Access the full text.

Sign up today, get DeepDyve free for 14 days.

Photonics
, Volume 8 (7) – Jun 24, 2021

/lp/multidisciplinary-digital-publishing-institute/robust-four-wave-mixing-and-double-second-order-optomechanically-3GZ1lvV1MY

- Publisher
- Multidisciplinary Digital Publishing Institute
- Copyright
- © 1996-2021 MDPI (Basel, Switzerland) unless otherwise stated Disclaimer The statements, opinions and data contained in the journals are solely those of the individual authors and contributors and not of the publisher and the editor(s). MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Terms and Conditions Privacy Policy
- ISSN
- 2304-6732
- DOI
- 10.3390/photonics8070234
- Publisher site
- See Article on Publisher Site

hv photonics Article Robust Four-Wave Mixing and Double Second-Order Optomechanically Induced Transparency Sideband in a Hybrid Optomechanical System Huajun Chen School of Mechanics and Photoelectric Physics, Anhui University of Science and Technology, Huainan 232001, China; hjchen@aust.edu.cn Abstract: We theoretically research the four-wave mixing (FWM) and second-order sideband genera- tion (SSG) in a hybrid optomechanical system under the condition of pump on-resonance and pump off-resonance, where an optomechanical resonator is coupled to another nanomechanical resonator (NR) via Coulomb interaction. Using the standard quantum optics method and input–output theory, we obtain the analytical solution of the FWM and SSG with strict derivation. According to the numerical simulations, we ﬁnd that the FWM can be controlled via regulating the coupling strength and the frequency difference of the two NRs under different detuning, which also gives a means to determine the coupling strength of the two NRs. Furthermore, the SSG is sensitive to the detuning, which shows double second-order optomechanically induced transparency (OMIT) sidebands via controlling the coupling strength and frequencies of the resonators. Our investigation may increase the comprehension of nonlinear phenomena in hybrid optomechanics systems. Keywords: hybrid optomechanical systems; four-wave mixing; second-order sideband generation Citation: Chen, H. Robust Four-Wave Mixing and Double 1. Introduction Second-Order Optomechanically Cavity optomechanics (COM) systems [1,2] (as a milestone [3] in optics history), which Induced Transparency Sideband in a investigates the interaction of electromagnetic ﬁelds and micromechanical motion, have Hybrid Optomechanical System. witnessed signiﬁcant progress over the past decade both in fundamental studies and prac- Photonics 2021, 8, 234. https:// tical applications including ground state cooling [4–8], mass sensing [9,10], high-precision doi.org/10.3390/photonics8070234 measurements [11–17], and quantum information processing [18–21]. The mechanical mo- tions in COM systems, due to the radiation pressure forces, are tunable by optomechanical Received: 18 May 2021 interactions, which in turn inﬂuence the optical medes resulting in prominent quantum Accepted: 22 June 2021 interference effects. There are many famous phenomena that have been obtained in COM Published: 24 June 2021 systems, such as phonon lasers [22–25], squeezing [26,27], entanglement [20,21], nonre- ciprocity [28–30], exceptional point (EP) devices [24,25,31,32], optomechanically induced Publisher’s Note: MDPI stays neutral transparency (OMIT) [33–40], and OMIT induced slow and fast light [36,41–44]. We notice with regard to jurisdictional claims in that many above phenomena remain in the linear optical regime. published maps and institutional afﬁl- COM systems also present a medium to research the nonlinear phenomena between iations. the electromagnetic ﬁeld light and matter. Optical bistability [45–49], as a representative nonlinear phenomena, has been extensively investigated in some kinds of COM systems. In the COM systems, if the cavity is driven by a strong pump laser ﬁeld (with frequency w ) and a weak probe laser ﬁeld (with frequency w ), then when the two pump photons mix Copyright: © 2021 by the authors. with a probe photon, an idler photon at frequency 2w w will emerge, as a result the four- p s Licensee MDPI, Basel, Switzerland. wave mixing (FWM) appear in the ouput ﬁeld, which has been investigated in different This article is an open access article optomechanical systems [50–53]. Except nonlinear phenomena of optical bistability and distributed under the terms and FWM, recently, another remarkable nonlinear optomechanical effect, i.e., the second-order conditions of the Creative Commons sideband generation (SSG), has also been demonstrated in COM systems [54–64], where Attribution (CC BY) license (https:// the SSG will appear in the output ﬁelds with frequencies w 2d (d = w w is the creativecommons.org/licenses/by/ p s p probe-pump detuning) and w + 2d (w 2d) is the second-order upper (lower) sideband 4.0/). p p Photonics 2021, 8, 234. https://doi.org/10.3390/photonics8070234 https://www.mdpi.com/journal/photonics Photonics 2021, 8, 234 2 of 12 frequency component. However, the FWM and SGG in a hybrid optomechanical system, where a typical optomechanical cavity coupled to another NR via Coulomb interaction has not been demonstrated until now. In this paper, we investigate the FWM and SSG in a hybrid optomechanical system as shown in Figure 1 under the condition of pump on-resonance and off-resonance. The location of sideband peaks both in the FWM spectrum and SSG depends on the resonator frequencies and the Coulomb interaction of the two NRs. In particular, the different frequencies of the resonators also alter the location of the peaks in the FWM spectrum and SSG. Interestingly, in the pump off-resonance regime, the FWM spectrum can indicate a means to measure the Coulomb interaction, and we obtain double second-order OMIT sideband, which is sensitive to the Coulomb interaction and different resonator frequencies. Figure 1. (a) Schematic diagram of the hybrid coupled optomechanical system including two coupled nanomechanical resonators (NRs) via Coulomb interaction. (b) Schematic of the energy-level diagram of the system. n , jn i, and jn i denote the number states of the cavity photon, the phonons of p m1 m2 NR and NR , respectively. V is the coupling strength of the two NRs. 1 2 2. System and Method The hybrid optomechanical system is shown in Figure 1, which includes a Fabry– Perot (FP) optomechanical cavity coupling to another NR via Coulomb interaction, and the Hamiltonian can be given by [65–67] † † † † † † † H = h ¯ w c c + h ¯ w b b + h ¯ w b b h ¯ gc c(b + b ) h ¯ V(b b + b b ) m1 1 m2 2 1 2 1 1 2 1 1 2 p p † iw t iw t † iw t iw t p p s s +ih ¯ k # (c e ce ) + ih ¯ k # (c e ce ). (1) ex p ex s where the ﬁrst term gives the cavity ﬁeld (with frequency w ) and we use the creation (annihilation) operator c (c) to describe the optical cavity. The second and third terms † † shows two NRs with the frequencies w and w , b (b ) and b (b ) are the creation and m2 2 m1 1 1 2 annihilation operators of the two NRs, respectively. The fourth term gives the optomechan- w h ¯ ical coupling with coupling strength g = , where M is the effective mass of L 2M w 1 m1 NR and L is the cavity length. The ﬁfth term describes the interaction of two charged NRs with coupling strength V via the Coulomb interaction [65–67]. As the cavity is driven by p p a two-tone ﬁelds, we deﬁne # = P /h ¯ w and # = P /h ¯ w as the amplitudes of the p c p s s s Photonics 2021, 8, 234 3 of 12 two laser ﬁelds, and P and P are their powers. k is extra loss rate and here we consider c s ex k = k , where k is the intrinsic loss rate of photon with the relation of k = k + k . ex 0 0 ex 0 We can rewrite Equation (1) in the following in a frame rotating with the frequency w [65–67] † † † † † † † H = h ¯D c c + h ¯ w b b + h ¯ w b b h ¯ gc c(b + b ) h ¯ V(b b + b b ) c m1 1 m2 2 1 2 1 1 2 1 1 2 p p † † idt idt +ih ¯ k # (c c) + ih ¯ k # (c e ce ). (2) ex p ex s where D = w w is the cavity-pump detuning. We then obtain the Langevin equations c c p (LEs) as follows with adding the corresponding damping and input noise terms [33,35,65–67] idt c ˙ = (iD + k)c + igcQ + k (# + # e ) + c , (3) c a ex p s in 2 2 † ¨ ˙ Q + g Q + (w + V )Q + V(w + w )Q = 2gw c c + x , (4) a a a m2 m1 m1 m1 b m1 1 2 2 † ¨ ˙ Q + g Q + (w + V )Q + V(w + w )Q = 2gVc c + x , (5) m2 m1 m2 a 2 b b m2 b where the input vacuum noise is denoted by c with zero mean value, and x and x are in 1 2 Langevin force arising from the environment. g and g are the decay rates of the two m1 m2 † † NRs. Q = b + b and Q = b + b are the position operators of the two NRs. a 1 2 1 2 Due to the pump ﬁeld being stronger than the probe ﬁeld, we use the conversion of O = O + dO (O = c, Q , Q ), i.e., the operators are divided into the steady-state mean s a value and a small ﬂuctuation. For the steady-state values, which is determined by the following equations (iD + k)c = k # , (6) s ex p 0 0 2 2 w Q + V Q = 2gw c , (7) j j as bs m1 s m1 0 0 w Q + V Q = 2gVjc j , (8) as s bs m2 0 0 0 2 2 2 2 2 2 where D = D gQ , w = w + V , w = w + V , and V = V(w + w ). In the c as m2 m1 m1 m1 m2 m2 condition of mean-ﬁeld approximation, the operators can be replaced by their expectation valueshxci = hxihci [33], For the ﬂuctuation operators, we also use their expectation values with neglecting nonlinear terms, and we obtain the expectation values of the LEs [33] idt dc ˙ = (iD + k ) dc + igc dQ + ig dc dQ + k # e , (9) h i h i h i h ih i c s a a ex s D E D E 0 0 2 † † ¨ ˙ dQ + g dQ + w hdQ i + V hdQ i = 2gw (c hdci + c dc + dc hdci), (10) a m1 a a b m1 s m1 s D E D E 0 0 2 † † ¨ ˙ dQ + g dQ + w hdQ i + V hdQ i = 2gV(c hdci + c dc + dc hdci). (11) b m2 b b a s m2 s In order to solve Equations (9)–(11), we use the ansatz as [68] idt idt i2dt i2dt hdOi = O e + O e + O e + O e , (12) 1+ 1 2+ 2 and substituting Equation (12) into Equations (9)–(11), we can obtian two group equations with ignoring the terms higher than the second-order. The ﬁrst group describe the ﬁrst- order sideband as following (iD + k id)c = igc Q + k # , 1+ s a1+ ex s (iD + k + id)c = igc Q , 1 a1 Q + V Q = 2gc(d)(c c + c c ), a1+ 1 1+ s b1+ s 1 Q + V Q = 2gl(d)(c c + c c ). (13) b1+ 3 a1+ s 1+ 1 Photonics 2021, 8, 234 4 of 12 0 0 02 2 02 2 02 where V = V /(w ig d d ), V = V /(w ig d d ), c(d) = w /(w 1 m1 3 m2 m1 m1 m2 m1 2 02 2 ig d d ), and l(d) = V /(w ig d d ). Solving the equations, we can obtain m1 m2 m2 (L + iglP jc j ) k # 1 s ex s c = , (14) 1+ 2 2 L (L + iglP jc j ) igL P jc j 1 1 s 1 s 2 2 igc P L k # ex s s 1 c = , (15) 2 2 L [L (L + iglP jc j ) igL P jc j ] s s 1 1 1 2 2 2 P L c k # 1 ex s 2 s Q = , (16) a1+ 2 2 L (L + iglP jc j ) igL P jc j 1 1 s 1 s 2 2 where L = i(D d) + k, L = i(D + d) + k, and P = 2g(c(d) V l(d))/(1 V V ). 2 3 1 1 1 1 The second group gives the SSG progress as (iD + k 2id)c = igc Q + igc Q , 2 s a2+ 1+ a1+ (iD + k + 2id)c = igc Q + igc Q , 2+ a2 1 a1 Q + V Q = 2gc(2d)(c a + c c + c c ). a2+ 2 b2+ 2+ s 1+ s 2 1 Q + V Q = 2gl(2d)(c a + c c + c c ). (17) b2+ 4 a2+ s 2+ 1+ 2 1 0 0 02 2 02 2 02 where V = V /(w 2ig d 4d ), V = V /(w 2ig d 4d ), c(2d) = w /(w 2 m1 4 m2 m1 m1 m2 m1 2 02 2 2ig d 4d ), and l(2d) = V /(w 2ig d 4d ). Solving the equations, we can obtain m1 m2 m2 2 2 g P c c Q + igP L c c c + igc Q (L + igP jc j ) + igc Q 3 a1+ 3 s 1+ 1+ a1+ 3 s 1+ a1+ 1 4 1 4 c = , (18) 2+ 2 2 L (L + igP jc j ) igP jc j L 3 3 s 3 s 4 4 where L = i(D 2d) + k, L = i(D + 2d) + k, and P = 2g(c(2d) V l(2d))/(1 V V ). 3 4 3 2 2 4 According to the optical cavity input and output theory [69] c (t) = c (t) 2kc(t), out in we then reach the following relation p p iw t iw t p s hc (t)i = (# k c )e + (# k c )e out p ex 0 s ex 1+ p p p i(2w w )t i(2w w )t i(3w 2w )t p s s p p s k c e k c e k c e (19) ex 1 ex 2+ ex 2 where the ﬁrst term is the output ﬁeld with the frequency w , the second one indicates output ﬁeld with the frequency w , and the third term denotes FWM process with the requency 2w w . The forth and ﬁfth terms shows the SSG process where 2w w p s s p (i.e., w + 2d) is the second-order upper sideband and 3w 2w (i.e., w 2d) is the p p s p second-order lower sideband. We introduce a dimensionless FWM intensity to study the FWM process [70] k c ex FWM = . (20) In order to research the second-order process conveniently, we use a dimensionless efﬁciency of SSG as [54–57,62,64] k c ex 2+ h = . (21) The parameter values used in the paper [71] are: l = 1064 nm, L = 25 mm, w = m1 w = 2p 947 kHz, Q = w /g = 6700, Q = w /g = Q , m = m = 145 ng, m2 1 m1 m1 2 m2 m2 1 1 2 k = 2p 215 kHz, P = 0.5 W, and # = 0.05 # . c s p Photonics 2021, 8, 234 5 of 12 3. Results and Discussion 3.1. The FWM Process The FWM process has been demonstrated in optomechanical systems [50,51,70], which depends on the intracavity photon number c µ P in Equation (20). However, in our hybrid optomechanical system, we concentrate on another two parameters, i.e., the coupling strength V and the frequency difference of the two NRs. Figure 2a plots the FWM spectrum as a function of the probe-cavity detuning D = w w for several different coupling s s c strength V with the parameters of the pump power P = 0.5 W and the same frequencies w = w under the condition of pump on-resonance (D = 0). In the case of V = 0, m1 m2 c i.e., the typical FP optomechanical system without considering another NR, we ﬁnd two sharp sideband peaks (the black curve in Figure 2a) in the FWM spectrum accurately locating at the resonator frequency D = w , which can be attributed to the quantum s m1 interference of the phonon mode and the beat frequency d of two optical ﬁelds. Then, if the beat frequency d is close to the resonator frequency w , the resonator starts to oscillate m1 coherently leading to Stokes (w = w w ) scattering of light from the optomechanical p m system. However, when another NR is taken into consideration (V 6= 0), the sideband peaks locates at D = w splits into two peaks, and with increasing V from V = 0.1 w m1 m1 to V = 0.9 w , the width of the splitting in the FWM spectra is broadening at the expense m1 of intensity as shown the color curves in Figure 2a. When we measure the sideband peaks splitting width of the FWM spectrum under V 6= 0, we ﬁnd that the relation between the splitting width and the coupling strength V of the two NRs is linear as shown in Figure 2b, and the inset in Figure 2b gives the FWM spectrum at a ﬁxed coupling strength V = 0.3 w . The result will give a method the measure the coupling strength V of the two m1 NRs and we will discuss the result in the following. V=0 (a) (b) V=0.1 m 1 V=0.2 m 1 V=0.3 m 1 V=0.4 1.5 m 1 V=0.5 m 1 V=0.6 m 1 V=0.7 m 1 V=0.8 m 1 V=0.9 m 1 1.0 V=0.3 m1 2 3 0.5 1 1 -1 0 1 s m 1 0.0 -2 -1 0 1 2 0.0 0.2 0.4 0.6 0.8 1.0 s m1 V/ m1 Figure 2. (a) The FWM spectrum versus the probe-cavity detuning D = w w for several different s s c coupling strengths V with the parameters P = 0.5 W and w = w at D = 0. (b) the peak c m1 m2 c splitting of the two sideband peaks versus the coupling strength V, and the inset in (b) give the FWM spectrum at a ﬁxed V = 0.3 w m1 Switching the condition to pump off-resonance, i.e., D = w , we investigate the FWM c m1 process under different parametric regimes. In Figure 3a, we plot the FWM spectra for five different coupling strengths V at P = 5 W and w = w in the case of red sideband c m1 m2 (D = w ). We can see that there are four sharp sideband peaks in the FWM spectra, which m1 presents mirror symmetry for D = 0, and the splitting of both sides of the peaks is broaden- ing with enhanced intensity of the FWM spectra for increasing the coupling strength V from FWM intensity (a.u.) Peak-Splitting(/ ) m1 FWM intensity (a.u.) Photonics 2021, 8, 234 6 of 12 V = 0.1 w to V = 0.9 w . In addition, the location of the sideband peaks is related to the m1 m1 coupling strength V. We take V = 0.3 w as an example as shown the left inset in Figure 3a, we m1 see that the left two peaks locate at1.3 w and0.7w , i.e.,w V; while the right two m1 m1 m1 peaks locate at 0.7 w and 1.3 w , i.e., w V. When measuring the width of the splitting of m1 m1 m1 both of the two sideband peaks in the FWM spectra, we find the width of the splitting is related to the coupling strength V as shown the right inset in Figure 3a, which plots the splitting of both of the two sideband peaks versus the coupling strength V. Obviously, the splitting width relies linearly on the coupling strength V and reaches to 0 in the absence of the coupling, which presents an effective means to determine the coupling strength V of the two NRs. Thus, we can measure the coupling strength V of the two NRs via only simply measuring the splitting distance of two sideband peaks in the FWM spectrum. Moreover, we also study the FWM spectrum for different resonator frequencies at a fixed coupling strength V. In Figure 3b, we show the FWM spectra as a function of D for several different resonator frequencies with the parameters of P = 5 W and V = 0.3 w in the case of D = w . We find that the peaks in the left part is c m1 c m1 left-shift and the peaks in the right part is right-shift with increasing the frequency w from m2 w = 0.2 w to w = 2.0 w , and both the intensities of two sideband peaks experience m2 m1 m2 m1 2 2 2 the process of enhancement to decrease. As in Equations (9)–(11), where w = w + V and m1 m1 2 2 2 w = w + V , i.e., the effective frequencies of the two NRs are modulated by the coupling m2 m2 strength V of the two NRs. In Figure 3b, we set a fixed coupling strength V = 0.3 w , if the two m1 NRs are identical resonators, i.e., the two NRs own the same frequencies w = w , which is m1 m2 demonstrated in Figure 3a and the splitting width of the sideband peaks in the FWM spectrum 0 0 is proportional to the coupling strength V. In the case of w > w , i.e., w > w , the FWM m1 m2 m1 m2 spectra is squeezed and both the two sideband peaks move to D = 0. In the case of w < w , s m1 m2 0 0 i.e., w < w , the FWM spectra is expanded and the sideband peaks move to both sides. m1 m2 Compared with FWM process in the optomechanical system only including one mechanical mode and one optical mode [72,73], the intensity of the FWM in our hybrid system is enhanced significantly modulated by another NR. 0.15 0.015 V=0.1 (a) 2.0 m1 (-0.7) (0.7) V=0.3 m1 V=0.3 m1 1.5 0.010 V=0.5 m1 (-1.3) (1.3) 0.10 1.0 V=0.7 m1 0.005 V=0.9 0.5 m1 0.0 0.000 0.0 0.2 0.4 0.6 0.8 1.0 0.05 -2 -1 0 1 2 V/ m1 s m1 0.00 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 s m1 =0.2 0.025 m2 m1 (b) =0.4 m2 m1 =0.6 0.020 m2 m1 =0.8 m2 m1 =1.0 0.015 m2 m1 =1.2 m2 m1 =1.4 m2 m1 0.010 =1.6 m2 m1 =1.8 0.005 m2 m1 =2.0 m2 m1 0.000 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 s m1 Figure 3. (a) The FWM spectra for ﬁve different coupling strengths V at P = 5 W and w = w c m2 m1 in the case of D = w , and the left inset is V = 0.3w , the right inset plots the splitting of the two c m1 m1 sideband peaks as a function of the coupling strength V. (b) The FWM spectra as a function of D for several different resonator frequencies with the parameters P = 5 mW and V = 0.3 w in the case c m1 of D = w . m1 FWM intensity (a.u.) FWM intensity (a.u.) Peak-Splitting(/ ) m1 Photonics 2021, 8, 234 7 of 12 3.2. The SSG Process On the other hand, the SSG is also a distinguished nonlinear phenomenon in optome- chanical systems, and Equation (21) gives the dimensionless efﬁciency of the SSG process. As we know, if the optomechanical system is driven by a two-tone ﬁeld, the output ﬁeld with frequency 2w w (i.e., w + 2d) is the second-order upper sideband and the output s p p ﬁeld with frequency 3w 2w (i.e., w 2d) is the second-order lower sideband. Here, we p s p concentrate on the second-order upper sideband (w + 2d) and study the coupling strength V and the frequency difference of the two NRs the affect the SSG process for different detuning. In Figure 4, we show the efﬁciency h of SSG as a function of the normalized probe detuning D /w for several different coupling strengths V with the parameters of s m1 the pump power P = 1.5 W and the same frequencies w = w under the condition of c m1 m2 pump on-resonance D = 0. If the system is the typical FP optomechanical system without considering another NR, i.e., V = 0, as shown the black curve in Figure 4, it is clear that four sideband peaks emerge in the efﬁciency h of SSG, where two principal sideband peaks locate at D = w and another two secondary sideband peaks locate at D 0.5 w s s m1 m1 with a small shift. We accurately identify the location of the principal sideband peaks at 1.0 w and secondary sideband peaks at0.46 w and 0.53 w , respectively, as shown m1 m1 m1 the black curve in Figure 4. When another NR is considered (V 6= 0), we ﬁnd that both the principal sideband peaks locating at 1.0 w and the secondary sideband peaks around m1 0.5 w split into two peaks, and then eight sideband peaks appear in efﬁciency h of SSG. m1 Here, we take V = 0.1 w as an example as shown the inset in Figure 4. The splitting of m1 principal sideband peaks locate at 1.1 w and 0.9 w (i.e., w V), and 0.9 w m1 m1 m1 m1 and 1.1 w (i.e., w V) with the width of splitting of 2 V. The splitting of secondary m1 m1 sideband peaks are located at (0.53, 0.43) w and (0.46, 0.56) w , i.e., w 0.07 V m1 m1 m1 and w + 0.03 V, respectively, with the width of splitting of V. m1 0.025 (-1.1) V=0.0 m1 V=0.10 m1 0.020 V=0.05 m1 V=0.10 (-0.9) m1 0.015 (1.1) 0.020 V=0.1 m1 (-0.43) 0.010 V=0.20 (-1.0) (0.9) m1 (-0.53) 0.005 (0.56) 0.015 (0.46) 0.000 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 s m1 (1.0) (-0.46) 0.010 0.005 (0.53) 0.000 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 s m1 Figure 4. The efﬁciency h of SSG versus D /w for several different coupling strengths V with the s m1 parameters P = 1.5 W and w = w under the condition of D = 0, and the inset gives the c c m1 m2 condition of V = 0.1 w . m1 Moreover, we also investigate the different resonator frequencies of the two NRs that inﬂuence the SSG process as shown in Figure 5. Figure 5b gives the efﬁciency h of SSG ( ) ( ) Photonics 2021, 8, 234 8 of 12 versus D /w for the same frequencies of the two NRs at the parameters of P = 1.5 W s m1 c and V = 0.1 w under the pump on-resonance D = 0, which manifests four principal m1 c sideband peaks locating at w V and four secondary sideband peaks locating around m1 D 0.5 w , and the precise locations are marked in Figure 5b. If w < w as shown s m2 m1 m1 in Figure 5a, we ﬁnd the spectrum of the SSG is squeezed, while if w > w as shown in m2 m1 Figure 5c, we can see that the spectrum of the SSG is stretched, and the accurate locations are identiﬁed in Figure 5a,c, respectively. (-1.06) (a) 0.02 =0.9 m2 m1 (-0.84) (-0.4) (1.06) (-0.5) 0.01 (0.84) (0.55) (0.43) 0.00 (-1.1) (b) =1.0 m2 m1 0.02 (-0.9) (1.1) (-0.43) 0.01 (0.9) (-0.53) (0.56) (0.46) 0.00 0.03 (c) (-1.16) =1.1 m2 m1 0.02 (1.16) (-0.94) (-0.44) (0.94) 0.01 (0.49) (-0.57) (0.59) 0.00 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 s m1 Figure 5. (a) The efﬁciency h of SSG as a function of D /w for w = 0.9w . (b) The efﬁciency h s m1 m2 m1 of SSG as a function of D /w for w = 1.0w . (c) The efﬁciency h of SSG as a function of D /w s s m1 m2 m1 m1 for w = 1.1w . The other parameters are P = 1.5 mW, V = 0.1w , and D = 0. c c m2 m1 m1 Then, we further research the SSG for several different parametric regimes in the condition of D = w . Figure 6a presents the efﬁciency h of SSG versus D /w with four c s m1 m1 different coupling strengths V at P = 2.0 W and the same frequencies w = w . When c m1 m2 V = 0 as shown the black curve in Figure 6a, i.e., without considering another NR, we ﬁnd that not only a second-order sharp single peak locating at D = 0.5 w but also a s m1 second-order OMIT sideband locating at D = 1.0 w with symmetrical splitting appear in m1 the SSG spectrum. Once another NR is taken into consideration, i.e., V 6= 0, we can obtain two signiﬁcant phenomena in the SSG spectra: (a) the single peak locating at D = 0.5 w s m1 splits into two peaks; (b) the second-order OMIT sideband locating at D = 1.0 w splits s m1 into double second-order OMIT sideband with asymmetrical splitting. The phenomenon originates from the coupling between NR and NR which not only adds a fourth level, as shown in Figure 1b, but also breaks down the symmetrical second-order OMIT sideband due to quantum interference and, therefore, induces sharp bright resonance within the second-order OMIT sideband. It is obvious that due to the interaction of the two NRs via Coulomb interaction, the coupled number states of the photon and phonons are induced. Then, the symmetrical second-order OMIT sideband is split into two asymmetrical double second-order OMIT sideband. Here, we take V = 0.1 w as an example as shown in the m1 inset of Figure 6a, we ﬁnd the two splitting peaks located at 0.45 w and 0.55 w with m1 m1 ( ) Photonics 2021, 8, 234 9 of 12 the width V of the splitting, i.e., the two sharp peaks locate at w V /2. While another m1 double asymmetrical second-order OMIT sidebands locate at 0.9w and 1.1 w with the m1 m1 width 2 V of the splitting, i.e., the double second-order OMIT sidebands locate at w V. m1 With further increasing the coupling strengths V of the two NRs, both the splitting of the two sharp peaks and the double second-order OMIT sidebands are enhanced. Moreover, we also investigate the frequency difference of the two NRs that affect the efﬁciency h of SSG. In Figure 6b, we show the efﬁciency h of SSG as a function of D /w for three s m1 different frequencies at P = 2.0 W and V = 0.1 w . It is obvious that the SSG spectra m1 show two splitting peaks around D = 0.5 w and double asymmetrical second-order s m1 OMIT sidebands around D = 1.0 w , and if w < w , the SSG spectrum shifts to the s m1 m2 m1 left, if w > w , the SSG spectrum shifts to the right. Therefore, the double second-order m2 m1 OMIT sidebands can be controlled with manipulating the coupling strength V and the frequency difference of the two resonators. 0.004 (0.55 ) (0.5 ) V=0 V=0.10 (a) m1 0.002 V=0.05 m1 (0.45 ) 0.003 V=0.10 0.001 m1 V=0.15 m1 0.002 0.000 (0.9 ) (1.1 ) 2V 0.3 0.6 0.9 1.2 0.001 s m1 0.000 (1.0 ) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 s m1 0.004 (b) =0.9 m2 m1 =1.0 0.003 m2 m1 =1.1 m2 m1 0.002 0.001 0.000 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 s m1 Figure 6. (a) The efﬁciency h of SSG four different coupling strengths V with the parameters of P = 2.0 W and w = w at D = w , and the inset gives V = 0.1 w as an example. (b) The c m1 m2 c m1 m1 efﬁciency h of SSG as a function of D /w for three different frequencies at P = 2.0 W and s m1 c V = 0.1 w . m1 Compared with the SSG in the typical FP optomechanical system without considering another NR, there are several advantages of the SSG in our hybrid optomechanical system. In the condition of D = 0, if only one NR in the system (V = 0), the SSG spectrum presents the sharp principal sideband peaks at 1.0 w and secondary sideband peaks around m1 0.5 w , while if another NR is considered (V 6= 0) in the case of identical NRs with the m1 same frequencies (w = w ), both the sharp principal sideband peaks and secondary m2 m1 sideband peaks are split into two peaks, and the splitting width of the principal sideband peaks is linear with respect to the coupling strength V of the two NRs, which indicates a method to precisely determine the coupling strength V of the two NRs. In addition, if the two NRs are different with different frequencies (w 6= w ), the location of sharp m2 m1 sideband peaks in the SSG spectra are tunable. On the other hand, in the condition of pump off-resonance, i.e., D = w , the SSG shows a second-order OMIT sideband [64] at m1 V = 0, while if another NR is taken into consideration (V 6= 0), the SSG displays double second-order OMIT sidebands which is very different from the case of V = 0. Therefore, the SSG process can be tunable via controlling the coupling of the two NRs. ( ) ( ) ( ) Photonics 2021, 8, 234 10 of 12 4. Conclusions We theoretically demonstrated the FWM and SSG in a hybrid COM system which is driven by a two-tone ﬁeld for different detuning conditions, where an optomechanical res- onator is coupled to another NR via Coulomb interaction. We ﬁrst studied the FWM under the condition of pump on-resonance and pump off-resonance, when the coupling strength V of the two NRs is considered, the FWM presents four modes splitting, which gives a method to determine the coupling strength V. In addition, the frequency difference of the two NRs also alter the FWM process. On the other hand, another nonlinear phenomena of the SSG process has also demonstrated both in pump on-resonance and red sideband with controlling the parameters of the coupling strength V and the frequency difference of the two NRs. Moreover, the SSG is sensitive to the detuning, which displays the dou- ble asymmetrical second-order OMIT sidebands via controlling V and frequencies of the resonators, which may indicate a further insight of nonlinear optomechanical phenomena and may ﬁnd important applications fot manipulating light propagation and quantum communications based on the hybrid optomechanical system. Funding: This research was funded by National Natural Science Foundation of China (Nos:11647001 and 11804004), Project funded by China Postdoctoral Science Foundation (No:2020M681973) and Anhui Provincial Natural Science Foundation (No:1708085QA11). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: Huajun Chen is supported by the National Natural Science Foundation of China (Nos:11647001 and 11804004), Project funded by China Postdoctoral Science Foundation (No:2020M681973) and Anhui Provincial Natural Science Foundation (No:1708085QA11). Conﬂicts of Interest: The authors declare that they have no known competing ﬁnancial interests or personal relationships that could have appeared to inﬂuence the work reported in this paper. References 1. Aspelmeyer, M.; Kippenberg, T.J.; Marquardt, F. Cavity optomechanics. Rev. Mod. Phys. 2014, 86, 1391. [CrossRef] 2. Metcalfe, M. Applications of cavity optomechanics. App. Phys. Rev. 2014, 1, 031105. [CrossRef] 3. Milestones: Photons. Available online: https://www.nature.com/milestones/milephotons/timeline.html (accessed on 24 June 2021). 4. Schliesser, A.; Riviere, R.; Anetsberger, G.; Arcizet, O.; Kippenberg, T.J. Resolved-sideband cooling of a micromechanical oscillator. Nat. Phys. 2008, 4, 415–419. [CrossRef] 5. Teufel, J.D.; Donner, T.; Li, D.; Harlow, J.W.; Allman, M.S.; Cicak, K.; Sirois, A.J.; Whittaker, J.D.; Lehnert, K.W.; Simmonds, R.W. Sideband cooling of micromechanical motion to the quantum ground state. Nature 2011, 475, 359–363. [CrossRef] 6. Chan, J.; Alegre, T.P.M.; Safavi-Naeini, A.H.; Hill, J.T.; Krause, A.; Groblacher, S.; Aspelmeyer, M.; Painter, O. Laser cooling of a nanomechanical oscillator into its quantum ground state. Nature 2011, 478, 89–92. [CrossRef] 7. Chen, X.; Liu, Y.-C.; Peng, P.; Zhi, Y.; Xiao, Y.-F. Cooling of macroscopic mechanical resonators in hybrid atomoptomechanical systems. Phys. Rev. A 2015, 92, 033841. [CrossRef] 8. Lai, D.-G.; Zou, F.; Hou, B.-P.; Xiao, Y.-F.; Liao, J.-Q. Simultaneous cooling of coupled mechanical resonators in cavity optome- chanics. Phys. Rev. A 2018, 98, 023860. [CrossRef] 9. Li, J.J.; Zhu, K.D. All-optical mass sensing with coupled mechanical resonator systems. Phys. Rep. 2013, 525, 223–254. [CrossRef] 10. Jiang, C.; Cui, Y.; Zhu, K.D. Ultrasensitive nanomechanical mass sensor using hybrid opto-electromechanical systems. Opt. Express 2014, 22, 13773. [CrossRef] 11. Schliesser, A.; Arcizet, O.; Riviere, R.; Anetsberger, G.; Kippenberg, T.J. Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit. Nat. Phys. 2009, 5, 509. [CrossRef] 12. Basiri-Esfahani, S.; Akram, U.; Milburn, G.J. Phonon number measurements using single photon opto-mechanics. New J. Phys. 2012, 14, 085017. [CrossRef] 13. Gavartin, E.; Verlot, P.; Kippenberg, T.J. A hybrid on-chip optomechanical transducer for ultrasensitive force measurements. Nat. Nanotechnol. 2012, 7, 509–514. [CrossRef] [PubMed] 14. Krause, A.G.; Winger, M.; Blasius, T.D.; Lin, Q.; Painter, O. A high-resolution microchip optomechanical accelerometer. Nat. Photon. 2012, 6, 768–772. [CrossRef] Photonics 2021, 8, 234 11 of 12 15. Schreppler, S.; Spethmann, N.; Brahms, N.; Botter, T.; Barrios, M.; Stamper-Kurn, D.M. Optically measuring force near the standard quantum limit. Science 2014, 344, 1486–1489. [CrossRef] [PubMed] 16. Xiong, H.; Si, L.G.; Wu, Y. Precision measurement of electrical charges in an optomechanical system beyond linearized dynamics. Appl. Phys. Lett. 2017, 110, 171102. [CrossRef] 17. Matsumoto, N.; Catano-Lopez, S.B.; Sugawara, M.; Suzuki, S.; Abe, N.; Komori, K.; Michimura, Y.; Aso, Y.; Edamatsu, K. Demonstration of Displacement Sensing of a mg-Scale Pendulum for mm- and mg-Scale Gravity Measurements. Phys. Rev. Lett. 2019, 122, 071101. [CrossRef] 18. Wang, Y.D.; Clerk, A.A. Using interference for high ﬁdelity quantum state transfer in optomechanics. Phys. Rev. Lett. 2012, 108, 153603. [CrossRef] 19. Tian, L. Adiabatic state conversion and pulse transmission in optomechanical systems. Phys. Rev. Lett. 2012, 108, 153604. [CrossRef] 20. Tian, L. Robust photon entanglement via quantum interference in optomechanical interfaces. Phys. Rev. Lett. 2013, 110, 233602. [CrossRef] 21. Wang, Y.D.; Clerk, A.A. Reservoir-engineered entanglement in optomechanical systems. Phys. Rev. Lett. 2013, 110, 253601. [CrossRef] [PubMed] 22. Grudinin, I.S.; Lee, H.; Painter, O.; Vahala, K.J. Phonon laser action in a tunable two-level system. Phys. Rev. Lett. 2010, 104, 083901. [CrossRef] 23. Zhang, J.; Peng, B.; Ozdemir, S.K.; Pichler, K.; Krimer, D.O.; Zhao, G.; Nori, F.; Liu, Y.; Rotter, S.; Yang, L. A phonon laser operating at an exceptional point. Nat. Photonics 2018, 12, 479–484. [CrossRef] 24. Jing, H.; Ozdemir, S.K.; Lü, X.Y.; Zhang, J.; Yang, L.; Nori, F. PT-symmetric phonon laser. Phys. Rev. Lett. 2014, 113, 053604. [CrossRef] 25. Lü, H.; Ozdemir, S.K.; Kuang, L.M.; Nori, F.; Jing, H. Exceptional points in random-defect phonon lasers. Phys. Rev. Appl. 2017, 8, 044020. [CrossRef] 26. Safavi-Naeini, A.H.; Groeblacher, S.; Hill, J.T.; Chan, J.; Aspelmeyer, M.; Painter, O. Squeezed light from a silicon micromechanical resonator. Nature 2013, 500, 185–189. [CrossRef] 27. Agarwal, G.S.; Huang, S.M. Strong mechanical squeezing and its detection. Phys. Rev. A 2016, 93, 043844. [CrossRef] 28. Manipatruni, S.; Robinson, J.T.; Lipson, M. Optical nonreciprocity in optomechanical structures. Phys. Rev. Lett. 2009, 102, 213903. [CrossRef] [PubMed] 29. Xu, X.W.; Li, Y.; Chen, A.X.; Liu, Y.X. Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems. Phys. Rev. A 2016, 93, 023827. [CrossRef] 30. Jiang, C.; Song, L.N.; Li, Y. Directional ampliﬁer in an optomechanical system with optical gain. Phys. Rev. A 2018 , 97, 053812. [CrossRef] 31. Lü, X.-Y.; Jing, H.; Ma, J.-Y.; Wu, Y. PT -symmetry-breaking chaos in optomechanics. Phys. Rev. Lett. 2015, 114, 253601. [CrossRef] 32. Xu, H.; Mason, D.; Jiang, L.; Harris, J.G.E. Topological energy transfer in an optomechanical system with exceptional points. Nature 2016, 537, 80–83. [CrossRef] [PubMed] 33. Agarwal, G.S.; Huang, S. Electromagnetically induced transparency in mechanical effects of light. Phys. Rev. A 2010, 81, 041803. [CrossRef] 34. Liu, Y.-C.; Li, B.-B.; Xiao, Y.-F. Electromagnetically induced transparency in optical microcavities. Nanophotonics 2017, 6, 789–811. [CrossRef] 35. Weis, S.; Riviere, R.; Deleglise, S.; Gavartin, E.; Arcizet, O.; Schliesser, A.; Kippenberg, T.J. Optomechanically induced transparency. Science 2010, 330, 1520–1523. [CrossRef] [PubMed] 36. Safavi-Naeini, A.H.; Alegre, T.P.M.; Chan, J.; Eichenﬁeld, M.; Winger, M.; Lin, Q.; Hill, J.T.; Chang, D.E.; Painter, O. Electromag- netically induced transparency and slow light with optomechanics. Nature 2011, 472, 69–73. [CrossRef] [PubMed] 37. Teufel, J.D.; Li, D.; Allman, M.S.; Cicak, K.; Sirois, A.J.; Whittaker, J.D.; Simmonds, R.W. Circuit cavity electromechanics in the strong-coupling regime. Nature 2011, 471, 204–208. [CrossRef] [PubMed] 38. Fan, L.; Fong, K.Y.; Poot, M.; Tang, H.X. Cascaded optical transparency in multimode-cavity optomechanical systems. Nat. Commun. 2015, 6, 5850. [CrossRef] 39. Dong, C.; Zhang, J.; Fiore, V.; Wang, H. Optomechanically induced transparency and self-induced oscillations with Bogoliubov mechanical modes. Optica 2014, 1, 425. [CrossRef] 40. Lü, H.; Jiang, Y.; Wang, Y.Z.; Jing, H. Optomechanically induced transparency in a spinning resonator. Photonics Res. 2017, 5, 367–371. [CrossRef] 41. Jiang, C.; Liu, H.X.; Cui, Y.S.; Li, X.W.; Chen, G.B.; Chen, B. Electromagnetically induced transparency and slow light in two-mode optomechanics. Opt. Express 2013, 21, 12165. [CrossRef] 42. Zhou, X.; Hocke, F.; Schliesser, A.; Marx, A.; Huebl, H.; Gross, R.; Kippenberg, T.J. Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics. Nat. Phys. 2013 , 9, 179–184. [CrossRef] 43. Fiore, V.; Dong, C.; Kuzyk, M.C.; Wang, H. Optomechanical light storage in a silica microresonator. Phys. Rev. A 2013, 87, 023812. [CrossRef] 44. Chen, H.-J.; Hou, B.-C.; Yang, J.-Y. Controllable coherent optical response in a ring cavity optomechanical system. Phys. E 2021, 125, 114394. [CrossRef] Photonics 2021, 8, 234 12 of 12 45. Sete, E.A.; Eleuch, H. Controllable nonlinear effects in an optomechanical resonator containing a quantum well. Phys. Rev. A 2012, 85, 043824. [CrossRef] 46. Kanamoto, R.; Meystre, P. Optomechanics of a quantum-degenerate Fermi gas. Phys. Rev. Lett. 2010, 104, 063601. [CrossRef] 47. Purdy, T.P.; Brooks, D.W.C.; Botter, T.; Brahms, N.; Ma, Z.-Y.; Stamper-Kurn, D.M. Tunable cavity optomechanics with ultracold atoms. Phys. Rev. Lett. 2010, 105, 133602. [CrossRef] [PubMed] 48. Yan, D.; Wang, Z.H.; Ren, C.N.; Gao, H.; Li, Y.; Wu, J.H. Duality and bistability in an optomechanical cavity coupled to a Rydberg superatom. Phys. Rev. A 2015, 91, 023813. [CrossRef] 49. Xiong, W.; Jin, D.Y.; Qiu, Y.; Lam, C.H.; You, J.Q. Cross-Kerr effect on an optomechanical system. Phys. Rev. A 2016, 93, 023844. [CrossRef] 50. Huang, S.; Agarwal, G.S. Normal-mode splitting and antibunching in Stokes and anti-Stokes processes in cavity optomechanics: Radiation-pressure-induced four-wave-mixing cavity optomechanics. Phys. Rev. A 2010, 81, 033830. [CrossRef] 51. Jia, W.Z.; Wei, L.F.; Li, Y.; Liu, Y. Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator. Phys. Rev. A 2015, 91, 043843. [CrossRef] 52. Xu, X.W.; Li, Y. Controllable optical output ﬁelds from an optomechanical system with mechanical driving. Phys. Rev. A 2015, 92, 023855. [CrossRef] 53. Chen, H.J.; Wu, H.W.; Yang, J.Y.; Li, X.C.; Sun, Y.J.; Peng, Y. Controllable Optical Bistability and Four-Wave Mixing in a Photonic-Molecule Optomechanics. Nanoscale Res. Lett. 2019 , 14, 73–82. [CrossRef] [PubMed] 54. Xiong, H.; Si, L.G.; Zheng, A.S.; Yang, X.; Wu, Y. Higher-order sidebands in optomechanically induced transparency. Phys. Rev. A 2012, 86, 013815. [CrossRef] 55. Suzuki, H.; Brown, E.; Sterling, R. Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation. Phys. Rev. A 2015, 92, 033823. [CrossRef] 56. Jiao, Y.; Lu, H.; Qian, J.; Li, Y.; Jing, H. Nonlinear optomechanics with gain and loss: amplifying higher-order sideband and group delay. New J. Phys. 2016, 18, 083034. [CrossRef] 57. Li, J.; Li, J.; Xiao, Q.; Wu, Y. Giant enhancement of optical high-order sideband generation and their control in a dimer of two cavities with gain and loss. Phys. Rev. A 2016, 93, 063814. [CrossRef] 58. Xiong, H.; Si, L.-G.; Lu, X.-Y.; Wu, Y. Optomechanically induced sum sideband generation. Opt. Express 2016, 24, 5773. [CrossRef] [PubMed] 59. Xiong, H.; Fan, Y.-W.; Yang, X.; Wu, Y. Radiation pressure induced difference-sideband generation beyond linearized description. Appl. Phys. Lett. 2016, 109, 061108. [CrossRef] 60. Kong, C.; Xiong, H.; Wu, Y. Coulomb-interaction-dependent effect of high-order sideband generation in an optomechanical system. Phys. Rev. A 2017, 95, 033820. [CrossRef] 61. Si, L.-G.; Guo, L.-X.; Xiong, H.; Wu, Y. Tunable high-order-sideband generation and carrier-envelope-phase–dependent effects via microwave ﬁelds in hybrid electro-optomechanical systems. Phys. Rev. A 2018, 97, 023805. [CrossRef] 62. Jiao, Y.-F.; Lu, T.-X.; Jing, H. Optomechanical second-order sidebands and group delays in a Kerr resonator. Phys. Rev. A 2018, 97, 013843. [CrossRef] 63. Yellapragada, K.C.; Pramanik, N.; Singh, S.; Lakshmi, P.A. Optomechanical effects in a macroscopic hybrid system. Phys. Rev. A 2018, 98, 053822. [CrossRef] 64. Chen, B.; Shang, L.; Wang, X.-F.; Chen, J.-B.; Xue, H.-B.; Liu, X.; Zhang, J. Atom-assisted second-order sideband generation in an optomechanical system with atom-cavity-resonator coupling. Phys. Rev. A 2019, 99, 063810. [CrossRef] 65. Ma, P.-C.; Zhang, J.-Q.; Xiao, Y.; Feng, M.; Zhang, Z.-M. Tunable double optomechanically induced transparency in an optome- chanical system. Phys. Rev. A 2014, 90, 043825. [CrossRef] 66. Wang, Q.; Zhang, J.Q.; Ma, P.C.; Yao, C.M.; Feng, M. Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed ﬁeld. Phys. Rev. A 2015, 91, 063827. [CrossRef] 67. Wu, S.-C.; Qin, L.-G.; Jing, J.; Yan, T.-M.; Lu, J.; Wang, Z.-Y. Microwave-controlled optical double optomechanically induced transparency in a hybrid piezo-optomechanical cavity system. Phys. Rev. A 2018, 98, 013807. [CrossRef] 68. Boyd, R.W. Nonlinear Optics; Academic: San Diego, CA, USA, 1992; p. 225. 69. Gardiner, C.W.; Zoller, P. Quantum Noise; Springer: Berlin, Germany, 2000. 70. Jiang, C.; Cui, Y.; Liu, H. Controllable four-wave mixing based on mechanical vibration in two-mode optomechanical systems. Europhys. Lett. 2013, 104, 34004. [CrossRef] 71. Groblacher, S.; Hammerer, K.; Vanner, M.R.; Aspelmeyer, M. Observation of strong coupling between a micromechanical resonator and an optical cavity ﬁeld. Nature 2009, 460, 724–727. [CrossRef] [PubMed] 72. Jiang, C.; Chen, B.; Zhu, K.D. Controllable nonlinear responses in a cavity electromechanical system. J. Opt. Soc. Am. B 2012, 29, 220–225. [CrossRef] 73. Xing, H.W.; Chen, B.; Xing, L.L.; Chen, J.B.; Xue, H.B.; Guo, K.X. Controllable four-wave mixing based on quantum dot-cavity coupling system. Commun. Theor. Phys. 2021, 73, 055101. [CrossRef]

Photonics – Multidisciplinary Digital Publishing Institute

**Published: ** Jun 24, 2021

**Keywords: **hybrid optomechanical systems; four-wave mixing; second-order sideband generation

Loading...

You can share this free article with as many people as you like with the url below! We hope you enjoy this feature!

Read and print from thousands of top scholarly journals.

System error. Please try again!

Already have an account? Log in

Bookmark this article. You can see your Bookmarks on your DeepDyve Library.

To save an article, **log in** first, or **sign up** for a DeepDyve account if you don’t already have one.

Copy and paste the desired citation format or use the link below to download a file formatted for EndNote

Access the full text.

Sign up today, get DeepDyve free for 14 days.

All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.