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OPTICS LETTERS / Vol. 39, No. 14 / July 15, 2014
Optomechanically induced transparency in the mechanical-mode splitting regime Jinyong Ma, Cai You, Liu-Gang Si, Hao Xiong, Xiaoxue Yang, and Ying Wu* Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China *Corresponding author: [email protected] Received March 5, 2014; revised June 6, 2014; accepted June 8, 2014; posted June 9, 2014 (Doc. ID 207583); published July 10, 2014 We employ a decoupled Heisenberg–Langevin equation for the observation and physical interpretation of mechanical-mode splitting (MMS) of the movable mirror in a generic optomechanical system. Then we identify some observable and significant features of MMS in a two-mode cavity. That is, the second control field coupled to another optical mode is input to the system to modify the mechanical mode, leading to the suppression of transmission, the appearance of the doublet in the spectrum of the anti-Stokes field, and the emergence of optomechanically induced transparency in corresponding new mechanical modes. Furthermore, we open two transparent windows in virtue of MMS and find the second splitting of the mechanical mode in this two-mode optomechanical system. © 2014 Optical Society of America OCIS codes: (140.0140) Lasers and laser optics; (120.4880) Optomechanics; (270.1670) Coherent optical effects. http://dx.doi.org/10.1364/OL.39.004180
The research of cavity optomechanical systems [1], in which the optical mode is coupled to the mechanical mode via radiation pressure [2], has made spectacular advances in recent years, deriving abundant applications [3,4] and effects [5]. Among the numerous relevant topics, the sideband cooling [6,7] holds remarkable promise for the exploration of the quantum-classical boundary and fundamental quantum nature. Another highlight in the field of cavity optomechanics is optomechanically induced transparency (OMIT) [8–11], which is an analog of electromagnetically induced transparency [12,13]. The transmission of the probe laser is drastically modulated by a control laser, leading to a sharp signature in the transmission signal. The extensive applications of this emerging subject sufficiently attest to its potential significance, including slow light [14], precision measurement [15,16], enhanced Kerr nonlinearities [17], and so on. The OMIT signal is investigated in a generic optomechanical system [8,9], in which the mechanical-mode splitting (MMS) of the movable mirror (mechanical resonator) is achieved for a strong optical power of control field [18]. Then we give a decoupled equation analogous to that of the forced oscillation of a generic harmonic resonator, which provides a clear physical interpretation for MMS. It is worth noting that one does not find an observable phenomenon reflecting MMS in this optomechanical system. The coupling of optical modes and mechanical modes has been achieved in many ways, such as using gradient optical force [19,20], electrostrictive pressure [21,22], and optical angular momentum [23,24]. Some previous works have also investigated the coupling of two optical modes and a mechanical mode in an optomechanical system both theoretically [25,26] and experimentally [27]. In this Letter, we present a scheme to reflect the effect of MMS in a two-mode cavity, in which two optical modes are coupled to two mechanical modes (obtained from splitting). MMS is demonstrated by the appearance of the doublet in the spectrum of the anti-Stokes field 0146-9592/14/144180-04$15.00/0
and the occurrence of OMIT in corresponding new mechanical modes. The presence of these effects suggests that MMS is an observable and available process rather than an appearance that cannot be employed. Otherwise, the emergence of two transparent windows may open up the potential application on the operation of light. And the observation of a second splitting of the mechanical mode illuminates the further cooling of the mechanical resonator via the manipulation of two optical modes [28]. We start from a generic optomechanical system [8] with a strong control field (frequency ωl , pump power p P l , and amplitude εl P l ∕ℏωl ) and a weak probe field p (frequency ωp , power P p , and amplitude εp P p ∕ℏωp ), whose Hamiltonian in a frame rotating at ωl reads [9]: H pˆ 2 ∕2m mω2m qˆ 2 ∕2 − ℏΔcˆ † cˆ ℏGqˆ cˆ † cˆ p iℏ ηc κ εl εp e−iΩt ˆc† − H:c:, with Δ ωl − ωc and Ω ωp − ωl , where pˆ and qˆ are, respectively, the momentum and position operators of the movable mirror with effective mass m and eigenfrequency ωm . ωc is an eigenfrequency of the cavity coupled to the mechanical oscillator via a coupling constant G. κ denotes the total loss rate of the cavity, including an external loss rate κ ex and an intrinsic loss rate κ0 . Otherwise, ηc κ ex ∕κex κo (ηc 1∕2 is chosen in this Letter) is for the coupling parameter [9]. In our work, we follow a classical theory by neglecting the quantum noise of the movable mirror and the cavity, and consider the expectation values of the operators. In the case without probe field, the optomechanical system is in the steady state controlled by the pump field with an appropriate optical power. Then, by taking account of the perturbation induced by the probe field (i.e., q q0 δq and c c0 δc) and the decay rates of the movable mirror (γ m ) and cavity field (κ), we give a decoupled equation describing the motion of the mechanical resonator from Heisenberg–Langevin equations within the first-order approximation (the higher-order terms are omitted) as follows: © 2014 Optical Society of America
July 15, 2014 / Vol. 39, No. 14 / OPTICS LETTERS
d2 d δq mγ 0m δq mω02 m δq 2jf Ωj cosΩt ϕf dt dt2 (1)
f Ω : jQj 2 −Ω − iγ 0 Ω ω02 m
(2)
m
With respect to the forced oscillation of a generic harmonic oscillator [whose behaviors are described by Eq. (2) in the case that γ 0m and ω0m are constants, independently of Ω], it will oscillate coherently when p Ω ω02 − γ 02 m m ∕2. However, in this optomechanical sys0 0 tem, ωm , γ m and jf Ωj are the functions of Ω. Then multiple resonance frequencies (mechanical modes) become possible, which cannot be achieved in a generic harmonic oscillator. When the pump power is strong enough, MMS is observed, that is, there are two peaks and a dip in the displacement spectrum [Fig. 1(a)]. For the sake of illuminating MMS clearly and comprehensively, the dependence of the peak and dip position of the displacement spectrum on P l is exhibited in Fig. 1(b). We separate the pump power into two regions in terms of the splitting of the mechanical mode. Even though MMS is not observed in Region I due to finite width of the peak in the displacement spectrum [18], the position of the resonant peak (viz., mechanical mode) shifts, which may be the reason for asymmetric and weak cooling. In Region II, on account of the effect of strong cooling near Ω ωm , MMS begins to appear. Otherwise, we note that the right peak is higher than the left one in the displacement spectrum when P l 60 mW [Fig. 1(a)]. In sum, MMS of the oscillator emerges when the pump power is greater than the critical value (about 26 mW). The presence of MMS in an optomechanical system may imply potential importance on OMIT signals. We therefore take account of a two-mode cavity with two strong control fields and a weak probe field to exploit the observable effects of MMS. The coupling of the first control field (frequency ωl1 ) and the cavity is realized
4 2
0.5 1.4
1
1.5
(b) dip
1.2 m
2 with γ 0m γ m jηΩj sin ϕη ∕Ω and ω02 m ωm 2 2 jηΩj cos ϕη , where ηΩ iℏG jc0 j 1∕−iΩ β − 1∕ p −iΩ β, f Ω −ℏGc0 ηc κεp ∕−iΩ β, and β κ∕2 i−Δ Gq0 . ϕη and ϕf are the arguments of ηΩ and f Ω, respectively. This decoupled equation (which can also be achieved within the higher-order approximation according to the solution of higher-order optical sidebands [29]) shares the similar form as the equation of the forced oscillation of a harmonic resonator. γ 0m and ω0m are the effective damping rate and oscillation frequency, respectively, which vary with the frequency of the probe field. Moreover, Eq. (1) implies that the movable mirror is driven by an effective harmonic force with frequency Ω and varied amplitude 2jf Ωj. From the ansatz δq Qe−iΩt Q eiΩt , it is indicated that 2jQj denotes the amplitude of oscillation of the movable mirror. Hence, the frequencies of the driving force 2jf Ωj cosΩt ϕf to make jQj reach the maximum are the effective resonance frequencies (defined as mechanical mode, i.e., Ωeff ) of the resonator. The displacement spectrum (jQj) is obtained readily from Eq. (1) as follows:
x 10 6
1
/
m
4181
0.8 resonantt peak peaks ks (mechanical mode)
0
20
40
Pl (mw)
60
80
100
Fig. 1. (a) Calculation result of jQj varies with Ω in different optical power P l of the control field with frequency ωl . (b) The dependence of the peak and dip position of jQj on P l . We use m 20 ng, ωm 2π × 51.8 MHz, γ m 2π × 41.0 kHz, G −2π × 12 GHz∕nm, κ 2π × 15.0 MHz, Δ −ωm throughout the work, which are taken from the experiments in [9]. And we fix εp εl ∕100.
through the optical mode ωc1 , while the probe field (frequency ωp ) and the second control field (frequency ωl2 ) are coupled to the cavity via another one (ωc2 ). The control field with frequency ωl1 is used to manipulate the mechanical mode of the movable mirror, the other control field and probe field have similar roles as those in a generic traditional system [8,9]. We start from the following rotating-frame Hamiltonian with Δ1 ωl1 − ωc1 , Δ2 ωl2 − ωc2 , and Ω ωp − ωl2 : mω2m qˆ 2 pˆ 2 − ℏΔ1 cˆ †1 cˆ 1 − ℏΔ2 cˆ †2 cˆ 2 H 2m 2
ℏGqˆ cˆ †1 cˆ 1 ℏGqˆ cˆ †2 cˆ 2 εin ;
(3)
p with εin iℏ ηc κ εl1 cˆ †1 εl2 cˆ †2 εp e−iΩt cˆ †2 − H:c:, where cˆ 1 and cˆ 2 are for the bosonic annihilation operators of two different optical modes, and cˆ †1 and cˆ †2 denote the creation operators of them. Moreover, εl1 and εl2 are the amplitudes of the two control lasers, respectively. Other parameters are the same as the generic optomechanical system. We also follow the Langevin equations by considering the mean values of the operators and ignoring the quantum fluctuation of the system, and employ the following ansatz in the case of considering only the first-order sidebands: q q0 Q0 e−iΩt Q0 eiΩt , iΩt , and c2 c02 C −2 e−iΩt c1 c01 C −1 e−iΩt C 1 e iΩt C 2 e . If β1 and β2 are defined as β1 κ∕2 i−Δ1 Gq0 and β2 κ∕2 i−Δ2 Gq0 , we can obtain the solution with the perturbation method as follows: p 1 ifΩ ηc κ εp ; 2Δ2 − Gq0 f Ω − iΩ β2 p −iΩ β2 C −2 − ηc κεp −iGc01 Q0 ; C −1 ; Q0 0 −iΩ β1 −iGc2
C −2
(4)
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with f Ω ℏG2 jc02 j2 χΩ∕−iΩ β2 , χΩ 1∕ 2 2 mωm − ω − iγ m ω ξΩ, and ξΩ iℏG2 jc01 j2 1∕ −iΩ β1 − 1∕−iΩ β1 . We have the output field (whose frequencies are near ωc2 ) by applying the input– p output relation of the cavity: S out εl2 − ηc κc20 e− p − −iωp t p −iωl −Ωt iωl2 εp − ηc κ C 2 e − ηc κ C 2 e . Consep quently, we define tp εp − ηc κC −2 ∕εp as the transmission of the probe laser, which behaves like the properties of OMIT. It is written as tp 1 − 1 if Ωηc κ∕ 2Δ2 − Gq0 f Ω − iΩ β2 . We note this equation will reduce to the generic form defined by previous work [9] when P l1 0 mW since ξΩ 0 at the same time. When we input the second control field with frequency ωl1 , the traditional OMIT signal is suppressed as a result of the mechanical-mode shift induced by this new control field [Fig. 2(a)]. The stronger P l1 is, the more the mechanical mode shifts, leading to greater suppression of the OMIT signal. Otherwise, the effect of mechanical-mode shift and splitting can be observed in the p behaviors (which are described by ρ j ηc κC −1 ∕εp j, a dimensionless quantity) of the cavity field with frequency ωl1 Ω [Fig. 2(b)], which is produced by the anti-Stokes scattering of light from the control field with frequency ωl1 . We find a symmetric structure in the spectrum of the anti-Stokes field when P l1 10 mW, but it is broken when P l1 is strong enough, that is, there is an asymmetric peak due to the mechanical-mode shift when P l1 20 mW and a dip due to MMS when P l1 60 mW
1 (a)
(b) 0.4
ρ
p
|t |
2
0.3 0.5
0.2 0.1
0 0.5
1 Ω/ω
0 0
1.5
1 Ω/ω
m
(d)
(c) 0.02
0.8
0.015 p
|t |2
|Q' |(pm)
2
m
0.01
0.6 0.4
0.005
0.8
0.8
0.6
0.6
p
|t |2
p
|t |2
0.2 1
ω/ωm
1
0.4
0.4
0.2
0.2 0.5
1 Ω/ω
m
1.5
0.5
1 Ω/ω
1.5
m
Fig. 2. (a) Suppression of OMIT signal in the case of Δ2 −ωm . (b) Spectrum of anti-Stokes field from the scattering of light from the control field with frequency ωl1 . (c) OMIT signals by considering corresponding new mechanical modes. (d) OMIT signals by considering left mechanical mode (top figure) and right mechanical mode (bottom figure)in cases of different P l2 .
[Fig. 2(b)]. These behaviors agree with the descriptions about MMS in the previous discussion. Now, we focus on such a case that the control field (P l1 60 mW) with frequency ωl1 is strong enough to achieve MMS and the other control field (P l2 24 mW) is weaker so that we ignore its effect on the mechanical mode temporarily. In view of this, we achieve OMIT signals by considering corresponding new mechanical modes [Fig. 2(c)], which are obtained from the process of MMS. This feature is the result of the coupling of two optical modes and two mechanical modes. We can consider the following process to obtain more physical insight. From the analysis above, we find that the intensity of the anti-Stokes field induced by the scattering of light from the control field with frequency ωl2 will be resonantly enhanced twice with the variation of the frequency of the probe laser. Then, we tune the ¯ 2 (defined by Δ ¯ 2 Δ2 − Gq 0 ) to effective detuning Δ make the frequency of the probe laser match the resonant frequency of the cavity and the anti-Stokes field simultaneously. Finally, destructive interference between the anti-Stokes field and the near-resonant probe ¯ 2, field weakens the intracavity probe field for Ω ≈ −Δ leading to the OMIT signals in corresponding mechanical modes. It is worth noting that another resonance of the anti-Stokes field near the transparent window is subdued due to off resonance of the probe laser with the cavity. Moreover, the transmission of the probe laser can be enhanced [Fig. 2(d)] by increasing P l2 . At the same time, the peak position of the transparent window shifts due to the weak effect of P l2 on the mechanical mode. Therefore, we conclude that the transparent window is modulated by the two control fields, that is, one contributes to the peak position by tuning the mechanical mode and the other one primarily controls the transmission. Furthermore, we will show that Δ2 is a significant parameter to control the transmission of the probe field and the mechanical mode of the movable mirror. If Δ2 is modulated to be off resonance with the anti-Stokes field, the obvious transparent window disappears in the case of ¯ 2 [Fig. 3(a)]. However, some structures similar to Ω ≈ −Δ the OMIT signal are observed by considering other frequencies of the probe laser due to its interference with the anti-Stokes field. What is more, we open two transparent windows [Fig. 3(b)] by modulating Δ2 for a strong P l2 . Apparently, these transparent windows are the result of the combined effect between the generic OMIT signal and another resonant anti-Stokes field. We should note that the transmission of the probe laser can reach only about 80% under two windows operation since comparatively high pump powers will drive this optomechanical system into a nonperturbative regime, where the perturbation method employed in our work is invalid. Moreover, when Δ2 is changed to a large value, we observe three resonant peaks in the displacement spectrum [Fig. 3(c)], which indicates that the mechanical mode splits again. This phenomenon acts as a positive role in the cooling of the movable mirror, which can be demonstrated by the observation of the spectrum of the anti-Stokes field excited from the control field with frequency ωl1 [Fig. 3(d)].
July 15, 2014 / Vol. 39, No. 14 / OPTICS LETTERS 0.01 (c)
(a)
|Q'|(pm)
p
|t |2
1
0.5
0.006
0.002
0 0
1 Ω/ω
2
m
(d) 0.2 ρ
0.8
p
|t |2
1
(b)
0.1 0.6 0.4
0.5
1 Ω/ωm
1.5
0 0
0.5
1 Ω/ωm
1.5
2
Fig. 3. (a) Transmission of the probe field in corresponding Δ2 . (b) Two transparent windows induced by modulating Δ2 for a strong P l2 . (c) The second splitting of the mechanical mode of the movable mirror. (d) The spectrum of anti-Stokes field excited from the control field with frequency ωl1 .
To conclude, unlike the work that successfully explores normal-mode splitting by following semiclassical theory [18], we give a clear physical picture of MMS in virtue of an effective equation similar to the oscillation of a generic resonator. Further, since observable effects of MMS cannot be found in the generic optomechanical system, we presented a specific approach for describing the novel features of MMS in a two-mode cavity, identifying the OMIT signals in corresponding new mechanical modes, the emergence of two transparent windows, and the second splitting of the mechanical mode, which may deepen the comprehension of MMS and inspire potential application on the operation of light. The work is supported in part by the National Fundamental Research Program of China (grant 2012CB922103) and the National Science Foundation (NSF) of China (grants 11375067, 11275074, 11204096, and 11374116). References 1. T. J. Kippenberg and K. J. Vahala, Opt. Express 15, 17172 (2007). 2. A. Ashkin, Phys. Rev. Lett. 24, 156 (1970). 3. T. Antoni, A. G. Kuhn, T. Briant, P.-F. Cohadon, A. Heidmann, R. Braive, I. Abram, L. Le Gratiet, I. Sagnes, and I. R. Philip, Opt. Lett. 36, 3434 (2011).
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4. A. Bonakdar, J. Kohoutek, D. Dey, and H. Mohseni, Opt. Lett. 37, 3258 (2012). 5. H. Xiong, L.-G. Si, X.-Y. Lü, X. Yang, and Y. Wu, Opt. Lett. 38, 353 (2013). 6. T. W. Hansch and A. L. Schawlow, Opt. Commun. 13, 68 (1975). 7. A. Mari and J. Eisert, Phys. Rev. Lett. 108, 120602 (2012). 8. G. S. Agarwal and S. Huang, Phys. Rev. A 81, 041803 (2010). 9. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, Science 330, 1520 (2010). 10. A. Kronwald and F. Marquardt, Phys. Rev. Lett. 111, 133601 (2013). 11. K. Brøkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, Phys. Rev. Lett. 111, 053603 (2013). 12. Y. Wu and X. Yang, Phys. Rev. A 76, 013832 (2007). 13. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005). 14. A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, Nature 472, 69 (2011). 15. S. Basiri-Esfahani, U. Akram, and G. J. Milburn, New J. Phys. 14, 085017 (2012). 16. W. He, J.-J. Li, and K.-D. Zhu, Opt. Lett. 35, 339 (2010). 17. X.-Y. Lü, W.-M. Zhang, S. Ashhab, Y. Wu, and F. Nori, Sci. Rep. 3, 2943 (2013). 18. J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, Phys. Rev. Lett. 101, 263602 (2008). 19. A. Ashkin, J. Dziedzic, J. Bjorkholm, and S. Chu, Opt. Lett. 11, 288 (1986). 20. P. C. Chaumet and A. Rahmani, Nat. Nanotechnol. 9, 252 (2014). 21. I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” arXiv:0805.0803 (2008). 22. M. Tomes and T. Carmon, Phys. Rev. Lett. 102, 113601 (2009). 23. A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, Nat. Photonics 8, 234 (2014). 24. X. Zhang, M. Tomes, and T. Carmon, Opt. Express 19, 9066 (2011). 25. K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rab, Phys. Rev. Lett. 109, 013603 (2012). 26. M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, Phys. Rev. Lett. 109, 063601 (2012). 27. J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, Nat. Phys. 6, 707 (2010). 28. M. Bhattacharya and P. Meystre, Phys. Rev. Lett. 99, 073601 (2007). 29. H. Xiong, L.-G. Si, A.-S. Zheng, X. Yang, and Y. Wu, Phys. Rev. A 86, 013815 (2012).
OPTICS LETTERS / Vol. 39, No. 14 / July 15, 2014
Optomechanically induced transparency in the mechanical-mode splitting regime Jinyong Ma, Cai You, Liu-Gang Si, Hao Xiong, Xiaoxue Yang, and Ying Wu* Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China *Corresponding author: [email protected] Received March 5, 2014; revised June 6, 2014; accepted June 8, 2014; posted June 9, 2014 (Doc. ID 207583); published July 10, 2014 We employ a decoupled Heisenberg–Langevin equation for the observation and physical interpretation of mechanical-mode splitting (MMS) of the movable mirror in a generic optomechanical system. Then we identify some observable and significant features of MMS in a two-mode cavity. That is, the second control field coupled to another optical mode is input to the system to modify the mechanical mode, leading to the suppression of transmission, the appearance of the doublet in the spectrum of the anti-Stokes field, and the emergence of optomechanically induced transparency in corresponding new mechanical modes. Furthermore, we open two transparent windows in virtue of MMS and find the second splitting of the mechanical mode in this two-mode optomechanical system. © 2014 Optical Society of America OCIS codes: (140.0140) Lasers and laser optics; (120.4880) Optomechanics; (270.1670) Coherent optical effects. http://dx.doi.org/10.1364/OL.39.004180
The research of cavity optomechanical systems [1], in which the optical mode is coupled to the mechanical mode via radiation pressure [2], has made spectacular advances in recent years, deriving abundant applications [3,4] and effects [5]. Among the numerous relevant topics, the sideband cooling [6,7] holds remarkable promise for the exploration of the quantum-classical boundary and fundamental quantum nature. Another highlight in the field of cavity optomechanics is optomechanically induced transparency (OMIT) [8–11], which is an analog of electromagnetically induced transparency [12,13]. The transmission of the probe laser is drastically modulated by a control laser, leading to a sharp signature in the transmission signal. The extensive applications of this emerging subject sufficiently attest to its potential significance, including slow light [14], precision measurement [15,16], enhanced Kerr nonlinearities [17], and so on. The OMIT signal is investigated in a generic optomechanical system [8,9], in which the mechanical-mode splitting (MMS) of the movable mirror (mechanical resonator) is achieved for a strong optical power of control field [18]. Then we give a decoupled equation analogous to that of the forced oscillation of a generic harmonic resonator, which provides a clear physical interpretation for MMS. It is worth noting that one does not find an observable phenomenon reflecting MMS in this optomechanical system. The coupling of optical modes and mechanical modes has been achieved in many ways, such as using gradient optical force [19,20], electrostrictive pressure [21,22], and optical angular momentum [23,24]. Some previous works have also investigated the coupling of two optical modes and a mechanical mode in an optomechanical system both theoretically [25,26] and experimentally [27]. In this Letter, we present a scheme to reflect the effect of MMS in a two-mode cavity, in which two optical modes are coupled to two mechanical modes (obtained from splitting). MMS is demonstrated by the appearance of the doublet in the spectrum of the anti-Stokes field 0146-9592/14/144180-04$15.00/0
and the occurrence of OMIT in corresponding new mechanical modes. The presence of these effects suggests that MMS is an observable and available process rather than an appearance that cannot be employed. Otherwise, the emergence of two transparent windows may open up the potential application on the operation of light. And the observation of a second splitting of the mechanical mode illuminates the further cooling of the mechanical resonator via the manipulation of two optical modes [28]. We start from a generic optomechanical system [8] with a strong control field (frequency ωl , pump power p P l , and amplitude εl P l ∕ℏωl ) and a weak probe field p (frequency ωp , power P p , and amplitude εp P p ∕ℏωp ), whose Hamiltonian in a frame rotating at ωl reads [9]: H pˆ 2 ∕2m mω2m qˆ 2 ∕2 − ℏΔcˆ † cˆ ℏGqˆ cˆ † cˆ p iℏ ηc κ εl εp e−iΩt ˆc† − H:c:, with Δ ωl − ωc and Ω ωp − ωl , where pˆ and qˆ are, respectively, the momentum and position operators of the movable mirror with effective mass m and eigenfrequency ωm . ωc is an eigenfrequency of the cavity coupled to the mechanical oscillator via a coupling constant G. κ denotes the total loss rate of the cavity, including an external loss rate κ ex and an intrinsic loss rate κ0 . Otherwise, ηc κ ex ∕κex κo (ηc 1∕2 is chosen in this Letter) is for the coupling parameter [9]. In our work, we follow a classical theory by neglecting the quantum noise of the movable mirror and the cavity, and consider the expectation values of the operators. In the case without probe field, the optomechanical system is in the steady state controlled by the pump field with an appropriate optical power. Then, by taking account of the perturbation induced by the probe field (i.e., q q0 δq and c c0 δc) and the decay rates of the movable mirror (γ m ) and cavity field (κ), we give a decoupled equation describing the motion of the mechanical resonator from Heisenberg–Langevin equations within the first-order approximation (the higher-order terms are omitted) as follows: © 2014 Optical Society of America
July 15, 2014 / Vol. 39, No. 14 / OPTICS LETTERS
d2 d δq mγ 0m δq mω02 m δq 2jf Ωj cosΩt ϕf dt dt2 (1)
f Ω : jQj 2 −Ω − iγ 0 Ω ω02 m
(2)
m
With respect to the forced oscillation of a generic harmonic oscillator [whose behaviors are described by Eq. (2) in the case that γ 0m and ω0m are constants, independently of Ω], it will oscillate coherently when p Ω ω02 − γ 02 m m ∕2. However, in this optomechanical sys0 0 tem, ωm , γ m and jf Ωj are the functions of Ω. Then multiple resonance frequencies (mechanical modes) become possible, which cannot be achieved in a generic harmonic oscillator. When the pump power is strong enough, MMS is observed, that is, there are two peaks and a dip in the displacement spectrum [Fig. 1(a)]. For the sake of illuminating MMS clearly and comprehensively, the dependence of the peak and dip position of the displacement spectrum on P l is exhibited in Fig. 1(b). We separate the pump power into two regions in terms of the splitting of the mechanical mode. Even though MMS is not observed in Region I due to finite width of the peak in the displacement spectrum [18], the position of the resonant peak (viz., mechanical mode) shifts, which may be the reason for asymmetric and weak cooling. In Region II, on account of the effect of strong cooling near Ω ωm , MMS begins to appear. Otherwise, we note that the right peak is higher than the left one in the displacement spectrum when P l 60 mW [Fig. 1(a)]. In sum, MMS of the oscillator emerges when the pump power is greater than the critical value (about 26 mW). The presence of MMS in an optomechanical system may imply potential importance on OMIT signals. We therefore take account of a two-mode cavity with two strong control fields and a weak probe field to exploit the observable effects of MMS. The coupling of the first control field (frequency ωl1 ) and the cavity is realized
4 2
0.5 1.4
1
1.5
(b) dip
1.2 m
2 with γ 0m γ m jηΩj sin ϕη ∕Ω and ω02 m ωm 2 2 jηΩj cos ϕη , where ηΩ iℏG jc0 j 1∕−iΩ β − 1∕ p −iΩ β, f Ω −ℏGc0 ηc κεp ∕−iΩ β, and β κ∕2 i−Δ Gq0 . ϕη and ϕf are the arguments of ηΩ and f Ω, respectively. This decoupled equation (which can also be achieved within the higher-order approximation according to the solution of higher-order optical sidebands [29]) shares the similar form as the equation of the forced oscillation of a harmonic resonator. γ 0m and ω0m are the effective damping rate and oscillation frequency, respectively, which vary with the frequency of the probe field. Moreover, Eq. (1) implies that the movable mirror is driven by an effective harmonic force with frequency Ω and varied amplitude 2jf Ωj. From the ansatz δq Qe−iΩt Q eiΩt , it is indicated that 2jQj denotes the amplitude of oscillation of the movable mirror. Hence, the frequencies of the driving force 2jf Ωj cosΩt ϕf to make jQj reach the maximum are the effective resonance frequencies (defined as mechanical mode, i.e., Ωeff ) of the resonator. The displacement spectrum (jQj) is obtained readily from Eq. (1) as follows:
x 10 6
1
/
m
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0.8 resonantt peak peaks ks (mechanical mode)
0
20
40
Pl (mw)
60
80
100
Fig. 1. (a) Calculation result of jQj varies with Ω in different optical power P l of the control field with frequency ωl . (b) The dependence of the peak and dip position of jQj on P l . We use m 20 ng, ωm 2π × 51.8 MHz, γ m 2π × 41.0 kHz, G −2π × 12 GHz∕nm, κ 2π × 15.0 MHz, Δ −ωm throughout the work, which are taken from the experiments in [9]. And we fix εp εl ∕100.
through the optical mode ωc1 , while the probe field (frequency ωp ) and the second control field (frequency ωl2 ) are coupled to the cavity via another one (ωc2 ). The control field with frequency ωl1 is used to manipulate the mechanical mode of the movable mirror, the other control field and probe field have similar roles as those in a generic traditional system [8,9]. We start from the following rotating-frame Hamiltonian with Δ1 ωl1 − ωc1 , Δ2 ωl2 − ωc2 , and Ω ωp − ωl2 : mω2m qˆ 2 pˆ 2 − ℏΔ1 cˆ †1 cˆ 1 − ℏΔ2 cˆ †2 cˆ 2 H 2m 2
ℏGqˆ cˆ †1 cˆ 1 ℏGqˆ cˆ †2 cˆ 2 εin ;
(3)
p with εin iℏ ηc κ εl1 cˆ †1 εl2 cˆ †2 εp e−iΩt cˆ †2 − H:c:, where cˆ 1 and cˆ 2 are for the bosonic annihilation operators of two different optical modes, and cˆ †1 and cˆ †2 denote the creation operators of them. Moreover, εl1 and εl2 are the amplitudes of the two control lasers, respectively. Other parameters are the same as the generic optomechanical system. We also follow the Langevin equations by considering the mean values of the operators and ignoring the quantum fluctuation of the system, and employ the following ansatz in the case of considering only the first-order sidebands: q q0 Q0 e−iΩt Q0 eiΩt , iΩt , and c2 c02 C −2 e−iΩt c1 c01 C −1 e−iΩt C 1 e iΩt C 2 e . If β1 and β2 are defined as β1 κ∕2 i−Δ1 Gq0 and β2 κ∕2 i−Δ2 Gq0 , we can obtain the solution with the perturbation method as follows: p 1 ifΩ ηc κ εp ; 2Δ2 − Gq0 f Ω − iΩ β2 p −iΩ β2 C −2 − ηc κεp −iGc01 Q0 ; C −1 ; Q0 0 −iΩ β1 −iGc2
C −2
(4)
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OPTICS LETTERS / Vol. 39, No. 14 / July 15, 2014
with f Ω ℏG2 jc02 j2 χΩ∕−iΩ β2 , χΩ 1∕ 2 2 mωm − ω − iγ m ω ξΩ, and ξΩ iℏG2 jc01 j2 1∕ −iΩ β1 − 1∕−iΩ β1 . We have the output field (whose frequencies are near ωc2 ) by applying the input– p output relation of the cavity: S out εl2 − ηc κc20 e− p − −iωp t p −iωl −Ωt iωl2 εp − ηc κ C 2 e − ηc κ C 2 e . Consep quently, we define tp εp − ηc κC −2 ∕εp as the transmission of the probe laser, which behaves like the properties of OMIT. It is written as tp 1 − 1 if Ωηc κ∕ 2Δ2 − Gq0 f Ω − iΩ β2 . We note this equation will reduce to the generic form defined by previous work [9] when P l1 0 mW since ξΩ 0 at the same time. When we input the second control field with frequency ωl1 , the traditional OMIT signal is suppressed as a result of the mechanical-mode shift induced by this new control field [Fig. 2(a)]. The stronger P l1 is, the more the mechanical mode shifts, leading to greater suppression of the OMIT signal. Otherwise, the effect of mechanical-mode shift and splitting can be observed in the p behaviors (which are described by ρ j ηc κC −1 ∕εp j, a dimensionless quantity) of the cavity field with frequency ωl1 Ω [Fig. 2(b)], which is produced by the anti-Stokes scattering of light from the control field with frequency ωl1 . We find a symmetric structure in the spectrum of the anti-Stokes field when P l1 10 mW, but it is broken when P l1 is strong enough, that is, there is an asymmetric peak due to the mechanical-mode shift when P l1 20 mW and a dip due to MMS when P l1 60 mW
1 (a)
(b) 0.4
ρ
p
|t |
2
0.3 0.5
0.2 0.1
0 0.5
1 Ω/ω
0 0
1.5
1 Ω/ω
m
(d)
(c) 0.02
0.8
0.015 p
|t |2
|Q' |(pm)
2
m
0.01
0.6 0.4
0.005
0.8
0.8
0.6
0.6
p
|t |2
p
|t |2
0.2 1
ω/ωm
1
0.4
0.4
0.2
0.2 0.5
1 Ω/ω
m
1.5
0.5
1 Ω/ω
1.5
m
Fig. 2. (a) Suppression of OMIT signal in the case of Δ2 −ωm . (b) Spectrum of anti-Stokes field from the scattering of light from the control field with frequency ωl1 . (c) OMIT signals by considering corresponding new mechanical modes. (d) OMIT signals by considering left mechanical mode (top figure) and right mechanical mode (bottom figure)in cases of different P l2 .
[Fig. 2(b)]. These behaviors agree with the descriptions about MMS in the previous discussion. Now, we focus on such a case that the control field (P l1 60 mW) with frequency ωl1 is strong enough to achieve MMS and the other control field (P l2 24 mW) is weaker so that we ignore its effect on the mechanical mode temporarily. In view of this, we achieve OMIT signals by considering corresponding new mechanical modes [Fig. 2(c)], which are obtained from the process of MMS. This feature is the result of the coupling of two optical modes and two mechanical modes. We can consider the following process to obtain more physical insight. From the analysis above, we find that the intensity of the anti-Stokes field induced by the scattering of light from the control field with frequency ωl2 will be resonantly enhanced twice with the variation of the frequency of the probe laser. Then, we tune the ¯ 2 (defined by Δ ¯ 2 Δ2 − Gq 0 ) to effective detuning Δ make the frequency of the probe laser match the resonant frequency of the cavity and the anti-Stokes field simultaneously. Finally, destructive interference between the anti-Stokes field and the near-resonant probe ¯ 2, field weakens the intracavity probe field for Ω ≈ −Δ leading to the OMIT signals in corresponding mechanical modes. It is worth noting that another resonance of the anti-Stokes field near the transparent window is subdued due to off resonance of the probe laser with the cavity. Moreover, the transmission of the probe laser can be enhanced [Fig. 2(d)] by increasing P l2 . At the same time, the peak position of the transparent window shifts due to the weak effect of P l2 on the mechanical mode. Therefore, we conclude that the transparent window is modulated by the two control fields, that is, one contributes to the peak position by tuning the mechanical mode and the other one primarily controls the transmission. Furthermore, we will show that Δ2 is a significant parameter to control the transmission of the probe field and the mechanical mode of the movable mirror. If Δ2 is modulated to be off resonance with the anti-Stokes field, the obvious transparent window disappears in the case of ¯ 2 [Fig. 3(a)]. However, some structures similar to Ω ≈ −Δ the OMIT signal are observed by considering other frequencies of the probe laser due to its interference with the anti-Stokes field. What is more, we open two transparent windows [Fig. 3(b)] by modulating Δ2 for a strong P l2 . Apparently, these transparent windows are the result of the combined effect between the generic OMIT signal and another resonant anti-Stokes field. We should note that the transmission of the probe laser can reach only about 80% under two windows operation since comparatively high pump powers will drive this optomechanical system into a nonperturbative regime, where the perturbation method employed in our work is invalid. Moreover, when Δ2 is changed to a large value, we observe three resonant peaks in the displacement spectrum [Fig. 3(c)], which indicates that the mechanical mode splits again. This phenomenon acts as a positive role in the cooling of the movable mirror, which can be demonstrated by the observation of the spectrum of the anti-Stokes field excited from the control field with frequency ωl1 [Fig. 3(d)].
July 15, 2014 / Vol. 39, No. 14 / OPTICS LETTERS 0.01 (c)
(a)
|Q'|(pm)
p
|t |2
1
0.5
0.006
0.002
0 0
1 Ω/ω
2
m
(d) 0.2 ρ
0.8
p
|t |2
1
(b)
0.1 0.6 0.4
0.5
1 Ω/ωm
1.5
0 0
0.5
1 Ω/ωm
1.5
2
Fig. 3. (a) Transmission of the probe field in corresponding Δ2 . (b) Two transparent windows induced by modulating Δ2 for a strong P l2 . (c) The second splitting of the mechanical mode of the movable mirror. (d) The spectrum of anti-Stokes field excited from the control field with frequency ωl1 .
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