Abstract

We reveal novel features of nonlinear optomechanical interactions in coupled suspended nanocavities that are driven by two detuned laser frequencies. Such driving enables simultaneous excitation of odd and even optical supermodes, which induce gradient forces of opposite signs, and the competition between these forces enables the realization of optomechanical potentials with large barriers and narrow wells. These types of potentials were suggested for precise displacement control, or “spectral bonding,” in the static regime. However we find that self-induced oscillations appear even at the deep global potential minima when the mechanical damping rate is below a certain threshold, including a new regime of chaotic switching between mechanical deformations of opposite signs.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  10. F. Marino and F. Marin, Phys. Rev. E 87, 052906 (2013).
    [CrossRef]
  11. M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, Nature 459, 550 (2009).
    [CrossRef]
  12. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Westview, 1994).

2013 (1)

F. Marino and F. Marin, Phys. Rev. E 87, 052906 (2013).
[CrossRef]

2011 (1)

G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, Phys. Rev. Lett. 107, 043603 (2011).
[CrossRef]

2009 (3)

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, Nature 459, 550 (2009).
[CrossRef]

I. Favero and K. Karrai, Nat. Photonics 3, 201 (2009).
[CrossRef]

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, Nature 462, 633 (2009).
[CrossRef]

2007 (2)

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, Nat. Photonics 1, 658 (2007).
[CrossRef]

T. Carmon, M. C. Cross, and K. J. Vahala, Phys. Rev. Lett. 98, 167203 (2007).
[CrossRef]

2006 (1)

F. Marquardt, J. G. E. Harris, and S. M. Girvin, Phys. Rev. Lett. 96, 103901 (2006).
[CrossRef]

2005 (1)

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef]

2001 (1)

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, Phys. Lett. A 287, 331 (2001).
[CrossRef]

1991 (1)

S. Chu, Science 253, 861 (1991).
[CrossRef]

Braginsky, V. B.

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, Phys. Lett. A 287, 331 (2001).
[CrossRef]

Camacho, R.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, Nature 459, 550 (2009).
[CrossRef]

Carmon, T.

T. Carmon, M. C. Cross, and K. J. Vahala, Phys. Rev. Lett. 98, 167203 (2007).
[CrossRef]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef]

Chan, J.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, Nature 459, 550 (2009).
[CrossRef]

Chen, L.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, Nature 462, 633 (2009).
[CrossRef]

Chu, S.

S. Chu, Science 253, 861 (1991).
[CrossRef]

Cross, M. C.

T. Carmon, M. C. Cross, and K. J. Vahala, Phys. Rev. Lett. 98, 167203 (2007).
[CrossRef]

Eichenfield, M.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, Nature 459, 550 (2009).
[CrossRef]

Favero, I.

I. Favero and K. Karrai, Nat. Photonics 3, 201 (2009).
[CrossRef]

Girvin, S. M.

F. Marquardt, J. G. E. Harris, and S. M. Girvin, Phys. Rev. Lett. 96, 103901 (2006).
[CrossRef]

Gondarenko, A.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, Nature 462, 633 (2009).
[CrossRef]

Harris, J. G. E.

F. Marquardt, J. G. E. Harris, and S. M. Girvin, Phys. Rev. Lett. 96, 103901 (2006).
[CrossRef]

Heinrich, G.

G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, Phys. Rev. Lett. 107, 043603 (2011).
[CrossRef]

Ippen, E. P.

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, Nat. Photonics 1, 658 (2007).
[CrossRef]

Karrai, K.

I. Favero and K. Karrai, Nat. Photonics 3, 201 (2009).
[CrossRef]

Kippenberg, T. J.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef]

Kubala, B.

G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, Phys. Rev. Lett. 107, 043603 (2011).
[CrossRef]

Lipson, M.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, Nature 462, 633 (2009).
[CrossRef]

Ludwig, M.

G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, Phys. Rev. Lett. 107, 043603 (2011).
[CrossRef]

Marin, F.

F. Marino and F. Marin, Phys. Rev. E 87, 052906 (2013).
[CrossRef]

Marino, F.

F. Marino and F. Marin, Phys. Rev. E 87, 052906 (2013).
[CrossRef]

Marquardt, F.

G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, Phys. Rev. Lett. 107, 043603 (2011).
[CrossRef]

F. Marquardt, J. G. E. Harris, and S. M. Girvin, Phys. Rev. Lett. 96, 103901 (2006).
[CrossRef]

Painter, O.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, Nature 459, 550 (2009).
[CrossRef]

Popovic, M. A.

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, Nat. Photonics 1, 658 (2007).
[CrossRef]

Qian, J.

G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, Phys. Rev. Lett. 107, 043603 (2011).
[CrossRef]

Rakich, P. T.

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, Nat. Photonics 1, 658 (2007).
[CrossRef]

Rokhsari, H.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef]

Scherer, A.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef]

Soljacic, M.

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, Nat. Photonics 1, 658 (2007).
[CrossRef]

Strigin, S. E.

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, Phys. Lett. A 287, 331 (2001).
[CrossRef]

Strogatz, S. H.

S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Westview, 1994).

Vahala, K. J.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, Nature 459, 550 (2009).
[CrossRef]

T. Carmon, M. C. Cross, and K. J. Vahala, Phys. Rev. Lett. 98, 167203 (2007).
[CrossRef]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef]

Vyatchanin, S. P.

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, Phys. Lett. A 287, 331 (2001).
[CrossRef]

Wiederhecker, G. S.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, Nature 462, 633 (2009).
[CrossRef]

Nat. Photonics (2)

I. Favero and K. Karrai, Nat. Photonics 3, 201 (2009).
[CrossRef]

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, Nat. Photonics 1, 658 (2007).
[CrossRef]

Nature (2)

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, Nature 462, 633 (2009).
[CrossRef]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, Nature 459, 550 (2009).
[CrossRef]

Phys. Lett. A (1)

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, Phys. Lett. A 287, 331 (2001).
[CrossRef]

Phys. Rev. E (1)

F. Marino and F. Marin, Phys. Rev. E 87, 052906 (2013).
[CrossRef]

Phys. Rev. Lett. (4)

F. Marquardt, J. G. E. Harris, and S. M. Girvin, Phys. Rev. Lett. 96, 103901 (2006).
[CrossRef]

T. Carmon, M. C. Cross, and K. J. Vahala, Phys. Rev. Lett. 98, 167203 (2007).
[CrossRef]

G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, Phys. Rev. Lett. 107, 043603 (2011).
[CrossRef]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef]

Science (1)

S. Chu, Science 253, 861 (1991).
[CrossRef]

Other (1)

S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Westview, 1994).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Schematic of an optomechanical cavity supporting even and odd optical modes that induce a mechanical deformity ξ to the transverse separation. (b) Dependence of two optical mode resonant angular frequencies on the mechanical deformity, ωp± mark the angular frequencies of optical even and odd modes’ excitations.

Fig. 2.
Fig. 2.

(a) Optomechanical potential Veff versus displacement ξ and optical frequency detuning a shown in color map with local minima (white lines) and maxima (black lines). (b)–(e) Optomechanical potential profiles (solid black lines) and separate contributions corresponding to different terms in Eq. (3) shown with the magenta, green, and cyan dashed lines. The frequency detunings a are (b) 0.1, (c) 0.5, (d) 0.58, and (e) 1, corresponding to the white dashed lines in (a). Parameters are Ωm=1, Γo=0.01, c=a1, b=0.8, d=0.7, and F±=1.

Fig. 3.
Fig. 3.

Results of stability analysis. (a)–(c) The growth rate of the most stable perturbations Im(γ) versus mechanical damping rate Γm for static solutions along the G3, G5. and G7 branches of local minima, horizontal white line marks Γm=3.5622 corresponding to (d), (e). (d) Self-sustained oscillation regions, colored in blue, red, and green. Lines show static states; local minima (solid gray lines) and local maxima (dashed gray lines), and the boundaries of static optomechanical potential barriers are indicated with thin dotted gray lines. (e) Frequency of the oscillatory states with the line colors corresponding to shadings in (d).

Fig. 4.
Fig. 4.

Aperiodic chaotic oscillations. (a)–(c) Time evolution of a small perturbation excited at a=0.9 in local minima branch G7. (a) mechanical deformity, (b) intensity of optical even and (c) odd modes, (d) spectrum of optical odd mode shown in (c), and (e) Lorenz map between the successive maxima of mechanical deformity, marked with circles in (a). Blue and red dashed lines show approximate mapping in the regular regions of negative and positive deformities, respectively. (f) Histogram of mechanical deformity with unit total counts at each optical frequency detuning a.

Equations (5)

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

ξ¨=Γmξ˙Ωm2ξ+F|A|2F+|A+|2,
A˙+=[i(a+bξ)Γo]A++s+,
A˙=[i(c+dξ)Γo]A+s.
A¯+=s+[i(a+bξ¯)+Γo]1,A¯=s[i(c+dξ¯)+Γo]1,
Veff(ξ¯)=[Ωm2ξ¯+F|A¯|2F+|A¯+|2]dξ¯=Ωm22ξ¯2+F+|s+|2bΓoarctanbξ¯+aΓoF|s|2dΓoarctandξ¯+cΓo.

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