Introduction to Optomechanics

Loıc Rondin

lrondin@ethz.ch

Photonics Group – ETH Zurich

December 2014

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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

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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

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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

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Challenges of optomechanics

Metrology

Macroscopic Quantum Physics

Signal processing

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Opto-mechanical systems

Kleckner & Bouwmeester Nature 444, 75(2006)

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

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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)

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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)

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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)

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Opto-mechanical systems

Teufel et al. Nature (2011)

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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

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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

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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

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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

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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

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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

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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

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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

]

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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

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Cavity cooling

|E0|2

ωω ω+ωmω-ωm

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Cavity cooling

|E0|2

ωω ω+ωmω-ωm

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Cavity cooling

|E0|2

ωω ω+ωmω-ωm

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Cavity cooling

|E0|2

ωω ω+ωmω-ωm

Metzger & Karrai Nature 432, 1002 (2004).

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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

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