Effects of laser linewidth on the back-action cooling of

optomechanical resonators.

Gregory A. Phelps

This work is sponsored by:

UA NASA Space Grant, NSF, ARO, ONR

Introduction

Gravitational Wave Detection (LIGO)• 4 kilometer interferometer

€

δx

L≈10−21 ⇒ δx ≈10−18

€

EGS =hΩm

2≈10−8eV

1000 times smaller than a proton!

€

TGS ≈ 0.0004K

• Quantum Mechanical ground state of a macroscopic object.

LIGO at Hanford [1]

(Hartle, 339-342)

[1] http://www.jb.man.ac.uk/research/pulsar/images/Ligo_hanford.jpg

Optomechanical System

From T. Kippenberg and K. VahalaScience 321, 1172 (2008)

Equations of Motion

Under-damped driven harmonic oscillator. Forcing terms are due to thermal motion of the mirror and the interaction with

the intra-cavity light field. Equations are derived from Quantum Mechanical Hamiltonian for the mirror

and light field.€

˙ ̇ x +Ωm

2Qm

˙ x + Ωm2x = α

2+ FL t( )

˙ α = i Δ + ωcavity

x

L

⎛

⎝ ⎜

⎞

⎠ ⎟−

1

2

⎡

⎣ ⎢

⎤

⎦ ⎥α + iSe iφ t( )

Thermal and Laser Noise

Thermal Noise:

€

FL t( ) = 0

FL t( )FL s( ) =2ΩmkbT

MQm

δ t − s( )

€

φ t( ) = 0

φ t( )φ s( ) =1

2τe− t−s /τ

Laser Noise (Frequency):

Monte Carlo Methods/Simulations

Temperature vs. Linewidth

Conclusions

The opto-mechanical system can be modeled as a damped, driven harmonic oscillator.

The final temperature of the mirror is linearly dependent on the linewidth of the laser, for small linewidth.

Laser noise places a limit on the temperatures attainable. Constructing the system to have certain parameters can help to

overcome laser noise. Feedback systems in the laser can reduce the laser noise.

These sources of noise place a limitation on the sensitivity of the interferometer at LIGO.

Acknowledgments

I would like to thank my advisor Pierre Meystre, Dan Goldbaum, Swati Singh, Ewan Wright for our lengthy discussions and their helpful insights.

This work is supported by the University of Arizona/NASA Space Grant, NSF, ARO, and ONR.

References

Hartle, James. GRAVITY, An Introduction to Einstein's General Relativity. 1st ed. 1. San Francisco: Addison Wesley, 2003. 339-342. Print.