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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 10 — May. 20, 2013
  • pp: 12165–12173
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Electromagnetically induced transparency and slow light in two-mode optomechanics

Cheng Jiang, Hongxiang Liu, Yuanshun Cui, Xiaowei Li, Guibin Chen, and Bin Chen  »View Author Affiliations


Optics Express, Vol. 21, Issue 10, pp. 12165-12173 (2013)
http://dx.doi.org/10.1364/OE.21.012165


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Abstract

We theoretically demonstrate the mechanically mediated electromagnetically induced transparency in a two-mode cavity optomechanical system, where two cavity modes are coupled to a common mechanical resonator. When the two cavity modes are driven on their respective red sidebands by two pump beams, a transparency window appears in the probe transmission spectrum due to destructive interference. Under this situation the transmitted probe beam can be delayed as much as 4 μs, which can be easily controlled by the power of the pump beams.

© 2013 OSA

1. Introduction

The emerging field of cavity optomechanics [1

1. T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15,17172–17205 (2007) [CrossRef] [PubMed] .

4

4. M. Aspelmeyer, P. Meystre, and K. Schwab, “Quantum optomechanics,” Phys. Today 65,29–35 (2012) [CrossRef] .

] studies the interaction between optical and mechanical modes via radiation pressure force, which enables to observe quantum mechanical behavior of macroscopic systems. Recent progress in fabrication and cooling techniques paves the way towards realizing strong coupling at the single-photon level in optomechanical systems [5

5. F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322,235–238 (2008) [CrossRef] [PubMed] .

10

10. B. He, “Quantum optomechanics beyond linearization,” Phys. Rev. A 85,063820 (2012) [CrossRef] .

] and cooling the nanomechanical resonators to their quantum ground state [11

11. J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature (London) 475,359–363 (2011) [CrossRef] .

,12

12. J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature (London) 478,89–92 (2011) [CrossRef] .

]. Moreover, the optical response of optomechanical systems is modified because of mechanical interactions, leading to the phenomenon of normal-mode splitting [13

13. J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett. 101,263602 (2008) [CrossRef] [PubMed] .

, 14

14. S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature (London) 460,724–727 (2009) [CrossRef] .

] and electromagnetically induced transparency (EIT) [15

15. G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010) [CrossRef] .

18

18. M. Karuza, C. Biancofiore, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a room temperature membrane-in-the-middle setup,” arXiv:1209.1352 (2012).

]. In EIT [19

19. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77,633–673 (2005) [CrossRef] .

] an opaque medium can be made transparent in the presence of a strong pump beam; the concomitant steep variation of the refractive index induces a drastic reduction in the group velocity of a probe beam, which can be used to slow and stop light [20

20. L.V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature (London) 397,594–598 (1999) [CrossRef] .

, 21

21. D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New J. Phys. 13,023003 (2011) [CrossRef] .

]. EIT has been first observed in atomic vapors [22

22. K.-J. Boller, A. Imamoğlu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66,2593–2596 (1991) [CrossRef] [PubMed] .

] and recently in various solid state systems such as quantum wells [23

23. M. C. Phillips, H. Wang, I. Rumyantsev, N. H. Kwong, R. Takayama, and R. Binder, “Electromagnetically Induced Transparency in Semiconductors via Biexciton Coherence,” Phys. Rev. Lett. 91,183602 (2003) [CrossRef] [PubMed] .

], metamaterial [24

24. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8,758–762 (2009) [CrossRef] [PubMed] .

] and nitrogen-vacancy centers [25

25. C. Santori, P. Tamarat, P. Neumann, J. Wrachtrup, D. Fattal, R. G. Beausoleil, J. Rabeau, P. Olivero, A. D. Greentree, S. Prawer, F. Jelezko, and P. Hemmer, “Coherent population trapping of single spins in diamond under optical excitation,” Phys. Rev. Lett. 97,247401 (2006) [CrossRef] .

]. In optomechanical systems, slowing and advancing of signals based on EIT have been observed both in optical [17

17. 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, “Electromagnetically induced transparency and slow light with optomechanics,” Nature (London) 472, 69–73 (2011) [CrossRef] .

, 18

18. M. Karuza, C. Biancofiore, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a room temperature membrane-in-the-middle setup,” arXiv:1209.1352 (2012).

] and microwave domains [26

26. X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9,179–184 (2013) [CrossRef] .

]. Recently, the phenomena of electromechanically induced amplification [27

27. F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351–354 (2011) [CrossRef] [PubMed] .

] and absorption [28

28. F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-electromechanical system,” New J. Phys. 14,123037 (2012) [CrossRef] .

] with a blue-detuned pump field have also been presented in the circuit nano-electromechanical system consisted of a superconducting microwave resonator and a nanomechanical beam.

Most recently, two-mode optomechanics in which two optical modes are coupled to a mechanical mode have received a lot of research interest. Dobrindt et al.[29

29. J. M. Dobrindt and T. J. Kippenberg, “Theoretical analysis of mechanical displacement measurement using a multiple cavity mode transducer,” Phys. Rev. Lett. 104,033901 (2010) [CrossRef] [PubMed] .

] have shown that the dual mode transducer leads to a dramatic reduction of the power to reach the standard quantum limit (SQL) for a high frequency resonator. Ludwig et al.[30

30. M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109,063601 (2012) [CrossRef] [PubMed] .

] and Kómár et al.[31

31. P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon n onlinearities in two-mode optomechanics,” Phys. Rev. A 87,013839 (2013) [CrossRef] .

] have theoretically shown that quantum nonlinearities can be enhanced significantly in two-mode optomechanical systems, which can be used in optomechanical quantum information processing with photons and phonons [32

32. K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109,013603 (2012) [CrossRef] [PubMed] .

]. However, the theoretical work of Dobrindt [29

29. J. M. Dobrindt and T. J. Kippenberg, “Theoretical analysis of mechanical displacement measurement using a multiple cavity mode transducer,” Phys. Rev. Lett. 104,033901 (2010) [CrossRef] [PubMed] .

], Ludwig [30

30. M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109,063601 (2012) [CrossRef] [PubMed] .

], and Kómár [31

31. P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon n onlinearities in two-mode optomechanics,” Phys. Rev. A 87,013839 (2013) [CrossRef] .

] require that the mechanical frequencies are nearly resonant to the optical level splitting, which is more demanding to realize experimentally. Recently, Qu and Agarwal [33

33. K. Qu and G. S. Agarwal, “Optical memories and transduction of fields in double cavity optomechanical systems,” arXiv:1210.4067 (2012).

] theoretically showed that double cavity optomechanical systems can be used both as memory elements as well as for the transduction of optical fields. Hill et al.[34

34. J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3,1196 (2012) [CrossRef] .

] and Dong et al.[35

35. C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, “Optical wavelength conversion via optomechanical coupling in a silica resonator,” arXiv:1205.2360 (2012).

] have experimentally demonstrated coherent wavelength conversion of optical photons between two different optical wavelengths in optomechanical crystal nanocavity and silica resonator, respectively. In the present paper, we investigate the optical response of the two-mode optomechanical system in the simultaneous presence of two strong pump beams and a weak probe beam. When the two cavities are pumped on their red sidebands (i.e., one mechanical frequency, ωm, below cavity resonances, ω1 and ω2), respectively, a transparency window appears in the probe transmission spectrum.

2. Model and theory

Fig. 1 Schematic of a two-mode optomechanical system where two optical cavity modes, a1 and a2, are coupled to the same mechanical mode b. The left cavity is driven by a strong pump beam EL in the simultaneous presence of a weak probe beam Ep while the right cavity is only driven by a pump beam ER.

Applying the Heisenberg equations of motion for operators a1, a2, and Q which is defined as Q = b + b and introducing the corresponding damping and noise terms [36

36. C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008) [CrossRef] .

], we derive the quantum Langevin equations as follows:
a˙1=i(Δ1g1Q)a1κ1a1+κe,1(EL+Epeiδt)+2κ1ain,1,
(2)
a˙2=i(Δ2g2Q)a2κ2a2+κe,2ER+2κ2ain,2,
(3)
Q¨+γmQ˙+ωm2Q=2g1ωma1a1+2g2ωma2a2+ξ,
(4)
where ain,1 and ain,2 are the input vacuum noise operators with zero mean value, ξ is the Brownian stochastic force with zero mean value [36

36. C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008) [CrossRef] .

].

Following standard methods from quantum optics, we derive the steady-state solution to Eqs. (2)(4) by setting all the time derivatives to zero. They are given by
as,1=κe,1ELκ1+iΔ1,as,2=κe,2ERκ2+iΔ2,Qs=2ωm(g1|as,1|2+g2|as,2|2),
(5)
where Δ′1 = Δ1g1Qs and Δ′2 = Δ2g2Qs are the effective cavity detunings including radiation pressure effects. We can rewrite each Heisenberg operator of Eqs. (2)(4) as the sum of its steady-state mean value and a small fluctuation with zero mean value,
a1=as,1+δa1,a2=as,2+δa2,Q=Qs+δQ.
(6)
Inserting these equations into the Langevin equations Eqs. (2)(4) and assuming |as,1| ≫ 1 and |as,2| ≫ 1, one can safely neglect the nonlinear terms δa1δa1, δa2δa2, δa1δQ, and δa2δQ. Since the drives are weak, but classical coherent fields, we will identify all operators with their expectation values, and drop the quantum and thermal noise terms [16

16. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010) [CrossRef] [PubMed] .

]. Then the linearized Langevin equations can be written as:
δa˙1=(κ1+iΔ1)δa1+ig1Qsδa1+ig1as,1δQ+κe,1Epeiδt,
(7)
δa˙2=(κ2+iΔ2)δa2+ig2Qsδa2+ig2as,2δQ,
(8)
δQ¨+γmδQ˙+ωm2δQ=2ωmg1as,1(δa1+δa1)+2ωmg2as,2(δa2+δa2).
(9)
In order to solve equations (7)(9), we make the ansatz [37

37. R. W. Boyd, Nonlinear Optics (San Diego, CA: Academic) (2008).

] 〈δa1〉 = a1+eiδt + a1−eiδt, 〈δa2〉 = a2+eiδt + a2−eiδt, and 〈δQ〉 = Q+eiδt + Qeiδt. Upon substituting the above ansatz into Eqs. (7)(9), we derive the following solution
a1+=κe,1Epκ1+iΔ1iδ1d(δ)ig12n1κe,1Ep(κ1+iΔ1iδ)2,
(10)
where
d(δ)=k=1,22Δkgk2nk(κkiδ)2+Δk2ωm2δ2iδγmωm,
(11)
and nk = |as,k|2. Here nk, approximately equal to the number of pump photons in each cavity, is determined by the following coupled equations
n1=κe,1EL2κ12+[Δ12g1/ωm(g1n1g2n2)]2,
(12)
n2=κe,2ER2κ22+[Δ22g2/ωm(g1n1g2n2)]2.
(13)

The output field can be obtained by employing the standard input-output theory [38

38. C. W. Gardiner and P. Zoller, Quantum Noise (Springer) (2004).

] aout(t)=ain(t)κea(t), where aout(t) is the output field operator. Considering the output field of the left cavity, we have
aout(t)=(ELκe,1as,1)eiωLt+(Epκe,1a1+)ei(δ+ωL)tκe,1a1ei(δωL)t
(14)
The transmission of the probe field, defined by the ratio of the output and input field amplitudes at the probe frequency, is then given by
t(ωp)=Epκe,1a1+Ep=1[κe,1κ1+iΔ1iδ1d(δ)ig12n1κe,1(κ1+iΔ1iδ)2].
(15)
The rapid phase dispersion ϕ = arg[t(ωp)] of the transmitted probe laser beam leads to a group delay τg expressed as
τg=dϕdωp|ωp=ω1.
(16)
Note that, if ER = 0 and g2 = 0, the Eqs (10)(16) lead to the well-known results for the single mode cavity optomechanical system, where electromagnetically induced transparency and slow light effect have been observed experimentally [16

16. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010) [CrossRef] [PubMed] .

, 17

17. 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, “Electromagnetically induced transparency and slow light with optomechanics,” Nature (London) 472, 69–73 (2011) [CrossRef] .

]. In what follows, we will investigate theoretically this phenomenon in the two-mode optomechanics we consider here.

3. Results and discussion

To illustrate the numerical results, we choose a realistic two-mode cavity optomechanical system to calculate the transmission spectrum of the probe field. The parameters used are [34

34. J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3,1196 (2012) [CrossRef] .

]: ω1 = 2π × 205.3 THz, ω2 = 2π × 194.1 THz, κ1 = 2π × 520 MHz, κ2 = 1.73 GHz, κe,1 = 0.2κ1, κe,2 = 0.42κ2, g1 = 2π × 960 kHz, g2 = 2π × 430 kHz, ωm = 2π × 4 GHz, Qm = 87 × 103, where Qm is the quality factor of the nanomechanical resonator, and the damping rate γm is given by ωmQm. We can see that ωm > κ1 and ωm > κ2, therefore the system operates in the resolved-sideband regime also termed good-cavity limit which is a prerequisite for the ground state cooling of a mechanical resonator [39

39. I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99,093901 (2007) [CrossRef] [PubMed] .

].

Characterization of the optomechanical cavity can be performed by using two strong pump beams combined with a weak probe beam. With both pump beams detuned a mechanical frequency to the red of their respective cavity modes (Δ1 = Δ2 = ωm), a weak probe beam is then swept across the left cavity mode. The resulting transmission spectra of the probe beam as a function of the probe-cavity detuning Δp = ωpω1 are plotted in Fig. 2, where PL = 0, 0.1, 1 and 10 μW, respectively, while the power of the right pump beam PR is kept equal to 0.1 μW. When PL = 0 μW, there is a transmission dip in the center of the probe transmission spectrum, as shown in Fig. 2(a). However, as PL = 0.1 μW, the broad cavity resonance splits into two dips and a narrow transparency window appears when the probe beam is resonant with the cavity frequency. As the left pump power increases, and hence effective coupling strength G1=g1n1, increase further, so does the probe transmission at the cavity resonance. The width of the transparency window also increases and is given by the modified mechanical damping rate γmeffγm(1+4g12n1κ1γm+4g22n2κ2γm)[16

16. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010) [CrossRef] [PubMed] .

, 34

34. J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3,1196 (2012) [CrossRef] .

, 39

39. I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99,093901 (2007) [CrossRef] [PubMed] .

]. This mechanically mediated electromagnetically induced transparency can be understood as a result of radiation pressure force oscillating at the beat frequency δ = ωpωL between the pump beam and the probe beam. If this driving force is close to the mechanical resonance frequency ωm, the vibrational mode is excited coherently, resulting in Stokes and anti-Stokes scattering of light from the strong pump field. If the cavity is driven on its red sideband, the highly off-resonant Stokes scattering is suppressed and only the anti-Stokes scattering builds up within the cavity. However, when the probe beam is resonant with the cavity, destructive interference with the anti-Stokes field suppresses its build-up and hence a transparency window appears in the probe transmission spectrum. These processes are captured by the linearized Langevin Eqs. (7)(9). In the resolved sideband regime (κ1, κ2 < ωm), when the pump beam detuning Δ′1 = Δ′2ωm, the lower sideband can be neglected, i.e., a1− ≈ 0 and a2− ≈ 0 [16

16. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010) [CrossRef] [PubMed] .

]. The solution for the left intracavity field reads
a1+κe,1Epix+κ1+g12n1/2ix+γm/2+g22n2/2ix+κ2,
(17)
where x(= δωm) represents the detuning of the probe frequency to the cavity frequency. When g2 = 0, Eq. (17) leads to the result for the single cavity [16

16. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010) [CrossRef] [PubMed] .

]. This solution has a form well known from the response of an EIT medium to a probe field [19

19. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77,633–673 (2005) [CrossRef] .

]. The role of the control laser’s Rabi frequency in an atomic system is taken by the parametrically enhanced optomechanical coupling rate G1=g1n1 and G2=g2n2, and the two-photon resonance condition is given by Δ′1 = Δ′2 = ωm.

Fig. 2 Probe transmission as a function of the probe-cavity detuning Δp = ωpω1 for left pump power PL equals to 0, 0.1, 1, and 10 μW, respectively. The right pump power is kept equal to 0.1 μW. Both the cavities are pumped on their respective red sidebands, i.e., Δ1 = ωm and Δ2 = ωm. Other parameters used are ω1 = 2π × 205.3 THz, ω2 = 2π × 194.1 THz, κ1 = 2π × 520 MHz, κ2 = 1.73 GHz, κe,1 = 0.2κ1, κe,2 = 0.42κ2, g1 = 2π × 960 kHz, g2 = 2π × 430 kHz, ωm = 2π × 4 GHz, Qm = 87 × 103.

Fig. 3 (a) Magnitude and (b) phase of the transmitted probe beam versus probe-cavity detuning Δp for PL = 10 μW and PR = 0.1 μW. Other parameters are the same with figure 2.
Fig. 4 Group delay τg of the (a)–(c) transmitted (d) reflected probe beam as a function of the left pump power PL with Δ1 = ωm and Δ2 = ωm considering the effects of κe,1 and PR. Other parameters are ω1 = 2π × 205.3 THz, ω2 = 2π × 194.1 THz, κ1 = 2π × 520 MHz, κ2 = 1.73 GHz, κe,2 = 0.42κ2, g1 = 2π × 960 kHz, g2 = 2π × 430 kHz, ωm = 2π × 4 GHz, Qm = 87 × 103.

4. Conclusion

In conclusion, we have demonstrated both electromagnetically induced transparency and amplification in a two-mode cavity optomechanics consisted of two optical cavity modes coupled to a common mechanical mode under different driving conditions. Destructive interference between the probe beam and the anti-Stokes field leads to a transparency window in the probe transmission spectrum in conjunction with a steep positive phase dispersion, giving rise to the corresponding slow light effect. Our theoretical results show an optically tunable delay of 4 μs of the transmitted probe beam.

Acknowledgments

The authors gratefully acknowledge support from National Natural Science Foundation of China (Grant Nos. 11074088 and 11174101) and Jiangsu Natural Science Foundation (Grant No. BK2011411).

References and links

1.

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15,17172–17205 (2007) [CrossRef] [PubMed] .

2.

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3.

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4.

M. Aspelmeyer, P. Meystre, and K. Schwab, “Quantum optomechanics,” Phys. Today 65,29–35 (2012) [CrossRef] .

5.

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322,235–238 (2008) [CrossRef] [PubMed] .

6.

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107,063601 (2011) [CrossRef] [PubMed] .

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A. Nunnenkamp, K. Borkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107,063602 (2011) [CrossRef] [PubMed] .

8.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature (London) 471, 204–208 (2011) [CrossRef] .

9.

E. Verhagen, S. Delglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482,63–67 (2012) [CrossRef] .

10.

B. He, “Quantum optomechanics beyond linearization,” Phys. Rev. A 85,063820 (2012) [CrossRef] .

11.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature (London) 475,359–363 (2011) [CrossRef] .

12.

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature (London) 478,89–92 (2011) [CrossRef] .

13.

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett. 101,263602 (2008) [CrossRef] [PubMed] .

14.

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature (London) 460,724–727 (2009) [CrossRef] .

15.

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010) [CrossRef] .

16.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010) [CrossRef] [PubMed] .

17.

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, “Electromagnetically induced transparency and slow light with optomechanics,” Nature (London) 472, 69–73 (2011) [CrossRef] .

18.

M. Karuza, C. Biancofiore, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a room temperature membrane-in-the-middle setup,” arXiv:1209.1352 (2012).

19.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77,633–673 (2005) [CrossRef] .

20.

L.V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature (London) 397,594–598 (1999) [CrossRef] .

21.

D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New J. Phys. 13,023003 (2011) [CrossRef] .

22.

K.-J. Boller, A. Imamoğlu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66,2593–2596 (1991) [CrossRef] [PubMed] .

23.

M. C. Phillips, H. Wang, I. Rumyantsev, N. H. Kwong, R. Takayama, and R. Binder, “Electromagnetically Induced Transparency in Semiconductors via Biexciton Coherence,” Phys. Rev. Lett. 91,183602 (2003) [CrossRef] [PubMed] .

24.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8,758–762 (2009) [CrossRef] [PubMed] .

25.

C. Santori, P. Tamarat, P. Neumann, J. Wrachtrup, D. Fattal, R. G. Beausoleil, J. Rabeau, P. Olivero, A. D. Greentree, S. Prawer, F. Jelezko, and P. Hemmer, “Coherent population trapping of single spins in diamond under optical excitation,” Phys. Rev. Lett. 97,247401 (2006) [CrossRef] .

26.

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9,179–184 (2013) [CrossRef] .

27.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351–354 (2011) [CrossRef] [PubMed] .

28.

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-electromechanical system,” New J. Phys. 14,123037 (2012) [CrossRef] .

29.

J. M. Dobrindt and T. J. Kippenberg, “Theoretical analysis of mechanical displacement measurement using a multiple cavity mode transducer,” Phys. Rev. Lett. 104,033901 (2010) [CrossRef] [PubMed] .

30.

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109,063601 (2012) [CrossRef] [PubMed] .

31.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon n onlinearities in two-mode optomechanics,” Phys. Rev. A 87,013839 (2013) [CrossRef] .

32.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109,013603 (2012) [CrossRef] [PubMed] .

33.

K. Qu and G. S. Agarwal, “Optical memories and transduction of fields in double cavity optomechanical systems,” arXiv:1210.4067 (2012).

34.

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3,1196 (2012) [CrossRef] .

35.

C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, “Optical wavelength conversion via optomechanical coupling in a silica resonator,” arXiv:1205.2360 (2012).

36.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008) [CrossRef] .

37.

R. W. Boyd, Nonlinear Optics (San Diego, CA: Academic) (2008).

38.

C. W. Gardiner and P. Zoller, Quantum Noise (Springer) (2004).

39.

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99,093901 (2007) [CrossRef] [PubMed] .

OCIS Codes
(270.1670) Quantum optics : Coherent optical effects
(140.3945) Lasers and laser optics : Microcavities

ToC Category:
Quantum Optics

History
Original Manuscript: February 25, 2013
Revised Manuscript: April 17, 2013
Manuscript Accepted: April 25, 2013
Published: May 10, 2013

Citation
Cheng Jiang, Hongxiang Liu, Yuanshun Cui, Xiaowei Li, Guibin Chen, and Bin Chen, "Electromagnetically induced transparency and slow light in two-mode optomechanics," Opt. Express 21, 12165-12173 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-12165


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References

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  11. J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature (London)475,359–363 (2011). [CrossRef]
  12. J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature (London)478,89–92 (2011). [CrossRef]
  13. J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett.101,263602 (2008). [CrossRef] [PubMed]
  14. S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature (London)460,724–727 (2009). [CrossRef]
  15. G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A81, 041803 (2010). [CrossRef]
  16. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science330, 1520–1523 (2010). [CrossRef] [PubMed]
  17. 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, “Electromagnetically induced transparency and slow light with optomechanics,” Nature (London)472, 69–73 (2011). [CrossRef]
  18. M. Karuza, C. Biancofiore, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a room temperature membrane-in-the-middle setup,” arXiv:1209.1352 (2012).
  19. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77,633–673 (2005). [CrossRef]
  20. L.V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature (London)397,594–598 (1999). [CrossRef]
  21. D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New J. Phys.13,023003 (2011). [CrossRef]
  22. K.-J. Boller, A. Imamoğlu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett.66,2593–2596 (1991). [CrossRef] [PubMed]
  23. M. C. Phillips, H. Wang, I. Rumyantsev, N. H. Kwong, R. Takayama, and R. Binder, “Electromagnetically Induced Transparency in Semiconductors via Biexciton Coherence,” Phys. Rev. Lett.91,183602 (2003). [CrossRef] [PubMed]
  24. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater.8,758–762 (2009). [CrossRef] [PubMed]
  25. C. Santori, P. Tamarat, P. Neumann, J. Wrachtrup, D. Fattal, R. G. Beausoleil, J. Rabeau, P. Olivero, A. D. Greentree, S. Prawer, F. Jelezko, and P. Hemmer, “Coherent population trapping of single spins in diamond under optical excitation,” Phys. Rev. Lett.97,247401 (2006). [CrossRef]
  26. X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys.9,179–184 (2013). [CrossRef]
  27. F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature480, 351–354 (2011). [CrossRef] [PubMed]
  28. F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-electromechanical system,” New J. Phys.14,123037 (2012). [CrossRef]
  29. J. M. Dobrindt and T. J. Kippenberg, “Theoretical analysis of mechanical displacement measurement using a multiple cavity mode transducer,” Phys. Rev. Lett.104,033901 (2010). [CrossRef] [PubMed]
  30. M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett.109,063601 (2012). [CrossRef] [PubMed]
  31. P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon n onlinearities in two-mode optomechanics,” Phys. Rev. A87,013839 (2013). [CrossRef]
  32. K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett.109,013603 (2012). [CrossRef] [PubMed]
  33. K. Qu and G. S. Agarwal, “Optical memories and transduction of fields in double cavity optomechanical systems,” arXiv:1210.4067 (2012).
  34. J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun.3,1196 (2012). [CrossRef]
  35. C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, “Optical wavelength conversion via optomechanical coupling in a silica resonator,” arXiv:1205.2360 (2012).
  36. C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A77, 033804 (2008). [CrossRef]
  37. R. W. Boyd, Nonlinear Optics (San Diego, CA: Academic) (2008).
  38. C. W. Gardiner and P. Zoller, Quantum Noise (Springer) (2004).
  39. I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett.99,093901 (2007). [CrossRef] [PubMed]

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