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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 7 — Mar. 26, 2012
  • pp: 7672–7684
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Optomechanically induced non-reciprocity in microring resonators

Mohammad Hafezi and Peter Rabl  »View Author Affiliations


Optics Express, Vol. 20, Issue 7, pp. 7672-7684 (2012)
http://dx.doi.org/10.1364/OE.20.007672


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Abstract

We describe a new approach for on-chip optical non-reciprocity which makes use of strong optomechanical interaction in microring resonators. By optically pumping the ring resonator in one direction, the optomechanical coupling is only enhanced in that direction, and consequently, the system exhibits a non-reciprocal response. For different configurations, this system can function either as an optical isolator or a coherent non-reciprocal phase shifter. We show that the operation of such a device on the level of single-photon could be achieved with existing technology.

© 2012 OSA

1. Introduction

The development of integrated photonic circuits is a rapidly progressing field which aims at the realization of micron scale photonic elements and the integration of these elements into a single chip-based device. Apart from conventional optical signal processing and telecommunication applications [1

1. R. B. Wehrspohn, H. S. Kitzerow, and K. Busch. Nanophotonic Materials: Photonic Crystals, Plasmonics, and Metamaterials (Wiley-VCH, 2008).

], this technology might eventually also provide the basis for applications on a more fundamental level such as optical quantum computation [2

2. J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009). [CrossRef]

5

5. A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science 325, 1221 (2009). [CrossRef] [PubMed]

] or photonic quantum simulation schemes [6

6. D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced mott transitions and xy spin models in coupled cavity arrays,” Phys. Rev. A 76, 31805 (2007). [CrossRef]

8

8. M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006). [CrossRef]

]. A remaining challenge in integrated photonic circuits is on-chip optical isolation, that is, filtering of photons propagating in different directions in the circuit, or more generally, the implementation of non-reciprocal optical elements on a micrometer scale. Standard approaches for optical isolation make use of magneto-optical properties (e.g. Faraday rotation), which however require large magnetic fields [9

9. R. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004). [CrossRef]

], and thus make it difficult for integration [10

10. R. L. Espinola, T. Izuhara, M. C. Tsai, R. M. Osgood Jr, and H. Dötsch, “Magneto-optical nonreciprocal phase shift in garnet/silicon-on-insulator waveguides,” Opt. Lett. 29, 941–943 (2004). [CrossRef] [PubMed]

12

12. T. R. Zaman, X. Guo, and R. J. Ram, “Faraday rotation in an InP waveguide,” App. Phys. Lett. 90, 023514 (2007). [CrossRef]

] on a small scale. To overcome this problem, other non-magnetic approaches have been proposed which, for example, rely on a dynamical modulation of the index of refraction [13

13. Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009). [CrossRef]

], stimulated inter-polarization scattering based on opto-acoustic effects [14

14. M. S. Kang, A. Butsch, and P. St. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics 5, 549–553 (2011). [CrossRef]

], modulated dielectric constant [15

15. L. Feng, M. Ayache, J. Huang, Y. -L. Xu, M. -H. Lu, Y. -F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011). [CrossRef] [PubMed]

,16

16. S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popovic, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, ”Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335, 38 (2011). [CrossRef]

] or on optical non-linearities that lead to an intensity dependent isolation [17

17. M. Soljačić, C. Luo, J. D. Joannopoulos, and S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. 28, 637–639 (2004). [CrossRef]

19

19. S. Manipatruni, J. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102, 213903 (2009). [CrossRef] [PubMed]

].

In this work, we propose a new approach for on-chip optical non-reciprocity which makes use of the recent advances in the fabrication of on-chip and micron sized optomechanical (OM) devices [26

26. E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” arXiv:1107.3761 (2011).

29

29. L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency gaas nano-optomechanical disk resonator,” Phys. Rev. Lett. 105, 263903 (2010). [CrossRef]

]. In our scheme, the non-linear coupling between light and a mechanical mode inside a ring resonator leads to a non-reciprocal response of the OM system, which is induced and fully controlled by an external driving field. We characterize the input-output relations of such a device and show that by choosing different configurations the same mechanism can be employed for optical isolation as well as non-reciprocal phase shifting and routing applications. We describe under which conditions non-reciprocity is optimized and in particular, we find that even in the presence of a finite intrinsic mode coupling inside the ring resonator, non-reciprocal effects remain large for a sufficiently strong OM coupling. In contrast to optical isolation based on a non-linear response of the OM system [19

19. S. Manipatruni, J. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102, 213903 (2009). [CrossRef] [PubMed]

], our schemes can in principle be applied on a single photon level, limited by the up-conversion of thermal phonons only [30

30. K. Stannigel, P. Rabl, A. S. Sørensen, P. Zoller, and M. Lukin, “Optomechanical transducers for long-distance quantum communication,” Phys. Rev. Lett. 105, 220501 (2010). [CrossRef]

32

32. K. Stannigel, P. Rabl, A. S. Sørensen, M. D. Lukin, and P. Zoller, “Optomechanical transducers for quantum information processing,” Phys. Rev. A 84, 042341 (2011). [CrossRef]

]. Our analysis shows that a noise level below a single photon can be achieved with present technology, which makes this device a suitable building block for various non-reciprocal applications in the classical as well as the quantum regime.

2. Optomechanically induced non-reciprocity: a toy model

Before starting with a more general treatment below, we first outline in this section the essence of the OM induced non-reciprocity for an idealized and slightly simplified setting. Specifically, we consider an OM ring resonator, for example a toroidal microresonator, which is side-coupled to a waveguide as shown in Fig. 1(a). This configuration is commonly referred to as an all-pass filter (APF). The ring resonator supports two degenerate right- and left-circulating optical modes with frequency ωc and bosonic operators aR and aL respectively. Radial vibrations of the resonator lead to a modulation of ωc which can be modeled by the standard OM Hamiltonian [33

33. C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49,1337–1343 (1994). [CrossRef] [PubMed]

36

36. A. Schliesser and T. J. Kippenberg, “Cavity optomechanics with whispering-gallery mode optical microresonators,” Adv. At., Mol., Opt. Phys. 58, 207–323 (2010). [CrossRef]

] ( = 1),
Hom=ωmbb+i=L,Rωcaiai+g0aiai(b+b).
(1)
Here b is the bosonic operator for the mechanical mode of frequency ωm and g0 is the OM coupling, which corresponds to the optical frequency shift per quantum of motion. Note that the mechanical mode is extended and varies slowly over the scale of the optical wavelength [36

36. A. Schliesser and T. J. Kippenberg, “Cavity optomechanics with whispering-gallery mode optical microresonators,” Adv. At., Mol., Opt. Phys. 58, 207–323 (2010). [CrossRef]

]. Therefore, the optomechanical coupling does not induce a mixing between the right- and left-circulating optical modes. In typical experiments g0 is very weak and to enhance OM interactions we now assume that the right-circulating resonator mode is excited by an external laser field of frequency ωL = ωc + Δ. In the limit |αR| ≫ 1, where αR is the classical field amplitude of the driven mode, we can make a unitary transformation aRaR +αR and linearize the OM coupling around αR. As a result, we obtain an effective Hamiltonian which in the frame rotating with ωL is given by [33

33. C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49,1337–1343 (1994). [CrossRef] [PubMed]

36

36. A. Schliesser and T. J. Kippenberg, “Cavity optomechanics with whispering-gallery mode optical microresonators,” Adv. At., Mol., Opt. Phys. 58, 207–323 (2010). [CrossRef]

]
Homlin=Δ(aRaR+aLaL)+ωmbb+(GRaR+GR*aR)(b+b),
(2)
where an additional OM frequency shift has been reabsorbed into the definition of Δ (see Sec. 3 for a more detailed derivation). In Eq. (2), we have introduced the enhanced OM coupling GR = g0αR, and in view of |GR| ≫ g0, neglected residual OM interactions ∼ 𝒪(g0). We see that the external driving field creates an asymmetry between left- and right-circulating modes, which we can exploit for generating non-reciprocal effects.

Fig. 1 Non-reciprocal optomechanical device. (a) A strong pump field enhances the optomechanical coupling between an isolated vibrational mode and the right-circulating optical mode inside a ring resonator. This results in different transmission properties for right-and left-moving fields in the waveguide. (b) Optical isolation. (c) Non-reciprocal phase shifter.

Optical isolation. Let us first consider a critically coupled ring resonator where κ = κin. In this case, we see that for frequencies around the ring optical resonance (δ ≈ 0),
(fR,out(δ)fL,out(δ))(1000)(fR,in(δ)fL,in(δ)).
(9)
Therefore, this configuration realizes an optical diode, where light passes unaltered in one direction, but is completely absorbed in the other direction, as schematically shown in Fig. 1(b). The frequency window over which this isolation is efficient is approximately GR2/κ in the weak coupling limit (GRκ) and κ in the strong coupling limit, where the width of the EIT window is 2GR and exceeds the resonator linewidth. A typical non-reciprocal transmission spectrum for the strong coupling regime is shown in Fig. 2(a), which is that of an optical diode for frequencies around δ ≈ 0. Note that in contrast to conventional optical isolation, no magnetic field is applied and instead the optical pump breaks the left-right symmetry.

Fig. 2 (a) Transmission |tR/L|2 of the OM system when operated as an optical isolator (κin = κ). Within the resonator bandwidth, the left-moving field is attenuated while the right-moving field is almost completely transmitted. For this plot GR = 5κ. (b) Non-reciprocal phase shifter (κin = 0.01κ). Both the left and the right input field are almost completely transmitted (> 98%), but acquire different phases, Δθ = θRθL. Black lines show the location of resonances. For these plots γm = 0.

Non-reciprocal phase shifter. Let us now consider the so-called over-coupled regime where the intrinsic loss is much smaller than the resonator-waveguide coupling (κinκ). In this case, the transmittance is close to unity in both directions. However, the left- and right-going fields experience a different dispersion and
(fR,out(δ)fL,out(δ))(eiθR(δ)00eiθL(δ))(fR,in(δ)fL,in(δ)).
(10)
In general, the phases θR and θL will be different, and therefore, in this configuration, our devices acts as a non-reciprocal phase shifter, as schematically shown in Fig. 1(c). Again, in contrast to conventional magnetic field induced non-reciprocal phases, e.g. Faraday rotation, our scheme does not require large magnetic fields. As shown in Fig. 2(b), the OM induced phase difference |θRθL| can easily be controlled by changing the pump intensity and can be tuned from zero to about π over a large range of frequencies. Therefore, a maximal non-reciprocal phase shift can already be achieved for light passing through a single device.

3. General formalism

In this section we present the general formalism for investigating OM induced non-reciprocity. In particular, we now include the effect of energy non-conserving terms as well as a finite coupling between the left- and right-circulating resonator modes which have been neglected in our simplified discussion above. For completeness, we will also extend our discussion to a slightly more general configuration shown in Fig. 3, where the ring resonator is side-coupled to two optical waveguides with rates κ and κ′. For κ′ = 0, this setting reduces to the resonator coupled to a single waveguide case, which was discussed above. Moreover, in the so-called add-drop configuration (κ′ = κ, κin ≈ 0), this device can be used for non-reciprocal routing of light.

Fig. 3 General add-drop configuration, which can be employed for non-reciprocal photon routing between the upper and lower waveguide. It reduces to the resonator coupled to a single waveguide, if the coupling to the lower waveguide is absent (κ′ = 0).

To account for a more realistic situation we now include the presence of intrinsic defects inside the ring resonator and model the system by the total Hamiltonian
H=Hom+βaLaR+β*aRaL.
(11)
Here, in addition to the OM interaction Hom given in Eq. (2), the second and third terms in this Hamiltonian represent a coherent scatting of strength β between the two degenerate optical modes, which is associated with bulk or surface imperfections [42

42. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. 27, 1669–1671 (2002). [CrossRef]

, 43

43. A. Mazzei, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99, 173603 (2007). [CrossRef] [PubMed]

]. The system dynamics is fully described by the set of quantum Langevin equations (i = L,R)
a˙i(t)=i[H,ai(t)]κtai(t)2κfi,in1(t)2κfi,in2(t)2κinfi,0(t),
(12)
b˙(t)=i[H,b(t)]γm2b(t)γmξ(t),
(13)
together with the relations fi,out1(t)=fi,in1(t)+2κai(t) and fi,out2(t)=fi,in2(t)+2κai(t) between the in- and out-fields. In these equations, κt = κ +κ′ +κin is the total ring resonator field decay rate and the operators fi,in1,2(t) are δ-correlated field operators for the in-fields in the upper and lower waveguide (see Fig. 3) and fi,0(t) is a vacuum noise operator associated with the intrinsic photon loss. Finally, γm = ωm/Qm is the mechanical damping rate for a quality factor Qm and ξ(t) is the corresponding noise operator. In contrast to the optical fields, the mechanical mode is coupled to a reservoir of finite temperature T such that [ξ(t),ξ(t′)] = δ(t − t′) and 〈ξ(t)ξ(t′)〉 = (Nth + 1)δ(t − t′) where Nth is the thermal equilibrium occupation number of the mechanical mode. Note that the Langevin equation for the mechanical mode, Eq. (13), is only valid for γmωm. For typical mechanical quality factors Qm ∼ 104 − 105 this condition is well satisfied and for most of the results discussed below we will consider the limit γm → 0, while keeping a finite thermal heating rate γmNthkBT/(h̄Qm).

As before, we assume that the clockwise mode of the resonator is driven by a strong classical field of frequency ωL = ωc + Δ0 and amplitude 𝒠. We make the transformation fR,in(t)fR,in(t)+2κ and write the average field expectation values in the frame rotating with ωL,
a˙R=(iΔ0ig0b+bκt)aRiβ*aL2κ,
(14)
a˙L=(iΔ0ig0b+bκt)aLiβaR,
(15)
b˙=iωmbig0(|aR|2+|aL|2).
(16)
In the steady-state, we find that 〈b〉 = −g0(|〈aR〉|2 + |〈aL〉|2)/ωm. By redefining the detuning to absorb the OM shift, Δ=Δ0+2g02(|aR|2+|aL|2)/ωm, we can rewrite the optical field equations in the steady state as
0=(iΔκt)aRiβ*aL2κ,
(17)
0=(iΔκt)aLiβaR.
(18)
In the absence of mode coupling (β = 0), the counter clockwise mode remains empty (〈aL〉 = 0), and we obtain 〈aR〉 = 2κ𝒠/(iΔ − κt). However, in the presence of mode coupling, we have
aR=2κ(iΔκt)(iΔκt)2+|β|2,aL=iβiΔκtaR,
(19)
and in general both optical modes are excited. As above, we proceed by making the unitary transformations aiai + 〈ai〉 and bb + 〈b〉 and after neglecting terms of 𝒪(g0), we arrive at the linearized OM Hamiltonian
H=ωmbbi=R,LΔaiai+βaLaR+β*aRaL+i=R,L(Giai+Gi*ai)(b+b),
(20)
where due to the mode coupling, both circulating modes exhibit an enhanced coupling (Gi = g0αi) to the mechanical mode. We are primarily interested in the case where the resonator is driven near the mechanical red sideband (Δ = −ωm), where the terms of the form aib+aib are dominant. However, small corrections due to the off-resonant couplings aib+aib are included in our general formalism.

4. Results and discussion

In the four port device shown in Fig. 3, we can study various different non-reciprocal effects and apart from the optical diode and phase-shifter settings outlined above the add-drop configuration (κ = κ′, κin = 0) could be used to realize a non-reciprocal optical router between the two waveguides where, e.g., fR,in1fR,out1 but fL,in1fL,out2. However, this situation is formally equivalent to the optical diode by interchanging the role of κ′ and κin and therefore we can restrict the following discussion to the transmission amplitudes tR,L(ω) as defined in the two port scattering matrix in Eq. (7).

Compared to the ideal situation described in Sec. 2, we are now in particular interested in OM non-reciprocity in the presence of a finite intrinsic mode coupling, β ≠ 0, where photons in the left- and right-circulating modes of both the probe and pump field can no longer propagate independently. Such a coupling is found in many experiments with high-Q micro-resonators and often attributed to bulk or surface imperfections [42

42. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. 27, 1669–1671 (2002). [CrossRef]

, 43

43. A. Mazzei, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99, 173603 (2007). [CrossRef] [PubMed]

]. As already mentioned above, a first consequence of this mode mixing is that the pump field is scattered into the left-circulating mode and we obtain enhanced OM couplings GR,LαR,L for both propagation directions (see Eq. (19)). More specifically, for a purely right-going pump field, the intra-resonator fields are given by
αR=2κ(iΔκt)(iΔκt)2+|β|2,αL=2iβκ|β|2+(iΔκt)2,
(26)
and these expressions are plotted in Fig. 4 as a function of the pump detuning Δ and for the case of large mode coupling (βκt). We see that in principle an asymmetric pumping can be achieved either for Δ = 0 or |Δ| ≫ β. However, to achieve a resonant OM coupling, we should choose Δ ≃ −ωm. Therefore, |GL|/|GR|β/ωm2+κ2 which means that the parasitic coupling can be suppressed by choosing high frequency mechanical modes. Further, we point out that a complete cancellation of GL could be achieved by adding a second pump beam in the left-circulating direction. In particular, if the strength of the left input pump is chosen as 𝒠′ = −/(iΔ − κt)𝒠, then 〈αR〉 = 2κ𝒠/(iΔ − κt) and 〈αL〉 = 0. In the following, we will simply assume that |GL| is suppressed either by a large detuning or by adding a reverse pumping field to cancel the coupling exactly.

Fig. 4 Mean photon number in the left and right circulating modes in the presence of a finite mode coupling β and as a function of the pump detuning Δ = ωL ωc. For this plot we have assumed that the pump field only drives the right-circulating mode and that the resonator is coupled to a single waveguide (κ′ = 0). The other parameters are (β,κin)/κ = (4,1). At the normal mode frequencies ω ≃ ±β, the left- and right-circulating modes are almost equally populated, while everywhere else, there is an intensity imbalance between left- and right-circulating modes. (b) The diagram shows the relation between the relevant frequencies in the system. In the presence of the mode coupling, the sidebands (±β) are located around the bare resonator frequency ωc and the resonator is pumped at the mechanical red sideband.

In addition to pump backscattering, the probe photons are also mixed by the coupling term ∼ β in Hamiltonian (Eq. (20)) and even for |GL| → 0 a degradation of the non-reciprocal response of the device will occur. Let us first consider the case of weak mode mixing, βκ, and assume that the system is pumped in the right-circulating mode at the OM red sideband (Δ = −ωm,ωmβ), as indicated in Fig. 4(b). In this regime, the rate of backscattering of photons inside the resonator is smaller than the decay rate, and therefore, the non-reciprocal response of the device is qualitatively the same as in the ideal case. This is shown in Fig. 5(a) where the mode coupling only slightly reduces the operational bandwidth, i.e., κκ(1β2/GR2).

Fig. 5 Transmittance for light propagating in a waveguide coupled to a resonator (AFP), in the presence of (a) weak (β = 2κ) and (b) strong (β = 8κ) mode mixing. For these plots we have assumed (ωm, GR,κin,γm)/κ = (20,5,1,0) and Δ = −ωm.

In contrast, when the mode coupling is strong (βκ), the backscattering strongly redistributes the probe field in between right- and left-circulating modes, and as shown in Fig. 5(b), the EIT width and the associated non-reciprocal effects can be significantly reduced. In Fig. 6, we have plotted the bandwidth of an optical diode as a function of the mode mixing and the strength of the OM coupling |GR|. While the bandwidth decrease with increasing β, we observe that this effect can be compensated for by using a stronger pump to achieve GR > β. Therefore, we conclude that the presence of a finite intrinsic mode mixing does not fundamentally limit the operation of our device, and even if this coupling exceeds the ring resonator linewidth, non-reciprocal effects can persist, provided that the OM coupling is sufficiently strong.

Fig. 6 Operational bandwidth of an optical diode in the presence of a finite mode coupling β and different values of the enhanced OM coupling GR. For this plot we have assumed GL = 0 and (ωminm)/κ = (20,1,0), Δ = −ωm. In the absence of the mode coupling the bandwidth is 4κ, which for a finite β can be recovered by using a strong pump to enhance |GR|.

5. Thermal noise and the single photon limit

So far we have only considered the scattering relations between the optical in- and out-fields, which due to the linearity of the equations of motion are the same for large classical fields as well as single photons. In practice additional noise sources will limit the operation of the device to a minimal power level, or equivalently to a minimal number of photons in the probe beams. A fundamental noise source in our system stems from the thermal Langevin force ξ(t) which excites the mechanical resonator. The OM coupling up-converts mechanical excitations into optical photons which then appear as noise in the output fields [30

30. K. Stannigel, P. Rabl, A. S. Sørensen, P. Zoller, and M. Lukin, “Optomechanical transducers for long-distance quantum communication,” Phys. Rev. Lett. 105, 220501 (2010). [CrossRef]

32

32. K. Stannigel, P. Rabl, A. S. Sørensen, M. D. Lukin, and P. Zoller, “Optomechanical transducers for quantum information processing,” Phys. Rev. A 84, 042341 (2011). [CrossRef]

]. To estimate the effect of this noise, we investigate the contribution of thermal phonons in the noise power of the right moving out-field
Pnoise=h¯ωc×Bdω2πf˜R,out1(ω)f˜R,out1(ω),
(27)
where B denotes frequency band of interest centered around the optical resonance. We can use Eq. (25) to express f˜R,out1(ω) in terms of the noise operator ξ(ω) and under the relevant conditions and Δ = −ωm, we obtain the approximate result
Pnoiseh¯ωcBdω2π2γmNthκGR2GR42GR2(ωωm)2+(κt2+(ωωm)2)(ωωm)2.
(28)
As described above, non-reciprocal effects are most effective in a small band around the mechanical frequency and we can set B = [ωm ΔB,ωm + ΔB] where ΔBωm is the operation bandwidth of the device. By assuming that ΔBGR2/κ in the weak coupling regime and ΔBκ for strong OM coupling we obtain – up to a numerical factor 𝒪(1) – the general relation
Pnoiseh¯ωc×γmNth×κΔBGR2.
(29)
For weak coupling and a maximal bandwidth ΔB=GR2/κ, the noise power is given by the rate γmNthkBT/(h̄Qm) at which phonons in the mechanical resonator are excited. This means, that if we send a signal pulse of length ΔB−1 through the device a number NnoiseγmNth/(GR2/κ) noise photons is generated during this time. Therefore, in this case the condition for achieving non-reciprocal effects on a single photon level, i.e. Nnoise < 1, is equivalent to OM ground state cooling [34

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

, 35

35. F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 93902 (2007). [CrossRef]

], which is achievable in a cryogenic environment [26

26. E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” arXiv:1107.3761 (2011).

, 27

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

]. Equation (29) also shows that the thermal noise level can be further reduced in the strong coupling regime. In this case the maximal operation bandwidth is ΔB = κ and the noise power is suppressed by an additional factor (κ/GR)2 ≪ 1. This is due to the fact that thermal noise is mainly produced at the two split mode frequencies ωm ± GR, while the non-reciprocal effects rely on the transparency window between those modes. Note that while OM cooling saturates at GRκ, the noise suppression in our device can always be improved with increasing ratio GR/κ, eventually limited by the onset of the OM instability at GR = ωm/2.

6. Conclusions and outlook

In summary, we have shown that optomechanics can induce non-reciprocity in the optical domain. In particular, an optomechanical ring resonator coupled to a waveguide induces a non-reciprocal phase in the under-coupled regime (κinκ) and forms an optical isolator in the critically coupled regime (κκin).

From an application perspective, this system provides an optical isolator that can be integrated on-chip. The bandwidth of such a device will be limited by the amount of pump power that the system can tolerate, before nonlinear effects become significant. In current experimental settings this amounts to bandwidths in the few MHz regime, which however could be further improved in optimized designs.

From a fundamental point of view, the relevant features of our technique are the possibility to implement coherent non-reciprocal phase shifts, to operate on the single photon level and the ability to dynamically control non-reciprocal effects by tuning the power of the pump beam. For example, one can consider a 2D array of optical resonators connected to each other via such non-reciprocal phase shifters. If the phase-shifts are chosen appropriately (e.g. according to the Landau gauge), then a tight-binding model of photons with an effective magnetic field can be simulated [24

24. M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011). [CrossRef]

]. In other words, one can simulate quantum Hall physics with photons where the time-reversal symmetry is broken. In future experiments, it might be possible to combine these techniques with single photon non-linearities which could be either induced by the intrinsic non-linearity of the OM interaction itself [46

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

] or by interfacing the OM system with other atomic [47

47. T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature , 443, 671 (2006). [CrossRef] [PubMed]

] or solid state qubits [30

30. K. Stannigel, P. Rabl, A. S. Sørensen, P. Zoller, and M. Lukin, “Optomechanical transducers for long-distance quantum communication,” Phys. Rev. Lett. 105, 220501 (2010). [CrossRef]

]. Combined with such strong interaction between photons, the implementation of magnetic Hamiltonians using micron-sized OM elements could pave the way for the exploration of fractional quantum Hall physics [48

48. R. E. Prange, S. M. Girvin, and M. E. Cage. The Quantum Hall Effect. (Springer-Verlag, 1986).

, 49

49. A. Comtet, T. Jolicoeur, S. Ouvry, and F. David, editors, The Quantum Hall Effect: Novel Excitations and Broken Symmetries (Spinger-Verlag, 2000).

] and various other exotic states of light.

A. Appendix: Phase sensitive transmission effects

Fig. 7 Ratio between phase sensitive squeezing terms (η) and the phase-insensitive transmission amplitudes (α). For this plot we have assumed Δ = −ωm and (G,κinm)/κ = (5,.5,0).

The authors gratefully thank A. Safavi-Naini, K. Srinvasan, J. Taylor, K. Stannigel and M. Lukin for fruitful discussions. This research was supported by the U.S. Army Research Office MURI award W911NF0910406, NSF through the Physics Frontier Center at the Joint Quantum Institute, the EU Network AQUTE and by the Austrian Science Fund (FWF): Y 591-N16.

References and links

1.

R. B. Wehrspohn, H. S. Kitzerow, and K. Busch. Nanophotonic Materials: Photonic Crystals, Plasmonics, and Metamaterials (Wiley-VCH, 2008).

2.

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009). [CrossRef]

3.

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009). [CrossRef]

4.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev. Lett. 105, 200503 (2010). [CrossRef]

5.

A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science 325, 1221 (2009). [CrossRef] [PubMed]

6.

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced mott transitions and xy spin models in coupled cavity arrays,” Phys. Rev. A 76, 31805 (2007). [CrossRef]

7.

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006). [CrossRef]

8.

M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006). [CrossRef]

9.

R. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004). [CrossRef]

10.

R. L. Espinola, T. Izuhara, M. C. Tsai, R. M. Osgood Jr, and H. Dötsch, “Magneto-optical nonreciprocal phase shift in garnet/silicon-on-insulator waveguides,” Opt. Lett. 29, 941–943 (2004). [CrossRef] [PubMed]

11.

M. Levy, “Nanomagnetic route to bias-magnet-free, on-chip faraday rotators,” J. Opt. Soc. Am. B 22, 254–260 (2005). [CrossRef]

12.

T. R. Zaman, X. Guo, and R. J. Ram, “Faraday rotation in an InP waveguide,” App. Phys. Lett. 90, 023514 (2007). [CrossRef]

13.

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009). [CrossRef]

14.

M. S. Kang, A. Butsch, and P. St. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics 5, 549–553 (2011). [CrossRef]

15.

L. Feng, M. Ayache, J. Huang, Y. -L. Xu, M. -H. Lu, Y. -F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011). [CrossRef] [PubMed]

16.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popovic, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, ”Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335, 38 (2011). [CrossRef]

17.

M. Soljačić, C. Luo, J. D. Joannopoulos, and S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. 28, 637–639 (2004). [CrossRef]

18.

K. Gallo, G. Assanto, K. Parameswaran, and M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001). [CrossRef]

19.

S. Manipatruni, J. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102, 213903 (2009). [CrossRef] [PubMed]

20.

J. Koch, A. A Houck, K. Le Hur, and S. M. Girvin, “Time-reversal symmetry breaking in circuit-QED based photon lattices,” Phys. Rev. A 82, 043811 (2010). [CrossRef]

21.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 13905 (2008). [CrossRef]

22.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461,772–775 (2009). [CrossRef] [PubMed]

23.

F. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 13904 (2008). [CrossRef]

24.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011). [CrossRef]

25.

R. O. Umucalilar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011). [CrossRef]

26.

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” arXiv:1107.3761 (2011).

27.

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

28.

T. Carmon and K. Vahala, “Modal spectroscopy of optoexcited vibrations of a micron-scale on-chip resonator at greater than 1 ghz frequency,” Phys. Rev. Lett. 98,123901 (2007). [CrossRef] [PubMed]

29.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency gaas nano-optomechanical disk resonator,” Phys. Rev. Lett. 105, 263903 (2010). [CrossRef]

30.

K. Stannigel, P. Rabl, A. S. Sørensen, P. Zoller, and M. Lukin, “Optomechanical transducers for long-distance quantum communication,” Phys. Rev. Lett. 105, 220501 (2010). [CrossRef]

31.

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]

32.

K. Stannigel, P. Rabl, A. S. Sørensen, M. D. Lukin, and P. Zoller, “Optomechanical transducers for quantum information processing,” Phys. Rev. A 84, 042341 (2011). [CrossRef]

33.

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49,1337–1343 (1994). [CrossRef] [PubMed]

34.

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

35.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 93902 (2007). [CrossRef]

36.

A. Schliesser and T. J. Kippenberg, “Cavity optomechanics with whispering-gallery mode optical microresonators,” Adv. At., Mol., Opt. Phys. 58, 207–323 (2010). [CrossRef]

37.

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985). [CrossRef] [PubMed]

38.

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

39.

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

40.

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

41.

A. H. Safavi-Naeini, T. P. M. 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 472, 69–73 (2011). [CrossRef] [PubMed]

42.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. 27, 1669–1671 (2002). [CrossRef]

43.

A. Mazzei, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99, 173603 (2007). [CrossRef] [PubMed]

44.

S. Mancini and P. Tombesi, “Quantum noise reduction by radiation pressure,” Phys. Rev. A 49, 4055–4065 (1994). [CrossRef] [PubMed]

45.

D. Brooks, T. Botter, N. Brahms, T. Purdy, S. Schreppler, and D. Stamper-Kurn, “Ponderomotive light squeezing with atomic cavity optomechanics,” arXiv:1107.5609 (2011).

46.

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

47.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature , 443, 671 (2006). [CrossRef] [PubMed]

48.

R. E. Prange, S. M. Girvin, and M. E. Cage. The Quantum Hall Effect. (Springer-Verlag, 1986).

49.

A. Comtet, T. Jolicoeur, S. Ouvry, and F. David, editors, The Quantum Hall Effect: Novel Excitations and Broken Symmetries (Spinger-Verlag, 2000).

OCIS Codes
(230.3240) Optical devices : Isolators
(270.1670) Quantum optics : Coherent optical effects
(120.4880) Instrumentation, measurement, and metrology : Optomechanics

ToC Category:
Integrated Optics

History
Original Manuscript: October 18, 2011
Revised Manuscript: January 12, 2012
Manuscript Accepted: February 24, 2012
Published: March 20, 2012

Citation
Mohammad Hafezi and Peter Rabl, "Optomechanically induced non-reciprocity in microring resonators," Opt. Express 20, 7672-7684 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7672


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References

  1. R. B. Wehrspohn, H. S. Kitzerow, K. Busch. Nanophotonic Materials: Photonic Crystals, Plasmonics, and Metamaterials (Wiley-VCH, 2008).
  2. J. C. F. Matthews, A. Politi, A. Stefanov, J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009). [CrossRef]
  3. J. L. O’Brien, A. Furusawa, J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009). [CrossRef]
  4. L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, R. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev. Lett. 105, 200503 (2010). [CrossRef]
  5. A. Politi, J. C. F. Matthews, J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science 325, 1221 (2009). [CrossRef] [PubMed]
  6. D. G. Angelakis, M. F. Santos, S. Bose, “Photon-blockade-induced mott transitions and xy spin models in coupled cavity arrays,” Phys. Rev. A 76, 31805 (2007). [CrossRef]
  7. A. D. Greentree, C. Tahan, J. H. Cole, L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006). [CrossRef]
  8. M. J. Hartmann, F. G. S. L. Brandao, M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006). [CrossRef]
  9. R. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004). [CrossRef]
  10. R. L. Espinola, T. Izuhara, M. C. Tsai, R. M. Osgood, H. Dötsch, “Magneto-optical nonreciprocal phase shift in garnet/silicon-on-insulator waveguides,” Opt. Lett. 29, 941–943 (2004). [CrossRef] [PubMed]
  11. M. Levy, “Nanomagnetic route to bias-magnet-free, on-chip faraday rotators,” J. Opt. Soc. Am. B 22, 254–260 (2005). [CrossRef]
  12. T. R. Zaman, X. Guo, R. J. Ram, “Faraday rotation in an InP waveguide,” App. Phys. Lett. 90, 023514 (2007). [CrossRef]
  13. Z. Yu, S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009). [CrossRef]
  14. M. S. Kang, A. Butsch, P. St. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics 5, 549–553 (2011). [CrossRef]
  15. L. Feng, M. Ayache, J. Huang, Y. -L. Xu, M. -H. Lu, Y. -F. Chen, Y. Fainman, A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011). [CrossRef] [PubMed]
  16. S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popovic, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, C. R. Doerr, ”Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335, 38 (2011). [CrossRef]
  17. M. Soljačić, C. Luo, J. D. Joannopoulos, S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. 28, 637–639 (2004). [CrossRef]
  18. K. Gallo, G. Assanto, K. Parameswaran, M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001). [CrossRef]
  19. S. Manipatruni, J. Robinson, M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102, 213903 (2009). [CrossRef] [PubMed]
  20. J. Koch, A. A Houck, K. Le Hur, S. M. Girvin, “Time-reversal symmetry breaking in circuit-QED based photon lattices,” Phys. Rev. A 82, 043811 (2010). [CrossRef]
  21. Z. Wang, Y. Chong, J. D. Joannopoulos, M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 13905 (2008). [CrossRef]
  22. Z. Wang, Y. Chong, J. D. Joannopoulos, M. Soljacic, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461,772–775 (2009). [CrossRef] [PubMed]
  23. F. Haldane, S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 13904 (2008). [CrossRef]
  24. M. Hafezi, E. A. Demler, M. D. Lukin, J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011). [CrossRef]
  25. R. O. Umucalilar, I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011). [CrossRef]
  26. E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” arXiv:1107.3761 (2011).
  27. J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, Simon Gröblacher, M. Aspelmeyer, O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011). [CrossRef] [PubMed]
  28. T. Carmon, K. Vahala, “Modal spectroscopy of optoexcited vibrations of a micron-scale on-chip resonator at greater than 1 ghz frequency,” Phys. Rev. Lett. 98,123901 (2007). [CrossRef] [PubMed]
  29. L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, I. Favero, “High frequency gaas nano-optomechanical disk resonator,” Phys. Rev. Lett. 105, 263903 (2010). [CrossRef]
  30. K. Stannigel, P. Rabl, A. S. Sørensen, P. Zoller, M. Lukin, “Optomechanical transducers for long-distance quantum communication,” Phys. Rev. Lett. 105, 220501 (2010). [CrossRef]
  31. D. E. Chang, A.H. Safavi-Naeini, M. Hafezi, O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New J. Phys. 13, 023003 (2011). [CrossRef]
  32. K. Stannigel, P. Rabl, A. S. Sørensen, M. D. Lukin, P. Zoller, “Optomechanical transducers for quantum information processing,” Phys. Rev. A 84, 042341 (2011). [CrossRef]
  33. C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49,1337–1343 (1994). [CrossRef] [PubMed]
  34. I. Wilson-Rae, N. Nooshi, W. Zwerger, T. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007). [CrossRef] [PubMed]
  35. F. Marquardt, J. P. Chen, A. A. Clerk, S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 93902 (2007). [CrossRef]
  36. A. Schliesser, T. J. Kippenberg, “Cavity optomechanics with whispering-gallery mode optical microresonators,” Adv. At., Mol., Opt. Phys. 58, 207–323 (2010). [CrossRef]
  37. C. W. Gardiner, M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985). [CrossRef] [PubMed]
  38. M. Fleischhauer, A. Imamoglu, J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys., 77, 633–673 (2005). [CrossRef]
  39. G. S. Agarwal, S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A, 81, 041803 (2010). [CrossRef]
  40. S. Weis, R. Riviere, S. Deleglise, E. Gavartin, O. Arcizet, A. Schliesser, T. J. Kippenberg, “Optomechanically induced transparency,” Science, 330,1520–1523 (2010). [CrossRef] [PubMed]
  41. A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011). [CrossRef] [PubMed]
  42. T. J. Kippenberg, S. M. Spillane, K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. 27, 1669–1671 (2002). [CrossRef]
  43. A. Mazzei, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99, 173603 (2007). [CrossRef] [PubMed]
  44. S. Mancini, P. Tombesi, “Quantum noise reduction by radiation pressure,” Phys. Rev. A 49, 4055–4065 (1994). [CrossRef] [PubMed]
  45. D. Brooks, T. Botter, N. Brahms, T. Purdy, S. Schreppler, D. Stamper-Kurn, “Ponderomotive light squeezing with atomic cavity optomechanics,” arXiv:1107.5609 (2011).
  46. P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011). [CrossRef] [PubMed]
  47. T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature, 443, 671 (2006). [CrossRef] [PubMed]
  48. R. E. Prange, S. M. Girvin, M. E. Cage. The Quantum Hall Effect. (Springer-Verlag, 1986).
  49. A. Comtet, T. Jolicoeur, S. Ouvry, F. David, editors, The Quantum Hall Effect: Novel Excitations and Broken Symmetries (Spinger-Verlag, 2000).

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