# A rotational-cavity optomechanical system with two revolving cavity mirrors:
optical response and fast-slow light mechanism

Amjad Sohail  Department of Physics, Government College University, Allama
Iqbal, Faisalabad 38000, Pakistan  Rameesa Arif  Department of Physics,
Government College University, Allama Iqbal, Faisalabad 38000, Pakistan  Naeem
Akhtar<EMAIL_ADDRESS>(First corresponding author) Department of
Physics, Zhejiang Normal University, Jinhua 321004, China  Ziauddin  Quantum
Optics Lab, Department of Physics, COMSATS University, Islamabad 45550,
Pakistan  Jia-Xin Peng<EMAIL_ADDRESS>(Second corresponding author)
State Key Laboratory of Precision Spectroscopy, Quantum Institute for Light
and Atoms, Department of Physics, East China Normal University, Shanghai
200062, China  Gao Xianlong  Department of Physics, Zhejiang Normal
University, Jinhua 321004, China  ZhiDong Gu  The Key Laboratory of Weak-Light
Nonlinear Photonics, Ministry of Education, School of Physics and TEDA Applied
Physics Institute, Nankai University, Tianjin 300071, China

###### Abstract

We investigate the optical behavior of a single Laguerre-Gaussian cavity
optomechanical system consisting of two mechanically rotating mirrors. We
explore the effects of various physical parameters on the double
optomechanically induced transparency (OMIT) of the system and provide a
detailed explanation of the underlying physical mechanism. We show that the
momentum is not the cause of the current double-OMIT phenomena; rather, it
results from the orbital angular momentum between the optical field and the
rotating mirrors. Additionally, the double-OMIT is simply produced using a
single Laguerre-Gaussian cavity optomechanical system rather than by
integrating many subsystems or adding the atomic medium as in earlier studies.
We also investigate the impact of fast and slow light in this system. Finally,
we show that the switching between ultrafast and ultraslow light can be
realized by adjusting the angular momentum, which is a new source of
regulating fast-slow light.

##  I Introduction

Cavity optomechanics has focused a lot of interest on the investigation of
coherent coupling between optical and mechanical modes via the radiation
pressure of photons trapped inside an optical cavity [1, 2, 3]. Additionally,
in the contemporary era of quantum technology, such cavities have achieved
significant advancements, including ultrahigh-precision measurement [4, 5],
gravitational wave detection [6], quantum information processing [7], quantum
teleportation [8, 9], nonclassical photon statistics [10, 11, 12], higher-
order sideband generation [13, 14, 15], optomagnonic frequency combs [16],
quantum entanglement [17, 18, 19], macroscopic quantum coherence [20, 21, 22],
ground state cooling of mechanical oscillator [23, 24], and optomechanically
induced transparency (OMIT) [25, 26, 27, 28].

The electromagnetically induced transparency (EIT) is a quantum interference
effect that has been shown in several quantum systems, including nitrogen
vacancy centers [31], quantum dots [32], metamaterial [33], and Bose Einstein
condensate [34]. The OMIT phenomenon in the cavity optomechanical system was
first reported theoretically [26] before being demonstrated experimentally in
[27]. The OMIT phenomena is similar to the extensively investigated EIT
phenomenon in the atomic medium [29, 30] because the radiation pressure
produced mechanical oscillations in OMIT significantly change the optical
response of a weak probing field. Moreover, single OMIT and double-OMIT
transparency windows have both been investigated in various quantum
optomechanical systems, such as hybrid optomechanical cavities [35, 36, 37,
38, 39, 40, 41, 42] and atomic-media assisted optomechanical systems [43, 44,
45, 46]. OMIT also has significant applications in the generation of fast/slow
light [50, 51, 52, 53, 54] including light storage [36, 55, 56] and the
precise measurement of a variety of physical quantities such as electrical
charge [47], mass sensor [48], and environmental temperature [49].

Most of the aforementioned research on OMIT is related to a typical cavity
optomechanical system, where the radiation pressure causes the interaction of
the mechanical mode with the cavity field. This optomechanical interaction is
mainly due to the exchange of linear momentum between the mechanical mode and
the cavity field. In particular, Bhattacharya and Meystre proposed a
rotational-cavity optomechanical system in 2007 in which a macroscopic
rotating mirror is coupled to a Laguerre-Gaussian (L-G) cavity mode through
the exchange of orbital angular momentum and studied the cooling of the
rotating mirror due to the action of the radiation torque [57]. Subsequently,
Liu et al. also accomplished the ground-state cooling of the rotating mirror
in a double L-G cavity with atomic ensemble [58]. These studies on the ground
cooling of the rotating mirror have provided many opportunities to further
investigate macroscopic quantum phenomena in rotational cavity optomechanical
systems, including bipartite quantum entanglement [57, 60, 61, 62] and
tripartite quantum entanglement [63], macroscopic quantum coherence [64], OMIT
phenomena [65, 66, 67, 68, 69, 70], and its applications for the measurement
of orbital angular momentum [65, 71, 72], higher order sideband generation
[73, 74] and phase-matching analysis [75]. However, compared to studies on
conventional optomechanical systems, the research on the double-OMIT
phenomenon in this type of rotating cavity optomechanical system is still
essentially very limited.

On the other hand, the realization of slow and fast light also makes the
propagation of a weak probe field in cavity optomechanical systems extremely
important [76, 77]. The slow and fast light in atomic vapours and solid-state
materials have been the subject of various investigations in the past. One
application of these methods is to change the group velocity of the probe
field, which results in the realization of either slow or fast propagation
[78]. For instance, slow light propagation occurs in atomic vaprs, the Bose-
Einstein condensate [79, 80], as well as in a medium of ultracold sodium
vapours [81] and crystal [83]. Similar fast and slow light propagation in
cavity optomechanical devices has also been demonstrated [84]. Based on these
previous studies, we are now interested to investigate the slow and fast light
generation in the L-G rotational cavity optomechanical system.

In the present work, we investigate the double-OMIT phenomena in a single mode
L-G cavity with two spiral phase elements serving as rotating mirrors, and a
cavity field controlled by a strong pump field. The effects of many
significant physical elements on OMIT have been thoroughly investigated, and
thorough physical justifications have been provided. In particular, our
approach opens up a brand-new opportunity to achieve double-OMIT in a single
mode L-G cavity optomechanical system without the addition of any atomic
medium or subsystem. Further, we turn to the study of group delay in the
system, which can be regarded as an analog of the fast-slow light in Fabry-
Pérot cavity. The results show that switching between fast and slow light can
be achieved by adjusting some parameters, such as orbital angular momentum. In
addition, we found that the group delay can even reach $-400$ $\mu$s and
$+400$ $\mu$s, which is a typical ultrafast-ultraslow light effect.

The paper is organized as follow. In Section II, we describe the model and
introduce the Hamiltonian of the system. We also derive the quantum Langevin
equations and provide the temporal evolution of the operator mean values in
the same section. In Section III, we obtain the expressions of the output
field at the probe frequency. The numerical results of the output field with
various input field power, cavity decay rate, mechanical quality factor, and
orbital angular momentum of the coupling field are discussed in Section IV. In
Section V, we study the fast and slow light effects related to our system.
Finally, we conclude our discussion in section VI.

Figure 1: Schematic diagram of the L-G cavity optomechanical with two rotating
mirrors, where a strong coupling field with zero topological charge at
frequency $\omega_{c}$ and a weak probe field at frequency $\omega_{p}$ are
input into the rotating cavity, they interact with two rotating mirrors at
angular frequency $\omega_{\phi_{1}}$ and $\omega_{\phi_{2}}$, respectively.
Besides, the angular displacement RM1 and RM2 are indicated by the angles
$\phi_{1}$ and $\phi_{2}$ under the optorotational interaction, respectively.

##  II Hamiltonian description

In this section, we present the model and introduce the Hamiltonian of our
system. The system we considered is a rotating optomechanical cavity with
length $\mathcal{L}$ which consists of two rotating mirrors [57, 61], as shown
in Fig. (1). The rotating mirror 1 (RM1) is partially transparent and the
rotating mirror 2 (RM2) is perfectly reflecting. The field of a rotating
cavity with frequency $\omega_{0}$ is driven by a strong coupling field with
amplitude $\varepsilon_{c}$ at frequency $\omega_{c}$. Meanwhile, a weak probe
field with amplitude $\varepsilon_{p}$ and frequency $\omega_{p}$ is used to
detect the optical behavior of the system. We assume a strong coupling field
with zero topological charge (Gaussian beam) as an input to the cavity. The
azimuthal structure of the rotating mirrors removes the charges from the beam
by reflecting [57, 58, 59, 60, 61]. When the coupling field is reflected by
the RM1 and RM2, respectively, it acquires a topological charge of $-2l$ and
$+2l$, demonstrating that the net topological charge increment of light
transmission back and forth in the cavity is zero, ensuring the stability of
the system. And this makes the strong coupling field which couples the two
rotating mirrors by the exchange of the orbital angular momentum. Here we
assume that the two rotating mirrors have the same effective mass $m$ and
radius $R$. Note that $\omega_{\phi_{j}}(j=1,2)$ and $\gamma_{j}(j=1,2)$
denote the angular frequency and intrinsic damping rate RM${j}(j=1,2)$,
respectively. In a frame rotating at the driving frequency $\omega_{c}$, the
Hamiltonian of the whole system takes the form

$\displaystyle H/\hbar$ $\displaystyle=$
$\displaystyle\Delta_{0}a^{\dagger}a+\sum_{j=1,2}\dfrac{\omega_{\phi_{j}}}{2}(\phi_{j}^{2}+L_{z_{j}}^{2})+(g_{1}\phi_{{}_{1}}-g_{2}\phi_{{}_{2}})a^{\dagger}a$
(1)
$\displaystyle+i\varepsilon_{c}(a^{\dagger}-a)+i\varepsilon_{p}\left(a^{\dagger}e^{-i\Delta
t}-ae^{i\Delta t}\right),$

where $\Delta_{0}=\omega_{0}-\omega_{c}$ ($\Delta=\omega_{p}-\omega_{c}$) is
the detuning of the cavity field (the probe field) from the strong coupling
field, $a$ and $a^{\dagger}$ ($\left[a,a^{\dagger}\right]=1$) are the
annihilation and creation operators of the cavity field, respectively. And
$\phi_{j}$ and $L_{z_{j}}$ with $[\phi_{j},L_{z_{k}}]=i\delta_{jk}(j,k=1,2)$
are representing the angular displacement and angular momentum of the rotating
mirrors, respectively. In Eq. (1), the first two terms are the free
Hamiltonian of the energy for the cavity field and the two rotating mirrors,
the last two terms give the coupling field and the probe field inside the
cavity, the third term describes the optomechanical coupling between the
cavity field and the two rotating mirrors with the coupling constant
$g_{i}=(cL/\mathcal{L})\sqrt{\hbar/I\omega_{\phi_{i}}}$ [57], where $c$ is the
velocity of light and $L$ is the orbital angular momentum quantum number,
$\mathcal{L}$ is the length of the cavity and $I=mR^{2}/2$ is the moment of
inertia of the two rotating mirrors about the central axis of the cavity. In
addition, the coupling field amplitude and the probe field are
$\varepsilon_{c}=\sqrt{2\kappa P/\hbar\omega_{c}},$
$\varepsilon_{p}=\sqrt{2\kappa P_{p}/\hbar\omega_{p}}$, respectively. Here,
$P$ ($P_{p}$) and $\omega_{c}$ ($\omega_{p}$) represent the power and
frequency of the coupling (probe) field.

Figure 2: The absorption spectra $\nu_{p}\ $of the output probe field as a
function of normalized $\Delta\diagup\omega_{m}$ for different power $P=0$, 2
mW and 5 mW, respectively. The angular frequency of rotating mirrors
$\omega_{\phi_{1}}=\omega_{\phi_{2}}=\omega_{m}$ in (a-b) and
$\omega_{\phi_{1}}=1.1\omega_{m}$, $\omega_{\phi_{2}}=0.9\omega_{m}$ in (c-d).

On the basis of the aforementioned, we examine the mean value Heisenberg
equation, which is derivable from the Hamiltonian in Eq. (1), while taking
into account the dissipation of the cavity field and the damping of the
revolving mirrors, and excluding quantum noise and thermal noise. In the
strong driving regime we have $\langle a\rangle\gg 1$, thus the factorization
assumption $\langle\phi_{i}a\rangle=\langle\phi_{i}\rangle\langle
a\rangle\left(i=1,2\right)$ and $\langle a^{\dagger}a\rangle=\langle
a^{\dagger}\rangle\langle a\rangle$ are valid. This implies that the time
evolution of the mean value equation of the system operators gets form as

$\displaystyle\langle\dot{\phi_{j}}\rangle$ $\displaystyle=$
$\displaystyle\omega_{{}_{\phi_{j}}}\langle L_{z_{j}}\rangle,$
$\displaystyle\langle\dot{L_{z_{j}}}\rangle$ $\displaystyle=$
$\displaystyle-\omega_{{}_{\phi_{j}}}\langle\phi_{j}\rangle+g_{\alpha{j}}\langle
a^{\dagger}\rangle\langle a\rangle-\gamma_{j}\langle L_{z_{j}}\rangle,$ (2)
$\displaystyle\langle\dot{a}\rangle$ $\displaystyle=$
$\displaystyle-i[\Delta_{0}+g_{1}\phi_{{}_{1}}-g_{2}\phi_{{}_{2}}]\langle
a\rangle-\kappa\langle a\rangle+\varepsilon_{c}+\varepsilon_{p}e^{-i\Delta
t},$

where $j=1,2$, $g_{\alpha{1}}=-g_{1}$ and $g_{\alpha{2}}=g_{2}$. In the weak
driving regime, especially in the single-photon strong coupling regime, the
response of the system to the probe field will be affected by the nonlinear
interaction between the cavity field and each rotating mirror. Since the probe
field is much weaker than the coupling field
$(\left|\varepsilon_{p}\right|\ll\varepsilon_{c})$, the steady-state solution
to Eqs. (II) can be approximated to the first order in the probe field
$\varepsilon_{p}$. In the long time limit, the solution to Eqs. (II) can be
written as [25, 26, 27, 28, 85]

$\langle s\rangle=s_{0}+s_{+}\varepsilon_{p}e^{-i\Delta
t}+s_{-}\varepsilon_{p}^{\ast}e^{i\Delta t},$ (3)

where $s=[\phi_{1},\phi_{2},L_{z_{1}},L_{z_{2}},a]$. Note that here we
retained only the components $\omega_{c},$ $\omega_{p},$
$2\omega_{c}-\omega_{p}$. Substituting the Eq. (3) into the Eqs. (II), we can
obtain the following expressions

$\displaystyle L_{z_{j}}$ $\displaystyle=$ $\displaystyle
0,\phi_{j0}=\frac{g_{\alpha{j}}\left|a_{0}\right|^{2}}{\omega_{\phi_{j}}},a_{0}=\frac{\varepsilon_{c}}{\kappa+i\Delta^{\prime}},$
$\displaystyle a_{+}$ $\displaystyle=$
$\displaystyle\frac{1}{d(\delta)}\left[A\ast\Lambda_{1}\ast\Lambda_{2}+i(G_{1}^{2}\omega_{\phi_{1}}\Lambda_{2}+G_{2}^{2}\omega_{\phi_{2}}\Lambda_{1})\right],$
$\displaystyle a_{-}$ $\displaystyle=$
$\displaystyle\frac{1}{d^{\ast}(\delta)}\left[\frac{a_{0}^{2}}{\left|a_{0}\right|^{2}}i(G_{1}^{2}\omega_{\phi_{1}}\Lambda_{2}+G_{2}^{2}\omega_{\phi_{2}}\Lambda_{1})\right],$
(4) $\displaystyle d(\delta)$ $\displaystyle=$ $\displaystyle A\ast
A^{\prime}\ast\Lambda_{1}\ast\Lambda_{2}-2\Delta^{\prime}(G_{1}^{2}\omega_{\phi_{1}}\Lambda_{2}+G_{2}^{2}\omega_{\phi_{2}}\Lambda_{1}).$

Here
$A=\kappa-i(\Delta^{\prime}+\Delta),A^{\prime}=\kappa+i(\Delta^{\prime}-\Delta),\Lambda_{j}=\omega_{\phi_{j}}^{2}-\Delta^{2}-i\gamma_{j}\Delta,G_{j}=g_{\alpha{j}}\left|a_{0}\right|(j=1,2)$.
Among $\Delta^{\prime}=\Delta_{0}+g_{1}\phi_{10}-g_{2}\phi_{20}$ is the
detuning between the effective frequency of L-G cavity and the frequency of
the coupling field, and it depends on $a_{0}$. Additionally, $G_{1}$ and
$G_{2}$ can be considered of as, respectively, the effective optomechanical
coupling rates to the cavity field and RM1 and RM2.

##  III The output field

In this section, we solve the properties of the components of the optical
field output at the probe field frequency $\omega_{p}$. According to the
input-output relation, we can get the magnitude of the output field as follows
[25, 26, 27, 28, 85],

$\varepsilon_{out}(t)+\varepsilon_{c}+\varepsilon_{p}e^{-i\Delta
t}=2\kappa\left\langle a\right\rangle.$ (5)

We write the output field $\varepsilon_{out}(t)$ in a similar form to Eq. (3),
then expand the amplitude of the output field by a small amount of
$\varepsilon_{p}$ with

$\varepsilon_{out}(t)=\varepsilon_{out0}+\varepsilon_{out+}\varepsilon_{p}e^{-i\Delta
t}+\varepsilon_{out-}\varepsilon_{p}^{\ast}e^{i\Delta t},$ (6)

where $\varepsilon_{out0}$, $\varepsilon_{out+}$, and $\varepsilon_{out-}$ are
the components of the output field at frequency $\omega_{c},$ $\omega_{p},$
$2\omega_{c}-\omega_{p}$, respectively. Here $\varepsilon_{out-}$ is known as
the Stokes field, and it is generated from the process where two photons at
angular frequency $\omega_{c}$ interact with a single photon at frequency
$\omega_{p}$ to create a new photon at frequency $2\omega_{c}-\omega_{p}$.
Substituting the Eq. (6), the part of Eqs. (II) and Eq. (3) into the Eq. (5),
the output field at the weak probe frequency is

$\displaystyle\varepsilon_{out+}$ $\displaystyle=$ $\displaystyle 2\kappa
a_{+}-1.$ (7)

The quadratures of the output field defined by

$\varepsilon_{T}=\varepsilon_{out+}+1=2\kappa a_{+}=\nu_{p}+iu_{p}.$ (8)

Here $\nu_{p}$ and $u_{p}$ are the two quadratures components of the output
field $\varepsilon_{T}$, where the real part $\nu_{p}$ and the imaginary part
$u_{p}$ represent the absorption characteristic and the dispersion
characteristic of the cavity optomechanical system, respectively. The homodyne
technique can be used to measure the field quadratures, as is widely known
[26, 27].

##  IV Analyzing the output probe field

Let us now discuss the effect of various physical factors on the output field,
such as the power of the pumping field, the mechanical frequencies of two
rotating mirrors, angular quantum number of L-G cavity modes, and damping of
the system. We utilize similar parameter values to those in the papers [57,
58, 60, 61, 62, 65, 66] for our numerical calculations. In particular, we
choose the wavelength of the strong coupling field $\lambda_{c}=810$ nm, the
orbital angular momentum number $L=100$, the masses and the radius of the
rotating mirrors $m_{1}=m_{2}=m=50$ ng and $R=0.1$ $\mu$m, the L-G cavity mode
decay rate $\kappa=15\pi\times 10^{6}$ Hz, the angular frequencies of the
rotating mirrors $\omega_{\phi_{1}}=1.1\omega_{m}$ and
$\omega_{\phi_{2}}=0.9\omega_{m}$, respectively, where
$\omega_{m}=160\pi\times 10^{6}$ Hz. The mechanical quality factors of the
rotating mirrors are $Q_{1}=Q_{2}=Q=1.2\times 10^{5}$ and the mechanical
damping rate is $\gamma_{i}=\omega_{\phi_{i}}/Q_{i}(i=1,2)$. In particular,
the system is placed in the resolved side-band regime
$(\kappa/\omega_{m}=0.187<1)$, which facilitates the study of anti-Stokes
processes. Besides, the strong coupling field is tuned close to the red side-
band of the cavity resonance, thus the effective detuning approximately
satisfies the relation $\Delta^{\prime}=\omega_{m}$ [25, 26, 27, 28].

###  IV.1 Pumping field versus the output probe field

In Fig. (2), we plot the absorption $\nu_{p}$ and dispersion $u_{p}$ of the
output probe field as a function of the normalized detuning
$\Delta/\omega_{m}$ for three different values of the power $P$ of the
coupling field. We consider both the rotating mirror RM1 and RM2 having the
same angular frequency, i.e.,
$\omega_{\phi_{1}}=\omega_{\phi_{2}}=\omega_{m}$. In Fig. 2(a), it is clear
that the absorption curve exhibits a standard Lorentzian line pattern when
$P=0$, which indicates that the probe field is strongly absorbed if
$\Delta=0$. However, as the power $P$ increases, it is evident that the
absorption peak transforms into a dip, as is typical for an EIT curve. In
particular, the greater the power of the pump field results into the dip of
the wider width and deeper depth, hence, causing the more significant the OMIT
effect. When the power is not zero, the probe light exhibits a dramatic
dispersion change at the dip position, as seen in Fig. 2 (b). The later study
of fast and slow light in this paper is built on this phenomena. It is
interesting to note that two revolving mirrors, with angular frequencies of
$\omega_{\phi_{1}}=1.1\omega_{m}$ and $\omega_{\phi_{2}}=0.9\omega_{m}$, have
distinct fundamental frequencies. Figure 2(c) demonstrates that the absorption
curve dips at $\Delta=0.9\omega_{m}$ and $\Delta=1.1\omega_{m}$ when the
pumping field power is not zero, which is a common double-OMIT phenomenon. The
left transparent window is caused by the coupling of RM2 and the cavity field,
while the right transparent window is caused by the coupling of RM1 and the
cavity field.

In particular, the two rotating mirrors in our system rotate about the cavity
axis; as a result, their motion can be explained by rotational modes of the RM
phonon states [57, 58]. By combining the photon state of the cavity field,
they create the energy-level diagram shown in Fig. (3), which has
$\Lambda$-type energy level structure. Therefore, the physical mechanism
behind the above phenomena can be easily explained by energy level diagram of
the system. In Fig. (3), when
$\omega_{\phi_{1}}=\omega_{\phi_{2}}=\omega_{m}$, it is clear that the
detuning $\Delta=\omega_{p}-\omega_{c}$ is equal to $\omega_{m}$,
$\left|a\right\rangle\leftrightarrow\left|b\right\rangle$ will have two
different transition channels (one $\Lambda$-type energy level structure) to
interfere. That is the anti-Stokes field at frequency $\omega_{c}+\omega_{m}$
and the input probe field at frequency $\omega_{p}$ cause the destructive
interference, and then it has one EIT dip at $\Delta=\omega_{m}$ on the
absorptive curve of output field, this is the standard model for appearing
single transparent window. However, when $\omega_{\phi_{1}}$ is not equal to
$\omega_{\phi_{2}}$, there is obviously one more transition channel than
before (two $\Lambda$-type energy level structures). Therefore, it leads to
the double-OMIT phenomenon. In addition, the position of the transparent
window is closely related to the frequency of the rotating mirror. It is
important to note that, unlike in the previous studies, where the double-OMIT
phenomena was caused by the integration of many subsystems or the addition of
an atomic medium, it is now only the outcome of a single L-G cavity system. So
the research model we currently use is simpler.

Figure 3: The cavity frequency of the excitation of
$\left|a\right\rangle\leftrightarrow\left|b\right\rangle$ and
$\left|a\right\rangle\leftrightarrow\left|c_{i}\right\rangle(i=1,2)$ are
$\omega_{p}$ and $\omega_{c}$, respectively. The frequency of the rotating
mechanical excitation
$\left|c_{1}\right\rangle\leftrightarrow\left|b\right\rangle$ is
$\omega_{\phi_{1}}$ and
$\left|c_{2}\right\rangle\leftrightarrow\left|b\right\rangle$ is
$\omega_{\phi_{2}}$. $\omega_{m}=(\omega_{\phi_{1}}+\omega_{\phi_{2}})/2$ is
normalized frequency. Figure 4: The contour plot of absorption spectra
$\nu_{p}\ $ of the output probe field as a function of normalized
$\Delta/\omega_{m}$ and angular momenta $L$ when (a)
$\omega_{\phi_{1}}=\omega_{\phi_{2}}=\omega_{m}$ and (b)
$\omega_{\phi_{1}}=1.1\omega_{m}$, $\omega_{\phi_{2}}=0.9\omega_{m}$.

###  IV.2 Orbital angular momentum of the coupling field

In this subsection, we examine the effect of the orbital angular momentum of
the coupling field on the absorption of the output probe field. In Fig. (4) we
have drawn diagrams of the relationship density plot of absorption spectra
versus scaled $\Delta/\omega_{m}$ and angular momenta $L$. Similar to the
previous results, when $\omega_{\phi_{1}}=\omega_{\phi_{2}}$, only one
transparent window appears, while when $\omega_{\phi_{1}}$ is not equal to
$\omega_{\phi_{2}}$, two transparent windows are displayed. Interestingly, we
can clearly see that with the increase of angular momentum, the width and
depth of the transparent window increases. This result is understood by the
optorotational coupling strength
$g_{i}=(cL_{i}/\mathcal{L})\sqrt{\hbar/I\omega_{\phi_{i}}}\left(i=1,2\right)$.
According to this formula, we know that, when $L$ is small, the effective
coupling strength between the cavity mode and the rotating mirrors will be
weak, which inevitably lead to the weakening of the anti-Stokes process in the
system. This make the destructive interference effect between the anti-Stokes
field and the probe field less significant. Hence, as a result, the OMIT
effect is not obvious, and vice versa. The current double-OMIT phenomenon
originates from the exchange of orbital angular momentum between the light
field and the rotating mirrors, rather than the momentum.

###  IV.3 Effects of dissipation on the output field

Here, we explore the effect of the dissipation channels on the output probe
field. For the currently studied optomechanical system, the main dissipation
channels are the thermal damping of the rotating mirrors and the cavity mode
decay. Figure (5) exhibits the role of the damping of the rotating mirrors on
the absorption spectra. In Fig. 5(a), we show that the mechanical quality
factor of RM1 decreases (corresponding to increase the damping), and the depth
of the right transparent window decreases. Moreover, here we also show that
the transparent window on the left is almost invariant. This happen due to the
transparent window on the right that is build by RM1, and thermal damping
harmful effect on the destructive interference between the anti-Stokes field
$\omega_{c}+\omega_{\phi_{1}}$ and the output probe field $\omega_{p}$.
Evidently, the influence of damping of RM2 on the absorption spectra lead to
identical results, as illustrated in Fig. 5(b).

We also study the influence of cavity field attenuation on the absorption
curve, and this is illustrated in Fig. (6). We find that the two transparency
windows and the center peak have no change with the increase of decay rate
$\kappa$, while sidebands on both sides of the transparent windows become
wider (this can be considered as the spectral line broadening caused by cavity
field attenuation). This phenomenon shows that the double-OMIT phenomenon is
robust against cavity field attenuation. Physically, the increase of cavity
field attenuation inevitably weaken the optomechanical coupling between the
cavity field and rotating mirrors. More importantly, when the angular
displacement of rotating mirrors is large then a large strain is needed to
reduce the optorotational effect and vice versa. Hence, the current research
system shows strong robustness to cavity field attenuation.

##  V Fast and slow light concept

In the previous section, we have investigated the optical response in the L-G
cavity optomechanical system consist of two rotating mirrors. In particular,
we have shown that the dispersion curve shows a sharp dispersion change at the
dip, which prompted the study of fast-slow light. In general, the phase of
output probe field is related to the group delays of the output field and is
given by [50, 51, 52, 53, 54]

$\tau_{g}=\frac{d\phi\left(t_{p}\right)}{d\omega_{p}},$ (9)

where $t_{p}=1-\varepsilon_{T}$ is transmitted field and
$\phi\left(t_{p}\right)=\arg\left[t_{p}\right]$ represents the phase of
$t_{p}.$ Note that $\tau_{g}>0$ (positive group delay) denotes slow light,
whereas $\tau_{g}<0$ (negative group delay) represents the fast light. In the
following, we first study the effects of physical parameters on the phase
dispersion $\phi\left(t_{p}\right)$, and then analyze their effects on group
delay $\tau_{g}$.

Figure 5: The plot of absorption spectra $\nu_{p}\ $ of the output probe field
as a function of normalized $\Delta/\omega_{m}$ for different values of
mechanical quality factors. Here, we set $Q=1.2\times 10^{5}$. Figure 6: The
plot of absorption spectra $\nu_{p}\ $ of the output probe field as a function
of normalized $\Delta/\omega_{m}$ for different values of cavity decay rate.

Figure 7: The phase dispersion $\phi\left(t_{p}\right)$ as a function of
normalized $\Delta/\omega_{m}$ for different the coupling field power, angular
momentum quantum number, cavity field attenuation and mechanical quality
factor. The four subgraphs above correspond to
$\omega_{\phi_{1}}=\omega_{\phi_{2}}$, while the four subgraphs below
correspond to $\omega_{\phi_{1}}=1.1\omega_{m}$,
$\omega_{\phi_{2}}=0.9\omega_{m}$. Here, we set $Q=1.2\times 10^{5}$, and the
statement in the first paragraph of Section IV for other parameters. Figure 8:
The density plot of the group delay $\tau_{g}$ (in $\mu$s) versus the coupling
field power and the angular momentum quantum number for the different values
of mechanical quality factors. We set $\omega_{\phi_{1}}=1.1\omega_{m}$,
$\omega_{\phi_{2}}=0.9\omega_{m}$, $P_{0}=1\times 10^{-3}$ mW, and
$\Delta=1.1\omega_{m}$. (a) $Q_{1}=Q$, (b) $Q_{1}=10^{-1}Q$, (c)
$Q_{1}=10^{-2}Q$, (d) $Q_{1}=10^{-3}Q$, here $Q_{2}=Q=1.2\times 10^{5}$.

Figure (7) is the phase dispersion $\phi\left(t_{p}\right)$ depending on the
normalized detuning $\Delta/\omega_{m}$ under different physical parameters,
such as the coupling power, the angular momentum quantum number, the cavity
field attenuation and quality factor of rotating mirror. One can clearly see
that when $\omega_{\phi_{1}}=\omega_{\phi_{2}}$, the phase dispersion
$\phi\left(t_{p}\right)$ presents a zigzag shape with two dips as
$\Delta=\omega_{m}$ changes. Because the derivative of
$\phi\left(t_{p}\right)$ with respect to $\Delta$ is the same as that with
respect to $\omega_{p}$. Equation (9) reveals the emergence of fast and slow
light. Physically, the sharp change of $\phi\left(t_{p}\right)$ results in a
huge change in the refractive index of the medium, here causing group advanced
and delay phenomena. In particular, we find that the distance between two dips
change with the adjustment of physical parameters, which is much similar to
the dispersion curve (see Fig. 2(b) and Fig. 2(d)). On the other hand, when
$\omega_{\phi_{1}}=1.1\omega_{m}$, $\omega_{\phi_{2}}=0.9\omega_{m}$, we see
that the phase dispersion $\phi\left(t_{p}\right)$ is still zigzag with the
change of $\Delta=\omega_{m}$, but the number of dips becomes three. Thus, in
the case of $\omega_{\phi_{1}}$ is not equal to $\omega_{\phi_{2}}$, the fast
and slow light effects can appear in more positions. This depends on the
number of transparent windows. In addition, when the mechanical quality factor
is very small, we find that the dip number decreases by one (see the dark blue
curve in the last subgraph (d)). The physical mechanism behind this is that
the thermal reservoir in contact with the vibrator will destroy the
interference process, which is similar to the mechanism in Fig. (5). Next we
will explore the influence of these physical factors on fast and slow light.

In Fig. (8), we draw the density plot of the group delay $\tau_{g}$ versus the
coupling field power and the angular momentum quantum number for different
values of mechanical quality factors. We can clearly see the dependence of
group delay $\tau_{g}$ on coupling power and the angular momentum quantum
number.More specifically, in Figs. 8(a), 8(b) and 8(c), we find that fixing
the coupling field power $P$ (the angular momentum quantum number) and
adjusting the angular momentum quantum number $L$ (the coupling field power)
can change group delay $\tau_{g}$ from positive to negative, or from negative
to positive, that is, adjusting the angular momentum can achieve the
adjustment of fast and slow light. This provides a new degree of freedom for
the adjustment of fast and slow light, which is not available in the
traditional Fabry-Pérot cavity. Interestingly, as illustrated in Fig. 8(a)
that the group delay can even reach $-400$ $\mu$s and $+400$ $\mu$s, which is
a typical ultrafast and ultraslow lights effect. However, with the increase of
the mechanical quality factor, we can clearly see that the fast and slow light
effects are not significant, and even only weak fast light effects are shown
in Fig. 8(d). The physical reason behind this is that the thermal reservoir
destroyed the OMIT effect caused by the RM1. Hence, we can speculate that if
set $\Delta=0.9\omega_{m}$, modifying $Q_{2}$ but keep $Q_{1}$ unchanged has
similar results.

Figure 9: The density plot of the group delay $\tau_{g}$ (in $\mu$s) versus
the coupling field power and the angular momentum quantum number for the
different cavity field attenuation. (a) $\kappa=2\pi\times 35$ MHz, (b)
$\kappa=2\pi\times 25$ MHz, (c) $\kappa=2\pi\times 15$ MHz,
(d)$\kappa=2\pi\times 5$ MHz. Other parameters are the same as those in Fig.
(8).

On the other hand, we also explore the influence of cavity field attenuation
on the group delay $\tau_{g}$. As expected, the group delay is closely related
to the attenuation of the cavity field. Specifically, with the increase of
cavity field attenuation, the effect of fast and slow light becomes less
significant. By Comparing Figs. 9(a) and 9(d), we found that the group delay
can differ to some extent. Interestingly, unlike the attenuation of the
oscillator, even if the cavity field attenuation is large, we can still get
ultrafast and ultraslow light (compare Fig. 8(d) and Fig. 9(a)). This
advantage comes from the “the double-OMIT phenomenon is robust against cavity
field attenuation” mentioned in the description of the Fig. (6). Note that
although the cavity field attenuation is not very destructive to the OMIT
phenomenon in the currently studied system (see Fig. (6)), and when it
increases then it may enhance the dip width of the phase dispersion
$\phi\left(t_{p}\right)$ (no longer sharp), which causes
$\phi\left(t_{p}\right)$ to change slowly with $\omega_{p}$ near the dip (see
Eq. (9)). Therefore, the cavity field attenuation leads to the weakening of
fast and slow light effects, and the mechanism and Fig. (8) are not completely
consistent.

##  VI Conclusion

We have presented a two rotating mirror single mode Laguerre-Gaussian cavity
optomechanical system. We have accomplished the twofold optomechanically
induced transparency of the Gaussian probe field in rotational degrees of
freedom. We have shown that the central absorption peak can become narrower
and the two OMIT dips can become wider if the strength of the coupling field
is increased. Furthermore, when the angular frequencies of the two rotating
mirrors coincide, the system exhibits properties similar to a normal
optomechanical Fabry-Pérot cavity system in which only one mirror is moving.
We also demonstrate the impact of the orbital angular momentum of the coupling
field and rotating cavity decay on the output field. This shows that as the
orbital angular momentum increases, the destructive interference increases in
intensity and the transparency windows of the output probe field’s absorption
grow larger. This is consistent with the larger orbital angular momentum being
associated with a larger optorotational coupling. We have shown that it is
possible to switch between ultrafast and ultraslow light by altering the
angular momentum, which is not possible in conventional optomechanical
systems. The study of our paper has bright prospects for the potential
application of the rotating cavity optomechanics.

###### Acknowledgements.

NA acknowledges the support of postdoctoral fund of the Zhejiang Normal
University under Grant No. ZC304021937. GX acknowledges support by the
National Natural Science Foundation of China under Grants Nos. 11835011 and
12174346.

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