JID:PLA AID:126153 /SCO Doctopic: Quantum physics [m5G; v1.261; Prn:26/11/2019; 14:59] P.

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Physics Letters A ••• (••••) ••••••

Contents lists available at ScienceDirect

Physics Letters A
www.elsevier.com/locate/pla

Double optomechanically induced transparency in a Laguerre-Gaussian

rovibrational cavity
Jia-Xin Peng, Zheng Chen, Qi-Zhang Yuan, Xun-Li Feng ∗
Department of Physics, Shanghai Normal University, Shanghai 200234, China

a r t i c l e i n f o a b s t r a c t

Article history: The optomechanically induced transparency (OMIT), an optomechanical analogue of electromagnetically
Received 19 June 2019 induced transparency, is a very interesting interference phenomenon. Recently, the studies on the OMIT
Received in revised form 15 November 2019 have been extended to double-OMIT by integrating more optical or mechanical subsystems such as
Accepted 16 November 2019
mechanical oscillators, coupled cavities, and atoms in vibrational cavity. In this paper, we demonstrate
Available online xxxx
Communicated by V.A. Markel
the double-OMIT can be observed in Laguerre-Gaussian (L-G) rovibrational cavity which was proposed
by Bhattacharya et al. (2008) [40], an analog of the double-OMIT in vibrational cavity. The double-
Keywords: OMIT in this research is naturally resulted from a single rovibrational mirror which vibrates and rotates
Double-OMIT simultaneously, rather than by integrating several subsystems as previously. We numerically examine the
Rovibrational cavity influence of the various factors on the double-OMIT and discuss its features and physics behind them
Stokes field in detail. In addition, we discuss the Stokes field generated via the four-wave mixing process in the L-G
Four-wave mixing rovibrational cavity.
© 2019 Elsevier B.V. All rights reserved.

1. Introduction tum router [23], ultraslow light propagation [24], four-wave mixing
[25], and precision measurement [26–28], etc. Afterwards, double-
Electromagnetically induced transparency (EIT) [1–4] is a quan- OMIT with two transparency windows, which also referred to as
tum interference phenomenon appearing in multiple-level atomic two-color OMIT, emerging in a coherent multichannels system, was
medium driven by strong laser fields, yielding the elimination of predicted by integrating more mechanical or optical modes, e.g., in
absorption of a weak signal light field. Since it was first theoret- the two coupled optomechanical systems [29–35], atomic-media
ically predicted [1] by Harris et al., and experimentally verified assisted optomechanical system [36], hybrid piezo-optomechanical
[2] in strontium vapor by Boller et al., EIT has been extensively cavity system [37].
studied and a lot of novel phenomena were revealed such as slow Double-OMIT provides more ways to control light and can po-
light [5,6], enhanced nonlinearity [7], quantum memories for pho- tentially be used for multichannels optical communication and
tons [8,9], narrower EIT resonance linewidth driven by L-G light precision measurement [38] in optomechanical systems. For more
[10,11], and the multiple-EIT with two or more transparency win- details, one can refer to a recent review article [39] in which
dow [12–19]. Generally, multiple-EIT emerge in a system with co- the latest research progress on the fundamentals and applications
herent multichannels, which can enable long-lived nonlinear inter- of OMIT and double-OMIT was summarized. While it is hard to
actions between weak fields, hence it was suggested to coherently
implement the double-OMIT in the usual optomechanical system.
control the photon-photon interactions [20].
Note that the double-OMIT phenomena in above-mentioned works
In 2010, OMIT, a phenomenon analogous to the EIT, was inves-
were almost obtained by integrating more mechanical or optical
tigated in the vibrational cavity optomechanical systems by Weis
subsystems, which makes the whole optomechanical system rel-
et al. [21]. The basic mechanism of OMIT is the destructive in-
atively complex. A natural idea is whether one can implement
terference between different pathways of the internal fields inside
double-OMIT without integrating more subsystems. The answer is
the optomechanical system, thereby leading to a transparency win-
positive. In 2008, Bhattacharya et al. proposed an L-G rovibrational
dow for the probe light in the otherwise strongly absorbed region
cavity [40], in which the rovibrational mirror can vibrate along as
[22], and it was suggested to apply in many fields including quan-
well as rotate about the cavity axis. Therefore, the L-G rovibra-
tional cavity provides a very promising platform for the study of
* Corresponding author. multi-mode macroscopic quantum phenomena in cavity optome-
E-mail address: xlfeng@shnu.edu.cn (X.-L. Feng). chanics.

https://doi.org/10.1016/j.physleta.2019.126153
0375-9601/© 2019 Elsevier B.V. All rights reserved.
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L 2z 1
H = h̄ωc a† a + + I ωφ2 φ 2
2I 2
 2 
p 1 2 2
+ + mω zq
2m 2
−h̄g φ a aφ − h̄g z a† aq
†
  
+ih̄ εl e −i ωl t + ε p e −i ω p t a† − H.c , (1)
Fig. 1. Schematic diagram of the present model studied is based on the L-G rovi-
brational cavity which was proposed by Bhattacharya et al. with a strong Gaussian
note that we have neglected ‘∧’ of all operators in Eq. (1), where
beam and a weak probe field. The partially transparent input mirror (IM) is the the first term describes the free Hamiltonian of the L-G cavity
rigidly fixed, the reflecting mirror (RM), perfectly reflective, is mounted in such a mode with the frequency ωc , the next two terms stand for, re-
way that it can vibrate along as well as rotate about the cavity axis. The deflection spectively, the free Hamiltonian of the rotational and vibrational
of the RM from its angular equilibrium position is indicated by the φ ; the z deflec-
modes of the RM, the fourth and the fifth represent the optoro-
tion of the RM has not been shown here for clarity. The topological charge on the
beams at various points has been indicated. tational and radiation pressure couplings between the L-G cavity
mode and the RM, respectively. The last term represents the in-
In this work, we show that the double-OMIT phenomenon can teraction between the cavity field with the pumping field and the
emerge in an L-G rovibrational cavity with a strong Gaussian beam probe field. In Eq. (1) a (a† ) is the annihilation (creation) operator
and a weak probe field, which is naturally resulted from the vi- for the cavity mode; φ and L z are, respectively, the angular dis-
bration and rotation modes of a single rovibrational mirror by placement and angular momentum of the RM, and I = mr 2 /2 is
exchanging the momentum and the orbital angular momentum moment of inertia about the cavity axis passing through its cen-
between the rovibrational mirror and the L-G cavity mode simul- ter; q and p are, respectively, the displacement and momentum
taneously, rather than by integrating several subsystems as previ- of the RM; ωφ (ωz ) is the angular rotation (linear vibration) fre-
ously done by others in vibrational cavity. We numerically examine quency of the RM; g φ = cl/ L ( g z = ωc / L ) is the optorotational
the influence of the power of the pumping field, the topological (radiation pressure) coupling strength  between the single photon
charge of L-G cavity mode, and the dissipation of system on the and the RM; εl = 2κ pl /h̄ωl (ε p = 2κ p p /h̄ω p ) is the strength
double-OMIT, and the physics behind them are clearly discussed. In of the pumping (probe) field, with κ being the cavity decay rate.
addition, we discuss the effect of the pumping field on Stokes field The nonzero commutation relations of the dynamical variables are
generated via the four-wave mixing process when the vibrational given by a, a† = 1, [φ, L z ] = i h̄, and [q, p] = i h̄.
and the rotational frequencies of rovibrational mirror are equal or In comparison to Ref. [40] where the cavity is driven by one
inconsistent. We believe that this research is of certain theoretical strong Gaussian beam only, while in the present model the cavity
significance and has potential applications in the rovibrational cav- is driven by both a strong Gaussian beam and a weak probe field.
ity systems such as slow and fast lights propagation of L-G light. This implies, in particular, that the Hamiltonian cannot be time-
This paper is organized as follows. In Sec. 2, we introduce the independent in any rotating frame if the strong and weak fields
theoretical model and Hamiltonian. In Sec. 3, we discuss the time have different frequencies. The above Hamiltonian can be rewrit-
evolutions of dynamical variables of system and solve them by ten in a frame rotating at frequency ωl as
Heisenberg-Langevin equations. In Sec. 4, the double-OMIT phe-
nomenon is studied in the homodyne spectrum of the output field,
 
† L 2z 1 2 2
and we investigate the features of the double-OMIT in the present H = h̄c a a + + I ωφ φ
2I 2
optomechanical system. In Sec. 5, the Stokes field is discussed by  
numerical examine. The last section concludes this paper. p2 1
+ + mω2z q2
2m 2
2. Model and Hamiltonian
−h̄g φ a aφ − h̄g z a† aq
†
  
The theoretical model adopted here is similar to that in
+ih̄ εl + ε p e −i  p t a† − H.c , (2)
Ref. [40], an extension of the rotational cavity system [41]. As
shown in Fig. 1, the two spiral phase plates, both acting as the where c = ωc − ωl is the detuning of the cavity mode from the
cavity mirrors, form a rovibrational cavity. The partially transpar- pumping field and  p = ω p − ωl the detuning of the probe field
ent input mirror (IM) is the rigidly fixed, the reflecting mirror from the pumping field.
(RM), perfectly reflective, is mounted in such a way that it can vi-
brate along as well as rotate about the cavity axis. When a strong 3. Solutions of the system
pumping field εl with frequency ωl , a Gaussian beam with zero
topological charge, is incident on the IM, where the IM does not In order to examine the mean response of the rovibrational cav-
change the topological charge of the light field passing through it, ity system to the probe field in the presence of the pumping field,
then the transmitted beam passing through IM still with a zero
starting from the Heisenberg equations of motion and consider-
topological charge gets charged to +2l after reflected from the RM.
ing the dissipations of the cavity field and the RM, we obtain the
Once again, when the light beam returning back to the IM, the
mean-value equations of the system operators:
transmission still does not change its topological charge, that is,
the output field from the cavity possesses the charge +2l, while dφ L z 
reflection gives the beam with the charge 0, as shown in Fig. 1 = ,
dt I
where the topological charge at each stage is expressed. The weak 
dL z
probe field ε p with frequency ω p is used to detect the mean op- = − I ωφ2 φ + h̄g φ a† a − γφ  L z  ,
tical response of the system, and the corresponding output field dt
is indicated by εout . The Hamiltonian of the system is of the form dq  p
[40] = , (3)
dt m
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dp  h̄g φ  
F φ∗ δφ− = as δa+ + a∗s δa− ,
†
= −mω2z q + h̄g z a† a − γz  p  ,
dt I
 
da G + δa+ = ias g z δq+ + g φ δφ+ + 1,
= −(κ + i ) a + εl + ε p e −i  p t ,  
dt G − δa− = ias g z δq− + g φ δφ− ,
where  = c − g φ φ − g z q is an effective detuning between the where F y = ω2y − 2p − i  p γ y ( y = z, φ), G ± = κ − i  p ± i ( g φ φs +
L-G cavity mode and the pumping field, γz and γφ are the in- † †
g z q s − c ). Note that δq− = δq+ , δφ− = δφ+ due to Hermite prop-
trinsic damping rates for vibrational and rotational modes of the
erty of δq and δφ , we can then obtain the solution of δa+ and
RM, respectively. Note that, in Eqs. (3) we have assumed the mean
δa−
values of the Langevin noise terms are zero and adopted the fac-
torization assumption (mean field approximation) because a  1 mI N
    δ a+ = , (8)
under the strong pumping field regime, viz, a† a = a† a. Thus, mI N G − + A ( Ia21 + ma12 ) − ma11 ξφ − Ia22 ξz
we are working in a classical regime and do not include quan-
mb11 ξφ + Ib22 ξz + A ( Ib21 − mb12 )
tum fluctuations [35,42]. In order to obtain the steady-state solu- δ a− = ,
tions, we can assume φ = φs + δφ because the pumping field is mI N  G ∗+ G ∗− + G ∗−2 A ( Ib21 + mb12 ) − Ib22 ξz − mb11 ξφ
 
much stronger than the probe field εl  ε p ; here φs describes (9)
the steady-state value governed by the pumping field, and δφ rep-
A A
resents the mean value of fluctuation which is proportional to the here a11 = 1 ( z, mG + ), a12 = , a21 = , a22 = 1 (φ, I G + ),
mG + I G+
weak probe field, the other dynamical variables do the same treat- A A
N = a11 a22 − a12 a21 , b11 = ∗ b12 =
, b21 =
ment. Under linearization condition, the steady-state values of the 2 ( z, mG − ), ,
mG ∗− I G ∗−
dynamical variables are easily found from Eq. (3) to be ∗ 
b22 = 2 (φ, I G − ), N = b 11 b 22 − b 12 b 21 , and ξ y = i h̄g y |a s | . More-
2 2

h̄g φ |as | 2
over, A = i h̄g φ g z |as |2 , η) = F y +
ξy
η) = F ∗y +
ξy
φs = and L zs = 0, 1 ( y, , and 2 ( y, .
I ωφ 2 η η
According to the standard input-output relation, the output
h̄g z |as |2 field satisfies relation
qs = and p s = 0, (4)
mω2z εl ε p −i  p t √
aout  + √ +√ e = 2κ a , (10)
εl |εl |2 2κ 2κ
as = ⇒ |as | = 2 , 2
κ + i κ + 2 hence we can obtain
where |as | represents the mean photon number in the rovibra-
2
εout = 2κ a − εl − ε p e−i p t , (11)
tional cavity. Then we can obtain the linearized Langevin equations
√
by retaining the first-order fluctuation term: where εout = 2κ aout . In addition, εout can be written as
dδφ δLz
= , εout = εouts + εout + ε p e−i p t + εout − ε∗p e i p t , (12)
dt I
dδ L z   where the first term corresponds to the output field at pumping
= − I ωφ2 δφ + h̄g φ a∗s δa + as δa† − γφ δ L z , field with frequency ωl , the second corresponds to the output field
dt
at probe field with frequency ω p , and the third corresponds to
dδ q δp
= , (5) the output field with frequency 2ωl − ω p which is called a Stokes
dt m field, and it is generated via the nonlinear four-wave mixing pro-
dδ p  
cess [25].
= − I ω2z δq + h̄g z a∗s δa + as δa† − γz δ p ,
dt The εout + and εout − are given by Eqs. (6), (11), and (12)
dδa
= − (κ + i ) δa + ig φ (φs δa + as δφ) εout + = 2κ δa+ − 1, (13)
dt
−i  p t εout − = 2κ δa− ,
+ig z (q s δa + as δq) + ε p e . (14)

To solve Eqs. (5) we divide each fluctuation into the two parts they can be measured by homodyne detecting technique [43]. The
as [26–35] amplitude of the rescaled εout + is
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
δφ δφ+ δφ− εT = εout + + 1 = u p + i v p , (15)
⎜ δ L z ⎟ ⎜ δ L z+ ⎟ ⎜ δ L z− ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ δq ⎟ = ⎜ δq+ ⎟ ε p e −i  p t + ⎜ δq− ⎟ ε ∗p e i  p t . (6) where u p and v p denote the in-phase and the out-of-phase
⎝ δp ⎠ ⎝ δp ⎠ ⎝ δp ⎠ quadratures of the output field associated with the absorption and
+ −
dispersion, respectively [42].
δa δ a+ δ a−
Substituting Eqs. (6) into Eqs. (5), we can obtain the following 4. Double-OMIT in the out probe field
equations:
In this section, we show that the double-OMIT can emerge in
h̄g z  ∗ †

the L-G rovibrational cavity. We will numerically investigate the
F z δq+ = a s δ a+ + a s δ a− ,
m various factors that influence the double-OMIT in the following
h̄g   subsections, such as the power of the pumping field, the mechani-
F z∗ δq− = as δa+ + a∗s δa− ,
z †
m cal frequencies of the RM, the topological charge of the L-G cavity
h̄g φ  ∗ †
 mode, the optomechanical couplings, and the dissipation of the
F φ δφ+ = a s δ a+ + a s δ a− , (7) system.
I
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Fig. 3. The absorption u p of the output field as a function of ( p − ω12 )/κ for
different mechanical frequencies. The mechanical frequencies are ωz = ωφ = 130π
kHz (blue dot-dashed), ωz = 120π kHz and ωφ = 140π kHz (green solid), and ωz =
110π kHz and ωφ = 150π kHz (red dashed), respectively. (For interpretation of the
colors in the figure(s), the reader is referred to the web version of this article.)
Fig. 2. The absorption curves (a) and dispersion curves (b) as function of the ( p −
ω12 )/κ for different the power of the pumping field. The powers of the pumping
field are pl = 0.1 mw, pl = 0.4 mw, and pl = 1 mw, respectively.

Unless stated otherwise, the parameter values chosen here are

mainly as follows [28,40,41,44]: L = 10 mm, κ = 25π kHz, λ =
1064 nm, pl = 0.5 mw, m = 10 mg, r = 16 μm, γz = γφ = 0.7π
kHz, ωz = 110π kHz and ωφ = 136π kHz, and the topological
charge l = 70. Note that we work in the resolved-side band regime
due to ωz , ωφ > κ , and we assume that ω12 = (ωz + ωφ )/2.

4.1. Relation to the power of the pumping field Fig. 4. Schematic of the energy-level diagram formed by the photon state of the
cavity field and phonon states of the RM in the rovibrational cavity system, where
| N , |n1  and |n2  denote the photon state of the cavity field, and rotational and vi-
Figs. 2(a) and 2(b) show, respectively, the absorption u p and
brational modes of the RM phonon states, respectively. | N , n1 , n2  ↔ | N + 1, n1 , n2 
dispersion v p plotted as functions of ( p − ω12 )/κ for different represents transition of the L-G cavity field, | N + 1, n1 , n2  ↔ | N , n1 + 1, n2  and
the power of the pumping field. From Fig. 2(a), we find that all | N + 1, n1 , n2  ↔ | N , n1 , n2 + 1 represent transition of the RM caused by the op-
the three absorption curves exhibit two basic symmetric narrow torotational and radiation pressure couplings between the L-G cavity field and the
EIT dips at ( p − ω12 )/κ = −0.5, 0.5 and an absorption peak at RM, respectively. Each -type energy level structure can individually form an EIT
dip.
( p − ω12 )/κ = 0. That is, two transparency windows emerge in
the profile of absorption curves, indicating the input probe field
could be simultaneously transparent at two symmetric frequencies, the two narrow EIT dips gradually move apart from each other and
the so-called double-OMIT. The central absorption peak implies the central absorption peak gradually becomes wider.
the probe field is almost fully absorbed. As shown in Fig. 2(a), Now let us explain the physical mechanism behind the above-
mentioned double-OMIT phenomenon. In our system the rovi-
with the increase of the power of the pumping field, each EIT
brational mirror can vibrate along as well as rotate about the
dip becomes deeper and broader, and the central absorption peak
cavity axis, thus its movement can be described by vibrational
becomes narrower, which can be understood by considering the
and  rotational modes of the RM phonon states [29–34,44], e.g.,
optomechanical interaction, including effective optorotational and
φ = h̄/2I ωφ (b + b† ), L z = i h̄I ωφ /2(b† − b), where b (b† ) is an-
radiation pressure couplings, becomes stronger with the increase
nihilation (creation) operator for the rotation mode, by combining
of pl . Therefore, by changing both g φ and g z to study the char-
with the photon state of the cavity field they form the energy-level
acteristics of double-OMIT, the similar conclusions can be drawn.
diagram as shown in Fig. 4 which contains two -type energy
Fig. 2(b) shows that the probe field undergoes a sharp dispersion
level structures. Each -type energy level structure can individ-
change when passing through the L-G rovibrational cavity. This
ually form an EIT dip for the probe field by the destructive in-
phenomenon plays a significant role in the investigate of slow and
terference between the probe field with frequency ω p and the
fast lights propagation of L-G light.
anti-Stokes field with the frequency ωl + ωz or ωl + ωφ . There-
fore, the double-OMIT dips appear at different frequencies of the
4.2. Relation to the mechanical frequencies of RM probe field ω p = ωl + ωz and ω p = ωl + ωφ , respectively. Specifi-
cally, when ωz = ωφ , the double-OMIT windows merge to a single
In Fig. 3, the absorption u p of the output field depending on deeper OMIT one.
( p − ω12 )/κ is showed for the mechanical frequencies. This fig-
ure indicates that the absorption curve has only one transparency 4.3. Relations to the topological charge of cavity mode and the
window when ωz = ωφ . This means that there is no obvious mode optomechanical couplings
splitting in u p of the output field in this case. However, when
ωz = ωφ the two transparency windows emerge, that is, normal Fig. 5(a) plots the absorption u p as a function of the ( p −
mode splitting appears, yielding the double-OMIT phenomenon. ω12 )/κ for different topological charge of the L-G cavity mode,
Furthermore, with the increase of difference between ωz and ωφ and it shows that, with the increase of the topological charge, the
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Fig. 6. The absorption u p as a function of ( p − ω12 )/ω12 for different the cavity
decay rate. The cavity decay rates are κ = 25π kHz, κ = 35π kHz, and κ = 45π
kHz, respectively.

Fig. 5. (a) The absorption u p as a function of ( p − ω12 )/κ for different topological
charge of the L-G cavity mode. The topological charges are l = 20, l = 40, and l = 70,
respectively. (b) The absorption u p as a function of ( p − ω12 )/κ for different ra-
diation pressure coupling. The radiation pressure coupling g z = 5.31 × 1016 Hz m−1
(blue dot-dashed), g z = 1.06 × 1017 Hz m−1 (green solid), and g z = 1.77 × 1017
Hz m−1 (red dashed), respectively.

right-hand EIT dip becomes deeper and wider while the left-hand
one keeps almost unchanged. That is to say, the topological charge
of the L-G cavity mode mainly affects the transparent window
corresponding to the optorotational coupling. This can be readily
understood because the increase of l only enhances the optorota-
tional coupling strength g φ ( g φ = cl/ L ) while has no effect on the
radiation pressure coupling g z . Therefore, by increasing g φ to study
the characteristics of double-OMIT in L-G rovibrational cavity, the
same results can be obtained. So we do not discuss the role of the
optorotational coupling in order to avoid the repeat of the figures.
On other hand, because g φ and g z are independent and equiva-
lent in status, hence by increasing g z to study the characteristics
of double-OMIT, the similar conclusions can be drawn. In Fig. 5(b),
the absorption u p is plotted as a function of ( p − ω12 )/κ for dif-
ferent radiation pressure coupling. It is easy to see that, with the Fig. 7. The absorption u p as a function of ( p − ω12 )/κ for the different damping of
increase of g z , the left-hand EIT dip becomes deeper and wider the RM. (a) Situation in which only the damping rate of vibrational mode changes:
while the right-hand one keeps almost unchanged. This result is γφ = 0.7π kHz, γφ = 7π kHz, and γφ = 42π kHz but keeping γz = 0.7π kHz un-
changed. (b) Situation in which both vibrational and rotating damping rates changes
consistent with our expectations. simultaneously: γz = γφ = 0.7π kHz, γz = γφ = 4.2π kHz, and γz = γφ = 42π kHz.

4.4. Influence of the dissipation of optomechanical system

are small, both EIT dips are very deep. While with the increase
Fig. 6 plots the absorption u p as a function of the ( p − of γφ the right-hand EIT dip gradually becomes shallow until dis-
ω12 )/ω12 for the different cavity decay rate κ . We find that the appears. This is because the rotational mode damping rate has a
two double-OMIT windows are scarcely affected by κ , that is, their harmful effect on the destructive interference between the probe
depth and width keep almost unchanged with the increase of κ . field and anti-Stokes field with frequency ωl + ωφ and completely
This result indicates that the double-OMIT phenomenon is robust damages such an interference for the very large γφ . At the same
against cavity decay. Increasing the value of κ causes to decreases time the depth of the left-hand EIT dip remains almost unchanged.
of the radiation pressure and optorotational couplings between the Our numerical calculation also indicates that the similar results to
cavity field and the rovibrational mirror. However, when both the vibrational mode can be obtained if keeping γφ unchanged while
vibration displacement and the angular displacement of the rovi- increasing γz , that is, the left-hand EIT dip gradually becomes shal-
brational mirror are large, a large strain is needed to reduce the low until disappears for very large γz .
radiation pressure and optorotational effect and vice versa. There- In addition, we consider both vibrational and rotating damping
fore, the system needs to be robust against cavity decay κ . Mean- rates change simultaneously. In Fig. 7(b), the absorption u p is plot-
while, the position and the height of the two absorption side peaks ted as a function of ( p − ω12 )/κ for the different damping rates
remain almost unchanged as well, while their width increases with with γz = γφ . For small damping rates the absorption u p turns out
κ , which can be regarded as broadening by cavity decay. to be a good double-OMIT with two deep EIT dips as shown by
In Fig. 7(a), the absorption u p is plotted as a function of the blue dot-dashed curve in Fig. 7(b). While with the increase of
( p − ω12 )/κ for different damping rate of rotational mode of the the damping rates the two EIT dips gradually become shallow un-
RM but keeping γz = 0.7π kHz unchanged. It is easy to see that til disappear, forming a standard Lorentzian-shaped peak for very
when γφ = 0.7π kHz, both vibrational and rotating damping rates large damping rate.
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Fig. 9. The intensity of the generated Stokes field |εout − |2 is plotted as a function of
Fig. 8. The intensity of the generated Stokes field |εout − |2 is plotted as a function
( p − ω12 )/κ for different powers of the pumping field pl = 0 mw, pl = 0.1 mw,
of ( p − ω12 )/κ for the different powers of the pumping field pl = 0 mw, pl = 0.1
and pl = 0.5 mw.
mw, and pl = 0.5 mw.

ω p = ωl + ωz and ω p = ωl + ωφ , respectively. In our study, the

5. Numerical results of the Stokes field double-OMIT is naturally resulted from the vibration and the rota-
tion modes of a single rovibrational mirror. Therefore, this research
In this section, we numerically examine the Stokes field gen- provides a simple and efficient way to implement double-OMIT
erated via the four-wave mixing process in the L-G rovibrational because it requires only a single rovibrational mirror, a single
cavity. pumping field, and a single probe field. We have numerically in-
As shown in Fig. 8, the intensity of the generated Stokes field vestigated the influence of the various factors on the double-OMIT,
|εout − |2 is plotted as a function of ( p − ω12 )/κ for the different and found the following main results: i) with the increase of pl ,
powers of the pumping field when ωz = 110π kHz and ωφ = 136π each EIT dip becomes deeper and broader, and the central absorp-
kHz, one can find that |εout − |2 = 0 in the absence of the pumping tion peak becomes narrower, this result may be applied to slow
field. However, two sharp peaks appear in each curve for certain and fast lights propagation of L-G light and high-resolution laser
power of the pumping field, that is the mode splitting in the Stokes spectroscopy; ii) the topological charge l mainly affects the trans-
field. The left peak of each curve represents maximal intensity of parency window corresponding to the rotation mode of RM, which
the Stokes field produced by the interaction between the pumping becomes deeper and wider with the increase of l; iii) the double-
field and the vibrational mode of the rovibrational mirror, and the OMIT phenomenon is robust to the cavity decay rate. In addition,
right one represents maximal intensity of the Stokes field produced we have discussed the Stokes field in the L-G rovibrational cavity
by the interaction between the pumping field and the rotational and found that the normal mode splitting appears in Stokes field
mode. Obviously, the maximal value of |εout − |2 is dependent on when ωz = ωφ , and the four-wave mixing process is completely
the power of the pumping field and it becomes larger with the in- suppressed on resonance. While when ωz = ωφ the normal mode
crease of the power of the pumping field. Particularly, the intensity splitting disappears in Stokes field and the four-wave mixing pro-
of the Stokes field becomes zero at ( p − ω12 )/κ = 0. Hence, the cess is not suppressed at ( p − ω12 )/κ = 0.
four-wave mixing process is completely suppressed on resonance Finally, as a remark, we should point out that the technology to
due to the destructive interference of the Stokes fields produced manufacture current optical micro-cavity is mature, and the spiral
by the interaction between the pumping field with vibrational and phase plates to generate L-G light has been widely used in optical
rotational modes the RM. Comparing |εout − |2 with u p shown in engineering. Therefore, it is permissible to realize L-G rovibrational
Fig. 2(a), u p also appears mode splitting when ωz = ωφ , while u p cavity in experiment, which paves a way to experimentally verify
has a maximum at ( p − ω12 )/κ = 0. this work.
In Fig. 9, the intensity of the generated Stokes field |εout − |2
is plotted as a function of ( p − ω12 )/κ for the different pow- Declaration of competing interest
ers of the pumping field when ωz = ωφ = 136π kHz. Similar to
the double-OMIT phenomenon in which both double-OMIT win- The authors declare that they have no known competing finan-
dows merge to a single one when ωz = ωφ , here the both inten- cial interests or personal relationships that could have appeared to
sity peaks merge to a single one at ( p − ω12 )/κ = 0. Therefore, influence the work reported in this paper.
when ωz = ωφ , the four-wave mixing process is not suppressed
at ( p − ω12 )/κ = 0 due to the constructive interference between Acknowledgements
the Stokes fields produced by the interaction between the pump-
ing field with vibrational and rotational modes of the RM at the This work is supported by National Natural Science Foundation
same mechanical frequency, so that there is no apparent normal of China under Grant Nos. 61475168, 11674231, 11074079 and
mode splitting in this case. Comparing |εout − |2 with u p shown in 11804225 and Ministry of Science and Technology of the People’s
blue dot-dashed of Fig. 3, u p also does not appear mode splitting Republic of China (2018YFA0404803).
when ωz = ωφ , while u p is almost zero at ( p − ω12 )/κ = 0.
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