text:
\section{Introduction and results}

Let $(M, \omega)$ be an open symplectic surface of finite type, possibly with boundary. Pick an open disk $\mathcal{D} \subset M$ of
$\Area(\mathcal{D}) = \int_\mathcal{D} \omega = A$
and a compactly supported Hamiltonian $\phi \in Ham(M)$ such that $\phi (\mathcal{D}) = \mathcal{D}$.
We define the \emph{trajectory} $\operatorname{traj}_{\mathcal{D}} (\phi) \in \pi_1(M)$ in the following way.
Pick a Hamiltonian isotopy $\phi_t$ which connects the identity to $\phi$. Roughly speaking,
$\operatorname{traj}_{\mathcal{D}} (\phi)$ is the free homotopy class of the trajectory of $\mathcal{D}$ under $\phi_t$.
More formally, pick $x \in \mathcal{D}$ and denote by $\gamma \subset \mathcal{D}$ a curve which connects $\phi (x)$ with $x$.
We catenate the trajectory of $x$ under $\phi_t$ with $\gamma$:
\[
  \operatorname{traj}_{\mathcal{D}} (\phi) = [\{\phi_t (x)\}_t * \gamma] \in \pi_1 (M).
\]
It is not difficult to see that $\operatorname{traj}_{\mathcal{D}} (\phi)$ does not depend on the choice of $x$, $\gamma$ and $\phi_t$.

Denote by $\| \cdot \|$ the Hofer norm on the group $Ham(M)$:
\[
	\| \phi \| = \inf \int_0^1 \max_{p \in M} H (p, t) - \min_{p \in M} H (p, t) \mathrm{d}t ,
\]
where the infimum goes over all compactly supported Hamiltonians $H: M \times [0, 1] \to \mathbb{R}$
such that $\phi$ is the time-$1$ map of the corresponding flow.

\medskip

Following F. Le Roux we consider the following invariant. Given $\alpha \in \pi_1 (M)$ and a disk $\mathcal{D}$ of area
$0 < A < \Area(M)$ we define
\[
  l_A (\alpha) = \inf \{ \|\phi\| \; \big| \; \phi \in Ham(M) \; \text{s.t.} \; \phi (\mathcal{D}) = \mathcal{D} \; \text{and} \;
			\operatorname{traj}_{\mathcal{D}} (\phi) = \alpha \}.
\]
Since all disks in $M$ of the same area are equivalent, $l_A$ depends only on the area $A$ and not on a choice of particular $\mathcal{D}$.

Clearly, $l_A$ satisfies the triangle inequality: let $a, b$ be two based loops with the same base point, denote $c = a * b$.
Then $l_A ([c]) \leq l_A ([a]) + l_A ([b])$. Using the argument from ~\cite{La-MD:L-infty-geom},
one shows that $l_A (\alpha) \geq A$ for all $\alpha \neq 0$. Namely, given $\phi \in Ham(M)$ with $\operatorname{traj}_{\mathcal{D}} (\phi) \neq 0$,
the lift of $\phi$ to the universal cover displaces
a lift of $\mathcal{D}$, hence $\| \phi \| \geq A$ by the energy-capacity inequality. Therefore $l_A$ is a nontrivial system of invariants
which behaves as a symplectic analog of the length spectrum.

Question 5 from ~\cite{LR:6-questions} addresses a computation of $l_A$ in the case when $M$ is an annulus and $\mathcal{D}$ is nondisplaceable.
Its solution appeared in ~\cite{Kh:disk-an}, this paper extends the results to general surfaces.

\begin{remnonum}
One may change the definition of $l_A$ and restrict attention to those $\phi$ that fix $\mathcal{D}$ pointwise. All estimates and results
presented in this article remain true also for this setting, the same ideas apply after minor adjustments of the argument.
\end{remnonum}

\medskip

In this article we discuss a weaker version of this invariant where we consider trajectories in a class $\alpha \in H_1 (M; \mathbb{Z})$.
We prove the following:
\begin{thm} \label{T:l_bound} given $0 < A < \Area(M)$ and $\alpha \in H_1 (M; \mathbb{Z})$, the following estimates hold:
  \begin{itemize}
	 \item
		If $A \leq \Area(M) / 2$, then $l_A (\alpha) \leq 2 A$.
	 \item
		If $genus(M) > 0$, then $l_A (\alpha) \leq 2 A$.
	 \item
		Otherwise ($genus(M) = 0$ and $A > \Area (M)/2$) $l_A : H_1 (M; \mathbb{Z}) \to \mathbb{R}$ is comparable to a homogeneous norm on $H_1 (M; \mathbb{Z})$.
		It is not important which norm we consider: $H_1 (M; \mathbb{R})$ is finitely generated, hence all homogeneous norms on $H_1 (M; \mathbb{Z})$ are equivalent.
  \end{itemize}
\end{thm}

Certain partial results are available also for the homotopical version $l_A :  \pi_1 (M) \to \mathbb{R}$, see the discussion in Section~\ref{S:disc_homo}.

\medskip

We also show:
\begin{thm} \label{T:quas}
  Let $(M, \omega)$ be a disk with $k$ punctures, pick $A \in \left(\frac{1}{2}\Area (M), \Area (M)\right)$.
  There exists a family $P_A = \{\rho_A\}$ of invariants $\rho_A : Ham (M) \to H_1 (M; \mathbb{R})$
  that satisfy:
  \begin{itemize}
	\item
	  $\rho_A$ is a homogeneous quasimorphism,
	\item
	  $\rho_A$ is Hofer-Lipschitz and $C_0$-continuous,
	\item
	  Suppose that $\phi \in Ham(M)$ has an invariant disk $D$ of $\Area(D) \geq A$. Then
	  $\rho_A (\phi) = [\operatorname{traj}_{D} (\phi)] \in H_1 (M; \mathbb{Z}) \subset H_1 (M; \mathbb{R})$.
  \end{itemize}
  $P_A$ contains continuum of such quasimorphisms that are linearly independent.
\end{thm}
These $\rho_A$ can be seen as one of many ways to generalize the rotation number to dimension two.
The Lipschitz property $|\rho_A (\phi)| \leq c \| \phi \|$ implies the lower bound for the third case of Theorem~\ref{T:l_bound}:
\begin{eqnarray*}
 l_A (\alpha) &=& \inf \{ \|\phi\| \; \big| \; \phi (\mathcal{D}) = \mathcal{D} \; \text{and} \;
 			[\operatorname{traj}_{\mathcal{D}} (\phi)] = \alpha)\} \geq \\
 			& \geq & \inf \left\{ \frac{|\rho_A(\phi)|}{c} \; \big| \; \phi (\mathcal{D}) = \mathcal{D} \; \text{and} \;
 			[\operatorname{traj}_{\mathcal{D}} (\phi)] = \alpha)\right\} = \frac{|\alpha|}{c}.
\end{eqnarray*}
Additional properties and applications of invariants $\rho_A$ are discussed in Section~\ref{S:disc_quas}.

\bigskip

This paper is organized as follows. In order to show the upper bounds for $l_A$ as in Theorem~\ref{T:l_bound} we
describe explicit Hamiltonian isotopies that achieve the desired energy bounds. Details are provided in Section~\ref{S:up}.
In Section~\ref{S:qm} we construct $\rho_A$ for Theorem~\ref{T:quas} from the Calabi quasimorphism $Ham(S^2) \to \mathbb{R}$ described in ~\cite{En-Po:calqm}.
(Note that the conditions on $M$ in Theorem~\ref{T:quas} are equivalent to the statement that $M$ embeds to a sphere.)
Section~\ref{S:disc} discusses possible
generalizations of $l_A$ and provides few applications of the quasimorphisms from Theorem~\ref{T:quas}.

\medskip

\emph{Acknowledgements:}
The author would like to thank D. Calegari, V. Humiliere, F. Le Roux and F. Zapolsky for useful discussions and comments.

\section{Upper bounds} \label{S:up}

In this section we show upper bounds for Theorem~\ref{T:l_bound}. We construct explicit Hamiltonian isotopies that translate a disk
along a given homology class and remain within the prescribed energy bound.

\subsection{Suppose $A = \Area{\mathcal{D}} \leq \Area (M) / 2$} \label{S:up-disp}
We assume first that $A < \Area (M) / 2$.

Consider the following Hamiltonian in $(\mathbb{R}^2, \mathrm{d}x \wedge \mathrm{d}y)$. We pick two disjoint disks $D_1$ and $D_2$ of area $A$
in the plane, connect them by two narrow nonintersecting strips (`pipes') as described in Figure~\ref{F:swap1}. Let $H$ be an autonomous
Hamiltonian which equals one in the area bounded by the two disks and the pipes, zero outside and linearly interpolated in between.
We apply to $H$ a $C_0$-small smoothing near the singular points.

\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{disk_swap.eps}
\caption{}
\label{F:swap1}
\end{center}
\end{figure}

The flux of the Hamiltonian flow through a curve $\gamma$ equals to the difference of values of $H$ at the endpoints of
$\gamma$. Hence if we choose $\gamma$ to be a cut going to the inner region (where $H = 1$) from the outside, the flux through $\gamma$ is one.
This implies that the resulting flow is relatively slow inside the disks and accelerates in the pipes where such a cut $\gamma$ can be very short.
For appropriate choice of the
smoothing of $H$ the time-($A+\varepsilon$) map of the flow of $H$ will move $D_1$ to $D_2$. (In fact, it is possible to arrange that
the two disks will be swapped.) The Hofer norm is at most $\int_0^{A + \varepsilon} (\max H - \min H) \mathrm{d}t = A + \varepsilon$.
$\varepsilon$ depends on the area of the pipes and the choice of smoothing of $H$ and can be made arbitrarily small.
Note that the energy cost of $(A+\varepsilon)$ is optimal since every Hamiltonian that displaces $D_1$ has energy
$\geq A$ by the energy-capacity inequality.

We augment this construction as in Figure~\ref{F:swap2}: we choose the first pipe to go along an interval connecting $D_1$ with $D_2$ and
let the second pipe lie in a small neighborhood of the two disks and the first pipe.

\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.65\textwidth]{disk_swap2.eps}
\caption{}
\label{F:swap2}
\end{center}
\end{figure}

This way the Hamiltonian $H$ is supported in a small neighborhood of the two disks and an interval. This construction can be
embedded to any surface given two disjoint copies of a disk and a simple path connecting boundary points of the disks. This allows
us to translate a disk along a simple path.

\medskip

Let $D_1, D_2$ be two disjoint copies of $\mathcal{D}$ in $M$ and let $\gamma_1$ be a simple path connecting $D_1$ to $D_2$
and $\gamma_2$ be a simple path connecting $D_2$ to $D_1$. $\gamma_1$ and $\gamma_2$ may intersect one another but not the disks. We apply the construction above
in order to move $D_1$ to $D_2$ along $\gamma_1$ and then move back to $D_1$ along $\gamma_2$. The total energy cost of such process
equals $2 \Area(\mathcal{D}) + 2 \varepsilon$ and the result is a translation of the disk $D_1$ along a trajectory in class
$[\gamma_1 * \gamma_2]$. (By abuse of notation we extend $\gamma_1, \gamma_2$ inside the disks so that they have the same endpoints
and catenation makes sense.)

\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{disk_swap3.eps}
\caption{}
\end{center}
\end{figure}

The upper bound follows from the following topological lemma:
\begin{lem} \label{l:loop2path}
  Let $M$ be a surface and $\alpha \in \pi_1 (M)$. Then there exist two simple curves $\gamma_1$, $\gamma_2$ with the same
  endpoints such that $\alpha = [\gamma_1 * -\gamma_2]$. ($-\gamma_2$ denotes $\gamma_2$ with reversed orientation.)
\end{lem}

Indeed, pick a class $\alpha \in \pi_1 (M)$. By this lemma there exist two points $p, q \in M$ and two simple curves
$\gamma_1, \gamma_2$ connecting them so that $[\gamma_1 * \gamma_2] = \alpha$. We replace $p$ and $q$ by small disks
$D_1, D_2$ which intersect both curves only in small intervals near the endpoints. Pick a symplectic form $\omega'$ on $M$
so that both disks have area $A$ and the total area is $\int_M \omega' = \int_M \omega$. $(M, \omega')$ is symplectomorphic to
$(M, \omega)$ by Moser's argument. We construct a Hamiltonian
in $(M, \omega')$: translate $D_1$ along $\gamma_1$ to $D_2$ and then back along $\gamma_2$. The resulting deformation $\phi$ satisfies
\[
  \operatorname{traj}_{D_1} (\phi) = [\gamma_1 * \gamma_2] = \alpha \in \pi_1 (M)
\]
at the cost of $2 \Area(\mathcal{D}) + 2 \varepsilon$, where $\varepsilon$ is arbitrarily small.

\medskip

Assume that $A = \Area (M) / 2$ and fix $\alpha = [\gamma_1 * \gamma_2] \in \pi_1(M)$.
Pick a smaller disk $D \subset \mathcal{D}$, suppose that $\Area (D) = A - \epsilon$.
We apply the previous argument and construct $\phi \in Ham(M)$ which translates $D$ along $\alpha$.
Then we deform $\phi (\mathcal{D})$ back to $\mathcal{D}$ by another Hamiltonian isotopy $f$ which fixes the smaller disk $D$.
Note that the annulus $\phi (\mathcal{D} \setminus D)$ is spread along $\gamma_1 \cup \gamma_2$ and its shape does not depend on the choice of $D$.
This implies that $f$ can be chosen with $\|f\| \leq c \epsilon$ where $c$ depends on $\gamma_1, \gamma_2$ but not on $\epsilon$.
Therefore
\[
  l_A (\alpha) \leq \| \phi \| + \| f \| \leq 2A + 2 \varepsilon + c \epsilon.
\]
The result follows by letting $\varepsilon, \epsilon \to 0$.

\begin{remnonum}
  Note that we proved the upper bound for $\alpha \in \pi_1 (M)$ which, in particular, implies the same bound for the homological spectrum.
\end{remnonum}

\medskip

\begin{proof}[Proof of lemma]
  Pick $\alpha \in \pi_1 (M)$. Let $\gamma$ be a loop representing $\alpha$ which has finite number of
  transverse intersections. We deform $\gamma$ so that it can be cut into a pair of simple curves. Pick arbitrary point $p \in \gamma$
  which will be a starting point of $\gamma_1$. We draw $\gamma_1$ by traversing $\gamma$ starting from $p$. We prevent self-intersections
  of $\gamma_1$ as follows. Once we arrive to a self-crossing we stretch the previously drawn path an push it in a
  small neighborhood in front of
  the pen (see Figure~\ref{F:pathlemma}). After passing several crossing we will have a number of strings pushed in this manner.
  During of this process we construct a simple path which is homotopic to the corresponding
  arc of $\gamma$ relative the endpoints. We stop the process when we arrive to a point $q \in \gamma$ such that the arc
  $[q, p] \subset \gamma$ is simple. Set $\gamma_2 = [q, p] \subset \gamma$.
  Lemma follows since $\gamma_1$ is homotopic to $[p, q] \subset \gamma$ relative the endpoints.

\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=\textwidth]{pathlemma.eps}
\caption{}
\label{F:pathlemma}
\end{center}
\end{figure}

\end{proof}

\subsection{Arbitrary $\mathcal{D}$}.
We show first that one can translate a disk along a simple loop using energy $\Area (\mathcal{D}) + \varepsilon$.
Then we consider separately the cases $genus(M) = 0$ and $genus (M) > 0$.

Consider the following Hamiltonian in $(\mathbb{R}^2, \mathrm{d}x \wedge \mathrm{d}y)$. We pick a disk $D \subset \mathbb{R}^2$ of area $A$
and connect two boundary points of the disk by a circular pipe. Let $H$ be an autonomous
Hamiltonian which equals one in the inner region, zero outside and linearly interpolated in between.
We apply to $H$ a $C_0$-small smoothing near the singular points (see Figure~\ref{F:loop}).

\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.4\textwidth]{disk_loop.eps}
\caption{}
\label{F:loop}
\end{center}
\end{figure}

The Hamiltonian flow of $H$ is relatively slow inside the disk $D$ and fast within the pipe. For an appropriate choice of a
smoothing of $H$ there exists $\varepsilon > 0$ such that $D$ is a fixed set of the time-$(A+\varepsilon)$ map of the flow of $H$.
The energy is at most $A+\varepsilon$ and $\varepsilon$ depends on the area of the pipe and a choice of smoothing and can be made arbitrarily small.

We cut $H$ off inside the circle so that it is supported in a small neighborhood of the disk and the pipe. Note that this cutoff
does not affect the flow of $H$ in the disk $D$ and the pipe. This construction can be copied to any symplectic surface
given a disk and a simple curve which connects boundary points of the disk and does not intersect it away from the endpoints.
This implies the following result.
\begin{lem}
  Suppose that $\alpha \neq 0$ is represented by a simple loop. Then $l_A (\alpha) = A$.
\end{lem}
\begin{proof}
  Let $\gamma$ be a simple loop representing $\alpha$. As in the previous subsection, we deform
  $\omega$ so that there exists a disk of area $A$ whose boundary points are connected by an arc of $\gamma$ and this arc does
  not intersect the disk except for the endpoints. An application of the Hamiltonian described above implies $l_A (\alpha) \leq A + \varepsilon$.
  The lemma follows by letting $\varepsilon \to 0$ and from the fact that $l_A (\alpha) \geq A$.
\end{proof}

\begin{cor}\label{C:w_length}
  Let $S \subseteq H_1 (M; \mathbb{Z})$ be the set of classes represented by simple loops. It is easy to see that $S$ generates $H_1 (M; \mathbb{Z})$.
  Denote by $\| \cdot \|_S$ the norm on $H_1 (M; \mathbb{Z})$ given by word length with respect to $S$. The triangle inequality implies
  $l_A (\alpha) \leq A \cdot \| \alpha \|_S$. The same argument holds also for the homotopical length spectrum.
\end{cor}

Given any surface $M$,  $H_1 (M; \mathbb{Z})$ admits a basis $B$ represented by simple loops. By the triangle inequality,
$l_A (\alpha) \leq A \cdot \| \alpha \|_B$ where $\| \cdot \|_B$ is the word length norm with respect to $B$.
$\| \cdot \|_B$ is a homogeneous norm on $H_1(M; \mathbb{Z})$. This shows the upper bound for the third case in Theorem~\ref{T:l_bound}.
A more careful argument may provide better constants. For example, in the case of an annulus $M$ the bound can be improved to
\[
  l_A (n \cdot \text{generator}) \leq (2 A - \Area(M)) \cdot |n| + \Area(M)
\]
(see ~\cite{Kh:disk-an}).

\medskip

Suppose that $genus(M) > 0$. The upper bound for Theorem~\ref{T:l_bound} follows from the lemma below and Corollary~\ref{C:w_length}.

\begin{lem}
  Let $M$ be a surface with positive genus. Then every homology class in $H_1 (M; \mathbb{Z})$ can be represented by a sum of two simple loops.
\end{lem}
\begin{proof}
  \underline{Step 1}: Suppose $M = S^1 \times S^1$ is a torus. $\alpha = [\{pt\} \times S^1]$ and $\beta = [S^1 \times \{pt\}]$
  generate $H_1 (M; \mathbb{Z}) = \mathbb{Z}<\alpha, \beta>$. Any class $n \alpha + m \beta$ can be decomposed into the sum
  \[
	n \alpha + m \beta = \big((n -1)\alpha + \beta\big) + \big( \alpha + (m-1) \beta \big)
  \]
  where each summand is represented by a simple loop (see Figure~\ref{F:torus}).
  \begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.7\textwidth]{torus.eps}
  \caption{}
  \label{F:torus}
  \end{center}
  \end{figure}

  \medskip

  \underline{Step 2}: Suppose $M = S^1 \times S^1 \setminus \{p_1, \ldots, p_k\}$ is a punctured torus.
  Without loss of generality we assume that all punctures lie on a horizontal meridian. $H_1 (M; \mathbb{Z})$ admits a basis
  of $k$ vertical meridians $\alpha_1, \ldots, \alpha_k$ and a horizontal meridian $\beta$ (see Figure~\ref{F:ptorus}).
  Then every
  \[
	\alpha = m_1 \alpha_1 + \ldots + m_k \alpha_k + n \beta \in H_1 (M; \mathbb{Z})
  \]
  admits a decomposition
  \[
	\alpha = \big((m_1-1) \alpha_1 + \ldots + m_k \alpha_k + \beta\big) + \big(\alpha_1 + (m-1) \beta \big)
  \]
  where each summand is represented by a simple loop.

  \begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.72\textwidth]{ptorus.eps}
  \caption{}
  \label{F:ptorus}
  \end{center}
  \end{figure}

  \medskip

  \underline{Step 3}: Suppose $genus(M) > 1$. $M$ can be decomposed into a connected sum of punctured tori
  \[
	M = T_1 \# \ldots \# T_l, \quad H_1 (M; \mathbb{Z}) = H_1 (T_1; \mathbb{Z}) + \ldots + H_1 (T_l; \mathbb{Z}).
  \]
  We identify $H_1 (T_i; \mathbb{Z})$ with their image in $H_1 (M; \mathbb{Z})$ by abuse of notation.
  \begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.5\textwidth,natwidth=693,natheight=426]{surfcut.jpg}

  \label{F:cut}
  \end{center}
  \end{figure}

  Pick $\alpha \in H_1 (M; \mathbb{Z})$. There exist $\alpha_i \in H_1 (T_i; \mathbb{Z})$ such that $\alpha = \sum \alpha_i$.
  By the previous step we choose homology classes $\beta_i, \beta'_i \in H_1 (T_i; \mathbb{Z})$
  represented by simple loops such that $\alpha_i = \beta_i + \beta'_i$.
  Now
  \[
	\alpha = \sum \alpha_i = \sum (\beta_i + \beta'_i) = \sum \beta_i + \sum \beta'_i.
  \]
  Both $\sum \beta_i$ and $\sum \beta'_i$ are represented by simple loops: consider a connected sum of the corresponding simple loops in each $T_i$.
  The result is a simple loop in $M$ since $T_i$ are disjoint.
\end{proof}

\section{Construction of quasimorphisms $\rho_A$}\label{S:qm}

Let $G$ be a group. A function $r : G \to \mathbb{R}$ is called a \emph{quasimorphism} if there exists
a constant $R$ such that $|r(fg) - r(f) - r(g)| \leq R$ for all $f, g \in G$.
$R$ is called the \emph{defect} of $r$. The quasimorphism $r$
is \emph{homogeneous} if it satisfies $r(g^m) = mr(g)$ for all $g \in G$ and $m \in \mathbb{Z}$.
Any homogeneous quasimorphism satisfies $r(fg) = r(f) + r(g)$ for commuting elements $f, g$ and is invariant under conjugations.

\medskip

We construct quasimorphisms $\rho_A$ described in Theorem~\ref{T:quas}. Recall that $M$ is a surface of finite type of genus zero.
$M$ has finite area, so without loss of generality we rescale $\omega$ to get $\Area(M) = 1$. Such $M$ is symplectomorphic to the interior of
a $k$-times punctured disk $U \setminus \{p_1, \ldots , p_k\}$ where $k$ is the rank of $H_1 (M; \mathbb{Z})$ and $\Area (U) = 1$.

\begin{prop} \label{P:quas}
  Let $(\mathbb{A}, \omega)$ be an annulus (disk with one puncture). Suppose that $\Area(\mathbb{A}) = 1$.
  Pick $\frac{1}{2} < A < 1$ and choose a generator $\alpha$ for $H_1 (M, \mathbb{Z})$.
  There exists a family $P_A = \{\rho_A\}$ of invariants $\rho_A : Ham (\mathbb{A}) \to \mathbb{R}$
  that satisfy:
  \begin{itemize}
	\item
	  $\rho_A$ is a homogeneous quasimorphism,
	\item
	  $\rho_A$ is Hofer-Lipschitz and $C_0$-continuous,
	\item
	  Suppose that $\phi \in Ham(\mathbb{A})$ has an invariant disk $D$ of $\Area(D) \geq A$. Then
	  $[\operatorname{traj}_{D} (\phi)] = \rho_A (\phi) \cdot \alpha$.
  \end{itemize}
  $P_A$ contains continuum of such quasimorphisms that are linearly independent.
\end{prop}

The proposition implies Theorem~\ref{T:quas}.
\begin{proof}[Proof of Theorem~\ref{T:quas}]
Pick such a quasimorphism $\rho_\mathbb{A} : Ham (\mathbb{A}) \to \mathbb{R}$ and denote by
\[
  i_j : U \setminus \{p_1, \ldots , p_k\} \hookrightarrow U \setminus \{p_j\} \simeq \mathbb{A}, \quad 1 \leq j \leq k
\]
the embeddings obtained by gluing all punctures except for $p_j$ and identifying the resulting once punctured disk with $\mathbb{A}$.
The pullback quasimorphisms $\rho_j = i_j^* \rho_\mathbb{A} : Ham (U \setminus \{p_1, \ldots , p_k\}) \to \mathbb{R}$ satisfy the Lipschitz and
continuity properties. Given a Hamiltonian $\phi$ with a fixed disk $\mathcal{D}$ of area $\geq A$, $\rho_j (\phi)$ counts
the winding number of $\operatorname{traj}_{\mathcal{D}} (\phi)$ around $p_j$. Denote by $\gamma_j$ a small positively oriented circle in $U$ around $p_j$
which does not encircle other punctures. $\{[\gamma_j]\}_{j=1}^k$ is a basis for $H_1(U \setminus \{p_1, \ldots , p_k\}; \mathbb{Z})$.
The dual basis $\{[\gamma_j]^*\}$ allows us to write
\[
  \rho_j (\phi) = [\gamma_j]^*[\operatorname{traj}_{\mathcal{D}} (\phi)]
\]
for all Hamiltonians $\phi$ with a fixed disk $\mathcal{D}$ as above. This implies
\[
  [\operatorname{traj}_{\mathcal{D}} (\phi)] = \sum_{j=1}^k [\gamma_j]^*[\operatorname{traj}_{\mathcal{D}} (\phi)] \cdot [\gamma_j] = \sum_{j=1}^k \rho_j (\phi) [\gamma_j].
\]
\[
  \rho_A = \sum_{j=1}^k \rho_j (\cdot) [\gamma_j] : Ham (U \setminus \{p_1, \ldots , p_k\}) \to H_1 (U \setminus \{p_1, \ldots , p_k\}; \mathbb{R})
\]
is a homogeneous quasimorphism which satisfies all properties of Theorem~\ref{T:quas}. Continuum of linearly independent choices of
$\rho_\mathbb{A}$ implies that for a choice of $\rho_A$.
\end{proof}

\medskip

Proof of Proposition~\ref{P:quas} takes the rest of this section. It can be found in ~\cite{Kh:disk-an}, we give it below
for the sake of completeness.
We use notation $\mathbb{A} = S^1 \times (0, 1) \simeq \mathbb{R} / \mathbb{Z} \times (0, 1)$ equipped with the
standard symplectic form $\omega = \mathrm{d}\theta \wedge \mathrm{d}h$, $\Area (\mathbb{A}) = 1$. Without loss of generality we assume
that the generator $\alpha \in H_1 (\mathbb{A}; \mathbb{Z})$ is represented by the positively oriented circle $S^1 \times \{0.5\}$.
Let $\mathcal{D} \subset \mathbb{A}$ be a disk with $\Area (\mathcal{D}) = A$, denote $L = \partial \mathcal{D}$.
Put $S = \{ \phi \in Ham (\mathbb{A}) \, | \, \phi (\mathcal{D}) = \mathcal{D}\}$ to be the stabilizer of $\mathcal{D}$ in
$Ham(\mathbb{A})$. We denote $S_n = \{\phi \in S \, | \, [\operatorname{traj}_\mathcal{D} (\phi)] = n \alpha\}$.

Pick a function $\widehat{H} : \mathbb{A} \to \mathbb{R}$ with compact support such that $\widehat{H}(\theta, h) = h$
away from a small neighborhood of $\partial \mathbb{A}$. Denote by $\widehat{\Phi}$ the time-$1$ map of the
Hamiltonian flow generated by $\widehat{H}$. It is easy to see that $\widehat{\Phi} \in S$ with trajectory
$[\operatorname{traj}_\mathcal{D} (\widehat{\Phi})] = \alpha$. Note that $S = \bigcup_{n \in \mathbb{Z}} S_n = \bigcup_{n \in \mathbb{Z}} \widehat{\Phi}^n S_0$.
We construct a quasimorphism $\rho_A$.

\underline{Step I:} it is sufficient to construct a homogeneous quasimorphism $\rho: Ham(\mathbb{A}) \to \mathbb{R}$ which satisfies:
\begin{itemize}
  \item
	$\rho$ is Hofer-Lipschitz,
  \item
	$\rho (S_0) = 0$, $\rho (\widehat{\Phi}) = c > 0$,
  \item
	$\rho$ vanishes on $Ham(U)$ for all disks $U \subset \mathbb{A}$.
\end{itemize}

\begin{proof} $\rho$ vanishes on Hamiltonians supported in a disk. It follows from the results of ~\cite{En-Po-Py:qm-continuity} that
such $\rho$ is continuous in the $C^0$-topology.
Any $\phi \in S_n$ decomposes as
$\phi = \widehat{\Phi}^n \circ s$ for some $s \in S_0$. Hence
$$|\rho (\phi) - c n| = |\rho (\widehat{\Phi}^n \circ s) - n \rho(\widehat{\Phi}) - 0| =
|\rho (\widehat{\Phi}^n \circ s) - \rho(\widehat{\Phi}^n) - \rho(s)| < R.$$
($R$ denotes the defect of $\rho$). It follows that
\[
	\rho (\phi) = c \cdot n + \delta_{\phi}
\]
where $|\delta_{\phi}| \leq R$. Note that $\phi^k \in S_{nk}$, hence by homogenuity
\[
	\rho (\phi) = \frac{\rho (\phi^k)}{k} = \frac{c \cdot n k + \delta_{\phi^k}}{k} = c \cdot n + \frac{\delta_{\phi^k}}{k}
\]
which implies $\rho (\phi) = c \cdot n$ in the limit as $k \to \infty$.

We set $\rho_A = \frac{\rho}{c}$. Suppose that $\phi \in Ham(\mathbb{A})$ has an invariant disk $D$ of $\Area(D) \geq A$.
\begin{enumerate}
  \item
	We consider the case $D = \mathcal{D}$. Then $\phi \in S_n$ for some $n \in \mathbb{Z}$ and
	\[
	  \rho_A (\phi) \cdot \alpha = \frac{\rho(\phi)}{c} \cdot \alpha = n \alpha =[\operatorname{traj}_D (\phi)].
	\]
  \item
	The case $\Area(D) = A$. Pick $g \in Ham(\mathbb{A})$ such that $g (D) = \mathcal{D}$. Then $g \phi g^{-1} \in S$ hence
	\[
	  \rho_A (\phi) \cdot \alpha = \rho_A (g \phi g^{-1}) \cdot \alpha =[\operatorname{traj}_\mathcal{D} (g \phi g^{-1})] = [\operatorname{traj}_D (\phi)].
	\]
  \item
	The case $\Area(D) > A$. Pick a smaller disk $D_1 \subset D$ with $\Area(D_1) = A$. $\phi (D) = D$, choose $g \in Ham(D)$
	such that $g (\phi (D_1)) = D_1$. Then $D_1$ is a fixed disk of area $A$ for $g \phi$ hence
	\[
	  \rho_A (g \phi) \cdot \alpha = [\operatorname{traj}_{D_1} (g \phi)] = [\operatorname{traj}_D (\phi)].
	\]
	We claim that $\rho_A (g \phi) = \rho_A (\phi)$. Indeed, the disk $D$ is fixed for $g \phi$, $\phi$ and their iterates.
	Moreover, $(g \phi)^n$ differs from $\phi^n$ by a Hamiltonian $f_n \in Ham(D)$.
	Hence by the homogenuity of $\rho_A$
	\[
	  |\rho_A ((g \phi)^n) - \rho_A(\phi^n) - \rho_A (f_n)| = |n \rho_A (g \phi) - n\rho_A(\phi) - 0| = n \cdot |\rho_A (g \phi) - \rho_A(\phi)|
	\]
	is bounded by the defect of $\rho_A$. Letting $n \to \infty$ we show $\rho_A (\phi) = \rho_A (g \phi)$.
\end{enumerate}
\end{proof}

\medskip

\underline{Step II:} in order to build $\rho$ we use the Calabi quasimorphism
on $Ham (S^2)$ which was constructed by M. Entov and L. Polterovich
in ~\cite{En-Po:calqm}. We give a brief recollection of the relevant facts.

Let $U$ be an open disk equipped with a symplectic form $\omega$.
Let $F_t : U \to \mathbb{R}$, $t \in [0, 1]$ be a time-dependent smooth function with compact support. We define
$$\widetilde{Cal} (F_t) = \int_0^1 \left( \int_U F_t \omega \right) \text{d} t.$$ As $\omega$ is exact on $U$,
$\widetilde{Cal}$ descends to a homomorphism $Cal_U: Ham(U) \to \mathbb{R}$ which is called
the Calabi homomorphism. Clearly, for $\phi \in Ham(U)$, $| Cal_U(\phi) | \leq \Area(U) \cdot \| \phi \|$.

Let $S^2$ be a sphere equipped with a symplectic form $\omega$. Suppose $\Area (S^2) = 2 A$.
For a smooth function $F: S^2 \to \mathbb{R}$ the Reeb graph $T_F$ is defined as the set of connected
components of level sets of $F$ (for a more detailed definition we refer the reader to ~\cite{En-Po:calqm}).
For a generic Morse function $F$ this set, equipped with the topology
induced by the projection $\pi_F: S^2 \to T_F$, is homeomorphic to a tree.
We endow $T_F$ with a measure given by $\mu (X) = \int_{\pi_F^{-1}(X)} \omega$
for any $X \subseteq T_F$ with measurable $\pi_F^{-1}(X)$. $x \in T_F$ is called a \emph{median} of $T_F$
if the measure of each connected component of $T_F \setminus \{x\}$ does not exceed $A$. By ~\cite{En-Po:calqm}
a median exists and is unique.
This construction can be extended to functions $F$ such that $F \big|_{supp (F)}$ is Morse.

~\cite{En-Po:calqm} describes construction of a homogeneous quasimorphism $Cal_{S^2} : Ham (S^2) \to \mathbb{R}$.
It has the following properties: $Cal_{S^2}$ is Hofer-Lipschitz ($|Cal_{S^2}(\phi)| \leq 2A \cdot \| \phi \|$).
In the case when $\phi \in Ham(S^2)$ is supported in a disk $U$ which is displaceable
in $S^2$, $Cal_{S^2} (\phi) = Cal_U (\phi \big|_U)$.
Moreover, for $\phi \in Ham(S^2)$ generated by an autonomous function $F: S^2 \to \mathbb{R}$,
$Cal_{S^2} (\phi)$ can be computed in the following way.
Let $x$ be the median of $T_F$ and $X = \pi_F^{-1} (x)$ be the corresponding subset of $S^2$.
Then
\[
	Cal_{S^2}(\phi) = \int_{S^2} F \omega - 2A \cdot F(X).
\]

Given a symplectic embedding $j : \mathbb{A} \to S^2$ into a sphere of area $2 A$,
consider the pullback $Cal_{j} = j^* (Cal_{S^2}) : Ham(\mathbb{A}) \to \mathbb{R}$.
Namely, given $\phi \in Ham(\mathbb{A})$, extend $j_* (\phi)$ to $\tilde{\phi} \in Ham (S^2)$ by identity
on the complement of $j (\mathbb{A})$. Then $Cal_j (\phi) = Cal_{S^2} (\tilde{\phi})$.
Clearly, $Cal_{j}$ is a homogeneous quasimorphism.
It has the following properties:
\begin{itemize}
	\item
		$Cal_j (\phi) = Cal_D(\phi \big|_D)$ for any $\phi$ supported in a disk $D$ of area $A$.
		To see that note that the corresponding $\tilde{\phi} \in Ham(S^2)$
		is supported in a displaceable disk $j(D)$ in $S^2$.
	\item
		\[
			\left|Cal_j(\phi)\right| = \left|Cal_{S^2}\left(\tilde{\phi}\right)\right| \leq
			2A \cdot \left\| \tilde{\phi} \right\|_{S^2} \leq 2A \cdot \left\| \phi \right\|_{\mathbb{A}}.
		\]
	\item for an autonomous $\phi$ generated by a compactly supported function $H : \mathbb{A} \to \mathbb{R}$,
		\[
			Cal_j (\phi) = \int_\mathbb{A} H \omega - 2A \cdot H(X)
		\]
		where $X \subseteq \mathbb{A}$ is the level set component which is sent by $j$ to the median set
		of $j_* (H)$ in $S^2$.
\end{itemize}

\medskip

Consider the embeddings $j_s : \mathbb{A} \to S^2$ ($0 \leq s \leq 2 A - 1$) into a sphere of area $2 A$
that are given by gluing a disk of area $s$ to $S^1 \times \{0\}$ and a disk of area
$(2 A - 1 - s)$ to $S^1 \times \{1\}$.
This construction ensures that $j_s (L) = j_s (\partial \mathcal{D})$ bisects $S^2$
into two displaceable disks.

We pick $0 \leq s_1 < s_2 \leq 2 A - 1$ and set
\begin{equation}\label{eq:qm_def}
	\rho = \rho_{s_1, s_2} = Cal_{j_{s_2}} - Cal_{j_{s_1}}.
\end{equation}

\medskip

We claim that $\rho$ satisfies conditions of Step I.
Obviously, $\rho$ is a homogeneous quasimorphism on $Ham(\mathbb{A})$ which satisfies
the Lipschitz property:
\[
	|\rho (\phi)| \leq |Cal_{j_{s_2}} (\phi)| + |Cal_{j_{s_1}} (\phi)| \leq 4 A \cdot \|\phi\|.
\]
Consider the function $\widehat{H}$ which was used to define the Hamiltonian $\widehat{\Phi}$ described
above. It is easy to see that the ``median'' level set $X_s$ of $\widehat{H}$
which is relevant for the computation of $Cal_{j_s} (\widehat{\Phi})$ is $S^1 \times \{A-s\}$.
Hence
\[
	Cal_{j_s} (\widehat{\Phi}) = \int_\mathbb{A} \widehat{H} \omega - 2A \cdot \widehat{H}(X_s) =
		\int_\mathbb{A} \widehat{H} \omega - 2A \cdot (A-s).
\]
This implies
\[
	\rho(\widehat{\Phi}) = Cal_{j_{s_2}} (\widehat{\Phi}) - Cal_{j_{s_1}} (\widehat{\Phi})= 2A \cdot [- (A-s_2) + (A-s_1) ] = 2A \cdot (s_2 - s_1) > 0.
\]
Suppose that $\phi \in Ham(U)$ for some disk $U \subset \mathbb{A}$. Then
\[
  \rho(\phi) = Cal_{j_{s_2}} (\phi) - Cal_{j_{s_1}} (\phi) = Cal_{S^2} (j_{s_2, *} (\phi)) - Cal_{S^2} (j_{s_1, *} (\phi)) = 0
\]
since $j_{s_2, *} (\phi)$ is conjugate to $j_{s_1, *} (\phi)$.

\medskip

It is left to show that $\rho$ vanishes on $S_0$. Denote by
$S'_0$ the subgroup of $S_0$ which fixes a neighborhood of $L$ pointwise.

\begin{lem}\label{lm:fix_neigh}
	Let $q$ be a homogeneous quasimorphism which is Hofer-continuous and vanishes on $S'_0$.
	Then $q$ vanishes on $S_0$.
\end{lem}
\begin{proof}
Pick an open disk $U \subset \mathbb{A} \setminus L$. $Ham (U) \subset S'_0$, therefore
$q$ vanishes on $Ham (U)$. It follows from the results of ~\cite{En-Po-Py:qm-continuity} that
$q$ is continuous in the $C^0$-topology.

Suppose $\phi \in S_0$. This implies $\phi (L) = L$ (as a set). Applying an appropriate isotopy $\psi \in S_0$ supported near $L$,
we may ensure that $\psi \circ \phi = {{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l$ on $L$. Moreover, $\psi$ may be chosen with arbitrarily small Hofer norm.
Further, we may find a $C^0$-small diffeomorphism $h \in Ham(\mathbb{A})$ such that
$h \circ \psi \circ \phi = {{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l$ in a neighborhood of $L$.
It follows that $h \circ \psi \circ \phi \in S'_0$ and $q (h \circ \psi \circ \phi) = 0$.
Hofer- and $C^0$-continuity of $q$ imply that $q (\phi) = 0$.
\end{proof}

Let $\phi \in S'_0$. We show that $\rho (\phi) = 0$. $\phi$ splits to a composition
$\phi = \phi_\mathcal{D} \circ \phi_P$ where $\phi_\mathcal{D}$ is supported in
$\mathcal{D}$ and $\phi_P$ is supported in the pair of pants
$P = \mathbb{A} \setminus \mathcal{D}$. $\phi_\mathcal{D}, \phi_P$ have disjoint supports,
therefore they commute. Hence $\rho (\phi) = \rho(\phi_\mathcal{D}) + \rho (\phi_P)$.
Note that $\phi_\mathcal{D}$ is supported in a disk, so $\rho (\phi_\mathcal{D}) = 0$.

$\phi_P$ is Hamiltonian in $\mathbb{A}$, but after the restriction to $P$
we have just $\bar{\phi}_P = \phi_P\big|_P \in Symp_c(P)$. In the argument below we apply a sequence of
deformations to $\phi_P$ in order to get $\phi'_P$ whose restriction $\bar{\phi}'_P \in Ham (P)$. All deformations
involved in the process preserve the value $\rho (\phi_P)$. Finally, we show that
$\rho (\phi'_P) = 0$ by explicit computation.

The mapping class group
$\pi_0 (Symp_c (P))$ is isomorphic to $\mathbb{Z}^3$ and is generated by Dehn twists near the
three boundary components. For the proof of this fact we refer the reader to
~\cite{F-M:mapping-class} where the authors show that $\pi_0 (Diff_c (P)) \simeq \mathbb{Z}^3$
and is generated by Dehn twists. Note that $\phi, \psi \in Symp_c (P)$ are isotopic in
$Symp_c$ if and only if they are isotopic in $Diff_c$. As Dehn twists belong to
$Symp_c (P)$, the statement for $\pi_0 (Symp_c (P))$ follows.

Denote by $T_1, T_0$ Dehn twists near $S^1 \times \{1\}$, $S^1 \times \{0\}$
and by $T_L$ a Dehn twist in
$P$ near $L = \partial \mathcal{D}$.
There are Hamiltonians $\psi_L$ in $S'_0$ with arbitrary small Hofer norm
whose restriction to $P$ realizes the Dehn twist $T_L$. For example, consider a bump function supported near $\mathcal{D}$
which has small height but is very steep in an annulus near $L$.
$\bar{\phi}_P$ is isotopic in $Symp_c (P)$
to some $T_1^{k_1} T_0^{k_0} T_L^{k_L}$ ($k_i \in \mathbb{Z}$).
If $k_L \neq 0$ we replace the original $\phi \in S'_0$ by $\psi_L^{-k_L} \circ \phi \in S'_0$.
As $\|\psi_L\|$ can be chosen to be arbitrarily small, by continuity of $\rho$ it is enough to show
the desired statement for the deformed $\phi$.
After the replacement $k_L$ vanishes, hence the modified $\bar{\phi}_P \sim T_1^{k_1} T_0^{k_0}$.
Note that $\bar{\phi}_P$ is induced by a Hamiltonian $\phi \in S$. The definition of
$\operatorname{traj}_{\mathcal{D}} (\phi)$ implies that $k_1 \alpha = [\operatorname{traj}_{\mathcal{D}} (\phi)] = -k_0 \alpha$
(recall that $\alpha$ is the chosen generator of $H_1 (\mathbb{A}; \mathbb{Z})$).
The minus sign appears because opposite orientations of the boundary components
result in the opposite directions of the corresponding Dehn twists.
Moreover, as $\phi \in S'_0$, $[\operatorname{traj}_{\mathcal{D}} (\phi)] = 0$ hence $k_1 = k_0 = 0$.
Therefore the restriction $\bar{\phi}_P$ belongs to the identity component of $Symp_c (P)$.

Pick $K : \mathbb{A} \to \mathbb{R}$ supported in a small neighborhood of $\mathcal{D}$
such that $K = 1$ in a neighborhood of the closure $\overline{\mathcal{D}}$.
Denote by $\chi^t$ the time-$t$ map generated by the Hamiltonian flow of $K$.
$\chi^t$ is supported in a disk in $\mathbb{A}$, hence $\rho (\chi^t) = 0$ for all $t$.

Consider the homomorphism $i_* : H^1_c (P; \mathbb{R}) \to H^1_c (\mathbb{A}; \mathbb{R})$ induced by inclusion $i : P \to \mathbb{A}$.
Both $\chi^t, \phi_P$ are Hamiltonian in $\mathbb{A}$, hence their fluxes are zero in $H^1_c(\mathbb{A}; \mathbb{R})$.
After restriction to $P$, $flux (\chi^t\big|_P), flux (\bar{\phi}_P)$ belong to one-dimensional subspace $\ker i_* \subset H^1_c (P; \mathbb{R})$.
$flux (\chi^t\big|_P) \neq 0$, therefore one can find an appropriate $t_\phi \in \mathbb{R}$
such that $\phi'_P = \chi^{t_\phi} \circ \phi_P$ restricts to $\bar{\phi}'_P = \chi^{t_\phi}\big|_P \circ \bar{\phi}_P$
with zero flux in $P$.
The results of ~\cite{Ba:structure-groups} imply $\bar{\phi}'_P \in Ham (P)$.

Pick a compactly supported function $F_t : P \times [0, 1] \to \mathbb{R}$ whose flow generates $\bar{\phi}'_P$.
Denote by $U_s$ the complement of the closed disk $\overline{j_s(\mathcal{D})}$ in $S^2$, it is a displaceable disk.
$j_{s,*} (\phi'_P) \in Ham(S^2)$ and it is supported in $U_s$, therefore
\[
	Cal_{j_s} (\phi'_P) = Cal_{U_s} (j_{s,*} (\phi'_P)) =
				\int_0^1 \left( \int_{U_s} j_{s,*} F_t \omega \right) \text{d} t = \int_0^1 \left( \int_P F_t \omega \right) \text{d} t
\]
is independent of $s$. Hence $\rho(\phi'_P) = 0$. From $\phi'_P = \chi^{t_\phi} \circ \phi_P$
we obtain that $$|\rho (\phi'_P) - \rho (\chi^{t_\phi}) - \rho (\phi_P)| = |\rho (\phi_P)|$$
is bounded by the defect of $\rho$.
It follows that $\rho$ is bounded on the subgroup $S'_0$. As $\rho$ is homogeneous, it vanishes there.

\begin{remnonum}
  Choice of different parameters $s_1, s_2$ in ~\eqref{eq:qm_def} gives rise to
  different quasimorphisms $\rho_A = \rho_{A, s_1, s_2}$. If we fix $s_1$ and let $s_2$ vary, we get a linearly independent family.
  We show that by an argument from ~\cite{Bi-En-Po:Calabi-qm}:
  let $H (\theta, h) = H (h): \mathbb{A} \to \mathbb{R}$ be a function which depends only on $h$ and denote by $\phi_H \in Ham(\mathbb{A})$
  the corresponding Hamiltonian.
  The median level set relevant for $Cal_{j_s} (\phi_H)$ is $S^1 \times \{A - s\}$, hence
  \[
	\rho_{A, s_1, s_2} (\phi_H) = Cal_{j_{s_2}} (\phi_H) - Cal_{j_{s_1}} (\phi_H) = 2A \cdot (H (A-s_1) - H (A-s_2)).
  \]
  If we consider the set of all Hamiltonians generated by functions $H = H(h)$,
  the values of the family $\{\rho_{A, s_1, s_2}\}_{s_2}$ applied to this set are linearly independent.
\end{remnonum}

\section{Discussion} \label{S:disc}

\subsection {Comparison with the Riemannian length spectrum}
In the case of a Riemannian manifold $(M, g)$ the marked length spectrum $l : \pi_1 (M) \to \mathbb{R}$ contains a lot of information
regarding the metric $g$. For example, in a closed negatively curved surface $(M, g)$, $g$ is completely determined by $l$ (~\cite{Otal:spec}).

The symplectic analogue looks much less rewarding. The only case when $l_A : H_1 (M; \mathbb{Z}) \to \mathbb{R}$ is not bounded is the case of
a punctured disk. Therefore Hofer's length spectrum is able to detect genus zero surfaces of bounded area. It may happen that more
accurate estimates of $l_A$ provide additional information regarding the topology of $M$.

On the other side, the symplectic setting is
less rich than the Riemannian one. The equivalence class of $(M, \omega)$ is determined by the genus, $\text{rk} (H_1 (M; \mathbb{Z}))$ and $\Area(M)$.
In the case of finite area $\omega$ is just a scaling factor, hence asymptotic behavior behavior of $l_A$ depends only on the topology of $M$.
Clearly, $l_A$ is able to extract some of the topological information of $M$. This result may be interpreted as the fact that the geometry of $Ham (M)$ sees some of
the topology of $M$.

\subsection {Homotopical spectrum} \label{S:disc_homo}
It would be interesting to provide similar estimates for $l_A : \pi_1 (M) \to \mathbb{R}$.
When we consider the asymptotical behavior of $l_A$, it is more convenient to discuss the stabilized version
\[
  \bar{l}_A (\alpha) = \lim_{n \to \infty} \frac{l_A (n \alpha)}{n}.
\]
The homological version of $\bar{l}_A$ is a norm on $H_1$ when $2 A > \Area(M)$ and $genus(M) = 0$. In all other cases $\bar{l}_A \equiv 0$.

If we replace $H_1$ by $\pi_1$, the argument from the previous sections gives the following partial information:
\begin{itemize}
  \item
	Suppose that $2 A \leq \Area(M)$. Then $\bar{l}_A \equiv 0$. This is true since the argument of Section~\ref{S:up-disp}
	applies for $\pi_1$ and not just $H_1$.
  \item
	Suppose $2 A > \Area(M)$ and $genus(M)>0$. Assume that $\alpha$ is represented by a simple nonseparating loop. Then
	$n \alpha$ can be represented by catenation of two simple loops, hence $l_A (n \alpha) \leq 2 A$. Therefore $\bar{l}_A (\alpha) = 0$
	and $\bar{l}_A (k \alpha) = 0$ for all $k \in \mathbb{Z}$.
  \item
	Suppose $2 A > \Area(M)$ and $genus(M)=0$. If $\alpha$ corresponds to a nonzero homology class
	then $\bar{l}_A (\alpha) > 0$.
\end{itemize}

The author does not know estimates for most of the remaining cases. For example, let $2 A > \Area(M)$ and $genus(M) geq 2$.
For a primitive self intersecting class $\alpha$ the number of simple curves needed to represent $n \alpha$ is not bounded as $n \to \infty$
(see ~\cite{Ca:wlen}), so the argument for the upper bound does not give a lot. On the other hand, we do not know any tools that may give a nontrivial
lower bound. There are several constructions of quasimorphisms on positive genus surfaces, but none of them is known to respect Hofer's metric.
When $genus(M) = 0$ and $\alpha$ belongs to the commutator subgroup $[\pi_1 (M), \pi_1 (M)]$, the argument used in this paper
fails as translation by $\alpha$ cannot be realized by an autonomous flow. It may happen that Entov-Polterovich quasimorphisms
used in our argument still imply a nontrivial lower bound. Unfortunately, the author was not able to compute them on appropriate
non-autonomous examples.

D. Calegari observed that if one takes the word length norm with
respect to the generating set of simple loops, then $A$ times its stabilized version  $A \| \cdot \|_S$ satisfies all properties of $\bar{l}_A$ described above
(we assume $2A > \Area(M)$). Indeed, the discussion in Section~\ref{S:up} implies $\bar{l}_A \leq A \| \cdot \|_S$ and we do not know any example
where the inequality is strict.

\subsection{Higher dimensions}
Most arguments and constructions used in this paper completely fail when one considers balls or tori in higher dimension manifolds.
For example, upper bounds for $l_A (\alpha)$ were induced by decomposition of $\alpha$ into simple loops.
In dimension $\geq 4$ any loop can be perturbed into a simple one. However, we cannot translate a ball through a pipe containing
such simple loop because of the `symplectic camel' phenomenon.

\subsection{The quasimorphism $\rho_A$} \label{S:disc_quas}
Theorem~\ref{T:quas} describes quasimorphisms $\rho_A : Ham (M) \to H_1 (M; \mathbb{R})$ for a $k$ times punctured disk $M$.
$\rho_A$ can be seen as a generalization of the rotation number $\rho : \widetilde{Homeo}_+ (S^1) \to \mathbb{R}$. Indeed, $\rho$
satisfies the following: it is Lipschitz with respect to the $C_0$-norm on $\widetilde{Homeo}_+ (S^1)$. If $f \in \widetilde{Homeo}_+ (S^1)$
has a fixed point $x$, then $\rho (f)$ computes the degree of the trajectory of $x$ under $f$. In fact, $\rho$ is determined by the
second property, while there is a continuous family if linearly independent homogeneous quasimorphisms satisfying same properties as $\rho_A$.

Let $f_t$ be a Hamiltonian isotopy and $\mathcal{D}$ a disk of area $A$ or greater.
The computation in Section~\ref{S:qm} shows that $\rho_A$ vanishes on those isotopies $f_t$ that fix $\partial \mathcal{D}$ for all $t$.
This implies that the value $\rho_A (f_1)$ is determined up to a bounded defect by the trajectory $f_t (\partial \mathcal{D})$.
That is, for any $g_t$ such that $g_t (\partial \mathcal{D}) = f_t (\partial \mathcal{D})$, $\rho_A (g_1) - \rho_A (f_1)$ is bounded.
Therefore $\rho_A$ can be thought as an invariant of Hamiltonian isotopies of $\partial \mathcal{D}$.

In particular, $\rho_A$ nearly ignores those dynamical attributes that do not affect a large enough disk in $M$, it sees only `global' features that
deform all large disks. In this sense it is different from most other generalizations of rotation number to dimension two.

A possible application is the ability to detect non-existence of large fixed or periodic disks. Namely, if we pick two
such quasimorphisms $\rho_A, \rho'_B$ and $\rho_A(f) \neq \rho'_B (f)$, then
$f$ has neither fixed nor periodic disks of area $\geq \max\{A, B\}$. More than that, if one wishes to perturb $f$ in order to have such a periodic
disk, $\rho_A(f) - \rho'_B (f)$ allows to estimate a lower bound on Hofer's energy of such perturbation. However, current techniques
allow to compute the value of $\rho_A$ in very limited number of scenarios (like the case of autonomous Hamiltonians, locally supported and etc.)
which limits practical use of this observation.

Another application is as follows. Pick a non-displaceable disk $\mathcal{D}$ in $M$ and two linearly independent quasimorphisms $\rho_A, \rho'_B$
where $A, B \leq \Area{\mathcal{D}}$. Denote by $\mathcal{L}$ the space of Lagrangians Hamiltonian isotopic to $\partial \mathcal{D}$.
Suppose that $f_t$ is a Hamiltonian isotopy. A simple computation shows that $\tau = \rho_A - \rho'_B$ depends (up to a bounded defect) just on
$f_1 (\partial \mathcal{D}) \in \mathcal{L}$. This way $\tau$ gives rise to an invariant on $\mathcal{L}$ which is Lipschitz with respect to the induced Hofer's norm.
$\tau$ is not bounded since it is a nonzero homogeneous quasimorphism, therefore by Lipschitz property the space $\mathcal{L}$ is not bounded in Hofer's norm as well
(a similar argument with more details can be found in ~\cite{Kh:thesis}).


text:
\section{Introduction}

It is well believed that entanglement is a very important ingredient for
quantum information processing and quantum communication tasks that cannot
be done efficiently classically \cite{Nielsen}. It plays a central role in
several tasks such as, quantum teleportation, quantum key distribution, and
superdense coding \cite{Loock,HorodeckiS}. Previously, any correlation
without entanglement was thought to be purely classical as they can be
produced with local operations and classical communications (LOCC) \cit
{Kavan Modi}. However, it has been shown in many theoretical \cit
{Braunstein-} and experimental \cite{Lanyon} works that
separable(unentangled) states, traditionally referred to as \textit
classically correlated,} might retain some signatures of quantumness with
potential operational applications for quantum tasks \cite{Kavan Modi,G(1)}.
For example, quantum correlations without entanglement were shown to be
useful to characterize resources in \cite{RahimiKeshari}: a quantum
computational model (DQC1); quantum state merging; remote state preparation;
encoding information onto a quantum state; quantum phase estimation; and
quantum key distribution. In general, entanglement is difficult to generate,
more fragile and rapidly decays to zero in dissipative-noisy systems \cit
{ESD}. Conversely, it has been pointed out that quantum correlations without
entanglement are robust, exhibiting some peculiar features against Markovian
decoherence \cite{Kavan Modi}. We can cite for instance \cite{NSP}: they can
never suffer the sudden death; within a dissipative-noisy systems, they can
reach an asymptotic nonzero value (i.e., freezing behavior) even for high
temperatures; they can exhibit a sudden change from the \textit{classical
decoherence} regime to the \textit{quantum decoherence} regime. Such
interesting properties make, therefore, this kind of quantum correlations
more desirable resource in quantum information processing, opening new
avenues for theoretical exploration and practical applications \cite{Kavan
Modi}.

Having a \textit{quantum separable state}, the question is then how to
quantify its quantum correlations ? In fact, quantifying quantum
correlations of separable states,i.e.,\textit{the quantumness of correlation
}, is one of the crucial tasks related to the understanding and efficient
exploitation of such states in different quantum information processing
schemes \cite{Kavan Modi}. In this sense, the quantification of the
quantumness of correlations, was carried out originally by means of the
quantum discord \cite{ZA}, defined and evaluated mainly for discrete
variables systems \cite{ZA} and subsequently extended to the continuous
variable systems \cite{G(1)}. Recently, a number of other related
quantifiers of the \textit{quantumness} of correlations were proposed. As
examples, one may cite the relative entropy of quantumness \cite{ERQ}, the
quantum deficit \cite{Marcio}, the negativity of quantumness \cite{Takafumi
, the Gaussian quantum discord \cite{G(1)}, the Gaussian geometric discord
\cite{AG}, the operational Gaussian Discord \cite{Rahimi}, the Gaussian
Hellinger distance \cite{Marian}, and also the Gaussian R\'{e}nyi-2 discord
\cite{AGS}.

In a double-cavity optomechanical system fed by broadband two-mode squeezed
light, we propose in this paper to investigate both entanglement and quantum
discord between two mechanical modes (the mechanical subsystem) on one hand,
and between two optical modes (the optical subsystem) on the other hand. To
accomplish this, we shall use respectively the Gaussian R\'{e}nyi-2
entanglement and the Gaussian R\'{e}nyi-2 discord to quantify entanglement
and quantum discord \cite{AGS}.

During the last few years, optomechanical devices based on the interaction
between light and mechanical degrees of freedom, have sparkled the interest
of a vast scientific community as very promising platform for demonstrating
various quantum phenomena \cite{Meystre}. Proposals include decoherence \cit
{Ulrich}, quantum back-action evading \cite{Korppi}, macroscopic quantum
superposition \cite{MAbdi}, optomechanically induced transparency \cit
{GAgarwal,OIT}, the cooling of an harmonic oscillator near its quantum
ground state \cite{Teufel,Chan,Yu}, quantum states transfer \cit
{Rakhubovsky}, and also the study of various aspects of quantum
correlations: quantum entanglement \cite{entanglement, OET,MET}, quantum
discord \cite{discord} and quantum steering \cite{steering}.

The remainder of this paper is organized as follows. In Sec \ref{sec2}, we
introduce the optomechanical system at hand and we exactly solve the quantum
Langevin equations in Fourier space, which allows us to obtain analytical
formulas for the covariance matrices fully describing the subsystems under
investigation. Sec \ref{sec3} is devoted to the quantification of
nonclassical correlations in the two studied subsystems. This quantification
is performed using the Gaussian R\'{e}nyi-2 entropy. Finally, in Sec \re
{sec4} we draw our conclusions.

\section{ The optomechanical model and Hamiltonian \label{sec2}}

\subsection{ The model}

\begin{figure}[tbh]
\centerline{\includegraphics[width=14cm]{fig1.eps}}
\caption{Sketch of two optomechanical Fabry-P\'{e}rot cavities fed by
non-classical light from \textit{self-phase-locked OPO}. Each cavity is
pumped by a coherent laser field of power ${\wp }_{j}$ and frequency
\protect\omega _{L_{j}}$ ($j=1,2$). The $j^{th}$ movable mirror will be
treated as an harmonic oscillator with frequency $\protect\omega _{\protec
\mu _{j}}$, a damping rate $\protect\gamma _{j}$ and an effective mass $m_
\protect\mu _{j}}$. $\digamma _{j}^{in}$ is the $j^{th}$ noise operator
corresponding to the $j^{th}$ squeezed mode.}
\label{Fig.1}
\end{figure}
\noindent We consider two optomechanical Fabry-P\'{e}rot cavities (see Fig.
\ref{Fig.1}). Each cavity having length $l_{j}$ and frequency $\omega
_{c_{j}}$, is constituted by two mirrors. The first one is fixed and
partially transmitting. Whereas the second is a totally reflecting movable
mirror. This end is modeled as single-mode quantum harmonic oscillator with
effective mass $m_{\mu _{j}}$, oscillating with the frequency $\omega _{\mu
_{j}}$ and damped with a rate $\gamma _{j}$ \cite{mmm}. In addition, the
j^{th}$ cavity is pumped by a laser field with the input power ${\wp }_{j}$,
phase $\varphi _{j}$ and frequency $\omega _{L_{j}}$ ($j=1,2$). Finally, the
two cavities are jointly pumped by a two-mode squeezed light produced for
example by a self-phase-locked optical parametric operator (OPO) below
threshold \cite{Keller}.

\subsection{ The Hamiltonian}

The Hamiltonian of the two cavities, in a frame rotating at the frequency of
the lasers, reads \cite{Law}
\begin{equation}
H=\sum_{j=1}^{2}\left[ \hbar\left( \omega _{c_{j}}-\omega _{L_{j}}\right)
c_{j}^{\dag }c_{j}+\hbar\omega _{\mu _{j}}b_{j}^{\dag }b_{j}+\hbar
g_{j}c_{j}^{\dag }c_{j}(b_{j}^{\dag }+b_{j})+\hbar\varepsilon
_{j}(e^{i\varphi _{j}}c_{j}^{\dag }+e^{-i\varphi _{j}}c_{j})\right],
\label{E1}
\end{equation
where $b_{j}$($c_{j}$) is the annihilation operator of the $j^{th}$
mechanical(optical) mode. Moreover, $g_{j}=\left( \omega
_{c_{j}}/l_{j}\right) \sqrt{\hbar /m_{\mu _{j}}\omega _{\mu _{j}}}$ is the
vacuum optomechanical coupling rate between single photon and single phonon.
The parameter $\varepsilon _{j}$ is related to the input laser power ${\wp
_{j}$ by $\varepsilon _{j}=\sqrt{2\kappa _{j}{\wp }_{j}/\hbar \omega _{L_{j}
}$, where $\kappa _{j}$ is the photon decay rate by leaking out of the
j^{th}$ cavity.

\subsection{Quantum Langevin equation}

A proper analysis of the system at hand must include the dissipations and
the noises in the cavities. This can be accomplished by considering the
following set of nonlinear quantum Langevin equations (QLEs)
\begin{eqnarray}
\dot{b}_{j} &=&-(\gamma _{j}/2+i\omega _{\mu _{j}})b_{j}-ig_{j}c_{j}^{\dag
}c_{j}+\sqrt{\gamma _{j}}\zeta _{j}^{in},  \label{E2} \\
\dot{c}_{j} &=&-\left( \kappa _{j}/2-i\Delta _{j}\right)
c_{j}-ig_{j}c_{j}(b_{j}^{\dag }+b_{j})-i\varepsilon _{j}e^{i\varphi _{j}}
\sqrt{\kappa _{j}}\digamma _{j}^{in},  \label{E3}
\end{eqnarray
where $\Delta _{j}=\omega _{L_{j}}-\omega _{c_{j}}$ is the $j^{th}$ detuning
\cite{mmm}. Moreover, $\zeta _{j}^{in}$ is the $j^{th}$ Brownian noise
operator with zero mean value ($\langle \zeta _{j}^{in}\rangle =0$),
describing the coupling of the $j^{th}$ movable mirror with its own
environment. In general, $\zeta _{j}^{in}$ is not a Markovian Gaussian noise
(not $\delta $-correlated). However, for large mechanical quality factors
\mathcal{Q}_{j}=\omega _{\mu _{j}}/\gamma _{j}\gg 1$, which is a necessary
condition to achieve quantum effects \cite{Vitali(1)}, the operator $\zeta
_{j}^{in}$ becomes $\delta $-correlated \cite{Girish}
\begin{equation}
\left( \langle \zeta _{j}^{in\dag }(\omega )\zeta _{j}^{in}(\omega ^{\prime
})\rangle ;\langle \zeta _{j}^{in}(\omega )\zeta _{j}^{in\dag }(\omega
^{\prime })\rangle \right) =2\pi \Big(n_{th,j};(n_{th,j}+1)\Big)\delta
(\omega +\omega ^{\prime }),  \label{E4-}
\end{equation
with $n_{th\mathrm{,}j}=1/\left( \exp (\hbar \omega _{\mu
_{j}}/k_{B}T_{j})-1\right) $ is the $j^{th}$ mean thermal photons number at
the mechanical frequency $\omega _{\mu _{j}}$. $T_{j}$ and $k_{B}$ are
respectively the temperature of the environment surrounding the $j^{th}$
movable mirror and the Boltzmann constant. Another kind of noise affecting
the $j^{th}$ cavity is the input squeezed vacuum noise operator $\digamma
_{j}^{in}$, with zero mean value ($\langle \digamma _{j}^{in}\rangle =0$). In what follows, we assume that the squeezed field is resonant with the
cavities, which is required in order to achieve optimal quantum states transfer from
squeezed light to either mechanical degrees of freedom or optical modes \cite{Hammerer}. The operator $\digamma _{j}^{in}$ has the following nonzero frequency-domain correlation functions \cite{JieLi}
\begin{eqnarray}
\left( \langle \delta \digamma _{j}^{in^{\dag }}(\omega )\delta \digamma
_{j}^{in}(\omega ^{\prime })\rangle ;\langle \delta \digamma
_{j}^{in}(\omega )\delta \digamma _{j}^{in^{\dag }}(\omega ^{\prime
})\rangle \right)  &=&2\pi \Big(N;(N+1)\Big)\delta (\omega +\omega ^{\prime
})\text{ ,}  \label{E7} \\
\left( \langle \delta \digamma _{j}^{in}(\omega )\delta \digamma
_{k}^{in}(\omega ^{\prime })\rangle ;\langle \delta \digamma _{j}^{in^{\dag
}}(\omega )\delta \digamma _{k}^{in^{\dag }}(\omega ^{\prime })\rangle
\right)  &=&2\pi \Big(M;M\Big)\delta (\omega +\omega ^{\prime }-\omega _{\mu
_{1}}-\omega _{\mu _{2}})\text{ },  \label{E9}
\end{eqnarray
for $j\neq k=1,2$ with $N=\mathrm{sinh}^{\mathrm{2}}r$ and $M=\mathrm{sinh}
\mathrm{cosh}r$, $r$ being the squeezing parameter.

\subsection{Linearization of quantum Langevin equations}

To study quantum correlations in the mechanical and optical subsystems under
investigation, we need to calculate their fluctuations. Assuming that the
nonlinear coupling between the $j^{th}$ cavity mode and its corresponding
mechanical mode is weak \cite{Girish}, the fluctuation $\delta c_{j}$(
\delta b_{j}$) of the operator $c_{j} $($b_{j}$) is much smaller than its
steady-state mean value $\langle c_{j}\rangle $($\langle b_{j}\rangle $).
Thus we can linearize the system around the steady-states by writing the
operator $c_{j}$($b_{j}$) as the sum of its steady-state value $\langle
c_{j}\rangle $($\langle b_{j}\rangle $) and a small linear displacement
\delta c_{j}$ ($\delta b_{j} $), i.e., $c_{j}=\langle c_{j}\rangle +\delta
c_{j}$ and $b_{j}=\langle b_{j}\rangle +\delta b_{j}$. The mean values
\langle c_{j}\rangle \equiv c_{js}$ and $\langle b_{j}\rangle \equiv b_{js}$
can be easily calculated by setting the time derivatives to zero and
factorizing the averages in Eqs. (\ref{E2})-(\ref{E3}). One obtains
therefore
\begin{equation}
\langle c_{j}\rangle =c_{js}=\frac{-i\varepsilon _{j}e^{i\varphi _{j}}}
\kappa _{j}/2-i\Delta _{j}^{\prime }}\text{ \ \ \ and \ \ \ }\langle
b_{j}\rangle =b_{js}=\frac{-ig_{j}\left\vert c_{js}\right\vert ^{2}}{\gamma
_{j}/2+i\omega _{\mu _{j}}},\qquad  \label{E10}
\end{equation
with $\Delta _{j}^{\prime }$ $=$ $\Delta _{j}$ $-g_{j}(b_{js}^{\ast
}+b_{js}) $ is the $j^{th}$ effective cavity detuning including the
radiation pressure effects \cite{mmm}. We suppose henceforth that the two
cavities are strongly driven $\left\vert c_{js}\right\vert \gg 1$, which can
be achieved through a large input power $\wp _{j}$ \cite{Vitali(1)}. So, the
second-order product fluctuations terms $\delta c_{j}^{\dag }\delta c_{j}$,
\delta c_{j}\delta b_{j}$ and $\delta c_{j}\delta b_{j}^{\dag }$ can be
safely ignored. Hence, we get
\begin{eqnarray}
\delta \dot{b}_{j} &=&-\left( \gamma _{j}/2+i\omega _{\mu _{j}}\right)
\delta b_{j}+\mathcal{G}_{j}\left( \delta c_{j}-\delta c_{j}^{\dag }\right)
\sqrt{\gamma _{j}}\zeta _{j}^{in},  \label{E11} \\
\delta \dot{c}_{j} &=&-\left( \kappa _{j}/2-i\Delta _{j}^{\prime }\right)
\delta c_{j}-\mathcal{G}_{j}\left( \delta b_{j}^{\dag }+\delta b_{j}\right)
\sqrt{\kappa _{j}}\delta \digamma _{j}^{in},  \label{E12}
\end{eqnarray
$\mathcal{G}_{j}$ $=g_{j}\left\vert c_{js}\right\vert =g_{j}\sqrt{\bar{n
_{cav}^{j}}$ and $\bar{n}_{cav}^{j}$ are respectively the $j^{th}$
many-photon optomechanical coupling and the average number of photons
circulating inside the $j^{th}$ cavity \cite{Marquardt}. We note that Eqs.
\ref{E11})-(\ref{E12}) are obtained by choosing the phase $\varphi_{j}$ of
the $j^{th}$ input laser field as $\varphi _{j}=-\arctan(2\Delta
_{j}^{\prime }/\kappa _{j})$ or equivalently $c_{js}=-i\left\vert
c_{js}\right\vert $. Moving into an interaction picture by introducing the
operators $\delta \tilde{b}_{j}=\delta b_{j}e^{i\omega _{\mu _{j}}t}$ and
\delta \tilde{c}_{j}=\delta c_{j}e^{-i\Delta _{j}^{\prime }t}$, Eqs. (\re
{E11})-(\ref{E12}) become
\begin{eqnarray}
\delta \dot{\tilde{b}}_{j} &=&-\frac{\gamma _{j}}{2}\delta \tilde{b}_{j}\
\mathcal{G}_{j}\left( \delta \tilde{c}_{j}e^{i\left( \Delta _{j}^{\prime
}+\omega _{\mu _{j}}\right) t}-\delta \tilde{c}_{j}^{\dag }e^{-i\left(
\Delta _{j}^{\prime }-\omega _{\mu _{j}}\right) t}\right) +\sqrt{\gamma _{j}
\tilde{\zeta}_{j}^{in},  \label{E13} \\
\delta \dot{\tilde{c}}_{j} &=&-\frac{\kappa _{j}}{2}\delta \tilde{c}_{j}
\mathcal{G}_{j}\left( \delta \tilde{b}_{j}e^{-i\left( \Delta _{j}^{\prime
}+\omega _{\mu _{j}}\right) t}+\delta \tilde{b}_{j}^{\dag }e^{-i\left(
\Delta _{j}^{\prime }-\omega _{\mu _{j}}\right) t}\right) +\sqrt{\kappa _{j}
\delta \tilde{\digamma}_{j}^{in}.  \label{E14}
\end{eqnarray
Next, we focus on the red sideband,i.e., $\Delta _{j}^{\prime }=-\omega
_{\mu _{j}}$, which is relevant for quantum states transfer \cite{steering}.
In addition, in the resolved-sideband regime, where $\omega _{\mu _{j}}\gg
\kappa _{j}$, one can perform the rotating wave approximation (RWA) \cit
{SGHofer}, which allows us to drop fast-rotating terms oscillating with $\pm
2\omega _{\mu _{j}}$ in Eqs. (\ref{E13})-(\ref{E14}). We then obtain
\begin{eqnarray}
\quad \delta \dot{\tilde{b}}_{j} &=&-\frac{\gamma _{j}}{2}\delta \tilde{b
_{j}+\mathcal{G}_{j}\delta \tilde{c}_{j}+\sqrt{\gamma _{j}}\tilde{\zeta
_{j}^{in},  \label{E15} \\
\delta \dot{\tilde{c}}_{j} &=&-\frac{\kappa _{j}}{2}\delta \tilde{c}_{j}
\mathcal{G}_{j}\delta \tilde{b}_{j}+\sqrt{\kappa _{j}}\delta \tilde{\digamma
_{j}^{in}.  \label{E16}
\end{eqnarray}

\subsection{Covariance matrix}

Equations (\ref{E15})-(\ref{E16}) can be solved in the frequency domain
using the Fourier transform of each equation. So, one obtains
\begin{eqnarray}
\delta \tilde{c}_{j}(\omega ) &=&-\frac{\mathcal{G}_{j}}{\Xi _{j}(\omega )
\sqrt{\gamma _{j}}~\tilde{\zeta}_{j}^{in}(\omega )+\frac{\left( \gamma
_{j}/2+i\omega \right) }{\Xi _{j}(\omega )}\sqrt{\kappa _{j}}~\delta \tilde
\digamma}_{j}^{in}(\omega ),  \label{deltactilde} \\
\delta \tilde{b}_{j}(\omega ) &=&\frac{\left( \kappa _{j}/2+i\omega \right)
}{\Xi _{j}(\omega )}\sqrt{\gamma _{j}}~\tilde{\zeta}_{j}^{in}(\omega )+\frac
\mathcal{G}_{j}}{\Xi _{j}(\omega )}\sqrt{\kappa _{j}}~\delta \tilde{\digamma
_{j}^{in}(\omega ),  \label{deltabtilde}
\end{eqnarray
with $\Xi _{j}(\omega )=\mathcal{G}_{j}^{2}+\left( \gamma _{j}/2+i\omega
\right) \left( \kappa _{j}/2+i\omega \right) $. \newline
The correlation function of any pair of fluctuation operators is acquired as
\begin{equation}
\mathcal{V}_{kk^{\prime }}=\frac{1}{4\pi ^{2}}\int \int d\omega d\omega
^{\prime }e^{-i(\omega +\omega ^{\prime })t}\mathcal{V}_{kk^{\prime
}}(\omega ,\omega ^{\prime }),  \label{CM elements}
\end{equation
$\mathcal{V}_{kk^{\prime }}(\omega ,\omega ^{\prime })=\langle \tilde{u
_{k}(\omega )\tilde{u}_{k^{\prime }}(\omega ^{\prime })+\tilde{u}_{k^{\prime
}}(\omega ^{\prime })\tilde{u}_{k}(\omega )\rangle /2$ (for $k,k^{\prime
}=1,..,8$) is the frequency-domain correlation function between elements $k$
and $k^{\prime }$ of $\tilde{u}=(\delta \tilde{q}_{1}^{m},\delta \tilde{p
_{1}^{m},\delta \tilde{q}_{2}^{m},\delta \tilde{p}_{2}^{m},\delta \tilde{q
_{1}^{op},\delta \tilde{p}_{1}^{op},\delta \tilde{q}_{2}^{op},\delta \tilde{
}_{2}^{op})$, where $\delta \tilde{q}_{j}^{m}=(\delta \tilde{b}_{j}^{\dagger
}+\delta \tilde{b}_{j})/\sqrt{2}$ and $\delta \tilde{p}_{j}^{m}=i(\delta
\tilde{b}_{j}^{\dagger }-\delta \tilde{b}_{j})/\sqrt{2}$ \Big($\delta \tilde
q}_{j}^{op}=(\delta \tilde{c}_{j}^{\dag }+\delta \tilde{c}_{j})/\sqrt{2}$
and $\delta \tilde{p}_{j}^{op}=i(\delta \tilde{c}_{j}^{\dag }-\delta \tilde{
}_{j})/\sqrt{2}$\Big) are the quadratures position and momentum of the
j^{th}$ mechanical(optical) mode. Using Eq. (\ref{CM elements}) and the
correlation properties of the noise operators given by Eqs. [(\ref{E4-})-
\ref{E9})], the covariance matrix (CM) of the whole system (four-mode
Gaussian states) can be evaluated explicitly. In this work, we are
interested in the non-classical properties of the steady state of two-mode
Gaussian states. So, we shall focus onto the entanglement and the quantum
discord of two different bipartite subsystems, which can be formed by
tracing over the remaining degree of freedom. \newline
The $4\times 4$ CM $\mathcal{V}^{i}$ of the selected bipartition that
obtained by neglecting the rows and columns of the uninteresting mode, can
be written in the following standard form with diagonal sub-blocks,
\begin{equation}
\mathcal{V}^{i}=\left(
\begin{array}{cc}
\mathcal{V}_{1}^{i} & \mathcal{V}_{3}^{i} \\
\mathcal{V}_{3}^{i\mathrm{T}} & \mathcal{V}_{2}^{i
\end{array
\right) ,\text{ for }i \in \{m,op\}\text{,}  \label{CM}
\end{equation
with $\mathcal{V}_{1}^{i}=\mathrm{diag}(\nu _{1}^{i},\nu _{1}^{i})$,
\mathcal{V}_{2}^{i}=\mathrm{diag}(\nu _{2}^{i},\nu _{2}^{i})$ and $\mathcal{
}_{3}^{i}=\mathrm{diag}(\nu _{3}^{i},\nu _{3}^{^{\prime }i})$, where
i\equiv m$ ($op$) refers the two-mode mechanical(optical) subsystem. The
matrix $\mathcal{V}^{i}$ is a real, symmetric and positive definite matrix.
The $2\times 2$ matrices $\mathcal{V}_{1}^{m}$ and $\mathcal{V}_{2}^{m}$(
\mathcal{V}_{1}^{op}$ and $\mathcal{V}_{2}^{op}$) represent the first and
second mechanical(optical) modes, while the correlations between them are
described by $\mathcal{V}_{3}^{m}$($\mathcal{V}_{3}^{op}$). Being a physical
state, the $\mathcal{V}^{i}$-(CM) should satisfy the Heisenberg-Robertson
uncertainty principle \ $\mathcal{V}^{i}+i\Omega _{3}/2\geqslant 0$, where
\Omega _{3}=\oplus _{1}^{3}i\sigma _{y}$ and $\sigma _{y}$ are respectively
the so-called symplectic matrix and the $y$-Pauli matrix \cite{SSM}. \newline
Considering identical cavities, i.e., $\omega _{\mu _{1,2}}=\omega _{\mu }, $
$\gamma _{1,2}=\gamma ,$ $\kappa _{1,2}=\kappa $, etc.., the elements of the
matrices $\mathcal{V}^{i}$ ($i\in \{m,op\}$) defined by Eq.(\ref{CM}) are
\begin{eqnarray}
\nu _{1}^{m} &=&\nu _{2}^{m}=\frac{\left( 2n_{th}+1\right) \left( 1+\Gamma
+\Gamma \mathcal{C}\right) +\mathcal{C}\cosh 2r}{2(1+\Gamma )\left( 1
\mathcal{C}\right) },\text{ \ }\nu _{3}^{m}=-\nu _{3}^{^{\prime }m}=\frac
\mathcal{C}\sinh 2r}{2(1+\Gamma )\left( 1+\mathcal{C}\right) },\text{\ \ \ \
\ \ \ \ }  \label{a} \\
\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ }\nu _{1}^{op} &=&\nu _{2}^{op}=\frac
\left( 2n_{th}+1\right) \Gamma \mathcal{C}+\left( 1+\Gamma +\mathcal{C
\right) \cosh 2r}{2(1+\Gamma )\left( 1+\mathcal{C}\right) },\text{ }\nu
_{3}^{op}=-\nu _{3}^{^{\prime }op}=\frac{\left( 1+\Gamma +\mathcal{C}\right)
\sinh 2r}{2(1+\Gamma )\left( 1+\mathcal{C}\right) },  \label{c}
\end{eqnarray
where $\Gamma $ $=\gamma /\kappa $ and $\mathcal{C}=4\mathcal{G}^{2}/\gamma
\kappa =\frac{8\omega _{c}^{2}\wp }{\gamma m_{\mu }\omega _{\mu }\omega
_{L}l^{2}\left[ \left( \frac{\kappa }{2}\right) ^{2}+\omega _{_{\mu }}^{2
\right] }$ are respectively the damping ratio and the optomechanical
cooperativity \cite{OET}. \newline
In the following, using the approach sketched above, one can analysis both
entanglement and quantum discord in the mechanical and optical subsystems.
For this, we shall use the Gaussian R\'{e}nyi-$2$ entropy \cite{AGS}.

\section{Gaussian R\'{e}nyi-2 quantum correlations \label{sec3}}

R\'{e}nyi-$\alpha $ entropies were first introduced by Alfred R\'{e}nyi as a
generalisation of the usual concept of entropy \cite{Renyi}. These entropies
encompass several entropic measures, including the Shannon entropy and the
von Neumann entropy \cite{AGS}. Motivated by the mathematical foundations of
entropy, the R\'{e}nyi-$\alpha $ entropies of a given quantum state
\mathcal{\varrho }$ are defined as \cite{AGS}
\begin{equation}
\mathcal{S}_{\alpha }\mathcal{(\varrho )\equiv (}1-\alpha \mathcal{)
^{-1}\ln \mathrm{Tr}\left( \varrho ^{\alpha }\right) ,  \label{Renyi-alpha}
\end{equation
where $\alpha \in (0,1)\cup (1,+\infty )$, which usually written as $\alpha
\in (0,1,+\infty )$ \cite{Berta2015}. These entropies are continuous,
non-negatives, invariants under the action of the unitary operations,
additive on tensor-product states, and reduce in the limit $\alpha
\rightarrow 1$ to the conventional von Neumann entropy defined by $\mathcal
S(\varrho )=}-\mathrm{Tr}\left( \varrho \ln \varrho \right) $ \cite{Martin}.
In quantum information theory, this famous entropy captures the degree of
quantum information possessed in an ensemble of a large number of
independent and identically distributed (i.i.d.) copies of the state \cit
{Schumacher}, where it has been intensively used in different fields of
sciences such as communication and coding theory \cite{CCT}, statistical
physics \cite{Sta}, quantum computation \cite{VVedral}, statistics and
related fields \cite{Statistique}. In addition, the von Neumann entropy has
been widely used with the well known name \textit{entropy of entanglement}
to quantify entanglement in miscellaneous areas of physics like theory of
black holes \cite{DKT}, relativistic quantum field theory \cite{QFT}, spins
of noninteracting electron gases \cite{Oh}, three-level atom under influence
of a Kerr-like medium \cite{Aty}, harmonic lattice systems \cite{P and E},
the Lipkin-Meshkov-Glick model \cite{Rico}, and also spin-orbital models
\cite{Horsch}. We notice that in the case of \textit{pure states}, many
entanglement measures (including entanglement cost \cite{Wootters,VDC},
entanglement of formation (EoF) \cite{Wootters,EoF}, squashed entanglement
\cite{Christandl}, and also the quantum discord \cite{G(1),ZA}) coincide
with the entropy of entanglement as the only measure of entanglement in this
case \cite{HorodeckiS}. In contrast, some other quantifiers of entanglement
such as the negativity \cite{Zyc} and the logarithmic negativity \cit
{MBPlenio} can not coincide with the entropy of entanglement even on pure
states. \newline
On the other hand, for $\alpha \neq 1$, the R\'{e}nyi-$\alpha $ entropies
are relevant in situations other than the i.d.d. setting \cite{Tomamichel}.
For example, they are useful to characterize many tasks of information
processing with either a single or finite number of resource utilizations
\cite{Hayashi}, and also to establish strong converse theorems \cite{Konig}.
Now, replacing $\alpha $ by the special value $2$ in Eq. (\ref{Renyi-alpha
), we obtain the simple expression $\mathcal{S}_{2}\mathcal{(\varrho )=}-\ln
\mathrm{Tr}\left( \varrho ^{2}\right) $ corresponding to the R\'{e}nyi-2
entropy \cite{AGS}. It has been proven that for arbitrary Gaussian states,
the R\'{e}nyi-2 entropy fulfills the strong subadditivity inequality;
\mathcal{S}_{2}\left( \mathcal{\varrho }_{AB}\right) +\mathcal{S}_{2}\left(
\mathcal{\varrho }_{BC}\right) \geqslant \mathcal{S}_{2}\left( \mathcal
\varrho }_{ABC}\right) +\mathcal{S}_{2}\left( \mathcal{\varrho }_{B}\right) $
\cite{AGS}, which made it possible to define bona fide Gaussian measures of
entanglement \cite{G(1),AL} and quantum discord \cite{Kavan Modi,G(1),ZA}.

\subsection{ Gaussian R\'{e}nyi-2 entanglement}

For generally mixed two-mode Gaussian states $\mathcal{\varrho
_{A^{i}B^{i}} $, the Gaussian R\'{e}nyi-2 entanglement (GR2E) $\mathcal{E
_{2}$ admits a complicated parametric formula \cite{AGS, AL}, that can be
computed numerically only by programming \cite{AGS, AL}. However, for all
two-mode Gaussian pure states as well as for several important classes of
mixed states (including squeezed thermal states STS \cite{G(1)}), closed
formulas for the GR2E have been obtained \cite{AGS,AL}. \newline
On one hand, the covariance matrix $\mathcal{V}^{i}$ (\ref{CM}) is in the
standard form \cite{G(1)}. On the other hand, the $\mathcal{V}_{3}^{i}
-block matrix has the form $\mathcal{V}_{3}^{i}=\mathrm{diag}(\nu
_{3}^{i},-\nu _{3}^{i})$ (see Eqs. [(\ref{CM})-(\ref{c})]), which
corresponds to the squeezed-thermal states STS \cite{G(1)}. Thus, the GR2E
\mathcal{E}_{2}^{i}$ ($i\in \{m,op\}$) for two-mode Gaussian states with CM
\mathcal{V}^{i}$ (\ref{CM}) is given by \cite{AL}
\begin{equation}
\mathcal{E}_{2}^{i}=\left\{
\begin{array}{c}
\frac{1}{2}\ln \left[ \frac{(4f^{i}+1)s^{i}-\sqrt{\left[ (4f^{i}-1)^{2}-1
\left( d^{i}\right) ^{2}\right] \left[ \left( s^{i}\right) ^{2}-\left(
d^{i}\right) ^{2}-f^{i}\right] }}{4(\left( d^{i}\right) ^{2}+f^{i})}\right]
^{2}\text{if \ \ \ \ }4|d^{i}|+1\leq 4f^{i}<4s^{i}-1, \\
0\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if \ \ \ \ \ \ }
4f^{i}\geqslant 4s^{i}-1
\end{array
\right.  \label{-E33-}
\end{equation
where $s^{i}=\left( \nu _{1}^{i}+\nu _{2}^{i}\right) /2$, $d^{i}=\left( \nu
_{1}^{i}-\nu _{2}^{i}\right) /2$ and $f^{i}=\sqrt{\det \mathcal{V}^{i}}$ for
$i\in \{m,op\}$.\newline
We note that, another computable quantifier of entanglement in two-mode
Gaussian states called Gaussian intrinsic entanglement (GIE), was introduced
more recently by Mi\u{s}ta and Tatham \cite{Tatham}. It has been \textit
surprisingly} found that GIE coincides exactly with the GR2E \cite{Tatham}
for all pure two-mode Gaussian states as well as for relevant subclasses of
Gaussian states, including symmetric states, squeezed thermal states, and
so-called GLEMS Gaussian states of partial minimum uncertainty \cite{AL
,leading to conjecture that GIE and GR2E are equal on all Gaussian states
\cite{Tatham}. \newline
\begin{figure}[t]
\centerline{\includegraphics[width=0.5\columnwidth,height=4.5cm]{fig2.eps
\includegraphics[width=0.5\columnwidth,height=4.5cm]{fig3.eps}}
\centerline{\includegraphics[width=0.5\columnwidth,height=4.5cm]{fig4.eps
\includegraphics[width=0.5\columnwidth,height=4.5cm]{fig5.eps}}
\caption{GR2E $\mathcal{E}_{2}^{i}$ versus the thermal bath temperature $T$
for various squeezing $r$. [(a)-(b)] GR2E $\mathcal{E}_{2}^{m}$ of the two
mechanical modes, [(c)-(d)] GR2E $\mathcal{E}_{2}^{op}$ of the two optical
modes. The optomechanical cooperativity $\mathcal{C}$ and the damping ratio
\Gamma =\frac{\protect\gamma }{\protect\kappa }$ are respectively $\mathcal{
}=34$ and $\Gamma =\frac{\protect\gamma }{\protect\kappa }\approx 0.01$. }
\label{fig.2}
\end{figure}
In our simulations, we have used parameters taken from experiment \cit
{Groblacher}. As mentioned above, we take identical mechanical modes with
frequency $\omega _{\mu }=2\pi \times 947~\mathrm{KHz}$, mass $m_{\mu} =145
\mathrm{ng}$, and damping rate $\gamma =2\pi \times 140~\mathrm{Hz}$, and
identical cavities having length $l=25~\mathrm{\ mm}$, frequency $\omega
_{c}=2\pi \times 5.26\times 10^{14}~\mathrm{Hz}$, decay rate $\kappa =2\pi
\times 172$ $\mathrm{KHz}$ and pumped by laser fields of frequency $\omega
_{L}=2\pi \times 2.82\times 10^{14}$ $\mathrm{Hz}$ and power $\wp =1.5
\mathrm{mW}$ (for the parameters $\kappa $ and $\wp $, see the supplementary
information in \cite{Groblacher}). We note that, the situation $\omega _{\mu
}\gg \kappa $ ($\omega _{\mu }=2\pi \times 947~\mathrm{KHz}$, $\kappa =2\pi
\times 172~\mathrm{KHz}$) corresponding to the resolved sideband regime \cit
{SGHofer}, justifies the use of the RWA in section \ref{sec2}. Finally, the
values of the squeezing parameter $r$ are chosen of the same order of
magnitude as in \cite{JieLi}. \newline
Fig. \ref{fig.2} shows that for a fixed value of the thermal bath
temperature $T$, the GR2E $\mathcal{E}_{2}^{m}$($\mathcal{E}_{2}^{op}$) of
the mechanical(optical) modes increases as the squeezing parameter $r$
increases, which means that the squeezing enhances the quantum correlations
in the two considered subsystems. On the other hand, in the case of no
injection of the squeezed light ($r=0$), we remark that $\mathcal{E
_{2}^{op}=\mathcal{E}_{2}^{m}=0$, which indicates that the two
non-interacting mechanical(optical) modes cannot be entangled without
squeezed light. However, for $r\neq 0$, one can see that both mechanical and
optical entanglement are detectable, meaning that there are \textit
inseparable quantum correlations} between the two mechanical(optical) modes,
even though they are spatially disjoint. These results can be explained by
two types of quantum fluctuations transfer. The first one from the two-mode
squeezed light to the optical subsystem, which corresponds to \textit
light-light quantum fluctuations transfer}, and the second one from the
two-mode squeezed light to the mechanical subsystem, which corresponds to
\textit{light-matter quantum fluctuations transfer} \cite{discord}. Fig. \re
{fig.2} shows also, that the entanglement $\mathcal{E}_{2}^{m}$ and
\mathcal{E}_{2}^{op}$ decay rapidly to zero when the thermal bath
temperature $T$ increases, remarking that the mechanical entanglement
\mathcal{E}_{2}^{m}$ is more affected by thermal effects than the optical
entanglement $\mathcal{E}_{2}^{op}$. In fact, for a given squeezing $r\neq 0
, Fig. \ref{fig.2} shows that always $T_{c}^{op}>T_{c}^{m},$ where
T_{c}^{op}$($T_{c}^{m}$) is the optical(mechanical) critical temperature
defined by $T\geqslant T_{c}^{i}\Rightarrow \mathcal{E}_{2}^{i}=0$ ($i\in
\{m,op\}$), from which the entangled optical(mechanical) modes become
separable. This can be well understood relying on the recent work \cit
{YaoYao}. Indeed, the two optical modes are subject to the zero-point
fluctuation from their vacuum environments, while the mechanical modes are
subject to the thermal noise from their baths. Consequently, the two optical
modes, which are in a more coherent environment than that of the mechanical
modes, should have a robust entanglement with larger degree, knowing that
coherences are a genuine resource of quantum entanglement \cite{C and E}
\newline
\begin{figure}[t]
\centerline{\includegraphics[width=0.5\columnwidth,height=4.5cm]{fig6.eps
\includegraphics[width=0.5\columnwidth,height=4.5cm]{fig7.eps}}
\caption{GR2E $\mathcal{E}_{2}^{i}$ versus the dimensionless optomechanical
cooperativity $\mathcal{C}$ for various values of the mean thermal photons
number $n_{th}$. (a) Mechanical GR2E $\mathcal{E}_{2}^{m}$, (b) optical GR2E
$\mathcal{E}_{2}^{op}$. The squeezing degree $r$ and the damping ratio
\Gamma =\frac{\protect\gamma }{\protect\kappa }$ are respectively $r=2$ and
\Gamma =\frac{\protect\gamma }{\protect\kappa }\approx 0.01$.}
\label{fig.3}
\end{figure}
Next, fixing the squeezing parameter as $r=2$, Fig. \ref{fig.3} shows the
behavior of the GR2E $\mathcal{E}_{2}^{m}$ and $\mathcal{E}_{2}^{op}$ versus
the optomechanical cooperativity $\mathcal{C}$ for different values of the
mean thermal photons number $n_{\mathrm{th}}$. Unlike the optical
entanglement $\mathcal{E}_{2}^{op}$ illustrated in Fig. \ref{fig.3}(b), Fig.
\ref{fig.3}(a) shows that non-zero optomechanical coupling ($\mathcal{C}\neq
0$) is required to entangle the two mechanical modes. This can be understood
using Eq. (\ref{a}), where $\det \mathcal{V}_{3}^{m}=0$ when $\mathcal{C}=0
, knowing that $\det \mathcal{V}_{3}^{i}<0$ ($i\in \{m,op\}$) is a necessary
condition for a two-mode Gaussian state to be entangled\ \cite{Keller2}. In
addition, Fig. \ref{fig.3} shows that the entanglement $\mathcal{E}_{2}^{m}$
and $\mathcal{E}_{2}^{op}$ behave oppositely under the same circumstances.
Indeed, for a given value of the mean thermal photons number $n_{\mathrm{th}}
$, it is clear that when the optomechanical cooperativity $\mathcal{C}$
increases, the mechanical entanglement $\mathcal{E}_{2}^{m}$ increases,
while the optical entanglement $\mathcal{E}_{2}^{op}$ decreases. This can be
explained by the quantum fluctuations transfer from the optical subsystem to
the mechanical subsystem (\textit{light-matter quantum fluctuations transfer
). Finally, one can see also from Fig. \ref{fig.3} that the optical
subsystem requires high values of $n_{\mathrm{th}}$ to switch from entangled
to separable states (see Fig. \ref{fig.3}(b)), which reflects the robust
behavior of the optical modes versus thermal noises. This is consistent with
the results obtained in Fig. \ref{fig.2}.

\subsection{Gaussian R\'{e}nyi-2 discord}

We now turn our attention to the quantification of quantum correlations
beyond entanglement which are commonly known as \textit{the quantumness of
correlations} \cite{ZA}. As mentioned previously, we will use the Gaussian
\'{e}nyi-2 discord (GR2D) $\mathcal{D}_{2}^{i}(\varrho _{A^{i}/B^{i}})$
defined in \cite{AGS}, as the difference between mutual information
\mathcal{I}_{2}(\varrho _{A^{i}/B^{i}})$ and classical correlations
\mathcal{J}_{2}(\varrho _{A^{i}/B^{i}})$
\begin{equation}
\mathcal{D}_{2}^{i}(\varrho _{A^{i}/B^{i}})\doteq \mathcal{I}_{2}(\varrho
_{A^{i}:B^{i}})-\mathcal{J}_{2}(\varrho _{A^{i}/B^{i}})\text{ for }i\in
\{m,op\}\text{.}  \label{GRQD}
\end{equation
The GR-2 mutual information $\mathcal{I}_{2}(\varrho _{A^{i}:B^{i}})$ for an
arbitrary bipartite Gaussian state $\varrho _{A^{i}B^{i}}$ is given by
\mathcal{I}_{2}(\varrho _{A^{i}:B^{i}})=\mathcal{S}_{2}\mathcal{(}\varrho
_{A^{i}}\mathcal{)+S}_{2}\mathcal{(}\varrho _{B^{i}}\mathcal{)-S}_{2
\mathcal{(}\varrho _{A^{i}B^{i}}\mathcal{)}$, where $\varrho _{A^{i}}$ and
\varrho _{B^{i}}$ are the two marginals of $\varrho _{A^{i}B^{i}}$, while a
GR-2 measure of one-way classical correlations $\mathcal{J}_{2}(\varrho
_{A^{i}/B^{i}})$ can be obtained as the maximum decrease in the R\'{e}nyi-2
entropy of subsystem $A^{i}$, given that a Gaussian measurement has been
performed on the subsystem $B^{i}$, where the maximization is over all
Gaussian measurements, which map Gaussian states into Gaussian states \cit
{G(1)}.\newline
Interestingly, we note that Gaussian states have been shown to play a
privileged role in quantum optics and bosonic field theories, essentially
since the very early steps of such theories, partly due to the associated
reduced mathematical complexity, and partly due to their immense importance
for quantum information processing and their easy experimental production
and detection \cite{Lami,Madsen}.\newline
\begin{figure}[t]
\centerline{\includegraphics[width=0.5\columnwidth,height=4.5cm]{fig8.eps
\includegraphics[width=0.5\columnwidth,height=4.5cm]{fig9.eps}}
\centerline{\includegraphics[width=0.5\columnwidth,height=4.5cm]{fig10.eps
\includegraphics[width=0.5\columnwidth,height=4.5cm]{fig11.eps}}
\caption{GR2D $\mathcal{D}_{2}^{i}$ versus the thermal bath temperature $T$
for various values of the squeezing parameter $r$. [(a)-(b)] Mechanical GR2D
$\mathcal{D}_{2}^{m}$, [(c)-(d)] optical GR2D $\mathcal{D}_{2}^{op}$. The
optomechanical cooperativity $\mathcal{C}$ and the damping ratio $\Gamma
\frac{\protect\gamma }{\protect\kappa }$ are the same as in Fig. \protect\re
{fig.2}.}
\label{fig.4}
\end{figure}
For a two-mode Gaussian state $A^{i}$ and $B^{i}$ with CM (\ref{CM}), the
\'{e}nyi-2 measure of one-way quantum discord (\ref{GRQD}) admits the
following expression \cite{AGS}
\begin{equation}
\mathcal{D}_{2}^{i}(\varrho _{A^{i}/B^{i}})=\ln (2\nu _{2}^{i})-\frac{1}{2
\ln \left( 16\det \mathcal{V}^{i}\right) +\frac{1}{2}\ln \left( \epsilon
^{i}\right) \text{ for }i\in \{m,op\}\text{,}  \label{GR2QD}
\end{equation
with
\begin{equation}
\epsilon ^{i}=\left\{
\begin{array}{c}
4\nu _{1}^{i}\left( \nu _{1}^{i}\text{ }-\frac{\left( \nu _{3}^{i}\right)
^{2}}{\nu _{2}^{i}}\right) \text{\ \ if\ \ }\left[ 4\nu _{1}^{i}\left( \nu
_{2}^{i}\right) ^{2}\left( \nu _{3}^{^{\prime }i}\right) ^{2}-\left( \nu
_{3}^{i}\right) ^{2}\left( \nu _{1}^{i}+4\nu _{2}^{i}\left( \nu
_{3}^{^{\prime }i}\right) ^{2}\right) \right] \text{ }\times \\
\left[ 4\nu _{1}^{i}\ (\nu _{2}^{i})^{2}\left( \nu _{3}^{i}\right) ^{2}-\ \
\left( \nu _{3}^{^{\prime }i}\right) ^{2}\left( \nu _{1}^{i}+4\nu
_{2}^{i}\left( \nu _{3}^{i}\right) ^{2}\right) \right] <0, \\
\left[ \frac{4\left\vert \nu _{3}^{i}\nu _{3}^{^{\prime }i}\right\vert +
\sqrt{\left[ \nu _{1}^{i}\left( 4(\nu _{2}^{i})^{2}-1\right) -4\nu
_{2}^{i}(\nu _{3}^{^{\prime }i})^{2}\right] \left[ \nu _{1}^{i}\left[ 4(\nu
_{2}^{i})^{2}-1\right] -4\nu _{2}^{i}(\nu _{3}^{i})^{2}\right] }}{4(\nu
_{2}^{i})^{2}-1}\right] ^{2}\text{ \ otherwise}
\end{array
\right.  \label{GR2D-}
\end{equation
Fig. \ref{fig.4} shows the robustness of the mechanical(optical) GR2D
\mathcal{D}_{2}^{m}$ ($\mathcal{D}_{2}^{op}$) against the temperature $T$
for various squeezing $r$. The damping ratio $\Gamma $ and the
optomechanical cooperativity $\mathcal{C}$ are the same as in Fig. \re
{fig.2}. As expected for quantum discord in a dissipative system, we see
that the GR2D $\mathcal{D}_{2}^{m}$ and $\mathcal{D}_{2}^{op}$ decrease
monotonically when the temperature increases, reaching an asymptotic regime.
In addition, comparing between the results reported in Figs. \ref{fig.2}-\re
{fig.4}, we remark that the two quantifiers of nonclassicality $\mathcal{E
_{2}^{i}$ and $\mathcal{D}_{2}^{i}$ show rather different behaviors against
the temperature. Indeed, after the total disappearance of the entanglement
\mathcal{E}_{2}^{m}$ and $\mathcal{E}_{2}^{op}$, the discord $\mathcal{D
_{2}^{m}$ and $\mathcal{D}_{2}^{op}$ still persist and are significantly
non-zero over a wide range of the temperature $T$. We thus conclude that
entanglement is not the only resource comprising quantum discord,i.e.,
two-mode mixed separable states may also contain quantum discord ensuring
the \textit{quantumness of correlations} in such states. More importantly,
Fig. \ref{fig.4} reveals a very special behavior of the Gaussian discord
under thermal noises. In fact, in both mechanical and optical subsystems,
the quantum discord associated with the entangled states has a tendency to
diminish drastically under thermal noises. However the discord associated
with the separable states exhibits a robust behavior,i.e., the asymptotic
value of Gaussian discord may be small but never vanishes remaining almost
constant (\textit{frozen}) even for temperature $T$ up to $0.1~\mathrm{K}
(this has been checked for the squeezing values $r=1$ and $r=1.5$). This
behavior reminds us of the freezing phenomenon of quantum correlations
beyond entanglement, where an insightful physical interpretation of such
phenomenon has been discussed in \cite{Kavan Modi,FQC}. Remarkably, the GR2D
$\mathcal{D}_{2}^{m}$ and $\mathcal{D}_{2}^{op}$ are zero only for $r=0.$
This is a consequence of the fact that quantum discord vanishes only if the
Gaussian state is a product state and therefore if and only if the
determinant of the $\mathcal{V}_{3}^{i}$ matrices (see (\ref{CM})) is zero
\cite{Keller2}. Such a condition is well verified for $r=0$, where $c_{\pm
}^{i}=0$ and consequently $\det \mathcal{V}_{3}^{i}=0$ (see Eqs. [(\ref{CM
)-(\ref{c})]).\newline
Finally, Fig. \ref{fig.5} shows the GR2D $\mathcal{D}_{2}^{\mathrm{m}}$ and
\mathcal{D}_{2}^{op}$ dependence on the optomechanical cooperativity
\mathcal{C}$. The mean thermal photons number $n_{\mathrm{th}}$, the damping
ratio $\Gamma $ and the squeezing parameter $r$ are the same as in Fig. \re
{fig.3}.
\begin{figure}[t]
\centerline{\includegraphics[width=0.5\columnwidth,height=4.5cm]{fig12.eps
\includegraphics[width=0.5\columnwidth,height=4.5cm]{fig13.eps}}
\caption{The GR-2 discord $\mathcal{D}_{2}^{i}$ versus the dimensionless
optomechanical cooperativity $\mathcal{C}$ for various values of the thermal
photons number $n_{th}$. (a) Mechanical GR2D $\mathcal{D}_{2}^{m}$, (b)
optical GR2D $\mathcal{D}_{2}^{op}$. The squeezing degree $r$ and the
damping ratio $\Gamma =\frac{\protect\gamma }{\protect\kappa }$ are the same
as in Fig. \protect\ref{fig.3}.}
\label{fig.5}
\end{figure}
Comparing Figs. \ref{fig.3} and \ref{fig.5}, it is clear that when the GR2E
\mathcal{E}_{2}^{m}$ and $\mathcal{E}_{2}^{op}$ vanish, the GR2D $\mathcal{D
_{2}^{m}$ and $\mathcal{D}_{2}^{op}$ remain non-zero. This result witnesses
the existence of non-classical correlations even in separable states
\textit{\ the quantumness of correlations}) of both mechanical and optical
subsystems. \newline
From Fig. \ref{fig.5}, one sees also that like entanglement, quantumness of
correlations can be transferred from light to light or from light to matter.
This transfer is clearly observed from Figs. \ref{fig.4}(a)-\ref{fig.4}(c),
where $\mathcal{D}_{2}^{op}=\mathcal{D}_{2}^{m}=0$ when $r=0$. In contrast
\mathcal{D}_{2}^{op}>0$ and $\mathcal{D}_{2}^{m}>0$ for $r\neq 0$, even when
$\mathcal{E}_{2}^{op}$ and $\mathcal{E}_{2}^{m}$ are zero.

\section{Conclusions \label{sec4}}

In two optomechanical Fabry-P\'{e}rot cavities fed by two-mode squeezed
light, the quantumness of correlations of two non-interacting mechanical
modes as well as two non-interacting optical modes have been investigated.
For this, we used respectively the Gaussian R\'{e}nyi-2 entanglement (GR2E)
and the Gaussian R\'{e}nyi-2 discord (GR2D) to quantify bipartite
entanglement and non-classical correlations far beyond entanglement
(quantumness of correlations). The GR2E and GR2D have been analysed under
the same circumstances in terms of the parameters characterizing both the
environment (temperature and squeezing) and the cavities (coupling and
damping). Solving the linearized quantum Langevin equations in the resolved
sideband regime around the classical steady state, we provided analytical
formulas for both mechanical and optical covariances matrices, fully
describing the Gaussian stationary states of the two considered subsystems.
We have seen that in an experimentally accessible parameter regime, that it
is possible to create both quantum entanglement and quantum discord in the
two studied subsystems by quantum fluctuations transfer from either light to
light or light to matter.

The results presented in Sec. \ref{sec3} show clearly that both mechanical
and optical entanglement ($\mathcal{E}_{2}^{m}$ and $\mathcal{E}_{2}^{op}$)
are strongly sensitive to the thermal noises. In particular, the optical
entanglement $\mathcal{E}_{2}^{op}$ is found more robust against thermal
noises than the mechanical entanglement $\mathcal{E}_{2}^{m}$. This makes
optical modes extremely useful in quantum information networks, where they
can be used to transfer states (i.e., information) through dissipative and
noisy channels. In contrast, both mechanical and optical discord $\mathcal{D
_{2}^{m}$ and $\mathcal{D}_{2}^{op}$ show a robust behavior under influences
of the thermal baths. Indeed, as shown in Figs. \ref{fig.4}(a)-\ref{fig.4
(c), the asymptotic values of the mechanical and optical discord $\mathcal{D
_{2}^{m}$ and $\mathcal{D}_{2}^{op}$ may be small but never vanish.
Interesting enough, it has been observed in both mechanical and optical
subsystems that under thermal noises, the discord associated with the
entangled states was found to decay drastically, while the Gaussian discord
measured in the separable states (quantumness of correlations) was
exhibiting a robust behavior. In fact, when the GR2E $\mathcal{E}_{2}^{m}$
and $\mathcal{E}_{2}^{op}$ are non-zero, which corresponds to the entangled
states, the GR2D $\mathcal{D}_{2}^{m}$ and $\mathcal{D}_{2}^{op}$ decrease
aggressively (see Figs. \ref{fig.2} and \ref{fig.4}), whereas when $\mathcal
E}_{2}^{m}=\mathcal{E}_{2}^{op}=0$, which corresponds to the separable
states, the GR2D $\mathcal{D}_{2}^{m}$ and $\mathcal{D}_{2}^{op}$ seem to
remain almost constant (frozen) over a wide range of the environmental
temperature (see also Figs. \ref{fig.2} and \ref{fig.4}), meaning that the
quantumness of the correlations can be captured in both mechanical and
optical subsystems even for high temperatures (up to $0.1K$). This has been
checked using squeezing $r=1$ and $r=1.5$.

We notice that, the quantum correlations generated here (entanglement and
quantum discord), could be experimentally verified by homodyning the two
cavities outputs \cite{JieLi}, while the position and momentum of the two
mechanical modes can be measured with the strategy proposed in \cit
{Vitali(1)}. Finally, the study of stationary evolution of Gaussian R\'{e
nyi-2 entanglement and the Gaussian R\'{e}nyi-2 discord may contribute to
the understanding of the quantification of correlations in noisy
environments, which is very interesting for applications in quantum
information processing and communication.

redpajama_set_name: RedPajamaArXiv