Normal mode splitting due to quadratic reactive coupling in a microdisk-waveguide optomechanical system

Physics Letters A 377 (2012) 133–137

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

Normal mode splitting due to quadratic reactive coupling in a microdisk-waveguide optomechanical system Chang-Bao Fu a,b , Kai-Hui Gu a , Xiao-Bo Yan a , Xin Yang a , Cui-Li Cui a , Jin-Hui Wu a,∗ a b

College of Physics, Jilin University, Changchun 130012, PR China College of Physics, Tonghua Normal University, Tonghua 134000, PR China

a r t i c l e

i n f o

Article history: Received 6 May 2012 Received in revised form 25 October 2012 Accepted 4 November 2012 Available online 9 November 2012 Communicated by R. Wu Keywords: Normal mode splitting Quadratic reactive coupling Four-wave mixing

a b s t r a c t We study the effects of quadratic reactive coupling on normal mode splitting in a microdisk-waveguide optomechanical system, whose stability can be determined by the Routh–Hurwitz criterion. We find that the quadratic reactive coupling is important for the mode splitting of a Stokes field only in the presence of a strong pump field. For a large enough pump power, the quadratic reactive coupling will lead to larger and asymmetric frequency shifts for the two Stokes modes, which are however smaller and symmetric when only the linear reactive coupling is considered. Such a unique normal mode splitting happens in an efficient four-wave-mixing process of optomechanical interaction. This is why an anti-Stokes field is seen to arise in the presence of an excited waveguide oscillation. The anti-Stokes field has a much stronger interaction with the lower-frequency Stokes mode because their frequencies and amplitudes change remarkably in a similar way when the quadratic coupling constant is gradually increased. © 2012 Elsevier B.V. All rights reserved.

1. Introduction

Cavity optomechanics is a rapidly advancing field that explores the interaction of optical and mechanical degrees of freedom in a composite system. A number of optomechanical phenomena have been demonstrated in a variety of coherent optical systems combined with micromechanical devices [1–10]. Cooling a micromechanical device to its quantum ground state is a key goal in the field of optomechanics and has attracted great attention to pursue essential progresses both in theory and in experiment [11–14]. Accompanied with the cooling of mechanical oscillators in the resolved sideband regime, an interesting phenomenon, normal mode splitting [15–18], appears as a result of nonlinear four-wave mixing with the micromechanical vibration included. In these works on normal mode splitting, only dispersive coupling was taken into account that manifests itself in the sensitivity of the eigen-frequency of a cavity mode on the displacement of a micromechanical device. Very recently, Huang et al. [19] showed that the reactive-couplinginduced normal mode splitting may be attained in a microdisk cavity coupled to a nanomechanical waveguide [20]. Here the reactive coupling indicates the linear dependence of the decay rate of the microdisk cavity on the small displacement of the waveguide. Note that the optomechanical coupling may also be quadratic instead of linear in certain situations as discussed in Refs. [21,22].

*

Corresponding author. Tel.: +86 431 85168401; fax: +86 431 85168822. E-mail address: [email protected] (J.-H. Wu).

0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.11.007

In this Letter, we study the effects of a quadratic coupling term on the reactive-coupling-induced normal mode splitting in a microdisk-waveguide optomechanical system. This nonlinear system may be either stable or unstable depending on real values of relevant parameters as determined by the Routh–Hurwitz criterion [23,24]. In the stable regime, we find that the quadratic term of reactive optomechanical coupling is important to be considered only when the pump field is strong enough and it can result in an asymmetric and enhanced normal mode splitting of the Stokes field. It is of particular interest that the incident Stokes field and the generated anti-Stokes field may attain rather large amplitudes due to the efficient energy transfer from the pump field for a specific quadratic coupling constant. Our manuscript is structured as follows: we describe in Section 2 the optomechanical system under consideration by first introducing a quadratic term into the reactive coupling expression, then giving the equation of motion for expectation values of relevant operators, and finally obtaining the output field with three different frequencies; we discuss in Section 3 the effects of quadratic reactive coupling on normal mode splitting by examining the output field at both Stokes and anti-Stokes frequencies with a series of numerical calculations for various realistic parameters. 2. The theoretical model We consider an optomechanical system consisting of a fixed microdisk cavity and a movable nanomechanical waveguide along the horizontal axis as shown in Fig. 1. A strong pump field with frequency ω p and a weak Stokes field with frequency ωs

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damping rate κi ≈ κ /2 of the cavity, we obtain the following mean value equations

˙q =

 p m

,

  2  p˙  = −h¯ χ c † c − mωm q − γm  p        − ih¯ ε p + εs e −i δt c † − ε p + εs∗ e i δt c  

2 κom κom 2g + √ q , × √ + − √

κ

κ κ κ     ˙c  = −i ωc − ω p + χ q c  − κ + κom q + g q2 c  







√



2 κom κom 2g q2  κ + √ q + − √ +√ κ κ κ κ 2 

2 √ κ κ 2g q2  om om , +√ + εs e −i δt κ + √ q + − √ κ κ κ κ 2

+ εp

Fig. 1. The sketch of a microdisk-waveguide optomechanical system. A pump field with frequency ω p and a Stokes field with frequency ωs are incident upon one end of the waveguide. An anti-Stokes field with frequency 2ω p − ωs can be generated inside the composite system and found from the other end of the waveguide together with the Stokes and pump fields.

simultaneously enter the free standing waveguide and propagate inside the composite optomechanical system. Because there exists phonon–photon interactions between the waveguide and the cavity, the cavity eigen-frequency and the cavity decay rate, ωc (q) and κe (q), depend on the displacement q of the waveguide from its equilibrium position. The two q-dependent quantities represent the so-called dispersive optomechanical coupling and reactive optomechanical coupling, respectively. In the following, we denote the annihilation (creation) operator of the microdisk cavity by c (c † ) and the momentum (position) operator of the waveguide oscillator with mass m by p (q). We also introduce ampli tudes of the pump field and the Stokes field, ε p = ℘ p /¯hω p and

|ε s | =



℘s /¯hωs , with ℘ p (℘s ) being the power of the incident pump (Stokes) field. Then we can write down the relevant Hamiltonian [20]





H = h¯ ωc (q)c † c + ih¯ 2κe (q)ε p e −i ω p t c † − e i ω p t c

 1 L  2 2 q + h¯ n g ω p ε 2p + ωs |εs |2 + mωm c˜ 2   p2  + ih¯ 2κe (q) εs e −i ωs t c † − εs∗ e i ωs t c + , 2m



(1)

(3)

where γm denotes momentum decay rate of the waveguide oscillator while δ = ωs − ω p is the detuning between the Stokes field and the pump field. In addition, we have assumed κe = κi ≈ κ /2 for simplicity. Since the Stokes field εs is much weaker than the pump field ε p , the steady-state solution of Eqs. (3) can be expressed with a perturbation theory as

q  p c 



=

q0 p0 c0



+

q+ p+ c+



εs e

−i δ t

+

q− p− c−



εs∗ e iδt .

(4)

Substituting Eq. (4) into Eqs. (3), equating coefficients of terms with the same frequency, and keeping constant terms and those terms containing εs e −i δt and εs∗ e i δt , we then obtain

c0 = q0 = c+ =

Aεp

κ + κom q0 + i Δ + gq20

,

−h¯ χ |c 0 |2 − ih¯ ε p κ√omκ (c 0∗ − c 0 ) ,  2 2g  ∗ 2 om √ ( mωm + ih¯ ε p − κκ√ + c0 − c0 ) κ κ 1 



J F ∗ A + F B∗ N + B∗ E A ,

d(δ)  F  ∗ ∗ c− = ∗ V N −AJ , d (δ)  B∗  q+ = V N − A J∗ , d(δ) q− = (q+ )∗ ,





(5)

where L is the length of the waveguide, c˜ is the speed of light in vacuum, n g is the group index of the waveguide optical mode. In the typical case of a small displacement, ωc (q) and κe (q) can be expanded to the power series of q. If the pump field is very weak, such an expansion can be truncated to the first order of q so that the linear dispersive and reactive couplings are attained. When the pump field is strong enough, however, the quadratic reactive coupling should be also important and could result in a large modification of relevant results. This means that we have

d(δ) = V B E + J F

ωc (q) = ωc + χ q,

The output field observed at one end of the waveguide can √ be derived by using the input–output relation c out  = 2κe (q)c  [19], from which and Eq. (4) we have

2

κe (q) = κe + κom q + gq ,

(2)

where χ (κom ) describes the linear dispersive (reactive) coupling strength while g denotes the quadratic reactive coupling constant. When the cavity field is transformed to a rotating frame at the pump frequency ω p , by using the Heisenberg equations of motion, adding the corresponding damping rates, and assuming an intrinsic

∗

∗

∗ ∗

+ FB J ,

with Δ = ωc − ω p + χ q0 , A =

√

(6) 2

q20

κ + κ√omκ q0 + (− κκ√omκ + √2gκ ) 2 , B = 2

κ + κom q0 + i (Δ + δ) + gq20 , E = m(ωm2 − δ 2 − i γm δ) + ih¯ ε p (− κκ√omκ + 2g √

κ

2 κom )(c 0∗ − c 0 ), F = −c 0 (κom + i χ ) + κ√omκ ε p + ε p (− κ √ + κ

2g √ )q0

−

κ 2g √ 2gq0 c 0 , J = h¯ χ c 0 + i h¯ κ√om ε p + i h¯ ε p (− + ) q0 , V = κ + κ κ κ κ 2 κom q0 + i (Δ − δ) + gq20 , and N = −ih¯ c0∗ κ√omκ − ih¯ (− κκ√omκ + √2gκ )q0 c0∗ . 2 κom √

c out  = c p + c s εs e −i δt + c as εs∗ e i δt ,

(7)

with c p , c s , and c as being the optical responses at the pump frequency ω p , the Stokes frequency ωs , and the new (anti-Stokes) frequency 2ω p − ωs , respectively. Thus we finally obtain

C.-B. Fu et al. / Physics Letters A 377 (2012) 133–137

√









2 κom 1 κom 2g − √ q0 c 0 + + √ c 0 q20 , κ 2 κ κ κ

 √ κom κom c s = √ q+ c 0 + κ 1 + q0 c + κ κ 



2 κom 2g 1 + − √ +√ c + q20 + c 0 q0 q+ , 2 κ κ κ

 √ κom κom c as = √ q− c 0 + κ 1 + q0 c − κ κ 



2 κom 2g 1 + − √ +√ c − q20 + c 0 q0 q− . 2 κ κ κ

cp =

κ 1+

(8)

To investigate the effects of quadratic reactive coupling on normal mode splitting, we first define two phase quadratures υs = (c s + c ∗s )/2 and υ˜ s = (c s − c ∗s )/2i of the Stokes component, one of which shows the normal mode splitting when plotted as a function of the normalized detuning δ/ωm . Then we introduce the output power G as of the anti-Stokes component

G as =

135

h¯ (2ω p − ωs )|c as εs |2

℘s

,

Fig. 2. The positive real roots of d(δ) as a function of the pump power ℘ p for g = 0 (thin solid), 3|κom | nm−1 (thick solid), 5|κom | nm−1 (dash-dotted), 9|κom | nm−1 (dotted), and 10|κom | nm−1 (dashed). Other parameters are given at the beginning of Section 3 in the main text.

(9)

which is generated in a nonlinear process of four-wave mixing with the pump field, the Stokes field, and the waveguide oscillator involved. It is worth stressing that an optomechanical system generally works in the stable regime, which can be assessed by applying the Routh–Hurwitz criterion [23]. To examine the system stability, Eqs. (3) for the mean values q,  p , and c  are not enough and should be compensated with dynamical equations of the fluctuation operators δq, δ p, δ c, and δ c † . These nonlinear dynamic equa√ tions can be linearized by setting δ X = (δ c † + δ c )/ 2 and δ Y = √ i (δ c † − δ c )/ 2 after neglecting all second-order smaller terms. It is well known that a nonlinear system goes into one steady state only when the coefficient matrix A of its linearized dynamic equations have roots λi with negative real parts [Re(λi ) < 0 for all λi ] [24]. Instead of trying to find all λi with i ∈ [1, 4], here we first attain a characteristic polynomial a0 λ4 + a1 λ3 + a2 λ2 + a3 λ + a4 = 0 from the determinant of matrix A and then construct a sequence T 0 = a0 ≡ 1, T 1 = a1 , T 2 = a1 a2 − a0 a3 , T 3 = a1 a2 a3 − a21 a4 − a0 a23 , and T 4 = a1 a2 a3 a4 − a21 a24 − a0 a23 a4 . Considering that the number of roots with positive real parts is equal to the number of sign changes in the sequence T i , we can say that our optomechanical system has a steady state only when all T i are positive for certain parameters (κom , g, ℘ p , etc.). 3. Results and discussions In this section, we evaluate the output field at the Stokes frequency ωs and the anti-Stokes frequency 2ω p − ωs to demonstrate various effects of quadratic reactive coupling on normal mode splitting. The microdisk-waveguide coupling is strongest when δ = ±ωm or δ = ±Δ, so we consider here only the most efficient case of Δ = ωm as far as nonlinear four-wave mixing is concerned. In numerical calculations, most parameters are adopted from Ref. [19]: λ p = 1564.25 nm, χ = 2π × 2 MHz/nm, m = 2 pg, κ = 0.2ωm , ωm = 2π × 25.45 MHz, κom = 2π × 26.6 MHz/nm, and the mechanical quality factor Q = ωm /γm = 5000. In Fig. 2, we show positive real roots of d(δ) as a function of the pump power ℘ p for different quadratic coupling constants g = 0, 3|κom | nm−1 , 5|κom | nm−1 , 9|κom | nm−1 , and 10|κom | nm−1 . As we can see, only when the pump power is very small, the two real roots of d(δ) are almost indistinguishable to result in an unclear normal mode splitting. If the pump power is large enough, we can find two very different real roots of d(δ) as a signature of obvious

Fig. 3. The frequency separation of two split Stokes modes as a function of the scaled quadratic coupling constant g /|κom | for ℘ p = 50 μW (a), 300 μW (b), and 700 μW (c). Other parameters are the same as in Fig. 2.

normal mode splitting. Note also that, with the increasing of g, the frequency separation between two split Stokes modes becomes more and more asymmetric. Relevant results are further confirmed in Fig. 3 where the frequency separation between two split Stokes modes is plotted as a function of the quadratic coupling constant g for different pump powers ℘ p . In Fig. 2 and Fig. 3, the pump power has been restricted in the range of [0, 700 μW] to make sure that our optomechanical system dose not enter the unstable regime. With the Routh–Hurwitz criterion, by calculating the T i sequence, we find that our optomechanical system works in the stable regime only for ℘ p ∈ [0, 790 μW] and [0, 700 μW] when g = 9|κom | nm−1 and 10|κom | nm−1 . As to g = 0, 3|κom | nm−1 , and 5|κom | nm−1 , much larger ranges of ℘ p are expected to guarantee the stability of our optomechanical system. To have a deeper insight into the quadratically reactivecoupling-induced normal mode splitting, we now examine one phase quadrature of the Stokes component and the output power of the anti-Stokes component. In Fig. 4, we show υs and G as as a function of the normalized detuning δ/ωm for ℘ p = 50 μW and g = 0, 3|κom | nm−1 , 5|κom | nm−1 , and 9|κom | nm−1 , respectively. We find from Fig. 4(a) and Fig. 4(b) that all four curves of υs and G as are almost indistinguishable, which once again verifies that the quadratic reactive coupling is trivial for a small pump power. That is, the quadratic coupling term gq2 in Eq. (2) can be safely omitted when ℘ p is not too large. This is in agreement with Fig. 2 and Fig. 3 where the small normal mode splitting is not sensitive to g for ℘ p = 50 μW. In Fig. 5, we show υs and G as as a function of the normalized detuning δ/ωm for ℘ p = 300 μW and g = 0, 3|κom | nm−1 ,

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C.-B. Fu et al. / Physics Letters A 377 (2012) 133–137

Fig. 4. The Stokes quadrature υs (a) and the anti-Stokes power G as (b) as a function of the scaled detuning δ/ωm for g = 0 (solid), 3|κom | nm−1 (dashed), 5|κom | nm−1 (dash-dotted), and 9|κom | nm−1 (dotted). Other parameters are the same as in Fig. 2 except ℘ p = 50 μW.

Fig. 5. The Stokes quadrature υs (a) and the anti-Stokes power G as (b) as a function of the scaled detuning δ/ωm for g = 0 (solid), 3|κom | nm−1 (dashed), 5|κom | nm−1 (dash-dotted), and 9|κom | nm−1 (dotted). Other parameters are the same as in Fig. 2 except ℘ p = 300 μW.

Fig. 6. The Stokes quadrature υs (a) and the anti-Stokes power G as (b) as a function of the scaled detuning δ/ωm for g = 0 (solid), 3|κom | nm−1 (dashed), 5|κom | nm−1 (dash-dotted), and 9|κom | nm−1 (dotted). Other parameters are the same as in Fig. 2 except ℘ p = 700 μW.

5|κom | nm−1 , and 9|κom | nm−1 , respectively. We find from Fig. 5(a) that the dip between two Stokes peaks gradually broadens because one moves left and becomes higher while the other changes little with the increasing of g. In addition, the only anti-Stokes peak in Fig. 5(b) is found to depend critically on g in a similar way as the left Stokes peak in Fig. 5(a). The above results are also in agreement with Fig. 2 and Fig. 3 where one mode (lower frequency) is more sensitive to g than the other mode (higher frequency) for ℘ p = 300 μW. Thus, for a strong pump field, the quadratic coupling constant becomes so important that it has to be taken into account. It is the quadratic reactive coupling that results in two asymmetric Stokes peaks with a larger frequency separation. The higher Stokes peak indicates a more efficient nonlinear interaction with the pump field, the anti-Stokes field, and the waveguide oscillator. In Fig. 6, we show υs and G as as a function of the normalized detuning δ/ωm for ℘ p = 700 μW and g = 0, 3|κom | nm−1 , 5|κom | nm−1 , and 9|κom | nm−1 , respectively. It is clear that, for a larger pump power, the quadratic coupling constant has a more important influence on the Stokes mode splitting and the antiStokes mode generation. That is, the two Stokes modes become more asymmetric in position and in amplitude; the only antiStokes mode attains a much higher amplitude together with the left Stokes mode for ℘ p = 700 μW. The prominent amplification of both Stokes and anti-Stokes modes at δ/ωm ≈ 0.5 can be attributed to the efficient energy transfer from a very strong pump

field in the presence of a quite large quadratic coupling constant. Finally, we note that the vibration amplitude of a nanomechanical waveguide is usually very small (i.e. of the order of nanometers) and the reactive coupling rate depends linearly on such a small displacement for typically small pump powers. The quadratic term gq2 is essential to be considered in the reactive coupling rate only when the pump field is strong enough as shown before. We now briefly discuss how large could be the quadratic coupling constant g in a potential experiment. In Ref. [25], the authors show via an appropriate analysis that the quadratic coupling constant is about 1023 Hz/m2 for realistic parameters in another optomechanical system. The quadratic coupling constant in our optomechanical system has been given several different values, among which g 9|κom | nm−1 is about 1024 Hz/m2 . This value is one-order larger than that in Ref. [25] but is also in the reach of a potential experiment. One main reason is that the reactive coupling can be strengthened by adjusting the size ratio between the microdisk and the waveguide as discussed in a relevant experiment [20]. 4. Conclusions In summary, we have shown how normal mode splitting is affected by the quadratic reactive coupling in a microdisk-waveguide system driven by a Stokes field and a pump field. We find that the quadratic reactive coupling cannot be neglected in a four-wavemixing process characterized by mode splitting of the Stokes field and nonlinear generation of an anti-Stokes field only when the pump field is strong enough. For a small pump power, the two split Stokes modes are not sensitive to the quadratic term gq2 of reactive optomechanical coupling in the presence of little energy transfer from the pump field. For a large pump power, however, the two split Stokes modes have a larger and larger frequency separation and the lower-frequency mode changes more in its frequency shift to result in an asymmetric double-peak structure when the quadratic coupling constant is increased. In addition, the generated anti-Stokes field changes in its frequency and amplitude just like the lower-frequency Stokes mode when the quadratic coupling constant is increased for a given pump power. This indicates that the four-wave mixing mentioned above is more efficient for the lower-frequency Stokes mode than the higher-frequency Stokes mode. Last but not least, both Stokes and anti-Stokes fields may be amplified due to the efficient energy transfer from the pump field in the presence of an excited waveguide oscillation. Acknowledgements We would like to thank the support from the National Natural Science Foundation of China under grant 11174110, the National Basic Research Program of China under grant 2011CB921603 and the Graduate Innovation Fund of Jilin University under grant 20121027. References [1] T. Carmon, H. Rokhsari, L. Yang, T.J. Kippenberg, K.J. Vahala, Phys. Rev. Lett. 94 (2005) 223902. [2] T.J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, K.J. Vahala, Phys. Rev. Lett. 95 (2005) 033901. [3] J.D. Teufel, J.W. Harlow, C.A. Regal, K.W. Lehnert, Phys. Rev. Lett. 101 (2008) 197203. [4] A. Schliesser, P. Del’Haye, N. Nooshi, K.J. Vahala, T.J. Kippenberg, Phys. Rev. Lett. 97 (2006) 243905. [5] I. Wilson-Rae, P. Zoller, A. Imamoglu, Phys. Rev. Lett. 92 (2004) 075507. [6] L. Tian, P. Zoller, Phys. Rev. Lett. 93 (2004) 266403. [7] K.R. Brown, J. Britton, R.J. Epstein, J. Chiaverini, D. Leibfried, D.J. Wineland, Phys. Rev. Lett. 99 (2007) 137205.

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