## Optomechanically induced non-reciprocity in microring resonators |

Optics Express, Vol. 20, Issue 7, pp. 7672-7684 (2012)

http://dx.doi.org/10.1364/OE.20.007672

Acrobat PDF (1695 KB)

### 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

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. 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]

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]

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]

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]

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

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]

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]

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]

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

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. 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]

## 2. Optomechanically induced non-reciprocity: a toy model

*ω*and bosonic operators

_{c}*a*and

_{R}*a*respectively. Radial vibrations of the resonator lead to a modulation of

_{L}*ω*which can be modeled by the standard OM Hamiltonian [33

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

*h̄*= 1), Here

*b*is the bosonic operator for the mechanical mode of frequency

*ω*and

_{m}*g*

_{0}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]

*g*

_{0}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

*ω*=

_{L}*ω*+ Δ. In the limit |

_{c}*α*| ≫ 1, where

_{R}*α*is the classical field amplitude of the driven mode, we can make a unitary transformation

_{R}*a*→

_{R}*a*+

_{R}*α*and linearize the OM coupling around

_{R}*α*. As a result, we obtain an effective Hamiltonian which in the frame rotating with

_{R}*ω*is given by [33

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

*enhanced*OM coupling

*G*=

_{R}*g*

_{0}

*α*, and in view of |

_{R}*G*| ≫

_{R}*g*

_{0}, neglected residual OM interactions ∼

*𝒪*(

*g*

_{0}). 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.

*Optical isolation.*Let us first consider a critically coupled ring resonator where

*κ*=

*κ*. In this case, we see that for frequencies around the ring optical resonance (

_{in}*δ*≈ 0), 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

*G*≪

_{R}*κ*) and

*κ*in the strong coupling limit, where the width of the EIT window is 2

*G*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

_{R}*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.

*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 (

*κ*≪

_{in}*κ*). In this case, the transmittance is close to unity in both directions. However, the left- and right-going fields experience a different dispersion and In general, the phases

*θ*and

_{R}*θ*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 |

_{L}*θ*−

_{R}*θ*| can easily be controlled by changing the pump intensity and can be tuned from zero to about

_{L}*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

*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*,

*κ*≈ 0), this device can be used for non-reciprocal routing of light.

_{in}*ω*=

_{L}*ω*+ Δ

_{c}_{0}and amplitude

**. We make the transformation

*ω*, In the steady-state, we find that 〈

_{L}*b*〉 = −

*g*

_{0}(|〈

*a*〉|

_{R}^{2}+ |〈

*a*〉|

_{L}^{2})/

*ω*. By redefining the detuning to absorb the OM shift,

_{m}*β*= 0), the counter clockwise mode remains empty (〈

*a*〉 = 0), and we obtain 〈

_{L}*a*〉 = 2

_{R}*κ*

**/(

*i*Δ −

*κ*). However, in the presence of mode coupling, we have and in general both optical modes are excited. As above, we proceed by making the unitary transformations

_{t}*a*→

_{i}*a*+ 〈

_{i}*a*〉 and

_{i}*b*→

*b*+ 〈

*b*〉 and after neglecting terms of

*𝒪*(

*g*

_{0}), we arrive at the linearized OM Hamiltonian where due to the mode coupling, both circulating modes exhibit an enhanced coupling (

*G*=

_{i}*g*

_{0}

*α*) to the mechanical mode. We are primarily interested in the case where the resonator is driven near the mechanical red sideband (Δ = −

_{i}*ω*), where the terms of the form

_{m}## 4. Results and discussion

*κ*=

*κ*′,

*κ*= 0) could be used to realize a non-reciprocal optical router between the two waveguides where, e.g.,

_{in}*κ*′ and

*κ*and therefore we can restrict the following discussion to the transmission amplitudes

_{in}*t*(

_{R,L}*ω*) as defined in the two port scattering matrix in Eq. (7).

*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. 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]

*G*∼

_{R,L}*α*for both propagation directions (see Eq. (19)). More specifically, for a purely right-going pump field, the intra-resonator fields are given by and these expressions are plotted in Fig. 4 as a function of the pump detuning Δ and for the case of large mode coupling (

_{R,L}*β*≫

*κ*). We see that in principle an asymmetric pumping can be achieved either for Δ = 0 or |Δ| ≫

_{t}*β*. However, to achieve a resonant OM coupling, we should choose Δ ≃ −

*ω*. Therefore,

_{m}*G*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

_{L}**′ = −

*iβ*/(

*i*Δ −

*κ*)

_{t}**, then 〈

*α*〉 = 2

_{R}*κ*

**/(

*i*Δ −

*κ*) and 〈

_{t}*α*〉 = 0. In the following, we will simply assume that |

_{L}*G*| is suppressed either by a large detuning or by adding a reverse pumping field to cancel the coupling exactly.

_{L}*β*in Hamiltonian (Eq. (20)) and even for |

*G*| → 0 a degradation of the non-reciprocal response of the device will occur. Let us first consider the case of weak mode mixing,

_{L}*β*≪

*κ*, and assume that the system is pumped in the right-circulating mode at the OM red sideband (Δ = −

*ω*

_{m}*,*

*ω*≫

_{m}*β*), 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.,

*β*≫

*κ*), 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 |

*G*|. While the bandwidth decrease with increasing

_{R}*β*, we observe that this effect can be compensated for by using a stronger pump to achieve

*G*>

_{R}*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.

*ω*/(2

_{m}*π*) = 78 MHz. In this system the single-photon OM coupling is

*g*

_{0}/(2

*π*) = 3.4 kHz and the directional enhanced coupling can reach

*G*/(2

*π*) = 11.4 MHz. The resonator decay rate is

*κ*/(2

_{t}*π*) = 7.1 MHz. Therefore, this system can be operated in the strong coupling regime |

*G*| >

*κ*, and assuming that intrinsic defects can be reduced to a level |

_{t}*β*| < |

*G*| ∼ 10 MHz, this device can be used for implementing the different non-reciprocal effects described in this work. In particular, if

*κ*≃

*κ*, then the optical isolation can be observed within the resonator bandwidth. Note that recents experiments have demonstrated OM systems supporting optical whispering gallery modes with mechanical frequencies

_{in}*ω*∼ GHz [28

_{m}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]

## 5. Thermal noise and the single photon limit

*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. 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]

*B*denotes frequency band of interest centered around the optical resonance. We can use Eq. (25) to express

*ξ*(

*ω*) and under the relevant conditions and Δ = −

*ω*, we obtain the approximate result As described above, non-reciprocal effects are most effective in a small band around the mechanical frequency and we can set

_{m}*B*= [

*ω*

_{m}*−*Δ

*B,*

*ω*+ Δ

_{m}*B*] where Δ

*B*≪

*ω*is the operation bandwidth of the device. By assuming that

_{m}*B*≤

*κ*for strong OM coupling we obtain – up to a numerical factor

*𝒪*(1) – the general relation For weak coupling and a maximal bandwidth

*γ*

_{m}*N*≃

_{th}*k*

_{B}*T*/(

*h̄Q*) at which phonons in the mechanical resonator are excited. This means, that if we send a signal pulse of length Δ

_{m}*B*

^{−1}through the device a number

*N*

_{noise}< 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. 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]

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]

*B*=

*κ*and the noise power is suppressed by an additional factor (

*κ*

*/G*)

_{R}^{2}≪ 1. This is due to the fact that thermal noise is mainly produced at the two split mode frequencies

*ω*±

_{m}*G*, while the non-reciprocal effects rely on the transparency window between those modes. Note that while OM cooling saturates at

_{R}*G*≈

_{R}*κ*, the noise suppression in our device can always be improved with increasing ratio

*G*

_{R}*/*

*κ*, eventually limited by the onset of the OM instability at

*G*=

_{R}*ω*/2.

_{m}## 6. Conclusions and outlook

*κ*≪

_{in}*κ*) and forms an optical isolator in the critically coupled regime (

*κ*≃

*κ*).

_{in}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]

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]

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]

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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 |

42. | T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. |

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. |

44. | S. Mancini and P. Tombesi, “Quantum noise reduction by radiation pressure,” Phys. Rev. A |

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. |

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 , |

48. | R. E. Prange, S. M. Girvin, and M. E. Cage. |

49. | A. Comtet, T. Jolicoeur, S. Ouvry, and F. David, editors, |

**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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