[email protected] Photonics Group – ETH Zurich¨ … · Introduction to Optomechanics Lo¨ıc Rondin [email protected] Photonics Group – ETH Zurich¨ December 2014 () Introduction

Introduction to Optomechanics

Loıc 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

x(t)

k

m

EinE0(t)

Cavity Optomechanics setup

I Mirror motion impacting

the light phase

I Light gives momentum to

the mirror through

radiation pressure

(http://photonics.ethz.ch) Introduction to Optomechanics 4 of 15

Challenges of optomechanics

Metrology

Macroscopic Quantum Physics

Signal processing

(http://photonics.ethz.ch) Introduction to Optomechanics 5 of 15

Opto-mechanical systems

Kleckner & Bouwmeester Nature 444, 75(2006)

Groblacher et al. Nat. Phys. 5, 485 (2009)

(http://photonics.ethz.ch) Introduction to Optomechanics 6 of 15

Opto-mechanical systems

Kleckner & Bouwmeester Nature 444, 75(2006)

Groblacher et al. Nat. Phys. 5, 485 (2009) Schliesser et al. Nat. Phys. 4, 415 (2008)

(http://photonics.ethz.ch) Introduction to Optomechanics 6 of 15

Opto-mechanical systems

Kleckner & Bouwmeester Nature 444, 75(2006)

Groblacher et al. Nat. Phys. 5, 485 (2009) Schliesser et al. Nat. Phys. 4, 415 (2008)

Chan et al. Nature (2011)

(http://photonics.ethz.ch) Introduction to Optomechanics 6 of 15

Opto-mechanical systems

Kleckner & Bouwmeester Nature 444, 75(2006)

Groblacher et al. Nat. Phys. 5, 485 (2009) Schliesser et al. Nat. Phys. 4, 415 (2008)

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(2006)

Groblacher et al. Nat. Phys. 5, 485 (2009) Schliesser et al. Nat. Phys. 4, 415 (2008)

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

x(t)

k

m

Fopt

FBrownian

Langevin equation

mx+mΓx+mωmx = Ffluct(t)+Fopt

Fluctuation dissipation theorem

For non correlated noise

〈Ffluct(t)Ffluct(t′)〉= 2mΓkBT

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

Impulse response

Mechanical susceptibility χ

x =1

m(ω2−ω2m + iΓω)

F

we note χ =1

m(ω2−ω2m + iΓω)

, the mechanical susceptibility.

Power-spectra density

x is a random stationary signal

I we note x(ω) =1√T

∫ T0 x(t)eiωtdt

I The power spectral density is : Sxx(ω) = limT→+∞

1T|x(ω)|2

(http://photonics.ethz.ch) Introduction to Optomechanics 9 of 15

Power spectral density PSD

Wiener–Khinchin Theorem

〈x(t)x(t+ τ)〉=∫

Sxx(ω)e−iωτ dω

2π

Interesting results related tothe PSD

I Fluctuation-Dissipation

Theorem

Sxx(ω) =2kBT

ωIm(χ)

I Equipartition theorem∫R

Sxx(ω)dω = 〈x2〉∝ Teff

x(t)

t

≈1/Γ

(http://photonics.ethz.ch) Introduction to Optomechanics 10 of 15

Power spectral density PSD

Wiener–Khinchin Theorem

〈x(t)x(t+ τ)〉=∫

Sxx(ω)e−iωτ dω

2π

Interesting results related tothe PSD

I Fluctuation-Dissipation

Theorem

Sxx(ω) =2kBT

ωIm(χ)

I Equipartition theorem∫R

Sxx(ω)dω = 〈x2〉∝ Teff

x(t)

t

≈1/Γ

≈√Teff

(http://photonics.ethz.ch) Introduction to Optomechanics 10 of 15

Power spectral density PSD

Wiener–Khinchin Theorem

〈x(t)x(t+ τ)〉=∫

Sxx(ω)e−iωτ dω

2π

Interesting results related tothe PSD

I Fluctuation-Dissipation

Theorem

Sxx(ω) =2kBT

ωIm(χ)

I Equipartition theorem∫R

Sxx(ω)dω = 〈x2〉∝ Teff

x(t)

t

≈1/Γ

≈√Teff

PSD(ω)

ωωm-ωm

(http://photonics.ethz.ch) Introduction to Optomechanics 10 of 15

Optical Resonator : Cavity

x(t)

Ein

E0(t)

γ0

L(t)

ω

γ0

ρ

ωq ωq+1 ωq+2ωq-1

Cavity

I E0 obeys

∇2E− 1c2

∂ 2E∂ t2 = 0

I Cavity decays γ0

I generate a radiation

pressure on the mirror

Fopt ∝ |E0|2

(http://photonics.ethz.ch) Introduction to Optomechanics 11 of 15

Cavity Opto-mechanics

Finally, coupled equations systemmx+mΓx+mωmx = Ffluct(t)+

ε0

2|E0|2nA(1+R)

E0 =

[i(ω−ω0

(1− x(t)

L

)− γ0

]E0 +κEin

Rewrote :

mx+m(Γ+δΓ)x+m(ωm +δω)x = Ffluct(t)

δΓ =

π2R(1−R)2

8n2ω0

m2cωm

γexγ0Pin

(ω−ω0)2 + γ20

[γ2

0

(ω−ω0 +ωm)2 + γ20+

γ20

(ω−ω0−ωm)2 + γ20

]

δω =π2R

(1−R)24n2ω0

m2cωm

γexγ0Pin

(ω−ω20 + γ2

0

[(ω−ω0 +ωm)γ0

(ω−ω0 +ωm)2 + γ20+

(ω−ω0−ωm)γ0

(ω−ω0−ωm)2 + γ20

]

(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

|E0|2

ωω ω+ωmω-ωm

(http://photonics.ethz.ch) Introduction to Optomechanics 14 of 15

Cavity cooling

|E0|2

ωω ω+ωmω-ωm

(http://photonics.ethz.ch) Introduction to Optomechanics 14 of 15

Cavity cooling

|E0|2

ωω ω+ωmω-ωm

(http://photonics.ethz.ch) Introduction to Optomechanics 14 of 15

Cavity cooling

|E0|2

ωω ω+ωmω-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