# Negative cavity photon spectral function in an optomechanical system with
two parametrically-driven mechanical oscillators

Ali Motazedifard  Quantum Optics Group, Department of Physics, University of
Isfahan, Hezar-Jerib, 81746-73441, Isfahan, Iran  Quantum Sensing Lab, Quantum
Metrology Group, Iranian Center for Quantum Technologies (ICQT), Tehran, Iran
<EMAIL_ADDRESS>A. Dalafi  Laser and Plasma Research Institute,
Shahid Beheshti University, Tehran, Iran  M. H. Naderi  Quantum Optics Group,
Department of Physics, University of Isfahan, Hezar-Jerib, 81746-73441,
Isfahan, Iran  Department of Physics, University of Isfahan, Hezar-Jerib,
81746-73441, Isfahan, Iran

###### Abstract

We propose an experimentally feasible optomechanical scheme to realize a
negative cavity photon spectral function (CPSF). The system under
consideration is an optomechanical system (OMS) consisting of two mechanical
(phononic) modes which are linearly coupled to a common cavity mode via the
radiation pressure while parametrically driven through the coherent time-
modulation of their spring coefficients. We find that, in the red-detuned and
weak-coupling regimes, a frequency-dependent effective cavity damping rate
(ECDR) is induced in the system. Furthermore, using the equations of motion
for the cavity retarded Green’s function obtained in the framework of a
generalized linear response theory (GLRT), we show that a negative ECDR
corresponding to a negative CPSF can be realized by controlling the
cooperativities and modulation parameters while the system still remains in
the stable regime. Nevertheless, such a negativity never occurs in a standard
cavity optomechanical system. Besides, we find that the presence of two
modulated mechanical degrees of freedom provides more controllability on the
negativity of CPSF with a smaller parametric drive in comparison to the setup
with a single modulated mechanical oscillator (MO). Interestingly, the
introduced negativity may open a new platform to realize an extraordinary
(modified) optomechanically induced transparency (OMIT) leading to perfect
tunable optomechanical filters, and a negative effective temperature (NET)
corresponding, respectively, to the probe reflection above the unity and the
coupled qubit-cavity population inversion.

###### keywords:

Optomechanics, Parametric amplification, Green’s function, Spectral function

## Introduction

In the recent two decades, the field of quantum optomechanics [1], in which a
mechanical oscillator (MO) is linearly or quadratically coupled to an
optical/microwave mode via the radiation pressure, using state-of-the-art
technologies has been significantly progressed both in theory and experiment.
The optomechanical systems (OMSs) play a key role in quantum technologies and
also can be employed as a controllable setup for the emergence of quantum
effects on macroscopic scales or probing the fundamentals of physics [2, 3].

OMSs have been applied in a variety range of applications including
ultraprecision quantum sensing and measurement [4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16], quantum illumination or quantum radar [17, 18], cooling of
the MO [19, 20, 21], generation of entanglement [22, 23, 24, 25, 26, 27],
synchronization of MOs [28, 29, 30, 31], generation of quadrature squeezing
and amplification [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45,
46], realizing the dynamical Casimir effect-based nonclassical radiation
sources [47, 48, 49, 50, 51, 52], optomechanically induced transparency (OMIT)
[53, 54, 55, 56, 57, 58, 59, 60], which is analogous to the familiar
phenomenon of electromagnetically induced transparency (EIT) [61], and also
the realization of microwave nonreciprocity, unidirectional transport,
isolator and circulator [62].

Very recently, by applying a generalized linear response theory (GLRT) to a
standard driven-dissipative OMS [63] as an open quantum system, the Green’s
function equations of motion have been obtained from quantum Langevin
equations (QLEs) in the Heisenberg picture to deeply explain a wealth of
phenomena, including the anti-resonance, normal mode splitting, and OMIT. On
the other hand, driven-dissipative quantum systems have recently attracted
intensive research interests, particularly in the context of quantum control
(for a very recent, comprehensive review, the reader is referred to Ref.
[64]). Generally, the steady state of such systems can be far from equilibrium
because of the unavoidable competition between driving and dissipation. The
steady-state behaviors of such systems have been investigated theoretically by
many authors (see, for example, [65, 66, 67]). Besides the steady-state
behavior, also the description of the system’s response to weak external
perturbations is of great interest. It is well-known that the Green’s function
method is a powerful technique for this purpose. To determine the Green’s
functions of open quantum systems there are some approaches in the literature
[68, 69, 70, 71, 72, 73, 74, 75] which are mainly based on the Lindblad master
equation, i.e., presented in the Schrödinger picture.

In this paper, we will follow the method introduced in Ref. [63] to obtain the
cavity retarded Green’s function (CRGF) of an OMS composed of two
parametrically-driven mechanical modes which are coupled to a common
electromagnetic mode. We show that by controlling the system parameters a
negative CPSF may be achieved in the stable regime of the system. In other
words, we show that the coherent time-modulation of the spring coefficients of
the mechanical modes leads to a mechanical parametric amplification which in
turn induces a negative frequency-dependent effective cavity damping rate
(ECDR) that corresponds to a negative cavity photon spectral function (CPSF).
It should be pointed out that unlike other standard systems such as the
degenerate parametric amplifier (DPA), the occurrence of a negative CPSF in
the present system takes place in the stable regime. Furthermore, we briefly
address how such a negative CPSF may result in the so-called extraordinary
(modified) OMIT which corresponds to a probe reflectivity above the unity.
Also, it may lead to the definition of a negative effective temperature (NET)
for the intracavity mode which can emerge in qubit-cavity population
inversion. These two important issues will be discussed in detail in our
future paper.

The paper is organized as follows. In Sec. 1, we describe the basic model of
the system under study and its effective Hamiltonian. In Sec. 2, by solving
the QLEs in frequency space we find the self-energies and induced-parametric
squeezing coefficients corresponding to the induced-effective damping rates.
Using the solutions in the frequency space, we calculate the CPSF in Sec. 3,
and show that by controlling the cooperativities and modulation parameters it
can be negative when the ECDR becomes negative. In Sec. 4, we present a
physical interpretation of the CPSF negativity in our system together with a
comparison to an ordinary OPA. Finally, we summarize our concluding remarks
and outlooks in Sec. 5. In particular, the manifestation of the negative CPSF
in the modified OMIT and NET are briefly discussed in Sec. 5.

##  1 System Hamiltonian

Figure 1: (Color online) Schematic of an optomechanical system consisting of
two similar MOs which are linearly coupled to the common electromagnetic (EM)
mode via the radiation pressure. The spring coefficients of the MOs are
parametrically driven at twice their natural frequencies so that
$k(t)=k+\delta k\cos(2\omega_{m(d)}t+\varphi_{m(d)})$.

As depicted in Fig. (1), we consider an OMS consisting of two MOs with the
natural frequencies $\omega_{m}$ and $\omega_{d}$, and damping rates
$\gamma_{m}$ and $\gamma_{d}$ interacting with the radiation pressure of the
single optical mode of the cavity while the spring coefficients of the MOs are
coherently modulated at twice their natural frequencies. Besides, the natural
frequency of the optical mode inside the cavity and the cavity damping rate
are, respectively, $\omega_{c}$ and $\kappa$. It is also assumed that the
cavity is driven at the rate of $E_{L}=\sqrt{\kappa P_{L}/\hbar\omega_{L}}$ by
an external laser with frequency $\omega_{L}$ and the input power $P_{L}$. The
total Hamiltonian of the system in the frame rotating at the driving laser
frequency, $\omega_{L}$, can be written as

$\displaystyle\hat{H}=\hbar\Delta_{c}\hat{a}^{\dagger}\hat{a}+\hbar\omega_{m}\hat{b}^{\dagger}\hat{b}+\hbar\omega_{d}\hat{d}^{\dagger}\hat{d}+i\hbar
E_{L}(\hat{a}^{\dagger}-\hat{a})-\hbar
g_{0}\hat{a}^{\dagger}\hat{a}(\hat{b}+\hat{b}^{\dagger})-\hbar
G_{0}\hat{a}^{\dagger}\hat{a}(\hat{d}+\hat{d}^{\dagger})$
$\displaystyle\qquad\qquad+\frac{i\hbar}{2}(\lambda_{m}\hat{b}^{{\dagger}2}e^{-2i\omega_{m}t}-\lambda_{m}^{\ast}\hat{b}^{2}e^{2i\omega_{m}t})+\frac{i\hbar}{2}(\lambda_{d}\hat{d}^{{\dagger}2}e^{-2i\omega_{d}t}-\lambda_{d}^{\ast}\hat{d}^{2}e^{2i\omega_{d}t}).$
(1)

The first four terms in the Hamiltonian describe, respectively, the free
energy of the optical mode, the free energy of the MOs and the coupling
between the optical mode and the classical driving laser. Here, $\hat{a}$, and
$\hat{b}$($\hat{d}$) are the annihilation operators of the optical and
mechanical modes, and $\Delta_{c}=\omega_{c}-\omega_{L}$ is the detuning of
the optical mode from the driving laser frequency. The fifth and sixth terms
denote, respectively, the optomechanical interactions between the optical mode
and the two mechanical modes $\hat{b}$ and $\hat{d}$ with the single-photon
optomechanical coupling strengths $g_{0}$ and $G_{0}$.

The seventh and eighth terms account for the parametric driving of the MOs
spring coefficients at the twice of their natural frequencies giving rise to
the time-dependent spring coefficient as $k_{m(d)}(t)=k+\delta
k\cos(2\omega_{m(d)}t+\varphi_{m(d)})$ where $\varphi_{m(d)}$ is the phase of
external modulation. These terms have been written in the rotating wave
approximation (RWA) which is valid over the time scales longer than
$\omega_{m(d)}^{-1}$. Here,
$\lambda_{m(d)}=|\lambda_{m(d)}|e^{i\varphi_{m(d)}}$ is the mechanical
modulation amplitude with $|\lambda_{m(d)}|=\delta kx_{zp}^{2}/2\hbar$ [49,
35] where $x_{zp}=\sqrt{\hbar/2m\omega_{m(d)}}$ is the zero-point position
fluctuation. Note that by fixing the phase of modulation, it is always
possible to take $\lambda_{m(d)}$ as a real number. It is worth mentioning
that the parametric mechanical driving term can be considered as the
mechanical phonon analog of the degenerate parametric amplification which may
lead to the dynamical Casimir effect of mechanical phonons [49]. Furthermore,
the parametric driving has been recently used for the preparation of an
optomechanical system in a two-mode squeezed thermal state [33].It is pointed
out that the parametric mechanical driving can be realized in a
superconducting microwave optomechanical circuit by applying a combination of
a static voltage and an oscillating voltage (ac-voltage) in order to simulate
the modulation of the spring coefficient [34].

##  2 Dynamics of the system

By the assumption that the two MOs have equal natural frequencies, i.e.,
$\omega_{m}=\omega_{d}$, and in the red-detuned regime where
$\Delta_{0}=\omega_{m}=\omega_{d}$, in which
$\Delta_{0}=\Delta_{c}-2g_{0}\bar{b}-2G_{0}\bar{d}$ is the effective cavity
detuning and $\bar{b}\approx g_{0}\bar{a}^{2}/\omega_{m}$ and $\bar{d}\approx-
G_{0}\bar{a}^{2}/\omega_{d}$ are the mean mechanical fields with
$\bar{a}=E_{L}/\sqrt{\kappa^{2}/4+\Delta_{0}^{2}}$ being the steady-state mean
value of the optical mode, the linearized QLEs for the quantum fluctuations of
the system operators can be obtained in the RWA as the following set of
equations [49]

$\displaystyle\\!\\!\\!\\!\\!\delta\dot{\hat{a}}=-\frac{\kappa}{2}\delta\hat{a}+ig\delta\hat{b}+iG\delta\hat{d}+\sqrt{\kappa}\hat{a}_{in},$
(2a)
$\displaystyle\\!\\!\\!\\!\\!\delta\dot{\hat{b}}=-\frac{\gamma_{m}}{2}\delta\hat{b}+ig\delta\hat{a}+\lambda_{m}\delta\hat{b}^{\dagger}+\sqrt{\gamma_{m}}\hat{b}_{in},$
(2b)
$\displaystyle\\!\\!\\!\\!\\!\delta\dot{\hat{d}}=-\dfrac{\gamma_{d}}{2}\delta\hat{d}+iG\delta\hat{a}+\lambda_{d}\delta\hat{d}^{\dagger}+\sqrt{\gamma_{d}}\hat{d}_{in},$
(2c)

where $g(G)=g_{0}(G_{0})\bar{a}$ is the enhanced-optomechanical coupling
strength. Moreover, the optical and the Brownian mechanical input noises
$\hat{o}_{in}$ ($o=a,b,d$) satisfy the Markovian correlation functions
$\langle\hat{o}_{in}(t)\hat{o}_{in}^{\dagger}(t^{\prime})\rangle=(1+\bar{n}_{j}^{T})\delta(t-t^{\prime})$
and
$\langle\hat{o}_{in}^{\dagger}(t)\hat{o}_{in}(t^{\prime})\rangle=\bar{n}_{j}^{T}\delta(t-t^{\prime})$
($j=c,m,d$) where $\bar{n}_{j}^{T}=[{\rm exp}(\hbar\omega_{j}/k_{B}T)-1]^{-1}$
is the mean number of thermal excitations of the cavity and mechanical modes
at temperature $T$.

The linearized QLEs of motion, i.e., Eqs.(2a-2c) together with their Hermitian
conjugates can be rewritten in the following compact form

$\frac{d}{dt}\hat{\boldsymbol{u}}(t)=\boldsymbol{\chi}_{0}\hat{\boldsymbol{u}}(t)+\hat{\boldsymbol{u}}_{in}(t),$
(3)

where
$\hat{\boldsymbol{u}}(t)=(\delta\hat{a},\delta\hat{a}^{\dagger},\delta\hat{b},\delta\hat{b}^{\dagger},\delta\hat{d},\delta\hat{d}^{\dagger})^{\rm
T}$, and
$\hat{\boldsymbol{u}}_{in}(t)=(\sqrt{\kappa}\hat{a}_{in},\sqrt{\kappa}\hat{a}^{{\dagger}}_{in},\sqrt{\gamma_{m}}\hat{b}_{in},\sqrt{\gamma_{m}}\hat{b}^{{\dagger}}_{in},\sqrt{\gamma_{d}}\hat{d}_{in},\sqrt{\gamma_{d}}\hat{d}^{{\dagger}}_{in})^{\rm
T}$ are the vector of the quantum field fluctuations and the vector of quantum
noises, respectively. The drift matrix $\boldsymbol{\chi}_{0}$ can be found
easily from the equations of motion as follows

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\boldsymbol{\chi}_{0}\\!=\\!\left(\begin{matrix}{-\frac{\kappa}{2}}&{0}&{ig}&{0}&{iG}&{0}\\\
{0}&{-\frac{\kappa}{2}}&{0}&{-ig}&{0}&{-iG}\\\
{ig}&{0}&{-\frac{\gamma_{m}}{2}}&{\lambda_{m}}&{0}&{0}\\\
{0}&{-ig}&{\lambda_{m}^{\ast}}&{-\frac{\gamma_{m}}{2}}&{0}&{0}\\\
{iG}&{0}&{0}&{0}&{-\frac{\gamma_{d}}{2}}&{\lambda_{d}}\\\
{0}&{-iG}&{0}&{0}&{\lambda_{d}^{\ast}}&{-\frac{\gamma_{d}}{2}}\\\
\end{matrix}\right).$ (4)

It is obvious that one can solve Eq.(3) in the Fourier space as follows

$\displaystyle\hat{\boldsymbol{u}}(\omega)=\boldsymbol{\chi}(\omega)\hat{\boldsymbol{u}}_{in}(\omega),$
(5)

where the susceptibility matrix $\boldsymbol{\chi}(\omega)$ is generally
defined as

$\displaystyle\boldsymbol{\chi}(\omega)=\Big{(}-i\omega\boldsymbol{I}-\boldsymbol{\chi}_{0}\Big{)}^{-1},$
(6)

in which $\boldsymbol{I}$ is the $6\times 6$ idendity matrix. Now, based on
Eq.(5) the Fourier transform of the optical field can be obtained as follows

$\delta\hat{a}(\omega)=\sum_{o=a,b,d}\sqrt{\kappa_{o}}\Big{[}\chi_{ao}(\omega)\hat{o}_{in}(\omega)+\chi_{ao^{\dagger}}(\omega)\hat{o}^{\dagger}_{in}(\omega)\Big{]},\\\
$ (7)

with $\chi_{ij}(\omega)$ being the susceptibility matrix elements, the
expressions of which will be given later [Eqs.(14a-14f)].

In the following, in order to compare the present modulated optomechanical
system with the well-known DPA we write the solution to the Eq. (3) for the
the optical field noise operator and its Hermitian conjugate as

$\displaystyle-i\omega\delta\hat{a}(\omega)$ $\displaystyle=$
$\displaystyle-\Big{(}\kappa/2+i\Sigma_{a}(\omega)\Big{)}\delta\hat{a}(\omega)+\tilde{\lambda}_{a}(\omega)\delta\hat{a}^{\dagger}(\omega)+\sqrt{\kappa}\hat{A}_{in}(\omega),$
(8a) $\displaystyle-i\omega\delta\hat{a}^{\dagger}(\omega)\\!$
$\displaystyle=$
$\displaystyle-\Big{(}\kappa/2-i\Sigma_{a}^{\ast}(-\omega)\Big{)}\delta\hat{a}^{\dagger}(\omega)+\tilde{\lambda}_{a}^{\ast}(-\omega)\delta\hat{a}(\omega)+\sqrt{\kappa}\hat{A}_{in}^{\dagger}(\omega),$
(8b)

where the cavity self-energy $\Sigma_{a}(\omega)$ and the induced frequency-
dependent parametric amplification coefficient $\tilde{\lambda}_{a}(\omega)$,
which is analogous to the gain factor in the conventional OPA, are,
respectively, given by [49]

$\displaystyle\\!\\!\\!\\!\\!\\!i\Sigma_{a}(\omega)=\\!g^{2}\frac{\frac{\gamma_{m}}{2}-i\omega}{(\frac{\gamma_{m}}{2}-i\omega)^{2}-|\lambda_{m}|^{2}}\\!+\\!G^{2}\frac{\frac{\gamma_{d}}{2}-i\omega}{(\frac{\gamma_{d}}{2}-i\omega)^{2}-|\lambda_{d}|^{2}},$
(9a)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\tilde{\lambda}_{a}(\omega)=\frac{g^{2}\lambda_{m}}{(\gamma_{m}/2-i\omega)^{2}-|\lambda_{m}|^{2}}+\frac{G^{2}\lambda_{d}}{(\gamma_{d}/2-i\omega)^{2}-|\lambda_{d}|^{2}}.$
(9b)

Furthermore, the generalized cavity input-noise in the frequency space is
given by (see Ref. [49])

$\displaystyle\hat{A}_{in}=$
$\displaystyle\hat{a}_{in}+ig\sqrt{\frac{\gamma_{m}}{\kappa}}\frac{\gamma_{m}/2-i\omega}{(\gamma_{m}/2-i\omega)^{2}-|\lambda_{m}|^{2}}\left[\hat{b}_{in}+\frac{\lambda_{m}}{\gamma_{m}/2-i\omega}\hat{b}^{\dagger}_{in}\right]$
$\displaystyle\qquad+iG\sqrt{\frac{\gamma_{d}}{\kappa}}\frac{\gamma_{d}/2-i\omega}{(\gamma_{d}/2-i\omega)^{2}-|\lambda_{d}|^{2}}\left[\hat{d}_{in}+\frac{\lambda_{s}}{\gamma_{d}/2-i\omega}\hat{d}^{\dagger}_{in}\right].$
(10)

It should be noted that the susceptibility matrix of the system,
$\boldsymbol{\chi}$, can be found by rewriting the self-energies as [40]
$\boldsymbol{\Sigma}[\omega]=i(\boldsymbol{\chi}_{0}^{-1}[\omega]-\boldsymbol{\chi}^{-1}[\omega])$
with $\boldsymbol{\chi}_{0}^{-1}[\omega]=(\kappa/2-i\omega)\boldsymbol{I}$. As
is seen, the cavity self-energy satisfies the relation
$\Sigma_{a}^{\ast}(-\omega)=-\Sigma_{a}(\omega)$. As is evident, by turning on
the modulations, i.e., with $\lambda_{m,d}\neq 0$, the two mechanical modes do
indeed mediate a parametric amplifier-like effective squeezing interaction for
the optical mode which is also frequency-dependent (see Eq. (9b)), unlike the
case of the conventional DPA. Recently, it has been shown [40] that the
mechanical parametric drive can lead to a strong and robust optomechanical
squeezing and phase-sensitive amplification which can provide a situation for
an ultraprecision single-quadrature force sensing [12]. Moreover, as is clear
from Eq. (9b), for $\lambda_{m(d)}^{\ast}=\lambda_{m(d)}$ (real parametric
amplification (paramp)) one has
$\tilde{\lambda}_{a}^{\ast}(-\omega)=\tilde{\lambda}_{a}(\omega)$.

By solving the coupled Eqs. (8a) and (8b), one can find the intracavity
fluctuation operator $\delta\hat{a}(\omega)$ in the following form

$\displaystyle\delta\hat{a}(\omega)=\sqrt{\kappa}\chi_{aa}(\omega)\left[\hat{A}_{in}(\omega)+\mathcal{M}(\omega)\hat{A}_{in}^{\dagger}(\omega)\right],$
(11)

where

$\displaystyle\mathcal{M}(\omega)=\frac{\tilde{\lambda}_{a}(\omega)}{\kappa/2-i\Big{(}\omega+\Sigma_{a}^{\ast}(-\omega)\Big{)}},$
(12)

and the matrix element $\chi_{aa}(\omega)$ is given by Eq.(14a) in which the
modified cavity self-energy is as follows

$\displaystyle\tilde{\Sigma}_{a}(\omega)=\Sigma_{a}(\omega)-\frac{\tilde{\lambda}_{a}(\omega)\tilde{\lambda}_{a}^{\ast}(-\omega)}{i\kappa/2+\Big{(}\omega+\Sigma_{a}^{\ast}(-\omega)\Big{)}}\quad.$
(13)

It is straightforward to show that for $\lambda_{m(d)}^{\ast}=\lambda_{m(d)}$,
we have $\tilde{\Sigma}_{a}^{\ast}(-\omega)=-\tilde{\Sigma}_{a}(\omega)$ and
$\mathcal{M}^{\ast}(-\omega)=\mathcal{M}(\omega)$.

In the following, all the matrix elements of the susceptibility matrix
$\boldsymbol{\chi}(\omega)$, which are derived from the solution to Eq. (3) in
the frequency space, have been presented as

$\displaystyle\chi_{aa}(\omega)=\frac{i}{i\kappa/2+\Big{(}\omega-\tilde{\Sigma}_{a}(\omega)\Big{)}}\quad,$
(14a)
$\displaystyle\chi_{aa^{\dagger}}(\omega)=\chi_{aa}(\omega)\mathcal{M}(\omega),$
(14b)
$\displaystyle\chi_{ab}(\omega)=\chi_{aa}(\omega)\frac{ig\left[(\frac{\gamma_{m}}{2}-i\omega)-\lambda_{m}^{\ast}\mathcal{M}(\omega)\right]}{(\frac{\gamma_{m}}{2}-i\omega)^{2}-|\lambda_{m}|^{2}},$
(14c)
$\displaystyle\chi_{ab^{\dagger}}(\omega)=\chi_{aa}(\omega)\frac{ig\left[\lambda_{m}-(\frac{\gamma_{m}}{2}-i\omega)\mathcal{M}(\omega)\right]}{(\frac{\gamma_{m}}{2}-i\omega)^{2}-|\lambda_{m}|^{2}},$
(14d)
$\displaystyle\chi_{ad}(\omega)=\chi_{aa}(\omega)\frac{iG\left[(\frac{\gamma_{d}}{2}-i\omega)-\lambda_{d}^{\ast}\mathcal{M}(\omega)\right]}{(\frac{\gamma_{d}}{2}-i\omega)^{2}-|\lambda_{d}|^{2}},$
(14e)
$\displaystyle\chi_{ad^{\dagger}}(\omega)=\chi_{aa}(\omega)\frac{iG\left[\lambda_{d}-(\frac{\gamma_{d}}{2}-i\omega)\mathcal{M}(\omega)\right]}{(\frac{\gamma_{d}}{2}-i\omega)^{2}-|\lambda_{d}|^{2}}.$
(14f)

It can be easily shown that for real paramps,
$\lambda_{m(d)}=\lambda^{\ast}_{m(d)}$, one has
$\chi_{aa}^{\ast}(-\omega)=\chi_{aa}(\omega)$.

##  3 Cavity retarded Green’s function: negative cavity photon spectral
function

In this section we proceed to calculate the cavity retarded Green’s functions
(CRGFs) which are defined as [63]

$\displaystyle
G_{aa}^{R}(t)=-i\theta(t)\langle[\delta\hat{a}(t),\delta\hat{a}(0)]\rangle,$
(15a) $\displaystyle
G_{aa^{\dagger}}^{R}(t)=-i\theta(t)\langle[\delta\hat{a}(t),\delta\hat{a}^{\dagger}(0)]\rangle,$
(15b)

where $\theta(t)$ it the Heaviside step function. It should be noted that the
time evolution of the quantum field fluctuations in the Green’s function
definitions of Eqs.(15a-15b) are obtained from the QLEs given by Eqs. (2a-2c)
and the expectation values have been defined in the steady state of the
system.

In the following we will calculate the CRGFs using their equations of motion
which are obtained through the QLEs. To drive the equation of motion for
$G_{aa^{\dagger}}^{R}(t)$ ($G_{aa}^{R}(t)$), one should multiply each of the
equations in Eq. (3) by $\hat{a}^{\dagger}(0)$ $\big{(}\hat{a}(0)\big{)}$ on
the left and on the right, subtract them from each other and then taking their
mean values. In this way, the Green’s functions equations of motion can be
obtained as the following compact forms

$\displaystyle\frac{d}{dt}\boldsymbol{G}^{R}_{a^{\dagger}}(t)=-i\delta(t)\boldsymbol{V}_{a^{\dagger}}+\boldsymbol{\chi}_{0}\boldsymbol{G}^{R}_{a^{\dagger}}(t),$
(16a)
$\displaystyle\frac{d}{dt}\boldsymbol{G}^{R}_{a}(t)=+i\delta(t)\boldsymbol{V}_{a}+\boldsymbol{\chi}_{0}\boldsymbol{G}^{R}_{a}(t),$
(16b)

in which the six-dimensional Green’s functions vectors are defined as follows

$\displaystyle\boldsymbol{G}^{R}_{a^{\dagger}}(t)=\Big{(}G^{R}_{aa^{\dagger}}(t),G^{R}_{a^{\dagger}a^{\dagger}}(t),G^{R}_{ba^{\dagger}}(t),G^{R}_{b^{\dagger}a^{\dagger}}(t),G^{R}_{da^{\dagger}}(t),G^{R}_{d^{\dagger}a^{\dagger}}(t)\Big{)}^{\rm
T},$
$\displaystyle\boldsymbol{G}^{R}_{a}(t)=\Big{(}G^{R}_{aa}(t),G^{R}_{a^{\dagger}a}(t),G^{R}_{ba}(t),G^{R}_{b^{\dagger}a}(t),G^{R}_{da}(t),G^{R}_{d^{\dagger}a}(t)\Big{)}^{\rm
T}.$

Besides, $\boldsymbol{V}_{a^{\dagger}}:=(1,0,0,0,0,0)^{\rm T}$ and
$\boldsymbol{V}_{a}:=(0,1,0,0,0,0)^{\rm T}$ are fixed six-dimensional vectors
in Eqs.(16a,16b).

Now, by taking the Fourier transforms of Eqs.(16a) and (16b) one can find the
Green’s function vectors in the Fourier space as

$\displaystyle\boldsymbol{G}^{R}_{a^{\dagger}}(\omega)=-i\boldsymbol{\chi}(\omega)\boldsymbol{V}_{a^{\dagger}},$
(17a)
$\displaystyle\boldsymbol{G}^{R}_{a}(\omega)=+i\boldsymbol{\chi}(\omega)\boldsymbol{V}_{a},,$
(17b)

where $\boldsymbol{\chi(\omega)}$ is the susceptibility matrix defined by
Eq.(6). As is seen from Eqs.(17a) and (17b) the CRGFs in the frequency space,
i.e., the Fourier transform of Eqs.(15a) and (15b) can be obtained as

$\displaystyle
G^{R}_{aa^{\dagger}}(\omega)=-i\chi_{aa}(\omega)=\frac{1}{i\kappa/2+\Big{(}\omega-\tilde{\Sigma}_{a}(\omega)\Big{)}},$
(18) $\displaystyle
G^{R}_{aa}(\omega)=-i\chi_{aa^{\dagger}}(\omega)=G^{R}_{aa^{\dagger}}(\omega)\mathcal{M}(\omega).$
(19)

Note that for $\lambda_{m(d)}^{\ast}=\lambda_{m(d)}$, it is easy to show that
$G^{R\ast}_{aa^{\dagger}}(-\omega)=-G^{R}_{aa^{\dagger}}(\omega)$ and
$G^{R\ast}_{aa}(-\omega)=-G^{R}_{aa}(\omega)$.

The CPSF plays the role of an effective density of photon states and is
defined as [35, 63]

$\displaystyle\mathcal{A}(\omega):=-2{\rm Im}G^{R}_{aa^{\dagger}}(\omega).$
(20)

We will show in the following that in the present parametrically driven OMS
the CPSF, $\mathcal{A}(\omega)$, can be negative by controlling the modulation
parameters and cooperativities due to the parametric modulations of the
phononic mechanical modes. The on-resonance CPSF can be easily calculated as
follows

$\displaystyle\mathcal{A}(0)=\frac{4}{\kappa}\frac{(\xi_{m}^{2}-1)(\xi_{d}^{2}-1)-\mathcal{C}_{0}(\xi_{d}^{2}-1)-\mathcal{C}_{1}(\xi_{m}^{2}-1)}{(\xi_{m}^{2}-1)(\xi_{d}^{2}-1)-2\mathcal{C}_{0}\mathcal{C}_{1}(\xi_{m}\xi_{d}-1)-\mathcal{C}_{0}(\mathcal{C}_{0}+2)(\xi_{d}^{2}-1)-\mathcal{C}_{1}(\mathcal{C}_{1}+2)(\xi_{m}^{2}-1)}$
$\displaystyle\qquad\qquad:=\frac{4}{\kappa}\frac{A_{N}}{A_{D}}:=\frac{4}{\kappa}M^{\prime}=\frac{1}{\kappa}M,$
(21)

where $\mathcal{C}_{0}=4g^{2}/\kappa\gamma_{m}$ and
$\mathcal{C}_{1}=4G^{2}/\kappa\gamma_{d}$ are the optomechanical
cooperativities associated with the two mechanical modes $\hat{b}$ and
$\hat{d}$, and $\xi_{d(m)}=2\lambda_{d(m)}/\gamma_{d(m)}$ plays the role of an
effective dimensionless amplitude of modulation. As is clearly seen, the on-
resonance cavity photon spectral function ($\mathcal{A}(0)$) can be controlled
by the effective amplitude of modulations $\xi_{m,d}$ as well as the
optomechanical cooperativities $\mathcal{C}_{0}$ and $\mathcal{C}_{1}$. It can
be easily shown that the denominator $A_{D}$ in Eq. (21) is always positive
while the numerator $A_{N}$ can be negative in the stable regime. As has been
shown in Ref. [49], based on the Routh-Hurwitz criterion for the
optomechanical stability condition, the modulation parameters $\lambda_{m}$
and $\lambda_{d}$ should satisfy the condition

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\lambda_{m(d)}\leq\gamma_{m(d)}\\!+\\!\Gamma_{op}^{m(d)}=\\!\frac{\gamma_{m(d)}}{2}\left[1+\mathcal{C}_{m(d)}\right],$
(22)

where $\Gamma_{op}^{m(d)}=-2{\rm Im}\Sigma_{b(d)}(0)$ is the maximum
optomechanically induced damping rate in which the self-energies $\Sigma_{b}$
and $\Sigma_{d}$ are, respectively, given by Eqs. (26b) and (26c) of Ref.
[49]. Furthermore, $\mathcal{C}_{m}$($\mathcal{C}_{d}$) is the collective
optomechanical cooperativity which is given by

$\displaystyle\mathcal{C}_{m(d)}=\mathcal{C}_{0(1)}\frac{1+\mathcal{C}_{1(0)}-\xi_{d(m)}^{2}}{(1+\mathcal{C}_{1(0)}-\xi_{d(m)}^{2})^{2}-\xi_{d(m)}^{2}\mathcal{C}_{1(0)}^{2}}\leavevmode\nobreak\
.$ (23)

The maximum values of modulation amplitudes are obtained self-consistently
from Eq. (22), as
$\lambda^{max}_{m(d)}=\frac{\gamma_{m(d)}}{2}\left[1+\mathcal{C}_{m(d)}\right]$.
It should be emphasized that so far as the modulation amplitudes are less than
their maximum values ($\xi_{m(d)}^{max}=2\lambda^{max}_{m(d)}/\gamma_{m(d)}$)
the system is stable and the approximations we have made are valid because the
system is in the weakly interacting regime.

To find the negativity condition or find the optimized paramps $\xi_{m}^{\rm
opt}$ and $\xi_{d}^{\rm opt}$, for example in on-resonance frequency, one
should simultaneously solve inequality $4A_{N}/A_{D}=M<0$ together with the
stability condition of Eq. (22); $\xi_{m(d)}\leq 1+\mathcal{C}_{m(d)}$. In
other words, to find the paramps corresponding to the negative CPSF, one
should require both inequalities

$\displaystyle
4\frac{A_{N}(\xi_{m},\xi_{d})}{A_{D}(\xi_{m},\xi_{d})}=M<0,\quad\&\qquad\xi_{m(d)}\leq
1+\mathcal{C}_{m(d)}.$ (24)

Note that $M$ ($M<M_{\rm max}$ where $M_{\rm max}$ is the maximum achievable
negativity) is an arbitrary negative value which should satisfy the negativity
conditions. Furthermore, the maximum negativity $M_{\rm max}$ should also
satisfy the negativity conditions (24).

Let us consider a special case where just one of the mechanical modes is
parametrically driven while the other one remains unmodulated, i.e.,
$\xi_{d}=0$. In this case, the on-resonance CPSF reads as

$\displaystyle\mathcal{A}(0)|_{\xi_{d}=0}=\frac{4}{\kappa}\frac{1}{1+\mathcal{C}_{1}}\frac{\xi_{m}^{max}-\xi_{m}^{2}}{(\xi_{m}^{max})^{2}-\xi_{m}^{2}},$
(25)

where $\xi_{m}^{max}=1+\mathcal{C}_{0}/(1+\mathcal{C}_{1})$. As is evident
from Eq. (21) and (25), the denominators are always positive while the
numerators can take negative values. For example, in Eq. (25) if the
modulation amplitude lies in the interval
$\sqrt{\xi_{m}^{max}}\leq\xi_{m}\leq\xi_{m}^{max}$ where the strong-coupling
regime is avoided, the system is stable while the CPSF becomes negative near
the resonance frequency. Note that in the absence of the parametric
modulation, the cavity photon spectral function is given by

$\displaystyle\mathcal{A}(0)|_{\xi_{m,d}=0}=\frac{4}{\kappa}\frac{1}{1+\mathcal{C}_{0}},$
(26)

which never becomes negative.

Figure 2: (Color online) Behavior of the CPSF, $\kappa\mathcal{A}[\omega]$, vs
normalized frequency $\omega/\kappa$ for the case of standard OMS with a
single MO ($\mathcal{C}_{1}=\lambda_{d}=\gamma_{d}=0$) which is parametrically
driven via the coherent-time modulation of the spring coefficient. (a) For the
fixed cooperativity $\mathcal{C}_{0}=2$ and different values of relative
amplitude paramp $\xi_{m}=0,\sqrt{\xi_{m}^{\rm max}},0.9\xi_{m}^{\rm max}$,
respectively, corresponding to the green dot-dashed, red solid and blue dashed
lines. (b) for the fixed relative amplitude paramp $\xi_{m}/\xi_{m}^{\rm
max}=0.9$ and different values of cooperativity $\mathcal{C}_{0}=0.5,2,6,8$
corresponding, respectively, to the black-dotted, green-dot-dashed, blue-
dashed and red-solid lines. Here, we have set $\kappa/\gamma_{m}=10^{4}$. Note
that in the case of a single driven MO we have $\xi_{m}^{\rm
max}=1+\mathcal{C}_{0}$.

Figure (2) shows the effect of modulation on the CPSF in the case of an OMS
with a single MO, when $\mathcal{C}_{1}=\lambda_{d}=\gamma_{d}=0$. In Fig.
2(a), we have fixed the cooperativity and plotted the CPSF for different
values of the relative amplitude $\xi_{m}/\xi_{m}^{\rm max}$. As is evident,
in the absence of the modulation (green dot-dashed curve), $\xi_{m}=0$, the
CPSF remains positive at all frequencies. But, by turning on the parametric
drive the CPSF can become negative at frequencies near the resonance frequency
(blue dashed line). This negativity can be controlled by adjusting the
amplitude of modulation such that for $\xi_{m}\geq\sqrt{\xi_{m}^{\rm max}}$
the CPSF becomes negative, $\kappa\mathcal{A}(\omega)\leq 0$, and the amount
of the negativity increases when the strength of the paramp increases until
both conditions (24) are satisfied in the stable regime. To examine the effect
of cooperativity on the behavior of the CPSF, we have plotted in Fig. 2(b) the
CPSF for different values of cooperativity and a fixed value of the relative
amplitude $\xi_{m}/\xi_{m}^{\rm max}$. As is seen, the larger the
cooperativity $\mathcal{C}_{0}$, the more negative the CPSF occurs over a
broader range of frequencies. Moreover, as the cooperativity increases the two
symmetric sidebands in the CPSF, arising from the normal mode splitting,
become more prominent. The peaks and widths of the sidebands can be controlled
by the cooperativity $\mathcal{C}_{0}$ as well as the modulation amplitude.
Another feature illustrated in Fig. 2(b) is that, with increasing
$\mathcal{C}_{0}$, the maximum negative value of the CPSF occurs at off-
resonance frequencies (see blue-dashed and red-solid lines in Fig. 2(b)).

Figure 3: (Color online) The effects of both modulation amplitudes on the CPSF
for an OMS with two parametrically driven mechanics. Here, we have set
$\mathcal{C}_{0}=2$. (a) CPSF for the fixed value of $M=-3$, and for different
values of the modulation amplitudes ($\xi_{m},\xi_{d}$) and the cooperativity
$\mathcal{C}_{1}$. The black double-dot-dashed, blue-solid, and the red-dashed
lines are corresponding, respectively, to ($\mathcal{C}_{1}=0$,
$\xi_{m}=0.786\xi_{m}^{\rm max}=2.358$, $\xi_{d}=0$); ($\mathcal{C}_{1}=0.25$,
$\xi_{m}=\xi_{m}^{\rm op}=0.15\xi_{m}^{\rm max}=1.390$, $\xi_{d}=\xi_{d}^{\rm
op}=0.971\xi_{d}^{\rm max}=0.931$); and ($\mathcal{C}_{1}=0.5$,
$\xi_{m}=\xi_{m}^{\rm op}=0.25\xi_{m}^{\rm max}=1.353$, $\xi_{d}=\xi_{d}^{\rm
op}=0.975\xi_{d}^{\rm max}=0.879$). Also, the green dot=dashed curve shows the
non-optimized case referred to ($\mathcal{C}_{1}=0.25$,
$\xi_{m}=0.98\xi_{m}^{\rm op}$, $\xi_{d}=098\xi_{d}^{\rm op}$). (b) CPSF for
different values of negativity $M=-3$ ($\xi_{m}=\xi_{m}^{\rm op}=1.390$,
$\xi_{d}=\xi_{d}^{\rm op}=0.931$); $M=-10$ ($\xi_{m}=\xi_{m}^{\rm op}=1.393$,
$\xi_{d}=\xi_{d}^{\rm op}=0.935$); $M=-15$ ($\xi_{m}=\xi_{m}^{\rm op}=1.394$,
$\xi_{d}=\xi_{d}^{\rm op}=0.936$) and the fixed value of cooperativity
$\mathcal{C}_{1}=0.25$, corresponding, respectively, to the black dot-dashed,
blue-solid and the red-dashed lines. Here, in all curves we have set
$\kappa/\gamma_{m}=10^{4}$ and $\gamma_{m}/\gamma_{d}=1$. Note that the
optimized paramps are obtained by numerically solving Eqs. (24) which
determine the optimized negativity conditions.

Figure (3) shows the effect of modulation when there are two parametrically
driven mechanics in the system under consideration. To examine the effects of
the amplitude of the modulation and the cooperativity on the CPSF, we have
fixed the negativity at $M=-3$ in Fig. 3(a), and then, we have obtained the
optimum paramps ($\xi_{m(d)}$) to achieve this negativity through solving Eqs.
(24). As is evident, when both paramps are turned on and fixed at their
optimized values [see blue-solid and red-dashed curves], the same negativity
with smaller paramps is achievable with respect to the case in which only one
of the MOs is driven [see black double-dot-dashed curve]. This can be
considered as an advantageous feature of two parametrically driven mechanics
with respect to one parametrically driven MO, as far as the negativity of the
CPSF is concerned. In other words, one can conclude that by increasing the
number of parametrically driven modes, as the system degrees of freedom, the
CPSF negativity can be amplified. This is in agreement with the results
obtained in Refs. [12, 40] , based on the theory of linear quantum amplifiers
[76], for enhancing the functionality of a linear amplifier via introducing
additional controllable degrees of freedom to the system. Moreover, at the
fixed negativity, with increasing the cooperativity $\mathcal{C}_{1}$ smaller
paramps are required to achieve that negativity [compare the blue-solid and
red-dashed curves in Fig. 3(a)]. Also, when both modulated phononic modes are
present, the system is very sensitive to the values of paramps needed to
realize the desired negativity [see the green dot-dashed curve in Fig. 3(a)
which has only $2\%$ mismatch in its paramps with respect to the optimized
one]. In addition, when both mechanical modes are parametrically driven the
negative values of the spectral function occur over a very narrow range of
frequencies around resonance. Figure 3(b) shows that for the fixed values of
both cooperativities, one can achieve different values of negativity in the
stable regime by slowly tuning and increasing both paramps. This implies the
advantage of the present parametrically driven OMS with two phononic mode with
respect to an OMS with a single driven MO.

Figure 4: (Color online) The effect of damping rate ratio
$\gamma_{m}/\gamma_{d}$ on the CPSF in the case of an OMS with two
parametrically driven mechanics. The green-solid, red-dashed, and blue-dotted
curves are, respectively, corresponding to $\gamma_{m}/\gamma_{d}=0.2,1,5$.
All curves have plotted for the optimized paramps $\xi_{m}=1.390$,
$\xi_{d}=0.931$ corresponding to the negativity $M=-3$ through solving Eq.
(24) for the fixed cooperativities $\mathcal{C}_{0}=2$ and
$\mathcal{C}_{1}=0.25$. Here, we have set $\kappa/\gamma_{m}=10^{4}$.

Let us now explore the effect of the mechanical damping rates on the CPSF. For
this purpose, we have illustrated in Fig. (4) the CPSF against the normalized
frequency, $\omega/\kappa$, for different values of the ratio
$\gamma_{m}/\gamma_{d}$. As is seen, for the fixed cooperativities and the
optimized paramps, $\xi_{m(d)}^{\rm opt}$, obtained through Eqs. (24)
corresponding to an arbitrary fixed value of the negativity $M$, the
negativity bandwidth is increased around the on-resonance frequency by
decreasing the ratio $\gamma_{m}/\gamma_{d}$. In other words, one can see that
in the off-resonance frequency region, larger negativity occurs for higher
$\gamma_{m}/\gamma_{d}$. It means that in an OMS with two parametrically
driven mechanics, one can have more negativity for off-resonance frequencies
by controlling the mechanical damping rates ratio.

Here, it is worthwhile to mention that in an experiment by measuring the
output cavity spectra, one can directly reconstruct the scattering matrix
$\boldsymbol{s}_{cavity}=1-\kappa\boldsymbol{\chi}_{cavity}$ [70, 40] which
consequently yields $\mathcal{A}(\omega)=2{\rm Re}\chi_{aa}(\omega)$.

To sum up so far, it can be concluded that the OMS with two parametrically
driven MOs provides more controllability to achieve the more negative CPSF in
the stable regime with weaker paramps, as compared to the system with a single
driven MO. It is worth mentioning that the occurrence of negative CPSF has
also been predicted recently in a driven-dissipative nonlinear quantum van-der
Pol oscillator, describing a single-mode cavity with Kerr nonlinearity subject
to incoherent driving and two-photon losses [70]. By using a generalized
Lehmann representation of the Green’s function of Markovian open quantum
systems governed by a Lindblad-type master equation, it has been shown that
the interplay of interaction and non-equilibrium effects can give rise to a
negative density of states, or equivalently a negative CPSF, which can be
associated with an effective frequency-dependent negative temperature for
cavity photons, even in the absence of steady-state population inversion in
the system density matrix.

In the two following sections, firstly, we present the physical interpretation
of the negative CPSF in our system by comparing to the OPA or DPA. After that,
we shortly present some physical consequences of the negative CPSF, i.e.,
modified OMIT with probe reflectivity above the unity and negative effective
temperature for the intracavity photons.

##  4 comparison to the OPA

###  4.1 negativity conditions for OPA

We are interested in comparison between the CPSF of the present system and
that of an OPA since our system is effectively analogous to an OPA with a
time-dependent amplitude which is responsible for the frequency-dependent
squeezing coefficient in the Fourier space [49, 40, 12, 35]. For a detuned
OPA, where the pump field frequency ($\omega_{p}$) is not equal to the cavity
frequency, the Hamiltonian is given by

$\displaystyle\hat{H}^{(\rm OPA)}_{(\rm
det)}=-\Delta_{p}\hat{a}^{\dagger}\hat{a}+\frac{i}{2}(\lambda\hat{a}^{{\dagger}2}-\lambda^{\ast}\hat{a}^{2}),$
(27)

in a frame rotating at $\omega_{p}/2$ with
$\Delta_{p}=\omega_{p}/2-\omega_{c}$. By solving the equations of motion in
Fourier space we find that our system resembles an OPA with an effective
frequency-dependent parametric drive $\lambda\to\tilde{\lambda}_{a}(\omega)$
and with a non-zero cavity self-energy $\Sigma_{a}(\omega)$ [see Eqs. (8) and
(9)].

The susceptibility matrix for the detuned OPA is given by

$\displaystyle\boldsymbol{\chi}^{(\rm OPA)}_{(\rm
det)}(\omega)=\frac{1}{(\kappa/2-i\omega_{+})(\kappa/2-i\omega_{-})-|\lambda|^{2}}\left(\begin{matrix}{\kappa/2-i\omega_{-}}&{\lambda}\\\
{\lambda^{\ast}}&{\kappa/2-i\omega_{+}}\\\ \end{matrix}\right),$

with $\omega_{\pm}=\omega\pm\Delta_{p}$, which leads to the following retarded
Green’s function

$\displaystyle G^{(\rm OPA)}_{aa^{\dagger}}(\omega)=-i\chi^{(\rm
OPA)}_{aa}(\omega)=-i\frac{[(\kappa/2-i(\omega-\Delta_{p})][\kappa^{2}/4-|\lambda|^{2}-(\omega^{2}-\Delta_{p}^{2})+i\omega\kappa]}{[\kappa^{2}/4-|\lambda|^{2}-(\omega^{2}-\Delta_{p}^{2})]^{2}+\omega^{2}\kappa^{2}}.$
(29)

Therefore, the CPSF for the detuned OPA can be found as follows

$\displaystyle\mathcal{A}^{(\rm OPA)}_{(\rm det)}(\omega)=-2{\rm Im}G^{(\rm
OPA)}_{aa^{\dagger}}(\omega)=+\kappa\frac{[\kappa^{2}/4-|\lambda|^{2}-(\omega^{2}-\Delta_{p}^{2})]+2\omega(\omega-\Delta_{p})}{[\kappa^{2}/4-|\lambda|^{2}-(\omega^{2}-\Delta_{p}^{2})]^{2}+\omega^{2}\kappa^{2}}.$
(30)

The stability condition, $\rm det\boldsymbol{\chi}^{(\rm OPA)}_{0}>0$,
requires that the poles of the susceptibility lie in the lower half-plane. For
the detuned-OPA the stability condition becomes
$|\lambda|^{2}<\kappa^{2}/4+\Delta_{p}^{2}$. In order to examine the
negativity of the CPSF for an OPA, let us rewrite Eq. (30) as follows

$\mathcal{A}^{(\rm
OPA)}_{\rm(det)}(\omega)=\kappa\frac{F(\omega)}{[\kappa^{2}/4-|\lambda|^{2}-(\omega^{2}-\Delta_{p}^{2})]^{2}+\omega^{2}\kappa^{2}},$
(31)

with $F(\omega)=\omega^{2}-2\Delta_{p}\omega+S$ and
$S=\kappa^{2}/4+\Delta_{p}^{2}-|\lambda|^{2}>0$ according to the the stability
condition. It follows from Eq. (31) that the stable negativity in CPSF occurs
if $F(\omega)<0$ for any positive frequency $\omega>0$. Evidently, for the on-
resonance condition we have $F(0)=S$ which is always positive in the stability
region and thus the spectral function never becomes negative in on-resonance
frequency ($\mathcal{A}^{(\rm OPA)}_{\rm(det)}(\omega=0)>0$).

Let us now see what happens with the off-resonance frequencies for the OPA or
detuned OPA. In this case, to find the negativity condition in off-resonance
frequencies for the CPSF, one must solve inequality $F(\omega>0)<0$ which
yields the constraint
$\kappa^{2}/4<|\lambda|^{2}<\kappa^{2}/4+\Delta_{p}^{2}$. It is clear that the
CPSF for nondetuned OPA, i.e., when $\Delta_{p}=0$, never becomes negative in
off-resonance frequencies. In other words, as the detuned OPA is stable the
CPSF can be negative in off-resonance frequencies
$\Delta_{p}-\Delta\omega\leq\omega\leq\Delta_{p}+\Delta\omega$ where
$\Delta\omega=\sqrt{\Delta_{p}^{2}-S}=\sqrt{|\lambda|^{2}-\kappa^{2}/4}$ when
$\kappa^{2}/4<|\lambda|^{2}<\kappa^{2}/4+\Delta_{p}^{2}$.

In Fig. (5), we have plotted the CPSF for the nondetuned and detuned OPA
(Eq.(30)). As is evident for the case of nondetuned OPA (see the green thick-
solid and purple loosely-dashed lines), the CPSF never becomes negative and
has a positive peak at the resonance frequency which decreases with increasing
the effective paramp ($\xi_{k}=2\lambda/\kappa$) to its maximum value
$\xi_{k}=1$. Note that the case of zero detuning and absence of paramp
corresponds to an empty cavity subjected to dissipation. However, for the
detuned OPA, the CPSF can be negative in off-resonance frequencies as
expected.

For a fixed value of the effective paramp $\xi_{k}$ but opposite values of
effective detuning $\Delta_{k}$ (see red densely-dotted and blue double-dot
dashed lines in Fig. (5)) the height of the peaks in CPSF is the same, while
their locations are dependent on the sign of the detuning $\Delta_{k}$.
Moreover, the spectral function exhibits negativity over a range of positive
(negative) frequencies for positive (negative) detuning $\Delta_{k}$. On the
other hand, for a fixed value of the effective detuning $\Delta_{k}$ the
spectral function negativity is enhanced with increasing the paramp $\xi_{k}$
(see red densely-dotted and black densely-dashed lines in Fig. (5)). It is
also seen from Fig. (5) (see orange-solid and blue double-dot dashed lines)
that for a fixed value of the effective paramp $\xi_{k}$ a larger value of
detuning $\Delta_{k}$ leads to smaller negativity of the spectral function.

Here, it is worth remarking that unlike the detuned OPA, the parametrically
driven OMS considered in this paper can exhibit stable negative spectral
function at on-resonance frequency ($\omega=0$). This is because by
controlling the system parameters, in particular the effective induced paramp,
the spectral function negativity can be achieved even at $\omega=0$ due to the
negative ECDR, which we will explain in the next subsection.

Figure 5: (Color online) CPSF for the OPA vs normalized frequency
$\omega/\kappa$ for different values of $\xi_{k}=\frac{\lambda}{\kappa/2}$ and
$\Delta_{k}=\Delta_{p}/\kappa$. The green thick-solid and purple loosely-
dashed lines refer to zero detuning $\Delta_{p}=0$, respectively for
$\xi_{k}=0$ and $\xi_{k}=1$. Red densely-dotted and blue double-dot dashed
lines refer to the same paramp $\xi_{k}=1.2$, respectively, for
$\Delta_{k}=0.5$ and $\Delta_{k}=-0.5$. Black densely-dashed and orange-solid
lines refer to ($\xi_{k}=1.25$, $\Delta_{k}=0.5$) and ($\xi_{k}=1.2$,
$\Delta_{k}=-0.6$), respectively.

###  4.2 physical interpretation of the negative CPSF

In what follows, we are going to illustrate the physics behind the spectral
function negativity in the stable regime in OPA and in our parametrically-
driven OMS, and thus, compare them together.

As was shown in the previous subsection, the stable negativity in the cavity
spectral function of an OPA occurs only in the detuned regime ($\Delta_{p}\neq
0$) due to the extended stability region. However, we show that the negativity
of the CPSF in our system originates from the negativity of the frequency
dependent ECDR defined as

$\displaystyle\kappa_{\rm eff}(\omega):=\kappa-2{\rm
Im}\tilde{\Sigma}_{a}(\omega)=\kappa+\kappa_{\rm opt}(\omega).$ (32)

We first determine the ECDR for a detuned OPA whose self-energy matrix,
$\boldsymbol{\Sigma}_{(\rm det)}^{(\rm
OPA)}(\omega)=i(\boldsymbol{\chi}_{0}^{-1}-(\boldsymbol{\chi}_{(\rm
det)}^{{\rm(OPA)}})^{-1})$, on using Eq. (4.1), is obtained as

$\displaystyle\boldsymbol{\Sigma}_{\rm(det)}^{\rm(OPA)}(\omega)=i\left(\begin{matrix}{0}&{\lambda}\\\
{\lambda^{\ast}}&{0}\\\ \end{matrix}\right),$ (33)

which is clearly frequency-independent and purely off-diagonal. Therefore,
$(\kappa_{\rm eff})^{(\rm OPA)}_{(\rm det)}=\kappa>0$ indicating that the EDCR
for a detuned OPA is always positive.

Despite of the OPA, we will show that in our system the ECDR can be negative,
$\kappa_{\rm eff}(\omega)<0$, which reveals the physics behind this negativity
in comparison to detuned OPA. In fact, the negative $\kappa_{\rm eff}(\omega)$
in our system is because of its frequency dependency through frequency
dependency of the cavity self-energy ${\rm Im}\Sigma_{a}(\omega)$ due to
$\tilde{\lambda}_{a}(\omega)$ which originates from the coherent time-
modulations of the mechanical spring coefficients of the MOs.

Now, we return to our system and calculate its ECDR. For this purpose, let us
first make a comparison between the paramps for an ordinary OPA and the
optomechanical system under consideration. In an ordinary OPA when its
parametric drive strength satisfies the relation
$(\lambda)^{\ast}=\lambda^{\ast}$, i.e., the real paramp, the interaction is
called a coherent interaction [35, 49]. Generally in our parametrically-driven
system, a frequency-dependent effective parametric drive like
$\tilde{\lambda}_{a}(\omega)$ satisfies the relation
$(\tilde{\lambda}_{a}(\omega))^{\ast}=\tilde{\lambda}_{a}^{\ast}(-\omega)$. If
$\tilde{\lambda}_{a}(\omega)$ satisfies the relation
$(\tilde{\lambda}_{a}(\omega))^{\ast}=\tilde{\lambda}_{a}(\omega)$ the system
behaves like an ordinary OPA in a coherent regime. In this situation,
$\tilde{\lambda}_{a}(\omega)$ can be considered as an effective coherent
interaction strength whose real and imaginary parts might lead to the coherent
and dissipative behaviors. In this way, two different regimes can be
distinguished depending on which part is larger than the other. In the so-
called coherent regime where [49]

$\displaystyle\lvert\frac{{\rm Im}\tilde{\lambda}_{a}(\omega)}{{\rm
Re}\tilde{\lambda}_{a}(\omega)}|\ll 1$ (34)

is satisfied [for example, it can be satisfied in the largely different
cooperativities regime with nonzero modulation such that $\xi_{m(d)}>1$] [49],
the coherent term is dominant otherwise the dissipative term is dominant.

Because of the frequency dependence of $\tilde{\lambda}_{a}(\omega)$ and
$\Sigma_{a}(\omega)$, we can not directly map our system to an OPA. However,
in the on-resonance case, the effective susceptibility matrix becomes exactly
similar to an OPA but with the effective parametric strength
$\tilde{\lambda}_{a}(0)$ instead of $\lambda$ and the effective cavity damping
rate $\kappa_{\rm eff}(0)=\kappa-2{\rm Im}\tilde{\Sigma}_{a}(0)$ (with
$\kappa_{\rm opt}=-2{\rm Im}\tilde{\Sigma}_{a}(0)$) instead of $\kappa$. The
imaginary parts of the cavity self-energy and modified cavity self energy in
on-resonance are, respectively, given by

$\displaystyle{\rm
Im\Sigma}_{a}(0)=-\Big{[}g^{2}\frac{\gamma_{m}/2}{\frac{\gamma_{m}^{2}}{4}-|\lambda_{m}|^{2}}+\textit{G}^{2}\frac{\gamma_{d}/2}{\frac{\gamma_{d}^{2}}{4}-|\lambda_{d}|^{2}}\Big{]}=-\frac{\kappa}{2}\left[\frac{\mathcal{C}_{0}}{1-\xi_{m}^{2}}+\frac{\mathcal{C}_{1}}{1-\xi_{d}^{2}}\right]:=-\frac{\kappa}{2}\mathcal{C}_{a}$
(35) $\displaystyle{\rm Im\tilde{\Sigma}}_{a}(0)={\rm
Im\Sigma}_{a}(0)-\frac{[\tilde{\lambda}_{a}(0)]^{2}}{i\kappa/2-\Sigma_{a}(0)}=-\frac{\kappa}{2}\Big{[}\mathcal{C}_{a}-\frac{\mathcal{C}^{\prime
2}_{a}}{1+\mathcal{C}_{a}}\Big{]}:=-\frac{\kappa}{2}\mathcal{C}^{\prime\prime}_{a},$
(36)

where we have defined

$\displaystyle\tilde{\lambda}_{a}(0)=\frac{\kappa}{2}\Big{[}\frac{2\mathcal{C}_{0}\xi_{m}}{1-\xi_{m}^{2}}+\frac{2\mathcal{C}_{1}\xi_{d}}{1-\xi_{d}^{2}}\Big{]}:=\frac{\kappa}{2}\mathcal{C}^{\prime}_{a}.$
(37)

Note that the optomechanical cooperativities $\mathcal{C}^{\prime}_{a}$,
$\mathcal{C}^{\prime\prime}_{a}$, $\mathcal{C}_{a}$ are real parameters. In
the stable regime and when the condition (34) is satisfied the parameters
$\mathcal{C}_{a}$, $\mathcal{C}^{\prime}_{a}$ and
$\mathcal{C}^{\prime\prime}_{a}$ become negative (for $\xi_{m(d)}>1$ we have
$-0.5<\mathcal{C}_{a}<0$ and $-2<\mathcal{C}^{\prime}_{a}<-1$) which in turn
lead to a negative optomechanically-induced cavity damping rate, $\kappa_{\rm
opt}=-2{\rm Im}\tilde{\Sigma}_{a}(0)<0$. Consequently, the effective cavity
damping rate $\kappa_{\rm eff}=\kappa+\kappa_{\rm
opt}(\omega)=\kappa(1+\mathcal{C}^{\prime\prime}_{a})$ decreases, and can even
become negative by controlling the system parameters. As Eq. (36) shows, the
parameter $\mathcal{C}_{a}$ plays the key role for the occurrence of this
negativity. As a numerical example, for the parameters given in Fig. (3)
corresponding to $M=-3$, we obtain $\kappa_{\rm eff}/\kappa\simeq-1.3$. Thus,
in this analogy between our system and OPA in the on-resonance frequency the
negative CPSF ($\mathcal{A}<0$) is responsible for the negative ECDR
($\kappa_{\rm eff}<0$). Therefore, the on-resonance CPSF ($\mathcal{A}(0)$)
for our system can be found by directly substituting $\kappa\to\kappa_{\rm
eff}(0)$ and $\lambda\to\tilde{\lambda}_{a}(0)$ into $A^{(\rm
OPA)}_{\rm(det)}(0)|_{\Delta_{p}=0}$. Note that this analogy cannot be seen as
a coincidence, it only shows that the CPSF in our system looks exactly like
that of a non-equilibrium OPA which can provide the link between the negative
CPSF and associated negative photon temperature in our system.

##  5 concluding remarks, discussion and outlooks

###  5.1 summary and conclusion

In summary, as a novel feature of the OMSs we have theoretically proposed and
investigated an experimentally feasible scheme to generate negative CPSF in an
OMS with two parametrically driven MOs in the red-detuned and weak coupling
regimes through the coherent modulation of their spring coefficients.

Using the the equations of motion for the CRGF, we have calculated the CPSF
and have shown when the induced-frequency-dependent ECDR becomes negative, the
CPSF becomes negative while the system still remains in the stable regime
unlike the conventional OPA. This negativity can be controlled by the
optomechanical cooperativities and amplitudes of modulation of MOs. In
addition, the OMS with two parametrically driven MOs provides more
controllability to achieve the more negative CPFS in the stable regime with
weaker paramps, as compared to the system with a single parametrically driven
MO.

The comparison of our parametrically-driven OMS with the conventional OPA and
detuned OPA shows that the CPSF negativity never occurs in the conventional
OPA in the stable regime while occurs in the detuned OPA in the stable regime.
However, the nature of the negativity in our system is totally different. This
negativity originates from the negative ECDR because of its frequency
dependence while the ECDR in the detuned OPA is zero and the CPSF negativity
stems from the extension of the stability condition of the susceptibility
matrix due to the presence of the cavity detuning.

###  5.2 outlooks and discussion

As an interesting outlook of the continuation of the present work, let us now
briefly discuss how the negative CPSF may lead to to some experimentally
observable signatures such as modified OMIT and NET which are, respectively,
corresponding to the reflectivity above the unity and qubit-cavity population
inversion.

Optomechanically induced suppression of the effective density of photon states
at the cavity resonance frequency, whose role is played by the CPSF
$\mathcal{A}$, is known as the standard OMIT which can be interpreted as an
effective density of single-particle states [1, 35, 63]. This means that
incident near-resonant photons ($\mathcal{A}_{\rm std}(0)\to 0$) in the weakly
coupled auxiliary mode are perfectly reflected ($\mathcal{R}\simeq 1$).
Therefore, when probe photons go into the cavity there is no available states,
and, thus, all photons perfectly reflect, i.e., a reflective narrow-band
optomechanical filter is realized. In other words, in the standard-OMIT (when
$\xi_{m,d}=0$), the simultaneous injection of a strong control driving laser
and a weak probe laser with coherency times longer than the effective
mechanical damping rate into the red-detuned sideband of an OMS provides
cavity resonance which leads to the transparency of the weak probe leaser [1,
53, 54, 55, 56, 57, 58, 63, 60]. On the other hand, the beat of the probe
field and the driving laser induces a time-dependent radiation pressure force.
If the beat frequency matches the frequency of the MO, then the MO is driven
resonantly which in turn creates sidebands on the intracavity field where by
considering the strong coupling laser in the resolved sideband regime, the
lower sideband is far off cavity resonance and can be neglected. In contrast,
the upper sideband of the driving laser, created by the mechanical motion, has
precisely the same frequency as the probe field and is moreover phase-coherent
with the probe field. This leads to an interference that yields a cancellation
of the intracavity field on resonance, giving rise to the transparency window.
The phenomenon thereby results from the destructive interference between
reflection amplitudes for photons scattered from the driving laser and photons
of the probe field [1, 63]. But, in the OMS presented here we assume that in
addition to the red-detuned input/output driving laser port, the cavity is
extremely weakly coupled to an auxiliary waveguide mode $\hat{c}$ at the small
rate $\kappa^{\prime}\ll\kappa$ with the output field [35, 63, 60]
$\hat{c}_{out}(\omega)=r_{p}(\omega)\hat{c}_{in}(\omega)-i\kappa^{\prime}G^{R}_{aa}(\omega)\hat{c}_{in}^{\dagger}(-\omega)$
with the amplitude of reflection as $r_{p}\equiv
1-\kappa^{\prime}G^{R}(\omega)=|r_{p}(\omega)|e^{i\Phi_{p}(\omega)}$. If we
consider a detuning between the frequency of the probe laser and that of the
cavity, $\delta_{p}=\omega-\omega_{c}$ in the lab frame, one can ignore the
second term proportional to $\chi_{aa^{\dagger}}$ because of
$\langle\hat{c}_{in}^{\dagger}(-\omega)\rangle\approx 0$ [35] due to the RWA.
Therefore, the reflection power of the probe mode from the cavity in the limit
of $\kappa^{\prime}\ll\kappa$ is $\mathcal{R}(\omega)\approx
1-\kappa^{\prime}\mathcal{A}(\omega)$ which also holds for $\omega=0$ if one
averages over the phase of the incident drive in the waveguide, that in this
case, the contribution of the off-diagonal susceptibility in output field
averages away. It is clear that if $\mathcal{A}(\omega)<0$, the probe signal
can be reflected with a coefficient larger than 1 ($\mathcal{R}>1$) which
never occurs in the standard-OMIT or even in the standard OPA which their CPSF
is always positive. As we have shown, the OMS with two parametrically driven
MOs offers more controllability to achieve the more negative CPSF compared to
a system with a single driven MO. Therefore, we may expect that the
parametrically driven OMS considered in this work provides the possibility of
engineering a tunable optomechanical filter with the probe reflection above
the unity. This issue will be addressed in detail in our future work. As is
clear, the zero CPSF is responsible for the standard-OMIT or probe reflection
near the unity at near resonance frequency, $\mathcal{R}(\omega\sim 0)\simeq
1$. Interestingly, in the presented model by turning on two parametric drives
through the time-modulation and by controlling the system parameters such as
cooperativities, amplitudes of modulation and also the mechanical damping
rates ratio, when CPSF becomes negative one can attain the probe reflection
above the unity at near resonance frequency, i.e., extraordinary or modified
OMIT. Let us interpret what happens in extraordinary OMIT: as the CPSF
decreases, the available states for the incident probe mode to occupy
decreases such that when $\mathcal{A}\to 0$, there are no available states for
incident photons and thus perfectly reflection occurs, $\mathcal{R}\to 1$.
While if the CPSF becomes negative, $\mathcal{A}<0$, not only there are no
available states for the incident probe photons but also there are not enough
states for the intracavity photons. It means that in addition to the incident
probe photons, some intracavity photons return and reflect yielding to
reflection above the unity, $\mathcal{R}\gtrsim 1$.

Let us now return to the CRGF in the lab frame where the negativity in CPSF
and also OMIT phenomenon occur at near resonance $\omega\approx\omega_{c}$.
Negative CPSF at the positive frequencies, i.e., $\mathcal{A}(\omega>0)<0$, is
responsible for a stationary population inversion between stationary
eigenstates separated by $\hbar\omega$ for a time-independent Hamiltonian in a
time-independent state [35, 76, 70]. Although our system as an open quantum
system with time-dependent Hamiltonian interacts with environment, but the
negative CPSF is still responsible for the population inversion. Therefore, it
is worthwhile to address the effective temperature of the intracavity photons
which can be obtained via the quantum noise properties of the system. As
discussed in Refs. [35, 76, 70], the effective temperature can be defined
using the quantum noise properties of photons field by comparing the size of
the classical symmetrized photon correlation function to the size of the
spectral function as $T_{\rm
eff}(\omega)=\frac{\omega}{2\coth^{-1}\left(-iG^{k}(\omega)/\mathcal{A}(\omega)\right)}$
where $G^{k}(\omega)=i\int d\tau
e^{i\omega\tau}\langle\\{\hat{a}(t+\tau),\hat{a}^{\dagger}(t)\\}\rangle$
stands for the so-called Keldysh Green’s function [77, 70], and
$\langle\\{\hat{a}(t+\tau),\hat{a}^{\dagger}(t)\\}\rangle$ is a classical
symmetrized photon correlation function. If the system is not in equilibrium,
$T_{\rm eff}$ depends explicitly on frequency, i.e., $T_{\rm eff}\equiv T_{\rm
eff}(\omega)$. In thermal equilibrium, effective temperature is exactly the
temperature of the system at all frequencies, i.e., $T_{\rm
eff}(\omega)=T\quad\forall\omega$ . Out of equilibrium, for positive
frequencies ($\omega>0$) as $-iG^{k}(\omega)$ is always positive definite, a
negative CPSF at positive frequency ($\mathcal{A}[\omega>0]<0$) necessarily
leads to a frequency-dependent NET for the intracavity photons, $T_{\rm
eff}[\omega>0]<0$ [35]. As discussed in Refs. [70, 76, 35] the effect of NET
can be directly probed by coupling the cavity field weakly to an auxiliary
probe qubit with a splitting frequency equal to the cavity frequency. Then,
the intracavity photons act as an effective engineered bath for the qubit. The
steady-state population inversion of the qubit depends on the effective cavity
temperature at cavity frequency as
$\langle\hat{\sigma}_{z}\rangle_{s}=-\tanh\left(\hbar\beta_{\rm
eff}(\omega_{c})\omega_{c}/2\right)$ with $\beta_{\rm eff}(\omega_{c})=k_{\rm
B}^{-1}T_{\rm eff}^{-1}(\omega)$ [76]. Thus, a NET for the intracavity photons
can directly translate into a simple population inversion of the qubit. In
light of this issue, a particular focus in our future work will be the
investigation of realizing a controllable NET for the intracavity photons in
the stable regime of a hybrid qubit-OMS with two parametrically driven
mechanical modes.

## Acknowledgements

AMF would like to thank the Office of the Vice President for Research of the
University of Isfahan and ICQT for their support. AD gratefully acknowledges
support from the Iran National Science Foundation (INSF) under Grant No.
99020597.

## Author contributions statement

The authors contribution is as follows: AMF proposed and developed the primary
idea, carried out all analytical calculations and numerical codes, and
provided the first draft of the manuscript. AD and MHN checked the results,
and participated in writing and editing the manuscript.

## Competing Interests

The authors declare no competing interests.

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