Introduction to Optomechanics Lo¨ıc Rondin [email protected] Photonics Group – ETH Zurich¨ December 2014 (http://photonics.ethz.ch) Introduction to Optomechanics 1 of 15 Content Introdution Cavity optomechanics Challenges Opto-mechanical systems Physics of optomechanics Mechanical resonator Optical Resonator Cooling of the centre of mass motion Cavity cooling Feedback cooling Alternative cooling Ground State of the mechanical resonator Standard Quantum limit Applications of OM (http://photonics.ethz.ch) Introduction to Optomechanics 2 of 15 Content Introdution Cavity optomechanics Challenges Opto-mechanical systems Physics of optomechanics Mechanical resonator Optical Resonator Cooling of the centre of mass motion Cavity cooling Feedback cooling Alternative cooling Ground State of the mechanical resonator Standard Quantum limit Applications of OM (http://photonics.ethz.ch) Introduction to Optomechanics 3 of 15 Cavity Optomechanics Cavity Optomechanics setup m k I Mirror motion impacting E0(t) the light phase Ein I Light gives momentum to x(t) the mirror through radiation pressure (http://photonics.ethz.ch) Introduction to Optomechanics 4 of 15 Challenges of optomechanics Signal processing Metrology Macroscopic Quantum Physics (http://photonics.ethz.ch) Introduction to Optomechanics 5 of 15 Opto-mechanical systems Kleckner & Bouwmeester Nature 444, 75 Groblacher¨ et al. Nat. Phys. 5, 485 (2009) (2006) (http://photonics.ethz.ch) Introduction to Optomechanics 6 of 15 Opto-mechanical systems Kleckner & Bouwmeester Nature 444, 75 Groblacher¨ et al. Nat. Phys. 5, 485 (2009) Schliesser et al. Nat. Phys. 4, 415 (2008) (2006) (http://photonics.ethz.ch) Introduction to Optomechanics 6 of 15 Opto-mechanical systems Kleckner & Bouwmeester Nature 444, 75 Groblacher¨ et al. Nat. Phys. 5, 485 (2009) Schliesser et al. Nat. Phys. 4, 415 (2008) (2006) Chan et al. Nature (2011) (http://photonics.ethz.ch) Introduction to Optomechanics 6 of 15 Opto-mechanical systems Kleckner & Bouwmeester Nature 444, 75 Groblacher¨ et al. Nat. Phys. 5, 485 (2009) Schliesser et al. Nat. Phys. 4, 415 (2008) (2006) Chan et al. Nature (2011) Gieseler et al. Phys. Rev. Lett. (2012) (http://photonics.ethz.ch) Introduction to Optomechanics 6 of 15 Opto-mechanical systems Kleckner & Bouwmeester Nature 444, 75 Groblacher¨ et al. Nat. Phys. 5, 485 (2009) Schliesser et al. Nat. Phys. 4, 415 (2008) (2006) Chan et al. Nature (2011) Gieseler et al. Phys. Rev. Lett. (2012) (http://photonics.ethz.ch) Introduction to Optomechanics 6 of 15 Opto-mechanical systems Teufel et al. Nature (2011) (http://photonics.ethz.ch) Introduction to Optomechanics 6 of 15 Content Introdution Cavity optomechanics Challenges Opto-mechanical systems Physics of optomechanics Mechanical resonator Optical Resonator Cooling of the centre of mass motion Cavity cooling Feedback cooling Alternative cooling Ground State of the mechanical resonator Standard Quantum limit Applications of OM (http://photonics.ethz.ch) Introduction to Optomechanics 7 of 15 Langevin Equation of motion Langevin equation m FBrownian k mx¨ + mGx˙ + mwmx = Ffluct(t) + Fopt Fopt Fluctuation dissipation theorem For non correlated noise x(t) 0 hFfluct(t)Ffluct(t )i = 2mGkBT (http://photonics.ethz.ch) Introduction to Optomechanics 8 of 15 Impulse response Mechanical susceptibility c 1 ˜ x˜ = 2 2 F m(w − wm + iGw) 1 we note , the mechanical susceptibility. c = 2 2 m(w − wm + iGw) Power-spectra density x is a random stationary signal 1 R T iwt I we note x˜(w) = p x(t)e dt T 0 1 2 I The power spectral density is : Sxx(w) = lim jx˜(w)j T!+¥ T (http://photonics.ethz.ch) Introduction to Optomechanics 9 of 15 I Fluctuation-Dissipation Theorem 2k T S (w) = B Im(c) xx w I Equipartition theorem Z 2 Sxx(w)dw = hx i ∝ Teff R Power spectral density PSD Wiener–Khinchin Theorem Z dw hx(t)x(t + t)i = S (w)e−iwt xx 2p Interesting results related to ≈1/Γ the PSD x(t) t (http://photonics.ethz.ch) Introduction to Optomechanics 10 of 15 I Equipartition theorem Z 2 Sxx(w)dw = hx i ∝ Teff R Power spectral density PSD Wiener–Khinchin Theorem Z dw hx(t)x(t + t)i = S (w)e−iwt xx 2p Interesting results related to ≈1/Γ the PSD x(t) I Fluctuation-Dissipation ≈√T Theorem eff 2k T t S (w) = B Im(c) xx w (http://photonics.ethz.ch) Introduction to Optomechanics 10 of 15 Power spectral density PSD Wiener–Khinchin Theorem Z dw hx(t)x(t + t)i = S (w)e−iwt xx 2p Interesting results related to ≈1/Γ the PSD x(t) I Fluctuation-Dissipation ≈√T Theorem eff t 2kBT Sxx(w) = Im(c) w PSD(ω) I Equipartition theorem Z 2 Sxx(w)dw = hx i ∝ Teff R ω -ωm ωm (http://photonics.ethz.ch) Introduction to Optomechanics 10 of 15 Optical Resonator : Cavity L(t) Cavity E I E0 obeys in 2 2 1 ¶ E E0(t) ∇ E − = 0 c2 t2 x(t) ¶ I Cavity decays g0 γ0 I generate a radiation ρ pressure on the mirror 2 Fopt ∝ jE0j γ0 ωq-1 ωq ωq+1 ωq+2 ω (http://photonics.ethz.ch) Introduction to Optomechanics 11 of 15 Cavity Opto-mechanics Finally, coupled equations system 8 e0 2 > mx¨ + mGx˙ + mwmx = Ffluct(t) + jE0j nA(1 + R) < 2 x(t) > E˙ 0 = i(w − w0 1 − − g0 E0 + kEin : L Rewrote : mx¨ + m(G + dG)x˙ + m(wm + dw)x = Ffluct(t) 8 " # p2R 8n2w g g P g2 g2 > dG = 0 ex 0 in 0 + 0 > 2 2 2 2 2 2 2 2 < (1 − R) mc wm (w − w0) + g0 (w − w0 + wm) + g0 (w − w0 − wm) + g0 " # p2R 4n2w g g P (w − w + w )g (w − w − w )g > dw = 0 ex 0 in 0 m 0 + 0 m 0 > 2 2 2 2 2 2 2 2 : (1 − R) mc wm (w − w0 + g0 (w − w0 + wm) + g0 (w − w0 − wm) + g0 (http://photonics.ethz.ch) Introduction to Optomechanics 12 of 15 Content Introdution Cavity optomechanics Challenges Opto-mechanical systems Physics of optomechanics Mechanical resonator Optical Resonator Cooling of the centre of mass motion Cavity cooling Feedback cooling Alternative cooling Ground State of the mechanical resonator Standard Quantum limit Applications of OM (http://photonics.ethz.ch) Introduction to Optomechanics 13 of 15 Cavity cooling 2 |E0| ω ω-ωm ω ω+ωm (http://photonics.ethz.ch) Introduction to Optomechanics 14 of 15 Cavity cooling 2 |E0| ω ω-ωm ω ω+ωm (http://photonics.ethz.ch) Introduction to Optomechanics 14 of 15 Cavity cooling 2 |E0| ω ω-ωm ω ω+ωm (http://photonics.ethz.ch) Introduction to Optomechanics 14 of 15 Cavity cooling 2 |E0| ω ω-ωm ω ω+ωm Metzger & Karrai Nature 432, 1002 (2004). (http://photonics.ethz.ch) Introduction to Optomechanics 14 of 15 Content Introdution Cavity optomechanics Challenges Opto-mechanical systems Physics of optomechanics Mechanical resonator Optical Resonator Cooling of the centre of mass motion Cavity cooling Feedback cooling Alternative cooling Ground State of the mechanical resonator Standard Quantum limit Applications of OM (http://photonics.ethz.ch) Introduction to Optomechanics 15 of 15.