# Temperature gradient and asymmetric steady state correlations in
dissipatively coupled cascaded optomechanical systems

Claudio Pellitteri 1, G. Massimo Palma 1,2 and Salvatore Lorenzo1 1 Università
degli Studi di Palermo, Dipartimento di Fisica e Chimica - Emilio Segrè, via
Archirafi 36, I-90123 Palermo, Italy
2 NEST, Istituto Nanoscienze-CNR, Piazza S. Silvestro 12, 56127 Pisa, Italy

###### Abstract

The interaction between a light mode and a mechanical oscillator via radiation
pressure in optomechanical systems is an excellent platform for a multitude of
applications in quantum technologies. In this work we study the dynamics of a
pair of optomechanical systems interacting dissipatively with a wave guide in
a unidirectional way. Focusing on the regime where the cavity modes can be
adiabatically eliminated we derive an effective coupling between the two
mechanical modes and we explore both classical and quantum correlations
established between the modes in both in the transient and in the stationary
regime, highlighting their asymmmetrical nature due to the unidirectional
coupling, and we find that a constant amount of steady correlations can exist
at long times. Furthermore we show that this unidirectional coupling
establishes a temperature gradient between the mirrors, depending on the
frequencies’ detuning. We additionally analyze the power spectrum of the
output guide field and we show how, thanks to the chiral coupling, from such
spectrum it is possible to reconstruct the spectra of each single mirror.

##  I Introduction

Optomechanical systems, with light modes interacting with massive mechanical
oscillators, have attracted a considerable interest for their possible
application in quantum technologies [1, 2]. Depending on the working point,
the optomechanical interaction can be used to cool the mechanical mode near to
its ground state [3, 4, 5, 6, 7, 8] (a technique applied also to levitating
nanospheres [9]), to generate squeezing [10, 11, 12] or to create entanglement
between optical and mechanical modes [13, 14, 15]. These configurations can be
mixed in an appropriate way in order to generate purely quantum states of the
mechanical oscillators (e.g. generation of single phonon states [16]).

A natural extension of the standard single mode - single mirror oscillators
setups consists of several coupled modes. We can distinguish two major and
distinct setups. The first one is called Multimode optomechanical system [17,
18, 19, 20, 21], and consists of several mechanical oscillator interacting
with the same cavity. In Optomechanical array instead, each mechanical
oscillator interacts locally with its own cavity-mode but an effective
coupling between neighbouring mirrors is implemented by photons and/or phonons
tunneling [22, 23, 24].

In this work we propose the largely unexplored setup in which the cavity modes
are dissipatively coupled via a unidirectional waveguide [25] in a cascaded
configuration [26, 27, 28, 29, 30]. This arrangement induces a non-reciprocal
interaction, at first between the cavities and then between the mechanical
oscillators [31]. While previous studies have explored similar setups, none of
them has specifically addressed pure unidirectional coupling [32]. For
instance, in [33, 34], the authors study the synchronization between two
resonators driven by a blue detuned laser, which leads to a self-sustained
oscillatory dynamics and in [35] the possibility of creating non-reciprocal
devices that control the flow of thermal noise towards or away from specific
quantum devices in a network has been explored. However, in spite of the
above, the full potential of cascaded coupling between a pair of
optomechanical systems remains largely unexplored, especially when it comes to
the effective coupling between the two mechanical modes. To fill this gap, we
derive the equations describing the effective coupling between the two mirrors
by adiabatically eliminating the cavity modes. We then characterise the
correlations established between the mirrors in terms of mutual information
and quantum discord, the latter being the best quantifier when one is
interested in asymmetries between the two mirrors. We show that asymmetric
non-zero correlations persists even in the steady state. Additionally, we
explore the consequences of the asymmetry in the coupling, arising from the
unidirectionality, and its implications for establishing a temperature
gradient between the two modes. Notably, this temperature gradient vanishes in
the case of perfectly symmetrical bidirectional coupling.

This work is organized as follow: In section II we present our model, we
introduce its Hamiltonian. By employing Langevin equations we characterise the
evolution of the system in terms of mean values and fluctuations. The
fluctuations are further analysed using Lyapunov equation for the covariance
matrix. In section III, we derive an equation of motion for the effective
dynamics of the two mechanical oscillators. This is achieved by performing an
adiabatic elimination of the cavity modes. In section IV, we investigate the
correlations between the two mechanical modes through the evaluation of mutual
information and quantum discord. Interestingly, we find that these asymmetric
correlations (Discord) retain non-zero values even in the stationary state,
indicating the potential for establishing persistent correlations. In section
V, we show how, in the cooling regime, the unidirectional coupling leads to
the thermalization of the two mechanical modes at different temperatures,
depending on the frequency mismatch of the mirrors. This reveals the so far
unexplored setup to engineer a temperature gradient using the cascaded
configuration. Section VI focuses on the analysis of the power spectra of the
two mirrors and the output field mode, establishing the relationship between
them. In section VII we study the stability regions of the parameters space,
exploring when the system can exhibit multistability. Finally in section VIII
we draw our conclusions.

##  II The system: two cascaded optomechanical cavities

Our system consists of two optomechanical mirrors each coupled to the same
unidirectional waveguide. Such mediated indirect interaction (see fig. 1)
leads to a cascade scenario in which the first system drives the following one
without back action. Each subsystem consists of a mechanical harmonic
oscillator with mass $m$ and frequency $\Omega_{j}$ coupled to a cavity field
by means of its radiation pressure. As $\Omega_{j}$ is much smaller than
$c/2L$ ($c$ is the speed of light and $L$ stands for the cavity length) we can
consider a single cavity mode [36, 37] and write the following Hamiltonian for
both subsystems:

$\hat{H}_{S}=\sum_{j=1}^{2}\hat{H}_{j}=\sum_{j=1}^{2}\hbar\omega_{c}\hat{a}_{j}^{\dagger}\hat{a}_{j}{+}\dfrac{\hbar\Omega_{j}}{2}(\hat{q}_{j}^{2}{+}\hat{p}_{j}^{2})-\hbar
g_{j}\hat{a}_{j}^{\dagger}\hat{a}_{j}\hat{q}_{j}+i\hbar
E_{j}(\hat{a}_{j}^{\dagger}e^{-i\omega_{L}t}{-}\hat{a}_{j}e^{i\omega_{L}t})$
(1)

where $\hat{a}_{j}$ is the cavity mode annihilation operator with optical
frequency $\omega_{c}$ of the $j$-th subsystem and $\hat{q}_{j}$
($\hat{p}_{j}$) stands for dimensionless position (momentum) operators of
mechanical mode. The term proportional to
$g_{j}{=}\omega_{c}/L\sqrt{\hbar/(m\Omega_{j})}$ describes the optomechanical
coupling, while the last term is the coherent input field with frequency
$\omega_{L}$. The quantities $E_{j}$ are related to the input powers $P_{j}$
by $E_{j}=\sqrt{2\kappa P_{j}/(\hbar\omega_{L})})$ where $\kappa$ is the
cavity decay rate. In a rotating frame at laser frequency $\omega_{L}$, we
define $\Delta=\omega_{c}-\omega_{L}$ and eq. 1 becomes

$\hat{H}_{S}{=}\sum_{j}\hbar\left(\Delta-
g_{j}\hat{q}_{j}\right)\hat{a}_{j}^{\dagger}\hat{a}_{j}{+}\frac{\hbar\Omega_{j}}{2}(\hat{q}_{j}^{2}{+}\hat{p}_{j}^{2})+i\hbar
E_{j}(\hat{a}_{j}^{\dagger}-\hat{a}_{j})$ (2)

In the following we focus on the scenario in which the dynamics of the second
optomechanical system is driven only by the output field of the first cavity.
Therefore, henceforth, we assume that the external laser pumps the first
cavity only (i.e. $E_{2}=0$).

Due to the inherently open nature of our system, we consider that each
mechanical mode is coupled to its respective environment, which is assumed to
be at a finite temperature [38]. Additionally, we account for photon leakage
from the cavities. Specifically, we assume that the optical dissipation occurs
through a unidirectional waveguide, resulting in a cascade-like coupling
between the two optical modes mediated by the interaction with the guide [31].

Following the standard input-output prescription of optical quantum Langevin
equations [39], we introduce the radiation vacuum input noise operator
$\hat{a}^{\text{in}}$ [40, 41] and the Brownian noise operator $\hat{\xi}_{j}$
[38] with autocorrelation functions:

$\displaystyle\langle\hat{a}^{\text{in}}(t)\hat{a}^{\text{in}\dagger}(t^{\prime})\rangle=\delta(t-t^{\prime})$
(3a)
$\displaystyle\langle\\{\hat{\xi}_{j}(t),\xi_{j}(t^{\prime})\\}\rangle=\gamma_{j}(2\bar{n}_{j}+1)\delta(t-t^{\prime})$
(3b)

where $\gamma_{j}$ denotes the decay rate of the j-th mirror and
$\bar{n}_{j}=1/(\exp(\hbar\Omega_{j}/k_{B}T)+1)$ in which $k_{B}$ is the
Boltzman constant and $T$ is the temperature of the mechanical modes’ bath.
Although the cavity and the resonator are at the same temperature, however,
the cavity frequency is typically orders of magnitude larger than the
mechanical frequency, therefore the average number of photons in the optical
environment is negligible. Based on the preceding analysis, we can derive the
following quantum Langevin equations for the field operators $\hat{a}_{j}$:

$\displaystyle\frac{d\hat{a}_{1}}{dt}=-\frac{i}{\hbar}[\hat{a}_{1},H_{S}]-\frac{\kappa}{2}\hat{a}_{1}-\sqrt{\kappa}\hat{a}^{\text{in}}$
(4a)
$\displaystyle\frac{d\hat{a}_{2}}{dt}=-\frac{i}{\hbar}[\hat{a}_{2},H_{S}]-\frac{\kappa}{2}\hat{a}_{2}-{\kappa}\hat{a}_{1}-\sqrt{\kappa}\hat{a}^{\text{in}}$
(4b)

and the mirror operators $\hat{q}_{j}$, $\hat{p}_{j}$

$\displaystyle\frac{d\hat{q}_{j}}{dt}=-\frac{i}{\hbar}[\hat{q}_{j},H_{S}]$
(5a)
$\displaystyle\frac{d\hat{p}_{j}}{dt}=-\frac{i}{\hbar}[\hat{p}_{j},H_{S}]-\gamma_{j}\hat{p}_{j}-\sqrt{\gamma_{j}}\,\hat{\xi}_{j}$
(5b)

Figure 1:  $(\mathbf{a})$ Sketch of the model involving two optomechanical
systems: Each system consists of an optical mode and a mechanical mode,
mutually interacting through the radiation pressure force induced by external
laser power. We stress that these optomechanical systems are coupled to a
unidirectional waveguide. In this context, the operators ${\hat{q},\hat{p}}$
represent the mechanical position and momentum, while
${\hat{a},\hat{a}^{\dagger}}$ are the creation and annihilation operators for
the optical modes. The parameters $\gamma$ and $\kappa$ denote the mechanical
and optical decay rates, respectively. Furthermore, $\Omega_{1,2}$ refer to
the natural frequencies of the mechanical modes, and $g$ represents the
coupling strength between the optical and mechanical modes due to radiation
pressure. The amplitude of the coherent driving of the laser is denoted as
$E$. ($\mathbf{b}$) Effective model describing the coupling between the
mirrors, obtained by performing an adiabatic elimination of the optical modes.
Within this framework, the effective frequencies of the mechanical modes are
denoted as $\Omega^{\text{eff}}_{1,2}$, and $\Gamma^{\text{eff}}_{1,2}$
represents the effective decay rates. Moreover, $\Lambda$ (as defined in eq.
18) represents the cascaded effective coupling between the two mechanical
modes.

_Mean field equations and fluctuations Dynamics_ \- The combined dynamics of
the field-mirror system resulting from equations eq. 4 and eq. 5 is nonlinear.
To investigate the quantum characteristics of optomechanical systems, a common
method is to initially seek the mean field solution of the field and
mechanical operators, and subsequently analyze the linearized dynamics of
quantum fluctuations around these average values. Accordingly, we represent
the operators as the sum of their average value (a $c$ number) and a small
quantum fluctuation.

$\hat{o}=\langle\hat{o}\rangle+[\hat{o}-\langle\hat{o}\rangle]=O+\delta\hat{o}$
(6)

This leads to the following set of non linear differential equations for the
mean values

$\displaystyle\frac{dQ_{j}(t)}{dt}=\Omega_{j}P_{j}(t)$ (7a)
$\displaystyle\frac{dP_{j}(t)}{dt}=-\Omega_{j}Q_{j}(t)-\gamma_{j}P_{j}(t)+|G_{j}(t)|^{2}/g_{j}$
(7b)
$\displaystyle\frac{dA_{1}(t)}{dt}=-\frac{\kappa}{2}A_{1}(t)-i\Delta_{1}(t)A_{1}(t)+E$
(7c)
$\displaystyle\frac{dA_{2}(t)}{dt}=-\frac{\kappa}{2}A_{2}(t)-i\Delta_{2}(t)A_{2}(t)-\kappa
A_{1}(t)$ (7d)

where we defined $\Delta_{j}(t){=}\Delta{-}g_{j}Q_{j}(t)$ and
$G_{j}(t){=}g_{j}A_{j}(t)$. It is important to note the inherent asymmetry in
comparison to the bidirectional case (cfr. B). Solving eq. 7 and using eq. 6
into eqs. 4 and 5, we can write the following linearized set of equations for
the fluctuations, where we keep only terms $\mathcal{O}(\delta\hat{o})$:

$\displaystyle\frac{d\delta\hat{q}_{j}}{dt}=\Omega_{j}\delta\hat{p}_{j}$ (8a)
$\displaystyle\frac{d\delta\hat{p}_{j}}{dt}=-\Omega_{j}\delta\hat{q}_{j}+(G_{j}^{*}\delta\hat{a}_{j}+G_{j}\delta\hat{a}_{j}^{\dagger})-\gamma\delta\hat{p}_{j}-\xi_{j}$
(8b)
$\displaystyle\frac{d\delta\hat{a}_{1}}{dt}=iG_{1}\delta\hat{q}_{1}-i\Delta_{1}\delta\hat{a}_{1}-\frac{\kappa}{2}\delta\hat{a}_{1}-\sqrt{\kappa}\hat{a}^{\text{in}}$
(8c)
$\displaystyle\frac{d\delta\hat{a}_{2}}{dt}=iG_{1}\delta\hat{q}_{2}-i\Delta_{2}\delta\hat{a}_{2}-\frac{\kappa}{2}\delta\hat{a}_{2}-\kappa\delta\hat{a}_{1}-\sqrt{k}\hat{a}^{\text{in}}$
(8d)

_Covariance matrix and Lyapunov equation_ \- Given that the set of quantum
Langevin eq. 8 is linear and the quantum noise is Gaussian, we can fully
characterize the quantum fluctuations dynamics in terms of the covariance
matrix $\mathbf{C}$, whose elements are defined by
$C_{ij}=1/2\langle\hat{u}_{i}\hat{u}_{j}{+}\hat{u}_{j}\hat{u}_{i}\rangle$ with
$\vec{u}{=}\otimes_{j=1}^{2}\\{\hat{q}_{j},\hat{p}_{j},\hat{x}_{j},\hat{y}_{j}\\}$
and the cavity field quadratures as
$\hat{x}{=}1/\sqrt{2}(\hat{a}{+}\hat{a}^{\dagger})$,
$\hat{y}{=}-i/\sqrt{2}(\hat{a}{-}\hat{a}^{\dagger})$. It follows that the
$\mathbf{C}$ matrix obeys the following Lyapunov equation [5, 42]:

$\frac{d\mathbf{C}(t)}{dt}=\mathbf{S}(t)\mathbf{C}(t)+\mathbf{C}(t)\mathbf{S}(t){+}\mathbf{N}\;\;\text{with}\;\;\mathbf{S}{=}\begin{pmatrix}\mathbf{S}_{1}&\mathbf{0}\\\
\mathbf{S}_{R}&\mathbf{S}_{2}\end{pmatrix}\text{and}\;\;\mathbf{N}{=}\begin{pmatrix}\mathbf{N}_{1}&\mathbf{N}_{12}\\\
\mathbf{N}_{12}&\mathbf{N}_{2}\end{pmatrix}$ (9)

The block matrices entering in the drift ($\mathbf{S}$) and diffusion
($\mathbf{N}$) parts, reflect the unidirectionality of the model (as can be
seen comparing them to the ones of the bidirectional case cfr. B):

$\mathbf{S}_{j}{=}\begin{pmatrix}0&\Omega_{j}&0&0\\\
-\Omega_{j}&-\gamma&\text{Re}(G_{j})&\text{Im}(G_{j})\\\
-\text{Im}(G_{j})&0&-\kappa/2&\Delta_{j}\\\
\text{Re}(G_{j})&0&-\Delta_{j}&-\kappa/2\end{pmatrix}\qquad\mathbf{S}_{R}{=}\begin{pmatrix}0&0&0&0\\\
0&0&0&0\\\ 0&0&-\kappa&0\\\ 0&0&0&-\kappa\end{pmatrix}$ (10)

and

$\mathbf{N}_{j}{=}\begin{pmatrix}0&0&0&0\\\ 0&\gamma(2\bar{n}_{j}+1)&0&0\\\
0&0&\kappa&0\\\ 0&0&0&\kappa\\\
\end{pmatrix}\;\;\mathbf{N}_{12}{=}\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\
0&0&\kappa&0\\\ 0&0&0&\kappa\end{pmatrix}$ (11)

##  III Effective mirrors dynamics

The optomechanical coupling, combined with the coupling of the cavity modes to
the unidirectional waveguide, gives rise to an effective mediated interaction
between the two mechanical modes. In the weak coupling regime
($G_{j}\lesssim\kappa$), we can explicitly derive an effective interaction by
adiabatically eliminating the cavity field degrees of freedom associated with
the cavities. Indeed, if $G_{j}\lesssim\kappa$ we can focus on evolution of
mechanical operators $\bar{b}_{j}$ and $\bar{b}_{j}^{\dagger}$, defined
through
$\delta\hat{q}_{j}{=}(\bar{b}_{j}e^{-i\Omega_{j}t}+\bar{b}_{j}^{\dagger}e^{i\Omega_{j}t})/\sqrt{2}$
and
$\delta\hat{p}_{j}{=}i(\bar{b}_{j}^{\dagger}e^{i\Omega_{j}t}-\bar{b}_{j}e^{-i\Omega_{j}t})/\sqrt{2}$.
By considering these operators, we effectively eliminate the rapid timescale
associated with the evolution of the cavity field [43]. From eq. 8, dropping
counter-rotating terms, one obtains

$\displaystyle\frac{d\bar{b}_{j}}{dt}=\dfrac{ie^{i\Omega_{j}t}}{\sqrt{2}}(G_{j}^{*}\delta\hat{a}_{j}+G_{j}\delta\hat{a}_{j}^{\dagger})-\dfrac{\gamma}{2}\bar{b}_{j}-\dfrac{ie^{i\Omega_{j}t}}{\sqrt{2}}\xi_{j}$
(12)

The expressions for the cavity fields fluctuations can be found solving the
respective equations in the frequency domain by using
$\hat{O}(t)=1/\sqrt{2\pi}\int_{-\infty}^{+\infty}\hat{O}(\omega)e^{-i\omega
t}$. Therefore we rewrite the last two of the eq. 8 as

$\displaystyle\delta\hat{a}_{1}(\omega)=\chi_{a_{1}}(\omega)\left(\hat{a}^{\text{in}}(\omega)\sqrt{\kappa}+iG_{1}\delta\hat{q}_{1}(\omega)\right)$
(13a)
$\displaystyle\delta\hat{a}_{2}(\omega)=\chi_{a_{2}}(\omega)\left(\hat{a}^{\text{in}}(\omega)\sqrt{\kappa}+iG_{2}\delta\hat{q}_{2}(\omega)-\kappa\delta\hat{a}_{1}(\omega)\right)$
(13b)

where we introduced the natural susceptibility of the optical modes
$\chi_{a_{j}}$

$\displaystyle\chi_{a_{j}}(\omega)=\frac{1}{\kappa/2-i\left(\omega-\Delta_{j}\right)}.$
(14)

The optical input noise can be neglected as it is small compared to the
mechanical thermal noise, given that the system evolves at room-temperature.
So, back in time domain we obtain

$\displaystyle\delta\hat{a}_{1}(t)=\dfrac{iG_{1}}{\sqrt{2\pi}}\int
d\omega\chi_{a_{1}}(\omega)\delta\hat{q}_{1}(\omega)e^{-i\omega t}$ (15a)
$\displaystyle\delta\hat{a}_{2}(t)=\dfrac{iG_{2}}{\sqrt{2\pi}}\int
d\omega\chi_{a_{2}}(\omega)\left(\delta\hat{q}_{2}(\omega)-\kappa\delta\hat{a}_{1}(\omega)\right)e^{-i\omega
t}$ (15b)

Thanks to the properties of convolutions in Fourier transforms, assuming that
$\bar{b}_{j}$ and $\bar{b}^{\dagger}_{j}$ vary slowly in time, eq. 15 become

$\displaystyle\delta\hat{a}_{1}(t)=$
$\displaystyle\dfrac{iG_{1}}{\sqrt{2}}\left(\bar{b}_{1}\chi_{a_{1}}(\Omega_{1})e^{-i\Omega_{1}t}{+}\bar{b}^{\dagger}_{1}\chi_{a_{1}}^{*}(\text{-}{\Omega_{1}})e^{i\Omega_{1}t}\right)$
(16a) $\displaystyle\delta\hat{a}_{2}(t)=$
$\displaystyle\dfrac{iG_{2}}{\sqrt{2}}\left(\bar{b}_{2}\chi_{a_{2}}(\Omega_{2})e^{-i\Omega_{2}t}{+}\bar{b}^{\dagger}_{2}\chi_{a_{2}}^{*}(\text{-}{\Omega_{2}})e^{i\Omega_{2}t}\right)-$
$\displaystyle\dfrac{i\kappa
G_{1}}{\sqrt{2}}\left(\bar{b}_{1}\chi_{a_{1}}(\Omega_{1})\chi_{a_{2}}(\Omega_{1})e^{-i\Omega_{1}t}{+}\bar{b}^{\dagger}_{1}\chi_{a_{1}}^{*}(\text{-}{\Omega_{1}})\chi_{a_{2}}^{*}(\text{-}{\Omega_{1}})e^{i\Omega_{1}t}\right)$
(16b)

By substituting these results into eq. 12 and neglecting non-rotating terms,
we derive the following coupled equations of motion for the mirrors operators
only:

$\displaystyle\frac{d\bar{b}_{1}}{dt}=-i\Omega^{\text{eff}}_{1}\,\bar{b}_{1}-\left(\Gamma^{\text{eff}}_{1}+\dfrac{\gamma}{2}\right)\bar{b}_{1}-\dfrac{ie^{i\omega_{1}t}}{\sqrt{2}}\xi_{1}$
(17a)
$\displaystyle\frac{d\bar{b}_{2}}{dt}=-i\Omega^{\text{eff}}_{2}\,\bar{b}_{2}-\left(\Gamma^{\text{eff}}_{2}+\dfrac{\gamma}{2}\right)\bar{b}_{2}-\frac{\kappa}{2}\Lambda(\Omega_{1})\bar{b}_{1}-\dfrac{ie^{i\omega_{2}t}}{\sqrt{2}}\xi_{2}$
(17b)

in which $\Gamma^{\text{eff}}_{j}$ and $\Omega^{\text{eff}}_{j}$ are the
effective (induced by the adiabatically eliminated optical fields) decay rate
and mechanical frequencies, defined respectively as the real and imaginary
part of
$|G_{j}|^{2}\left(\chi_{a_{j}}(\Omega_{j}){-}\chi_{a_{j}}^{*}(\text{-}{\Omega_{j}})\right)/2$,
and $\Lambda(\Omega_{1})$ is the effective cascaded coupling between the two
mirrors (see fig. 1), defined by

$\Lambda(\omega){=}G_{2}G^{*}_{1}\chi_{a_{1}}^{*}(\text{-}{\omega})\chi_{a_{2}}^{*}(\text{-}{\omega}){-}G^{*}_{2}G_{1}\chi_{a_{1}}(\omega)\chi_{a_{2}}(\omega).$
(18)

These equations can be recast in a covariance matrix equation form (cfr. eq.
9) with $\vec{u}{=}\otimes_{j=1}^{2}\\{\bar{b}_{j},\bar{b}^{\dagger}_{j}\\}$.
In this case, the blocks entering in the drift $\mathbf{S}$ and diffusion
$\mathbf{N}$ matrices turn out to be

$\mathbf{S}_{j}{=}\begin{pmatrix}-i\Omega^{\text{eff}}_{j}-(\Gamma^{\text{eff}}_{j}+\dfrac{\gamma}{2})&0\\\
0&i\Omega^{\text{eff}}_{j}-(\Gamma^{\text{eff}}_{j}+\dfrac{\gamma}{2})\\\
\end{pmatrix}\qquad\mathbf{S}_{R}{=}\begin{pmatrix}-\Lambda(\Omega_{1})&0\\\
0&-\Lambda(\Omega_{1})^{*}\\\ \end{pmatrix}$ (19)

and

$\displaystyle\mathbf{N}_{j}{=}\begin{pmatrix}\gamma(2\bar{n}_{j}+1)&0\\\
0&\gamma(2\bar{n}_{j}+1)\\\
\end{pmatrix}\qquad\mathbf{N}_{12}{=}\begin{pmatrix}0&0\\\ 0&0\\\
\end{pmatrix}$

Notice that the noise matrix $\boldsymbol{N}$ is diagonal due to the fact that
we have dropped counter-rotating terms. So, as said in the beginning, we have
found that, looking on a timescale larger than the one describing the cavities
modes evolution, mirror’s interaction mediated by the cavities can be
described as an effective coupling.

##  IV Mutual Information and Quantum Discord

Once eq. 9 is solved, we can analyse and conveniently characterise the mirrors
correlations - both in the transient and in stationary regimes - by means of
the mutual information, which can be evaluated from the covariance matrix, as
shown in [44], in terms of its symplectic invariants and symplectic
eigenvalues (see A).

Figure 2: (Left) Mutual information between the two mirrors as a function of
time in the cooling regime ($\Delta=\Omega_{1}$). The dotted line refers to
the case $\Omega_{2}=0.75\Omega_{1}$, the dashed line refers to the case
$\Omega_{2}=\Omega_{1}$ and the dot-dashed one refers to the case
$\Omega_{2}=1.25\Omega_{1}$. The box in the right corner zooms on the long
time region where is shown the establishment of steady state correlations
(Right) Quantum discord $D(A|B)$ in the same cases considered for the mutual
information. The box in the right corner shows a zoom of the discord values on
the long time region. For these plots the following parameters’ values have
been considered: $m{=}150$ ng, $\Omega_{1}/(2\pi){=}1$ MHz,
$\gamma/(2\pi){=}1$ Hz ($\gamma_{j}$), $T{=}300$ K, $L{=}25$mm,
$\kappa{=}1.34$ MHz, $\lambda{=}1064$ nm, and $P_{1}{=}2$ mW and time is
expressed in units of $\tau=2\pi/\Omega_{1}$. These values are consistent with
the state of the art experiments and unless otherwise specified these are the
ones considered in all the shown results

As shown in fig. 2, the time evolution of the mutual information is
characterised by distinct phases. Initially, the two mirrors exhibit no
correlation, indicating an uncorrelated state. Following this, a short
transient of correlation between the mirrors occurs. Eventually, they return
to an uncorrelated state again, only to reach a state of steady-state
correlations. Comparing this result with the temperature behaviour reported in
fig. 3 can be observed that the rebirth of correlations occurs subsequent to
thermalization of the second mirror. The amount of quantumness and, most
importantly, the asymmetrical nature (due to the unidirectional coupling) of
such correlations can be characterized in terms of quantum Discord [45].

This different type of quantum correlations can be nonzero even in the case of
separable states which implies that some bipartite quantum states can show
correlations that are incompatible with classical physics. For our system we
can adopt the Gaussian quantum Discord [46, 47]. Quantum Gaussian Discord is
defined as the difference between mutual information and classical
correlations. Classical correlations are defined as the maximum amount of
information that one can gain on one subsystem by locally measuring the other
subsystem [44] and so, by this definition, quantum Discord is not symmetric
with respect to the interchange of the two subsystems. In particular in this
case, where the coupling is unidirectional, the Gaussian discord is maximally
asymmetrical due to the fact that measuring the first subsystem one cannot
recover any information about the second one.

The quantum Discord $D(A|B)$ plotted in fig. 2 refers only to the one relative
to the second subsystem conditioned to the first, indeed performing a measure
on the second mirror one can recover some information on the first one, but
the converse is not true. In fact $D(B|A)$ is identically zero at all times.
That is expected due to the unidirectionality of the coupling. Furthermore
also for the quantum discord there is a non zero value also in the stationary
state.

##  V Finite temperature gradient

Here we will show that in the cooling regime, due to the effective
unidirectional coupling, a temperature gradient is established between the two
mechanical modes. In fact in the cooling regime, i.e. $\Delta{=}\Omega_{1}$,
the optical field generates extra damping on the mechanical mode. Such optical
damping, caused by radiation pressure, depends on both the position $Q$ and
the speed with which the mirror changes its position. At $t=0$ the phonons
associated to the mechanical oscillator motion are in a thermal equilibrium
state. Then, the interaction between the photons and the phonons, as described
by the last term in eq. 2, leads to a change of the phonon number which
fluctuates due to the coupling to its environment, consisting of a hot phonon
bath at temperature $T$. The goal of optomechanical (sideband) cooling is to
reduce the amount of such fluctuations thereby cooling it down.

The mean energy of the mirrors is evaluated

$\displaystyle U_{j}(t)=\frac{\hbar\Omega_{j}}{2}\left(\langle\delta
q_{j}^{2}(t)\rangle+\langle\delta
p_{j}(t)^{2}\rangle\right)=\hbar\Omega_{j}\left(n^{\text{eff}}_{j}(t)-\frac{1}{2}\right)$
(20)

with $n^{\text{eff}}_{j}(t)$ obtained by the solution eq. 9 of covariance
matrix as $1/2(C_{11}+C_{22}-1)$ for the first mechanical mode and
$1/2(C_{55}+C_{66}-1)$ for the second one.

Figure 3: Temperatures of mechanical modes, as defined in eq. 21, as a
function of time. The solid line refers to the temperature of the first
mirror, the dotted one refers to the temperature of the second mirror in the
case $\Omega_{2}=0.75\Omega_{1}$, the dashed line represents the the
temperature of the second mirror in the case $\Omega_{2}=\Omega_{1}$, while
the dot-dashed one represents the temperature of the second mirror in the case
$\Omega_{2}=1.25\Omega_{1}$

The effective temperature of the movable mirrors are then given by

$\displaystyle
T^{\text{eff}}_{j}(t)=\frac{\hbar\Omega_{j}}{k_{B}\ln{(1+1/n^{\text{eff}}_{j}(t)})}$
(21)

When the two mirrors have the same frequency, i.e. $\Omega_{2}=\Omega_{1}$
(fig. 3), the steady state is characterised by a higher temperature of the
second mirror with respect to the first one. This is a consequence of the
unidirectionality of the coupling. Indeed, as shown in appendix, such
temperature gradient is absent in the bidirectional case.

To further investigate the properties of this temperature gradient, the
temperatures of the second mirror were evaluated varying its frequency. Due to
the mismatch between the optical detuning and the frequency of the mechanical
mode in the second optomechanical system, which corresponds to a variation in
the cooling efficiency, the time at which the second mirror reaches a steady
temperature value, $t_{s}$, increases as shown in fig. 4 as long as the
frequency is moving apart from the resonant case $\Omega_{2}=\Omega_{1}$

It can be seen in fig. 5 that for different values of $\Omega_{2}$, varying
the detuning between the pump and the first cavity, one can always tune it in
such a way that it creates a temperature gradient between the mirrors. The
stationary correlations between the two mirrors, evaluated as the mutual
information in the stationary regime, shows a peak in correspondence to the
minima of the second mirror temperatures.

Figure 4: (Left) Time at which the second mechanical mode reaches the
stationary state as a function of its frequency (Right) Temperature of the
second mechanical mode in the stationary regime as a function of its frequency
Figure 5: (Left) Temperature gradients of mechanical modes
$T^{\text{eff}}_{2}-T^{\text{eff}}_{1}$, where $T^{\text{eff}}_{j}$ is defined
as in eq. 21, in the stationary limit for different values of $\Omega_{2}$. In
particular, the curves refer to the cases $\Omega_{2}=0.75\Omega_{1}$,
$\Omega_{2}=\Omega_{1}$ and $\Omega_{2}=1.25\Omega_{1}$. (Right) Mutual
information between the two mechanical modes in the stationary limit for the
same values of $\Omega_{2}$

##  VI The steady state and the power spectra

One of the experimentally accessible quantity for the optomechanical systems
is the power spectrum of the cavity output field which allows to reconstruct
the spectrum (and so the dynamics and the temperature) of the mechanical
mirror [48]. We now show how in this cascaded configuration, the spectrum in
output from the last cavity contains informations about the two mirrors and
allows to reconstruct their dynamics. In order to evaluate the spectra of the
cavity output and the two individual mirrors, it’s necessary to evaluate the
stationary state of the system. As shown in the previous sections, the
linearized equations for the fluctuations eq. 8 can be solved in the frequency
domain. The correlation functions eq. 3 become

$\displaystyle\langle\hat{a}^{\text{in}}(\omega)\hat{a}^{\text{in}\dagger}(\omega^{\prime})\rangle=\delta(\omega+\omega^{\prime})$
(22a)
$\displaystyle\langle\\{\hat{\xi}_{j}(\omega),\xi_{j}(\omega^{\prime})\\}\rangle=2\gamma\coth\left(\frac{\hbar\Omega_{j}}{2k_{B}T}\right)\delta(\omega+\omega^{\prime}),$
(22b)

while the equations for the fluctuations of cavity field modes are eq. 13 and
the equations for the positions of the mirrors become

$\displaystyle\delta\hat{q}_{j}(\omega)=\chi_{j}(\omega)\left(G_{j}\delta\hat{a}_{j}^{\dagger}(\omega)+G^{*}_{j}\delta\hat{a}_{j}(\omega)+\xi_{j}(\omega)\right)$
(23)

where we have introduced the natural susceptibilities of the mechanical modes

$\displaystyle\chi_{j}(\omega)=\frac{\Omega_{j}}{\Omega_{j}^{2}-\omega^{2}-i\omega\gamma}$
(24)

The mirror’s position fluctuations can be expressed in terms of effective
susceptibilities and noise operators:

$\displaystyle\delta\hat{q}_{1}(\omega)=$
$\displaystyle\chi^{\text{eff}}_{1}(\omega)\xi_{1}(\omega)$ (25a)
$\displaystyle\delta\hat{q}_{2}(\omega)=$
$\displaystyle\chi^{\text{eff}}_{2}(\omega)\xi^{\text{eff}}_{2}(\omega)$ (25b)

with

$\displaystyle\chi_{j}^{\text{eff}}(\omega)=\frac{\chi_{j}(\omega)}{1-i|G_{j}|^{2}\chi_{j}(\omega)(\chi_{a_{j}}(\omega)-\chi_{a_{j}}^{*}(\text{-}{\omega}))}$
(26a)
$\displaystyle\xi^{\text{eff}}_{2}(\omega)=\xi_{2}(\omega)-i\kappa\chi^{\text{eff}}_{1}(\omega)\xi_{1}(\omega)\Lambda(\omega)$
(26b)

Note that the effective susceptibility of the mechanical oscillators
$\chi_{j}^{\text{eff}}(\omega)$ are modified by the radiation pressure [3].
Furthermore, in the second of eq. 25, the effective noise seen by the second
mirror, modified by the presence of the first, is made explicit. It is now
clear how the position fluctuations of the first ($\delta\hat{q}_{1}$) of the
two mechanical modes depends only on its local thermal bath, while the second
one ($\delta\hat{q}_{2}$) depends also on the thermal bath of the first via
the optical field.

In the same way, for the cavity field fluctuation we have

$\displaystyle\delta\hat{a}_{1}(\omega)$
$\displaystyle=iG_{1}\chi_{a_{1}}(\omega)\chi^{\text{eff}}_{1}(\omega)\xi_{1}(\omega)$
(27a) $\displaystyle\delta\hat{a}_{2}(\omega)$
$\displaystyle=iG_{2}\chi_{a_{2}}(\omega)\chi^{\text{eff}}_{2}(\omega)\xi^{\text{eff}}_{2}(\omega)-iG_{1}\kappa\chi_{a_{1}}(\omega)\chi_{a_{2}}(\omega)\chi^{\text{eff}}_{1}(\omega)\xi_{1}(\omega)$
(27b)

### Power Spectra

From eqs. 25 and 27, thanks to eq. 22, it is possible to evaluate the position
spectrum of the two mirrors defined by

$\displaystyle
S^{q}_{j}(\omega)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}d\Omega
e^{-i(\omega+\Omega)t}\langle\delta\hat{q}_{j}(\omega)\delta\hat{q}_{j}(\Omega)\rangle$
(28)

obtaining

$\displaystyle S^{q}_{1}(\omega){=}$
$\displaystyle\gamma(2\bar{n}_{1}{+}1)|\chi_{1}^{\text{eff}}(\omega)|^{2}$
(29a) $\displaystyle S^{q}_{2}(\omega){=}$
$\displaystyle\gamma(2\bar{n}_{2}{+}1)|\chi_{2}^{\text{eff}}(\omega)|^{2}{-}\kappa^{2}S^{q}_{1}(\omega)|\chi_{2}^{\text{eff}}(\omega)|^{2}|\Lambda(\omega)|^{2}$
(29b)

from which one can obtain the variances $\langle\delta\hat{q}_{j}^{2}\rangle$
trough

$\displaystyle\langle\delta\hat{q}_{j}^{2}\rangle=\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}S^{q}_{j}(\omega)$
(30)

We are also interested to the output power spectral density that would be
detected in an homodyne detection of the output fluctuations $\delta
x^{\text{out}}{=}1/\sqrt{2}(\delta\hat{a}^{\text{out}}+\delta\hat{a}^{\dagger\text{out}})$
where

$\displaystyle\delta\hat{a}^{\text{out}}(\omega)=\hat{a}^{\text{in}}(\omega)-\sqrt{\kappa}\delta\hat{a}_{1}(\omega)-\sqrt{\kappa}\delta\hat{a}_{2}(\omega)$
(31)

The spectrum of such fluctuations can be obtained as

$\displaystyle
P^{\text{out}}(\omega)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}d\Omega
e^{-i(\omega+\Omega)t}\braket{x^{\text{out}}(\omega)x^{\text{out}}(\Omega)}$
(32)

Using eq. 22 in the frequency domain one finds

$\displaystyle
P^{\text{out}}(\omega)\sim\sum_{j=1}^{2}K_{j}(\omega)S^{q}_{j}(\omega)$ (33)

with
$K_{j}(\omega)=\kappa|{G_{j}\chi_{a_{j}}(\omega){-}G_{j}^{*}\chi_{a_{j}}^{*}(\text{-}{\omega})}|^{2}/2$.
From eq. 33, it follows, as shown in fig. 6, that the output field from the
second cavity contains information on the power spectra of both mechanical
modes as it simply proportional to the sum of the two mechanical power
spectra. A similar result was obtained, for a single optomechanical system, in
[48]. Consequently, the peaks observed in the cavity’s output spectrum reflect
those of the individual mirrors’ spectra. Hence, one can infer the power
spectrum of each individual mirror by fitting the peaks of the spectrum, given
that the power spectrum of an individual optomechanical mirror is well
approximated by a Lorentzian curve [2].

Figure 6: (Left) _Mirror’s Spectra_ \- In this figure we plot the power
spectrum eq. 29 of the first mechanical mode and of the second one for three
different frequencies, from top to bottom, $\Omega_{2}=\Omega_{1}/2$,
$\Omega_{2}=\Omega_{1}$ and $\Omega_{2}=3\Omega_{1}/2$. (Right) _Output
Spectra_ \- Spectrum of the output field from the second cavity. In this case
$P=10^{-2}mW$

The appearance of two peaks in the power spectra of the first mirror, as
depicted in fig. 6, is contingent on the selected value for the first cavity’s
pump power. Specifically, these dual peaks manifest at a particular power pump
value and progressively move farther apart as the power value increases.

##  VII Self-induced oscillations and multistability

Up until this point, our analysis has primarily focused on stable states by
employing specific parameter values. However, we will now delve into the
nonlinear regime for the average values and demonstrate how to identify stable
states by tuning the system’s parameters. As the optomechanical coupling
becomes stronger and damping becomes weaker, the system’s nonlinearities
become significant and cannot be neglected any longer. In this regime, the
system exhibits instabilities, leading the mirror to enter a state of what is
known as ”self-sustained oscillations”. In the following, we will explore
these nonlinear dynamics and outline the conditions required to achieve stable
states amidst the presence of these oscillations[49, 50]. In this regime the
mean position of the mirrors can be written as
$Q_{j}(t)=\bar{Q}_{j}{+}\alpha_{j}cos(\Omega_{j}t)$. Putting this into eq. 7,
the exact solutions for the cavity modes amplitude $A_{j}$, in the long time
limit, can be written as

$\displaystyle A_{1}(t)=$
$\displaystyle\exp\left[ig_{1}\alpha_{1}\frac{\sin(\Omega_{1}t)}{\Omega_{1}}\right]\sum_{n}A_{1}^{n}e^{i\Omega_{1}nt}$
(34a) $\displaystyle A_{2}(t)=$
$\displaystyle\exp\left[ig_{2}\alpha_{2}\frac{\sin(\Omega_{2}t)}{\Omega_{2}}\right]\sum_{n,m,l}A_{2}^{nml}e^{i(\Omega_{1}(n+l)+\Omega_{2}m)t}$
(34b)

with

$\displaystyle
A_{1}^{n}=J_{n}\left(\frac{-g_{1}\alpha_{1}}{\Omega_{1}}\right)\chi_{a_{1}}(-\Omega_{1}n)E_{1}$
(35a) $\displaystyle
A_{2}^{nml}=-\kappa\;J_{n}\left(-\frac{g_{1}\alpha_{1}}{\Omega_{1}}\right)\;J_{m}\left(-\frac{g_{2}\alpha_{2}}{\Omega_{2}}\right)J_{l}\left(\frac{g_{1}\alpha_{1}}{\Omega_{1}}\right)\chi_{a_{2}}(-\Omega_{1}(n+l)-\Omega_{2}m)\;E_{1}$
(35b)

where $J_{n}(x)$ is the Bessel function of first kind and $\chi_{a_{j}}$ are
the susceptibilities defined in eq. 14. The stable states of the system are
those for which the total time-averaged force vanishes and the power due to
the radiation pressure $P_{rad_{j}}=G\langle|A_{j}|^{2}\dot{Q}_{j}\rangle$
equals the power dissipated $P_{fric}=\gamma\langle\dot{Q}^{2}_{j}\rangle$. By
plotting the ratio $P_{rad}/P_{fric}$ for the two subsystems as a function of
$A_{j}$ and detuning $\Delta$, we obtain diagrams that illustrate the
parameter regions corresponding to stable states. These diagrams provide
valuable insights into the values of $A_{j}$ and $\Delta$ where the system
exhibits stability.

Figure 7: (Left) Stability plot for the first mechanical oscillator. It shows
the ratio between the power due to radiation pressure and the power dissipated
as a function of the amplitude of oscillation and the detuning between the
pump and the cavity[50]. (Right) Stability Graph for the second mechanical
oscillator, it shows the ratio between the power due to radiation pressure and
the power dissipated as a function of the amplitude of oscillation and the
detuning between the pump and the cavity. In both the plots, the stable states
are those for which the power due to the radiation pressure
$P_{rad_{j}}=G\langle|A_{j}|^{2}\dot{Q}_{j}\rangle$ equals the power
dissipated $P_{fric}=\gamma\langle\dot{Q}^{2}_{j}\rangle$

_Multistability_ \- A characteristic feature of optomechanical systems is
that, in the regime in which $A_{j}=0$, they exhibit multistability. A given
intensity of the light pumped in the cavity can lead to different steady
states of both cavity photon number and mechanical position [2, 51]. From eq.
7, taking the stationary limit, we can find the equations for the average
number of photons in the two cavities $N_{j}$ i.e.

$\displaystyle\dfrac{g^{4}_{1}}{\Omega^{2}_{1}}N^{3}_{1}-\dfrac{2\Delta
g^{2}_{1}}{\Omega_{1}}N^{2}_{1}+\left(\Delta^{2}+\dfrac{\kappa^{2}}{2}\right)N_{1}-E^{2}_{1}=0$
(36a) $\displaystyle\dfrac{g^{4}_{2}}{\Omega^{2}_{2}}N^{3}_{2}-\dfrac{2\Delta
g^{2}_{2}}{\Omega_{2}}N^{2}_{2}+\left(\Delta^{2}+\dfrac{\kappa^{2}}{2}\right)N_{2}-\kappa^{2}N_{1}=0$
(36b)

and once found these, we can find the average cantilever positions as

$Q_{j}=\dfrac{g_{j}}{\Omega_{j}}N_{j}$ (37)

We note that the first of eq. 36 has three roots, but, as shown also in [51],
only two of these solutions are stable solutions, specifically the lower and
the higher ones while the middle one is unstable and can’t be observed
experimentally. Regarding the equation for the second cavity a richer
behaviour is obtained, as shown in fig. 8.

Figure 8: Steady-state position of the cantilevers as a function of detuning
for $\Omega_{2}{=}3/4\Omega_{1}$(left), $\Omega_{2}{=}\Omega_{1}$(center) and
$\Omega_{2}{=}5/4\Omega_{1}$(right). As known in literature [2, 51], the
positions of mirrors show multistability. In a single optomechanical system
the mechanical position have three stable solutions, but due to the
unidirectional coupling the second one shows three or more stable solutions
depending on the detuning between the two mirror’s frequencies

##  VIII Conclusion

In summary, this study has provided a comprehensive characterization of the
dynamics of two optomechanical systems indirectly coupled via a chiral
waveguide in a cascaded configuration. In the weak coupling regime, we
employed an adiabatic elimination technique to derive effective equations
governing the mirror dynamics.

By examining the evolution of correlations between the two mechanical modes,
we quantified their mutual information and quantum discord over time.
Remarkably, our results demonstrate that these correlations persist even in
the stationary state, indicating a non-zero value of steady correlations.
Furthermore, we investigated the steady-state temperatures of both mirrors for
different values of $\Omega_{2}$ and various $\Delta$ parameters. Our findings
revealed the existence of a finite temperature difference between the two
mirrors when employing this indirect effective coupling approach. This
suggests the exciting possibility of engineering a temperature gradient
between the mechanical modes using such a setup. Additionally, we analysed the
power spectra of the two mechanical modes and the output spectrum of the
second cavity. Remarkably, our study shows that by measuring the latter, it
becomes feasible to reconstruct the spectra of the mirrors accurately. To
ensure the stability of the mirror dynamics, we explored the regions of
stability as a function of the first cavity pump. Moreover, we delved into the
potential existence of multiple steady states for both cavity photon number
and mechanical position concerning a specific intensity of light pumped into
the first cavity.

The findings regarding correlations, temperature gradients, spectrum
reconstruction, stability regions, and multiple steady states open up new
possibilities for controlling and manipulating these complex systems for
various applications.

## Acknowledgement

SL and GMP acknowledge support by MUR under PRIN Project No. 2017 SRN-BRK
QUSHIP.

## Appendix A

### Two modes mutual information and quantum discord

As known by [44] given a two-mode Gaussian state covariance matrix

$\mathbf{C}=\begin{pmatrix}\mathbf{A}&\mathbf{D}\\\
\mathbf{D}^{T}&\mathbf{B}\\\ \end{pmatrix},$ (38)

where $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{D}$ are $2\times 2$ matrices one
can define four local symplectic invariants, $I_{1}=det(\mathbf{A})$,
$I_{2}=det(\mathbf{B})$, $I_{3}=det(\mathbf{D})$, $I_{4}=det(\mathbf{C})$ and
its symplectic eigenvalues

$d_{\pm}=\sqrt{\dfrac{I_{\Delta}\pm\sqrt{I_{\Delta}^{2}-4I_{4}}}{2}}$ (39)

where $I_{\Delta}=I_{1}+I_{2}+2I_{3}$.
Mutual information, defined for two quantum systems A and B as

$\mathcal{I}(AB)=S(\rho_{A})+S(\rho_{B})-S(\rho_{AB})$ (40)

(where $S(\rho)$ refers to Von Neumann entropy) can be evaluated, in the case
of continous variables gaussian states, in terms of those symplectic
invariants as

$\mathcal{I}(AB)=f(\sqrt{I_{1}})+f(\sqrt{I_{2}})-f(d_{+})-f(d_{-})$ (41)

with

$f(x)=\left(x+\dfrac{1}{2}\right)log\left(x+\dfrac{1}{2}\right)-\left(x-\dfrac{1}{2}\right)log\left(x-\dfrac{1}{2}\right).$
(42)

Mutual information quantifies the correlations between two quantum systems.
Quantum discord, interpreted as the amount on quantumness of these
correlations, is then defined as

$\mathcal{D}(A|B)=\mathcal{I}(AB)-\mathcal{C}(A|B)$ (43)

where

$\mathcal{C}(A|B)=\max_{\\{E_{a}\\}}\left[S(\rho_{A})-\sum_{a}p_{a}S\left(\frac{Tr_{B}[\rho_{AB}E_{a}]}{p_{a}}\right)\right]$
(44)

is the total amount of classical correlations and $\sum_{a}E_{a}=\mathbf{1}$
is a positive operator value measurement (POVM).
In terms of symplectic invariants of covariance matrix, quantum discord can be
evaluated as

$\mathcal{D}(A|B)=f(\sqrt{I_{2}})+f(d_{+})+f(d_{-})+f(\sqrt{W}),$ (45)

where

$W=\begin{cases}\left[\frac{2|I_{3}|+\sqrt{4I^{2}_{3}+(4I_{2}-1)(4I_{4}-1)}}{(4I_{2}-1)}\right]^{2}\text{
if }\frac{4(I_{1}I_{2}-I_{4})^{2}}{(I_{1}+4I_{4})(1+4I_{2})I^{2}_{3}}\leq
1,\\\ \\\
\frac{I_{1}I_{2}+I_{4}-I^{2}_{3}-\sqrt{(I_{1}I_{2}+I_{4}-I^{2}_{3})^{2}-4I_{1}I_{2}I_{4}}}{2I_{2}}\text{
otherwise.}\end{cases}$ (46)

## Appendix B

### Bidirectional case

In order to consider the most general case in which the two subsystems are
coupled through a bidirectional (non chiral) waveguide it’s necessary to
introduce two vacuum input noise operators, $\hat{a}^{in}_{R}$ and
$\hat{a}^{in}_{L}$, one for each of the direction of propagation in the
waveguide, with autocorrelation functions:

$\braket{\hat{a}^{in}_{i}(t)\hat{a}^{in\dagger}_{j}(t^{\prime})}=\delta_{ij}\delta(t-t^{\prime})\text{\quad
with\quad}i,j=R,L$ (47)

and the same Brownian noise operators $\hat{\xi}_{j}$ defined in section II.
Once defined these new operators, one can straightforwardly follow the
procedure described in the previous sections and find the new set of non
linear differential equations for the mean values

$\displaystyle\frac{dQ_{j}(t)}{dt}=\Omega_{j}P_{j}(t)$ (48a)
$\displaystyle\frac{dP_{j}(t)}{dt}=-\Omega_{j}Q_{j}(t)-\gamma_{j}P_{j}(t)+|G_{j}(t)|^{2}/g_{j}$
(48b)
$\displaystyle\frac{dA_{1}(t)}{dt}=-\frac{\kappa}{2}A_{1}(t)-i\Delta_{1}(t)A_{1}(t)-\kappa
A_{2}(t)+E_{1}$ (48c)
$\displaystyle\frac{dA_{2}(t)}{dt}=-\frac{\kappa}{2}A_{2}(t)-i\Delta_{2}(t)A_{2}(t)-\kappa
A_{1}(t)+E_{2}$ (48d)

and for the fluctuations one can again find a Lyapunov equation that the
covariance matrix $\mathbf{C}$ of the system must obey

$\frac{d\mathbf{C}(t)}{dt}=\mathbf{S}(t)\mathbf{C}(t)+\mathbf{C}(t)\mathbf{S}(t){+}\mathbf{N}$
(49)

in which the drift ($\mathbf{S}$) and diffusion ($\mathbf{N}$) matrices now
become (cfr. eq. 9)

$\displaystyle\mathbf{S}=\begin{pmatrix}\mathbf{S}_{1}&\mathbf{S}_{12}\\\
\mathbf{S}_{12}&\mathbf{S}_{2}\end{pmatrix}\qquad\text{and}\qquad\mathbf{N}=\begin{pmatrix}\mathbf{N}_{1}&\mathbf{N}_{12}\\\
\mathbf{N}_{12}&\mathbf{N}_{2}\end{pmatrix}$ (50)

with

$\displaystyle\mathbf{S}_{j}{=}\begin{pmatrix}0&\Omega_{j}&0&0\\\
-\Omega_{j}&-\gamma&\text{Re}(G_{j})&\text{Im}(G_{j})\\\
-\text{Im}(G_{j})&0&-\kappa&\Delta_{j}\\\
\text{Re}(G_{j})&0&-\Delta_{j}&-\kappa\end{pmatrix}\;\;\mathbf{S}_{12}{=}\begin{pmatrix}0&0&0&0\\\
0&0&0&0\\\ 0&0&-\kappa&0\\\ 0&0&0&-\kappa\end{pmatrix}$

and

$\displaystyle\mathbf{N}_{j}{=}\begin{pmatrix}0&0&0&0\\\
0&\gamma(2\bar{n}_{j}+1)&0&0\\\ 0&0&2\kappa&0\\\ 0&0&0&2\kappa\\\
\end{pmatrix}\;\;\mathbf{N}_{12}{=}\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\
0&0&2\kappa&0\\\ 0&0&0&2\kappa\end{pmatrix}$

It’s possible now to calculate the temperature as defined in eq. 21 and, as
shown in fig. 9, in the case in which the two optomechanical systems are
pumped (i.e. $E_{2}=E_{1}=E$) it can be seen that no gradient of temperature
is established between the two mirrors.

Figure 9:  In this figure is reported the temperature of the two mirrors in
the unidirectional (dashed and dot-dashed lines) and bidirectional (solid
line) as a function of time, for $E_{1}=E_{2}=E$ and $\gamma=2\pi\cdot
10^{4}$. In order to compare systems with constant total decay rate in the
unidirectional case the decay rates were fixed with values $\kappa_{R}=\kappa$
and $\kappa_{L}=0$, while in the bidirectional case
$\kappa_{R}=\kappa_{L}=\kappa/2$. In the first case one can see that a
temperature gradient is established while in the latter is not present
anymore.

## Appendix C

### Effective Lorenzian peak

The expression for the spectra in eq. 29 can be rearranged, expliciting the
expressions of $\chi_{\text{eff}_{j}}$, $\chi_{j}$ and $\chi_{a_{j}}$ in the
form that, in the limit of $\omega_{j}\simeq\Omega_{j}$, is a Lorenzian curve.
Indeed if we write $\chi_{\text{eff}_{j}}$ expliciting the susceptibilities we
obtain

$\displaystyle\chi_{j}^{\text{eff}}(\omega)=\frac{\chi_{j}(\omega)}{1-i|G_{j}|^{2}\chi_{j}(\omega)(\chi_{a_{j}}(\omega)-\chi_{a_{j}}^{*}(\text{-}{\omega}))}=$
(51)
$\displaystyle=\frac{\Omega_{j}}{\Omega_{j}-\omega^{2}-i\gamma\omega-i|G_{j}|^{2}\Omega_{j}\left(\frac{1}{\frac{\kappa}{2}-i(\omega-\Delta_{j})}-\frac{1}{\frac{\kappa}{2}-i(\omega+\Delta_{j})}\right)}.$

If we define $\tilde{\Omega}_{j}(\omega)$ and $\tilde{\gamma}_{j}(\omega)$ as

$\displaystyle\tilde{\Omega}_{j}(\omega)=\sqrt{\Omega^{2}_{j}-\frac{|G_{j}|^{2}2\Delta_{j}\Omega_{j}(\frac{\kappa^{2}}{4}+\Delta^{2}_{j}-\omega^{2})}{(\frac{\kappa^{2}}{4}+(\omega-\Delta_{j})^{2})(\frac{\kappa^{2}}{4}+(\omega+\Delta_{j})^{2})}}$
(52)
$\displaystyle\tilde{\gamma}_{j}(\omega)=\gamma+\frac{|G_{j}|^{2}2\kappa\Delta_{j}\Omega_{j}}{(\frac{\kappa^{2}}{4}+(\omega-\Delta_{j})^{2})(\frac{\kappa^{2}}{4}+(\omega+\Delta_{j})^{2})}$

we can rewrite eq. 51 as

$\chi^{\text{eff}}_{j}(\omega)=\frac{\Omega_{j}}{\tilde{\Omega}^{2}_{j}(\omega)-\omega^{2}-i\tilde{\gamma}_{j}(\omega)\omega}$
(53)

which in the neighborhood of $\omega=\Omega_{j}$ can be approximated by

$\chi^{\text{eff}}_{j}(\omega)=\frac{\Omega_{j}}{\tilde{\Omega}^{2}_{j}(\Omega_{j})-\omega^{2}-i\tilde{\gamma}_{j}(\Omega_{j})\omega}$
(54)

Being proportional to the squared modulus of $\chi^{\text{eff}}_{j}$, the
spectra will have the shape of a Lorenzian curve as shown in fig. 10.

Figure 10: Spectrum of the first mirror plotted using the exact expression
(solid line) and with the one obtained approximating the spectrum as a
lorenzian (dashed line)

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