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Laser Physics
PAPER
Bistability in a hybrid optomechanical system: effect of a gain medium
To cite this article: A Asghari Nejad et al 2017 Laser Phys. 27 115202
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This content was downloaded from IP address 129.8.242.67 on 25/10/2017 at 19:05
Laser Physics
Astro Ltd
Laser Phys. 27 (2017) 115202 (6pp)
https://doi.org/10.1088/1555-6611/aa8bac
Bistability in a hybrid optomechanical
system: effect of a gain medium
A Asghari Nejad, H R Baghshahi1 and H R Askari
Department of Physics, Faculty of Science, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran
E-mail: a.asghari@stu.vru.ac.ir, baghshahi@vru.ac.ir and hraskari@vru.ac.ir
Received 18 July 2017, revised 5 September 2017
Accepted for publication 5 September 2017
Published 26 October 2017
Abstract
In this paper, we investigate the optical bistability of a hybrid optomechanical system
consisting of two coupled cavities: a bare optomechanical cavity (with an oscillating mirror
at one end) and a traditional one. The traditional cavity is filled with an optical parametric
amplifier (OPA), and an input pump laser is applied to it. The Hamiltonian of the system is
written in a rotating frame. The dynamics of the system is driven by the quantum Langevin
equations of motion. We demonstrate that the presence of an OPA can dramatically affect the
type of stability of the optomechanical cavity. We show that it is possible to create a proper
optical bistability for the optomechanical cavity by changing the gain coefficient of the OPA.
Also, it is shown that changing the phase of the field driving the OPA has two different effects
on the bistability region of the optomechanical cavity. Moreover, we show that by choosing
a proper value for the detuning of the traditional cavity it is possible to observe a tristable
behavior in the optomechanical cavity.
Keywords: optomechanics, oscillating mirror, optical bistability, optical parametric amplifier
(Some figures may appear in colour only in the online journal)
1. Introduction
in optomechanical cavities with an oscillating mirror at one
end, the position of the oscillating mirror depends on the
intracavity intensity of the optomechanical cavity [11, 25,
26, 35]. Therefore, in these systems, optical bistability has
been investigated via the study of the position of the oscil­
lating mirror. To take perfect control of an optomechanical
system, a number of authors have proposed some systems
containing different physical options, such as atomic [3, 8,
11, 20, 33, 34, 36] and nonlinear mediums [35–38]. Recent
evidence suggests that a nonlinear medium can modify the
bistability of an optomechanical system [35]. Also, opti­
cal bistability in an optomechanical cavity can be changed
indirectly by the optical properties of another optical system
[11]. Because of the nonlinear nature of optical bistability,
enhancing the nonlinearity of an optomechanical system can
lead to the creation of a more controllable optical bistability
[39]. Experimentally, optical bistability has been observed
and employed in micro and nano cavities [40]. In this direc­
tion, Dorsel et al [28] have studied the output intensity of an
optomechanical cavity. The results obtained show that the
system behaves in a bistable regime. Also, optical bistabil­
ity of a system consisting of photonic-crystal nanocavities
Quantum optomechanics is a field of research in quantum
optics which investigates the interaction between light and
mechanical degrees of freedom [1–13]. In optomechanical
systems, in which light pressure affects mechanical oscilla­
tions, notable physical phenomena has been observed, such as
the cooling of mechanical oscillations [3, 4, 14–18], photon
blockade [19–23], single photon nonlinearity [24], optical
bistability [9, 11, 12, 25–30] and photon–photon interactions
[31]. Optomechanical systems have been studied both theor­
etically and experimentally. The findings of these studies
demonstrate a close relationship between the position of the
mechanical resonator of the system and the observed phe­
nomena [3, 5, 8, 10, 13, 20, 32, 33]. The key aspect of opto­
mechanical systems is the coupling between the mechanical
and optical modes, in which the oscillating part of the system
can modify the optical mode [34]. As mentioned previously,
optical bistability has been investigated in optomechanical
systems. Published studies in this area of research show that
1
Author to whom any correspondence should be addressed.
1555-6611/17/115202+6$33.00
1
© 2017 Astro Ltd Printed in the UK
A A Nejad et al
Laser Phys. 27 (2017) 115202
in =
2κc Pin
ωp
, where Pin is the power of the pump laser. The
Hamiltonian of the system reads:
H = H0 + HI ,
(1a)
p2
1
2 2
H0 = ωa a† a + ωc c† c + (
+ mωm
q ) − gom qa† a,
2m 2
(1b)
Figure 1. Schematic diagram of the proposed hybrid
optomechanical system. The system includes two coupled
optical cavities; one of them is traditional and the other is an
optomechanical cavity. The system is driven by a pump laser.
HI = iG(eiθ c†2 − e−iθ c2 ) + J(c† + c)(a† + a)
−iω t †
iω t
+ iin (e p c − e p c),
(1c)
where, a and c are the annihilation operators of cavities A
and C, respectively, which satisfy the commutation relation
[a, a† ] = [c, c† ] = 1. Also, p (q) denotes the momentum (posi­
tion) operator of the oscillating mirror. The first (second) term
of H0 depicts the free Hamiltonian of cavity A (C). The third
term of H0 is the energy of the movable mirror. The fourth
term of H0 arises from the coupling between the radiation
pressure of cavity A and the oscillating mirror. Respectively,
the first and second terms of HI present the Hamiltonian of the
OPA and the interaction between the cavities (with a coupling
strength of J). The last term of HI is denoted as the laser driven
term (i.e. the interaction between input pump laser and cavity
C). Applying the rotating wave approximation (i.e. neglecting
fast oscillating terms such as c† a† and ca), and also using a
rotating frame with a rotation frequency of ωp, one can obtain
has been utilized to configure an all-optical switching device
[41]. Therefore, investigation of optical bistability and other
aspects of it in optomechanical systems can lead to the fabri­
cation of more controllable all-optical switches. In our work,
we are interested in investigating the optical bistability of an
optomechanical cavity, where a nonlinear medium indirectly
affects the stability of it. Our proposed system opens the
way for better control of the bistability of an optomechani­
cal system.
In this paper, we consider a hybrid optomechanical sys­
tem which includes two coupled cavities. One of them is a
traditional cavity that encloses an OPA and the other is a cav­
ity with an oscillating mirror at one end. Due to the coupling
between cavities, optical stability of the optomechanical cav­
ity is affected by varying of optical parameters of the other
cavity. The Hamiltonian of the system is written in a rotat­
ing frame. The dynamics of the system is described by the
quant­um Langevin equations of motion and obtained equa­
tions are solved in a steady state regime. Results show that
in the presence of an OPA the behavior of proposed system
changes from a monostable regime (in the absence of an OPA)
to a bistable one (in the presence of an OPA). Also, we show
that, in our system there are different control parameters to
control and change the domain of bistability region.
The remainder of the paper is organized as follows.
Section 2 gives the Hamiltonian and the equations of motion
of the system. The results obtained from the steady state solu­
tions to the equations of the system and a discussion around
them are presented in section 3. Finally, section 4 gives a sum­
mary of the paper.
H0 = ∆a a† a + ∆c c† c + (
p2
1
2 2
+ mωm
q ) − gom qa† a,
2m 2
(2a)
HI = iG(eiθ c†2 − e−iθ c2 ) + J(c† a + ca† ) + iin (c† − c),
(2b)
where, ∆a = ωa − ωp and ∆c = ωc − ωp are defined as the
detunings of the cavities A and C, respectively.
2.2. Quantum Langevin equations of motion
The dynamics of the system is governed by the quantum
Langevin equations (QLEs) of motion as follows.
p
q̇ = ,
(3a)
m
2. Physical model
2
ṗ = −mωm
q − γm p + gom a† a + ξ(t),
(3b)
2.1. Model design and Hamiltonian
ȧ = −(κa + i(∆a − gom q))a − iJc + 2κa ain (t),
(3c)
The system under consideration is sketched in figure 1. The
proposed system is formed by two coupled optical cavities,
an optomechanical cavity (A) and a generic one (C). Cavity A
has a resonance frequency of ωa (when the oscillating mirror
is at its rest position) and a bandwidth of κa. The oscillating
mirror of cavity A is considered as a harmonic oscillator with
resonance frequency ωm , decay rate γm and effective mass m.
The generic cavity has a resonance frequency of ωc and corre­
sponding decay rate κc, which contains an optical paramet­
ric amplifier (OPA) with a gain coefficient of G. Cavity C is
driven by a pump laser of frequency ωp and an amplitude of
ȧ† = −(κa − i(∆a − gom q))a† + iJc† + 2κa a†in (t),
(3d)
ċ = −(κc + i∆c )c + 2Geiθ c† + in − iJa + 2κc cin (t),
(3e)
ċ† = −(κc − i∆c )c† + 2Ge−iθ c + in + iJa† + 2κc c†in (t),
(3f )
where ξ(t) is the Brownian noise force with zero mean value
and the following correlation function [42, 43]:
2
A A Nejad et al
Laser Phys. 27 (2017) 115202
γm
ω
ξ(t)ξ(t ) =
dω[1 + coth(
)]ωe−iω(t−t ) .
2πωm
2kB T
(4)
Here kB is the Boltzmann constant and ain (t) (cin (t)) is the
input vacuum noise operator to cavity A (C) with zero mean
value, where [42, 43]
ain (t)a†in (t ) = δ(t − t ),
(5a)
ain (t)ain (t ) = a†in (t)a†in (t ) = 0,
(5b)
cin (t)c†in (t ) = δ(t − t ),
(5c)
Figure 2. Intracavity intensity for cavity A as a function of input
power in both the presence (thick solid line) and in absence (thin
solid line) of the OPA. The set of parameters used are as ωm = 10
MHz, κa = κc = 0.9ωm , λ = 810 nm, ∆a = ∆c = 0 , J = 5ωm,
m = 10 ng, l = 1 mm, G = 3κa , and θ = 0.2π .
cin (t)cin (t ) = c†in (t)c†in (t ) = 0.
(5d)
In this paper we are interested in the investigation of
optical bistability for cavity A in a steady state regime.
Applying the steady state conditions (i.e. q̇ = 0 , ṗ = 0 ,
ȧ = a˙† = ċ = c˙† = 0 , ain (t) = cin (t) = 0 and ξ(t) = 0 )
on equations (3a)–(3f), one can obtain the following steady
state equations.
ps
= 0,
(6a)
m
|as |2
|in |2 = 2
,
(13a)
κn + ∆2n
where
κn =
2
−mωm
qs − γm ps + gom |as |2 = 0,
(6b)
J(J 2 + 2Gκa cos θ + 2G∆ sin θ + κa κc − ∆∆c )
,
A1 + B1
(13b)
J(2Gκa sin θ − 2G∆ cos θ − κa ∆c − κc ∆)
∆n =
.
(13c)
A1 + B1
−(κa + i(∆a − gom qs ))as − iJcs = 0,
(6c)
−(κa − i(∆a − gom qs ))a∗s + iJc∗s = 0,
(6d)
3. Results and discussion
−(κc + i∆c )cs + 2Geiθ c∗s + in − iJas = 0,
(6e)
To demonstrate optical bistability for cavity A, we con­
sider the following group of parameters: ωm = 10 MHz,
κa = κc = 0.9ωm , λ = 810 nm, ∆a = ∆c = 0 , J = 5ωm,
m = 10 ng, l = 1 mm, G = 3κa and θ = 0.2π . Figure 2 shows
the response of the intracavity mean photon number of cavity
A (na = |as |2) to the presence of an OPA. This figure repre­
sents na versus Pin for two different states: the presence (thick
solid line) and the absence (thin solid line) of an OPA. As
can be seen, optical bistability vanishes with the removal of
the OPA from the system. This behavior occurs because the
presence or absence of the OPA can change the mean pho­
ton number in cavity A, resulting in a different radiation
pressure on the oscillating mirror. Therefore, we observe a
different behavior from the system. So, the presence of the
OPA in the proposed system is important to achieve optical
bistability for the mean photon number of the optomechani­
cal cavity. Figure 3 presents another aspect of the response of
the proposed system to the presence of the OPA. As can be
seen, this figure shows na versus normalized detuning of the
optomechanical cavity (∆a /ωm ) in the presence (thick solid
line) and the absence (thin solid line) of the OPA. Obviously,
embedding the OPA inside the traditional cavity results in one
being able to observe a two-peak curve for na versus ∆a /ωm .
However, in the absence of the OPA, we face a wide, singlepeaked curve for na versus ∆a /ωm . Accordingly, for a proper
−(κc − i∆c )c∗s + 2Ge−iθ cs + in + iJa∗s = 0.
(6f)
Here, the subscripts s point out the steady state mean values
g2
of the operators. Also, ∆ = ∆a − mωom2 |as |2 is defined as the
m
effective detuning for cavity A. Therefore, the steady state
solutions to equations (6a)–(6f ) are obtained as follows.
ps = 0,
(7)
gom
q(8)
|as |2 ,
s =
2
mωm
iJ(J 2 + (κa − i∆)(2eiθ G + κc − i∆c ))in
as = −
,
(9)
A1 + B1
2iGeiθ (Ja∗s − iin ) − (iκc + ∆c )(Jas + iin )
cs =
,
(10)
κ2c + ∆2c − 4G2
where,
A1 = J 4 + 2J 2 (κa κc − ∆∆c ),
(11)
B1 = −(κ2a + ∆2 )(4G2 − (κ2c + ∆2c )).
(12)
Using equations (9) and (10), one can obtain the following
equation for |in |2:
3
A A Nejad et al
Laser Phys. 27 (2017) 115202
Figure 3. Effect of the OPA on the curve of na versus ∆a /ωm . The
Figure 5. Effect of G on the plot of na versus ∆a /ωm . The sketched
plots are for G = 2κa (thick solid line), G = 4κa (thin solid line)
and G = 6κa (dotted line), when Pin = 50 mW.
curves displayed are for the presence (thick solid line) and absence
(thin solid line) of the OPA. The rest of the parameters have values
as considered in figure 2.
Figure 6. Plot of na versus ∆a /ωm for θ = 0 (thick solid line),
Figure 4. Plot of na versus Pin for G = 3κa (thick solid line),
θ = 0.3π (thin solid line), and θ = 0.6π (dotted line). The rest of
the parameters are as considered in figure 2.
G = 4κa (thin solid line) and G = 5κa (dotted line). Here ∆a = 0
and the rest of the parameters are as assumed in figure 2.
value of the gain coefficient of the OPA, it is possible to make
a proper bistability in the system. For this reason, figure 4
shows na versus Pin for different values of G. It can seen from
this figure that increasing G decreases the domain of the bista­
bility region. Indeed, the OPA can change the mean photon
number in cavity C by changing G. This process changes the
mean photon number in cavity A, and then a different radia­
tion pressure acts on the oscillating mirror. Therefore, a differ­
ent plot is obtained for na versus Pin and ∆a /ωm .
Also, we can study the effect of changing G on the plot of
na versus ∆a /ωm . From figure 5 it can be seen that with an
increase of G, both peaks of na become smaller in width and
the separation distance between them decreases. This result
shows that we can obtain a controlled optical bistability for
cavity A by changing G when the detuning of this cavity has
a fixed value. The graphs presented in figure 5 show a close
relation between the value of G and the depth of the valley of
na versus ∆a /ωm . According to our results, with an increase
of G, the mean photon number in cavity A becomes smaller,
which leads to a decreased radiation pressure on the mechani­
cal mirror, resulting in a closer distance between the peaks
of the plot of na versus ∆a /ωm . In the limit of G → 20κa , the
system shows monostability and we observe just a single peak
for the curve of na versus ∆a /ωm .
Figure 7. Plot of na versus ∆a /ωm for θ = 0.6π (thick solid line),
θ = 0.8π (thin solid line), and θ = π (dotted line). The rest of the
parameters are as considered in figure 2.
Previously, we stated that an external field of the phase θ
drives the OPA. Figure 6 reveals the effect of increasing θ on
the plot of na versus ∆a /ωm . From the data presented in fig­
ure 6 it is apparent that with an increase of θ we observe a nar­
rower left-hand-side peak for the graph of na versus ∆a /ωm .
But, with an increase of θ from 0.6π to π, we observe that the
left-hand-side peak of the curve of na versus ∆a /ωm becomes
wider (figure 7). Also, for the right-hand-side peak of the plot
4
A A Nejad et al
Laser Phys. 27 (2017) 115202
Figure 8. Effect of θ on the bistability of intracavity intensity for
the optomechanical cavity. Other parameters used are ωm = 10
MHz, κa = κc = 0.9ωm , λ = 810 nm, ∆a = ∆c = 0 , J = 5ωm,
m = 10 ng, l = 1 mm, G = 2κa .
Figure 10. Plot of na versus input power for ∆c = −7ωm. Other
parameters used are as ωm = 10 MHz, κa = κc = 0.9ωm , λ = 810 nm,
∆a = 0, J = 5ωm, m = 10 ng, l = 1 mm and G = 2κa .
behavior is a periodic behavior which will be repeated even
after we increase θ from π. In comparison with [35], changing
θ in our system has a different effect on the optical bistabil­
ity of the optomechanical cavity rather than that as reported
in this reference. As reported in [35], changing of θ leads to
a change of the whole of the curve of na versus Pin, which is
different from our results. Also, the presented plots of na ver­
sus ∆a /ωm in [35] are different from the depicted plots of na
versus ∆a /ωm of our optomechanical cavity. In this reference,
the curve of na versus ∆a /ωm has just a single peak, while
in our system this curve has two separated peaks. Therefore,
our system shows a different behavior to that of the system
presented in [35].
In figures 2, 4 and 8 we assumed that ∆a = 0. Our results
show that the detuning of cavity A can dramatically influence
the stability of this cavity. In this direction, figure 9 presents
plots of na versus Pin for ∆a = 0 (thick solid line), ∆a = ωm
(thin solid line) and ∆a = 2ωm (dotted line). Apparently, with
an increase of ∆a , the domain of the bistability region becomes
wider. The data presented in figure 9 is a result of the coupling
between the cavities. For a bare cavity we observe a curve
with a single peak for the plot of the mean photon number of
it versus the detuning of the cavity. In our system the coupling
between cavities leads to a splitting of the spectrum to lower
and upper sidebands. Therefore, with an increase of ∆a , the
system moves toward the upper sideband of cavity A, resulting
in enhanced nonlinearity in the system. Therefore the bista­
bility region is broadened to higher input powers. A notable
result is obtained when we assume ∆c = −7ωm . In this situ­
ation, we observe tristability in the mean photon number in
cavity A. Indeed, in the range of −15ωm < ∆c < −5.4ωm , our
system exhibits tristability in the mean photon number in cav­
ity A. For this reason, the plot of na versus Pin for ∆c = −7ωm
is presented in figure 10. As mentioned previously, coupling
between the cavities results in a splitting of the spectrum of
both cavities. For ∆c = −7ωm , the system performs at the
lower sideband of cavity C. Therefore the nonlinearity in the
system extremely grows, which results in the observation of
tristability in the mean photon number in cavity A.
Figure 9. Plot of na versus input power for ∆a = 0 (thick solid
line), ∆a = ωm (thin solid line) and ∆a = 2ωm (dotted line). Other
parameters considered are ωm = 10 MHz, κa = κc = 0.9ωm ,
λ = 810 nm, ∆c = 0, J = 5ωm, m = 10 ng, l = 1 mm, G = 2κa .
of na versus ∆a /ωm we obtain reverse results in comparison
with obtained results for the left-hand-side peak. Comparing
with figure 5, we can see that the curve of na versus ∆a /ωm
has less sensitivity to the changing of θ, rather than that of G.
But it is possible to obtain a proper plot for na versus ∆a /ωm
by engineering the values of both θ and G. The data presented
in figures 6 and 7 are a result of the appearance of terms eiθ
and e−iθ in equations (3e) and (3f), which lead to a periodic
behavior for cs and then the mean photon number in cavity C.
Another remarkable result is depicted in figure 8. This
figure shows the graph of na versus Pin for different values
of θ. As can be seen, increasing θ from zero to 0.6π results
in broadening the domain of the bistability region. However,
with an increase of θ from 0.6π to π the domain of the bista­
bility region becomes smaller. Therefore, we see that there is
a critical value for θ to extending the domain of the bistability
region. According to our results, increasing θ from zero to 0.6π
leads to a decreased photon number in cavity A. Therefore,
due to this process we observe the first stable point at higher
input powers. But from θ = 0.6π to θ = π the mean photon
number in cavity A increases, which results in the observation
of the first stable point in lower input powers. The observed
5
A A Nejad et al
Laser Phys. 27 (2017) 115202
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4. Conclusions
In summary, we have presented a hybrid optomechanical sys­
tem consisting of two coupled cavities, an optomechanical
cavity and a traditional one. An OPA is embedded inside the
traditional cavity and an input pump laser is applied to it. The
Hamiltonian of the system has been exhibited in a rotating
frame with a rotation frequency of the frequency of the input
pump laser. The dynamics of the system has been obtained by
the quantum Langevin equations of motion in a steady state
framework. The results show a close relationship between the
optical bistability in the optomechanical cavity and the pres­
ence of the OPA. We observed that for reasonable values of
the parameters of the system, optical bistability vanishes in
the absence of the OPA. We have shown that with an increase
of the gain coefficient of the OPA the domain of the bistability
region becomes smaller. Also, we have studied the effect of
increasing the gain coefficient of the OPA on the plot of the
mean photon number in the optomechanical cavity versus its
detuning. The second major finding of this study is that with
an increase of the phase of the field driving the OPA (θ) from
zero to 0.6π , the domain of the bistability region becomes
wider. But with an increase θ from 0.6π to π the domain of the
bistability region becomes smaller. Also it can be seen that by
choosing the value of the detuning of a traditional cavity from
a certain range, the system shows tristability for the mean pho­
ton number in the optomechanical cavity. Experimentally, our
proposed system can be used to fabricate an all-optical switch,
which has different options (for instance, detunings of both
cavities and the gain coefficient of the OPA) to control the out­
put optical intensity of the system. Among the control options,
detunings of the cavities are appropriate to achieve the desired
output from the system, because one can completely control
the frequency of the input laser.
References
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