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Fluctuation-dissipation theorem for thermo-refractive noise. Formulation Some applications Proof Discussion. Y. Levin, Leiden University. Physics Letters A, 372, 1941 (2008). Thermo-refractive noise. Braginsky, Gorodetsky, & Vyatchanin 2000. medium. Index of refraction:.

Fluctuation-dissipation theorem for thermo-refractive noise

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## Fluctuation-dissipation theorem for thermo-refractive noise

• Formulation

• Some applications

• Proof

• Discussion

Y. Levin, Leiden University

Physics Letters A, 372, 1941 (2008)

### Thermo-refractive noise

Braginsky, Gorodetsky, & Vyatchanin 2000

medium

Index of refraction:

δn(r,t)=k δT(r,t)

beam

δφ(t)= ∫ I(r)k δT(r,t) d r

3

temperature

fluctuation

phaseshift

intensity

variable

Thermo-mechanical noise

Callen and Welton 51, L. 98

Interaction

1. oscillatingpressure

2. Compute/measuredissipatedpower

3.

Thermo-refractive noise

1. oscillatingentropy injection

2. Compute/measuredissipatedpower, e.g.

3.

Application 1: coating thermo-refractive noise.

heatwave

Mental experiment:

Identical to Braginsky, Gorodetsky, & Vyatchanin 2000

Application 2: coating thermo-optical noise

Evans, et al. 08

surface

displacement

δT

negative

correlation!

refractive

index

Proof:

• Thermometers-ensembles of identical harmonic oscillators, x , ω

• Choose ω >>ω

• Assume fast (<<1/ω) thermal coupling

• Consider dynamical coordinate:

i

0

0

for dense packing,

this becomes

density

but

so by

choosing

becomes

Proof (continued):

• Introduce interaction Hamiltonian

• This amounts to slow sinusoidal change in spring constant.