Novel cavity optomechanical systems at the micro- and nanoscale and quantum measurements of nanomechanical oscillators [Elektronische Ressource] / vorgelegt von Georg Anetsberger

Georg Anetsberger

Sujets

• Physik

Informations

• Publié par: ludwig-maximilians-universitat_munchen
• Publié le: 01 janvier 2010
• Nombre de lectures: 13
• Langue: English
• Poids de l'ouvrage: 21 Mo

Extrait

Novel Cavity Optomechanical Systems
at the Micro- and Nanoscale
and Quantum Measurements of
Nanomechanical Oscillators
Georg Anetsberger
Munchen 2010Novel Cavity Optomechanical Systems
at the Micro- and Nanoscale
and Quantum Measurements of
Nanomechanical Oscillators
Georg Anetsberger
Dissertation
an der Fakult at fur Physik
der Ludwig{Maximilians{Universit at
Munc hen
vorgelegt von
Georg Anetsberger
aus Landshut
Munc hen, den 9. November 2010Erstgutachter: Prof. Dr. T. W. H ansch
Zweitgutachter: Prof. Dr. J. P. Kotthaus
Tag der mundlic hen Prufung: 22. Dezember 2010In Erinnerung an
Johann Beck und
Georg Niedermaier.viContents
Table of Contents vii
List of Publications xi
List of Conference Contributions xiii
List of Figures xv
Zusammenfassung xvii
Abstract xix
1 Introduction 1
1.1 Cavity optomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Linear Hamiltonian of cavity optomechanical systems . . . . . . . . 3
1.1.2 Quantum Langevin equations . . . . . . . . . . . . . . . . . . . . . 5
1.1.3 Standard quantum limit of continuous position measurements . . . 9
1.1.4 Radiation pressure force - dynamical backaction . . . . . . . . . . . 15
1.1.4.1 Radiation pressure cooling . . . . . . . . . . . . . . . . . . 17
1.1.4.2 Heating and ampli cation . . . . . . . . . . . . . . . . . . 19
1.1.5 Optical spring e ect . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2 Toroid microresonators and their optical properties . . . . . . . . . . . . . 23
1.2.1 Optical mode pro le . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.2.2 mode volume . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.2.3 Tapered bre coupling . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 Near- eld cavity optomechanics 29
2.1 Theoretical characterization of the optomechanical coupling . . . . . . . . 35
2.1.1 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.2 Analytic approximation for toroid microresonators . . . . . . . . . . 36
2.1.2.1 Micromechanical membrane . . . . . . . . . . . . . . . . . 37
2.1.2.2 Nanomec string . . . . . . . . . . . . . . . . . . . . 39
2.1.3 Comparison to nite element modelling results . . . . . . . . . . . . 40
2.2 Static measurement of the optomechanical coupling . . . . . . . . . . . . . 43viii CONTENTS
2.2.1 Dispersive optomechanical coupling . . . . . . . . . . . . . . . . . . 43
2.2.2 Simultaneous coupling to several optical modes . . . . . . . . . . . 46
2.3 Transduction and actuation of nanomechanical motion . . . . . . . . . . . 49
2.3.1 E ective mass of the nanomechanical oscillators . . . . . . . . . . . 49
2.3.2 Transduction of Brownian motion . . . . . . . . . . . . . . . . . . . 52
2.3.3 Radiation pressure actuation . . . . . . . . . . . . . . . . . . . . . . 54
2.3.4 Frequency dispersion and mode patterns of nanomechanical oscillators 58
2.3.5 Vacuum optomechanical coupling rate . . . . . . . . . . . . . . . . 60
2.4 Dynamical backaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.4.1 Backaction ampli cation - parametric instability . . . . . . . . . . . 63
2.4.2 Radiation pressure cooling . . . . . . . . . . . . . . . . . . . . . . . 66
2.5 Quantum measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.5.1 Measurement techniques . . . . . . . . . . . . . . . . . . . . . . . . 69
2.5.1.1 Titanium-sapphire laser and homodyne measurement . . . 69
2.5.1.2 Fibre laser and Pound-Drever-Hallt . . . . . 72
2.5.2 Classical noise sources . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.5.2.1 Thermorefractive noise . . . . . . . . . . . . . . . . . . . . 72
2.5.2.2 Toroid mechanical modes . . . . . . . . . . . . . . . . . . 74
2.5.3 Sub-SQL imprecision . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.5.4 Backaction of the measurement . . . . . . . . . . . . . . . . . . . . 78
2.6 Optomechanical coupling to multiple optical modes . . . . . . . . . . . . . 81
2.6.1 Modal in whispering gallery microresonators . . . . . . . . 81
2.6.2 Transduction of motion and dynamical backaction using two optical
modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.7 Quadratic optomechanical coupling to nanomechanical oscillators . . . . . 87
2.8 Prospects for graphene sheets as mechanical oscillators . . . . . . . . . . . 91
2.8.1 Characterization of graphene membranes . . . . . . . . . . . . . . . 91
2.8.2 Optomechanical coupling of graphene . . . . . . . . . . . . . . . . . 92
3 Monolithic ultra-low dissipation optomechanical resonators 95
3.1 Optomechanical coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.2 Mechanical mode spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.3 General sources of mechanical dissipation . . . . . . . . . . . . . . . . . . . 101
3.3.1 Gas damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.3.2 Clamping losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.3.3 Two-level systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.3.4 Thermoelastic damping . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.3.5 Phonon-phonon interaction . . . . . . . . . . . . . . . . . . . . . . 104
3.4 Clamping losses in toroid microresonators . . . . . . . . . . . . . . . . . . 105
3.5 Design of ultra-low dissipation spoke-supported resonators . . . . . . . . . 109
3.5.1 Room-temperature quality factors . . . . . . . . . . . . . . . . . . . 109
3.5.2 Temperature dependence of mechanical Q . . . . . . . . . . . . . . 110Table of Contents ix
4 Summary 113
A Solution of the linearized quantum Langevin equations 115
A.1 Equations for the eld quadratures . . . . . . . . . . . . . . . . . . . . . . 115
A.2 Solutions for the intracavity eld quadratures . . . . . . . . . . . . . . . . 116
A.3 Solution for the output eld . . . . . . . . . . . . . . . . . . . 116
B Experimental techniques for near- eld cavity optomechanics 119
B.1 Microfabrication techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 119
B.1.1 Fabrication of toroid near- eld sensors . . . . . . . . . . . . . . . . 119
B.1.2 F of nanomechanical oscillators . . . . . . . . . . . . . . . 122
B.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
B.2.1 Compact tapered optical bres . . . . . . . . . . . . . . . . . . . . 124
B.2.2 Coupling setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
B.3 Homodyne interferometer signal and frequency noise calibration . . . . . . 127
B.3.1 Homodyne signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
B.3.2 Frequency noise calibration . . . . . . . . . . . . . . . . . . . . . . 128
B.3.2.1 Calibrated phase modulation . . . . . . . . . . . . . . . . 128
B.3.2.2 Calibration of frequency noise spectra . . . . . . . . . . . 129
C Fabrication of spoke-supported microresonators 133
D Code for FEM whispering gallery mode simulations 135
Bibliography 156
Danksagung 157x Table of Contents