Controllable optical bistability in an optomechanical system assisted by microwave

Optik 154 (2018) 139–144

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Original research article

Controllable optical bistability in an optomechanical system assisted by microwave Xiao-Fei Zhu a,b , Lei-Dong Wang a,b , Jia-Kai Yan a,b , Bin Chen a,b,∗ a b

Department of Physics, College of Physics and optoelectronics, Taiyuan University of Technology, Taiyuan, China Key Lab of Advanced Transducers and Intelligent Control System, Ministry of Education and Shanxi Province, Taiyuan, China

a r t i c l e

i n f o

Article history: Received 20 June 2017 Accepted 2 October 2017 Keywords: Optical bistability Microwave field Hybrid optomechanical system

a b s t r a c t We theoretically investigate bistable behaviour of the mean intracavity photon number in an optomechanical system, where a cavity mode and microwave field are coupled to a common mechanical resonator. The results of the numerical simulation demonstrate that by tuning the signal of microwave field, optical bistability can be controlled. More importantly, we found that by tuning the microwave input power or pump laser-exciton detuning, the optical bistability can be obviously modulated. These results can be used to find potential applications in optical switch in the quantum information processing. © 2017 Elsevier GmbH. All rights reserved.

1. Introduction During the past few years, we have witnessed a series of developments at the optomechanical system, and many theoretical and experimental analyses have been done [1–5]. Some remarkable and interesting phenomena, such as optomechanically induced transparency (OMIT) [6–8], quantum-state transfer [9,10], resolved sideband cooling to the quantum mechanical ground state [11–15], slow light [16–20], and so on, have successively been discovered in the cavity optomechanical system. Among all the nonlinear phenomena in a quantum optomechanical system, optical bistability is one of the focuses of research interest. Recently, the bistable behaviour of the mean intracavity photon number in optomechanical system with a Bose–Einstein condensate (BEC) [21,22], ultracold atoms [23–25], and a quantum well [26] has been studied systematically. The photon number in the optical cavity with a BEC allow bistable behaviour is usually low and even below unity due to the collective atomic motion. However, in the generic optomechanical system, optical bistability occurs at high photon numbers. Then, Jiang et al. [27] have shown that bistable behaviour of the intracavity photon number in a two-mode optomechanical system in the simultaneous presence of the two strong pump laser beams and a weak probe laser beam can effective modulate the optical bistability (OB) [28]. By modulating the pump laser power or cavity-pump beam detuning, the mean intracavity photon number has changed. So the image of the bistable behaviour has changed. They also have discussed in detail the red-sideband and blue-sideband in their two-mode optomechanical system and found some interesting phenomena. Motivated by their work, we investigate the OB in a cavity optomechanical system assisted by microwave. The numerical results demonstrate that in such a hybrid system which consisted by optical cavity, mechanical oscillator and microwave field, optical bistability can be modulated, and that the microwave input power, microwave decay rate, coupling rate or microwave cavity-pump detuning play an important role in controlling the optical bistable behaviour. Besides, by modulating these parameters of microwave to control optical beam, we can simplify process to modulate the optical threshold. And this paper

∗ Corresponding author at: Department of Physics, College of Physics and optoelectronics, Taiyuan University of Technology, Taiyuan, China E-mail address: [email protected] (B. Chen). https://doi.org/10.1016/j.ijleo.2017.10.001 0030-4026/© 2017 Elsevier GmbH. All rights reserved.

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Fig. 1. Schematic diagram for a generic optomechanical system assisted by the microwave field.

is organized as follows. In Section 2, we describe the total Hamiltonian of the system and derive the quantum Heisenberg Lagrange equations for the system operators [29]. Section 3 is devoted to the analysis of bistable behavior shown by the mean intracavity optical field in the optomechanical cavity, and some results are discussed in detail. In Section 4, we make a brief summary. 2. Model and method We consider the hybrid optomechanical system that is consisting of an optical cavity, mechanical oscillator and a microwave cavity [30,31], as shown in Fig. 1. The optomechanical system is composed of an optical cavity and a microwave cavity, and these two cavities are simultaneously coupled to the same mechanical resonator. The microwave cavity with frequency ω1 is driven by a microwave signal with frequency ωp1 and amplitude Ep1 , while the optical cavity with frequency ω2 is driven by a pump laser with frequency ωp2 and amplitude Ep2 . The effective Hamiltonian of this hybrid system by transferring into the rotating frame at the pump frequency can be read as follows [31–33] H

=

1 ωm (P 2 + X 2 ) + a a† a + c c † c 2

+g1 a† aX − g2 c † cX +i



1 k1 Ep1 (a† − a) + i

(1)



2 k2 Ep2 (c † − c),

where a = ω1 − ωp1 and c = ω2 − ωp2 . The first term gives the energy of the mechanical resonator with frequency ωm , P and X are the momentum operators and dimensionless position [35–37] of the resonator respectively with the communication relation [X, P] = i. The second term is energy of the microwave field and the third is energy of the optical cavity, where c is the cavity-pump detuning and a, c are the annihilation operators of the microwave and optical modes [38–40]. The fourth and fifth terms respectively represent the coupling between the mechanical resonator and the microwave cavity as well as the optical cavity by the coupling rate gj (j = 1, 2). The last two terms represent the interaction between the cavity and the driving field. The amplitude of the driving



field is related to the input power Pl by El = Pl /ωl , where l = p1, p2 represent the microwave field and the optical pump field, respectively. Relying on the Heisenberg Lagrange equation and the communication relation [a, a† ] = 1, [c, c† ] = 1 and [X, P] = i, we can obtain these operators a, c, and X. In our works, we focus on the mean response of the hybrid system, so all the operators can be reduced to their expectation values. We need to drop the thermal and quantum noise terms [40] out of consideration but we should take the damping terms into consideration, then we can obtain the Heisenberg Lagrange equations as follows,



k1 da = − ia + 2 dt



dc k2 = − ic + 2 dt

 

a − ig1 aX + c + ig2 cX +



1 k1 Ep1 ,

(2)

2 k2 Ep2 ,

(3)



d2 X dX 2 + ωm + m X = ωm (g2 c ∗ c − g1 a ∗ a), dt dt 2

(4)

ˆ and  m is the damping rate of the mechanical where a(t) ≡ ˆa(t), c(t) ≡ ˆc (t), a ∗ (t) ≡ ˆa† (t), c ∗ (t) ≡ ˆc † (t), X(t) ≡ X(t) resonator. Assuming a = a¯ + ıa, c = c¯ + ıc and X = X¯ + ıX, ignoring the second order and higher order small amount and setting all the time derivatives to zero, we can obtain the steady solutions of (2)–(4) as follows:



a¯ =

1 k1 Ep1

k1 /2 + i1

(5)

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Fig. 2. Mean intracavity photon number n2 as a function of optical pump power P2 . Other parameters of the system are m = 20 ng, G/2 =−12 GHz/nm, m /2 = 41.0 kHz, /2 = 15.0 MHz, m /2 = 51.8 MHz, and  =− m .

 c¯ =

2 k2 Ep2

(6)

k2 /2 + i2

X¯ =

1 ¯ 2 ), (g2 |¯c |2 − g1 |a| ωm

(7)

¯ a, ¯ c¯ and X¯ stand for the microwave interactivity field, optical interactivity field and mechanical here 2 = ω2 − ωp2 + g2 X, position. Combining Eqs. (5)–(7) as follows:



¯ 2 |a|

k12 4

 |¯c |2

k22 4

X¯ =



¯ + (1 + g1 X)

2

2 = 1 k1 Ep1

(8)

2 = 2 k2 Ep2

(9)

 ¯ + (2 − g2 X)

2

1 ¯ 2) (g2 |¯c |2 − g1 |a| ωm

(10)

¯ 2 is replaced by n1 , |¯c |2 is replaced by n2 , nj means the intracavity photon number. Rearranging Eqs. (8)–(10), the here |a| resulting equations are n1 [ n2 [

k12 4 k22 4

¯ 2 ] = 1 k1 E 2 , + (1 + g1 X) p1 2

¯ ] = 2 k2 E 2 . + (2 − g2 X) p2

(11) (12)

The strong optomechanical interaction [42–44] between the microwave and the mechanical mode also facilitates the study of nonlinear dynamics in the optomechanical cavity. We can clearly see from Eqs. (11) and (12) that mean intracavity photon number n1 and n2 are interconnected, in which one can be tuned by the power or frequency of the pump laser beams via changing parameters Ep1 , Ep2 , 1 , and 2 . This enables us to control the mean intracavity photon number in various ways. For instance, the systems mean intracavity photon number n2 in the optical cavity can be controlled directly by the right optomechanical pump beam or indirectly by the left microwave field. The effect of two kinds of control mode is not the same. And in this paper, we focus on the parameters of microwave modulating the mean intracavity photon number. 3. Results and discussion In order to illustrate the numerical results, we choose the realistic parameters of the system as follows [45,46]: ωm = 2 × 10.69 MHz,  m = 2 × 30 Hz, ωp1 = 2 × 7.47 GHz, ωp2 = 2 × 1.77 THz, k1 = 2 × 170 kHz, k2 = 2 × 215 kHz, 1 = 2 = 0.5. Fig. 2 is the OB hysteresis curve for the mean intracavity photon number under different input power. This hysteresis curve clearly indicates the bistable behaviour of the intracavity photon intensity. Right now, the mean intracavity photon number follows the upper power if we start scanning with a low driving power and increase the power from zero gradually. When it reaches the first bistable point p1 , it jump down to the lower stable branch and continues to follow that branch for further increasing laser power. In return, if we start reducing the input laser the photon number will be found to be increasing by following the lower branch [47,48]; however by decreasing the power even further, when it reaches the second bistable point p2 , it will jump up to the upper stable branch and continue to increase further along that branch.

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Fig. 3. Optical cavity mean intracavity photon number n1 in the microwave field as a function of the cavity-pump detuning 1 (a) for microwave coupling rate g1 equal to 2 × 230 Hz, 2 × 250 Hz, 2 × 260 Hz, and (b) for microwave input power P1 equal to 0.2, 0.5, 0.8 nW (from bottom to top).

Fig. 4. Optical cavity mean intracavity photon number n2 of the optomechanical cavity as a function of (a) the microwave input power P1 for coupling rate g1 equal to 2 × 230 Hz, 2 × 240 Hz and 2 × 260 Hz and (b) the microwave input power P1 for decay rate k1 is equal to 2 × 170 kHz, 2 × 230 kHz and 2 × 270 kHz.

The optomechanical system assisted by microwave field we consider here enables more controllability in the bistable behaviour of the mean intracavity photon number. In Fig. 3(a), the optical bistability of mean intracavity photon number n1 in the microwave field as a function of the cavity-pump detuning 1 for different microwave coupling rates g1 as 2 × 230 Hz, 2 × 250 Hz, 2 × 260 Hz. The microwave input power (left cavity) is kept 0.8 nW. From this figure, it can be clearly seen that the optical bistability can be effectively modulated by tuning the microwave coupling rate g1 . With increasing coupling rate, for larger value of g1 , optical bistable range is seen to occur for larger in the optomechanical cavity. Fig. 3(b) shows the optical bistability of photon number n1 in the microwave field as a function of the cavity-pump detuning 1 for various microwave input powers P1 with 1 = 2 = ωm . Power plays a significant role in controlling the bistable behaviour of microwave field in left cavity. When we slightly modulate the input power, the curve (OB) of the mean intracavity photon number changes will be very obvious. So with the increasing of the microwave input power P1 , the mean intracavity photon number n1 increases along the branch of the curve and as we can see from the curve, the larger the microwave input power, the larger the bistable range. Fig. 4 plot the mean intracavity photon number n2 of the optomechanical cavity as a function of the microwave input power P1 . For the cavity detuning in red sideband regime, the cavity-pump detuning 1 is constant which is equal to ωm . In Fig. 4(a), the optical pump power P2 (right cavity) is equal to 4.0 mW and the optical cavity-pump detuning 2 (right cavity) is equal to ωm . As seen, continuing increasing the microwave input power P1 , the mean intracavity photon number n2 from the branch of the curve jump to the next branch and n2 decreases with increasing the microwave input power P1 . The process is the same as the modulating the cavity-pump detuning 1 . With the increasing of the coupling rate g1 , the region of the optical behaviour become small. But the optical bistable threshold reaches our expectation. Because we are hoping to get the low value of the optical bistable threshold when developing the bistable devices which has the practical value. So it is very important to modulate the parameters of the system reducing the value of the optical bistable threshold. Here we can modulate the coupling rate to control the bistable threshold. And we also can modulate the decay rate to modulate the value of bistable threshold. As we can see from Fig. 4(b), with increasing the decay rate k1 , the bistable region is reduced, and the bistable threshold decreases which also can reach our expectation. Fig. 5 plots the mean intracavity photon number n1 of the optomechanical cavity as a function of the optical pump power P2 (right cavity). For the cavity detuning in red sideband regime, 1 = ωm . As shown in Fig. 5(a), continuing increasing the optical pump power P2 , the mean intracavity photon number n1 from the branch of the curve jumps to the next branch and the n1 decreases with increasing the optical pump power P2 at the same time. And with the increasing of the coupling rate g2 , the region of the optical behaviour become small. But the optical bistable threshold reaches our expectation. Therefore we get the low value of the optical bistable threshold when developing the bistable devices which has the practical value. That

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Fig. 5. Optical cavity mean intracavity photon number n1 of the optomechanical cavity as a function of (a) the optical pump power P2 for coupling rate g2 equal to 2 × 4400 Hz, 2 × 4500 Hz and 2 × 4600 Hz and (b) the optical pump power P2 for decay rate k2 is equal to 2 × 215 kHz, 2 × 275 kHz and 2 × 315 kHz.

is to say, it is very important to modulate the parameters of the system reducing the value of the optical bistable threshold. Here we can modulate the coupling rate g2 to control the bistable threshold. And we also can modulate the decay rate k2 to modulate the value of bistable threshold. As we can see from Fig. 5(b), with increasing the decay rate k2 , the bistable region is reduced, and the bistable threshold decreases. Through these images, we can know that the coupling rate g1 , g2 or the decay rate k1 , k2 will play a vital role in the optical mean intracavity photon number, and then controlling the optical bistability. That is to say, these parameters play important roles in control of the intracavity photon number well. In discussions, we have demonstrated controlling the optical bistability behaviour in the optomechanical cavity assisted by microwave field, and the mean intracavity photon number is often very large, which can reach tens of thousands of quantity. In order to reach the bistable regime, the optomechanical system (g2 = 0) would need more photon number in the hybrid field. The follows, we shall show that the microwave–optomechanical system allows for optical bistability at extremely low cavity photon number. Generally, such low photon number cannot generate OB in an empty cavity optomechanical system [28–30]. Then OB can still exist whatever in the optomechanical cavity or in the microwave cavity at the extremely low intracavity photon number. More importantly, the OB of mean intracavity photon number in the optomechanical system under consideration also provides a candidate for realizing a controllable optical switch. So the two stable branches of photon number in the mechanical cavity act as the optical switch. When the frequency and pump power of the optical mechanical pump beam are fixed, the switch between the lower stable branch and the upper stable branch can easily be realized by controlling the cavity-pump detuning or input power of the microwave pump beam. Furthermore, the microwave pump beam can be used as a control parameter to realize this switch. So, as a hybrid optomechanical system, microwave signal takes an important part by modulating parameter value to the bistability. 4. Conclusion In conclusion, we have investigated the bistable behaviour of the intracavity photon number for an optomechanical resonator containing a microwave field. As a result of the nonlinearity induced by the optomechanical coupling, this nonclassical property can be controlled by tuning the power or cavity-pump detuning of the microwave. Our work focus on investigating the parameters of this hybrid system assisted by microwave on the OB. Therefore, motivated by the formation of the OB, the microwave introduced will exist a great effect on the optical cavity. So, mediated by the microwave signal, this hybrid optomechanical system can serve as a better optical switch. According to analyze these transformation, we can obtain that the bistable behaviour and the optical bistable threshold can be controlled easily by modulating the value of decay rate, coupling rate or input power of microwave. These results may have potential applications in the quantum information processing. Acknowledgements This work is supported by the National Science Foundation of China (Grant Nos. 11504258 and 11347181), Natural Science Foundation of Shanxi Province (Grant No. 201421011-1), and the Qualified personnel Foundation of Taiyuan University of Technology (Grant No. tyutrc201245a). References [1] [2] [3] [4] [5]

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