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 View the article online for updates and enhancements. 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. 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A 63 023812 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. 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