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¨ + mΓx˙ + mωmx = Ffluct(t) + Fopt

Fopt Fluctuation dissipation theorem For non correlated noise x(t) 0 hFfluct(t)Ffluct(t )i = 2mΓkBT

(http://photonics.ethz.ch) Introduction to Optomechanics 8 of 15 Impulse response

Mechanical susceptibility χ

1 ˜ x˜ = 2 2 F m(ω − ωm + iΓω) 1 we note , the mechanical susceptibility. χ = 2 2 m(ω − ωm + iΓω) Power-spectra density x is a random stationary signal

1 R T iωt I we note x˜(ω) = √ x(t)e dt T 0 1 2 I The power spectral density is : Sxx(ω) = lim |x˜(ω)| T→+∞ T

(http://photonics.ethz.ch) Introduction to Optomechanics 9 of 15 I Fluctuation-Dissipation Theorem 2k T S (ω) = B Im(χ) xx ω

I Equipartition theorem Z 2 Sxx(ω)dω = hx i ∝ Teff R

Power spectral density PSD

Wiener–Khinchin Theorem

Z dω hx(t)x(t + τ)i = S (ω)e−iωτ xx 2π

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(ω)dω = hx i ∝ Teff R

Power spectral density PSD

Wiener–Khinchin Theorem

Z dω hx(t)x(t + τ)i = S (ω)e−iωτ xx 2π

Interesting results related to ≈1/Γ the PSD x(t)

I Fluctuation-Dissipation ≈√T Theorem eff 2k T t S (ω) = B Im(χ) xx ω

(http://photonics.ethz.ch) Introduction to Optomechanics 10 of 15 Power spectral density PSD

Wiener–Khinchin Theorem

Z dω hx(t)x(t + τ)i = S (ω)e−iωτ xx 2π

Interesting results related to ≈1/Γ the PSD x(t)

I Fluctuation-Dissipation ≈√T Theorem eff

t 2kBT Sxx(ω) = Im(χ) ω PSD(ω) I Equipartition theorem Z 2 Sxx(ω)dω = 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 γ0 γ0 I generate a radiation ρ pressure on the mirror 2 Fopt ∝ |E0| 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

 ε0 2  mx¨ + mΓx˙ + mωmx = Ffluct(t) + |E0| nA(1 + R)  2   x(t)   E˙ 0 = i(ω − ω0 1 − − γ0 E0 + κEin  L

Rewrote :

mx¨ + m(Γ + δΓ)x˙ + m(ωm + δω)x = Ffluct(t)

 " # π2R 8n2ω γ γ P γ2 γ2  δΓ = 0 ex 0 in 0 + 0  2 2 2 2 2 2 2 2  (1 − R) mc ωm (ω − ω0) + γ0 (ω − ω0 + ωm) + γ0 (ω − ω0 − ωm) + γ0 " # π2R 4n2ω γ γ P (ω − ω + ω )γ (ω − ω − ω )γ  δω = 0 ex 0 in 0 m 0 + 0 m 0  2 2 2 2 2 2 2 2  (1 − R) mc ωm (ω − ω0 + γ0 (ω − ω0 + ωm) + γ0 (ω − ω0 − ωm) + γ0

(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