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Introduction to Optomechanics
Loıc Rondin
lrondin@ethz.ch
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
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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
(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
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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)
(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
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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
(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
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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
(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
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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
]
(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
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
(http://photonics.ethz.ch) Introduction to Optomechanics 15 of 15