This paper deals with sequences of random variables $X_n$ only taking values in $\{0,\ldots,n\}$.
The probability generating functions of such random variables are polynomials of degree $n$.
Under the assumption that the roots of these polynomials are either all real or all lie on the unit circle in the complex plane, a quantitative normal approximation bound for $X_n$ is established in a unified way.
In the real rooted case the result is classical and only involves the variances of $X_n$, while in the cyclotomic case the fourth cumulants or moments of $X_n$ appear in addition.
The proofs are elementary and based on the Stein–Tikhomirov method.

Berry–Esseen bound,
central limit theorem,
cyclotomic polynomial,
fourth moment theorem,
generating function,
real rooted polynomial,
Stein–Tikhomirov method

MSC. 05A15, 60F05

§ INTRODUCTION AND RESULTS

Let $n\in\N$ be a natural number and $X_n$ a random variable that only takes integer values in $\{ 0, \dots , n \}$.
We abbreviate its expectation $\mathbb{E}X_n$ by $\mu_n$,
its variance $\Var(X_n)$ by $\sigma_n^2$
and its fourth cumulant $\Cum_4(X_n) = \E X_n^4-3(\E X_n^2)^2$ by $\kappa_n$. In what follows we shall implicitly assume that $\sigma_n^2 > 0$ to avoid discussion of degenerate cases.
Moreover, by $\gf_n \colon \C \to \C$ we denote the (probability) generating function of $X_n$, that is,

\gf_n(z) := \E[z^{X_n}]= \sum_{k=0}^n \PP(X_n=k) z^k,\qquad z\in\C.

Clearly, $\gf_n$ is a polynomial of degree less than or equal to $n$.
The complex roots of $\gf_n$ will be denoted by $\nst^{(n)}_1, \dots, \nst^{(n)}_n \in \C$, where we do not necessarily assume them to be pairwise distinct.

Starting with the seminal work of Harper [6] it has become an active and fruitful direction of research to deduce probabilistic limit theorems for the sequence of random variables $X_n$ as $n\to\infty$ from information about the location of the zeros $\nst^{(n)}_1, \dots, \nst^{(n)}_n$ of $\gf_n$.
Thereby the following two cases have attracted particular interest:

$\gf_n$ is real rooted, that is, all roots $\nst^{(n)}_1, \dots, \nst^{(n)}_n$ are non-positive real numbers;

$\gf_n$ is cyclotomic, meaning that all roots $\nst^{(n)}_1, \dots, \nst^{(n)}_n$ are located on the unit circle in the complex plane.
In this case, one also says that $g_n$ is root unitary.

It is a remarkable observation that a large number of random variables arising in problems having their origin in statistical mechanics or in algebraic and probabilistic combinatorics fall into these two classes.
For example, the number of descents or the number of cycles in a uniform random permutation in finite Coxeter groups (such as the classical groups of type $A$, $B$ or $D$) lead to real rooted generating functions. Also, the matching polynomial associated with an arbitrary graph is real rooted by the celebrated Heilmann–Lieb theorem [8]. As special cases this yields real rootedness of the monomer-dimer partition function, the $r$-Touchard polynomials or the $r$-Lah number polynomials, for example. We also mention that real rootedness of a polynomial has deep connections with unimodality and log-concavity properties of its coefficient sequence. On the other hand, cyclotomic generating functions arise in the context of the number of inversions in a uniform random permutation in finite Coxeter groups and are prevalent in the theory of polynomials whose coefficients are quotients of so-called $q$-integers. Examples include the $q$-Catalan or the $q$-Narayana numbers, the $q$-analogues of the Hook length formula for standard Young tableaux or the $q$-analogue of the Weyl dimension formula for highest weight modules of semi-simple Lie algebras. Limit theorems for sequences of random variables under assumption <ref> or <ref> can be found in [1] or [9], respectively.
For more examples, background material and further references we refer the reader to the survey articles [2, 3, 12].

Under assumption <ref> it is a folklore result in probability theory that the sequence of random variables $X_n$ satisfies a central limit theorem if and only if the variances $\sigma_n^2$ satisfy

\begin{align}
\sigma_n^2\to\infty&\qquad\text{as }n\to\infty,
\label{iff:realrooted}
\end{align}

see [1, 3, 12].
On the other hand, in the framework of set-up <ref> it was shown in [9] that a necessary and sufficient condition for $X_n$ to satisfy a central limit theorem is that the fourth cumulants $\kappa_n$ satisfy

\begin{align}
\frac{\kappa_n}{\sigma_n^4}\to 0&\qquad\text{as }n\to\infty.
\label{iff:cyclotomic}
\end{align}

The goal of this article is to provide an elementary and unified approach to the following Berry-Esseen type quantitative central limit theorem under assumption <ref> or <ref>. It is an important observation that the bounds we obtain match the necessary and sufficient conditions presented in (<ref>) and (<ref>).
In what follows we let

\begin{align*}
\frac{1}{\sqrt{2\pi}}\int_{-\infty}^w e^{-x^2/2}\,\textrm{d}x,\qquad w\in\R,
\end{align*}

be the distribution function of a standard normal random variable.

For $n\in\N$ let $X_n$ be a random variable only taking values in $\{ 0, \dots , n \}$.
Let $\mu_n=\E X_n$, $\sigma_n^2=\Var(X_n)>0$, $\kappa_n=\Cum_4(X_n)$ and $g_n$ be the generating function of $X_n$. Then, there are absolute constants $c_1>0$ and $c_2>0$ such that the following assertions are valid.

*  If $g_n$ is real rooted as defined in <ref>, it holds that

\begin{align*}
\sup_{w\in\R} \Big\vert \PP\Big( \frac{X_n-\mu_n}{\sigma_n} \leq w \Big)-F(w) \Big\vert
&\leq \frac{c_1}{\sigma_n}.
\end{align*}

*  If $g_n$ is cyclotomic as defined in <ref>, one has that

\begin{align*}
\sup_{w\in\R} \Big\vert \PP\Big( \frac{X_n-\mu_n}{\sigma_n} \leq w \Big)-F(w) \Big\vert
&\leq c_2 \frac{\vert\kappa_n\vert^{\frac14}}{\sigma_n}.
\end{align*}

In principle, the result of Theorem <ref> is known and can be regarded as a special case of the main result of [11].
In that paper, the roots of the generating function are assumed to be outside a circle of radius $d_n \in (0,1)$ around $1$ or outside a cone pointed at the origin with central angle $\delta_n \in (0,\pi]$, see Figure <ref>.
Under these assumptions, Berry–Essen type bounds of order $\log n/(d_n\sigma_n)$ or $1/(\delta_n \sigma_n)$, respectively, were derived.
Moreover, for random variables whose generating functions are of the form

\begin{align}
g_n(z) &= \frac{f(z)}{f(1)}\qquad\text{with}\qquad f(z)=\prod_{j} \frac{1-z^{b_j}}{1-z^{a_j}},\quad a_j,b_j\in\N,
\label{eq:ass:HST}
\end{align}

a quantitative central limit theorem was obtained in [7], among other results, by the method of cumulants.
In this case, the rate of convergence was shown to be of the order $M_n/\sigma_n$, where $M_n:=\max(b_j-a_j)$.
Even though it is known from [9] that in the cyclotomic case <ref> the behaviour of the fourth cumulant regulates whether or not a central limit theorem holds, see (<ref>),
our result is the first to provide a Berry–Esseen type bound including the fourth cumulant. We remark in this context that for all examples treated in [2, 7, 9] our result leads to a speed of convergence of the same order and differs only by an absolute constant.
Our computation we present in Section <ref> below can in principle be combined with the main result in [11] to derive a result similar to Theorem <ref>. However, the computations and arguments that are carried out in [11] require substantial theoretical and technical background, though. For this reason, we believe that a self-contained and elementary proof  of  Theorem <ref> is of independent interest.

Our proof also delivers estimates for the constants $c_1$ and $c_2$ in Theorem <ref>:

\begin{align*}
c_1 \leq 169
\qquad\text{and}\qquad
c_2 \leq 447.
\end{align*}

This should be compared to the constants $2^{3261}$ and $2^{3257}$ from [11] for the two already mentioned more general cases,
and to the constant $324\sqrt{2} \cdot (\pi \sqrt{7/24})^{-1} \leq 271$ from [7] for the special case satisfying (<ref>).

As we already pointed out above, our contribution is an elementary and unified route to Theorem <ref>,
which is based on the so-called Stein–Tikhomirov method.
The classical Stein's method for normal approximation is based on the observation
that a real-valued random variable $Z$ is standard normally distributed
if and only if

\begin{equation}\label{eq:SteinCharcaterization}
\E[ f^\prime(Z) - Zf(Z) ]=0
\end{equation}

for all sufficiently regular functions $f \colon \R\to\R$.
If $Z$ is not standard normally distributed, the expectation $\E[ f^\prime(Z) - Zf(Z) ]$ can be used to quantify how `close' the distribution of $Z$ is to the standard normal distribution.
Eventually, this leads to bounds for common probability distances,
such as the Kolmogorov distance or the Wasserstein distance.
For a detailed introduction to Stein's method, see [4].
Tikhomirov [14] combined this core idea of Stein's method with the more traditional theory of characteristic functions.
Choosing for fixed $t\in\R$ the function $f$ in (<ref>) as $f(x) = e^{itx}$, $x \in \R$, leads to the ordinary differential equation

\begin{align}
t \cf_Z(t) + \cf_Z^\prime(t) = 0 \label{eq:STM}
\end{align}

for all $t \in \R$,
where $\cf_Z(t) := \E[e^{itZ}]$ denotes the characteristic function of $Z$.
The differential equation in (<ref>) indeed characterizes the characteristic function of the standard normal distribution, which is $e^{-\frac12 t^2}$, $t \in \R$.
This motivates the idea of evaluating the left-hand side of (<ref>) even if $Z$ is not exactly standard normally distributed,
in order to derive a non-uniform bound for the difference $\vert \cf_Z(t) - e^{-\frac12 t^2} \vert$ of characteristic functions.
This bound can then be combined with Esseen's classical smoothing lemma, which states that

\begin{align*}
\sup_{w\in\R} \big\vert \PP(Z \leq w) - F(w) \big\vert
&\leq \frac1\pi \int_{-T}^T \bigg\vert \frac{\cf_Z(t) - e^{-\frac12 t^2}}{t} \bigg\vert\,\textrm{d}t + \frac{12\sqrt{2}}{\sqrt{\pi^3}T}
\end{align*}

for all $T>0$, see <cit.>. Many important aspects of this strategy are carried out rather implicitly in the original paper [14],
but a lemma summarizing and partly generalizing the essential ingredients of this approach is the content of <cit.>. A version of it can be rephrased as follows:
Suppose there are continuous functions $a,b \colon \R \to \C$ and non-negative constants $A_0, A_1, B_0, B_1, B_2,C \in \R$
satisfying $A_0 < \frac12$,
$\cf_Z^\prime(t) = -t (1 + a(t)) \cf_Z(t) + b(t)$,
$\vert a(t) \vert \leq A_0 + A_1 \vert t \vert$, and
$\vert b(t) \vert \leq B_0 + B_1 \vert t \vert + B_2 t^2$
for all $t\in(-C,C)$.

\begin{align}
\nonumber\sup_{w\in\R} \big\vert \PP(Z \leq w) - F(w) \big\vert
\leq{}&
\frac2\pi A_0
+ \frac{4}{3\sqrt{\pi}} A_1
+ \sqrt{\pi} B_0 \\
\nonumber&+ \frac2\pi \big( 1 + 2 \max \{ 0, -\log 2s \} \big) B_1 \\
&+ \frac4{\pi s} B_2
+ \frac{24s}{\pi \sqrt{2\pi}} \label{eq:SteinTikBound}
\end{align}

for all $s \geq \max \big\{ \frac{2A_1}{1-2A_0} , \frac{1}{C} \bigr\}$. In fact, this follows from Lemma 2.3 in [13] by realizing that the assumptions on $a(t)$ and $b(t)$ in said lemma actually are only required to hold in an interval $(-C,C)$ around zero and not necessarily for the whole real line.

The Berry-Esseen type bound  (<ref>) is the starting point of our proof of Theorem <ref>, which in turn is the content of the remaining parts of this paper. More precisely, in Section <ref> we collect some preliminary computations and introduce some further notation. The proof of Theorem <ref> is the content of the final Section <ref>.

§ PRELIMINARY COMPUTATIONS

To simplify our presentation, in this section and the next one we fix $n\in\N$ and suppress the index $n$ in what follows.
In particular, we write
$X$ for a random variable $X_n$ taking values in $\{0,\ldots,n\}$,
$\mu$ for the expectation,
$\sigma^2$ for the variance,
$\kappa$ for the fourth cumulant,
$\gf$ for the generating function of $X$
$\nst_1, \dots, \nst_n$ for the (not necessarily distinct) complex roots of $g$. We also put $\roots := \{ \nst_1, \dots , \nst_n \}$.

We start by noting that for all $z \in \C$,

\begin{align}
\gf(z)
&= \E[z^X]
= \sum_{k=0}^n \PP(X=k) z^k\nonumber\\
&= \PP(X=n) \prod_{k=0}^n ( z - \nst_k )
, \nonumber
\intertext{and}
\gf^\prime(z)
&= \PP(X=n) \sum_{k=0}^n \prod_{\substack{ \ell=0 \\ \ell \neq k}}^n ( z - \nst_\ell )
\end{align}

Since all coefficients of $\gf$ are real-valued and positive or zero, there cannot be any root on the positive real axis $\{z\in\C:\Im(z)=0,\Re(z)>0\}\subset\C$.
Further, if there is a root with non-zero imaginary part, then its complex conjugate is a root as well and it also has the same multiplicity.
Excluding the case of $z$ being a root of $\gf$, we can rewrite (<ref>) as follows:

\begin{align}
\gf^\prime(z)
&= \gf(z)  \sum_{k=0}^n \frac1{z - \nst_k}
\label{gf-prime-relation-to-gf}
\end{align}

for all $z \in \C \setminus \roots$.

For every $k\in\{ 1, \dots , n \}$,
let $(\nstrad_k,\nstarg_k) \in [0, \infty) \times (-\pi, \pi]$
be the polar coordinates of $\nst_k$,
i.e. $\nst_k = \nstrad_k  e^{i\nstarg_k}$.
If $0 \in \C$ is a root, we assign to it the polar coordinates $(0,\pi)$.
We define the angle set $\rootsarg := \{ \nstarg_1, \dots , \nstarg_n \}$
and denote $\delta := \min_{\nstarg \in \rootsarg} \vert\nstarg\vert$.
Since $0 \not\in \rootsarg$, we have that $\delta>0$, see Figure <ref> for an illustration.

Let $\cf \colon \R \to \C$ be the characteristic function of $X$ given by $\cf(t) := \E[e^{itX}]$, $t\in\R$.
In particular, $\cf$ arises from the generating function $\gf$ via the change of variables $t\mapsto e^{it}$,
so that

\begin{align}
\cf(t)
&= g(e^{it})
\nonumber
\intertext{and}
\cf^\prime(t)
&= i  e^{it} \, g^\prime(e^{it})
\label{cf-prime}
\end{align}

for all $t \in \R$.

(180+:) coordinate (LU) – (:) coordinate (RO);
(0,0) coordinate (CC)
– (LU |- RO) coordinate (LO)
– (RO |- LU) coordinate (RU)
– (LU |- CC) coordinate (LC)
– (RO |- CC) coordinate (RC)
– (CC |- RO) coordinate (CO)
– (CC |- LU) coordinate (CU)

[color=green!8!white,domain=0:1.9,samples=100] (CC) – (:) – (LO) – (LU) – (-:) – cycle;
[color=red!8!white,domain=0:1.9,samples=100] (CC) – (:) – (-:) – cycle;

[Stealth-] (RC) ++ (0.2,0) – (LC);
[Stealth-] (CO) ++ (0,0.2) – (CU);

(LU) rectangle (RO);
[color=orange,very thick] (RU) – (CC) – (RO);
(0.5,0) arc (0::0.5);
(/2:0.33) node[font=] $\delta$;
[color=blue,very thick] (CC) circle(1);
[color=black,dashed] (0,0) – (200:1) node[pos=0.5,sloped,below,font=,inner sep=2] $\vert z\vert=1$;
(  :1.1),
( -:1.1),
(  50:1.0),
( -50:1.0),
(  53:1.4),
( -53:1.4),
(  65:1.2),
( -65:1.2),
(  80:0.8),
( -80:0.8),
( 105:1.1),
( 120:1.4),
( 125:0.7),
( 135:1.6),
( 150:0.8),
( 160:1.3),
( 180:1.4)

[fill=green!75!black,radius=0.03] circle;

Roots of the generating function $\gf$.
The angle parameter $\delta$ is chosen such that no root (green dots) of $\gf$ is in the interior of the cone pointed at the origin and having central angle $\delta$ (red).
Evaluation of $\gf$ around the unit circle (blue) yields the characteristic function $\cf$.
Note that we currently do not necessarily assume $\gf$ to be real rooted or cyclotomic as in <ref> or <ref>, respectively.

Similar to (<ref>),
relation (<ref>) can be rewritten as

\begin{align*}
\cf^\prime(t)
&= i \sum_{k=0}^n \frac{e^{it}}{e^{it} - \nst_k}\, g(e^{it})
\end{align*}

for all $t \in \R \setminus \rootsarg$.
Therefore, defining the function $\quot \colon \R \setminus \rootsarg \to \C$ by

\begin{align}\label{eq:defQ}
\quot(t) := i \sum_{k=0}^n \frac{e^{it}}{e^{it} - \nst_k},
\end{align}

we arrive at

\begin{align*}
\cf^\prime(t)
&= \quot(t) \, g(e^{it}) \\
&= \quot(t) \, \cf(t) ,
\intertext{and}
\cf^{\prime\prime}(t)
&= \quot^\prime(t) \, \cf(t) + \quot(t) \, \cf^\prime(t)\\
&= \bigl( \quot^\prime(t) + (\quot(t))^2 \bigr) \, \cf(t)
\end{align*}

for all $t \in \R \setminus \rootsarg$.

In preparation for later calculations,
we compute the first three derivatives of $\quot$.
For $t \in \R \setminus \rootsarg$ we have

\begin{align}
\quot^\prime(t)
&= i\sum_{k=0}^n \frac{ie^{it}(e^{it} - \nst_k) - e^{it} ie^{it}}{(e^{it} - \nst_k)^2}\nonumber\\
&=  \sum_{k=0}^n \frac{ \nst_k e^{it} }{(e^{it} - \nst_k)^2}
\nonumber \\
&=  \sum_{k=0}^n \biggl(
\frac{ -1 }{1 - \nst_k e^{-it}}
+ \Bigl( \frac{ -1 }{1 - \nst_k e^{-it}} \Bigr)^2
\biggr)
\nonumber
\intertext{and}
\quot^{\prime\prime}(t)
&= \sum_{k=0}^n \biggl(
\frac{ (-\nst_k)(-ie^{-it}) }{(1 - \nst_k e^{-it})^2}
+ 2
\cdot \frac{ - 1 }{1 - \nst_k e^{-it}}
\cdot \frac{ (-\nst_k)(-ie^{-it}) }{(1 - \nst_k e^{-it})^2}
\biggr)
\nonumber \\
&= i \sum_{k=0}^n
\frac{ \nst_k e^{-it} }{(1 - \nst_k e^{-it})^2}
\biggl( 1 - \frac{ 2 }{1 - \nst_k e^{-it}} \biggr)
\intertext{whereas the third derivative satisfies}
\quot^{\prime\prime\prime}(t)
&= i \sum_{k=0}^n \biggl(
\frac{
-i \nst_k e^{-it} (1 - \nst_k e^{-it})^2
- \nst_k e^{-it} 2 (1 - \nst_k e^{-it}) i\nst_k e^{-it}
}{(1 - \nst_k e^{-it})^4}
\nonumber\\
-i \nst_k e^{-it} (1 - \nst_k e^{-it})^3
- \nst_k e^{-it} 3 (1 - \nst_k e^{-it})^2 i\nst_k e^{-it}
}{(1 - \nst_k e^{-it})^6}
\biggr)
\nonumber \\
&= \sum_{k=0}^n
\frac{ \nst_k e^{-it} }{(1 - \nst_k e^{-it})^3}
\biggl( 1 + \nst_k e^{-it} - 2 \frac{ 1 + 2 \nst_k e^{-it} }{1 - \nst_k e^{-it}} \biggr)
\nonumber \\
&= \sum_{k=0}^n
\frac{ -\nst_k e^{-it} }{(1 - \nst_k e^{-it})^4}
\Bigl( (1 + \nst_k e^{-it})^2 + 2 \nst_k e^{-it} \Bigr)
\end{align}

By definition,
the function $\quot$ is closely related to the cumulants of $X$.
In particular, for the first, the second and the fourth cumulant we have that

\begin{align}
\mu
&= \E[X]
= -i \cf^\prime(0)
= -i \quot(0) \, \cf(0) \nonumber\\
&= -i \quot(0)
= \sum_{k=0}^n \frac{1}{1-\nst_k}
, \label{expectation}\\
\sigma^2
&= \Var(X)
= - \cf^{\prime\prime}(0) - \mu^2
= - \bigl( (\quot(0))^2 + \quot^\prime(0) \bigr) + (\quot(0))^2 \nonumber\\
&= - \quot^\prime(0)
= \sum_{k=0}^n \frac{-\nst_k}{(1-\nst_k)^2}
\label{variance}
\intertext{and}
\kappa
&= \Cum_4(X) = \quot^{\prime\prime\prime}(0) \nonumber\\
&= \sum_{k=0}^n
\frac{ -\nst_k }{(1 - \nst_k )^4}
\bigl( (1 + \nst_k )^2 + 2 \nst_k \bigr)
. \label{fourth-cumulant}
\end{align}

Regarding the variance in (<ref>),
it is interesting to note that for any root $\nst\in\roots$,

\begin{align*}
\frac{-\nst}{(1-\nst)^2}
&= \frac{-\nst \cdot (1-\bar\nst)^2}{(1-\nst)^2(1-\bar\nst)^2}
= \frac{-\nst + 2 \vert\nst\vert^2 - \bar\nst\vert\nst\vert^2}{\vert 1-\nst \vert^4}
\end{align*}

Straightforward calculations lead now to the crucial observation that

\begin{align}
\Re\Bigr( \frac{-\nst}{(1-\nst)^2} \Bigr) &\geq 0
& \iff &&
&\Im(\nst)^2 \geq \frac{\Re(\nst) (\Re(\nst)-1)^2}{2-\Re(\nst)},
\label{sigma-re}\\
\Im\Bigr( \frac{-\nst}{(1-\nst)^2} \Bigr) &= 0
& \iff &&
&\Im(\nst)=0 \text{ or } \vert\nst\vert=1
\end{align}

Therefore, the expression $\frac{-\nst}{(1-\nst)^2}$ is real and positive or zero if and only if $\nst$ is real and negative or zero or $\nst$ has modulus $1$,
see Figure <ref>.
These are precisely the cases <ref> and <ref> we consider in Theorem <ref>.

(-3.5,-1.8) coordinate (LU) – (4.0,1.8) coordinate (RO);
(0,0) coordinate (CC)
– (LU |- RO) coordinate (LO)
– (RO |- LU) coordinate (RU)
– (LU |- CC) coordinate (LC)
– (RO |- CC) coordinate (RC)
– (CC |- RO) coordinate (CO)
– (CC |- LU) coordinate (CU)

(LU) rectangle (RO);
plot (, sqrt(*(-1)^2/(2-)) – (LO) – (LC) – cycle;
plot (,-sqrt(*(-1)^2/(2-)) – (LU) – (LC) – cycle;
plot (, sqrt(*(-1)^2/(2-)) – (RO) – (RC) – cycle;
plot (,-sqrt(*(-1)^2/(2-)) – (RU) – (RC) – cycle;

[Stealth-,overlay] (RC) ++ (0.2,0) node[below] $x$ – (LC);
[Stealth-] (CO) ++ (0,0.2) node[right] $y$ – (CU);

(LU) rectangle (RO);
[color=orange,very thick,domain=0:1.9,samples=100]
plot (,sqrt(*(-1)^2/(2-))
plot (,-sqrt(*(-1)^2/(2-));
(1.7,0 |- CU) node[above left,font=,inner sep=2,color=orange] $y^2 = \frac{x (x-1)^2}{2-x}$;
[color=blue,very thick]
(LC) – (CC) circle(1) – (RC);
(-1,0) node[above left,font=,inner sep=2,color=blue] $y  (\vert\nst\vert-1)=0$;
(2,0 |- CU) – (2,0 |- CO) node[pos=0.25,sloped,below,font=,inner sep=2] $x=2$;
(0,0) – (200:1) node[pos=0.5,sloped,below,font=,inner sep=2] $\vert\nst\vert=1$;

(LO) – (CC) node[pos=0.5,above,color=green!66!black] $\Re\bigr( \frac{-\nst}{(1-\nst)^2} \bigr) \geq 0$;
(RO) – (2,0) node[pos=0.5,above,color=red] $\Re\bigr( \frac{-\nst}{(1-\nst)^2} \bigr) < 0$;
(225:1) node[below left,color=blue] $\Im\bigr( \frac{-\nst}{(1-\nst)^2} \bigr) = 0$;

Behaviour of $\frac{-\nst}{(1-\nst)^2}$;
$x$ denotes $\Re(\nst)$, $y$ denotes $\Im(\nst)$.
According to (<ref>),
the real part of $\frac{-\nst}{(1-\nst)^2}$ is positive or zero for all $\nst$ in the green area.
According to (<ref>),
the imaginary part of $\frac{-\nst}{(1-\nst)^2}$ is zero for all $\nst$ on the blue line and the blue circle.
The orange curve is the
right strophoid
of the curve given by $x=2$
with respect to the pole $0+0i$
and the fixed point $1+0i$,
see page 96 in [10].

§ PROOF OF THEOREM <REF>

§.§ Common Ground

We use the same notation as in the previous section and let $X$ be a random variable taking values in $\{0,\ldots,n\}$ for some $n\in\N$, $\mu$ and $\sigma^2>0$ stand for its expectation and variance, respectively.
Let $\widehat X$ be the standardization of $X$, that is,

\widehat X = \frac{1}{\sigma} (X - \mu).

We write $\widehat \cf$ for the characteristic function of $\widehat X$.

\begin{align*}
\widehat\cf(t)
&= \cf \Bigl( \frac{t}{\sigma} \Bigr) \, e^{-i\frac{\mu}{\sigma}t}
\intertext{and}
\widehat\cf^\prime(t)
&= \frac1\sigma
\, \cf^\prime\Bigl( \frac{t}{\sigma} \Bigr)
\, e^{-i\frac{\mu}{\sigma}t}
- i \frac\mu\sigma
\,\cf\Bigl( \frac{t}{\sigma} \Bigr)
\, e^{-i\frac{\mu}{\sigma}t}
\end{align*}

for all $t \in \R$.
Using the function $\quot$ defined by (<ref>) we find

\begin{align*}
\widehat\cf^\prime(t)
&= \frac1\sigma
\, \Bigl( \quot\Bigl( \frac{t}{\sigma} \Bigr) - i\mu \Bigr)
\, \widehat\cf(t)
\end{align*}

for $t \in \R\setminus\sigma\rootsarg$ with $\sigma\rootsarg=\{\sigma\varphi:\varphi\in\Phi\}$.

Following the Stein–Tikhomirov method as explained in Section <ref>, we are interested for $t \in \R \setminus \sigma\rootsarg$ in the differential expression

\begin{align*}
\widehat\cf^\prime(t) + t \widehat\cf(t)
&= \biggl( \frac{ \quot\bigl( \frac{t}{\sigma} \bigr) - i\mu }\sigma + t \biggr) \,\widehat\cf(t)
= \alpha(t) \, \widehat\cf(t)
\end{align*}

with the continuous function

\begin{align*}
\alpha(t) :={}& \frac{ \quot\bigl( \frac{t}{\sigma} \bigr) - i\mu }\sigma + t.
\end{align*}

We aim to show that under the assumptions <ref> or <ref> on the location of the roots of $\gf$ there are constants $A > 0$ and $C > 0$ such that

\begin{align}
\vert \alpha(t) \vert &\leq A \, t^2 \qquad \text{for all}\qquad t \in (-C,C).
\label{assumption}
\end{align}

We can then apply the Stein–Tikhomirov bound (<ref>) with the functions $a(t)$ and $b(t)$ there given by $-ta(t)=\alpha(t)$ and $b(t)\equiv 0$.
This leads to $A_0=B_0=B_1=B_2=0$ and $A_1=A$ and with the choice $s=\max\bigl\{ 2A, \frac1C \bigr\}$, (<ref>) then leads to

\begin{align}
\sup_{w \in\R} \vert \PP(\widehat X \leq w) - F(w)\vert
\frac{4}{3\sqrt\pi} \, A
+ \frac{24}{\pi\sqrt{2\pi}} \, \max\Bigl\{ 2A, \frac1C \Bigr\}.
\label{Stein-Tikho}
\end{align}

In order to verify (<ref>)
we use (<ref>) to write $i\mu = \quot(0)$ and
we use (<ref>) to see that
$t = -\frac{1}{\sigma} \cdot \quot^\prime(0) \cdot \frac{t}{\sigma}$.

\begin{align*}
\alpha(t)
&= \frac{
\quot\bigl( \frac{t}{\sigma} \bigr)
- \quot(0)
- \quot^\prime(0) \cdot \frac{t}{\sigma}
\end{align*}

for $t \in \bigl( -\frac{\sigma\delta}{2} , \frac{\sigma\delta}{2} \bigr) \subset \R \setminus \sigma\rootsarg$.
This equals the remainder term of a Taylor polynomial, which implies that $\alpha(t)$ satisfies the estimate

\begin{align}
\vert \alpha(t) \vert
&\leq \frac{1}{2\sigma}
\sup_{s \in ( -\frac{\sigma\delta}{2} , \frac{\sigma\delta}{2} )}
\Bigl\vert \quot^{\prime\prime} \Bigl( \frac{s}{\sigma} \Bigr) \Bigr\vert
\cdot
\frac{t^2}{\sigma^2}
= \frac{t^2}{2\sigma^3}
\sup_{s \in ( -\frac{\delta}{2} , \frac{\delta}{2} )}
\vert \quot^{\prime\prime}(s) \vert
\label{alpha-estimate}
\end{align}

for all $t \in ( -\frac{\sigma\delta}{2} , \frac{\sigma\delta}{2} )$.

To bound $\quot^{\prime\prime}(s)$, we recall from (<ref>) that

\begin{align*}
\quot^{\prime\prime}(s)
&= i \sum_{k=0}^n
\frac{ \nst_k e^{-is} }{(1 - \nst_k e^{-is})^2}
\biggl( 1 - \frac{ 2 }{1 - \nst_k e^{-is}} \biggr)
\end{align*}

for $s \in ( -\frac{\delta}{2} , \frac{\delta}{2} )$.
Now, take a root $\nst \in \roots$ with polar representation $\nst = \nstrad e^{i \nstarg}$.

\begin{align}
&(1 - \nst e^{-is})^2 e^{i(s-\nstarg)} \nonumber \\
={}& e^{i(s-\nstarg)} - 2 \nstrad + \nstrad^2 e^{i(\nstarg-s)} \nonumber \\
={}& (1+\nstrad^2) \cos(\nstarg-s) - 2 \nstrad - i (1-\nstrad^2) \sin(\nstarg-s) \nonumber\\
={}&  -\frac12 (1+\nstrad)^2 \bigl( 1-\cos(\nstarg-s) \bigr)
+\frac12 (1-\nstrad)^2 \bigl( 1+\cos(\nstarg-s) \bigr) \nonumber \\
&	- i (1-\nstrad^2) \sin(\nstarg-s) \nonumber \\
={}&  \frac12 \Bigl[
(1+\nstrad) \sqrt{ 1-\cos(\nstarg-s) }
- i \cdot \sgn_{\nstarg-s} \cdot
(1-\nstrad) \sqrt{ 1+\cos(\nstarg-s) }
\Bigr]^2,
\label{quot-primeprime-exact}
\end{align}

where $\sgn_x \in \{ -1, 0, +1 \}$ denotes for $x \in \R$ the sign of $\sin(x)$.

\begin{align*}
\vert 1 - \nst e^{-is} \vert
&= \sqrt{ \bigl\vert (1 - \nst e^{-is})^2 e^{i(s-\nstarg)} \bigr\vert }
\\
&= \sqrt{
\frac12 (1+\nstrad)^2 \bigl( 1-\cos(\nstarg-s) \bigr)
\frac12 (1-\nstrad)^2 \bigl( 1+\cos(\nstarg-s) \bigr)
\\
&\geq \frac1{\sqrt{2}} (1+\nstrad) \sqrt{ 1-\cos(\nstarg-s) }
\\
&\geq \frac1{\sqrt{2}} (1+\nstrad) \frac{\sqrt{2}}{\pi} \bigl\vert \vert\nstarg\vert - \vert s \vert \bigr\vert
\\
&\geq \frac1{2\pi} (1+\nstrad) \vert\nstarg\vert,
\end{align*}

where we used that
$1-\cos(x) \geq \frac2{\pi^2} x^2$ for $x \in [-\pi, \pi]$ and that
$\bigl\vert \vert\nstarg\vert - \vert s \vert \bigr\vert \geq \frac{\vert\nstarg\vert}{2}$
since $\vert s \vert < \frac{\delta}{2} \leq \frac{\vert\nstarg\vert}{2}$.
Recalling the representation (<ref>) of $q^{\prime\prime}$ we can now estimate separately both factors appearing there.
The first factor satisfies

\begin{align}
\biggl\vert \frac{ -\nst e^{-is} }{(1 - \nst e^{-is})^2} \biggr\vert
\frac{4\pi^2 \nstrad}{(1+\nstrad)^2\,\nstarg^2}
\leq
\frac{2\pi^2 \nstrad}{(1+\nstrad)^2\,\bigl( 1 - \cos(\nstarg) \bigr)}
\label{quot-primeprime-estimate}
\end{align}

where we used that $1-\cos(x) \leq \frac12 x^2$ for $x \in [-\pi, \pi]$.
For the second factor we get

\begin{align*}
\biggl\vert 1 - \frac{ 2 }{1 - \nst e^{-is}} \biggr\vert
1 + \frac{4\pi}{(1+\nstrad)\,\vert\nstarg\vert}
\leq
1 + \frac{4\pi}{(1+\nstrad)\,\delta}
\end{align*}

where we used that $\delta \leq \vert\nstarg\vert$ by definition of $\delta$.
As an intermediate result, we get

\begin{align*}
\vert \quot^{\prime\prime}(s) \vert
&= \Bigg\vert
i \sum_{k=0}^n
\frac{ \nst_k e^{-is} }{(1 - \nst_k e^{-is})^2}
\biggl( 1 - \frac{ 2 }{1 - \nst_k e^{-is}} \biggr)
\Bigg\vert
\\
&\leq \sum_{k=0}^n
\frac{2\pi^2 \nstrad_k}{(1+\nstrad_k)^2\,\bigl( 1 - \cos(\nstarg_k) \bigr)}
\biggl( 1 + \frac{4\pi}{(1+\nstrad_k)\,\delta} \biggr)
\end{align*}

for $s \in \bigl( -\frac{\delta}{2} , \frac{\delta}{2} \bigr)$.
We emphasize that up to here all estimates hold for every possible root $\nst$, regardless of its location.
To proceed, we shall now specialize to the set-up of <ref> or <ref>.

§.§ Specialization

If a root $\nst$ of $\gf$ is real and negative or zero or if it satisfies $|\nst|=1$, then

\begin{align*}
\biggl\vert \frac{-\nst}{(1-\nst)^2} \biggr\vert
\frac{-\nst}{(1-\nst)^2},
\end{align*}

compare with (<ref>), (<ref>) and with Figure <ref>.
Further, in this case, (<ref>) for $s=0$ reduces to

\begin{align*}
(1 - \nst)^2 e^{-i\nstarg}
&=  \frac12 (1+\nstrad)^2 \bigl( 1-\cos(\nstarg) \bigr).
\end{align*}

Together with (<ref>) this implies

\begin{align*}
\biggl\vert \frac{ -\nst e^{-is} }{(1 - \nst e^{-is})^2} \biggr\vert
\frac{2\pi^2 \nstrad}{(1+\nstrad)^2\,\bigl( 1 - \cos(\nstarg) \bigr)}
\pi^2
\biggl\vert \frac{-\nst}{(1-\nst)^2} \biggr\vert
\pi^2 \cdot
\frac{-\nst}{(1-\nst)^2}
\end{align*}

so that

\begin{align*}
\sup_{s \in ( -\frac{\delta}{2} , \frac{\delta}{2} )}
\vert \quot^{\prime\prime}(s) \vert
\sup_{s \in ( -\frac{\delta}{2} , \frac{\delta}{2} )}
\Biggl\vert
i \sum_{k=0}^n
\frac{ \nst_k e^{-is} }{(1 - \nst_k e^{-is})^2}
\biggl( 1 - \frac{ 2 }{1 - \nst_k e^{-is}} \biggr)
\Biggr\vert
\\
\pi^2\sum_{k=0}^n \frac{-\nst_k}{(1-\nst_k)^2}
\biggl( 1 + \frac{4\pi}{(1+\nstrad_k)\cdot\delta} \biggr)
\\
\sigma^2
\, \frac{5\pi^3}{\delta}
\end{align*}

Combining this with (<ref>) yields

\begin{align*}
\vert \alpha(t) \vert
&\leq \frac{t^2}{2\sigma^3}
\sup_{s \in ( -\frac{\delta}{2} , \frac{\delta}{2} )}
\vert \quot^{\prime\prime}(s) \vert
\leq \frac{5\pi^3}{2\sigma\delta} \, t^2
\end{align*}

for all $t \in ( -\frac{\sigma\delta}{2} , \frac{\sigma\delta}{2} )$.
Finally, (<ref>) leads to the bound

\begin{align}
\sup_{w \in\R} \vert \PP(\widehat X \leq w) - F(w) \vert
\frac{4}{3\sqrt\pi} \, \frac{5\pi^3}{2\sigma\delta}
+ \frac{24}{\pi\sqrt{2\pi}}
\max\biggl\{ \frac{5\pi^3}{\sigma\delta} , \frac{2}{\sigma\delta} \biggr\}
\nonumber\\
&= \frac{c_0}{\sigma\delta},
\label{main-result}
\end{align}

with a constant $c_0$ satisfying $c_0 := \frac{20\pi^3}{6\sqrt\pi} + 60\pi\sqrt{2\pi} < 531$. We are now in the position to complete the proof of Theorem <ref>.

§.§ Proof of Theorem <ref> <ref>

First, we assume the set-up <ref> of part <ref> of Theorem <ref>.
Then, since all roots are real, we have that $\delta = \pi$.
The result thus follows from (<ref>) with the constant $c_1$ given by $c_1:= \frac{c_0}{\pi} = \frac{20\pi^2}{6\sqrt\pi} + 60\sqrt{2\pi} < 169$.

§.§ Proof of Theorem <ref> <ref>

Next, we assume the set-up <ref> of part <ref> of Theorem <ref>.
In this case, all roots $\nst\in\roots$ satisfy $|\nst|=1$ and $\delta$ is in relation with the fourth cumulant $\kappa$ of $X$.
Namely, in this case we get from (<ref>) that

\begin{align*}
\kappa
&= \sum_{k=0}^n
\frac{ -\nst_k (1 + \nst_k )^2 - 2 \nst_k^2 }{(1 - \nst_k )^4}
\\
&= -\sum_{k=0}^n
\frac{
\bigl( \nst_k (1 + \nst_k )^2 + 2 \nst_k^2 \bigr)
}{\vert 1 - \nst_k \vert^8}
\\
&= -\sum_{k=0}^n
\frac{
\bigl( \nst_k + 4 \nst_k^2 + \nst_k^3 \bigr)
\bigl( 1 - 4\bar\nst_k + 6\bar\nst_k^2 - 4\bar\nst_k^3 + \bar\nst_k^4 \bigr)
}{\vert 1 - \nst_k \vert^8}
\\
&= -\sum_{k=0}^n
\frac{
(\nst_k^3 + \bar\nst_k^3)
%+ 0 (\nst_k^2 + \bar\nst_k^2)
- 9 (\nst_k + \bar\nst_k)
+ 16
}{\vert 1 - \nst_k \vert^8}
\\
&= -\sum_{k=0}^n
\frac{
- 18\cos(\nstarg_k)
+ 16
}{\vert 1 - \nst_k \vert^8}
\end{align*}

From (<ref>) we know that
$\vert 1 - \nst \vert = \sqrt{2} \sqrt{1 - \cos(\nstarg)}$
for every root $\nst = e^{i\nstarg} \in \roots$.
Further, we have that $\cos(3x) = 4\cos(x)^3 - 3\cos(x)$ for all $x \in \R$.
This yields

\begin{align*}
\kappa
&= -\sum_{k=0}^n
\frac{
- 24\cos(\nstarg_k)
+ 16
}{2^4 (1-\cos(\nstarg_k))^4}
\\
&= -\sum_{k=0}^n
\frac{
}{16 (1-\cos(\nstarg_k))^4}
\\
&= -\sum_{k=0}^n
\frac{
}{2 (1-\cos(\nstarg_k))^2}
\end{align*}

Note that if $n$ is even, this result already appeared as Equation (8) in [9].
Finally, we use once again the fact that $1-\cos(x) \leq \frac12 x^2$ for $x \in [-\pi, \pi]$,
which implies the bound

\begin{align*}
\vert\kappa\vert
= -\kappa
&\geq \sum_{k=0}^n
\frac{2}{\nstarg_k^4}
\geq \frac{2}{\delta^4}
\end{align*}

Regarding the estimate (<ref>), this yields

\begin{align*}
\sup_{w \in\R} \vert \PP(\widehat X \leq w) - F(w) \vert
\frac{c_0}{\sigma\delta}
\leq
c_2\, \frac{\vert\kappa\vert^{1/4}}{\sigma}
\end{align*}

with the constant $c_2$ given by  $c_2 := \frac{c_0}{2^{\frac14}} = \frac{10\pi^3 2^{\frac34} }{6\sqrt\pi} + 60 \pi^{\frac32} 2^{\frac14} < 447$. This completes the proof of Theorem <ref>.

§ ACKNOWLEDGMENT

BR has been supported by the German Research Foundation (DFG) under project number 459731056. CT was supported by the DFG through SPP 2265 Random Geometric Systems and CRC/TRR 191 Symplectic Structures in Algebra, Geometry and Dynamics.

[1]
E.A. Bender:
Central and local limit theorems applied to asymptotic enumeration.
Journal of Combinatorial Theory. Series A 15 (1973),

[2]
S.C. Billey, J.P. Swanson:
Cyclotomic generating functions.
arXiv: 2305.07620.

[3]
P. Brändén: Unimodality, log-concavity, real–rootedness and beyond.
In Handbook of Enumerative Combinatorics. Chapman and Hall/CRC (2015).

[4]
L.H.Y. Chen, L. Goldstein, Q.-M. Shao:
Normal Approximation by Stein's Method.
Berlin and Heidelberg: Springer (2011).

[5]
W. Feller:
An Introduction to Probability and its Applications, Vol. 2.
Wiley (1966).

[6]
L.H. Harper:
Stirling behavior is asymptotically normal.
Annals of Mathematical Statistics 38 (1967),

[7]
N. Heerten, H. Sambale, C. Thäle:
Probabilistic limit theorems induced by the zeros of polynomials.
To appear in Mathematische Nachrichten (2023+).

[8]
O.J. Heilmann, E.H. Lieb:
Theory of monomer-dimer systems.
Communications in Mathematical Physics, 25 (1972), 190–232.

[9]
H.-K. Hwang, V. Zacharovas:
Limit Distribution of the Coefficients of Polynomials With Only Unit Roots.
Random Structures & Algorithms 46 (2015),

[10]
E.H. Lockwood:
Book of Curves.
Cambridge University Press (1961).

[11]
M. Michelen, J. Sahasrabudhe:
Central limit theorems and the geometry of polynomials.
arXiv: 1908.09020.

[12]
J. Pitman:
Probabilistic bounds on the coefficients of polynomials with only real zeros.
Journal of Combinatorial Theory. Series A 77 (1997),

[13]
A. Röllin:
Kolmogorov bounds for the normal approximation of the number of triangles in the Erdős-Rényi random graph.
Probability in the Engineering and Informational Sciences 36 (2022),

[14]
A.N. Tikhomirov:
On the rate of convergence in the central limit theorem for weakly dependent random variables.
Theory of Probability and Its Applications 25 (1980),


# Bimodal Approach for Noise Figures of Merit Evaluation in Quantum-Limited
Josephson Traveling Wave Parametric Amplifiers ††thanks: This work is
supported by the Italian Institute of Nuclear Physics (INFN) within the
Technological and Interdisciplinary research commission (CSN5), by the
European Union’s H2020-MSCA Grant Agreement No.101027746, by the Institute for
Basic Science (IBS-R017-D1) of the Republic of Korea, by the University of
Salerno - Italy under the projects FRB19PAGAN and FRB20BARON, by the
SUPERGALAX project in the framework of the H2020-FETOPEN-2018-2020 call, and
the Joint Research Project PARAWAVE of the European Metrology Programme for
Innovation and Research (EMPIR). This project (PARAWAVE) received funding from
the EMPIR programme co-financed by the Participating States and from the
European Union Horizon 2020 research and innovation programme.

(Corresponding author: Emanuele Enrico<EMAIL_ADDRESS>
L. Fasolo and A.Greco are with INRiM, Istituto Nazionale di Ricerca
Metrologica, I-10135 Torino, Italy and with Politecnico di Torino, I-10129
Torino, Italy<EMAIL_ADDRESS>a.greco@inrim.it). E. Enrico and
L.Oberto are with INRiM, Istituto Nazionale di Ricerca Metrologica, I-10135
Torino, Italy and with INFN - Trento Institute for Fundamental Physics and
Applications, I-38123, Povo, Trento, Italy<EMAIL_ADDRESS>l.oberto@inrim.it). C. Barone, G. Carapella, and S. Pagano are with University
of Salerno, Department of Physics, I-84084 Fisciano, Salerno, Italy, with INFN
- Napoli, Salerno group, I-84084 Fisciano, Salerno, Italy, and with CNR-SPIN
Salerno section, I-84084 Fisciano, Salerno, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>spagano@unisa.it). M. Borghesi, M. Faverzani, E. Ferri,
A. Giachero, and A. Nucciotti are with University of Milano Bicocca,
Department of Physics, I-20126 Milano, Italy and with INFN - Milano Bicocca,
I-20126 Milano, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>angelo.nucciotti@unimib.it).
A. P. Caricato, A. Leo, G. Maruccio, A. G. Monteduro, and S. Rizzato are with
University of Salento, Department of Physics, I-73100 Lecce, Italy and with
INFN - Lecce, I-73100 Lecce, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>silvia.rizzato@unisalento.it). I.
Carusotto is with INO-CNR BEC Center, I-38123 Povo, Trento, Italy, with
University of Trento, Department of Physics, I-38123, Povo, Trento, Italy and
with Fondazione Bruno Kessler, I-38123, Povo, Trento, Italy
(iacopo.carusotto@ino.cnr.it). W. Chung, Ç. Kutlu, A. Matlashov, Y. K.
Sermertzidis, and S. Uchaikin are with Center for Axion and Precision Physics
Research, Institute for Basic Science (IBS), KR-34051, Daejeon, Republic of
Korea. Ç. Kutlu and Y. K. Sermertzidis are also with Korea Advanced Institute
of Science and Technology (KAIST), Department of Physics, KR-34051, Daejeon,
Republic of Korea<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>uchaikin@ibs.re.kr). A. Cian, P. Falferi, D. Giubertoni,
B. Margesin, and A. Vinate are with Fondazione Bruno Kessler, I-38123, Povo,
Trento, Italy and with INFN - Trento Institute for Fundamental Physics and
Applications, I-38123, Povo, Trento, Italy. P. Falferi and A. Vinante are also
with IFN-CNR, I-38123 Povo, Trento, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>andrea.vinante@ifn.cnr.it). D. Di
Gioacchino, C. Gatti, C. Ligi, G. Maccarrone, L. Piersanti, and A. Rettaroli
are with INFN - Laboratori Nazionali di Frascati, I-00044, Frascati, Rome,
Italy. A. Rettaroli is also with University of Roma Tre, Departement of
Physics, I-00146, Roma, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>alessio.rettaroli@uniroma3.it). G. Filatrella is with University of Sannio,
Department of Science and Technology, I-82100, Benevento and with INFN -
Napoli, Salerno group, I-84084 Fisciano, Salerno, Italy (filatr@unisannio.it).
P. Livreri is with University of Palermo, Department of Engineering, I-90128,
Palermo, Italy and with CNIT - Interuniversity Consortium for
Telecommunication, National Laboratory for Radar and Surveillance Systems,
I-56124, Pisa, Italy (patrizia.livreri@unipa.it). C. Mauro is with INFN -
Napoli, Salerno group, I-84084 Fisciano, Salerno, Italy (cmauro@na.infn.it).
R. Mezzena is with University of Trento, Department of Physics, I-38123, Povo,
Trento, Italy and with INFN - Trento Institute for Fundamental Physics and
Applications, I-38123, Povo, Trento, Italy (renato.mezzena@unitn.it). V.
Pierro is with University of Sannio, Department of Engineering, I-82100
Benevento, Italy and with INFN - Napoli, Salerno group, I-84084 Fisciano,
Salerno, Italy (pierro@unisannio.it). M. Rajteri is with INRiM - Istituto
Nazionale di Ricerca Metrologica, I-10135 Torino, Italy and with INFN -
Torino, I-10125 Torino, Italy (m.rajteri@inrim.it).

L. Fasolo, C. Barone, M. Borghesi, G. Carapella, A. P. Caricato, I. Carusotto,
W. Chung, A. Cian,
D. Di Gioacchino, E. Enrico, P. Falferi, M. Faverzani, E. Ferri, G.
Filatrella, C. Gatti, A. Giachero,
D. Giubertoni, A. Greco, Ç. Kutlu, A. Leo, C. Ligi, P. Livreri, G. Maccarrone,
B. Margesin, G. Maruccio,
A. Matlashov, C. Mauro, R. Mezzena, A. G. Monteduro, A. Nucciotti, L. Oberto,
S. Pagano, V. Pierro,
L. Piersanti, M. Rajteri, A. Rettaroli, S. Rizzato, Y. K. Semertzidis, S.
Uchaikin, and A. Vinante

###### Abstract

The advent of ultra-low noise microwave amplifiers revolutionized several
research fields demanding quantum-limited technologies. Exploiting a
theoretical bimodal description of a linear phase-preserving amplifier, in
this contribution we analyze some of the intrinsic properties of a model
architecture (i.e., an rf-SQUID based Josephson Traveling Wave Parametric
Amplifier) in terms of amplification and noise generation for key case study
input states (Fock and coherents). Furthermore, we present an analysis of the
output signals generated by the parametric amplification mechanism when
thermal noise fluctuations feed the device.

Index Terms \- Microwave photonics, Noise figure, Superconducting microwave
devices.

##  I Introduction

Nowadays, the technological progress in several fields of research, spanning
from quantum computation and communication [1, 2, 3], radio-astronomy [4],
radio detection and ranging [5, 6, 7, 8], up to fundamental physics
experiments [9, 10, 11], led to the demand for ultra-low noise microwave
amplifiers for broadband high-fidelity readout. All these applications are
deeply affected by the noise performances of such amplifiers. In this
contribution, the standard Haus-Caves description of noise generation in an
ideal bosonic phase-preserving linear amplifier is taken into account [12, 13,
14]. In this representation the amplifier is considered as a two-ports black-
box driven at a pump frequency $\omega_{\text{p}}$ that amplifies a bosonic
input mode at frequency $\omega$. The amplification is associated with the
creation of a second mode at frequency
$\omega^{\prime}=\omega_{\text{p}}-\omega$ (the so-called idler mode of a
three-wave mixing parametric amplification [15]) that is commonly considered
as an internal mode of the amplifier that causes the onset of noise at the
output port. Here, we extend and give a different perspective of this
description considering the case in which an uncorrelated idler mode is
already present at the input port (i.e., considering a bimodal input field),
analyzing the effect of the interaction between these modes inside the
amplifier in terms of typical noise estimators.This operative condition may
arise in real measurement setups where the amplifier is exploited, for
instance, for the multiplexed readout of broadband signals [16] or for the
joint detection and amplification of probing signals in a microwave quantum
illumination experiment [8].
The theoretical framework presented in this manuscript is supported with
numerical simulations of the noise estimators for a realistic implementation
of a quantum-limited amplifier[13, 14] such as the rf-SQUID based Josephson
Travelling Wave Parametric Amplifier (JTWPA) [17, 18, 19, 20, 21], which
represents a promising realization of a microwave amplifier with high gain,
large bandwidth and quantum-limited added noise [22]. The core of this device
is a repetition of rf-SQUIDs, embedded in a coplanar waveguide, that generate
a non-dissipative and highly non-linear superconducting metamaterial.

##  II Gain, Quantum Efficiency, and Noise Figure

Within the Heisenberg picture, a generic two-ports phase-preserving linear
amplifier driven at frequency $\omega_{\text{p}}$ combines two uncorrelated
input modes at frequency $\omega$ and
$\omega^{\prime}=\omega_{\text{p}}-\omega\neq\omega$, described through their
dimensionless complex-amplitude operator $\hat{a}_{\omega,\text{in}}$ and
$\hat{a}_{\omega^{\prime},\text{in}}$, in an output mode at frequency $\omega$

$\hat{a}_{\omega,\text{out}}=u(\omega)\hat{a}_{\omega,\text{in}}+iv(\omega)\hat{a}^{\dagger}_{\omega^{\prime},\text{in}}$
(1)

with
$[\hat{a}_{\omega,\text{in}},\hat{a}^{\dagger}_{\omega^{\prime},\text{in}}]=\delta_{\omega,\omega^{\prime}}$.
A pictorial representation of the system is given in Fig. 1.

Figure 1: Pictorial representation of a linear amplifier, driven at
$\omega_{\text{p}}$, having at the input port two uncorrelated fields at
frequency $\omega$ and $\omega^{\prime}=\omega_{\text{p}}-\omega$.

The output mode operator fulfils the common bosonic commutation relation only
if the complex functions $u(\omega)$ and $v(\omega)$ respect the unitary
condition

$|u(\omega)|^{2}-|v(\omega)|^{2}=1$ (2)

As reported in [21] for an rf-SQUID based JTWPA, an expression of these
functions can be analytically derived exploiting a circuit quantum
electrodynamics description of the system. These quantities depend both on the
design parameters of the device and on the driving conditions (i.e., on the
pump frequency $\omega_{\text{p}}$ and on the amplitude of this tone, that can
be quantified in terms of an intensity of current $I_{\text{p}}$). In this
derivation, the ideality condition of the phase-preserving linear amplifier is
guaranteed neglecting all the possible dissipation sources, such as the
dielectric losses or the side-band generation [23]. All the numerical
simulations given below are evaluated for the circuital parameters described
in [21] for an amplifier operating in the three-wave mixing regime and pumped
with a current intensity $I_{\text{p}}$ at frequency
$\omega_{\text{p}}=$12\text{\,}\mathrm{GHz}$$.

The expectation value for the photon number of the output field is given by

$\displaystyle\braket{\hat{n}_{\omega,\text{out}}}=\braket{\hat{a}^{\dagger}_{\omega,\text{out}}\hat{a}_{\omega,\text{out}}}=$
$\displaystyle=|u(\omega)|^{2}\braket{\hat{n}_{\omega,\text{in}}}+|v(\omega)|^{2}\braket{\hat{n}_{\omega^{\prime},\text{in}}}+$
$\displaystyle+i\left(u^{*}(\omega)v(\omega)\braket{\hat{a}^{\dagger}_{\omega,\text{in}}\hat{a}^{\dagger}_{\omega^{\prime},\text{in}}}-u(\omega)v^{*}(\omega)\braket{\hat{a}_{\omega^{\prime},\text{in}}\hat{a}_{\omega,\text{in}}}\right)+$
$\displaystyle+|v(\omega)|^{2}$ (3)

where the first term is the input field at $\omega$ frequency being amplified
of a factor $|u(\omega)|^{2}$, while the second term represents the
contribution to the amplification of the input field at $\omega$ frequency
given by the input field at $\omega^{\prime}$. The third and fourth terms
represent respectively the contributions deriving from the spontaneous
annihilation or creation of a pump photon, that respectively implies the
creation or the annihilation of a couple of photons at $\omega$ and
$\omega^{\prime}$. Eventually the last term, independent from the input
fields, represents the number of photons with frequency $\omega$ added by the
amplifier (the so-called added noise photons).

Supposing a vacuum state
$\ket{\text{vac}}=\ket{0}_{\omega}\ket{0}_{\omega^{\prime}}$ as the input
state of the amplifier, one can easily derive from (II) that the expectation
value for the photon number of the output field is

$\braket{\hat{n}^{\text{vac}}_{\omega,\text{out}}}=\bra{\text{vac}}\hat{n}_{\omega,\text{out}}\ket{\text{vac}}=|v(\omega)|^{2}$
(4)

This means that the added noise photons can be equivalently seen as the
product of the amplification of an input vacuum state. Furthermore, from (II)
derives that the spectral gain distribution of the amplifier $G(\omega)$,
meant as the scale factor of the sole input mode at frequency $\omega$, is
given by

$G(\omega)\equiv|u(\omega)|^{2}$ (5)

whereas, in order to take into consideration the contribution given by the
input field at $\omega^{\prime}$ frequency, it is possible to define a bimodal
gain $G_{\text{b}}(\omega)$ as

$G_{\text{b}}(\omega)\equiv\frac{\braket{\hat{n}_{\omega,\text{out}}}-\braket{\hat{n}^{\text{vac}}_{\omega,\text{out}}}}{\braket{\hat{n}_{\omega,\text{in}}}}$
(6)

This latter quantity reduces to $G(\omega)$ when the $\omega^{\prime}$ input
field is in the vacuum state.

Figure 2: Gain spectrum $(G(\omega))$, expressed in dB, as a function of
$I_{\text{p}}$ normalized on the critical current $I_{\text{c}}$ of the
Josephson junctions composing the referenced device. The dashed lines identify
the bandwidth of this device at the 3dB threshold level

.

Fig. 2 reports the spectral distribution of the gain $G(\omega)$ as a function
of $I_{\text{p}}$ normalized on the critical current of the referenced
Josephson junctions $I_{\text{c}}$. It has to be noticed that, being the rf-
SQUIDs the nonlinear core of the modeled JTWPA, the driving pump current
$I_{\text{p}}$ flowing through the amplifier can exceed the critical current
of the Josephson junctions $I_{\text{c}}$ (i.e., the pump current can be
unevenly divided between the two branches of the rf-SQUIDs, maintaining the
junctions in their superconductive state). The gain results negligible for
small values of $I_{\text{p}}$, whereas an increase in the pumping current
increases the intensity of coupling between tones and thus the gain of the
amplifier, with a corresponding reduction of the bandwidth. The dashed lines
in Fig. 2 represent the edges of this bandwidth, defined exploiting a gain
threshold of 3 dB.

To evaluate the performance of an ideal phase-preserving linear amplifier it
is possible to define its quantum efficiency spectrum as the ratio between the
vacuum fluctuations in the input field and the fluctuations of the output
field, this latter quantity normalized on the bimodal gain of the amplifier
[16]

$\eta(\omega)\equiv\frac{\braket{(\Delta\hat{a}^{\text{vac}}_{\omega,\text{in}})^{2}}}{\braket{(\Delta\hat{a}_{\omega,\text{out}})^{2}}/G_{\text{b}}(\omega)}=\frac{1/2\cdot
G_{\text{b}}(\omega)}{\braket{(\Delta\hat{a}_{\omega,\text{out}})^{2}}}$ (7)

where $\braket{(\Delta\hat{a}_{\omega,\text{in
(out)}})^{2}}=\braket{\hat{a}^{2}_{\omega,\text{in
(out)}}}-\braket{\hat{a}_{\omega,\text{in (out)}}}^{2}$ is the variance of the
input (output) annihilation operator with frequency $\omega$. Another commonly
exploited figure of merit to evaluate the performance of an amplifier is the
noise figure spectrum $\mathcal{F}(\omega)$, defined as the ratio between the
input signal to noise ratio (SNR) and the output SNR

$\mathcal{F}(\omega)\equiv\frac{\text{SNR}_{\text{in}}(\omega)}{\text{SNR}_{\text{out}}(\omega)}$
(8)

with

$\text{SNR}_{\text{in}}(\omega)\equiv\frac{\braket{\hat{n}_{\omega,\text{in}}}^{2}}{\braket{\left(\Delta\hat{n}_{\omega,\text{in}}\right)^{2}}}$
(9)

and

$\text{SNR}_{\text{out}}(\omega)\equiv\frac{\braket{\hat{n}_{\omega,\text{out}}}^{2}-\braket{\hat{n}^{\text{vac}}_{\omega,\text{out}}}^{2}}{\braket{\left(\Delta\hat{n}_{\omega,\text{out}}\right)^{2}}}$
(10)

where $\braket{(\Delta\hat{n}_{\omega,\text{in
(out)}})^{2}}=\braket{\hat{n}^{2}_{\omega,\text{in
(out)}}}-\braket{\hat{n}_{\omega,\text{in (out)}}}^{2}$ is the variance of the
input (output) photon number with frequency $\omega$. In the definition of the
output SNR given in (10) the contribution given by the added noise photons has
been excluded from the amplified output field [24].

The definitions given so far hold for any bimodal input states. In the
following subsections the quantum efficiency and the noise figure associated
to two particular classes of them are presented.

###  II-A Bimodal Fock input states

Considering a generic bimodal Fock state
$\ket{\Psi_{\text{F}}}=\ket{n_{\omega}}_{\omega}\ket{n_{\omega^{\prime}}}_{\omega^{\prime}}$
as the input state of the amplifier, it can be easily demonstrated that the
bimodal gain is

$G_{\text{b},\text{F}}(\omega)=|u(\omega)|^{2}+\frac{n_{\omega^{\prime}}}{n_{\omega}}|v(\omega)|^{2}$
(11)

and the variance of the output annihilation operator at $\omega$ frequency is
$\braket{(\Delta\hat{a}_{\omega,\text{out}})^{2}}_{\text{F}}=1/2\cdot[(1+2n_{\omega})|u(\omega)|^{2}+(1+2n_{\omega^{\prime}})|v(\omega)|^{2}]$.
Exploiting (2) and (5) the quantum efficiency results in

$\eta_{\text{F}}(\omega)=\frac{G(\omega)+(G(\omega)-1)\cdot(n_{\omega^{\prime}}/n_{\omega})}{2G(\omega)[1+n_{\omega}+n_{\omega^{\prime}}]-2n_{\omega^{\prime}}-1}$
(12)

In the high gain limit (i.e., $G(\omega)\gg 1$) this quantity tends to

$\eta_{\text{F}(\omega)}\rightarrow\frac{n_{\omega}+n_{\omega^{\prime}}}{2n_{\omega}(1+n_{\omega}+n_{\omega^{\prime}})}$
(13)

In this limit, supposing the input $\omega^{\prime}$ mode in the vacuum state
(i.e., $n_{\omega^{\prime}}=0$) the quantum efficiency reduces to
$\eta_{\text{F}}(\omega)=1/[2(1+n_{\omega})]$, at most equal to the standard
quantum limit (SQL) $\eta_{\text{SQL}}(\omega)=1/2$. The introduction of an
$\omega^{\prime}$ field at the input port (i.e., $n_{\omega}\neq 0$) induces
an increase of the quantum efficiency of the amplifier, up to the limit
$1/(2n_{\omega})$.

Regarding the noise figure spectrum, it can be derived that
$\mathcal{F}_{\text{F}}(\omega)=\infty$, being
$\text{SNR}_{\text{in}}(\omega)=\infty$, for any value of $n_{\omega}$ and
$n_{\omega^{\prime}}$. Fig. 3 presents the $\text{SNR}_{\text{out}}(\omega)$
for three different bimodal Fock input states as a function of
$I_{\text{p}}/I_{\text{c}}$. For a given input state, across all the frequency
spectrum, the $\text{SNR}_{\text{out}}(\omega)$ decreases with the increase of
$I_{\text{p}}$ (thus with the raise of the gain of the amplifier, as shown in
Fig. 2). On the contrary, when $I_{\text{p}}$ tends to zero, the amplifier
slightly perturbs the input state, and the $\text{SNR}_{\text{out}}(\omega)$
diverges, recovering the value of $\text{SNR}_{\text{in}}(\omega)$. In
accordance with the gain profile, a drastic reduction of the
$\text{SNR}_{\text{out}}(\omega)$ occurs for $I_{\text{p}}$ values that
increase with the increase of the distance of the considered mode $\omega$
from the center of the band of the amplifier. Furthermore, for a fixed value
of $I_{\text{p}}$, an increase of the occupancy of the input $\omega$ mode
induces a raise of the $\text{SNR}_{\text{out}}(\omega)$, while an increase of
the occupancy of the input $\omega^{\prime}$ mode induces a reduction.
Therefore, the drawback of the enhancement of the output power at $\omega$
frequency promoted by an $\omega^{\prime}$ input field is the degradation of
the $\text{SNR}_{\text{out}}(\omega)$.

Figure 3: Signal to Noise Ratios of the output field at $\omega$ frequency
$(\text{SNR}_{\text{out}}(\omega))$, expressed in dB, for three different
bimodal Fock state inputs as a function of $I_{\text{p}}$ normalized on the
critical current $I_{\text{c}}$ of the Josephson junctions composing the
referenced device.

###  II-B Bimodal Coherent input states

In this subsection a bimodal coherent state
$\ket{\Psi_{\text{C}}}=\ket{\alpha_{\omega}}_{\omega}\ket{\alpha_{\omega^{\prime}}}_{\omega^{\prime}}=D(\alpha_{\omega})\ket{0}_{\omega}D(\alpha_{\omega^{\prime}})\ket{0}_{\omega^{\prime}}$,
with $D$ the displacement operator, is considered as the input state of the
amplifier. For this input state the bimodal gain results to be

$\displaystyle G_{\text{b,C}}(\omega)=$
$\displaystyle=|u(\omega)|^{2}+\frac{\alpha^{2}_{\omega^{\prime}}}{\alpha^{2}_{\omega}}|v(\omega)|^{2}+i\frac{\alpha_{\omega^{\prime}}}{\alpha_{\omega}}\left(v(\omega)u^{*}(\omega)+u(\omega)v^{*}(\omega))\right)$
(14)

while the variance of the output annihilation operator is
$\braket{(\Delta\hat{a}_{\omega,\text{out}})^{2}}_{\text{C}}=1/2\cdot(|u(\omega)|^{2}+|v(\omega)|^{2})$.

Figure 4: Quantum efficiency spectrum $(\eta_{C}(\omega))$ for six different
bimodal coherent input states as a function of $I_{\text{p}}$ normalized on
the critical current $I_{\text{c}}$ of the Josephson junctions composing the
referenced device.

Fig. 4 presents the spectral distribution of the quantum efficiency for six
different bimodal coherent input states as a function of
$I_{\text{p}}/I_{\text{c}}$. The shape of the distributions recalls once again
the one of the gain $G(\omega)$. It can be observed that, when the
$\omega^{\prime}$ mode of the bimodal coherent input state is in the vacuum
state (i.e., $\alpha_{\omega^{\prime}}=0$), the quantum efficiency in the high
gain limit tends to the SQL $\eta_{\text{SQL}}=1/2$. Differently from what
described in the previous subsection, this limit is exceeded when the average
occupancy of the input $\omega^{\prime}$ mode is non-zero. The former result
clearly indicates that going beyond the SQL is non limited to a linear
amplifier working in the phase-sensitive regime (i.e., when
$\omega=\omega^{\prime}$), but it can also be obtained in the phase-preserving
regime, in presence of a bimodal coherent input state with a non-zero average
occupancy of the $\omega^{\prime}$ mode. In a complementary way, the exceed of
the SQL can also be shown evaluating the noise figure for different coherent
input states. Being for these states
$\text{SNR}_{\text{in}}(\omega)=|\alpha_{\omega}|^{2}$, the noise figure can
be written as
$\mathcal{F}_{\text{C}}(\omega)=|\alpha_{\omega}|^{2}/\text{SNR}_{\text{out}}(\omega)$.

Fig. 5 reports the spectral distribution of the noise figure for six different
bimodal coherent input states as a function of $I_{\text{p}}/I_{\text{c}}$.
Similarly to what previously discussed, for a fixed value of $I_{\text{p}}$,
an increase in the average occupancy of the input $\omega$ mode (i.e.,
$\alpha_{\omega}$) induces an increase of the noise figure. In the high gain
limit, when $\alpha_{\omega^{\prime}}=0$, this quantity tends to the SQL
$\mathcal{F}_{\text{SQL}}(\omega)=2$, whereas an increase in the average
occupancy of the input $\omega^{\prime}$ mode (i.e.,
$\alpha_{\omega^{\prime}}$) allows to reach values beyond this limit.

Figure 5: Noise figure $(\mathcal{F}_{\text{C}}(\omega))$, expressed in dB,
for six different bimodal coherent states input as a function of
$I_{\text{p}}$ normalized on the critical current $I_{c}$ of the Josephson
junctions composing the referenced device.

##  III Effective Temperature

The quantification of the noise spectrum generated by an amplifier commonly
requires the definition of its effective temperature $T_{\text{eff}}(\omega)$
as the temperature that a Bose-Einstein distribution should have to equal the
output $\omega$ mode occupancy generated by a vacuum input state (see (4))
[22]

$\frac{1}{e^{\hbar\omega/k_{\text{B}}T_{\text{eff}}(\omega)}-1}=|v(\omega)|^{2}$
(15)

where $\hbar$ is the reduced Planck constant and $k_{\text{B}}$ is the
Boltzmann constant. The expression for the effective temperature derives from
(15) exploiting the unitary condition given in (2) and the gain definition
given in (5)

$\displaystyle T_{\text{eff}}(\omega)$
$\displaystyle=\frac{\hbar\omega}{k_{\text{B}}}\left[\ln{\left(1+\frac{1}{|v(\omega)|^{2}}\right)}\right]^{-1}=$
$\displaystyle=\frac{\hbar\omega}{k_{\text{B}}}\left[\ln{\left(1+\frac{1}{G(\omega)-1}\right)}\right]^{-1}$
(16)

which implies that an ideal phase-preserving linear amplifier acting as a
passive element (i.e., $G=1$) has an effective temperature equal to zero
while, when acting as an active element (i.e., $G>1$), its effective
temperature increases with the increase of the gain. Fig. 6 reports the
numerical evaluation of the spectral distribution of the normalized effective
temperature $T_{\text{eff}}(\omega)/G(\omega)$ for the referenced device as a
function of $I_{\text{p}}/I_{\text{c}}$. Here it can be noticed that, in the
high gain limit, the ratio $T_{\text{eff}}(\omega)/G(\omega)$ tends to the SQL
$\hbar\omega/k_{\text{B}}$, corresponding to the sum of the equivalent
temperature of a half of quantum of noise associated to the signal mode, plus
the equivalent temperature of the minimal added noise that the idler mode has
to inject in the $\omega$ mode in order to preserve the bosonic commutation
relations at the output [25].

Figure 6: Spectral distribution of the normalized effective temperature
$T_{\text{eff}}(\omega)/G(\omega)$,

as a function of $I_{\text{p}}$ normalized on the critical current
$I_{\text{c}}$ of the Josephson junctions composing the referenced device.

##  IV Environment contribution to the Output Noise

In this section the behaviour of the amplifier when inserted in a realistic
measurement setup is taken into account. In particular the case in which the
input field of the amplifier is a bimodal thermal-noise state generated by the
surrounding environment at a given temperature $T$ is considered. The density
operator for this state can be written as

$\rho_{\text{th}}=\sum_{n=0}^{\infty}\frac{e^{n\beta\hbar\omega}}{1-e^{\beta\hbar\omega}}\ket{n}_{\omega}\bra{n}_{\omega}\sum_{m=0}^{\infty}\frac{e^{m\beta\hbar\omega^{\prime}}}{1-e^{\beta\hbar\omega^{\prime}}}\ket{m}_{\omega^{\prime}}\bra{m}_{\omega^{\prime}}$
(17)

where $\beta=1/k_{\text{B}}T$. The expectation value for the photon number of
the output field at frequency $\omega$ is obtained tracing over both the two
modes Fock basis

$\displaystyle\braket{\hat{n}_{\omega,\text{out}}}_{\text{th}}$
$\displaystyle=\text{Tr}\left[{\rho_{\text{th}}\;\hat{n}_{\omega,\text{out}}}\right]=$
$\displaystyle=\sum_{j,k=0}^{\infty}\bra{j}_{\omega}\bra{k}_{\omega^{\prime}}\cdot\rho_{\text{th}}\;\hat{n}_{\omega,\text{out}}\cdot\ket{j}_{\omega}\ket{k}_{\omega^{\prime}}$
(18)

To isolate the contribution to the output field given by the amplification of
the thermal-noise and compare it with the contribution given by the added
noise generated by the device itself, we define the ratio
$r(\omega)=(\braket{\hat{n}_{\omega,\text{out}}}_{\text{th}}-\braket{\hat{n}^{\text{vac}}_{\omega,\text{out}}})/\braket{\hat{n}^{\text{vac}}_{\omega,\text{out}}}$.
Fig. 7 presents this quantity as a function of both the mode frequency and the
temperature of the surrounding environment for the referenced device. It can
be appreciated that in the low temperature regime the main contribution to the
output noise, across the entire frequency spectrum, is given by the added
noise ($r(\omega)<1$), while an increase of $T$ induces an increase of
$r(\omega)$ as fast as the considered mode $\omega$ is far from the center of
the band of the amplifier.

Figure 7: Spectral distribution of the adimensional ratio $(r(\omega))$
between the expectation value of the output photon number produced by the
amplification of an input thermal-state and the added noise photons, as a
function of the temperature $T$ of the environment. For this numerical
computation the pump current was set to $I_{\text{p}}=I_{\text{c}}$.

##  V Conclusion

The behaviour of a generic phase-preserving linear amplifier having at the
input port a bimodal signal composed by two uncorrelated fields at frequency
$\omega$ and $\omega^{\prime}=\omega_{\text{p}}-\omega$ was investigated. For
this, a definition of the spectra of gain, quantum efficiency, noise figure,
and effective temperature was given. Afterwards, the behaviour in terms of
noise estimators of a rf-SQUID based JTWPA driven at different conditions and
with different input states was simulated, as a case study of the theoretical
framework. It was demonstrated that feeding the amplifier with bimodal Fock
input states leads to a quantification for the noise estimators that is within
the SQLs, while when exploiting bimodal coherent input states these limits can
be exceeded. The results represent a key model to predict the behaviour of rf-
SQUID based JTWPA [18, 19] and other Josephson-based metamaterials [23] for
quantum computing (via quantum limited amplification of complex signals) [17,
26, 27] and for quantum information (via the generation of non-classical
radiation [5, 6, 7, 28, 8]) in non-ideal cryogenic environment [29].

## References

  * [1] M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information”, 10th Anniversary Edition (Cambridge University Press, 2010)
  * [2] P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, “A quantum engineer’s guide to superconducting qubits”, App. Phys. Rev. 6, 021318 (2019). DOI:10.1063/1.5089550.
  * [3] A. Blais, S. M. Girvin, and W. D. Oliver, “Quantum information processing and quantum optics with circuit quantum electrodynamics”, Nat. Phys. 16, 247-256 (2020). DOI:10.1038/s41567-020-0806-z
  * [4] D. M. P. Smith, L. Bakkerr, R. H. Witvers, B. E. Woestenburg, and K. D. Palmer, “Low noise amplifier for radio astronomy”, Int. J. Microwave Wireless Technol. 5, 453–461 (2013). DOI:10.1017/S1759078712000840.
  * [5] S. Loyd, “Enhanced sensitivity of photo-detection via quantum illumination”, Science 321, 1463-1465 (2008). DOI: 10.1126/science.1160627.
  * [6] S.-H.Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Loyd, L. Maccone, S. Pirandola, and J. H. Shapiro, “Quantum illumination with Gaussian States”, Phys. Rev. Lett. 101, 253601 (2008). DOI:10.1103/PhysRevLett.101.253601.
  * [7] S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination” Phys. Rev. Lett 114, 080503 (2015). DOI:10.1103/PhysRevLett.114.080503.
  * [8] L.Fasolo, A. Greco, E. Enrico, F. Illuminati, R. Lo Franco, D. Vitali, P. LIvreri “Josephson Traveling Wave Parametric Amplifiers as non-classical light source for Microwave Quantum Illumination”, Measurements: Sensors 18, 100349 (2021). DOI:10.1016/j.measen.2021.100349.
  * [9] A. Caldwell, G. Dvali, B. Majorovits, A. Millar, G. Raffelt, J. Redondo, O. Reimann, F. Simon, and F. Steffen, “Dielectric haloscopes: A new way to detectaxion dark matter”, Phys. Rev. Lett. 118, 091801 (2017). DOI:10.1103/PhysRevLett.118.091801
  * [10] J. Jeong, S. W. Youn, S. Bae, J. Kim, T. Seong, J. E. Kim, and Y. K. Semertzidis, “Search for invisible axion dark matter with a multiple-cell haloscope”, Phys.Rev. Lett. 125, 221302 (2020). DOI:10.1103/PhysRevLett.125.221302.
  * [11] K. M. Backes, D. A. Palken, S. A. Kenany, B. M. Brubaker, S. B. Cahn, A.Droster, G. C. Hilton, S. Ghosh, H. Jackson, S. K. Lamoreaux, A. F. Leder, K. W. Lehnert, S. M. Lewis, M. Malnou, R. H. Maruyama, N. M. Rapidis, M.Simanovskaia, S. Singh, D. H. Speller, I. Urdinaran, L. R. Vale, E. C. van Assendelft, K. van Bibber, and H. Wang, “A quantum enhanced search fordark matter axions”, Nature 590, 238–242 (2021). DOI:10.1038/s41586-021-03226-7
  * [12] H. A. Haus and J. A. Mullen, “Quantum Noise in Linear Amplifiers”, Phys. Rev. 128, 2407 (1962). DOI:10.1103/PhysRev.128.2407.
  * [13] C. M. Caves, “Quantum limits on noise in linear amplifiers”, Phys. Rev. D 26 1817 (1982). DOI:10.1103/PhysRevD.26.1817.
  * [14] C. M. Caves, J. Combes, Z. Jiang, S. Pandey,“Quantum limits on phase-preserving linear amplifiers”, Phys. Rev. A 86, 063802 (2012). DOI:10.1103/PhysRevA.86.063802.
  * [15] R. W. Boyd, Nonlinear Optics, 3rd ed. (Elsevier, 2008).
  * [16] M. Renger, S. Pogorzalek, Q. Chen et al., “Beyond the standard quantum limit for parametric amplification of broadband signals”, npj Quantum Inf 7, 160 (2021). Doi:10.1038/s41534-021-00495-y.
  * [17] C. Macklin, K. O’Brien, D. Hover, M. E. Schwartz, V. Bolkhovsky, X. Zhang, W. D. Oliver, and I. Siddiqi, “A near quantum-limited Josephson traveling- wave parametric amplifier”, Science 350, 6258, pp. 307–310 (2015). DOI:10.1126/science.aaa8525.
  * [18] A. B. Zorin, “Josephson traveling-wave parametric amplifier with three-wave mixing, Phys. Rev. Appl. 6, 034006 (2016). DOI:10.1103/PhysRevApplied.6.034006.
  * [19] A. B. Zorin, M. Khabipov, J. Dietel, and R. Dolata, “Traveling-wave parametric amplifier based on three-wave mixing in a Josephson metamaterial,” in 2017 16th International Superconductive Electronics Conference (ISEC) (Institute of Electrical and Electronics Engineers Inc., 2018), pp. 1–3.
  * [20] L. Fasolo, A. Greco and E. Enrico, “Superconducting Josephson-Based Metamaterials for Quantum-Limited Parametric Amplification: A Review”, Advances in Condensed-Matter and Materials Physics - Rudimentary Research to Topical Technology, Ed., IntechOpen, (2020). DOI:10.5772/intechopen.89305
  * [21] A. Greco, L. Fasolo, A. Meda, L. Callegaro, and E. Enrico, “Quantum model for rf-SQUID-based metamaterials enabling three-wave mixing and four-wave mixing traveling-wave parametric amplification”. Phys. Rev. B 104, 184517 (2021). DOI:10.1103/PhysRevB.104.184517.
  * [22] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification”, Rev. Mod. Phys. 82, 1155 (2010). DOI:10.1103/RevModPhys.82.1155.
  * [23] M. Esposito, A. Ranadive, L. Planat, and N. Roch, “Perspective on traveling wave microwave parametric amplifiers”, Appl. Phys. Lett. 119, 120501 (2021). DOI:10.1063/5.0064892.
  * [24] Z. Shi, K. Dolgaleva, R.W. Boyd, “Quantum Noise properties of non-ideal optical amplifiers and attenuators”, Journal of Optics 13, 125201 (2011). DOI:10.1088/2040-8978/13/12/125201.
  * [25] K. Peng, M. Naghiloo, J. Wang, G. D. Cunningham, Y. Ye, and K. O’Brien, “Near-Ideal Quantum Efficiency with a Floquet Mode Traveling Wave Parametric Amplifier”. Preprint at arXiV:2104.08269 [quant-ph].
  * [26] B. Abdo, F. Schackert, M. Hatridge, C. Rigetti, and M. Devoret, “Josephson amplifier for qubit readout”, Appl. Phys. Lett. 99, 162506 (2011). DOI:10.1063/1.3653473.
  * [27] J. Aumentado, “Superconducting parametric amplifiers: The state of the art in Josephson parametric amplifiers”, IEEE Microwave Mag. 21, pp. 45–59 (2020). DOI:10.1109/MMM.2020.2993476.
  * [28] A. L. Grimsmo and A. Blais, “Squeezing and quantum state engineering with Josephson travelling wave amplifiers”, npj Quantum Inf. 3, 20 (2017). DOI:10.1038/s41534-017-0020-8.
  * [29] S. Krinner, S. Storz, P. Kurpiers, P. Magnard, J. Heinsoo, R. Keller, J. Lütolf, C. Eichler, and A. Wallraff, “Engineering cryogenic setups for 100-qubit scale superconducting circuit systems”, EPJ Quantum Technology 6 (2019). DOI:10.1140/epjqt/s40507-019-0072-0.