition~\ref{farc}, the argument based on Mann's Theorem used in order to settle Case~1 of its proof can be extended to $T(a_0,\ldots, a_r)$, however, in the situation of Case~2 we were not able to prove that the orders of the occurring roots of unity are bounded by a reasonable bound. We expect that a generalization of this case requires new ideas.

\bigskip

{\bf Acknowledgment}.
The authors are indebted to the referee for insightful comments on the first version
of this paper.

\def$'${$'$}


\section{Introduction}

\label{intro}

The essential achievements in nanoscience and nanotechnology during the past
decade lead to great interest in studying nanoscale optical fields. The
phenomenon of surface plasmon amplification by stimulated emission of
radiation (spaser) was proposed in Ref.~\onlinecite{Bergman_Stockman} (see
also Refs.~\onlinecite{Stockman_review,Protsenko}). Spaser generates
coherent high-intensity fields of selected surface plasmon (SP) modes that
can be strongly localized on the nanoscale. The properties of localized
plasmons are reviewed in Refs.
\onlinecite{Agranovich,Klyuchnik,Zayats,Bozhevolnyi}. The spaser consists of
an active medium formed by two-level systems (semiconductor quantum dots
(QDs) or organic molecules) and a plasmon resonant nanosystem where the
surface plasmons are excited. The emitters transfer their excitation energy
by radiationless transitions through near fields to a resonant plasmon
nanosystem.

By today theoretical and experimental studies are focused on
metal-based spasers, where surface plasmons are excited in different
metallic nanostructures of different geometric shapes. A spaser
consisted of the nanosystem formed by the V-shaped silver
nanoinclusion embedded in a dielectric host with the embedded PbS
and PbSe QDs was considered ~in Ref. \cite{Bergman_Stockman}.
  A spaser formed by a silver spherical nanoshell on a
dielectric core with a radius of $10-20\ \mathrm{nm}$, and
surrounded by two dense monolayers of nanocrystal QDs  was
considered in Ref.~\onlinecite{Stockman}. The SPs propagating along
the bottom of a groove (channel) in the metal surface were studied
in Ref.~\onlinecite{Lisyansky}. The SPs are assumed to be coherently
excited by a linear chain of QDs at the bottom of the channel. It
was shown that for realistic values of the system parameters, gain
can exceed loss and plasmonic lasing in
 a ring or linear channels in the silver surface
surrounded by a linear chain of CdSe QDs can occur. In Refs.
\cite{Andrianov_ol,Andrianov_oe,Andrianov_prb1,Andrianov_prb2} the
spaser formed by the metal sphere surrounded by the two-level
quantum dot was studied theoretically. The spaser consisting of the
spherical gain core, containing two-level systems, coated with a
metal spherical plasmonic shell was theoretically analyzed in Ref.
\cite{Baranov}. The experimental study of the spaser formed by $44
 \ \mathrm{nm}$ diameter nanoparticles with the gold spherical core surrounded
by dye-doped silica shell was performed in
Refs.~\cite{Noginov_2007,Noginov}. In this experiment the emitters
were formed by dye-doped silica shell instead of QDs. It was
demonstrated that a two-dimensional array of a certain class of
plasmonic resonators supporting coherent current excitations with
high quality factor can act as a planar source of spatially and
temporally coherent radiation~\cite{Zheludev}. This structure
consists of a gain medium slab supporting a regular array of
silver asymmetric split-ring resonators. The spaser formed by $55\ \mathrm{n
}$-thick gold film with the nano-slits located on the silica
substrate surrounded by PbS QDs was experimentally studied in
Ref.~\onlinecite{Plum}. Room temperature spasing of surface plasmon
polaritons at $1.46\ \mathrm{\mu m}$ wavelength has been
demonstrated by sandwiching a gold-film plasmonic
waveguide between optically pumped InGaAs quantum-well gain media~\cit
{Flynn}.

Since plasmons can be excited also in graphene, and damping in
graphene is much less than in
metals~~\cite{Varlamov,Falkovsky_prb,Falkovsky_conf}, we propose to
use graphene nanoribbon surrounded by semiconductor QDs as the
nanosystem for the spaser. Plasmons in graphene provide a suitable
alternative to plasmons in noble metals, because they exhibit much
tighter confinement and relatively long propagation distances, with
the advantage of being highly tunable via electrostatic
gating~\cite{Koppens}. Besides, the graphene-based spaser can work
in THz frequency regime. Recently there were many experimental and
theoretical studies devoted to graphene known by
unusual properties in its band structure~\cite{Castro_Neto_rmp,Das_Sarma_rmp
. The properties of plasmons in graphene were discussed in Refs.~\cit
{Hwang_Das_Sarma,Lozovik_u,Mikhailov,Geim_plasmons}. The electronic
properties of graphene nanoribbons depend strongly on their size and
geometry~\cite{Brey_Fertig_01,Brey_Fertig}. The frequency spectrum
of oblique terahertz plasmons in graphene nanoribbon arrays was obtained~\cit
{Popov}. Besides, graphene-based spaser seems to meet the new
technological needs, since it works at the infrared  (IR)
frequencies, while the metal-based spaser works at the higher
frequencies. Let us mention that the graphene-based photonic two-
and one-dimensional crystals proposed in Refs.~\onlinecite{BBKKL,BK}
also can be used effectively as the frequency filters and waveguides
for the far infrared region of electromagnetic spectrum.

In this Paper we propose the graphene nanoribbon based spaser
consisting of a graphene nanoribbon surrounded by semiconductor QDs.
The QDs excited by the laser pumping nonradiatively transfer their
excitation to the SPs localized at the graphene nanoribbon, which
results in an increase of intensity of the SP field. We calculate
the minimal population inversion that is the difference between the
surface densities of QDs in the excited and ground states needed for
the net SP amplification and study its dependence on the surface
plasmon wavevector, graphene nanoribbon width  at fixed temperature
for different doping and damping parameters for the armchair
graphene nanoribbon.

The paper is organized in the following way. In Sec.~\ref{dev} the minimal
population inversion for the graphene-based spaser is obtained. The
discussion of the results and conclusions follow in Sec.~\ref{disc}.

\section{Surface plasmon amplification}

\label{dev}

The system under consideration is the graphene nanoribbon, which is
the stripe of graphene at $z=0$ in the plane $(x,y)$, that is
infinite in $x$ direction and has the width $W$ in $y$ direction.
This stripe is surrounded by the deposited dense manolayers of
nanocrystal quantum dots with the dielectric constant $\varepsilon
_{d}$ at $z<0$ and $z>0$. When the quantum dots are optically
pumped, the resonant nonradiative transmission occurs by creating a
surface plasmon localized in the graphene nanoribbon. Our goal is to
show that amplification by QDs can exceed absorption of the surface
plasmon in the graphene nanoribbon. As a result we obtain an
increase of intensity of the surface plasmon field. In other words,
the competition between gain and loss of the surface plasmon field
in the graphene nanoribbon will result in favor of the gain.

Below we derive the expression for the minimal population inversion
per unit area $N_{c}$, needed for the net amplification of SPs in a
doped graphene nanoribbon from the condition that for the regime of
the plasmon amplification the rate $\partial \bar{U}/\partial t$ of
the transfer of the average energy of the QDs is greater than the
heat released per unit time $\partial Q/\partial t$ due to the
absorption of the energy of the plasmon field in the graphene
nanoribbon.

 Let us start from the Poynting theorem for the rate of the
transfer of the energy density from a region of space $\partial \mathcal{W
/\partial t=-\mathrm{div}\vec{S}$, where $\vec{S}$ is the Poynting
vector and assume that the plasmon frequency equals the QD
transition frequency. From the other side the rate of the
transferred energy related to the rates of the average energy of the
QDs and the heat released due to the absorption of the energy by the
graphene nanoribbon can be presented as
\begin{eqnarray}
\label{wuq}
-\frac{\partial }{\partial t}\int \mathcal{W}dV=\frac{\partial \bar{U}}
\partial t}-\frac{\partial Q}{\partial t}\ ,
\end{eqnarray}
where  $V$ is the volume of the system. Therefore, from the Poynting
theorem  we have the following expression
\begin{eqnarray}
\label{S01}
\int \nabla \cdot \vec{S}dV=\frac{\partial \bar{U}}{\partial t}-\frac
\partial Q}{\partial t}\ \ .
\end{eqnarray}

Let us consider now each term in the left  hand side
 of Eq.~(\ref{S01}) separately. The excitation
causing  the generation of plasmons in the graphene nanoribbon comes
from the transitions in the QDs between the excited and ground
states. The average
energy $\bar{U}$ of the QDs characterized by the dipole moment is given by
\cite{Tamm}
\begin{eqnarray}
\label{uav1}  \bar{U}=\frac{1}{2}\int \vec{P}\cdot \vec{E}dV\ ,
\end{eqnarray}
where $\vec{E}$ is the electric field of the graphene nanoribbon
plasmon, and $\vec{P}$ is the polarization of QDs, which is the
average total dipole moment of the unit of the volume $V$. When the
plasmon frequency $\omega $ equals the QD transition frequency, and,
$\vec{E}\sim \exp (-i\omega t)$ and $\vec{P}\sim \exp (-i\omega t)$,
the relation between the polarization of QDs $\vec{P}$ and electric
field of the graphene nanoribbon plasmon $\vec{E}$ has the
form~\cite{Lisyansky}
\begin{eqnarray}
\label{Lis} \vec{P}=-ik\frac{\tau _{p}|\mu |^{2}n_{0}}{\hbar
}\vec{E}\ ,
\end{eqnarray}
where $k=9\times 10^{9}\ \mathrm{N\times m^{2}/C^{2}}$, $n_{0}$ is
the difference between the concentrations of quantum dots in the
excited and ground states, $\tau _{p}$ is the inverse line width,
and $\mu $ is the average off-diagonal element of the dipole moment
of a single QD.

Substituting Eq.~(\ref{Lis}) into Eq.~(\ref{uav1}), we obtain the rate of
the transfer of the average energy of the QDs
\begin{eqnarray}  \label{uav11}
\frac{\partial \bar{U}}{\partial t} = \int \omega \mathrm{Im} \left(\vec{E
\cdot\vec{P}^{\ast}\right) d V = \omega k\frac{\tau _{p}|\mu |^{2}}{\hbar
\int n_{0} |\vec{E}|^{2} d V \ .
\end{eqnarray}

We assume that the distances between the quantum dots are small, so
their effect on a plasmon is the same as that of a continuous
(constant) gain distribution along the graphene nanoribbon.  We
consider the two-dimensional graphene nanoribbon at $z=0$ and assume
it is infinite in $x$ direction, has the width $W$ in $y$ direction
and therefore, $n_{0}=N_{0}\eta (y,-W/2,W/2)\delta (z)$, where
$N_{0}$ is the difference between the numbers of the excited and
ground state quantum dots per unit area of the graphene nanoribbon,
and $\eta (y,-W/2,W/2)=1$ at $-W/2\leq y\leq W/2$, $\eta
(y,-W/2,W/2)=0$ at $y<-W/2$ and $y>W/2$. Then, taking into account
 mentioned above, we obtain from Eq.~(\ref{uav11})
\begin{eqnarray}\label{dudt}
\frac{\partial \bar{U}}{\partial t} &=&\omega k\frac{\tau _{p}|\mu |^{2}}
\hbar }\int_{-\infty }^{+\infty }dx\int_{-\infty }^{+\infty }dy\int_{-\infty
}^{+\infty }dzN_{0}\eta (y,-W/2,W/2)\delta (z)|\vec{E}(x,y,z)|^{2}=
\nonumber  \label{uav111} \\
&=&\omega k\frac{\tau _{p}|\mu |^{2}N_{0}}{\hbar }\int_{-W/2}^{+W/2}d
\int_{-\infty }^{+\infty }dx|\vec{E}(x,y,0)|^{2}\ .
\end{eqnarray

Taking into account the spatial dispersion of the dielectric
function in the graphene
nanoribbon~\cite{Brey_Fertig_01,Brey_Fertig}, we use the following
expression for the rate of the heat  $\partial Q/\partial t$
released due to the absorption of the energy of the plasmon field in
the graphene nanoribbon~\cite{Agranovich,Landau}
\begin{eqnarray}  \label{uav2}
\frac{\partial Q}{\partial t} &=& \int \omega \mathrm{Im} \varepsilon
\omega,q_{x})\eta(y,-W/2,W/2)|\vec{E}|^{2} d V  \nonumber \\
&=& \omega \mathrm{Im} \varepsilon(\omega,q_{x}) \int_{-\infty}^{+\infty} d
x \int_{-W/2}^{+W/2} d y \int_{-\infty}^{+\infty} d z |\vec{E}(x,y,z)|^{2} \
.
\end{eqnarray}
where $\mathrm{Im} \varepsilon(\omega,q_{x})$ is the imaginary part of the
dielectric function $\varepsilon(\omega,q_{x})$ of graphene nanoribbon given
by Eq.~(\ref{epsilon}).

The plasmons in a graphene nanoribbon are excited due to the
radiation caused by the transitions from the excited state to the
ground state on the QDs. Therefore, according to the conservation of
energy, the regime of the amplification of the plasmons in the
graphene nanoribbon is established, if the rate of the transfer of
the average energy  $\partial \bar{U}/\partial t$  of the QDs
 given by Eq.~(\ref{dudt}) is greater than the heat
released rate $\partial Q/\partial t$ due to the absorption of the
energy of the plasmon field in the graphene nanoribbon:
\begin{eqnarray}
\label{ineq} \frac{\partial \bar{U}}{\partial t}>\frac{\partial
Q}{\partial t}\ .
\end{eqnarray}
Substituting Eqs.~(\ref{uav111}) and~(\ref{uav2}) into Eq.~(\ref{ineq}), we
get
\begin{eqnarray}
\label{comb}\omega k\frac{\tau _{p}|\mu |^{2}N_{0}}{\hbar }\int_{-W/2}^{+W/2}dy\int_{
\infty }^{+\infty }dx|\vec{E}(x,y,0)|^{2}>\omega \mathrm{Im}\varepsilon
(\omega ,q_{x})\int_{-\infty }^{+\infty }dx\int_{-W/2}^{+W/2}dy\int_{-\infty
}^{+\infty }dz|\vec{E}(x,y,z)|^{2}\ .
\end{eqnarray}
From Eq.~(\ref{comb}), one can obtain the condition for the
difference between the surface densities of the quantum dots in the
excited and ground state corresponding to the amplification of
plasmons:
\begin{eqnarray}
\label{N0} N_{0}>N_{c}=\frac{\mathrm{Im}\varepsilon (\omega
,q_{x})\int_{-\infty
}^{+\infty }dx\int_{-W/2}^{+W/2}dy\int_{-\infty }^{+\infty }dz|\vec{E
(x,y,z)|^{2}}{k\frac{\tau _{p}|\mu |^{2}}{\hbar }\int_{-\infty
}^{+\infty }dx\int_{-W/2}^{+W/2}dy|\vec{E}(x,y,0)|^{2}}\ ,
\end{eqnarray}
where $N_{c}$ is the critical density of the QDs required for the
amplification of the plasmons. The evaluation of the integrals in
Eq.~(\ref{N0}) requires the knowledge of the electric field of a
plasmon in a graphene nanoribbon. The electric field of a plasmon in
a graphene nanoribbon is derived in Appendix~\ref{ap.el}. Using
Eq.~(\ref{E}) for the electric field of a plasmon, we have:
\begin{eqnarray}  \label{E0}
|\vec{E}(x,y,0)|^{2}=E_{0}^{2}\left( 2q_{x}^{2}\cos
^{2}(q_{y}y)+q_{y}^{2}\right) \ ,
\end{eqnarray}
\begin{eqnarray}  \label{Exyz}
|\vec{E}(x,y,z)|^{2}=E_{0}^{2}e^{-2\alpha |z|}\left( 2q_{x}^{2}\cos
^{2}(q_{y}y)+q_{y}^{2}\right) \ ,
\end{eqnarray}
where $\alpha =\sqrt{q_{x}^{2}+q_{y}^{2}}$ and for the armchair nanoribbon
we have $q_{yn}=2\pi /(3a_{0})\left( (2M+1+n)/(2M+1)\right) $ at the width
W=(3M+1)a_{0}$ \cite{Brey_Fertig_01}, where $a_{0}$ is the graphene
lattice constant, $M$ is the integer. We will use $n=1$.
Substituting Eqs.~(\ref{E0}) and~(\ref{Exyz}) into Eq.~(\ref{N0}),
we obtain
\begin{eqnarray}
\label{N01} N_{0}>N_{c}=\frac{2\hbar \mathrm{Im}\varepsilon (\omega
,q_{x})\int_{-\infty }^{+\infty
}dx\int_{-W/2}^{+W/2}dy\int_{0}^{+\infty }dze^{-2\alpha z}\left(
2q_{x}^{2}\cos ^{2}(q_{y}y)+q_{y}^{2}\right) }{k\tau _{p}|\mu
|^{2}\int_{-\infty }^{+\infty }dx\int_{-W/2}^{+W/2}dy\left(
2q_{x}^{2}\cos ^{2}(q_{y}y)+q_{y}^{2}\right) } \ .
\end{eqnarray}
 Finally from Eq.~(\ref{N01}) we obtain
\begin{eqnarray}
\label{N02} N_{0}>N_{c}=\frac{\hbar \mathrm{Im}\varepsilon (\omega
,q_{x})}{\alpha k\tau _{p}|\mu |^{2}}\ .
\end{eqnarray}
Using Eqs.~(\ref{epsilon}) and (\ref{pi}) one can find $\mathrm{Im
\varepsilon (q_{x},\omega )$:
\begin{eqnarray}  \label{ime}
\mathrm{Im}\varepsilon (q_{x},\omega
)=-\frac{V_{00}(q_{x})f_{1}(q_{x},\beta ,\mu )g_{s}v_{F}q_{x}\omega
\gamma }{\pi \hbar \left( \left( \omega
^{2}-v_{F}^{2}q_{x}^{2}\right) ^{2}+\omega ^{2}\gamma ^{2}\right) }
\ ,
\end{eqnarray}
where $v_{F}$ is the Fermi velocity of electrons in graphene. The
plasmon frequency $\omega$ can be obtained at $\gamma = 0$ from the
condition $\mathrm{Re} \varepsilon (q_{x},\omega) = 0$ using Eqs.~(\re
{epsilon}) and~(\ref{pi}):
\begin{eqnarray}  \label{pl}
\omega^{2} = v_{F}^{2}q_{x}^{2} - \frac{V_{0,0}(q_{x})f_{1}(q_{x},\beta
\mu)g_{s}v_{F}q_{x}}{\pi\hbar} \ .
\end{eqnarray}

To perform the calculations, one should calculate the critical density $N_{c}
$ using Eq.~(\ref{N02}). $N_{c}$ is a function of the wave vector $q_{x}$,
the graphene nanoribbon width $W$, temperature $T$, and electron
concentration $n_{0}$ determined by the doping.

\section{Results and discussion}

\label{disc}

For our calculations we use the following parameters for the PbS and PbSe
QDs. Since the typical energy corresponding to the transition between the
ground and excited electron states for PbS and PbSe QDs synthesized with the
radii from $1 \ \mathrm{nm}$ to $8 \ \mathrm{nm}$ can be $0.7 \ \mathrm{eV}
, we use $\tau_{p} \approx 5.9 \ \mathrm{fs}$, and $|\mu| = 1.9 \times
10^{-17} \ \mathrm{esu} = 19 \ \mathrm{Debye}$ ($1 \ \mathrm{Debye} =
10^{-18} \ \mathrm{esu}$, $1 \ \mathrm{Debye} = 3.33564 \times 10^{-30} \
\mathrm{C\cdot m}$)~\cite{Stockman_QD}. Let us mention that the typical
frequency corresponding to the transition between the ground and excited
electron states for PbS and PbSe QDs, which is $f \approx 170 \ \mathrm{TH
}$, matches the resonance with the plasmon frequency in the armchair
graphene nanoribbon~\cite{Brey_Fertig}. Therefore, PbS and PbSe QDs can be
used for the spaser considered here. The damping in graphene $\gamma =
\tau^{-1}$ determined by $\tau$ is assumed to be either $\tau = 1 \ \mathrm
ps}$ or $\tau = 10 \ \mathrm{ps}$ or $\tau = 20 \ \mathrm{ps}$~\cit
{Neugebauer,Orlita,Dubinov,Emani}.

 The dependence of the critical density of the QDs $N_{c}$ required for
the amplification of the signal on the wave vector $q_{x}$ for the
different doping electron densities $n_{0}$ at the fixed width of
the nanoribbon, temperature and dissipation time $\tau $
corresponding to the damping, obtained using Eq.~(\ref{N02}) is
presented in Fig.~\ref{Fig1}. According to Fig.~\ref{Fig1}, $N_{c}$
decreases as $q_{x}$ and $n_{0}$ increase. Let us mention that at
$q_{x}$ larger than $0.6\ \mathrm{nm^{-1}}$ there is almost no
difference between the values of $N_{c}$ corresponding to the
different doping electron densities $n_{0}$, and for large $q_{x}$
$N_{c} $ converges to approximately $18\ \mathrm{\mu m^{-2}}$. In
Fig. \ref{Fig2} the dependence of the critical density of the QDs
$N_{c}$ required for the amplification of the signal on the wave
vector $q_{x}$ for the different dissipation time corresponding to
the damping at the fixed width of the nanoribbon, temperature and
doping electron densities obtained using
Eq.~(\ref{N02})   is shown. As it follows from Fig.~\ref{Fig2}, $N_{c}$ decreases as
q_{x}$ and $\tau $ increase. This means that higher damping
corresponds to higher $N_{c}$. According to Fig.~\ref{Fig2},
starting with $q_{x}\approx 1.0\ \mathrm{(nm)^{-1}}$, $N_{c}$
depends very weakly on $q_{x}$, converging to some constant values
that depend on the value of $\tau $. The dependence of the critical
density of the QDs $N_{c}$ required for the amplification of the
signal on the width of the nanoribbon $W$ at the different wave
vector  for the fixed dissipation time  corresponding to the
damping,
temperature  and doping electron densities  obtained using Eq.~
\ref{N02}) is  displayed in Fig.~\ref{Fig3}. From Fig.~\ref{Fig3} we
can conclude that $N_{c}$ increases as $W$ increases and decreases
as $q_{x}$
increases. When $W$ increases, the values of $N_{c}$ stronger depend on
q_{x}$. The dependence of the critical density of the QDs $N_{c}$
required for the amplification of the signal  on the frequency $f$
at the different dissipation time  corresponding to the damping for
the fixed temperature  and doping electron density obtained using
Eq.~(\ref{N02}) is shown in Fig.~\ref{Fig4}. As it is demonstrated in Fig.
\ref{Fig4}, $N_{c}$ increases as $f$ and $\tau $ decrease. According
to Fig.~\ref{Fig4}, starting with $f \approx 140\ \mathrm{THz}$,
$N_{c}$ depends very weakly on frequency and converges to some
constant values that depend on the value of $\tau $. The dependence
of the plasmon frequency $f$ on the width of the nanoribbon $W$, for
the different wave vectors  at the fixed dissipation time
corresponding to the damping,
temperature and doping electron density  obtained using Eq.~(\re
{pl}) is presented in Fig.~\ref{Fig5}. According to Fig.~\ref{Fig5},
the plasmon frequency $f$ increases as $q_{x}$ increases and the
width of the nanoribbon $W$ decreases. If in Eq.~(\ref{N02}), the
imaginary part of the dielectric function would not depend on the
width $W$, $N_{c}$ would  not depend on
 $W$. However, due to the complicated dependence of $\mathrm
Im}\varepsilon (\omega ,q_{x})$ on $W$ through $V_{0,0}(q_{x})$
given by
Eq.~(\ref{V002}), this dependence  exists. For the damping time we use
\tau =5\ \mathrm{ns}$, $\tau =10\ \mathrm{ns}$, and $\tau =20\
\mathrm{ns}$,
because such damping for graphene was obtained in the experimental studies
\cite{Neugebauer,Orlita,Dubinov,Emani}. One can conclude from Figs.~\re
{Fig2} and~\ref{Fig4}, that $N_{c}$ decreases when the damping time
$\tau $ increases.

\begin{figure}[tbp]
\includegraphics[width=10cm]{Fig1.eps}
\caption{The dependence of the critical density of the QDs $N_{c}$
required for the amplification of the signal on the wave vector
$q_{x}$ for the different doping electron densities $n_{0}$ at the
fixed width of the nanoribbon $W$, temperature $T$ and dissipation
time $\protect\tau$ corresponding to the damping.} \label{Fig1}
\end{figure}

\begin{figure}[tbp]
\includegraphics[width=10cm]{Fig2.eps}
\caption{The dependence of the critical density of the QDs $N_{c}$
required for the amplification of the signal on the wave vector
$q_{x}$ for the different dissipation time $\protect\tau$
corresponding to the damping at the fixed width of the nanoribbon
$W$, temperature $T$ and doping electron densitiy $n_{0}$.}
\label{Fig2}
\end{figure}

\begin{figure}[tbp]
\includegraphics[width=10cm]{Fig3.eps}
\caption{The dependence of the critical density of the QDs $N_{c}$
required for the amplification of the signal  on the width of the
nanoribbon $W$ at
the different wave vector $q_{x}$ for the fixed dissipation time $\protec
\tau$ corresponding to the damping, temperature $T$ and doping electron
densities $n_{0}$.}
\label{Fig3}
\end{figure}

\begin{figure}[tbp]
\includegraphics[width=10cm]{Fig4.eps}
\caption{The dependence of the critical density of the QDs $N_{c}$
required for the amplification of the signal  on the frequency $f$
at the different dissipation time $\protect\tau$ corresponding to
the damping for the fixed width $W$, temperature $T$ and doping
electron density $n_{0}$.} \label{Fig4}
\end{figure}

\begin{figure}[tbp]
\includegraphics[width=10cm]{Fig5.eps}
\caption{The dependence of the plasmon frequency $f$ on the width of
the nanoribbon $W$, for the different wave vectors $q_{x}$ at the
fixed dissipation time $\tau$ corresponding to the damping,
temperature $T$ and doping electron density $n_{0}$.} \label{Fig5}
\end{figure}

Let us mention that we used the parameters for PbS and PbSe QDs to
calculate $N_{c}$, because among different materials for the QDs the
PbS and PbSe QDs demonstrate the lowest transition
frequency~\cite{Auxier}, which can be in the resonance with the
plasmon in graphene nanoribbon in the IR region of spectrum.
According to Ref.~\onlinecite{Plum}, the transition frequency for
the QDs depends on the radius of the QDs. The PbS QDs with the radii
$2 \ \mathrm{nm}$---$5 \ \mathrm{nm}$ have the transition
frequencies $231$ and $194 \ \mathrm{THz}$~\cite{Plum}. For our
calculations we use PbS and PbSe QDs synthesized with the radii up
to $8 \ \mathrm{nm}$, which can provide the transition frequency $f
\approx 170 \ \mathrm{THz}$~\cite{Stockman_QD}. Let us mention that
changing the radius of the QDs, we can change the frequency of the
QDs resonant to the plasmon frequency in graphene nanoribbon
controlled by the wave vector $q_{x}$, and, therefore, we can
control $N_{c}$ by the radius of the $QDs$. The density of PbS QDs
with the diameter $3.2 \ \mathrm{nm}$ applied for the amplification
of plasmons in a gold film in the experiment~\cite{Plum} was $4\times 10^{6} \ \mathrm
\mu m^{-2}}$. According to Figs.~\ref{Fig1}-\ref{Fig4}, in the
graphene nanoribbon-based spaser there are possibilities to achieve
much less densities of the PbS QDs necessary for amplification than
in the gold film-based spaser.

Let us mention that in our calculations we take into account the
temporal and spatial dispersion of the dielectric function of
graphene nanoribbon in the random phase approximation
\cite{Brey_Fertig_01,Brey_Fertig}. The effects of spatial dispersion
are very important for the properties of spaser based on a flat
metal nanofilm~\cite{Larkin}. Taking into account the spatial
dispersion of the dielectric function of a metal surface in the
local random phase approximation allows to conclude that the strong
interaction of QD with unscreened metal electrons in the surface
nanolayer causes enhanced relaxation due to surface plasmon
excitation and Landau damping in a spaser based on a flat metal
nanofilm~\cite{Larkin}. And we assume that taking into account the
spatial dispersion of the dielectric function of graphene nanoribbon
is also very important to calculate the minimal population inversion
needed for the net SP amplification in the graphene nanoribbon based
spaser.

The advantages of the graphene nanoribbon based spaser are wide
frequency generation region from THz up to IR, small damping --- low
threshold for pumping, possibility of control by the gate. While we
perform our
calculations for IR radiation corresponding to the transition frequencies
170\ \mathrm{THz}$, the graphene-based spaser can work at the frequencies
much below that this one including THz regime.

\acknowledgments

The authors are grateful to M.~I. Stockman for valuable discussions.
The work was supported by PSC CUNY under Grant No.  65572-00 43.


\section*{Introduction}

Quantum particles were long thought to fall into one of two classes: fermions or bosons.
The two classes are distinguished by the effect on the many-particle wavefunction of
exchanging any pair of such particles: for fermions, the wavefunction acquires a minus
sign, whereas for bosons it does not.  It was later understood~\cite{LeinaasMyrheim}
that, while these are indeed the only two possibilities in three or more spatial dimensions,
the 2D case admits a remarkable generalization of the concept of bosons
and fermions known as anyons~\cite{Wilczek1,Wilczek2}.
Mathematically, it is useful to think of anyons in (2+1)-dimensional spacetime, where their exchanges are
viewed as ``braids"~\cite{Frohlich,Witten,NayakReview} in which the particles' worldlines
are wound around one another. In general, exchanging two anyons (i.e. performing a single braid)
can yield any phase between 0 (bosons) and $\pi$ (fermions)~\cite{LeinaasMyrheim}.
Even more strikingly, in certain scenarios the phase acquired after performing a series of exchanges can depend
on the order in which the exchanges occurred~\cite{Frohlich,Witten}. Aside from its fundamental scientific
importance, this ``non-Abelian" braiding of anyons has attracted substantial interest in the condensed
matter physics and quantum information communities as it can be used as a basis
for robust quantum information processing~\cite{Freedman,KitaevComputation,NayakReview}.

Anyons have been theoretically predicted to arise in a variety
of topological phases of matter, for example in fractional quantum Hall systems~\cite{Halperin,Arovas},
where they can exist as deconfined quasiparticle excitations, and in topological superconductors,
where Majorana bound states nucleate at topological defects such as domain walls and vortices.
Despite substantial effort and progress~\cite{Camino,An,Willett,Nakamura,Aasen,Karzig}, as of yet there has been no conclusive experimental evidence of anyonic braiding, Abelian or non-Abelian.

Perhaps counterintuitively, the non-Abelian phases acquired upon
braiding Majorana bound states in superconductors can be understood
from the viewpoint of noninteracting particles, wherein the
single-particle Schr\"odinger equation describes particles as
waves. In this picture, the braiding phases are geometric
phases~\cite{Pancharatnam,Berry1,Berry2} arising from the
the adiabatic variation of the phase texture of the bound-state
wavefunctions as the vortices are wound around one another~\cite{Ivanov}.
Therefore, non-Abelian braiding in its simplest
incarnation can be viewed as a universal wave phenomenon accessible
beyond electronic systems.  This implies that it can be realized
experimentally in the context of photonics, where a wide range of
topological phenomena have been predicted and observed recently
\cite{HaldaneRaghu, Soljacic2009, RechtsmanFloquet, Hafezi2013, RevModPhys.91.015006}.  In fact, non-Abelian gauge fields have recently been observed in a photonic device~\cite{yang2019synthesis}, hinting at the possibility of photonic braiding.  In general, photonic topological devices (as well as those in other bosonic systems) are expected to have entirely complementary applications to
their condensed matter analogues.

In this paper we report on the first measurement of the geometric phase arising from braiding topological defects in an array of photonic waveguides, fabricated using the femtosecond direct laser-writing technique \cite{Davis,Szameit2010}. Following the theoretical proposal of Ref.~\cite{Iadecola2016}, we realize topological defects as vortices in a vector field that encodes the displacements of each waveguide in the array.  The vortices realized in our experiment bind localized topological modes whose single-particle wavefunctions are identical to those of Majorana bound states in a 2D topological superconductor.\footnote{Note however that the model we realize is in symmetry class BDI, while the topological superconductor is in class D~\cite{Chiu}; thus the two systems have important fundamental differences.} Consequently, at the noninteracting level the effect of braiding these vortices is the same as what is expected for Majorana bound states. We experimentally realize such vortices in the waveguide array and measure the effect of braiding one such vortex with a second that resides outside the array at an effectively infinite distance from the first. In order to eliminate the effect of dynamical phases, a 180$^\circ$ braiding operation is performed in two adjacent arrays, each with a vortex at its center. If the sense of rotation of the two arrays is the same (opposite), the relative phase at the core is found to be $0$ ($\pi$).  This observation matches the theoretically predicted geometric phase, providing a clear signature of braiding.

We arrange the waveguides in a near-honeycomb lattice, with each waveguide displaced from its honeycomb position $\b{r}=(x,y)$ by an $\b{r}$- and $z$-dependent amount $\b{u}_{\b{r}}(z)$.
The diffraction of light through this waveguide array is governed by the paraxial wave equation
\begin{equation}
i\partial_{z}\psi(\textbf{r},z)=-\frac{1}{2k_{0}}\nabla^{2}_{\textbf{r}}\psi(\textbf{r},z)-\frac{k_{0}\Delta n(\textbf{r})}{n_{0}}\psi(\textbf{r},z),
\label{eq:propagation}
\end{equation}
where $\psi(\textbf{r},z)$ is the envelope function of the electric field $\textbf{E}(\textbf{r},z)=\psi(\textbf{r},z)e^{i(k_{0}z-\omega t)}\hat{x}$, $k_{0}=2\pi n_{0}/\lambda$ is the wavenumber within the medium, $\lambda$ is the wavelength of light, $\nabla^{2}_{\textbf{r}}$ is the Laplacian in the transverse $(x,y)$ plane, and $\omega=2\pi c/\lambda$. Here $n_{0}$ is the refractive index of the
ambient glass and $\Delta n$ is the refractive index relative to $n_{0},$ which acts as the potential in Eq.~\eqref{eq:propagation}.
Since the displacements are small compared to the lattice spacing $a$ ($|\b{u}| \leq .25 a$) and vary slowly in the $z$-direction, the propagation of light through the waveguide array can also be described by a coupled-mode (i.e. tight-binding) equation:
\begin{equation}\label{eq_paraxial}
i \partial_z c_{\b{r}}(z) = \sum_{\langle \b{r}' \rangle} [ t + \delta t_{\b{r},\b{r}'}(z) ] c_{\b{r}'}(z)
\end{equation}
where $c_{\b{r}}(z)$ denotes the amplitude of light in waveguide $\b{r}$, and the $z$-dependent hopping modification $\delta t_{\b{r},\b{r}'}$ arises from the change in waveguide separation due to the displacements $\b{u}_{\b{r}}, \b{u}_{\b{r}'}$.
Identical to electrons in graphene, the two sublattices of the honeycomb lattice give rise to two energy bands in the spectrum, which touch at two gapless ``Dirac'' points at crystal momenta $\b{K}_{\pm} = (\pm 4\pi / (3\sqrt{3}a) , 0)$.

As shown in Fig.~\ref{fig1}, we choose the displacements $\b{u}_{\b{r}}=(u^x_{\b{r}},u^y_{\b{r}})$ corresponding to a Kekul\'e distortion~\cite{Hou,Iadecola2016} controlled by the complex order parameter $\Delta_\b{r}(z)$.
The magnitude of the order parameter determines the displacement amplitude, while its phase controls the displacement angle according to $\text{arg}(u^x_{\b{r}}+i u^y_{\b{r}}) = \b{K}_{+} \cdot \b{r} + \text{arg}(\Delta_\b{r})$.
When $\Delta_\b{r}(z)$ is constant in space, the Kekul\'e distortion couples the two Dirac cones, leading to a band gap proportional to $| \Delta |$ in the energy spectrum.

Our work concerns order parameters that vary both throughout the lattice and in the $z$-direction.
In particular, we focus on vortex configurations of the order parameter, $\Delta_{\b{r}} = | \Delta | \exp(i [\alpha + q_{v}\,\text{arg}(\b{r} - \b{r}_v)])$, where the phase of $\Delta_{\b{r}}$ winds by $2\pi q_{v}$ about the vortex center $\b{r}_v$ for a vortex of charge $q_{v}$.
For $q_{v}=\pm 1$, such vortices are known to bind a single mid-gap photonic mode, localized near the vortex core \cite{JackiwRossi,Hou,Iadecola2016}.
The presence of this mode is protected by the nontrivial topology of the vortex configuration, as well as the time-reversal and chiral symmetries\footnote{Here, chiral symmetry arises in the effective coupled mode equation due to the bipartite coupling between the two sublattices of the honeycomb lattice, similar to graphene \cite{Semenoff}.} of the system \cite{TeoKane}.
A system with multiple vortices carries a bound mode at each vortex.
Away from the vortex centers, the order parameter varies slowly compared to the lattice scale, and the system remains gapped.
This opens up the possibility of \emph{braiding} the vortex modes as functions of $z$: if braiding is executed adiabatically in $z$, light bound in a vortex mode will not disperse into the gapped bulk.

Braiding has two effects on a vortex: moving its center $\b{r}_v$, and altering the offset $\alpha$ in the local order-parameter phase.
The latter arises due to the inherent nonlocality of vortices---as two vortices braid, each one changes its location in the space-dependent phase field of the other.
The first vortex therefore experiences an effective offset $\alpha + q_{v,2}\,\text{arg}(\b{r}_{v,1} - \b{r}_{v,2})$, which increases or decreases depending on the handedness of the braid and the charge of the second vortex.
The effect of braiding on light trapped in the vortex mode is captured by the offset-dependence of the vortex-mode wavefunction $c^v_\b{r}(\b{r}_v,\alpha)$.
Specifically, the wavefunction is \emph{double-valued} in the offset,
\begin{equation}\label{eq_doublevalue}
c^v_\b{r}(\b{r}_v,\alpha + 2\pi) = -c^v_\b{r}(\b{r}_v,\alpha),
\end{equation}
signifying that light trapped in the vortex mode gains a geometric phase of $\pi$ (i.e. a minus sign) after a full $2\pi$ braid.

The double-valued form of the wavefunction is reminiscent of Majorana wavefunctions at vortices in $p + ip$ superconductors~\cite{Ivanov, ReadGreen} or at superconductor-TI interfaces~\cite{FuKane}.
Indeed, due to this double-valuedness, the photonic zero modes in our system will gain the \emph{same} non-Abelian geometric phases as Majorana zero modes in solid state systems upon braiding~\cite{Iadecola2016}.
Crucially, this phase is gained by the \emph{bosonic} wavefunction describing photons in our system---not the Majorana wavefunction describing electrons in a superconductor---leading to a distinct, reduced class of operations realizable via braiding~\cite{Iadecola2016}.

In this work, we provide a robust verification of the geometric phase $\pi$ gained by the photonic vortex modes under a $2\pi$ rotation of the order parameter $\alpha$.
We detect this phase by performing two `on-chip' interferometry experiments, in which we interfere light from two vortex modes that have undergone different rotations of $\alpha$.
In each experiment, we fabricate a waveguide array containing two disconnected Kekul\'e-distorted honeycomb lattices, which constitute the `left' and `right' arms of an interferometer [see Fig.~\ref{fig1}(A)].
The left and right lattices are initially identical and contain a single vortex in the Kekul\'e order parameter.
As $z$ increases, the offset $\alpha$ of each lattice's order parameter is monotonically increased, or decreased, by $\pi$.
The first experiment serves as a control, with $\alpha\to\alpha+\pi$ in an identical sense of rotation in both lattices.
In the language of braiding, this increase of $\alpha$ can be interpreted as braiding a vortex `at infinity' counterclockwise around each lattice by an angle $\pi$ [as indicated in Fig.~\ref{fig1}(B)].
Since the left and right lattices are identical throughout, light trapped in the left vortex mode should gain the same phase as light trapped in the right (both dynamical and geometric phases), and they should interfere constructively after braiding.

The second experiment serves to detect the geometric phase from a $2\pi$ rotation of $\alpha$.
Here the left lattice undergoes a rotation $\alpha\to\alpha+\pi$, while the right undergoes $\alpha\to\alpha-\pi$.
This is analogous to braiding a vortex at infinity $180^\circ$ counterclockwise about the left lattice, but clockwise about the right [\cite{Supplementary}, Movies S1 and S2].
Although the final left and right lattices are identical, the paths they have traversed differ by a $2\pi$ rotation of $\alpha$, and the relative phase gained by light propagating through the two vortex modes will differ according to the geometric phase of this process.
We note that chiral symmetry fixes the energy of the vortex mode to occur at precisely the middle of the band gap~\cite{JackiwRossi, Hou, Iadecola2016}, such that, despite their different $z$-evolutions, light in the left and right lattices is expected to gain the same dynamical phase during braiding.

The entire experimental waveguide array consists of three stages, occurring in succession as a function of $z$ [see Fig.~\ref{fig1}(A)].
The total length of the sample is 10 cm and the radii of the major and minor axes of the waveguides are 5.35 and 3.5 $\mu$m, respectively.
The first stage prepares the light to be coupled in-phase into the vortex modes of the left and right lattices.
This is done using an `on-chip' Y-junction beam-splitter~\cite{Izutsu}. We first precisely couple light from a tunable-wavelength laser through a lens-tipped fiber into a single waveguide at the input facet of the sample.
We then split this waveguide into two in a symmetric manner, such that both waveguides trap light of nearly equal amplitude and phase.
The two waveguides are subsequently guided to the center of the vortex cores of the left and right lattices.

The middle stage contains the braiding operation, and begins by abruptly initializing all other waveguides in each lattice.
Light traveling in the waveguides of the first stage enters each lattice through a single waveguide near each vortex core, setting the initial condition for the braiding process.
The portion of the input light that overlaps with the vortex mode remains localized in the mode, while the rest diffracts throughout the lattice.
After initialization, the waveguide displacements are smoothly varied in $z$ to produce the desired change in order parameter for the experiment being performed.
We emphasize that the individual waveguides move little during this process: each waveguide only undergoes rotation of its displacement angle from the undistorted honeycomb position.

In Fig.~\ref{fig2}, we show the experimental output of a waveguide array containing only these first two stages.
As expected, a high fraction of the light is localized near each vortex core, indicating that the initialization stage succeeded and the braiding process was performed sufficiently adiabatically.
We also observe a nonzero intensity in nearly all waveguides of both lattices, indicating that light not bound to the vortex modes has diffracted throughout each lattice.

To detect the phase of the vortex mode light after braiding, we add a third, interferometric stage to the waveguide array~\cite{Izutsu}, depicted in Fig.~\ref{fig1}(A).
The left and right lattices are abruptly terminated except for one waveguide each, chosen for its high overlap with the final vortex-mode wavefunction.
Light localized in the left and right vortex modes continues to propagate through the remaining two waveguides. The intensity and phase of these waveguides' light are proportional to those of the respective vortex modes after braiding.
In order to read out the waveguides' relative phase, we first split each waveguide symmetrically into two, similar to the first stage.
The two innermost waveguides of the left-right pair are then combined into a single waveguide, while the two outermost remain separate.
The waveguide combination performs the interferometry: if the two input arms are in phase, they will excite the symmetric bound mode of the combined waveguide; if they are out of phase, they will not overlap the bound mode and thus the light will diffract away.
The combined waveguide intensity $I_C$ thus indicates the intensity of in-phase light.
The outermost left and right waveguide intensities $I_L$ and $I_R$ measure the waveguide intensities before combination.

Results of both experiments including the final stage are shown in Fig.~\ref{fig3}.
As anticipated, braiding the left and right vortices in the same direction results in a combined waveguide with higher intensity than the individual left and right waveguides, indicating constructive interference.
On the other hand, braiding the left and right vortices in opposite directions results in nearly zero combined waveguide intensity, indicating near-complete destructive interference and therefore a relative phase of $\pi$ between the left and right vortex modes.
To quantify these results, we define the \emph{contrast} $\eta$ as the ratio between the combined waveguide intensity and the sum of the individual left and right waveguide intensities, $\eta = I_C / (I_L + I_R)$.
In the ideal case, the contrast achieves the upper bound $1$ for perfect constructive interference and the lower bound $|I_L - I_R|/(I_L + I_R)$ for perfect destructive interference.
As plotted in Fig.~\ref{fig3}(A), the experimentally observed contrast is indeed near $1$ when braiding is performed in the same direction for both lattices,
and is close to zero (the lower bound for symmetric intensities, $I_L = I_R$) when it is performed in opposite directions.

We verify that this phase difference arises as a geometric, rather than a dynamical, phase by demonstrating that the interference is insensitive to the wavelength of light used.
The wavelength sets the `time' scale in the paraxial equation, Eq.~(\ref{eq_paraxial}), so that different-wavelength light acquires different dynamical phases during propagation~\cite{Guglielmon2018}.
If the relative phase between the left and right waveguides arose as a dynamical phase, we would expect it to oscillate as the wavelength of light was varied rather than being quantized at $\pi$.
Instead, as shown in Fig.~\ref{fig3}, we observe quite consistent, non-oscillatory values of $\eta$ for both experiments over a large range of wavelengths, 1450--1650 nm.  The principal sources of error/noise are interference from imperfectly in-coupled light in the braiding stage as well as diffracted light in the interferometric stage.  We note that the random error is significantly lower in the sample in which we observe the $\pi$ phase associated with braiding [red points in Fig.~\ref{fig3}(A)] compared to the `control' sample in which we do not [blue points in Fig.~\ref{fig3}(A)].
The measured $\eta$ values when the two vortices are braided in equal and opposite directions are 1.055$\,\pm\,$0.247 and 0.053$\,\pm\,$0.037, respectively.
This indicates that the relative phase is fixed to $\pi$ and supports its identification as a geometric phase.

In conclusion, we have used a photonic lattice of evanescently-coupled waveguides to directly measure the braiding of vortices.  The $\pi$ phase that we observe here directly implies that non-Abelian braiding operations can be carried out using similar ideas in this and many other platforms that are governed by classical wave equations.  We expect this work to motivate the exploration of braiding physics in interacting bosonic systems (via optical nonlinearity or mediation by Rydberg atoms, for example), as well as the search for applications beyond robust quantum information processing.

\bibliographystyle{Science}


\section{Introduction}
The equation of motion for a supernova remnant (SNR)
can be modeled by a single law of motion or multiple
laws of motion when the appropriate boundary
conditions are provided.
Examples of a single law of motion are:
the Sedov expansion
in
the presence of a circumstellar medium (CSM) with constant density
where  the radius, $r$,
scales as   $r \propto  t^{0.4}$, see \cite{Sedov1959},
and the momentum conservation in the
framework of the thin layer approximation
with CSM at constant density
where   $R \propto  t^{0.25}$,
see \cite{Dyson1997}.
Examples of piece-wise solutions for an SNR can be found
in \cite{Dalgarno1987}: a first energy conserving phase,
$r \propto  t^{0.4}$ followed by a second adiabatic phase
where  $r \propto  t^{0.285}$.
At the same time it has been shown
that in the first ten years of \sn1993j
$r \propto  t^{0.82}$, which means an observed exponent larger
than the previously suggested exponents,
see \cite{Zaninetti2011a}.
The previous analysis  allows posing
a basic question: `Is it possible to find an analytical
solution for SNRs given the three observable astronomical
parameters, age, radius and velocity ?'.
In order to answer the above question,
Section \ref{secprofiles} introduces three profiles for
the CSM,
Section \ref{secmotion} derives three
Pad\'e  approximated laws of motion for SNRs,
and
Section \ref{secastro}
closes the derived equations of motion  for four SNRs.

\section{Profiles of density}

This  section  introduces three density profiles
for the CSM:
an exponential profile,
a  Gaussian    profile,
and a  self-gravitating profile  of Lane--Emden type.

\label{secprofiles}
\subsection{The exponential profile}

This density  is  assumed to have the following
exponential dependence on $r$
in spherical  coordinates:
\begin{equation}
 \rho(r;r_0,b,\rho_0) =
\rho_0  \exp{(-\frac{(r-r_0)}{b})}
\quad ,
\label{profexponential}
\end{equation}
where $b$ represents the scale.
The piece-wise  density is
\begin{equation}
 \rho (r;r_0,b,\rho_0)  = \left\{ \begin{array}{ll}
            \rho_0                      & \mbox         {if $r \leq r_0 $ } \\
            \rho_0 \exp{-(\frac{(r-r_0)}{b})}   & \mbox {if $r >    r_0 $ }
            \end{array}
            \right.
\label{profile_exponential}
\end{equation}
The total mass swept,   $M(r;r_0,b,\rho_0) $,
in the interval $[0,r]$ is
\begin{eqnarray}
M(r;r_0,b,\rho_0) = \nonumber  \\
\frac{4}{3}\,\rho_{{0}}\pi\,{r_{{0}}}^{3}
\nonumber \\
-4\,b \left( 2\,{b}^{2}+2\,br+{r}^{2}
 \right) \rho_{{0}}{{\rm e}^{{\frac {r_{{0}}-r}{b}}}}\pi+4\,b \left( 2
\,{b}^{2}+2\,br_{{0}}+{r_{{0}}}^{2} \right) \rho_{{0}}\pi
\quad .
\end{eqnarray}

\subsection{The Gaussian  profile}

This density  has  the
Gaussian  dependence
\begin{equation}
 \rho(r;r_0,b,\rho_0) =
\rho_0  \exp{(-\frac{1}{2}\frac{r^2}{b^2})}
\quad ,
\label{prof_gaussian}
\end{equation}
and the piece-wise  density is
\begin{equation}
 \rho (r;r_0,b,\rho_0)  = \left\{ \begin{array}{ll}
            \rho_0                      & \mbox         {if $r \leq r_0 $ } \\
            \rho_0\exp{(-\frac{1}{2}\frac{r^2}{b^2})}   & \mbox {if $r >    r_0 $ }
            \end{array}
            \right.
\end{equation}

The total mass swept,   $M(r;r_0,b,\rho_0) $,
in the interval $[0,r]$ is
\begin{eqnarray}
M(r;r_0,b,\rho_0) = \nonumber  \\
\frac{4}{3}\,\rho_{{0}}\pi\,{r_{{0}}}^{3}+4\,\rho_{{0}}\pi\, \big  ( -{{\rm e}^
{-\frac{1}{2}\,{\frac {{r}^{2}}{{b}^{2}}}}}r{b}^{2}+\frac{1}{2}\,{b}^{3}\sqrt {\pi}
\sqrt {2}{\rm erf}   (\frac{1}{2}\,{\frac {\sqrt {2}r}{b}}  )  \big )\nonumber \\
 -
4\,\rho_{{0}}\pi\, \big  ( -{{\rm e}^{-\frac{1}{2}\,{\frac {{r_{{0}}}^{2}}{{b}^
{2}}}}}r_{{0}}{b}^{2}+\frac{1}{2}\,{b}^{3}\sqrt {\pi}\sqrt {2}{\rm erf}   (
\frac{1}{2}\,{\frac {\sqrt {2}r_{{0}}}{b}}  ) \big  )
\quad ,
\end{eqnarray}
where ${\rm erf}$ is the error function,
see \cite{NIST2010}.

\subsection{The Lane--Emden profile}

The Lane--Emden profile when $n=5$,
after \cite{Lane1870,Emden1907},
is
\begin{equation}
\rho(r;r_0,b,\rho_0) =\rho_0 {\frac {1}{{(1+ \frac{{r}^{2}}{3b^2})^{\frac{5}{2}}}} }
\label{profile_lane}
\quad ,
\end{equation}
\begin{equation}
 \rho (r;r_0,b,\rho_0)  = \left\{ \begin{array}{ll}
            \rho_0                      & \mbox         {if $r \leq r_0 $ } \\
            \rho_0 {\frac {1}{{(1+ \frac{{r}^{2}}{3b^2})^{\frac{5}{2}}}} }   & \mbox {if $r >    r_0 $ }
            \end{array}
            \right.
\end{equation}
The total mass swept,   $M(r;r_0,b,\rho_0) $,
in the interval $[0,r]$ is
\begin{eqnarray}
M(r;r_0,b,\rho_0) = \nonumber  \\
\frac{4}{3}\,\rho_{{0}}\pi\,{r_{{0}}}^{3}+4\,{\frac {{b}^{3}{r}^{3}\rho_{{0}}
\sqrt {3}\pi}{ \left( 3\,{b}^{2}+{r}^{2} \right) ^{\frac{3}{2}}}}-4\,{\frac {{
b}^{3}{r_{{0}}}^{3}\rho_{{0}}\sqrt {3}\pi}{ \left( 3\,{b}^{2}+{r_{{0}}
}^{2} \right) ^{\frac{3}{2}}}}
\quad .
\end{eqnarray}

\section{The equation of motion}
\label{secmotion}
The conservation of the  momentum in
spherical coordinates
in the framework of the thin
layer approximation  states that
\begin{equation}
M_0(r_0) \,v_0 = M(r)\,v
\quad ,
\end{equation}
where $M_0(r_0)$ and $M(r)$ are the masses swept at $r_0$ and $r$,
and $v_0$ and $v$ are the velocities of
the thin layer at $r_0$ and $r$.

\subsection{Motion with exponential profile}

Assuming an exponential profile as given by
Eq.~(\ref{profile_exponential})
the velocity is
\begin{equation}
\frac{dr}{dt} = \frac{NE}{DE}
\quad ,
\label{vel_exp}
\end{equation}
where
\begin{eqnarray}
NE =
{-{r_{{0}}}^{3}v_{{0}}}
\nonumber
\quad,
\end{eqnarray}
and
\begin{eqnarray}
DE=
6\,{{\rm e}^{{\frac {r_{{0}}-r}{b}}}}{b}^{3}+6\,{{\rm e}^{{\frac {r_{{0
}}-r}{b}}}}{b}^{2}r
\nonumber \\
+3\,{{\rm e}^{{\frac {r_{{0}}-r}{b}}}}b{r}^{2}-{r_{
{0}}}^{3}-3\,{r_{{0}}}^{2}b-6\,r_{{0}}{b}^{2}-6\,{b}^{3}
\quad .
\nonumber
\end{eqnarray}
In the above differential equation of the first order in $r$, the
variables can be separated and integration
gives the following non-linear equation:
\begin{eqnarray}
\frac {1}{{r_{{0}}}^{3}{\it v_0}}
\bigg (
18\,{{\rm e}^{{\frac {r_{{0}}-r}{b}}}}{b}^{4}+12\,{{\rm e}^{{\frac {r_
{{0}}-r}{b}}}}{b}^{3}r+3\,{{\rm e}^{{\frac {r_{{0}}-r}{b}}}}{b}^{2}{r}
^{2}-{r_{{0}}}^{4}-3\,{r_{{0}}}^{3}b
\nonumber\\
+{r_{{0}}}^{3}r-9\,{r_{{0}}}^{2}{b
}^{2}+3\,{r_{{0}}}^{2}br-18\,{b}^{3}r_{{0}}+6\,r_{{0}}{b}^{2}r-18\,{b}
^{4}+6\,{b}^{3}r
\bigg )
\nonumber \\
=\left( t-{\it t_0} \right)
\label .
\label{eqn_nl_exp}
\end{eqnarray}
In this case is not possible to find an analytical
solution for the radius, $r$,
as a function of  time.
We therefore apply
the Pad\'e rational polynomial
approximation of degree 2 in the numerator
and degree 1 in the denominator about the point $r=r_0$
to the  left-hand  side of
Eq.~(\ref{eqn_nl_exp}):
\begin{equation}
\frac
{
- \left( r_{{0}}-r \right)  \left( -5\,br-br_{{0}}-2\,rr_{{0}}+2\,{r_{
{0}}}^{2} \right)
}
{
2\,v_{{0}} \left( 2\,br-5\,br_{{0}}-rr_{{0}}+{r_{{0}}}^{2} \right)
}
= t-t_0
\quad .
\end{equation}
The resulting Pad\'e  approximant  for the radius $r_{2,1}$ is
\begin{eqnarray}
r_{2,1}=\frac{1}{2\,r_{{0}}+5\,b}
\Bigg  ( r_{{0}}tv_{{0}}-r_{{0}}{\it t_0}\,v_{{0}}-2\,btv_{{0}}+2\,b{\it t_0}\,v_
{{0}}+2\,{r_{{0}}}^{2}+2\,r_{{0}}b
\nonumber \\
+ \biggl ( {4\,{b}^{2}{t}^{2}{v_{{0}}}^{
2}-8\,{b}^{2}t{\it t_0}\,{v_{{0}}}^{2}+4\,{b}^{2}{{\it t_0}}^{2}{v_{{0}}
}^{2}-4\,b{t}^{2}r_{{0}}{v_{{0}}}^{2}}
\nonumber\\
{
+8\,bt{\it t_0}\,r_{{0}}{v_{{0}}}^
{2}-4\,b{{\it t_0}}^{2}r_{{0}}{v_{{0}}}^{2}+{t}^{2}{r_{{0}}}^{2}{v_{{0}
}}^{2}-2\,t{\it t_0}\,{r_{{0}}}^{2}{v_{{0}}}^{2}+{{\it t_0}}^{2}{r_{{0}}
}^{2}{v_{{0}}}^{2}
}
\nonumber \\
{
+42\,{b}^{2}tr_{{0}}v_{{0}}
-42\,{b}^{2}{\it t_0}\,r_{
{0}}v_{{0}}+6\,bt{r_{{0}}}^{2}v_{{0}}-6\,b{\it t_0}\,{r_{{0}}}^{2}v_{{0
}}+9\,{r_{{0}}}^{2}{b}^{2}} \biggl )^{\frac{1}{2}}
\Bigg )
\quad ,
\label{rmotionexp}
\end{eqnarray}
and the velocity is
\begin{equation}
v_{2,1}=\frac{dr_{2,1}}{dt} =\frac{NVE}{DVE}
\quad ,
\label{vmotionexp}
\end{equation}
\begin{eqnarray}
NVE =
4\,v_{{0}} \Big  \{    ( -b/2+1/4\,r_{{0}}   )\times  \nonumber \\
\sqrt {4\,   (
b-\frac{1}{2}\,r_{{0}}   ) ^{2}   ( t-t_{{0}}   ) ^{2}{v_{{0}}}^{2}
+42\,   ( b+1/7\,r_{{0}}   )    ( t-t_{{0}}   ) br_{{0}}
v_{{0}}+9\,{r_{{0}}}^{2}{b}^{2}}+\nonumber  \\  ( 3/4\,b+   ( t/4-1/4\,t_{{0
}}   ) v_{{0}}   ) {r_{{0}}}^{2}+{\frac {21\,r_{{0}}b}{4}
   ( v_{{0}}   ( -{\frac {4\,t}{21}}+{\frac {4\,t_{{0}}}{21}}
   ) +b   ) }
\nonumber  \\
+{b}^{2}v_{{0}}   ( t-t_{{0}}   )
  \Big \}
\quad ,
\end{eqnarray}
and
\begin{eqnarray}
DVE =   \nonumber \\
\sqrt {4\,   ( b-\frac{1}{2}\,r_{{0}}   ) ^{2}   ( t-t_{{0}}
   ) ^{2}{v_{{0}}}^{2}+42\,   ( b+1/7\,r_{{0}}   )    (
t-t_{{0}}   ) br_{{0}}v_{{0}}+9\,{r_{{0}}}^{2}{b}^{2}} \times
\nonumber \\
  ( 2\,r
_{{0}}+5\,b   )
\quad .
\end{eqnarray}

\subsection{Motion with Gaussian profile}

Assuming a   Gaussian  profile as given by
Eq.~(\ref{prof_gaussian})
the velocity is
\begin{equation}
\frac{dr}{dt} = \frac{NG}{DG}
\quad ,
\label{vel_gaussian}
\end{equation}
where
\begin{eqnarray}
NG= -2\,{r_{{0}}}^{3}v_{{0}}
\end{eqnarray}
and
\begin{eqnarray}
DG=
-3\,{b}^{3}\sqrt {\pi}\sqrt {2}{\rm erf} \left(\frac{1}{2}\,{\frac {\sqrt {2}r
}{b}}\right)
\nonumber \\
+3\,{b}^{3}\sqrt {\pi}\sqrt {2}{\rm erf} \left(\frac{1}{2}\,{
\frac {\sqrt {2}r_{{0}}}{b}}\right)+6\,{{\rm e}^{-\frac{1}{2}\,{\frac {{r}^{2}
}{{b}^{2}}}}}r{b}^{2}
\nonumber \\
-6\,{{\rm e}^{-\frac{1}{2}\,{\frac {{r_{{0}}}^{2}}{{b}^{2
}}}}}r_{{0}}{b}^{2}-2\,{r_{{0}}}^{3}
\quad .
\end{eqnarray}
The appropriate  non-linear equation  is
\begin{eqnarray}
\frac{1}{2\,{r_{{0}}}^{3}v_{{0}}}
\bigg (
  ( -12\,{b}^{4}+6\,r_{{0}}   ( r-r_{{0}}   ) {b}^{2}
   ) {{\rm e}^{-\frac{1}{2}\,{\frac {{r_{{0}}}^{2}}{{b}^{2}}}}}+12\,{b}^{4
}{{\rm e}^{-\frac{1}{2}\,{\frac {{r}^{2}}{{b}^{2}}}}}
\nonumber \\
-3\,\sqrt {\pi}{\rm erf}
  (\frac{1}{2}\,{\frac {\sqrt {2}r_{{0}}}{b}}  )\sqrt {2}{b}^{3}r+3\,{b
}^{3}\sqrt {\pi}\sqrt {2}{\rm erf}   (\frac{1}{2}\,{\frac {\sqrt {2}r}{b}}
  )r
\nonumber \\
+2\,{r_{{0}}}^{3}   ( r-r_{{0}}   )
\bigg )
= t-t_0 \, .
\label{eqn_nl_gaussian}
\end{eqnarray}
The Pad\'e rational polynomial
approximation of degree 2 in the numerator
and degree 1 in the denominator
about $r=r_0$
for the left-hand side of the above equation gives
\begin{eqnarray}
\frac
{
1
}
{
2\,v_{{0}} \left( 2\,{b}^{2}r-5\,r_{{0}}{b}^{2}-r{r_{{0}}}^{2}+{r_{{0}
}}^{3} \right)
}
\Bigg (
-   ( r-r_{{0}}   )   \bigg ( 9\,{{\rm e}^{-\frac{1}{2}\,{\frac {{r_{{0}}
}^{2}}{{b}^{2}}}}}{b}^{2}r
\nonumber \\
-9\,{{\rm e}^{-\frac{1}{2}\,{\frac {{r_{{0}}}^{2}}{{
b}^{2}}}}}r_{{0}}{b}^{2}-4\,{b}^{2}r+10\,r_{{0}}{b}^{2}+2\,r{r_{{0}}}^
{2}-2\,{r_{{0}}}^{3} \bigg  )
\Bigg )
= t-t_0 \, .
\end{eqnarray}
The resulting Pad\'e  approximant  for the radius $r_{2,1}$ is
\begin{eqnarray}
r_{2,1}=
\frac{1}
{
9\,{{\rm e}^{-\frac{1}{2}\,{\frac {{r_{{0}}}^{2}}{{b}^{2}}}}}{b}^{2}+2\,{r_{{0
}}}^{2}-4\,{b}^{2}
}
\Bigg \{
9\,{{\rm e}^{-\frac{1}{2}\,{\frac {{r_{{0}}}^{2}}{{b}^{2}}}}}r_{{0}}{b}^{2}-2
\,{b}^{2}tv_{{0}}
\nonumber \\
+2\,{b}^{2}{\it t_0}\,v_{{0}}+{r_{{0}}}^{2}tv_{{0}}-{r
_{{0}}}^{2}{\it t_0}\,v_{{0}}-7\,r_{{0}}{b}^{2}+2\,{r_{{0}}}^{3}
\nonumber \\
+ \bigg [
{54\,{b}^{4}r_{{0}}v_{{0}} \Big( t-{\it t_0} \Big) {{\rm e}^{-\frac{1}{2}\,{
\frac {{r_{{0}}}^{2}}{{b}^{2}}}}}
}
\nonumber \\
{
+4\, \Big(  \Big( t-{\it t_0}
 \Big)  \Big( {b}^{2}-\frac{1}{2}\,{r_{{0}}}^{2} \Big) v_{{0}}-\frac{3}{2}\,r_{{0
}}{b}^{2} \Big) ^{2}}
\bigg ]^{\frac{1}{2}}
\Bigg \}
\label{rmotiongauss}
\, ,
\end{eqnarray}
and the velocity is
\begin{equation}
v_{2,1}=\frac{dr_{2,1}}{dt} =\frac{NVG}{DVG}
\quad ,
\label{vmotiongauss}
\end{equation}
\begin{eqnarray}
NVG =
-   \Bigg ( -27\,{{\rm e}^{-\frac{1}{2}\,{\frac {{r_{{0}}}^{2}}{{b}^{2}}}}}r_{{0}
}{b}^{4}+   ( 2\,{b}^{2}-{r_{{0}}}^{2}   )    ( v_{{0}}
   ( t-t_{{0}}   ) {r_{{0}}}^{2}
\nonumber \\
+3\,r_{{0}}{b}^{2}
-2\,v_{{0}}{b
}^{2}   ( t-t_{{0}}   )
+ \bigg \{ {54\,{b}^{4}r_{{0}}v_{{0}}
   ( t-t_{{0}}   ) {{\rm e}^{-\frac{1}{2}\,{\frac {{r_{{0}}}^{2}}{{b}^{
2}}}}}}
\nonumber \\
{
+4\,   (    ( t-t_{{0}}   )    ( {b}^{2}-\frac{1}{2}\,{r_{{0
}}}^{2}   ) v_{{0}}-3/2\,r_{{0}}{b}^{2}   ) ^{2}}   )
  \bigg \} ^{\frac{1}{2}} \Bigg ) v_{{0}}
\quad ,
\end{eqnarray}
and
\begin{eqnarray}
DVG =
\Bigg \{ {54\,{b}^{4}r_{{0}}v_{{0}}   ( t-t_{{0}}   ) {{\rm e}^{-1
/2\,{\frac {{r_{{0}}}^{2}}{{b}^{2}}}}}+4\,   (    ( t-t_{{0}}
   )    ( {b}^{2}-\frac{1}{2}\,{r_{{0}}}^{2}   ) v_{{0}}
}
\nonumber \\
{
-3/2\,r_{{0
}}{b}^{2}   ) ^{2}}
\Bigg \}^{\frac{1}{2}}
( 9\,{{\rm e}^{-\frac{1}{2}\,{\frac {{r_{{0}}}^{2
}}{{b}^{2}}}}}{b}^{2}+2\,{r_{{0}}}^{2}-4\,{b}^{2}   )
\quad .
\end{eqnarray}

\subsection{Motion with the Lane--Emden profile}

Assuming a Lane--Emden profile, $n=5$,     as given by
Eq.~(\ref {profile_lane}),
the velocity is
\begin{equation}
\frac{dr}{dt} = \frac{NL}{DL}
\quad ,
\label{vel_lane}
\end{equation}
where
\begin{eqnarray}
NL = {r_{{0}}}^{3}v_{{0}} \left( 3\,{b}^{2}+{r}^{2} \right) ^{\frac{3}{2}} \left( 3
\,{b}^{2}+{r_{{0}}}^{2} \right) ^{\frac{3}{2}}
\end{eqnarray}
and
\begin{eqnarray}
DL = -3\, \left( 3\,{b}^{2}+{r}^{2} \right) ^{\frac{3}{2}}\sqrt {3}{r_{{0}}}^{3}{b}
^{3}+3\, \left( 3\,{b}^{2}+{r_{{0}}}^{2} \right) ^{\frac{3}{2}}\sqrt {3}{b}^{3
}{r}^{3}
\nonumber \\
+ \left( 3\,{b}^{2}+{r}^{2} \right) ^{\frac{3}{2}} \left( 3\,{b}^{2}+{
r_{{0}}}^{2} \right) ^{\frac{3}{2}}{r_{{0}}}^{3}
\, .
\end{eqnarray}
The connected  non-linear equation  is
\begin{eqnarray}
\frac {1}
{
{r_{{0}}}^{3}v_{{0}} \left( 3\,{b}^{2}+{r_{{0}}}^{2} \right) ^{\frac{3}{2}}
\sqrt {3\,{b}^{2}+{r}^{2}}
}
 \times
\nonumber \\
\bigg (
54\,   ( {b}^{2}+\frac{1}{3}\,{r_{{0}}}^{2}   )    ( \frac{1}{18}\,{r_{{0}}}
^{3}   ( r-r_{{0}}   ) \sqrt {3\,{b}^{2}+{r}^{2}}
\nonumber \\
+{b}^{3}\sqrt
{3}   ( {b}^{2}+\frac{1}{6}\,{r}^{2}   )    ) \sqrt {3\,{b}^{2}+{r_
{{0}}}^{2}}-54\,\sqrt {3\,{b}^{2}+{r}^{2}}\sqrt {3}{b}^{3}   ( {b}^
{4}
\nonumber \\
+\frac{1}{2}\,{b}^{2}{r_{{0}}}^{2}+\frac{1}{18}\,r{r_{{0}}}^{3}   )
\bigg ) =t-t_0
\nonumber
\quad .
\end{eqnarray}
The Pad\'e rational polynomial
approximation of degree 2 in the numerator
and degree 1 in the denominator for the left-hand side of the above equation gives
\begin{eqnarray}
\frac
{
NP
}
{
2\, \left( 3\,{b}^{2}+{r_{{0}}}^{2} \right) ^{\frac{3}{2}}v_{{0}} \left( 2\,r{
b}^{2}-5\,{b}^{2}r_{{0}}-r{r_{{0}}}^{2} \right)
}
=t-t_0
\, ,
\end{eqnarray}
where
\begin{eqnarray}
PN =
-27\,   ( r-r_{{0}}   )
  \Big ( \big  ( -\frac{4}{9}\,   ( r{b}^{2}-\frac{5}{2}\,{b}
^{2}r_{{0}}-\frac{1}{2}\,r{r_{{0}}}^{2} \big  ) \times
\nonumber \\
  \big ( {b}^{2}+\frac{1}{3}\,{r_{{0}}}
^{2}  \big ) \sqrt {3\,{b}^{2}+{r_{{0}}}^{2}}+{b}^{5}\sqrt {3}  \big (
r-r_{{0}} \big  )   \Big  )
\, .
\end{eqnarray}
The  Pad\'e  approximant  for the radius is
\begin{eqnarray}
r_{2,1}=\frac{NR}{DR}
\label{rmotionlaneemden}
\end{eqnarray}
where
\begin{eqnarray}
NR= -18\,   ( {b}^{2}+\frac{1}{3}\,{r_{{0}}}^{2}    ) ^{2}{b}^{2}
  ( -\frac{1}{2}\,{r_{{0}}}^{3}-\frac{1}{2}\,v_{{0}}   ( t-t_{{0}}    ) {r_{{0}}}^{2}
\nonumber \\
+ \frac{7}{2} \,{b}^{2}r_{{0}}+{b}^{2}v_{{0}}   ( t-t_{{0}}    )     )
\sqrt {3\,{b}^{2}+{r_{{0}}}^{2}}+   ( 81\,{b}^{9}r_{{0}}+27\,{b}^{7
}{r_{{0}}}^{3}    ) \sqrt {3}
\nonumber \\
+\sqrt {972}
\Bigg ( {   ( {b}^{2}+\frac{1}{3}
\,{r_{{0}}}^{2}    ) ^{4}{b}^{4}   ( \frac{9}{2}\,\sqrt {3}r_{{0}}{b}^{5
}v_{{0}}   ( t-t_{{0}}    ) \sqrt {3\,{b}^{2}
+{r_{{0}}}^{2}}
}
\nonumber \\
{
+
 \bigg  ( -\frac{1}{2}\,{r_{{0}}}^{3}-\frac{1}{2}\,v_{{0}}   ( t-t_{{0}}    ) {r_{
{0}}}^{2}-\frac{3}{2}\,{b}^{2}r_{{0}}
}
\nonumber \\
{
+{b}^{2}v_{{0}}   ( t-t_{{0}}    )
    ) ^{2}   ( {b}^{2}+\frac{1}{3}\,{r_{{0}}}^{2}    )  \bigg   ) } \Bigg )^{\frac{1}{2}}
\, ,
\end{eqnarray}
and
\begin{eqnarray}
DR=
{b}^{2}   ( 3\,{b}^{2}+{r_{{0}}}^{2}   ) \bigg   ( 27\,{b}^{5}
\sqrt {3}-12\,{b}^{4}\sqrt {3\,{b}^{2}+{r_{{0}}}^{2}}
\nonumber \\
+2\,{b}^{2}{r_{{0
}}}^{2}\sqrt {3\,{b}^{2}+{r_{{0}}}^{2}}+2\,{r_{{0}}}^{4}\sqrt {3\,{b}^
{2}+{r_{{0}}}^{2}} \bigg  )
\quad ,
\end{eqnarray}
and the velocity is
\begin{equation}
v_{2,1}=\frac{dr_{2,1}}{dt} =\frac{NVL}{DVL}
\quad ,
\label{vmotionlaneemden}
\end{equation}
where
\begin{eqnarray}
NVL =
-18\,\sqrt {3}   ( 3\,{b}^{2}+{r_{{0}}}^{2}   ) v_{{0}}
  \bigg  (
  \bigg  ( -243\,   ( {b}^{2}+\frac{1}{3}\,{r_{{0}}}^{2}   ) ^{2}{b}^{7}r_
{{0}}\sqrt {3}
\nonumber \\
+\sqrt {972}
{ \Bigg \{  ( {b}^{2}+\frac{1}{3}\,{r_{{0}}}^{2}
   ) ^{4}{b}^{4}   ( 9/2\,\sqrt {3}r_{{0}}{b}^{5}v_{{0}}
   ( t-t_{{0}}   ) \sqrt {3\,{b}^{2}+{r_{{0}}}^{2}}
}
\nonumber \\
{
+   ( {b}
^{2}+\frac{1}{3}\,{r_{{0}}}^{2}   ) \bigg   ( -\frac{1}{2}\,{r_{{0}}}^{3}-\frac{1}{2}\,v_{{0
}}   ( t-t_{{0}}   ) {r_{{0}}}^{2}-3/2\,{b}^{2}r_{{0}}
}
\nonumber  \\
{
+{b}^{2}v
_{{0}}   ( t-t_{{0}}   )    ) ^{2} \bigg  ) }
\Bigg \}^{\frac{1}{2}}
  ( 2\,{b}^
{2}-{r_{{0}}}^{2}   ) \Bigg   ) \sqrt {3\,{b}^{2}+{r_{{0}}}^{2}}
\nonumber  \\
-
108\,   ( {b}^{2}+\frac{1}{3}\,{r_{{0}}}^{2}   ) ^{3}{b}^{2}   ( -1/
2\,{r_{{0}}}^{3}-\frac{1}{2}\,v_{{0}}   ( t-t_{{0}}   ) {r_{{0}}}^{2}-3
/2\,{b}^{2}r_{{0}}
\nonumber  \\
+{b}^{2}v_{{0}}   ( t-t_{{0}}   )    )
   ( {b}^{2}-\frac{1}{2}\,{r_{{0}}}^{2} )
\quad ,
\end{eqnarray}
and
\begin{eqnarray}
DVL =
18\,\sqrt {972}\sqrt {3}
\Bigg
\{
  ( {b}^{2}+\frac{1}{3}\,{r_{{0}}}^{2}
   ) ^{4}{b}^{4}   ( 9/2\,\sqrt {3}r_{{0}}{b}^{5}v_{{0}}
\nonumber \\
   ( t-t_{{0}}   ) \sqrt {3\,{b}^{2}+{r_{{0}}}^{2}}+   ( {b}
^{2}+\frac{1}{3}\,{r_{{0}}}^{2}   )    ( -\frac{1}{2}\,{r_{{0}}}^{3}-\frac{1}{2}\,v_{{0
}}   ( t-t_{{0}}   ) {r_{{0}}}^{2}
\nonumber  \\
-3/2\,{b}^{2}r_{{0}}+{b}^{2}v
_{{0}}   ( t-t_{{0}}   )    ) ^{2}   )
\Bigg
\}
^{\frac{1}{2}}
  (
   ( -12\,{b}^{4}+2\,{b}^{2}{r_{{0}}}^{2}+2\,{r_{{0}}}^{4}   )
\sqrt {3\,{b}^{2}+{r_{{0}}}^{2}}
\nonumber \\
+27\,{b}^{5}\sqrt {3}   )
\quad .
\end{eqnarray}

\section{Astrophysical Applications}
\label{secastro}
In the previous section, we derived three equations of motion
in the form of non-linear  equations
and three Pad\'e approximated equations of motion.
We now check the reliability of the numerical and approximated
solutions on four SNRs: Tycho, see \cite{Williams2016},
Cas A, see \cite{Patnaude2009},  Cygnus loop,  see \cite{Chiad2015},
and  SN~1006, see \cite{Uchida2013}.
The three astronomical measurable parameters
are the time since the explosion in years, $t$,
the actual observed radius in pc, $r$,
and the present velocity of expansion in
km\,s$^{-1}$, see Table \ref{tablesnrs}.
\begin{table}[ht!]
\caption {
Observed astronomical parameters of SNRs
}
\label{tablesnrs}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
Name  & Age (yr)   &  Radius (pc) & Velocity (km\,s$^{-1}$)& References\\
\hline
Tycho        & 442        &  3.7         & 5300 & Williams~et~al.~2016    \\
Cas ~A       & 328        &  2.5         & 4700 & Patnaude~and~Fesen~2009 \\
Cygnus~loop  & 17000      &  24.25       & 250  & Chiad~et~al.~2015       \\
SN ~1006     & 1000       &  10.19       & 3100 & Uchida~et~al.2013       \\
\hline
\end{tabular}
\end{center}
\end{table}
The astrophysical  units have not yet been specified:
pc for length  and  yr for time
are the units most commonly used by astronomers.
With these units, the initial velocity is
$v_0(km s^{-1})= 9.7968 \, 10^5 v_0(pc\,yr^{-1})$.
The determination of the four  unknown parameters, which are
$t_0$, $r_0$, $v_0$ and $b$,
can be obtained by equating the observed astronomical velocities
and radius with those obtained with the
Pad\'e rational polynomial, i.e.
\begin{eqnarray}
\label{eqnnl1}
        r_{2,1}&= Radius(pc),\\
\label{eqnnl2}
        v_{2,1}&=Velocity(km s^{-1}).
\end{eqnarray}

In order to reduce the unknown parameters from four to two, we
fix $v_0$ and  $t_0$.
The two parameters $b$ and $r_0$ are found by solving the
two non-linear equations (\ref{eqnnl1})
and (\ref{eqnnl2}).
The results for the three types of profiles here adopted
are reported in Tables
\ref{tablesnrsexp},
\ref{tablesnrsgauss}
and
\ref{tablesnrslaneemden}.

\begin{table}[ht!]
\caption {
Theoretical  parameters of SNRs
for the Pad\'e  approximated equation of motion
with an exponential profile.
}
\label{tablesnrsexp}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Name         &$t_0$(yr)&$r_0$(pc)&$v_0(km\,s^{-1})$& b(pc)&$\delta\,(\%)$&
$\Delta\,v (km\,s^{-1}) $ \\
\hline
Tycho        &  0.1  &  1.203    & 8000   &  0.113  &   5.893  & -1.35   \\
Cas ~A       &  1    &  0.819    & 8000   &  0.1    &   6.668  & -3.29   \\
Cygnus~loop  &  10   &  12.27    & 3000   & 45.79   &   6.12   & -0.155  \\
SN ~1006     &  1    &  5.49     & 3100   & 2.332   &   1.455  & -12.34  \\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}[ht!]
\caption {
Theoretical  parameters of SNRs
for the Pad\'e  approximated equation of motion
with a Gaussian  profile.
}
\label{tablesnrsgauss}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Name         &$t_0$(yr)&$r_0$(pc)&$v_0(km\,s^{-1})$& b(pc)&$\delta\,(\%)$&
$\Delta\,v (km\,s^{-1}) $ \\
\hline
Tycho        &  0.1  &  1.022    & 8000   &  0.561   &   8.517  & -10.469   \\
Cas ~A       &  1    &  0.741    & 7000   &  0.406   &   7.571  & -13.16   \\
Cygnus~loop  &  10   &  11.92    & 3000   &  21.803  &   7.875  & -0.161  \\
SN ~1006     &  1    &  5.049    & 10000  &  4.311   &   4.568  & -18.58  \\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}[ht!]
\caption
{
Theoretical  parameters of SNRs
for the Pad\'e  approximated equation of motion
with a  Lane--Emden profile.
}
\label{tablesnrslaneemden}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Name         &$t_0$(yr)&$r_0$(pc)&$v_0(km\,s^{-1})$& b(pc)&$\delta\,(\%)$&
$\Delta\,v (km\,s^{-1}) $ \\
\hline
Tycho        &  0.1  &   0.971    & 8000   & 0.502   &    3.27   & -14.83   \\
Cas ~A       &  1    &   0.635    & 8000   & 0.35    &    4.769  & -23.454   \\
Cygnus~loop  &  10   &   11.91    & 3000   & 27.203  &    7.731  & -0.162  \\
SN ~1006     &  1    &   5        & 10000  & 4.85    &    3.297  & -19.334   \\
\hline
\end{tabular}
\end{center}
\end{table}
The    goodness of the approximation is evaluated
through the percentage error, $\delta$, which is
\begin{equation}
\delta = \frac{\big | r_{2,1} - r_E \big |}
{r_E} \times 100
\quad ,
\end{equation}
where $r_{2,1}$ is the
Pad\'e approximated radius and
$r_E$ is the exact solution which is obtained by solving numerically
the non-linear equation of motion, as an example Eq.~(\ref{eqn_nl_exp})
in the exponential case.
The numerical values of $\delta$ are reported
in  column 6 of
Tables
\ref{tablesnrsexp},
\ref{tablesnrsgauss}
and
\ref{tablesnrslaneemden}.
Another useful astrophysical variable  is the predicted decrease in velocity
on the basis of the
Pad\'e approximated velocity, $v_{2,1}$, in 10 years,
see column 7 of
Tables
\ref{tablesnrsexp},
\ref{tablesnrsgauss}
and
\ref{tablesnrslaneemden}.

\section{Conclusions}

The expansion of an SNR can be modeled by the conservation
of momentum in the presence of a decreasing density:
here we analysed an exponential, a Gaussian
and a Lane--Emden profile.
The three equations of motion have complicated left-hand sides
but simple left-hand sides, viz.,  $(t-t_0)$.
The application of the
Pad\'e  approximant to the left-hand side of the complicated equation of motion
allows finding three approximate laws of motion,
see Eqs~(\ref{rmotionexp}, \ref{rmotiongauss}, \ref{rmotionlaneemden}),
and three approximate velocities,
see Eqs~(\ref{vmotionexp}, \ref{vmotiongauss}, \ref{vmotionlaneemden}).
The astrophysical test is performed
on four spherical SNRs assumed to be spherical
and the four sets  of  parameters  are
reported in  Tables
\ref{tablesnrsexp},
\ref{tablesnrsgauss}
and
\ref{tablesnrslaneemden}.
The percentage of error of the Pad\'e  approximated
solutions for the radius is always less than
$10\%$ with respect to the numerical exact solution,
see column 6 of
the three last tables.
In order to produce an astrophysical prediction,
the theoretical decrease in velocity for the four SNRs here
analysed is evaluated,
see column 7 of
Tables
\ref{tablesnrsexp},
\ref{tablesnrsgauss}
and
\ref{tablesnrslaneemden}.

\noindent
{\bf REFERENCES}

\providecommand{\newblock}{}