# Transparency and enhancement in fast and slow light in $q$-deformed
optomechanical system

Akash<EMAIL_ADDRESS>Institute of Theoretical and Applied
Informatics, Polish Academy of Sciences, Bałtycka 5, 44-100 Gliwice, Poland
Joint Doctoral School, Silesian University of Technology, Akademicka 2A,
44-100 Gliwice, Poland  Jarosław A. Miszczak  Institute of Theoretical and
Applied Informatics, Polish Academy of Sciences, Bałtycka 5, 44-100 Gliwice,
Poland

###### Abstract

Nonclassical phenomena can be enhanced by introducing $q$-deformation in
optomechanical systems. This motivates investigation of the optical response
in a $q$-deformed linearly coupled optomechanical system. The system consists
of two deformed cavities that are linearly coupled to the motion of mechanical
mirrors, and the cavities are coupled to each other by transmission strength
parameter. This study shows that compared to non-deformed cases, the deformed
system exhibits more rapid phase transition at the positions of transparency
windows, causing stronger and enhanced fast and slow light phenomena.
Moreover, in the region $0<q<0.5$, the optomechanical system results in gain.
Additionally, for a fixed deformation of cavities, by tuning the tunneling
strength and optomechanical coupling, one can observe a delay and advancement
in probe field in the order of milliseconds and even above milliseconds for
fine-tuning of the coupling parameters. Finally, the bridge between
mathematical and physical models is built by assuming the deformation to be a
primitive root of unity, which indicates a class of anyon models. These
results demonstrate that $q$-deformation provides a novel method for
manipulating and significantly enhancing optical phenomena not only in
arbitrarily deformed optomechanical systems but also in anyon models.

Keywords: $q$-deformation, optomechanical system, group delay, anyon

##  1 Introduction

The concept of $q$-deformed quantum system lies in the SU${}_{q}(2)$ algebraic
group and was first introduced in [1, 2] in the context of Yang-Baxter (YB)
equations which known to play a very crucial role in classical (CYB) and
quantum (QYB) mechanical systems. Particularly QYB equations has a very
important role in solving variety of theoretic physical problems including
exactly solvable models in S-matrix theory [3], 2-dimension field theories
involving fields with statistics in-between bosons and fermions [4], conformal
field theories [5, 6, 7]. In Ref. [8] while describing the comparative
structure of $q$-deformation the author quoted “… just as the Jacobi identity
is an associativity condition for a Lie algebra, so does a QYB equation play a
similar role for an algebraic structure of a new type that is a generalization
of a Lie algebra. The structure sometimes describes as a $q$-deformation of a
Lie group.” Under the limit $q\rightarrow 1$, the $q$-deformed quantum group
SU${}_{q}(2)$ converges to the Lie algebra of SU$(2)$.

Significant research effort has been devoted to various applications of
$q$-deformation in physics. In [9] the authors presented a $q$-deformed Klein-
Gordon equation with clear physical motivation to get an equation which may
show nonlinear nature for higher intensities of the field. In [10] an
improvement in squeezing of the quadrature beyond the ordinary case has been
observed using $q$-deformed parameter. In [11] a numerical investigation of
the characteristics and stabilities of the self-localized solution of
nonlinear Schrödinger equation with a $q$-deformed Rosen-Morse potential has
been conducted. Finally, very recently a $q$-deformed formalism for qubits and
qutrits logic gates has been presented in [12, 13], suggesting the potential
application of $q$-deformed systems in quantum information processing.

In quantum optomechanics usually an optomechanical system contains information
about an interacting Febry-Perot cavity [14] and mechanical oscillator. The
impact of optomechanical systems can be seen in many fields such as, cooling
of mechanical resonators [15, 16, 17, 18], entanglement between macroscopic
oscillator and the cavity field [19, 20, 21, 22], optomechanical induced
transparency (OMIT) [23, 24, 25], electromagnetic induced transparency (EIT)
[26, 27, 28]. EIT appears due to the effect of quantum interference occurring
in three-level atoms. Observation of EIT involves two optical fields (probe
and coupling). The probe field is tuned near resonance between two of the
states and measures the absorption spectrum of the transition. The coupling
field is much stronger and tuned near resonance at a different transition. If
the states are chosen properly, the presence of coupling field will create a
transparency window (TW) which will be detected by the probe. In an
optomechanical system, the EIT is comparable to OMIT and can be used to tune
fast and slow light [29, 30, 31]. In recent years, a lot of work has been
conducted to study fast and slow light effects in a class of cavity
optomechanical systems. In [32] the authors used a two atom optomechanical
system to investigate fast and slow light, a comprehensive study of
nonreciprocal transmission, fast and slow light effects is presented in [33],
and in [34] slow or fast light effect in PT-symmetric mechanical systems is
conducted. A detailed study of fast and slow light effect in active-passive
optomechanical cavity is studied in [35].

Motivated by the these developments, we investigate optical response in a
$q$-deformed linearly coupled optomechanical system. The system consists of
two deformed cavities that are linearly coupled to motion of mechanical
mirrors, and the cavities are coupled to each other by transmission strength
parameter. Based on the OMIT investigation, we study the fast and slow light
phenomena under different deformation strength ($q$), and by manipulating
various optomechanical parameters. It has been observed that under the proper
selection of optomechanical parameters, the deformation strength in the system
induces gain (in the range $0<q<0.5$) by arising a negative absorption peak.
We also note that with the help of coupling and tunnneling parameters a clear
tunability in OMIT and later strong enhancement and manipulation in super and
subluminal light in deformed system can be achieved. Further when $q$ is
primitive root of unity, the cavity photons behaves as anyons. In this setting
we extend our investigation of phase transition and fast-slow light in case of
Ising and Fibonacci anyon models. The appearance in gain and strong
enhancement in super and subluminal behaviour in the probe field is also noted
in case of anyonic models.

The presented manuscript is arranged in the following way. In section 2 we
briefly describe the preliminaries of $q$-deformation. In section 3 a brief
description of the model Hamiltonian of the system is provided. In section 4
we present the dynamical behavior of the system at steady state and under
quantum fluctuations which is followed by a derivation of transmission
amplitude. An investigation of OMIT, phase transition and fast-slow light
phenomena is studied in section 5; further compare the non-deformed to
$q$-deformed cases. In section 6 we build the bridge between the $q$-deformed
mathematical formalism and physical models by considering the deformation
strength a primitive root of unity which opens up a class of anyon models.
Finally, in section 7 we conclude our results.

##  2 Preliminaries

###  2.1 $q$-deformed Harmonic Oscillator

The algebraic application of $q$-deformation has been developed during the
period $1987-89$ but the introduction of $q$-deformed harmonic oscillator [8,
36, 37] in terms of bosonic creation and annihilation operators triggered the
possibility of application of the $q$-deformation in quantum optical systems.
For a normal harmonic oscillator the commutator relation satisfies the
following set of equalities

$\displaystyle\hat{O}\hat{O}^{\dagger}-\hat{O}^{\dagger}\hat{O}=1\;\;;\textrm{i.e.}\;\;\left[\hat{O},\hat{O}^{\dagger}\right]=1,$
(1)
$\displaystyle\hat{O}^{\dagger}\hat{O}=\hat{N}_{O}\;\;\;;\;\;\;\hat{O}\hat{O}^{\dagger}=\hat{N}_{O}+1,$

where $\hat{O}\;(\hat{O}^{\dagger})$ define bosonic annihilation (creation)
operator.

Let the $q$-deformed bosonic annihilation (creation) operators defined by
$\hat{O}_{q}(\hat{O}_{q}^{\dagger})$. Under $q$-deformation the equation (1)
is modified as follows

$\displaystyle\hat{O}_{q}\hat{O}_{q}^{\dagger}-q^{\mp
1}\hat{O}_{q}^{\dagger}\hat{O}_{q}=q^{\pm{\hat{N}_{O}}},$ (2)
$\displaystyle\hat{O}_{q}^{\dagger}\hat{O}_{q}=\left[\hat{N}_{O}\right]\;\;\;;\;\;\;\hat{O}_{q}\hat{O}_{q}^{\dagger}=\left[\hat{N}_{O}+1\right],$

where $\left[\hat{N}_{O}\right]$ is the $q$-number representation of bosonic
number operator $\hat{N}_{O}$ which can be addressed in terms of bosonic
number operator as

$\left[\hat{N}_{O}\right]=\dfrac{q^{{\hat{N}_{O}}}-q^{-{\hat{N}_{O}}}}{q-q^{-1}}.$
(3)

The $q$-number representation can be modified under the considerations when
$q$ is a real entity and when $q$ defines phase factor as

$\left[\hat{N}_{O}\right]=\begin{cases}\dfrac{\textrm{sinh}(k{\hat{N}_{O}})}{\textrm{sinh}(k)},&q\rightarrow\textrm{real},\;k=\textrm{log}_{e}q,\\\
\\\
\dfrac{\textrm{sin}(k{\hat{N}_{O}})}{\textrm{sin}(k)},&q\rightarrow\textrm{phase},\;k=-i\textrm{log}_{e}q.\end{cases}$
(4)

In the limit $q\rightarrow 1$, we have

$\text{lim}_{q\rightarrow 1}\left[\hat{N}_{O}\right]=\hat{N}_{O}.$ (5)

A brief treatment on $q$-deformed algebra can be found in [38] where the
author explicitly defined $q$-deformed Hermite [39, 40, 41] and Laguerre
polynomials [39], $q$-deformed hyperbolic functions [42], and $q$-derivatives
and integrals [43, 44].

The commutator for $q$-deformed quantum system defined by equation (2) can be
re-written as,

$\left[\hat{O}_{q},\hat{O}_{q}^{\dagger}\right]=\left[\hat{N}_{O}+1\right]-\left[\hat{N}_{O}\right]=\dfrac{q^{{\hat{N}_{O}}+1}+q^{-{\hat{N}_{O}}}}{1+q}.$
(6)

This equation will turn out to be extremely helpful for further work.
Analogous to normal harmonic oscillator Hamiltonian

$\displaystyle\hat{H}$
$\displaystyle=\dfrac{\hslash\omega}{2}\left[\hat{O}\hat{O}^{\dagger}+\hat{O}^{\dagger}\hat{O}\right],$
$\displaystyle=\dfrac{\hslash\omega}{2}\left[2\hat{N}_{O}+1\right],$ (7)

the $q$-deformed Hamiltonian can written as follows

$\displaystyle\hat{H}_{q}$
$\displaystyle=\dfrac{\hslash\omega}{2}\left[\hat{O}_{q}\hat{O}^{\dagger}_{q}+\hat{O}^{\dagger}_{q}\hat{O}_{q}\right]$
$\displaystyle=\dfrac{\hslash\omega}{2}\left([\hat{N}_{O}+1]+[\hat{N}_{O}]\right),$
(8)

which can be re-written when $q$ is real and phase factor as follows

$\hat{H}_{q}=\begin{cases}\dfrac{\hslash\omega}{2}\dfrac{\text{sinh}\left[k\left(\hat{N}_{O}+\dfrac{1}{2}\right)\right]}{\text{sinh}\left(\dfrac{k}{2}\right)},&q\rightarrow\text{real},\\\
\\\ \dfrac{\hslash\omega}{2}\dfrac{\text{
{sin}}\left[k\left(\hat{N}_{O}+\dfrac{1}{2}\right)\right]}{\text{
{sin}}\left(\dfrac{k}{2}\right)},&q\rightarrow\text{phase}{\rightarrow
e^{\textrm{i}\theta}},\end{cases}$ (9)

where $\theta$ is a real number. If $q$ is primitive root of unity then we can
express write $\theta=\dfrac{2\pi i}{m}$ for $m=2,3,\ldots$, in this case, the
$q$-deformed annihilation and creation operator represents a discretize two-
anyon radial motion frozen [45].

(a) Real deformation parameter.

(b) Complex deformation parameter. The solid lines are real and the dot-dashed
lines are the complex part of the normalized $q$ frequency.

Figure 1: The variation of the ratio of $q$-deformed and non-deformed
frequency with $q$-parameter when (a) $q$ is real and (b)
$q=\alpha+\textrm{i}\beta$.

###  2.2 Physical interpretation $q$-deformation

Many attempt were made to produce a physically sensible explanation of
$q$-deformed oscillators which resulted in several publications in last few
decades. Some authors connect $q$-oscillators with relativistic oscillation
model [46], where other introduce $q$-oscillators and SU${}_{q}(2)$ quantum
group in the frame of generalized Jaynes-Cummins model [46]. The common thing
among these attempts had that they were based purely on the mathematical
properties of $q$-deformed oscillators. However, in [47] while observing the
classical limits of quantum q-oscillators the authors reached to a conclusion
that $q$-oscillator is a normal quantum oscillator with specific non-
linearity. They also observe that this turn out to be the classical nonlinear
oscillator with the frequency depending upon amplitude by particular
dependence. The statement is justified by solving the $q$-deformed Hamiltonian
of harmonic oscillator

$\hat{H}_{q}=\hslash\omega\hat{O}^{\dagger}_{q}\hat{O}_{q},$ (10)

which is $q$-deformed version of the state-of-the-art Hamiltonian

$\hat{H}=\hslash\omega\hat{O}^{\dagger}\hat{O}.$ (11)

The solution for $q$ $\rightarrow$ real returns $q$-deformed frequency

$\omega_{q}=\omega\dfrac{k\text{cosh}\left[k\left(\hat{O}^{\dagger}\hat{O}\right)\right]}{\text{sinh}k},$
(12)

a detailed derivation is provided in Appendix A.

The conclusion one can make from the Eq. (12) is that the frequency of a
normal harmonic oscillator is a constant entity but when it is acted by a
deformation strength $q$, then the frequency of the $q$-deformed harmonic
oscillator depends on the deformation factor $k=\text{log}_{e}q$. This leads
to the interpretation that, a $q$-oscillator system carries particular non-
linearity.

In normal harmonic oscillators we characterize angular frequency $\omega$ as
the evolution in any of the orbits, but when the same harmonic oscillator
turns into a $q$-deformed version then the angular frequency must be uniquely
defined when we move from one orbit to another; which indicates that the
evolution along any orbit can now be expressed as a functional constant for
each orbit individually. As initially $q$ considered as real, hence the
frequency of a $q$-oscillator instead of its phase depends on its amplitude.
And the degree of non-linearity enhances as we increase the amplitude.

A more generalized expression of Eq.(12) is given by considering $q$ as a
complex entity i.e. $q\rightarrow\alpha+i\beta$, which leads to the following
form

$\omega_{q}=\omega\times\dfrac{q^{N+1}+q^{-N}}{1+q},$ (13)

where $N$ is the number of particles in the state-of-the-art harmonic
oscillator. In Fig.(1), we illustrate the variation of normalized $q$-deformed
frequency under arbitrary deformation strength.

##  3 Model

Let us now describe the model used in the presented paper. As illustrated in
Fig.(2), we consider an optomechanical system consisting of two deformed
cavities, where each cavity’s external mirrors are movable but the internal
mirrors are fixed. The cavity-1 represented by deformed annihilation
(creation) operator $\hat{c}_{1}^{q_{\textrm{c}1}}$
($\hat{c}_{1}^{q_{\textrm{c}1}\dagger}$) is coupled to the cavity-2
represented by deformed annihilation (creation) operator
$\hat{c}_{2}^{q_{\textrm{c}2}}$ ($\hat{c}_{2}^{q_{\textrm{c}2}\dagger}$) by a
tunnelling parameter $g$. The $q_{\textrm{c}1},\;q_{\textrm{c}2}$ are the
deformation strength of the cavity-1 and cavity-2 respectively. The first
cavity is coupled to the mechanical oscillator MO-1 with deformed annihilation
(creation) operator
$\hat{b}^{q_{\textrm{m}1}}(\hat{b}^{q_{\textrm{m}1}\dagger})$ by coupling
strength $g_{\text{om}1}$. Meanwhile the second cavity is coupled with MO-2
with $q$-deformed annihilation (creation) operator
$\hat{b}^{q_{\textrm{m}2}}(\hat{b}^{q_{\textrm{m}2}\dagger})$ by
optomehcnaical coupling parameter $g_{\text{om}2}$. Where
$q_{\textrm{m}1},\;q_{\textrm{m}2}$ are the deformation strength of the MO-1
and MO-2 respectively. The loss rate and resonance frequency of the cavity-1
are $\gamma_{1}$, $\omega_{1}$ and for the cavity-2 are $\gamma_{2}$,
$\omega_{2}$. The decay rate and the resonance frequency of MO-1 are
$\gamma_{\text{m}1}$, $\omega_{\text{m}1}$ and for MO-2 are
$\gamma_{\text{m}2}$, $\omega_{\text{m}2}$ respectively.

In addition to the parameters motioned above the cavity-1 is driven by a
coupling field of strength $\varepsilon_{c}$, frequency $\omega_{c}$ and a
weak probe field $\varepsilon_{p}$ of frequency $\omega_{p}$. The total
Hamiltonian of the cavity optomechanical system in the rotating frame at
frequency $\omega_{c}$ is given as follows:

$\hat{H}=\sum_{i=1}^{2}\Delta_{i}\hat{c}^{q_{\textrm{c}i}\dagger}_{i}\hat{c}_{i}^{q_{\textrm{c}i}}+\sum_{i=1}^{2}\omega_{\textrm{m}i}\hat{b}^{q_{\textrm{m}i}\dagger}_{i}\hat{b}_{i}^{q_{\textrm{m}i}}-\sum_{i=1}^{2}g_{\textrm{om}i}\hat{c}^{q_{\textrm{c}i}\dagger}_{i}\hat{c}_{i}^{q_{\textrm{c}i}}\left(\hat{b}^{q_{\textrm{m}i}\dagger}_{i}+\hat{b}_{i}^{q_{\textrm{m}i}}\right)+\textrm{i}\varepsilon_{c}\left(\hat{c}^{q_{\textrm{c}1}\dagger}_{1}-\hat{c}_{1}^{q_{\textrm{c}1}}\right)\\\
+\textrm{i}\varepsilon_{p}\left(\hat{c}^{q_{\textrm{c}1}\dagger}_{1}e^{-\textrm{i}\Delta
t}-\hat{c}_{1}^{q_{\textrm{c}1}}e^{\textrm{i}\Delta
t}\right)+g\left(\hat{c}^{q_{\textrm{c}2}\dagger}_{2}\hat{c}^{q_{\textrm{c}1}}_{1}+\hat{c}^{q_{\textrm{c}1}\dagger}_{1}\hat{c}^{q_{\textrm{c}2}}_{2}\right).$
(14)

Figure 2: Illustration of the system where two interacting $q$\- deformed
cavities (cavity-1 and cavity-2), with tunnelling strength $g$ are coupled to
mechanical oscillators (MO-1 and MO-2) with coupling strength
$g_{\textrm{om}i}$. $\gamma_{1}$, and $\gamma_{2}$ are the loss rate of the
cavity-1 and cavity-2 respectively.

In Eq.(14) the first two terms, denote the free energy of the two cavities,
and the movable mirrors respectively, with $\Delta_{i}=\omega_{i}-\omega_{c}$
denoting the detuning of the cavities (for $i=1,\;2$), and
$\omega_{\text{m}i}$ are the oscillator frequencies. The third term represents
the linear coupling between the cavities and the respective mechanical
oscillators. The fourth and the fifth terms correspond to the interaction of
the cavity-1 with the coupling field of strength $\varepsilon_{c}$, and the
probing field of strength $\varepsilon_{p}$ respectively, where
$\Delta=\omega_{p}-\omega_{c}$. The last term describes the hopping between
the two cavities with hopping strength $g$. For linearly coupled system the
optomechanical coupling strength defined as

$g_{\text{om}i}=\dfrac{\delta\omega_{i}}{\delta
Q_{i}^{q_{\textrm{m}i}}}\hat{Q}^{2}_{\textrm{ZPF,
i}},\;\text{where}\;\;\hat{Q}_{\textrm{ZPF},i}=\sqrt{\dfrac{1}{m\omega_{\text{m}i}}},$
(15)

is the zero-point fluctuations of the mechanical membrane’s displacement and
$m$ is the effective mass. The quadratures of deformed system
$\hat{Q}_{i}^{q_{\textrm{m}i}}=\left(\hat{b}_{i}^{q_{\textrm{m}i}}+\hat{b}_{i}^{q_{\textrm{m}i}\dagger}\right)/\sqrt{2}$,
and
$\hat{P}_{i}^{q_{\textrm{m}i}}=\left(\hat{b}_{i}^{q_{\textrm{m}i}}-\hat{b}_{i}^{q_{\textrm{m}i}\dagger}\right)/\sqrt{2}\textrm{i}$.

##  4 Dynamics of the System

To get useful information about the system we are going to observe the
dynamics of the system by using quantum Langevin equation, which returns us a
set of dynamical equations as follows:

$\displaystyle{\frac{d\hat{c}_{1}^{q_{\textrm{c}1}}}{dt}}=\left(-i\Delta_{1}\hat{c}_{1}^{q_{\textrm{c}1}}+\varepsilon_{c}+\varepsilon_{p}e^{-i\Delta
t}+ig_{\textrm{om}1}\hat{c}_{1}^{q_{\textrm{c}1}}\left(\hat{b}_{1}^{q_{\textrm{m}1}\dagger}+\hat{b}_{1}^{q_{\textrm{m}1}}\right)-ig\hat{c}_{2}^{q_{\textrm{c}2}}\right)\left[\hat{c}_{1}^{q_{\textrm{c}1}},\hat{c}_{1}^{q_{\textrm{c}1}\dagger}\right]-\kappa_{1}\hat{c}_{1}^{q_{\textrm{c}1}}+\sqrt{\kappa_{1}}\hat{c}_{1,\text{in}}^{q_{\textrm{c}1}},$
$\displaystyle{\frac{d\hat{c}_{2}^{q_{\textrm{c}2}}}{dt}}=\left(-i\Delta_{2}\hat{c}_{2}^{q_{\textrm{c}2}}+ig_{\textrm{om}2}\hat{c}_{2}^{q_{\textrm{c}2}}\left(\hat{b}_{2}^{q_{\textrm{m}2}\dagger}+\hat{b}_{2}^{q_{\textrm{m}2}}\right)-ig\hat{c}_{1}^{q_{\textrm{c}1}}\right)\left[\hat{c}_{2}^{q_{\textrm{c}2}},\hat{c}_{2}^{q_{\textrm{c}2}\dagger}\right]-\kappa_{2}\hat{c}_{2}^{q_{\textrm{c}2}}+\sqrt{\kappa_{2}}\hat{c}_{2,\text{in}}^{q_{\textrm{c}2}},$
$\displaystyle{\frac{d\hat{b}_{1}^{q_{\textrm{m}1}}}{dt}}=\left(-i\omega_{\textrm{m}1}\hat{b}_{1}^{q_{\textrm{m}1}}+ig_{\textrm{om}1}\hat{c}_{1}^{q_{\textrm{c}1}\dagger}\hat{c}_{1}^{q_{\textrm{c}1}}\right)\left[\hat{b}_{1}^{q_{\textrm{m}1}},\hat{b}_{1}^{q_{\textrm{m}1}\dagger}\right]-\gamma_{\textrm{m}1}\hat{b}_{1}^{q_{\textrm{m}1}}+\sqrt{\gamma_{\text{m}1}}\hat{b}_{1,\text{in}}^{q_{\textrm{m}1}},$
$\displaystyle{\frac{d\hat{b}_{2}^{q_{\textrm{m}2}}}{dt}}=\left(-i\omega_{\textrm{m}2}\hat{b}_{2}^{q_{\textrm{m}2}}+ig_{\textrm{om}2}\hat{c}_{2}^{q_{\textrm{c}2}\dagger}\hat{c}_{2}^{q_{\textrm{c}2}}\right)\left[\hat{b}_{2}^{{q_{\textrm{m}2}}},\hat{b}_{2}^{q_{\textrm{m}2}\dagger}\right]-\gamma_{\textrm{m}2}\hat{b}_{2}^{q_{\textrm{m}2}}+\sqrt{\gamma_{\text{m}2}}\hat{b}_{2,\text{in}}^{q_{\textrm{m}2}}.$

After simplification we get,

$\displaystyle{\frac{d\langle\hat{c}_{1}^{q_{\textrm{c}1}}\rangle}{dt}}=\left[-\left(\kappa_{1}^{q_{\textrm{c}1}}+i\Delta_{1}\right)\langle\hat{c}_{1}^{q_{\textrm{c}1}}\rangle+ig_{\textrm{om}1}\langle\hat{c}_{1}^{q_{\textrm{c}1}}\rangle\left(\langle\hat{b}_{1}^{q_{\textrm{m}1}\dagger}\rangle+\langle\hat{b}_{1}^{q_{\textrm{m}1}}\rangle\right)-ig\langle\hat{c}_{2}^{q_{\textrm{c}2}}\rangle+\varepsilon_{c}+\varepsilon_{p}e^{-\Delta
t}\right]\chi_{N_{\textrm{c}1}}^{q_{\textrm{c}1}},$ (16a)
$\displaystyle{\frac{d\langle\hat{c}_{2}^{q_{\textrm{c}2}}\rangle}{dt}}=\left[-\left(\kappa_{2}^{q_{\textrm{c}2}}+i\Delta_{2}\right)\langle\hat{c}_{2}^{q_{\textrm{c}2}}\rangle+ig_{\textrm{om}2}\langle\hat{c}_{2}^{q_{\textrm{c}2}}\rangle\left(\langle\hat{b}_{2}^{q_{\textrm{m}2}\dagger}\rangle+\langle\hat{b}_{2}^{q_{\textrm{m}2}}\rangle\right)-ig\langle\hat{c}_{1}^{q_{\textrm{c}2}}\rangle\right]\chi_{N_{\textrm{c}2}}^{q_{\textrm{c}2}},$
(16b)
$\displaystyle{\frac{d\langle\hat{b}_{1}^{q_{\textrm{m}1}}\rangle}{dt}}=\left[-\left(\gamma_{\textrm{m}1}^{q_{\textrm{m}1}}+i\omega_{\textrm{m}1}\right)\hat{b}_{1}^{q_{\textrm{m}1}}+ig_{\textrm{om}1}\lvert\langle\hat{c_{1}^{q_{\textrm{c}1}}}\rangle\rvert^{2}\right]\chi_{N_{\textrm{m}1}}^{q_{\textrm{m}1}},$
(16c)
$\displaystyle{\frac{d\langle\hat{b}_{2}^{q_{\textrm{m}2}}\rangle}{dt}}=\left[-\left(\gamma_{\textrm{m}2}^{q_{\textrm{m}2}}+i\omega_{\textrm{m}2}\right)\hat{b}_{2}^{q_{\textrm{m}2}}+ig_{\textrm{om}2}\lvert\langle\hat{c}_{2}^{q_{\textrm{c}2}}\rangle\rvert^{2}\right]\chi_{N_{\textrm{m}2}}^{q_{\textrm{m}2}}.,$
(16d)

where
$\chi_{N_{\textrm{o}}}^{q_{\textrm{o}}}=\dfrac{q_{\textrm{o}}^{N_{\textrm{o}}+1}+q_{\textrm{o}}^{-N_{\textrm{o}}}}{1+q_{\textrm{o}}}$,
is the deformation coefficient for the optomechanical system and the substript
$``\textrm{o}"$ takes values $``\textrm{c}1",\;``\textrm{c}2"$ for the first
and second cavities respectively and $``\text{m}1",\;``\text{m}2"$ for the
mechanical oscillators respectively. Meanwhile
$\gamma_{\textrm{m}i}^{q_{\textrm{m}i}}=\gamma_{\textrm{m}i}/\chi_{N_{\text{m}i}}^{q_{\textrm{m}i}}$,
$\kappa_{i}^{q_{\textrm{c}i}}=\kappa_{i}/\chi_{N_{\textrm{c}i}}^{q_{\textrm{c}i}}$.
Also
$\hat{c}_{i,\text{in}}^{q_{\textrm{c}i}},\hat{b}_{i,\text{in}}^{q_{\textrm{c}i}}$
are the input vacuum noise of the cavity and mechanical oscillator that
satisfies the following relationships:
$\langle\hat{c}_{i,\text{in}}^{q_{\textrm{c}i}}\rangle=\langle\hat{b}_{i,\text{in}}^{q_{\textrm{m}i}}\rangle=0$,
due to the fact that
$\langle\hat{c}_{i,\text{in}}\rangle=\langle\hat{b}_{i,\text{in}}\rangle=0$.

###  4.1 Dynamics under fluctuations

To observe the dynamics of the system we decompose the expectation value of
the observable as

$\langle\hat{O}(t)\rangle=\bar{O}+\delta\hat{O}(t)\;,\;\hat{O}=\hat{c}_{i}^{q_{\textrm{c}i}},\;\hat{b}_{i}^{q_{\textrm{m}i}};\;\;\;\;i=1,2,$
(17)

where $\bar{O}$ corresponds to the steady state solution in the absence of
probing field i.e. $\omega_{p}=0$ and $\delta O(t)=\hat{O}_{-}e^{-i\Delta
t}+\hat{O}_{+}e^{i\Delta t}$, induces by the weak probing field. By using the
dynamical equation we can find out the steady state solutions of the modes of
cavity and mechanical oscillators. At the same time an array of equations can
be found out by expanding corresponding to $\delta\hat{O}(t)$, which will
eventually leads us to the solutions for $\hat{O}_{\pm}$. However, in this
paper we only focus on the the expression of $\hat{O}_{-}$ as we will be
focusing only the resolved sidebands regime i.e.
$\omega_{\text{m}i}\gg\kappa_{i}$, and
$\Delta_{i}=\tilde{\Delta}_{i}=\omega_{\text{m}i}$ where
$\tilde{\Delta}_{i}=\Delta_{i}-g\bar{\alpha}_{i}$, where $\bar{\alpha}_{i}$ is
defined in Appendix B and defined by

$\bar{\alpha}_{i}=\dfrac{2\omega_{\textrm{m}i}g_{\textrm{om}i}}{\left(\gamma_{\textrm{m}i}^{{q_{\textrm{m}i}}}\right)^{2}+\omega_{\textrm{m}i}^{{}^{2}}}\lvert\bar{c}_{i}^{q_{\textrm{c}i}}\rvert^{2}.$
(18)

###  4.2 Output field

In order to reveal the OMIT, we have to study the response of the system to
probing frequency, which can be detected by the output field. Based on the
input–output theory, we obtain the relationship

$\varepsilon_{\textrm{out},p}e^{-i\Delta t}+\varepsilon_{p}e^{-i\Delta
t}+\varepsilon_{c}=2\kappa_{1}\langle c_{1}\rangle.$ (19)

If we put Eq. (17) into the above equation we get

$\mathcal{A}_{T}=\dfrac{\varepsilon_{\textrm{out},p}}{\varepsilon_{p}}+1=\dfrac{2\kappa_{1}\hat{c}_{1,-}}{\varepsilon_{p}},$
(20)

where $\mathcal{A}_{T}$ is the output probing field. The $q$-deformed version
of $\hat{c}_{1,-}$ is give as follows:

$c_{1,-}^{q_{\textrm{c}1}}=\dfrac{\varepsilon_{p}}{\left(\kappa_{1}^{q_{\textrm{c}1}}-i\beta_{N_{\textrm{c}1}}^{q_{\textrm{c}1}}\right)+\dfrac{\lvert
G_{1}^{q_{\textrm{c}1}}\rvert^{2}}{\gamma_{\textrm{m}1}^{q_{\textrm{m}1}}-i\beta_{N_{\textrm{m}1}}^{q_{\textrm{m}1}}}+\dfrac{g^{2}}{\left(\kappa_{2}^{q_{\textrm{c}2}}-i\beta_{N_{c2}}^{q_{\textrm{c}2}}\right)+\dfrac{\lvert
G_{2}^{q_{\textrm{c}2}}\rvert^{2}}{\gamma_{\textrm{m}2}^{q_{\textrm{m}2}}-i\beta_{N_{\textrm{m}2}}^{q_{\textrm{m}2}}}}},$
(21)

which can decomposed into real an imaginary part. Further, we get the deformed
$\mathcal{A}^{T}$ using the deformed version of Eq.(20) as follows:

$\displaystyle\mathcal{A}_{T}^{q_{\textrm{c}1}}$ $\displaystyle=$
$\displaystyle\dfrac{2\kappa_{1}^{q_{\textrm{c}1}}}{\left(\kappa_{1}^{q_{\textrm{c}1}}-i\beta_{N_{\textrm{c}1}}^{q_{\textrm{c}1}}\right)+\dfrac{\lvert
G_{1}^{q_{\textrm{c}1}}\rvert^{2}}{\gamma_{\textrm{m}1}^{q_{\textrm{m}1}}-i\beta_{N_{\textrm{m}1}}^{q_{\textrm{m}1}}}+\dfrac{g^{2}}{\left(\kappa_{2}^{q_{\textrm{c}2}}-i\beta_{N_{c2}}^{q_{\textrm{c}2}}\right)+\dfrac{\lvert
G_{2}^{q_{\textrm{c}2}}\rvert^{2}}{\gamma_{\textrm{m}2}^{q_{\textrm{m}2}}-i\beta_{N_{\textrm{m}2}}^{q_{\textrm{m}2}}}}},$
(22) $\displaystyle=$
$\displaystyle\mu^{q_{\textrm{c}1}}+i\delta^{q_{\textrm{c}1}},$

where $\mathcal{A}_{T}^{q_{\textrm{c}1}}$ is the deformed output probing
field. The $\mu^{q_{\textrm{c}1}}$, and $\delta^{q_{\textrm{c}1}}$ are the
deformed in- and out-of phase quadratures of the deformed probing field. For
an elaborated derivation of Eq.(22) see Appendix B.

##  5 Results

We now demonstrate the behaviour of the studied system. Based on previous
studies on $N$ cavities which are connected through tunnelling parameter, the
properties of the output field for the considered system is investigated using
the parameters [48, 49, 50] as follows.

  * •

$\omega_{\textrm{m}1}=\omega_{\textrm{m}2}=2\pi\times 51.8$ MHz,

  * •

$\gamma_{\textrm{m}1}=\gamma_{\textrm{m}2}=2\pi\times 41$ KHz,

  * •

$\kappa_{1}=2\pi\times 15$ MHz,

  * •

$\kappa_{2}=2\pi\times 0.027$ MHz,

  * •

$g=\kappa_{1}$,

  * •

$G_{1}=G_{2}=2\pi\times 10$ MHz,

if not stated otherwise. We assume that
$N_{\textrm{m}1}=N_{\textrm{m}2}=N_{\textrm{c}1}=N_{\textrm{c}2}=1$.

###  5.1 Non-deformed case

Let us start by focusing in the case of the considered system when
$q_{\textrm{c}1}=q_{\textrm{c}2}=q_{\textrm{m}1}=q_{\textrm{m}2}=1$. All the
remaining optomechanical parameters are same as mentioned above.

####  5.1.1 Optomechanical induced transparency

In the non-deformed case we observe how the tunnelling parameter effects the
appearance of the transparency windows (TW). As illustrated in Fig.(3), both
the optomechanical couplings i.e. $G_{1}$ and $G_{2}$ are non-zero in the
system which causes appearance of an additional TW at $\beta/\kappa_{1}\approx
0$ where $\beta=\Delta-\omega_{m}$, along with the TWs appear at
$\beta/\kappa_{1}\approx\pm 0.80$. It should be noted that when $G_{1}=0$ the
central TW vanishes [50].

Figure 3: Illustration of variation of the $\mu$ with $\beta/\kappa_{1}$ when
$G_{1}=G_{2}$ and $q=1$.

For $g>\kappa_{1}$, as we increase the tunnelling parameter we observe the TW
at $\beta/\kappa_{1}\approx 0$ becomes narrower and the adjacent absorption
peaks turns smaller. Finally for $g/\kappa_{1}>4$, the absorption peaks that
was previously appearing both side of the $\beta/\kappa_{1}<1$, diminishes
finally we get a single wide TW. This is illustrated in the left picture of
Fig.(3).

Meanwhile when $g<\kappa_{1}$, and varrying $g$ in range $0\leq g<1$, helps us
observe that the TW that appeared at the position $\beta/\kappa_{1}=\pm 0.8$,
turns narrower and shallower. And at $g=0$, the TWs at $\beta/\kappa_{1}$
vanishes, leaving only the central TW. This is illustrated in the right hand
side picture of Fig.(3).

####  5.1.2 Fast-slow light

The fast and slow light effects in the weak probe field are illustrated and
discussed in this section. For an optomechanical system, in the region of
narrow TWs a rapid phase transition occurs using the relation

$\phi(\omega_{p})=\textrm{arg}[A_{T}(\omega_{p})],$ (23)

this rapid phase transition can cause the group delay in output field. In
respect with input-output relationship [51], the optical group delay of
transmitted light is related to $\phi$ as follows

$\tau=\dfrac{\partial\phi(\omega_{p})}{\partial\omega_{p}}=\dfrac{\partial\phi(\omega_{p})}{\partial\beta},$
(24)

where $\beta_{i}=\Delta-\omega_{mi}$. In Eq.(24), $\tau>0$ indicates slow-
light, and $\tau<0$ denotes fast-light.

Figure 4: The phase of the probe field as a function of $\beta/\kappa_{1}$.

Through Fig.(4) it can be observed that the rapid phase transition occurs at
$\beta/\kappa_{1}\approx\pm 0.8$ and $\beta/\kappa_{1}=0$, which are indeed
the positions where the transparency windows appear. Interestingly it has been
observed that the manipulation of phase transition near the origin is possible
for $g>\kappa_{1}$, illustrated in the left hand side picture of Fig.(4). On
the other hand phase transition appears at $\beta/\kappa_{1}\approx\pm 0.8$
can be manipulated by varying the tunnelling parameter in the range
$0<g<\kappa_{1}$, and at $g=0$, the phase transition in these regions
vanishes. The appearance of sharp enhancement and decrement in phase of the
probe field indicated that at that particular normalized detuning group delay
can be strongly manipulated. We plot the group delay, $\tau$ (defined in
Eq.(24)) with the normalized detuning $\beta/\kappa_{1}$ as illustrated in the
sub-figures of Fig.(5).

Figure 5: The group delay of the probe field as a function of
$\beta/\kappa_{1}$.

It is observed that when $g<\kappa_{1}$, strongest superluminal (fast light)
phenomena appears at the $\beta/\kappa_{1}\approx\pm 0.8$ (with pulse
advancement of the order of 450 $\mu s$) and $\beta/\kappa_{1}=0$ (pulse
advancement by $\approx$ 380 $\mu s$), but subluminal (slow light) phenomena
is weak. By varying the tunnelling parameter in range $0<g<1$ we can
manipulate the strength of fast light as shown on the right hand side of
Fig.(5). However, it does not put any impact on the central superluminal peak.

On the other hand, by varying $g>\kappa_{1}$, we can induce subluminal
phenomena, which can be seen though two symmetric peaks appears across the
central superluminal peak illustrated on the left hand side picture of
Fig.(5). Interestingly the strength of the sublimunal peak completely depends
upon the strength of the tunnelling between two non-deformed cavities. As the
tunnelling between cavities becomes stronger the subluminal peaks moves more
and more closer to $\beta/\kappa_{1}=0$, simultaneously the superluminal peak
deminishes. Finally, the two symmetric subluminal peaks gets merged and exists
as a single peak, which causes the pulse delay of the order of 380 $\mu s$ at
$g=5\times 10^{3}\kappa_{1}$.

Figure 6: Group delay around $\beta/\kappa_{1}=0$ for strong inter-cavity
tunnelling.

Experimentally the value of tunnelling strength is considered to be strong
when $g>(\kappa_{1}+\kappa_{2})/2$, and otherwise weak [31]. In our setup if
$g>7.5135\kappa_{1}$, we can define the inter-cavity transmission to be
strong. In this range even though we observe a strong superluminal phenomena
at the positions of the TWs but the subluminal phenomena is smaller than the
half compared to superluminal maxima. Our aim is to achieve strong super and
subluminal effect at experimentally achievable tunnelling strength. We, in the
following section show that it can be achieved by introducing a slight
deformation to the cavities.

###  5.2 $q$-deformed case

Figure 7: Real component of $\mathcal{A}_{T}^{q}$ for different deformation
amplitude of the cavities for $G_{1}=G_{2}$.

In this section we present the nontrivial nature and corresponding advantages
in the optomechanical system by introducing nonlinearity in the system by
tuning the deformation strength in the cavities. We compare and observe the
deformed OMIT, phase and fast-slow light with the nondeformed investigation.
Before discussing the results and advantages of introducing deformation we
formulate the deformed version of the cavity parameters in terms of the non-
deformed ones as follows

$\displaystyle\dfrac{G_{1}^{q_{\textrm{c}1}}}{G_{1}}=\dfrac{\bar{c}_{1}^{q_{\textrm{c}1}}}{c_{1}}=\dfrac{\kappa_{1}+i\omega_{\textrm{m}1}+\dfrac{g^{2}}{\kappa_{2}+i\omega_{\textrm{m}2}}}{\kappa_{1}^{q_{\textrm{c}1}}+i\omega_{\textrm{m}1}+\dfrac{g^{2}}{\kappa_{2}^{q_{\textrm{c}2}}+i\omega_{\textrm{m}2}}},$
(25)
$\displaystyle\dfrac{G_{2}^{q_{\textrm{c}2}}}{G_{2}}=\left(\dfrac{\kappa_{2}+i\omega_{\textrm{m}2}}{\kappa_{2}^{q_{\textrm{c}2}}+i\omega_{\textrm{m}2}}\right)\times\dfrac{\bar{c}_{1}^{q_{\textrm{c}1}}}{c_{1}},$
(26)
$\displaystyle\gamma_{\textrm{m}i}^{q_{\textrm{m}i}}=\gamma_{\textrm{m}i}\times\left(\chi_{N_{\textrm{m}i}}^{q_{\textrm{m}i}}\right)^{-1};\;\;\kappa_{i}^{q_{\textrm{c}i}}=\kappa_{i}\times\left(\chi_{N_{\textrm{c}i}}^{q_{\textrm{c}i}}\right)^{-1},$
(27)

where $\bar{c}_{i}$ represents the steady state value of $\hat{c}_{i}$, and
recalling
$\chi_{N_{\textrm{o}}}^{q_{\textrm{o}}}=\dfrac{q_{\textrm{o}}^{N_{\textrm{o}}+1}+q_{\textrm{o}}^{-N_{\textrm{o}}}}{1+q_{\textrm{o}}}$,
is the deformation coefficient for the optomechanical system and the substript
$``\textrm{o}"$ takes values $``\textrm{c}1",\;``\textrm{c}2",$ the cavity
modes and $``\text{m}1",\;``\text{m}2"$ for the mechanical modes. Throughout
the calculations we have used the the same optomechanical parameter values as
used in the case of non-deformed investigation. Meanwhile one can introduce
deformation to the system in the following ways (when the mechanical
oscillators are non-deformed i.e. $q_{\textrm{m}1}=q_{\textrm{m}2}=1$.):

  * •

Either one of the cavities ($q_{\textrm{c}1}$ or $q_{\textrm{c}2}$) can be
deformed independently, by keeping the other cavity non-deformed
($q_{\textrm{c}1}=1$ or $q_{\textrm{c}2}=1$),

  * •

Both cavities can be deformed simultaneously by tuning the deformation
strength either in the same ($q_{\textrm{c}1}=q_{\textrm{c}2}=q$) or distinct
($q_{\textrm{c}1}\neq q_{\textrm{c}2}$) manner.

In this article we will be focussing particularly on the deformation of the
first cavity keeping the second cavity non-deformed i.e. $q_{\textrm{c}1}\leq
1$ and $q_{\textrm{c}2}=1$, and the mechanical oscillators are non-deformed;
if not stated otherwise.

####  5.2.1 Deformed OMIT

By fixing $q_{\textrm{c}2}=q_{\textrm{m}1}=q_{\textrm{m}2}=1$, and varying
$q_{\textrm{c}1}=q$, in the range $0<q\leq 1$ we have observed the appearance
and behaviour of TWs. As a first observation it is noted that as we tend to
take the system further from the non-deformed condition towards $q\rightarrow
0$, the symmetrical nature of the probe field behavior on the both side of
$\beta=0$ gradually deminishes. The amplitude of the absorption peak exceeds
maximum absorption limit (which is $2$) twice. For first time at $q=0.7$ the
absorption coefficient reaches to $\approx 2.43$, $2.16$ at
$\beta/\kappa_{1}\approx-0.406$ and $-1.35$ respectively. For the second time
at $q=0.3$ where the absorption peak takes the value $\approx 8.5$ and $3.97$
at $\beta/\kappa_{1}\approx-0.3468$ and $-1.21$ respectively. On the other
hand the amplitude of absorption for $\beta/\kappa_{1}>0$ decreases. Another
observation in when we vary the deformation strength in range $0.5\leq q<1$ is
that, the width of the TWs which appear in between $\beta=\pm 2$ tends to
become wider towards $q\rightarrow 0.5$.

Interestingly as we deform the system further i.e. $q<0.5$ a negative
absorption peak appears in the region $-2<\beta/\kappa_{1}<0$. The absorption
peak takes a value of $\approx-1.00$ for $q=0.3$. The appearance of negative
absorption peaks in optomechanical systems is not new, in Ref.[52] the authors
detected the appearance of negative absorption that might be due the non-
Markovian nature of the environment. In our observation we detect the gain in
probe field which is caused due to the photon backflow from the environment,
leading to more transmitted energy than provided at $q=0.3$. The gain in probe
field might indicate that, although we initially started with Markovian
assumption at $q=1$; but as we vary the deformation parameter in range
$0<q<0.5$, the photon backflow becomes more prominent. Which is responsible
for the negative absorption peak.

Figure 8: The deformed phase of the probe field as a function of
$\beta/\kappa_{1}$.

(a)

(b)

Figure 9: In (a) we illustrate the deformed group delay of the probe field as
a function of $\beta/\kappa_{1}$ in range $0.3\leq q\leq 1$. In (b) we observe
the tunability of positive and negative group delay around $q=0.9$.

####  5.2.2 Deformed fast and slow light

As a consequence of the deformation introduced in the system, we expect to
observe a deformed phase and fast-slow light phenomena which is derived as
follows

$\tau^{q}=\dfrac{\partial\phi^{q}(\omega_{p})}{\partial\omega_{p}}=\dfrac{\partial\phi^{q}(\omega_{p})}{\partial\beta},$
(28)

where $\phi^{q}=\textrm{arg}\left[A_{T}^{q}\right]$ is the deformed phase of
the transmitted probe beam.

In Fig.(8) we illustrate the deformed phase diagram with normalized detuning
$\beta/\kappa_{1}$, for $q=0.9$. As it has already discussed while observing
the non-deformed phase a sharp change in phase indicates the possibility of
manipulation of super and subluminal light. In Fig.(8) this sharp phase change
is observed at $\beta\approx-0.8\kappa_{1}$, which indicates that at that
particular detuning value we can manipulate super and subluminal light which
is depicted in Fig.(9(a)). Compared to the highest possible subluminal peak
for non-deformed case at $g=5\times 10^{3}\kappa_{1}$, the subluminal peak for
$q=0.9$ at $g=\kappa_{1}$ is approximately 5 times stronger. The appearance of
the strong subluminal light at $q=0.9$ can be explained as follows. In
Fig.(9(a)) we can see that as we move from $q=1$ (the black line) to $q=0.9$
(the green line), due to induced nonlinearity the frequency of probe field
increases (as predicted and illustrated in Fig.(1(a))) and the absorption
decreases, indicating prominence of normal dispersion phenomena i.e. the sharp
enhancement of phase at $\beta/\kappa_{1}\approx-0.8$ in Fig.(9(a)). Meanwhile
in range $0<q<0.9$, the sharpness of the phase enhancement diminishes,
indicating to the fact that normal dispersion becomes weaker.

In Fig.(9(b)), at $q=0.92$, we see an enhancement in superluminal light. Which
indicates appearance of anomalous dispersion i.e. the phase of the probe field
shifts from positive to negative value. Our observation also indicates that a
slight change in deformation parameter may help us to tune from super to
subluminal light and vice versa. As the TWs at $\beta/\kappa_{1}\approx-0.8$
vanishes as we shifts towards $q=0$, the tunability of fast and slow light
becomes infeasible, which is illustrated in Fig.(9(b)).

Figure 10: Deformed group delay with normalized tunnelling strength for
$\beta=-0.8007\kappa_{1}$, and $q=0.9$.

Using the cumulative knowledge of deformed and non-deformed cases we can
expect an interplay of super and subluminal light is possible when we tune the
tunnelling paramerter $g$ around $\beta\approx-0.8\kappa_{1}$ for $g=0.9$
which is depicted in Fig.(10). In this figure using previous knowledge we fix
the $q$ value and the $\beta$ to study the nature of group delay with
tunnelling strength. The first observation one can infer from Fig.(10) is that
as the tunnelling strength increases the appearance of super and subluminal
light deminishes. At a first glance although the group delay seems to saturate
at null value when $g>2\kappa_{1}$, but through magnification a transition
from sub to superluminal light can be observe around $g\approx 3.2\kappa_{1}$.
The group delay saturates and the probe field shows an advancement of
$400\mu\text{s}$ in the strong inter-cavity transmission strength region.

Meanwhile in the weak inter-cavity transmission strength regime particularly
in the range $0.01<g/\kappa_{1}\approx 0.015$, a very sharpe transition from
super to subluminal light is observed. The probe field at $g\approx
0.011\kappa_{1}$, shows an advancement in the order of $9.7m\text{s}$ and at
$g\approx 0.0146\kappa_{1}$ a delay of probe field is observed in the order of
$32.3m\text{s}$. The observed characteristics of the group delay can be
explained in the following way. At $q=0.9$ when $g$ is weak, then the photon
in the second cavity impose a very weak perterbation to the photon in the
first cavity, so the non-linearity in first cavity, that is induced in the
from of deformation sustains as long as $g<3$. But when $g>3$, the influence
of the photon in the second cavity destroys the induced nonlinearity of the
first cavity. Hence the group delay tends to mimic the characteristics of an
non-deformed cavity and the group delay saturates at a probe field advancement
in the order of $400\mu\text{s}$.

Finally, we investigate the group delay under different values of
optomechanical coupling when $G_{1}/G_{2}\geq 0$ as depicted in Fig.(11).

Figure 11: Deformed group delay with optomechanical coupling and tunnelling
strength for $\beta^{q}=-0.809559\kappa_{1}^{q}$, and $q=0.9$.

Even when the first cavity is not interacting with the corresponding
mechanical oscillator, we get a strong super and subluminal light in the order
of milliseconds for $0<g/\kappa_{1}<0.02$. As $G_{1}$ approaches $G_{2}$, the
strength of subluminal light enhances but superluminal light weakens
gradually. Meanwhile when $G_{1}/G_{2}>1$, the sharpness of transition from
super to subluminal light as well as the absolute strength of group delay
deminishes.

####  5.2.3 Fast and slow light above milliseconds

By fixing the deformation strength at $q=0.9$ we show in Fig.(13(a)),
Fig.(13(b)) that the probe field can be delayed and advanced above
milliseconds respectively when fine tuning of normalized inter-cavity
tunnelling strength and the detuning parameter is possible.

##  6 Anyon models: $q$ as primitive root of unity

More than four decades ago it has been pointed out [53] that in systems
confined to two spatial dimensions, particles with general exchange statistics
more than those of bosons and fermions, are possible. Particles with exchange
statistics in between fermion and boson were later named as anyon by Wilczek
[54]. In a general $q$ deformed system when the deformation strength is
expressed as a primitive root of unity i.e.

$q=\textrm{exp}\left(\dfrac{2\pi\textrm{i}}{l+2}\right),$ (29)

then this specific type of nonlinear model belongs to a general family of
anyon models known as $SU(2)_{l}$ models. The mathematical framework that
describes the $SU(2)_{l}$ models are called topological quantum field theory
which provides tools to study physical processes in topological phases of
matter.

(a)

(b)

Figure 12: Illustration of variation of phase when the first cavity consists
of (a) Ising anyons i.e. $l=2$, and for (b) Fibonacci anyons i.e. $l=3$. From
the figure it is clearly visible that the appearance of rapid phase transition
is more frequency in (a), which indicates better manipulation super and
subluminal light.

In this section we will be particularly focusing on Ising and Fibonacci anyon
models by putting $l=2$ and $l=3$ respectively in Eq.(29). Although physical
realization of Ising anyons is proposed in the form of Majorana fermion zero
modes, but the evidence for Fibonacci anyons, has not been reported yet. The
Fibonacci anyons are believed to exist as quasiparticles in Kondo systems [55]
and in exotic quantum Hall fluids with filling fraction $12/5$ [56, 57].

By keeping the second cavity and the mechanical oscillators non-deformed, we
deform the first cavity setting
$q_{\textrm{c}1}=q=\textrm{exp}\left({\dfrac{2\pi\textrm{i}}{l+2}}\right)$.
This is indicating that the $q$-deformed particles in the first cavity are
behaving like Ising and Fibonacci anyons respectively. We observe the behavior
of phase transition, finally fast and slow light of the probe field.

Fig.(12) depicts the phase transition behavior of the probe field when the
first cavity consists of Ising (see Fig.(12(a))) and Fibonacci (see
Fig.(12(b))) anyons. The appearance of rapid phase transition in probe field
is more frequent and prominent when the first cavity consists of Ising anyons
compared to Fibonacci anyons. The reason behind this observation can be
described as follows. In case of the Ising anyons the increase in frequency of
the probe field due to induced nonlinearity is stronger than the case when we
Fibonacci anyons are considered (which can be deducted from Fig.(1(a))). This
observation directly indicates stronger normal dispersion, causing appearance
of subluminal light. And the strength of subluminal peak is stronger in case
of Ising anyons compared to Fibonacci ones as illustrated in the group delay
plot in Fig.(14) and Fig.(15) respectively.

A more elaborated discussion is given as follows. For Ising anyons which is
illustrated in figure Fig.(12(a)), when the normalized inter-cavity coupling
is $0<g/k_{1}<0.8$, and when the $G_{1}/G_{2}<0.2$, the phase of the probe
field increase sharply (transition from yellow to the red region), indicating
strong normal dispersion behavior and appearance of strong subluminal peak. A
slight increase in $g$, tends to steep decrease in the phase of the probe
field towards negative value (transition red to deep blue region), which in
turns indicates appearance of strong anomalous dispersion and we see strong
superluminal light. This description also holds for Fibonacci anyons
illustrated in Fig.(12(b)), but as the steepness of the variation of phase is
low compared to Ising anyons so we observe weaker normal and anomalous
dispersion, causing relatively weak super and subluminal light.

##  7 Summary and conclusions

As a summary, in this work, we present a comprehensive review on
$q$-deformation and analytically study the tunability of optomechanical
induced transparency in a $q$-deformed optomechanical system. The system
contains two $q$-deformed cavities which are coupled to deformed mechanical
oscillators, and with each other by tunneling parameter. We have particularly
examined the case when the mechanical membranes and the second cavity is non-
deformed but the first cavity is influenced by the deformation strength $q$.
Under the influence of $q$-deformation the dynamics of the cavity depends upon
$q$-number representation of bosonic number operator. When the deformation
strength is real and $q<1$, the deformation in the first cavity induces a
nonlinearity to the optomechanical system. As a result, in the region
$0<q<0.5$, the system experiences gain, which is caused by photon backflow
from the environment to the first cavity. Moreover, we achieve a significant
enhancement in the strength of fast and slow light phenomena in a scale of
milliseconds and even above by fixing the deformation of the first cavity at
$q=0.9$ , and by finely tuning the normalized inter-cavity coupling
$g/\kappa_{1}$ and detuning $\beta/\kappa_{1}$. In addition, to reduce the gap
between the mathematical $q$-deformation scheme and physically achievable
systems, we define $q$ as primitive root of unity, indicating a class anyon
models. We investigate the phase transition and the fast-slow light behavior
of the probe field under the two assumptions (1) the first cavity’s
$q$-deformed particles behave like Ising anyons at deformation strength
$q=\textrm{exp}\left(\pi\textrm{i}/2\right)$ and (2) when the $q$-deformed
particles mimics Fibonacci anyons at
$q=\text{exp}\left(2\pi\textrm{i}/5\right)$. In both cases the probe field
experiences enhanced rapidity in phase change, indicating towards more
tunability of group delay and enhancement in fast-slow light compared to the
non-deformed case. Notably Ising anyons under the influence of same
optomechanical parameters shows delay and advancement in probe field in the
order of milliseconds and even in seconds. Our study introduces a new tool to
not only investigate various cavity optomechanical systems, but to achieve
enhancement over systems which do not necessitate $q$-deformation setup.

As an extension to this work, it would be interesting to see how the
$q$-deformed system behaves when a two-level atomic system is introduced
inside one or multiple cavities. Introducing atomic ensemble could make the
width of the OMIT wider [50], which may allow better manipulation of the
deformed system. Meanwhile another obvious extension to this work could be to
observe how the deformed system responds when we consider the strength of
deformation in cavities are distinct (i.e. $q_{\textrm{c}1}\neq
q_{\textrm{c}2}$). Moreover as our work only study Ising and Fibonacci anyon
models when $l=2,3$ respectively, an investigation of the other classes of
anyon models when $l>3$ [58, 59] is yet to be explored.

## Acknowledgment

The authors would like to acknowledge fruitful discussions with Marek Gluza.
This work was supported by Polish National Center project 2019/33/B/ST6/02011.

## Funding

This work is supported by Polish National Center project 2019/33/B/ST6/02011.

## Conflict of Interest

The authors declare that they have no conflict of interest.

## Data availability

The datasets analysed during the current study are available from the
corresponding author on reasonable request.

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##  Appendix A Derivation of deformed frequency

The commutator relations used for the deformed system are as follows:

  * •

$\left[\hat{O}^{q},\hat{O}^{q\dagger}\right]=[\hat{N}_{O}+1]-[\hat{N}_{O}]=\dfrac{q^{N_{O}+1}+q^{-N_{O}}}{1+q}$,

  * •

$\left[\hat{Q}_{O^{q}},\hat{P}_{O^{q}}\right]=i\left[\hat{O}^{q},\hat{O}^{q\dagger}\right]=i\dfrac{q^{N_{O}+1}+q^{-N_{O}}}{1+q}$.

  * •

$[\hat{N}_{O}+1]+[\hat{N}_{O}]=\dfrac{q^{N_{O}+1}-q^{-N_{O}}}{q-1}$

From equation (11) the dynamics of the bosonic operators given by,

${\frac{d\hat{O}}{dt}}=-i\omega\hat{O}\;\;\;;\;\;\;{\frac{d\hat{O}^{\dagger}}{dt}}=-i\omega\hat{O}^{\dagger}$
(30)

following the same way using Heisenberg equation, the dynamics of $q$-deformed
bosonic operator defined by Hamiltonian (8),

$\displaystyle\frac{d\hat{O}}{dt}$
$\displaystyle=\hat{H}_{q\rightarrow\text{real}}\hat{O}-\hat{O}\hat{H}_{q\rightarrow\text{real}}=i\omega\dfrac{\text{sinh}\left[k\left(\hat{O}^{\dagger}\hat{O}\right)\right]}{\text{sinh}k}\hat{O}-i\omega\hat{O}\dfrac{\text{sinh}\left[k\left(\hat{O}^{\dagger}\hat{O}\right)\right]}{\text{sinh}k},$
$\displaystyle=\dfrac{i\omega}{\text{sinh}k}\left[\left(k\hat{O}^{\dagger}\hat{O}+\dfrac{k^{3}}{3!}\left(\hat{O}^{\dagger}\hat{O}\right)^{3}+\ldots\right)\hat{O}-\hat{O}\left(k\hat{O}^{\dagger}\hat{O}+\dfrac{k^{3}}{3!}\left(\hat{O}^{\dagger}\hat{O}\right)^{3}+\ldots\right)\right],$
$\displaystyle=\dfrac{i\omega
k}{\text{sinh}k}\left[-\hat{O}-\dfrac{k^{2}}{3!}3\left(\hat{O}^{\dagger}\hat{O}\right)^{2}\hat{O}+\ldots\right]=-i\omega\dfrac{k}{\text{sinh}k}\left[1+\dfrac{k^{2}}{2!}\left(\hat{O}^{\dagger}\hat{O}\right)^{2}+\ldots\right],$

form where we finally arrive at,

$\hat{O}=-i\omega_{q}\hat{O}\;\;\;;\;\;\;\hat{O}^{\dagger}=-i\omega_{q}\hat{O},$
(31)

where,

$\omega_{q}=\omega\dfrac{k\text{cosh}\left[k\left(\hat{O}^{\dagger}\hat{O}\right)\right]}{\text{sinh}k},$
(32)

which represents the orbital dependence frequency after passing through
$q$-deformation.

##  Appendix B Detailed derivation of dynamics

The steady state solutions by Equations (16a) to (16d) are given by,

$\displaystyle\bar{c}_{1}^{q_{\textrm{c}1}}=\varepsilon_{c}/\left[\left(\kappa_{1}^{q_{\textrm{c}1}}+\dfrac{g^{2}}{\kappa_{2}^{q_{\textrm{c}2}}+i\tilde{\Delta}_{2}}\right)+i\tilde{\Delta}_{1}\right],$
(33)
$\displaystyle\bar{c}_{2}^{q_{\textrm{c}2}}=-\dfrac{ig\bar{c}_{1}^{q_{\textrm{c}1}}}{\kappa_{2}^{q_{\textrm{c}2}}+i\tilde{\Delta}_{2}},$
(34)
$\displaystyle\tilde{\Delta}_{i}=\Delta_{i}-g_{\textrm{om}i}\bar{\alpha}_{i},$
(35)
$\displaystyle\bar{\alpha}_{i}=\dfrac{2\omega_{\textrm{m}i}g_{\textrm{om}i}}{\left(\gamma_{\textrm{m}i}^{{q_{\textrm{m}i}}}\right)^{2}+\omega_{\textrm{m}i}^{{}^{2}}}\lvert\bar{c}_{i}^{q_{\textrm{c}i}}\rvert^{2}.$
(36)

If we substitute $\langle\hat{O}(t)\rangle=\bar{O}+\delta\hat{O}(t)$
($\hat{O}=\hat{c}_{i}^{q_{\textrm{c}i}},\;\hat{b}_{i}^{q_{\textrm{m}i}},\;\;\;i=1,2$)
in equations (16a) to (16d) one get an array of time dependent equation
corresponding to $\delta\hat{O}(t)$ which should include the probing field
term $\varepsilon_{p}e^{-i\Delta t}$. These equations can be given as follows

$\displaystyle{\frac{d}{dt}\delta
c_{1}^{q_{\textrm{c}1}}}=\left[-\left(\kappa_{1}^{q_{\textrm{c}1}}+i\tilde{\Delta}_{1}\right)\delta\hat{c}_{1}^{q_{\textrm{c}1}}+iG_{1}^{q_{\textrm{c}1}}\left(\delta\hat{b}_{1}^{q_{\textrm{m}1}}+\left(\delta\hat{b}_{1}^{q_{\textrm{m}1}}\right)^{*}\right)-ig\delta\hat{c}_{2}^{q_{\textrm{c}2}}+\varepsilon_{p}e^{-i\Delta
t}\right]\chi_{N_{\textrm{c}1}}^{q_{\textrm{c}1}},$
$\displaystyle{\frac{d}{dt}\delta
c_{2}^{q_{\textrm{c}2}}}=\left[-\left(\kappa_{2}^{q_{\textrm{c}2}}+i\tilde{\Delta}_{2}\right)\delta\hat{c}_{2}^{q_{\textrm{c}2}}+iG_{2}^{q_{\textrm{c}2}}\left(\delta\hat{b}_{2}^{q_{\textrm{m}2}}+\left(\delta\hat{b}_{2}^{q_{\textrm{m}2}}\right)^{*}\right)-ig\delta\hat{c}_{1}^{q_{\textrm{c}1}}\right]\chi_{N_{c2}}^{q_{\textrm{c}2}},$
$\displaystyle{\frac{d}{dt}\delta
b_{i}^{q_{\textrm{m}i}}}=\left[-\left(\gamma_{mi}^{q_{\textrm{m}i}}+i\omega_{mi}\right)\delta\hat{b}_{i}^{q_{\textrm{m}i}}+i\left(G_{i}^{q_{\textrm{c}i}}\left(\delta\hat{c}_{i}^{q_{\textrm{c}i}}\right)^{*}+\left(G_{i}^{q_{\textrm{c}i}}\right)^{*}\delta\hat{c}_{i}^{q_{\textrm{c}i}}\right)\right]\chi_{N_{\textrm{m}i}}^{q_{\textrm{m}i}},$

for $i=1,2$,
$G_{i}^{q_{\textrm{c}i}}=g_{\textrm{om}i}\bar{c}_{i}^{q_{\textrm{c}i}}$ and
“$*$” denoted complex conjugate. To gain more physical we observe the dynamics
of the system in resolved sideband regime. In this regime
$\omega_{mi}\gg\kappa_{i}$ and $\tilde{\Delta}_{i}=\omega_{\text{m}i}$. In
such regime, the lower sideband far off resonance can be safely neglected
which means, $\delta\hat{O}(t)\approx\hat{O}_{-}e^{-i\Delta t}$ i.e.
$\hat{O}_{+}=0$. Thus the equation from (B) to (B) can be rewritten as

$\displaystyle
0=-\left(\kappa_{1}^{q_{\textrm{c}1}}-i\beta^{q_{\textrm{c}1}}_{N_{c1}}\right)\hat{c}_{1,-}^{q_{\textrm{c}1}}+iG_{1}^{q_{\textrm{c}1}}\hat{b}_{1,-}^{q_{\textrm{m}1}}-ig\hat{c}_{2,-}^{q_{\textrm{c}2}}+\varepsilon_{p},$
(37) $\displaystyle
0=-\left(\kappa_{2}^{q_{\textrm{c}2}}-i\beta^{q_{\textrm{c}2}}_{N_{c2}}\right)\hat{c}_{2,-}^{q_{\textrm{c}2}}+iG_{2}^{q_{\textrm{c}2}}\hat{b}_{2,-}^{q_{\textrm{m}2}}-ig\hat{c}_{1,-}^{q_{\textrm{c}1}},$
(38) $\displaystyle
0=-\left(\gamma_{\textrm{m}i}^{q_{\textrm{m}i}}-i\beta^{q_{\textrm{m}i}}_{N_{\text{m}i}}\right)\hat{b}_{i,-}^{q_{\textrm{m}i}}+i\left(G_{i}^{q_{\textrm{c}i}}\right)^{*}\hat{c}_{i,-}^{q_{\textrm{c}i}},$
(39)

where
$\beta^{q_{\textrm{c}i}}_{N_{\textrm{c}i}}=\Delta^{q_{\textrm{c}i}}_{N_{\textrm{c}i}}-\omega_{\text{m}i}$,
and
$\beta_{N_{\text{m}i}}^{q_{\textrm{m}i}}=\Delta_{N_{\text{m}i}}^{q_{\textrm{m}i}}-\omega_{\text{m}i}$;
where
$\Delta_{N_{\textrm{c}i}}^{q_{\textrm{c}i}}=\Delta/\chi_{N_{\textrm{c}i}}^{q_{\textrm{c}i}}$,
and
$\Delta_{N_{\textrm{m}i}}^{q_{\textrm{m}i}}=\Delta/\chi_{N_{\textrm{m}i}}^{q_{\textrm{m}i}}$.

##  Appendix C Illustrations

(a)

(b)

Figure 13: Illustration of variation of group delay in the probe field with
normalized detuning (x-axis) and normalized inter-cavity transmission
strength. In (a) we observe subluminal and in (b) we observe superluminal
light above milliseconds for $q$ = $0.9$, and
$\omega_{\text{m}1}=\omega_{\text{m}2}$. Figure 14: Illustrarion of group
delay in Ising anyons when the first cavity is deformed by strength of
$q=\textrm{exp}\left(\pi\textrm{i}/2\right)$, at
$\omega_{\text{m}1}=\omega_{\text{m}2}$, with normalized inter-cavity
transmission strength (y-axis) and ration of optomechanical coupling strength
of two cavities. The marked white regions in the density plot denotes region
of extremely enhanced fast-slow light. Figure 15: Illustrarion of group delay
in Fibonacci anyons when the first cavity is deformed by strength of
${q=\textrm{exp}\left(2\pi\textrm{i}/5\right)}$, at
$\omega_{\text{m}1}=\omega_{\text{m}2}$, with normalized inter-cavity
transmission strength (y-axis) and ration of optomechanical coupling strength
of two cavities. Although the fast and slow light is enhanced compared to non-
deformed case but the enhancement can not go beyond the milliseconds region.