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 Optical Society of America

1. Introduction

The emerging field of cavity optomechanics [14] 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 [510] and cooling the nanomechanical resonators to their quantum ground state [11,12]. Moreover, the optical response of optomechanical systems is modified because of mechanical interactions, leading to the phenomenon of normal-mode splitting [13, 14] and electromagnetically induced transparency (EIT) [1518]. In EIT [19] 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, 21]. EIT has been first observed in atomic vapors [22] and recently in various solid state systems such as quantum wells [23], metamaterial [24] and nitrogen-vacancy centers [25]. In optomechanical systems, slowing and advancing of signals based on EIT have been observed both in optical [17, 18] and microwave domains [26]. Recently, the phenomena of electromechanically induced amplification [27] and absorption [28] 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] 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] and Kómár et al.[31] 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]. However, the theoretical work of Dobrindt [29], Ludwig [30], and Kómár [31] 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] 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] and Dong et al.[35] 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

We consider an optomechanical system as shown in Fig. 1, where two optical cavity modes ak(k = 1, 2) are coupled to a common mechanical mode b. The left cavity is driven by a strong pump beam EL with frequency ωL and a weak probe beam Ep with frequency ωp simultaneously, and the right cavity is only driven by a strong pump beam ER with frequency ωR. In a rotating frame at the pump frequency ωL and ωR, the Hamiltonian of the two-mode optomechanical system reads as follows [34]:

H=k=1,2h¯Δkakak+h¯ωmbbk=1,2h¯gkakak(b+b)+ih¯κe,1EL(a1a1)+ih¯κe,2ER(a2a2)+ih¯κe,1Ep(a1eiδta1eiδt).
The first term describes the energy of the two optical cavity modes with resonance frequency ωk(k = 1, 2), where ak (ak) is the creation (annihilation) operator of each cavity mode. Δ1 = ω1ωL and Δ2 = ω2ωR are the corresponding cavity-pump field detunings. The second term gives the energy of the mechanical mode with creation (annihilation) operator b (b), resonance frequency ωm and effective mass m. The third term is the radiation pressure coupling rate gk=(ωk/Lk)h¯/(2mωm), where Lk is an effective length that depends on the cavity geometry. The last three terms represent the input fields, where EL, ER, and Ep are related to the power of the applied laser fields by |EL|=2PLκ1/h¯ωL, |ER|=2PRκ2/h¯ωR, and |Ep|=2Ppκ1/h¯ωp (κk the linewidth of the kth cavity mode), respectively. The total cavity linewidth κk = κi,k + κe,k, where κe,k is the cavity decay rate due to coupling to an external photonic waveguide, as presented in the realistic two-mode optomechanical nanocavity [34]. δ = ωpωL is the detuning between the probe filed and the left pump field.

 figure: Fig. 1

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.

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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], we derive the quantum Langevin equations as follows:

a˙1=i(Δ1g1Q)a1κ1a1+κe,1(EL+Epeiδt)+2κ1ain,1,
a˙2=i(Δ2g2Q)a2κ2a2+κe,2ER+2κ2ain,2,
Q¨+γmQ˙+ωm2Q=2g1ωma1a1+2g2ωma2a2+ξ,
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].

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),
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.
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]. Then the linearized Langevin equations can be written as:
δa˙1=(κ1+iΔ1)δa1+ig1Qsδa1+ig1as,1δQ+κe,1Epeiδt,
δa˙2=(κ2+iΔ2)δa2+ig2Qsδa2+ig2as,2δQ,
δQ¨+γmδQ˙+ωm2δQ=2ωmg1as,1(δa1+δa1)+2ωmg2as,2(δa2+δa2).
In order to solve equations (7)(9), we make the ansatz [37] 〈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,
where
d(δ)=k=1,22Δkgk2nk(κkiδ)2+Δk2ωm2δ2iδγmωm,
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,
n2=κe,2ER2κ22+[Δ22g2/ωm(g1n1g2n2)]2.

The output field can be obtained by employing the standard input-output theory [38] 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
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].
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.
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, 17]. 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]: 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].

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, 34, 39]. 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]. The solution for the left intracavity field reads

a1+κe,1Epix+κ1+g12n1/2ix+γm/2+g22n2/2ix+κ2,
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]. This solution has a form well known from the response of an EIT medium to a probe field [19]. 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.

 figure: Fig. 2

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.

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Much as in atomic EIT, this effect causes an extremely steep dispersion for the transmitted probe photons, leading to a group delay. Fig. 3 shows the magnitude and phase dispersion of the probe transmission as a function of probe-cavity detuning Δp with Δ1 = Δ2 = ωm for PL = 10 μW and PR = 0.1 μW. It can be seen clearly that there is a transparency window combined with a steep positive phase dispersion at the cavity resonance, which will result in a tunable group delay of the transmitted probe beam. In addition, the delay in transmission is directly related to the advance on reflection through the bare cavity transmission contrast. To verify this, we plot the corresponding transmission group delay τg(T) of the probe beam versus the left pump power PL with Δ1 = Δ2 = ωm for PR = 0.1 μW in Fig. 4(a). As can be seen from the figure, the maximum transmission delay is τg(T)4.5ns. However, when the right pump beam beam is turned off, the transmission delay can be significantly increased, with a maximum delay 4 μs. Therefore, the group delay in the two-mode optomechanical system is smaller than the delay in the single optical cavity system. The physical origin can be explained as follows. When the two cavities are pumped on their respective red sidebands, the mechanical oscillation is damped by both the left and right cavity mode, with the total damping rate γmeffγm(1+4g12n1κ1γm+4g22n2κ2γm). The width of the transparency window is larger than the one in the single cavity where g2 = 0, and the phase gradient in the probe transmission in the left cavity gets smaller. Therefore, the pump on the right cavity leads to the reduction of the group delay of the transmitted probe beam. However, much more flexible controllability of EIT and slow light effect in two-mode optomechanics can find potential applications in quantum memory [33] and coherent optical wavelength conversion [34], which is useful in communication system. In Fig. 4(c), we consider the effect of the external decay rate κe,1 on the group delay. If the external decay dominates the decay of the cavity, κe,1 = 0.6κ for example, the maximum transmission group delay can be increased further. Moreover, the group delay of the reflected probe beam as a function of the power of the left pump beam is plotted in Fig. 4(d) where the delay is negative, which represents the advancing of light pulses. Therefore, we can tune the group delay and advance of the probe beam by controlling the power of the pump beam.

 figure: Fig. 3

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.

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 figure: Fig. 4

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.

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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

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References

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  1. T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15,17172–17205 (2007).
    [Crossref] [PubMed]
  2. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321,1172–1176 (2008).
    [Crossref] [PubMed]
  3. F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2,40 (2009).
    [Crossref]
  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]
  7. 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]

2013 (2)

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]

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]

2012 (7)

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]

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]

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]

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]

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

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]

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

2011 (8)

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]

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]

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

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

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]

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]

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]

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]

2010 (3)

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]

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

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]

2009 (3)

F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2,40 (2009).
[Crossref]

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]

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]

2008 (4)

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]

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]

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

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321,1172–1176 (2008).
[Crossref] [PubMed]

2007 (2)

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

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]

2006 (1)

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]

2005 (1)

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

2003 (1)

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]

1999 (1)

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]

1991 (1)

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

Agarwal, G. S.

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

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

Alegre, T. P.

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]

Allman, M. S.

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]

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]

Arcizet, O.

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]

Aspelmeyer, M.

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

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]

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]

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]

Beausoleil, R. G.

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]

Behroozi, C. H.

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]

Bennett, S. D.

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]

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]

Biancofiore, C.

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).

Binder, R.

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]

Boller, K.-J.

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

Borkje, K.

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

Boyd, R. W.

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

Brennecke, F.

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

Chan, J.

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]

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]

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]

Chang, D. E.

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]

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]

Cho, S. U.

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]

Cicak, K.

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]

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]

Deléglise, S.

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]

Delglise, S.

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]

Di Giuseppe, G.

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).

Dobrindt, J. M.

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]

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]

Dong, C.

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).

Donner, T.

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]

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

Dutton, Z.

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]

Eichenfield, M.

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]

Esslinger, T.

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

Fattal, D.

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]

Fiore, V.

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).

Fleischhauer, M.

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]

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

Galassi, M.

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).

Gardiner, C. W.

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

Gavartin, E.

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]

Genes, C.

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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]

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]

Schwab, K.

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

Sillanpää, M. A.

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]

Simmonds, R. W.

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]

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]

Sirois, A. J.

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]

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]

Stannigel, K.

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]

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]

Takayama, R.

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]

Tamarat, P.

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]

Teufel, J. D.

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]

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]

Tian, L.

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).

Tombesi, P.

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]

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).

Vahala, K. J.

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321,1172–1176 (2008).
[Crossref] [PubMed]

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

Vanner, M. R.

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]

Verhagen, E.

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]

Vitali, D.

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]

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).

Wang, H.

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]

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).

Weis, S.

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]

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]

Weiss, T.

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]

Whittaker, J. D.

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]

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]

Wilson-Rae, I.

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]

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]

Winger, M.

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]

Wrachtrup, J.

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]

Zhou, X.

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]

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]

Zoller, P.

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]

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]

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

Zwerger, W.

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]

Nat. Commun. (1)

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]

Nat. Mater. (1)

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]

Nat. Phys. (1)

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]

Nature (1)

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]

Nature (London) (7)

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]

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]

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]

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]

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]

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]

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]

New J. Phys. (2)

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]

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]

Opt. Express (1)

Phys. Rev. A (4)

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

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

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]

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]

Phys. Rev. Lett. (10)

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]

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]

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]

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]

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

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]

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]

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

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

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]

Phys. Today (1)

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

Physics (1)

F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2,40 (2009).
[Crossref]

Rev. Mod. Phys. (1)

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

Science (3)

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]

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

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321,1172–1176 (2008).
[Crossref] [PubMed]

Other (5)

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).

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

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

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

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).

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Figures (4)

Fig. 1
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.
Fig. 2
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
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
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.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

H = k = 1 , 2 h ¯ Δ k a k a k + h ¯ ω m b b k = 1 , 2 h ¯ g k a k a k ( b + b ) + i h ¯ κ e , 1 E L ( a 1 a 1 ) + i h ¯ κ e , 2 E R ( a 2 a 2 ) + i h ¯ κ e , 1 E p ( a 1 e i δ t a 1 e i δ t ) .
a ˙ 1 = i ( Δ 1 g 1 Q ) a 1 κ 1 a 1 + κ e , 1 ( E L + E p e i δ t ) + 2 κ 1 a in , 1 ,
a ˙ 2 = i ( Δ 2 g 2 Q ) a 2 κ 2 a 2 + κ e , 2 E R + 2 κ 2 a in , 2 ,
Q ¨ + γ m Q ˙ + ω m 2 Q = 2 g 1 ω m a 1 a 1 + 2 g 2 ω m a 2 a 2 + ξ ,
a s , 1 = κ e , 1 E L κ 1 + i Δ 1 , a s , 2 = κ e , 2 E R κ 2 + i Δ 2 , Q s = 2 ω m ( g 1 | a s , 1 | 2 + g 2 | a s , 2 | 2 ) ,
a 1 = a s , 1 + δ a 1 , a 2 = a s , 2 + δ a 2 , Q = Q s + δ Q .
δ a ˙ 1 = ( κ 1 + i Δ 1 ) δ a 1 + i g 1 Q s δ a 1 + i g 1 a s , 1 δ Q + κ e , 1 E p e i δ t ,
δ a ˙ 2 = ( κ 2 + i Δ 2 ) δ a 2 + i g 2 Q s δ a 2 + i g 2 a s , 2 δ Q ,
δ Q ¨ + γ m δ Q ˙ + ω m 2 δ Q = 2 ω m g 1 a s , 1 ( δ a 1 + δ a 1 ) + 2 ω m g 2 a s , 2 ( δ a 2 + δ a 2 ) .
a 1 + = κ e , 1 E p κ 1 + i Δ 1 i δ 1 d ( δ ) i g 1 2 n 1 κ e , 1 E p ( κ 1 + i Δ 1 i δ ) 2 ,
d ( δ ) = k = 1 , 2 2 Δ k g k 2 n k ( κ k i δ ) 2 + Δ k 2 ω m 2 δ 2 i δ γ m ω m ,
n 1 = κ e , 1 E L 2 κ 1 2 + [ Δ 1 2 g 1 / ω m ( g 1 n 1 g 2 n 2 ) ] 2 ,
n 2 = κ e , 2 E R 2 κ 2 2 + [ Δ 2 2 g 2 / ω m ( g 1 n 1 g 2 n 2 ) ] 2 .
a out ( t ) = ( E L κ e , 1 a s , 1 ) e i ω L t + ( E p κ e , 1 a 1 + ) e i ( δ + ω L ) t κ e , 1 a 1 e i ( δ ω L ) t
t ( ω p ) = E p κ e , 1 a 1 + E p = 1 [ κ e , 1 κ 1 + i Δ 1 i δ 1 d ( δ ) i g 1 2 n 1 κ e , 1 ( κ 1 + i Δ 1 i δ ) 2 ] .
τ g = d ϕ d ω p | ω p = ω 1 .
a 1 + κ e , 1 E p i x + κ 1 + g 1 2 n 1 / 2 i x + γ m / 2 + g 2 2 n 2 / 2 i x + κ 2 ,

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