Opto-Mechanics in a Michelson-Sagnac Interferometer

Opto-mechanics in a Michelson-Sagnac interferometer

Von der QUEST-Leibniz-Forschungsschule der Gottfried Wilhelm Leibniz Universit¨at Hannover zur Erlangung des Grades

Doktor der Naturwissenschaften Dr. rer. nat.

genehmigte Dissertation von

Dipl.-Phys. Henning Kaufer

geboren am 18. November 1983 in Lehrte

2014 Referent: Prof. Dr. RomanSchnabel Korreferent: Prof. Dr. Karsten Danzmann Tag der Promotion: 30. August 2013 In any ﬁeld, ﬁnd the strangest thing and then explore it.

John Archibald Wheeler

Abstract

The quantum nature of light associates the process of optical reflection with the transfer of photon momentum. As the number of photonsina coherent and even in the vacuum state is subject to quantum fluctuations, this opto-mechanical interaction imposes a fundamental limit in high-precision laser interferometric setups such as gravitational wave detectors. In order to develop and test experimental strategies to minimise the disturbing back-action in these detectors, smaller, less complex experiments with perceivable measurement back-action effects are appropriate. Devoted opto-mechanical experiments are typically carried out in linear optical cavities. This thesis presents a novel approach to enable opto-mechanical interaction, using a signal-recycled Michelson-Sagnac laser interferometer. In this topology, the interferometer acts as a compound mirror in an effective cavity. Its reflectivity and therefore, the effective cavity finesse is modified by the displacement of a micro- mechanical oscillator inside the interferometer, giving rise to a dissipative coupling. In the framework of this thesis, a Michelson-Sagnac interferometer was realised and characterised within a high-vacuum system. A shot noise limited displacement sensitivity of 1.9 10−16 m/√Hz was achieved for · frequencies above 60 kHz. Subsequently, the interferometer was extended by a high-reflective mirror in the interferometer output. The resulting topology is described analytically and was investigated in the experiment. It features a complex optical spring with non-canonical properties such as the existence of a conditional instability, which could be observed in the experiments. Moreover, parametric heating and optical damping of the oscillator were observed, reducing the oscillator’s effective quality factor by more than two orders of magnitude. In second experiment, a Michelson-Sagnac interferometer was operated at cryogenic temperatures as low as 8 K. In this way, the oscillator thermal noise around the mechanical resonance frequency was reduced significantly and the interferometer displacement sensitivity could thus be enhanced. The results presented in this thesis demonstrate the potential of the Michelson-Sagnac topology and render it a unique addition to the toolbox of opto-mechanical experiments. Keywords: SiN membranes, opto-mechanics, cryogenics

Kurzfassung

Der Teilchencharakter des Lichts assoziiert den Prozess optischer Re- flektion mit einem Impulsübertrag des Photons. Die Anzahl der Pho- tonen in einem kohärenten Zustand unterliegt Quantenfluktuationen, was zu einer fundamentalen Empfindlichkeitsgrenze in hochpräzisen Laserinterferometern wie Gravitationswellendetektoren führt. Zur Ent- wicklung und zum Test experimenteller Strategien, um den störend- en Einfluss der Rückwirkung der Messung zu verringern sind weniger komplexe Experimente erforderlich, in welchen die Messrückwirkung ebenfalls beobachtbar ist. Solche, opto-mechanischen Experimente werden typischerweise mit linearen Resonantoren realisiert. Diese Arbeit präsentiert einen neuen Ansatz zur Erzeugung opto- mechanischer Wechselwirkungen - innerhalb eines signalüberhöhten Michelson-Sagnac Interferometers. Dieses übernimmt die Funktion eines Quasi-spiegels innerhalb eines effektiven Resonators. Dessen Fi- nesse wird durch die Auslenkung eines mikro-mechanischen Oszillators innerhalb des Interferometers modifiziert, was zu einer dissipativen Kopplung führt. In dieser Arbeit wurde ein Michelson-Sagnac Interferometer realisiert und innerhalb eines Hochvakuum-Systems charakterisiert. Eine Schrot- rausch-begrenzte Positionsempfindlichkeit von 1.9 10−16 m/√Hz kon- · nte für Frequenzen oberhalb von 60 kHz erreicht werden. Ein hoch reflektierender Spiegel wurde im Ausgang des Interferometers instal- liert und die entstandene Topologie analytisch sowie experimentell untersucht. Die resultierende optische Feder besitzt nicht-kanonische Eigenschaften, wie die Existenz einer bedingten Instabilität, die im Experiment beobachtet wurde. Neben parametrischem Heizen war es möglich, die mechanische Güte des Oszillators durch optische Dämpf- ung um mehr als zwei Größenordnungen zu reduzieren. In einem zweiten Experiment wurde ein Michelson-Sagnac Interfero- meter innerhalb eines kryogenen Systems bei einer Temperatur von 8 K betrieben. So konnte das thermisch getriebene Positionsrauschen um die mechanische Resonanzfrequenz des Oszillators erheblich ver- ringert und die Positionsgenauigkeit der Messung gesteigert werden. Die Ergebnisse dieser Arbeit demonstrieren das Potential der Michel- son-Sagnac Topologie, welche die Konzepte bisheriger opto-mechanischer Experimente in einzigartiger Weise erweitert. Schlagworte: SiN Membranen, Opto-mechanik, Kryogenik

VII

Contents

Contents IX

List of Figures XIII

List of Tables XVII

List of Abbreviations XIX

1 Introduction 1

2 Basic concepts of quantum interferometry 7 2.1 Spectraldensity...... 7 2.2 Quantisationoflightﬁelds...... 9 2.3 Performing measurements on light ﬁelds ...... 13 2.3.1 Directdetection ...... 13 2.3.2 Balancedhomodynedetection...... 13

3 The mechanical oscillator 15 3.1 Theoscillatortransferfunction ...... 15 3.2 Mechanicalqualityfactor ...... 18 3.3 Thermalnoise...... 19 3.4 SiliconNitridemembrane ...... 21 3.4.1 Opticalproperties ...... 21 3.4.2 Modesofmotionandeﬀectivemass ...... 22 3.4.3 Resonancefrequencies ...... 24 3.4.4 Membrane mechanical quality factor ...... 25

4 Michelson-Sagnac interferometer 27 4.1 Interferometertopology ...... 27 4.2 Interferometricsignalsandnoise ...... 32 4.2.1 Theshotnoiselimit ...... 32 4.2.2 Quantumback-actionnoise ...... 33 4.2.3 Standardquantumlimit ...... 35

IX CONTENTS

4.3 Inﬂuenceandsuppressionoflasernoise ...... 38 4.3.1 Interferometercontrast ...... 38 4.3.2 Interferometer arm length diﬀerences ...... 40 4.3.3 Laserfrequencynoise ...... 41 4.3.4 Laserintensitynoise ...... 43 4.3.5 Laserlightpreparation...... 46 4.4 Realisation of the Michelson-Sagnac interferometer . . 48 4.4.1 Interferometerdesign...... 50 4.4.2 Vibrationisolation ...... 51 4.4.3 Membraneholdering ...... 53 4.4.4 Interferometeralignment ...... 54 4.4.5 Vacuumrequirements ...... 56 4.4.6 Remotealignment ...... 57 4.5 Interferometercalibration ...... 58 4.5.1 Calibration via the interferometer fringe . . . . 58 4.5.2 Obtaining the membrane displacement spectrum 59 4.5.3 Calibrationusingamarkerpeak ...... 61 4.6 Achieveddisplacementsensitivity ...... 61 4.6.1 Readout with a single photo detector...... 61 4.6.2 Balancedhomodynereadout ...... 63 4.7 Chaptersummary ...... 66

5 Signal-recycled Michelson-Sagnac interferometer 69 5.1 Opticalcavity...... 69 5.1.1 Effectivecavityoutputpower ...... 73 5.1.2 Signal-Recycling ...... 74 5.1.3 Design considerations for the effective cavity . 78 5.1.4 Theimpactofopticallosses ...... 79 5.1.5 Lossinthecavity...... 81 5.2 Realisation of the recycled interferometer ...... 82 5.2.1 Cavity length stabilisation scheme ...... 83 5.2.2 Effective cavity optical linewidth ...... 86 5.2.3 Displacementsensitivity ...... 87 5.3 Opto-mechanicalinteraction ...... 89 5.3.1 Parametricinstability ...... 91 5.3.2 Opticalcooling ...... 92 5.3.3 Optical spring in the Michelson-Sagnac interferometer...... 95 5.3.4 Conditionalinstability ...... 98 5.3.5 Observation of optical damping ...... 102

X CONTENTS

5.3.6 Reduction of interferometer loss ...... 106 5.4 Chaptersummary ...... 107

6 Cryogenic cooling of the interferometer 109 6.1 Theneedforlowtemperatures ...... 109 6.2 Conceptsforcryogeniccooling ...... 110 6.3 TheGifford-McMahoncryocooler ...... 111 6.3.1 Thermodynamiccycle ...... 112 6.3.2 Cryocoolercomponents ...... 114 6.4 Interferometer prototype for cryogenic environment . . 115 6.4.1 Membrane alignment and positioning . . . . . 117 6.4.2 Theopticalsetup...... 119 6.4.3 Interferometer installation procedure ...... 120 6.5 Results...... 123 6.5.1 Cryogenic interferometer performance . . . . . 124 6.5.2 Influence of low temperatures on the membrane 126 6.5.3 Identifiedproblems...... 129 6.5.4 Nextsteps...... 130 6.6 Chaptersummary ...... 130

7 Summary 133

8 Appendix A 135 8.1 FinesseSimulationFile...... 135 8.2 MatlabFiles...... 138

Bibliography 141

List of publications 155

Acknowledgements 157

Curriculum Vitae 159

List of Figures

1.1 Second generation GW detector schematic and anticipated Advanced LIGO strain sensitivity ...... 2

2.1 Balancedhomodynedetectionscheme ...... 14

3.1 Oscillatortransferfunction ...... 19 3.2 Comparison of structural and viscous damping . . . . 20 3.3 SiNmembraneandsiliconframe ...... 21 3.4 SiN membrane power reﬂectivity for diﬀerent optical wavelengths ...... 22 3.5 Graphical illustration of the membrane fundamental os- cillationmode...... 23 3.6 Higher-ordermodespectrum ...... 24 3.7 Ring-downmeasurement...... 26

4.1 Michelson-SagnacInterferometer ...... 28 4.2 Interferometeroutputpower...... 30 4.3 Quantumandthermalnoise ...... 34 4.4 Noise budget at cryogenic temperatures, with signal- recycling...... 35 4.5 Illustration of interferometric contrast ...... 38 4.6 Interferometer contrast inﬂuence to achievable displacementsensitivity...... 39 4.7 Interferometer contrast with arm length diﬀerence . . 40 4.8 Mephisto NPRO laser frequency noise projection into apparent length noise for various interferometer arm lengthinequalities ...... 42 4.9 Laser intensity stabilisation performance ...... 45 4.10 Mach-Zehnder and EOAM power stabilisation schemes 46 4.11 Schematic of the laser light preparation ...... 47 4.12 Laser intensity stabilisation performance ...... 47 4.13 Interferometer output spectrum in 2010 ...... 49 4.14 Typical spectrum prior to interferometer isolation . . . 49 4.15 CAD Design for the Michelson-Sagnac Interferometer . 50

XIII LIST OF FIGURES

4.16 Passive isolation stages of the interferometer ...... 51 4.17 Performance of the passive isolation of the interferom- eterbaseplateinvertivaldirection ...... 52 4.18 Schematic of the membrane holder for the room tem- peratureexperiment ...... 53 4.19 Sagnacinterferometeralignment ...... 54 4.20 Alignment of the Michelson-Sagnac interferometer . . 55 4.21 Q-factor as function of environmental pressure . . . . 56 4.22 Vacuumsetupfortheexperiment ...... 58 4.23 The redesigned Michelson-Sagnac interferometer . . . 59 4.24 Interferometer fringe and output power derivative . . . 60 4.25 Spectrum around the fundamental mode ...... 62 4.26 Balanced homodyne detector locking scheme for the homodyne phase θ to the amplitude quadrature . . . . . 63 4.27 Homodyne detector DC Signal and demodulated error signal ...... 64 4.28 Homodyneoutputspectrum ...... 65 4.29 Broadband homodyne output spectrum ...... 66 4.30 Variation of the homodyne read out quadrature . . . . 67

5.1 Schematicofatwomirrorcavity ...... 70 5.2 Cavityresonancepeaks ...... 72 5.3 Schematic of the signal-recycled Michelson-Sagnac ef- fectivecavity ...... 73 5.4 Normalised output power of the recycled Michelson- Sagnac interferometer as function of membrane and mirrordetuning...... 74 5.5 Normalised transmitted power through signal recycling cavity ...... 75 5.6 Scans of the signal-recycling cavity length and the mem- braneposition...... 76 5.7 Displacement sensitivity for various signal-recycling mir- rorreﬂectivitiesanddetunings ...... 77 5.8 Signal-recycling gain as function of recycling mirror transmissivityincludingloss ...... 80 5.9 Opticallosschannelswithinthecavity ...... 82 5.10 Signal-recycled Michelson-Sagnac with balanced homo- dynereadout ...... 83 5.11 SR Cavity output power with Rsr = 99.97 %...... 84 5.12 Cavityoutputpowerandderivative...... 85

XIV LIST OF FIGURES

5.13 Schematic setup for measuring the effective cavity linewidth ...... 86 5.14 Minimum measured effective cavity linewidth . . . . . 87 5.15 Displacement sensitivity of the recycled interferometer and transmitted optical power fraction in case of loss andnoloss ...... 88 5.16 Single-ended, opto-mechanical cavity ...... 90 5.17 Scans of the signal-recycling cavity length with different velocities ...... 93 5.18 Signal-recycled Michelson-Sagnac interferometer . . . 95 5.19 Optical spring in the Michelson-Sagnac interferometer 97 5.20 Modification of the membrane mechanical quality factor 98 5.21 Conditional instability and parametric instability on two different sides of signal-recycling cavity resonance 99 5.22 Conditional instability threshold ...... 101 5.23 Interferometer output spectra with recycling-mirror . . 102 5.24 Numerical calculation of the optical damping for the experimentalparameters ...... 103 5.25 Calibrated output power spectrum with recycling mir- rorclosetocavityresonance...... 105 5.26 Modification of the mechanical quality factor with re- ducedopticallosses...... 107

6.1 GiﬀordMcMahoncycle ...... 113 6.2 Schematicofthecryogenicsystem ...... 116 6.3 CAD Design of the cryogenic interferometer ...... 117 6.4 Cryogenicmembranealignment ...... 118 6.5 RINmeasurementofthe1550nmlaser...... 120 6.6 Installation procedure into the cryo chamber . . . . . 121 6.7 Photosofthecryocoolersetup ...... 122 6.8 Cryocooler cool down sequence with installed interferometer...... 123 6.9 Switchingthecompressoroﬀ/on...... 124 6.10 Schematic of the cryogenic experiment ...... 125 6.11 Drift of mechanical resonance frequencies due to reduc- tionofoscillatortemperature ...... 126 6.12 SiN membrane ring-down measurements at cryogenic and close-to ambient temperatures ...... 127 6.13 Displacement spectrum at various temperatures . . . . 128 6.14 CCD output for the cryogenic interferometer . . . . . 129

List of Tables

3.1 Relevant SiN membrane parameters for the membrane used in the 1064 nm and 1550 nm cryogenic experiment 26

5.1 Eﬀective quality factors Qeﬀ due to optical damping of the oscillator fundamental mode for Pin = 10 mW, 50mWand200mW ...... 103

6.1 Attocube remote positioner data for membrane alignment and positioning at cryogenic temperatures . . . . 119

XVII

List of Abbreviations

AC AlternatingCurrent

AEI Albert Einstein Institute (in Hannover)

AR Anti-Reﬂective

BS BeamSplitter

CAD ComputerAidedDesign

CCD Charge-CoupledDevice

CLIO Cryogenic Laser Interferometer Observatory

DC DirectCurrent

DFT DiscreteFourierTransform

EOAM Electro-Optic Amplitude Modulator

EOM Electro-OpticModulator

FFT FastFourierTransform

FSR FreeSpectralRange

FWHM Full-Width at Half Maximum

GM Giﬀord-McMahon

GW GravitationalWaveDetector

HR High-Reﬂective

HV High Voltage

HWHM Half-Width at Half Maximum ifo Interferometer

KAGRA Kamioka Gravitational Wave Detector

XIX List of Abbreviations

LIGO Laser Interferometer Gravitational Wave Observatory

LO LocalOscillator

LP Low-Pass

MC ModeCleaner

Nd:YAG Neodymium-Doped Yttrium Aluminium Garnet

OB OpticalBench

OPD OpticalPathlengthDiﬀerence

PBS PolarisingBeamSplitter

PC PersonalComputer

PD Photodetector

PI PhysicsInstruments

PI Proportional-Integral (controller)

PID proportional-integral-derivative

PLL Phase-LockedLoop pp peak-to-peak ppm partspermillion

QPD QuadrantPhotodiode

RBW ResolutionBandwidth

RF RadioFrequency

RIN RelativeIntensityNoise

SiN SiliconNitride

SQL StandardQuantumLimit

SR Signal-recycling

SRM SignalRecyclingMirror

TMP TurboMolecularPump

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Chapter 1 1 Introduction

10 -22 End-mirror Total noise Quantum noise Radiation pressure noise ) Seismic noise 1/2 Suspension thermal Mirror thermal Arm cavities 10 -23

Laser Power-

recycling Strain sensitivity (1/Hz Signal-recycling

-24 Detection 10 10 100 1k Frequency (Hz)

Figure 1.1: Left: schematic of a second generation GW detector. The laser power is enhanced by a power-recycling mirror and resonant arm cavities. The signals are detected after resonant ampliﬁcation by a signal-recycling cavity. Right: anticipated strain sensitivity of the Advanced LIGO detector (Tuned signal recycling, no squeezing injection) simulated with GWINC [18]. Quantum noise dominates for high and low frequencies. Brownian thermal noise of the mirrors is dominant around 80 Hz.

Gravitational wave detection

A prominent examples for interferometric measurement for differential arm length changes are large scale Michelson-type gravitational wave (GW) detectors [8–10], such as LIGO [11, 12], GEO600 [13], Virgo [14] and TAMA300 [15]. Their sensitivity reaches the astonishing level of 10−19 m/√Hz around a frequency of 1 kHz. Vacuum fluctuations of the phase and amplitude of the electro-magnetic field entail two quantum noise contributions in the detectors described above. Phase fluctuations manifest in form of quantum read out noise, the so-called shot noise. It can be reduced by increasing the optical input power. However, amplitude fluctuations are thereby enhanced and induce a stronger, fluctuating back-action force on the suspended (free-falling) mirror test masses. It was first recognised by Caves [16] that this back-action actually results in a standard-quantum limit (SQL) for the measurement precision [17]. If amplitude and phase fluctuations are assumed uncorrelated, there exists an optimum optical power for every observation frequency, minimising the uncorrelated sum of both, quantum read out and back-action noise. In this case, the experiment operates at the SQL for this particular frequency.

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Chapter 1 1 Introduction

Their mass, mechanical quality factors and resonance frequencies span over several orders of magnitude. For recent reviews about opto- mechanical setups, the reader is kindly referred to [39–43]. The typical opto-mechanical setup consists of a single-ended Fabry- Perot cavity with one end-mirror (the oscillator) exhibiting one harmonic degree of freedom. In this case, the intra-cavity field and hence, the optical force acting on the oscillator depends on the cavity length, e.g. the oscillator position. This coupling gives rise to an optical spring [44], modifying the oscillator spring constant. Due to the finite cavity decay rate, its response to length variations is not instantaneous, causing optical (anti-) damping of the oscillator motion. This damping is associated with a reduction of the oscillator thermal motion, which has first been observed in the microwave [45] and later in the optical domain [46]. The ’good cavity’ limit describes the situation in which the optical linewidth γ fulfils γ < 2ωm, where ωm is the mechanical resonance frequency of the oscillator. In this situation, the cavity operates in the resolved side band regime. If the optical power is high enough, it can be shown [40, 41, 47, 48] that the minimal achievable mean phonon occupation number of the oscillator in such a setup is given by

κ 2 n = cav , (1.1) min 4ω m where κcav is the optical ring-down rate of the cavity and ωm is the mechanical resonance frequency of the oscillator. Thus, a high ﬁnesse (low decay rate) and a high mechanical resonance frequency will result in a small occupation number. Recently, several experiments achieved a mean occupation number n < 1, and thus prepared the oscillator in the mechanical ground state [49, 50]. The mechanical ground state has also been achieved by pure cryogenic cooling of a 6 GHz oscillator, as demonstrated in [51]. The achievement of the mechanical ground state with micro-mechanical oscillators is an important milestone for quantum-mechanical experiments. It provides the basis for fundamental research in follow-up experiments, allowing to investigate decoherence eﬀects in massive objects or to produce superposition of mechanical states. Other applications are expected in quantum information processing in photonic circuits [40] such as switching or storage.

However, macroscopic oscillators exhibit rather small resonance fre-

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Chapter 1 1 Introduction results in a anomalous, complex optical spring, which is described and observed in the experiment. In gravitational wave detectors, this mirror in the interferometer output port is used to resonantly enhance interferometric signals, and is therefore referred to as signal-recycling mirror [54–56]. Chapter 6 presents the ﬁrst application of cryogenic cooling to a full Michelson-Sagnac interferometer. A description of the experiment is given, including the cryogenic system and its principle of operation. An interferometer prototype for operation at cryogenic temperatures is presented. The chapter closes with the report of a successful operation of a Michelson-Sagnac interferometer at temperatures below 10 K. The research presented in this thesis is summarised in Chapter 7.

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Chapter 2 2 Basic concepts of quantum interferometry

x(t) = pixi , (2.4) h i X The so-called correlation-function

Sxy(t , t )= x(t )y(t ) (2.5) 1 2 h 1 2 i is a measure of how the measurement of y at time t2 relates to the measurement x at time t1. The correlation-function of a quantity with itself at a diﬀerent time t + τ is referred to as the auto-correlation function and is written as

Sx(t, τ)= x(t)x(t + τ) . (2.6) h i It expresses how much the measurement outcome x at time t is linked to measuring x again at some point τ later in time. It thus depicts the self-similarity of the quantity. The Fourier transform of the auto- correlation function

|G (ω)|2 = x(t)x(t + τ) e−iωtdt , (2.7) x ∞ Z− h i is referred to as the spectral∞ density. Equation 2.7 is symmetric in 2 2 ω, |Gx(ω)| = |Gx(−ω)| , such that the consideration can be re- stricted to positive frequencies (one-sided spectral density). In this case, an additional factor of 2 is required to maintain equality. Back- 2 transformation with Sx(t, 0)= ∆ x yields dω dω x2 = |G (ω)|2 = 2 |G (ω)|2 . (2.8) ∞ x ∞ x h i Z− 2π Z0 2π

It can be shown that∞ in case of stationary random processes [57], the following statement holds,

2 2 |xT (ω)| |Gx(ω)| = lim , (2.9) T→ T which is known as the Wiener-Khintchine∞ theorem [58, 59].

Optical shot noise The typical example for noise in optical experiments is photon shot noise. If a photodetector is measuring the number of arriving photons (which are assumed to have a mean ﬂux I), the arrival of each individual photon is completely random and unrelated to the arrival

8 htnhst ecniee nE.(.1,giving (2.11), Eq. in considered be to has photon pcrldniyo oe utain,tepwrcharge get power To the photons. fluctuations, of power number of powe the density the to and spectral proportional independent is frequency density is tral noise shot optical The faohrpoo.I h iedmi,ti a eepesdb expressed be can this domain, time the In [57] assuming photon. another of ie pcrldniy[0 cmaeSc 2.2) Sec. (compare yie [60] density function auto-correlation spectral its sided of transform Fourier The pe flight of speed Here, and itvt and mittivity h lsia aitnfnto n elcn h classica the replacing and q function Hamilton classical the fe,i smr ovnett osdrtelna spectral linear the consider to convenient by more given is it Often, etrstsyn ie onayconditions. boundary given satisfying vector qain 6] h n iesoa ouin nasec o [62] waves abscence sinusoidal in plane, solutions are dimensional currents one or The solu [61]. a as Equations described classically is field electro-magnetic An fields light of Quantisation 2.2 ( t ) q and ( V t ) steeetv uniainvolume, quantisation effective the is h oiin h uniaini efre yconsidering by performed is quantisation The position. the p B E ( t y x ) ( ( µ ,t z, ,t z, c ytecnnclpsto n oetmoperators momentum and position canonical the by 0 = h aumpreblt,wihaecnetdt the to connected are which permeability, vacuum the = ) = ) 1/ | G I ( q √ P t = ) ( | µ G ω 0 P ) ǫ µ X 2ω Vǫ | | | 2 G 0 2 k 0 k . ǫ = I = 0 2 ( 0 q δ ˙ ω 2 ( G ( t t hωP

1/2 ) P = ) | − 2 2ω = Vǫ = t q k p r . uniaino ih ﬁelds light of Quantisation 2.2 ( = 0 ) 2 2 t ( , . n ) t 4π 2 ) sin 1/2 h

λ stecnnclmomentum canonical the is I hc

2πc ( = λ kz q ˙ . P . n ( ) . P t ǫ k ) 0 cos , = stevcu per- vacuum the is ( ω/c kz ) ino Maxwell’s of tion d h one- the lds stewave the is hω

quantities l . charge f density, feach of spec- r othe to (2.11) (2.14) (2.10) (2.15) (2.12) (2.13) P y 9

Chapter 2 2 Basic concepts of quantum interferometry and Q. This yields the Hamilton operator H for a single optical mode of frequency ω

1 H(P, Q)= (P2 + ω2Q2) . (2.16) 2 The position and momentum operator fulﬁl the commutation relation

[Q, P]= QP − PQ = ih . (2.17)

Using the so-called raising and lowering operators A and A†, and 1/2 setting E0 = (hω/ǫ 0V) , the electric ﬁeld (now as an operator) writes [62] † Ex(z, t)= E0(A + A ) sin(kz) . (2.18) The raising and lowering operators arise as a combination of position and momentum operators

1 A = (ωQ + iP) . (2.19) √2hω 1 A† = (ωQ − iP) . (2.20) √2hω These operators also lead to an elegant expression for the Hamilton operator H

1 H = hω A†A + . (2.21) 2 The eigenstates of the Hamilton operators are thus eigenstates of the so-called number operator N = A†A. They are referred to as Fock states and labeled |n . A Fock state contains an exactly deﬁned i number of photons n of energy hω .

N |n = n |n . (2.22) i i The eﬀect of A† and A on a Fock state are given by

A |n = √n |n − 1 , A† |n = √n + 1 |n + 1 . (2.23) i i i i Raising and lowering operator do not commute, which is expressed in their commutator

A, A† := AA† − A†A = 1 . (2.24) h i

10 n hsavldrpeetto falsrotu ed h ei The field. output laser a ϕ electro-magnetic of representation sinusoidal valid a a thus for and approximation good a are ϕ A uhta h oeetsaei ie by given is state coherent the that such with oeetstate coherent A hc,frlarge for which, This states. Fock of sum weighted a as eigenstates de the expressed be for then can true state especially is optical It every basis in. complete a are states Fock opoosadteeoei utwitnas written con It just is state. vacuum therefore the and is state photons Fock no a of example prominent A h ennme fdtce htn ttu ie ytecoher the by given thus it photons detected excitation of number mean The ncnrs oaFc tt,tenme fpoosi coheren a in de photons of of probability The number defined. the exactly state, not Fock therefore is a state to contrast In htn nachrn state coherent a in photons a egnrtdb pligteriigoeao otegrou the to operator raising the n applying by generated be can n oeetsaecnb rte as written be can state coherent Any . | times r ope ngnrl and general, in complex are ϕ i D = ( ϕ ϕ = ) | | ϕ ϕ | i 2 exp hs ttsaerfre oas to referred are states These . h ainei endadgvnby given and defined is variance The . n | ϕ | ϕ ϕA saGusa distribution Gaussian a is P i | ϕ i n n = ∆ hsehbt nepcainvalue expectation an exhibits thus i ( = † 2 ϕ exp | = n N − h = ) i N n X ϕ := ∞ = = i ∗ | ϕ 0 | − A = h √ h C N i n 1 2 1 h n n aldthe called , ste ie yaPisndistribution Poisson a by given then is | 2 ϕ | ϕ ϕ ϕ ! i | ∗ n | i | | − N ϕ 2 | i A eoe h ope ojgt of conjugate complex the denotes 2 i h = | † . uniaino ih fields light of Quantisation 2.2 ϕ = N n ftelwrn operator lowering the of X ∞ D i = n i n 2 n = 0 ( | n 0 ϕ ! = √ displacement ϕ i | ϕ ) e n . n n − | | | 0 2 ! 0 n . i i | coherent n n okstate Fock Any . i . , , . prtr[62], operator n tts They states. ie by given genvalues ae[63] wave tecting dstate nd veloped A (2.26) (2.27) (2.25) (2.30) (2.28) (2.29) tains with ent | n 11 is n t i

Chapter 2 2 Basic concepts of quantum interferometry

1 (n − n)2 Pn(ϕ) exp − . (2.31) ≈ √ 2n 2πn Two other operators can be deﬁned using the raising and lowering operators. These are the so-called amplitude quadrature operator X1 and the phase quadrature operator X2, which are deﬁned as

† † X1 = A + A , X2 = i A − A . (2.32)

These operators obey the commutator relation [X1, X2] = 2i. Their expectation values for a coherent state |ϕ are given by i X = (ϕ + ϕ∗)= 2ℜ[ϕ] , X = 2ℑ[ϕ] . (2.33) h 1i h 2i For any coherent state |ϕ , the variances are i 2 2 ∆ X1 = ∆ X2 = 1 . (2.34) Coherent states exhibit a minimal uncertainty product [64]. Since the vacuum state is a coherent state, a measurement on the vacuum state will exhibit noise, as its variance is non-vanishing. Note that the amplitude and phase quadrature operators also allow to deﬁne arbitrary readout quadratures Xθ, [65]

† iθ −iθ Xθ = X1 cos(θ)+ X2 sin(θ)= A e + Ae . (2.35)

Operator linearisation

In the case of large coherent amplitudes ϕ that outweigh the ﬂuctua- tions, the raising and lowering operators write [66, 67]

A = ϕ + δA , A† = ϕ∗ + δA† . (2.36) The expectation values for the (time-depending) ﬂuctuating terms are δA = δA† = 0. This allows to rewrite the number operator. h i h i Assuming ϕ is real, it is given by

N = A†A = (ϕ∗ + δA†)(ϕ + δA) = |ϕ|2 + ϕ∗δA + ϕδA† + δA†δA |ϕ|2 + ϕδX , (2.37) ≈ 1 where in the last step the higher order terms δA†δA were neglected [65], and the operator is thus linearised.

12 ht eetr h niiulpoocret yield currents photo individual The detector. photo a sa lcr-pi ouao rapeoatae irr He mirror. dev as actuated to shifting piezo referred phase a be or a modulator by electro-optic adjusted an be as typically can fields light iersd h eetdpoocretcnit faconsta a ope term of Number fluctuating consists the time-dependent, current when photo detected e.g. The neglected, linearised. are terms order higher if excitation o)o aacdba pitr h eaiephase relative The int is splitter. beam (loca beam wavelength signal balanced optical a a same on scheme, the tor) of this beam In light another field. with optical arbitrar detect the to of allows tures [68] detection homodyne Balanced detection homodyne Balanced 2.3.2 re also is method This filtering. pass as low or high- electronic ubro riigpoos o oeetlsrba,tede the as beam, written laser be coherent can a current For proporti photo is q photons. diode amplitude arriving photo the single of a in number of fluctuations photocurrent the the meas as to a ture, such access that provide implies only detectio (2.37) can direct Equation a is photodiode. beam laser single a on measurement simplest The detection Direct 2.3.1 prese be shall measure and perform mode, briefly. to optical output used interferometer were an methods on two photons. thesis, arriving t this of of number Within intensity the The by to current. junction elec proportional electric free PIN is an energy, in or photon result p-n sufficient and a a created For of electr zone photons. diode, of depleted sorption the the Within within excited re diodes. be with photo performed semi-conductor typically are ased fields light on fields Measurements light on measurements Performing 2.3 self-homodyne ϕ wihi sue el.Ec emcnb cesdby accessed be can term Each real). assumed is (which θ detection. ohprso h emslte r edotwith out read are splitter beam the of ports Both . i PD . efrigmaueet nlgtfields light on measurements Performing 2.3 ∝ N = | ϕ δX | 2 1 + hc ssae ytecoherent the by scaled is which , ϕδX 1 , θ ewe both between ttr and term nt i current his nlt the to onal quadra- y tdhere nted e twill it re, rn are trons erdto ferred c,such ice, es bi- verse urement oscilla- l iha with n n can ons ao is rator erfered uadra- tected ments (2.38) ab- 13

Chapter 2 2 Basic concepts of quantum interferometry

Signal beam S = σ0 + δS Phase shifter θ Figure 2.1: Balanced homodyne detection scheme. The signal beam C is overlapped on a balanced beam splitter with a stronger second light Local oscillator beam D ﬁeld of the same wavelength, the so- A = λ0 + δA called local oscillator. The relative phase θ is adjusted by some phase shifting device.

1 C†C = (A† + S)(A + S†) , (2.39) 2 1 D†D = (−A† + S†)(−A + S) . (2.40) 2 The sum and diﬀerences of the individual photo currents are given by † † i± C C D D. They can be expressed according to [65] as ∝ ±

2 2 i+ λ + σ + λ δX + σ δX , (2.41) ∝ 0 0 0 1,A 0 1,S i− 2λ σ cos(θ)+ λ δX + σ δX− . (2.42) ∝ 0 0 0 θ,S 0 θ,A

Under the assumption that σ0 = 0 , which is fulﬁlled for the case of a vacuum state, the above expressions simplify to

2 i+ λ + λ δX , (2.43) ∝ 0 0 1,A i− 2λ σ cos(θ)+ λ δX . (2.44) ∝ 0 0 0 θ,S When σ = 0, e.g. when the signal beam is ’just’ weak instead of vac- 0 6 uum, then this approximation does not necessarily hold: The coherent amplitude σ0 of the signal beam enhances the fluctuations of the local oscillator. Especially for low frequencies, these (classical) fluctuations may be huge. Thus, σ δX− λ δX has to be fulfilled. 0 θ,S ≪ 0 θ,S

14 h prahi iia oteoegvni 5] ti conveni is It [57]. whi in transformation, Fourier given follows: by one as equation deﬁned the latter the to solve similar to is approach The trigwt etnslwfrasse ihapsto depen position a with system a force for restoring law Newton’s with Starting function transfer oscillator The 3.1 prope with mechanical and ends optical chapter membrane’s The the of description thesis. th is this which throughout described, os is investigated harmonic membrane damped nitride a silicon of the noise Next, thermal the and transf factor, mechanical quality a oscillator of the description introducing mathematical oscillator, brief a provides chapter This rr apn force damping trary utpiaino ohsdso q 31 with (3.1) Eq. of sides both of Multiplication vraltimes all over edrvdb ovn h ieeta equation diﬀerential the solving by derived be 3 x ( ω = ) Z − h ehncloscillator mechanical The ∞ m ∞ hpe 3 Chapter t d d F d yields t e t spring 2 2 − + iωt F F arb. − = arb. x ( ( ( t x x kx ) ( ( x , t t ( ) ) t , , ) x ˙ x ˙ ( ( ( fsrn constant spring of t t t )) = ) )+ )) h qaino motion of equation the , 2π k 1 Z x − ∞ ( t ∞ = ) d e e ω − F iωt ext iωt k ( t n integration and rfnto,the function, er n oearbi- some and ) x ( ω . oscillator e ) hcnbe can ch harmonic rties. x cillator. . short a ( t ) (3.1) (3.2) dent can ent 15

Chapter 3 3 The mechanical oscillator

d2 dt e−iωt m + F (x(t), x˙(t)) + mω2 x(t) ∞ 2 arb. 0 Z− dt (3.3) ∞ = dt e−iωtF (t) . ∞ ext Z−

∞ 2 Here, the resonance frequency ω0 = k/m is introduced. For physical functions x(t = )= 0, the Fourier transform of the time derivative ± x˙(ω) can be written as ∞ d x˙(ω) = dt e−iωt x(t) (3.4) Z ∞ dt + = e∞−iωtx(t) −(−iω) dt e−iωtx(t) . (3.5) − ∞ ∞ Z− =0 ∞ x(ω) | {z } ∞ | {z } d n n With dt x(ω) = (iω) x(ω), it directly follows from Eq. (3.3) that F (x(ω), x˙(ω)) −mω2 + i arb. + mω2 x(ω)= F (ω) . (3.6) ix(ω) 0 ext With the deﬁnition

F (x(ω), x˙(ω)) k (x(ω), x˙(ω))) = arb. , (3.7) arb. ix(ω) the algebraic Eq. (3.6) can be rearranged, such that if Fext(ω) is known, x(ω) can be retrieved easily 1 x(ω) = F (ω) (3.8) 2 2 ext m(ω0 − ω )+ ikarb.(x(ω), x˙(ω)) 2 1 − ω − i karb. 1 ω2 mω2 = 0 0 ( ) 2 2 2 Fext ω (3.9) mω 2  0 1 − ω + karb. ω2 mω2  0 0  = G(ω)Fext (ω) ,  (3.10) where G(ω) represents the complex oscillator transfer function. The modulus of this function gives the real displacement, whereas the phase diﬀerence φ(ω) between excitation and reaction is given by the imaginary part φ(ω)= arg(G(ω)) . (3.11)

16 tef tcnb oee yasmn eusv nenlfo internal repulsive a assuming by modeled be the can of It loss internal itself. to subject is mechanism damping Another damping Structural netn q 31)it q 37 yields (3.7) Eq. into (3.13) Eq. Inserting cigo h siltrsml d up add simply oscillator the on acting rrl o rqece.Frlna ytm,tedmigfo damping f the systems, fulﬁlled linear be For not frequencies. can low approximation trarily this although quencies, φ oprsnwt q 36 yields (3.6) Eq. with Comparison at nwihteoclao efrsmto.I hscs,teda the case, this In as written motion. be performs can residua oscillator force by damped the is which motion oscillator in the that basi idea It the assumed. from is damping dependent velocity a Typically, damping Viscous nielhroi siltrcnb ecie by described be can oscillator harmonic ideal An mechanisms Damping nti case, this In ( ω ω ) 0 . scmol sue ob osatoe ag ado fre- of band large a over constant be to assumed commonly is k arb. = F 0 int. n h siltrtase ucini singular is function transfer oscillator the and , F − = F arb. F arb. arb. ( k x ( ( = t ( 1 − = ) t ) X + . h siltrtase function transfer oscillator The 3.1 , i x ˙ iφ ( F t k γ )= )) arb. ( arb. x ω ˙ ( )) t ,i ( ) ω x . 0 k ( arb. = ) ω . . ) ( ω kφ . = ) ( ω γω ) h omof form The . . al arises cally rces oscillator rce a [69] gas l rarbi- or mping (3.13) (3.15) (3.14) (3.12) F arb. 17 ,

Chapter 3 3 The mechanical oscillator

3.2 Mechanical quality factor

An important quantity for a harmonic oscillator is the mechanical quality factor Q. It is a measure of energy dissipation

− ω0t E(t)= E0e Q . (3.16) In this deﬁnition it determines the time it needs for the oscillator’s energy to drop to a fraction of 1/e of any initial value E0, which is given by t = Q/ω0. For viscous and structural damping the quality factors are determined by

mω2 1 Q = 0 , Q = . (3.17) vis. γ st. φ

In the case of viscous or structural damping, the oscillator transfer function can be written as

1 1 Gvis.(ω) = , (3.18) mω2 ω2 1 ω 0 1 − 2 + i Q ω ω0 vis. 0 1 1 Gstruc.(ω) = . (3.19) mω2 ω2 1 1 0 1 − 2 + i Q 2 ω0 struc. mω0 It is remarkable that in case Q 1 (which is always fulﬁlled in the ≫ experiment), Gvis.(ω) Gstruc.(ω) and therefore will only be referred ≈ to as G(ω) in the following. For multiple damping mechanisms acting on the system, the mechanical quality factors add up inversely:

1 1 = . (3.20) Qtotal Qi Xi Another possibility to deﬁne the mechanical quality factor is [69]

G(ω ) ω Q = 0 = 0 , (3.21) G(0) ∆ω where ∆ω is the full width at half maximum (FWHM) of the oscillator transfer function. This also gives an intuitive meaning to the quality factor as can be seen in Fig. 3.1. Here, the quality factor is 20, which corresponds to the ratio of |G(ω 0)| and the transfer function peak → value |G(ω0)|.

18 rldniiso isptv ytmi hra equilibri thermal [74] t in by that system states given It dissipative are environment 73]. a [72, of Callen densities by proven tral was and [71] 1928 in formulated was (FDT) Theorem Fluctuation-Dissipation The theorem Fluctuation-Dissipation wi calculated be can [70]. excitation meas theorem thermal displacement Fluctuation-Dissipation by the caused for motion limit The sensitivity Brownian the a oscillator, imposes the of atoms temperature ﬁnite the to Due noise Thermal 3.3 mass. free a of function transfer the to iue3.1: Figure ult factor quality quency Here, esonta Im that shown be k q 31)yed h oc pcrldensity spectral force the yields (3.10) Eq.

arb. |G ( ω)| (m/N) 10 10 10 10 10 0 1 2 3 4 10 = k ω 3 B γω 0 steBlzancntn,and constant, Boltzmann the is = Q siltrtase ucinframs of mass a for function transfer Oscillator netn hsrsl noE.(.2 n oprsnwith comparison and (3.22) Eq. into result this Inserting . 0kzadasrn constant spring a and kHz 10 Q = ¯ ¯ ¯ ¯ s2.Frhg frequencies, high For 20. is G G ( | G ω ( 0 [ 0 G ) thermal x ) ¯ ¯ ¯ ¯ ( ω | G ]=− = )] thermal F ( ω ) Frequency | 2 k ( arb. − = ω 10 ) | | G 4 2 4k ω ( = k ω ω Q (Hz) | B arb. G 4k ) T = ( | ω 2 B = Im ∆ω ncs fvsosdamping viscous of case In . ω T ) γ. Tγ | 5.0 0 ∝ stetmeaue tcan It temperature. the is [ G = Magnitude 1/ · ( m 10 FWHM ω ( mω − ω = Phase )] 4 0 . hra noise Thermal 3.3 = 0 g eoac fre- resonance ng, 100 2 ) . os.Teresulting The const. hc corresponds which , ∝ mω mwt its with um 1 2 10 oinof motion urement. yNyquist by espec- he 5 - - 0 π π hthe th (3.22) (3.23) /2 19 arg( G ( ω ) ) (rad)

Chapter 3 3 The mechanical oscillator

-10

) 10 1/2 10 -11

10 -12

10 -13

10 -14

10 -15

-16 10 Viscous damping Structural damping Displacement spectral density (m/Hz Displacement 10 -17 10 2 10 3 10 4 10 5 Frequency f (Hz)

Figure 3.2: Comparison of viscous and structural damping mechanism and the corresponding displacement spectral density in m/√ Hz for m = 100 ng, ω0/(2π) = 10 kHz, T = 300 K and Q = 106.

The FDT theorem allows to calculate the thermal noise displacement spectral density, which, in case of viscous or structural damping is given by

x 2 ω0 2 |Gvis.(ω)| = 4mkBT |G(ω)| , (3.24) Qvis. 2 x 2 ω0 2 |Gstruc.(ω)| = 4mkBT |G(ω)| . (3.25) Qstr.ω The displacement spectral density in the case of structural and viscous damping is depicted in Fig. 3.2. Surprisingly, the choice of a constant (frequency independent) loss angle φ results in a more complex shaped thermal noise spectral density (additionally scaled with a factor ω−1/2), as can be seen from Eq. (3.25).

20 i.34frtpclwvlntsue tteAI h maximum The AEI. the at used wavelengths typical for 3.4 Fig. n[1 and [81] in rn xii netariaiyhg me- about of high factor quality extraordinarily chanical an exhibit mem- a brane such of eigenmodes mechanical The tahroi siltra h aetime. same the plan at membrane oscillator the harmonic to a perpendicular it forces Repulsive lator. n silicon by available manufactured commerically brane, a thesis, this Throughout membrane Nitride Silicon 3.4 oml sdrvdi hpe sE.(.) ee h ne o be index to the assumed Here, corresp is (5.5). nitride The silicon Eq. AEI. as of the 5 tion Chapter at in used derived dep wavelengths is is formula typical membrane for the of 3.4 reflectivity power Fig. The surface. each at 7] h eoac frequencies resonance The to increase [77]. even can 10 which temperature sn rse omls h mltd eetvt o h t the for (vacuum reflectivity one amplitude medium from the formulas, normal under Fresnel reflectivity Using amplitude the is case relevant The sub- the in membrane. silico interference thin of the refraction and wavelength) of particular index the the considering calculated be rcinrslsi napiuerflcinadtransmissio and reflection amplitude an in results fraction yial,tetikeso h ebaei eea eso n of tens several is membrane the 1 of to thickness the Typically, properties Optical 3.4.1 i 81]. implementation [76, in first reported their was 80], interferometer 37, [36, inter- before Fabry-Perot ferometers linear, in used been have r on sal nafeunyrnefrom range kHz frequency 10 a in usually found are iin(h ai ewe iio n nitro- and Si silicon gen between compo- ratio chemical (the membrane’s sition the on pend 7 tcygnctmeaue f30mK 300 of temperatures cryogenic at µ .Terpwrrflciiyfragvnlsrwavelength laser given a for reflectivity power Their m. x − N y H.Slcnntiemembranes nitride Silicon MHz. 1 7,7] hcns n ie but size, and thickness 79], [78, ) n SiN = r = 2.2 n n srpre n[7.Tersl sdpce in depicted is result The [77]. in reported as , SiN SiN Norcada + − n 1 1 = 10 1 and oamdu fhge ne fre- of index higher of medium a to ) 7] a sda ehncloscil- mechanical a as used was [75], 6 troom at f res . iio ird membrane Nitride Silicon 3.4 n t SiN de- = n = 2n SiN 1 a 3.3: Figure iei cm 1 frame is The size [76]. membrane 2.22 Silicon frame SiN + . swsdetermined was as , 5 1 × 1 . mm 5 2 . htgahof Photograph ird (at nitride n coefficient n tiemem- itride wavelength Sagnac a n 1.5 mm incidence. 2 ransition render e membrane SiN refrac- f ce in icted Norcada onding power (3.26) λ up m can 21

Chapter 3 3 The mechanical oscillator

0.45

0.4

0.35

0.3

0.25

0.2

Power re!ectivity 0.15

0.1 1064 nm 1550 nm 0.05 532 nm meas. value 0 10 20 30 50 75 100 150 200 Membrane thickness (nm)

Figure 3.4: SiN membrane power reflectivity for different optical wavelengths. The power reflectivity is a periodic function of membrane thickness and can be determined according to Eq. (5.5). The membrane thickness can be determined by measuring its power reflectivity under normal incidence at a particular laser wavelength for a given index of refraction. reflectivity is around 43 %, and is reached for a membrane thickness d that corresponds to an anti-resonance of the light field within the 2 membrane. In turn, for a given thickness, the power reflectivity rm depends on the laser wavelength.

3.4.2 Modes of motion and eﬀective mass

A SiN membrane is able to perform different deflection movements. Being fixed to the massive rectangular silicon frame which we assume to be in the xy- plane, possible eigenmotions u(x, y, t) can be calculated [76, 82] by solving the Helmholtz equation

T ∂2u(x, y, t)= (∂2 + ∂2 )u(x, y, t) . (3.27) t µ x y

Here, T is the intrinsic tension of the membrane, Lx and Ly are the membrane geometrical dimensions and µ is the mass per area, which is found by the density ρ and the membrane thickness d. The boundary condition u(x, 0, t)= u(0, y, t)= 0 t, results in a set of solutions ∀ mπ x nπ y um,n(x, y, t)= sin( ) sin( ) cos(ωm,nt) . (3.28) Lx Ly

22 h udmna silto mode. oscillation fundamental the iue3.5: Figure o h xeietlmmrn parameters, membrane experimental the For m o h udmna silto oe( mode oscillation fundamental the For hs nyacranrgo ftemmrn culymvs T moves. actually membrane the motion of of equation region oscillator certain harmonic a only Thus, moving. not is membrane nti aei 1 is case this in obe to where = n m =

ω Amplitude (a.U.) Amplitude (a.U.) eff 1 0 2 -0.5 0.5 0.2 0.4 0.6 0.8 -0.5 -0.5 rpia lutaino h ebaefnaetloscilla fundamental membrane the of illustration Graphical -1 uprpcue and picture) (upper 0 1 0 1 = = m/4 k/ × mm 1 ( m with x (mm) x (mm) eff 2 ) ncs fte(,)mcaia oetecnr fthe of centre the mode mechanical (2,2) the of case In . sflle o h orc ffciemass effective correct the for fulfilled is , m 0 0 0 en h ebaepyia as[6 81]. [76, mass physical membrane the being = m ∂ = t 2 z n + = ω 2 0.5 0.5 . iio ird membrane Nitride Silicon 3.4 m 0 2 lwrpcue.Temmrn size membrane The picture). (lower , z -0.5 -0.5 = n = m eff 1 ) saot10n for ng 100 about is m 0 0 eff a ederived be can y (mm) y (mm) inmode tion 0.5 0.5 (3.29) m eff he 23 .

Chapter 3 3 The mechanical oscillator

50 Output spectrum Modes by theory 40

10 Rel. noise power (dBm)

-10 10 100 k 200 k 500 k 1 M Frequency (kHz)

Figure 3.6: A typical interferometer output power spectrum. Various sharp resonances correspond to the thermally excited membrane oscillation modes. The fundamental oscillation mode is found at f11 = 133 kHz. Other resonances agree with the calculated frequencies, calculated by Eq. (3.30) and plotted as black crosses.

3.4.3 Resonance frequencies

Figure 3.6 shows a typical output power spectrum in transmission of the interferometer. It contains several sharp peaks at the membrane resonance frequencies. For each membrane mode m, n, the resonance frequency are given by [81]

T m2 n2 ω = π + . (3.30) m,n µ L2 L2 s x y

The calculated frequencies for modes m, n 6 5 for the membrane under investigation are depicted as black crosses in Fig. 3.6, and agree nicely with the experimental results. The height of the peaks originates from their eﬀective mass, which is lowest for the fundamental mode, and the readout position on the membrane. Fig. 3.5 depicts the fundamental mode (1, 1) for a square membrane. In case of the (2, 2) mode, no signal would be expected for case where the beam is perfectly centred on the membrane. However, the corresponding peak is non-vanishing in the spectrum, which is due to a beam oﬀset from the membrane centre.

24 mnspeetdi hstei sgvni al 3.1. table in given the is in thesis used this membranes in nitride presented silicon iments two the of comparison mu A measurement the repeating by times. confirmed is and inferred was yia esrmn ucm sdpce nFg ..Fo t From 3.7. Fig. in constant depicted is outcome measurement typical A hs rmosrigteepnnildcyo h neoeo envelope derived. be the can of factor decay quality mechanical exponential the the amplitude, observing from Thus, tiga prpit xoeta ea.I hscs,amec a case, this In of decay. factor quality exponential appropriate an fitting 31 a erewritten be can (3.1) h ehnclqaiyfco sdfie nE.(.7.Assum (3.17). Eq. in defined is damping factor sudden viscous quality a mechanical after observed The The then is frequency. excitation. by amplitude the resonance excited oscillation its hig the was with a of membrane transducer in the ceramic measurements Therefore, piezo ring-down by the environment. of determined vacuum mode been fundamental has the of brane factor quality mechanical The factor quality mechanical Membrane 3.4.4 h ouinfrlweeg ispto hg ehnclqu mechanical (high Q dissipation energy low for solution The x vis ( t ste ie by given then is ) = ) x 0 τ e fti ea,teqaiyfactor quality the decay, this of − ω 0 2 ⇔ t/2Q Q ln F arb d = d

vis t 2 x d 2 d 0.58 ( t cos = 2 t + 2 x + ( + γ t Q ( ω ωt ) · ∆t x ˙ vis 10 m γ 0 2 ( t ) ) 6 ) d d

ω , d d t t − = o h udmna silto mode oscillation fundamental the for n oectn force exciting no and + + 2 ω ω = ω 0 2 0 2 0 . iio ird membrane Nitride Silicon 3.4 ω ∆t/ 0 2 x x ( ( t t ( = ) = ) 1 Q Q − vis ol edrvdb a by derived be could 4Q ) ω . 0 0 0 4 vis 2 . F ≈ ext ω lt factor ality = 0 2 . etime he hanical tpof stop 0 exper- (3.34) (3.33) (3.32) (3.31) mem- decay Eq. , ltiple the f ing 25 h- a

Chapter 3 3 The mechanical oscillator | 0 1 0.8 Maxima Minima 0.6 excitation ring down 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 normalized amplitude |x/x 0 2 4 6 8 10 Time (s)

!t min 0 !t max |) 0 -2

ln(|x/x -4

-6 0 2 4 6 8 10 Time (s)

Figure 3.7: Ring-down measurement of the membrane mechanical quality factor for the fundamental mode of a SiN membrane at 133 kHz. Both, maxima and minima are plotted as function of time, and an exponential ﬁt was made in order to derive the membrane’s mechanical quality factor.

Table 3.1: Relevant SiN membrane parameters for the membrane used in the 1064 nm and 1550 nm cryogenic experiment

1064 nm 1550 nm Resonance frequency (300 K) fres 133 kHz 155 kHz Resonance frequency (8K) fres - 141.82 kHz Mechanical quality factor (300 K) Q 0.58 106 0.52 106 · · Mechanical quality factor (8K) Q - 6.8 106 · Effective mass meff 80 ng 130 ng Size Lx 1.5mm 1.5 mm Thickness d 45nm 75nm 2 Power reflectivity rm 0.17 0.25

26 a icse ntepeiu hpe.I xiisa amplit an exhibits It chapter. previous membran SiN flectivity the a in was mirror discussed two this was present, the thesis between the mirror In translucent mirrors. interfe a Michelson-Sagnac placing of The by power splitter. established form output beam an balanced The interferometer, ratio a splitter Sagnac beam for mirrors. the a steering on depends two is interferometer and Sagnac basis splitter The beam a discussed. is topo interferometer 86] Michelson-Sagnac the section, this In topology Interferometer 4.1 interferometric achieved the of presentation a described. with is closes interferometer Michelson-Sagnac experim a the of Next, sation topology. the within an contributions quantum relevant noise the about pr information is provides interferometer section Michelson-Sagnac the section, this In oiin h ih edicdn nteba pitrwt g with splitter beam the coefficients membran on transmission the incident and on field reflection depends light amplitude power The resp output their detected with position. the up add and sh amplitudes phases, is field interferometer all the recombining, of When schematic A 4.1. Fig. splitter. beam ar the beams displa all on membrane interferometer, a the through to circulation t proportional After as shift displacements, phase like membrane Michelson a to two sensitive accumulates to is rise mode gives This membrane mode. the of sides two the 4 r m ihlo-ancinterferometer Michelson-Sagnac hpe 4 Chapter n mltd transmissivity amplitude and t m h ih eetdby reflected light The . r ij and sne.The esented. overlapped e oy[1 83– [81, logy h section The sensitivity. na reali- ental vanishes d oee is rometer classical d t ij ,which e, steering cement. elight he d re- ude w in own optical is dby ed ective iven a the 27 in e .

Chapter 4 4 Michelson-Sagnac interferometer

la Membrane Figure 4.1: Michelson-Sagnac inter- (rm, tm) ferometer topology. The incident beam is split by a beam splitter 2 lb and folded by two steering mirrors. 1 A translucent SiN membrane with Beam splitter (rbs, tbs) amplitude reﬂectivity rm and transmissivity tm gives rise to a reﬂected 3 and transmitted optical mode, indi- 4 cated as black, dashed lines.

Here, i represents the input port and j the output port of the beam splitter respectively, i,j {1, 2, 3, 4} - compare Fig. (4.1), assuming ∈ rij = rji and tij = tji for reciprocity of light ways. To fulfil energy conservation, a phase flip of a total of π/2 at the beam splitter is required [87]. There is no constrain on where to introduce the phase flip. For convenience we write

iθij rij = rbse , (4.1) iθij tij = tbse . (4.2)

The phase relations for the beam splitter are chosen such that

θ12 = −θ34 , (4.3)

θ13 = −θ24 π . (4.4) ± A valid choice of phases is θ12 = θ34 = θ24 = 0 and θ13 = π.

Interferometer in transmission The output port 4 of the beam splitter is the sum of four electric ﬁeld amplitudes with their individually accumulated phases on their way to port 4.

ifo 2 iθSA1 2 iθSA2 aout = aintmrbse + aintmtbse + Sagnac (4.5) | iθMI1{z } iθMI2 ainrmrbstbse + ainrmrbstbse . Michelson | {z } The phases can be written as

28 o h ih hnrflce yo rnmte hog h mem the through transmitted or by k reflected when light the for hssfo q 43 n 44 no(.)aduigteident the using and (4.5) into (4.4) and (4.3) Eq. from phases ewe emslte n ebae and membrane) and splitter beam between nti case this In aacdba pitr( splitter beam balanced a t,ablne emslte sasmdfo o nadol t only and on now from assumed investigated. is is term splitter second beam balanced a ity, h atfco safnto ftedffrnilphases differential the of function a i is factor factor phase the last rendering The well), as position membrane the ntesmo h nefrmtramlengths arm interferometer the of sum the on edn ftemmrn oiin h phase The all position. of membrane First S the the of explanation. represents pendent still some interferom equation deserves the the and for of interferometer part expression first general The most field. the put is (4.11) Eq. h ouu qaeo q 41)i h omlsdinterfero normalised the is (4.11) transmission in Eq. power of output square modulus motion. The differentia membrane to some sensitive by actually caused is term respective the and eedu ihtefia xrsinfrteitreoee out interferometer the for expression final the with up end we ido ht detector photo of kind = a a 2π/λ out ifo in = stewv number, wave the is t ifo l a θ θ θ θ l , + = e SA2 SA1 MI2 MI1 b ix r | | t bs r h r egh fteitreoee (distance interferometer the of lengths arm the are m + t e bs = = = = iθ e r iy SA1 m Sagnac θ θ θ θ e = {z 13 12 13 12 ( 2 i r ( r e bs 2 θ + + + + bs 2 2 i MI1 θ ( − θ θ θ θ MI1 P λ x = 34 24 24 34 + out + t t stelsrwvlnt.Isrigthe Inserting wavelength. laser the is y bs 2 θ t Michelson + + + + + ) bs 2 MI2 hc a emntrdb some by monitored be can which , } ) · θ θ θ θ θ {z 2 = ) rm rm tm tm MI2 2 cos 0.5 cos . nefrmtrtopology Interferometer 4.1 + + + + scntn idpnetof (independent constant is k k 2k 2k tvnse.Frsimplic- For vanishes. it ) x ( ( θ l l ( ( θ − a a 2 rm l l MI1 θ b a + + y ) ) SA1 θ , k l l ( b b − l tm 2 , . a ) ) sol dependent only is θ + , r hs shifts phase are MI2 , , l b θ r length arm l ) } lo for Also, . MI1 agnac-like ti inde- is it . rrelevant. u field. put trout- eter t [88] ity − brane. (4.11) (4.10) meter (4.9) (4.8) (4.7) (4.6) θ MI2 he 29

Chapter 4 4 Michelson-Sagnac interferometer

0.3 Transmitted power RBS = 0.5 0.25 Transmitted power RBS = 0.45 0.2 in

/P 0.15 out P 0.1 0.05 0 0 λ /16 λ /8 λ /4 λ /2

1 0.95 0.9 in

/P 0.85 out P 0.8

0.75 Re!ected power RBS = 0.5 Re!ected power RBS = 0.45 0.7 0 λ /16 λ /8 λ /4 λ /2 Interferometer Detuning

2 Figure 4.2: Normalised interferometer output power for a membrane with rm = 0.17 2 2 for two diﬀerent beam splitter ratios rbs = 0.5 and rbs = 0.45, respectively. The bright-fringe detunings remain the same whereas to achieve a dark output port, the Michelson interferometer has to be detuned to compensate for the remaining Sagnac amplitude.

2 Pt aifo 1 out = out = r2 (1 + cos(4k∆x)) . (4.12) P a m 2 in in π In this case, a detuning ∆x = 4k 2n, n N corresponds to maximal π · ∈ output power, whereas ∆x = 4k (2n + 1) produces a dark interferometer output. These very tunings will be referred to as bright-fringe and dark-fringe condition from now on. Either tuning is also possible for an unbalanced beam splitter as depicted in Fig (4.2), as long as the membrane amplitude reﬂectivity allows for compensation of the uncancelled Sagnac amplitudes [76].

Interferometer in reﬂection The electric ﬁeld in the interferometer input port is again a sum of four electric amplitudes

30 sn h dniy[88] identity the Using eso h ebaetu lost oiintemmrn exa membrane the position to allows thus sub-waveleng membrane The the of port. ness output dark-interferometer a duces ie h oa oe eetvt fteitreoee as interferometer the of reflectivity power total the gives ebae sasproiino w one-rpgtn w counter-propagating two of phases superposition individual their a with is interferomete Sagnac membrane) a within a point each at field optical The anti-nodes and Nodes o aacdba splitter, beam balanced a For amplitude field a in ihteridvda phases individual their with pnst ebaepsto ntend fteeeti field electric the Positi of corr interferometer. node dark-fringe anti-node an Sagnac the a in the membrane in that within wave position finds standing membrane one the a (4.12), to Eq. sponds to compared When k ( l a − l b P a a P ( = ) θ θ θ θ out r refl ifo in in e SA2 r SA1 r MI2 r MI1 r ix = = 2n − r

a = = = = e ifo a + a iy reﬂ ifo = in + = 1 θ θ θ θ = r )

r 13 12 13 12 π bs 2 t ∆x bs 2 m e e = Θ + + + + 2 r r ⇔ i − m bs ( SA1 = 1 ikl x θ θ θ θ r e t bs + ∆x 13 12 12 13 − iθ bs 4k y a π = = r + + + + e ) MI1 r − = m 2 ( iθ · 2n kl t θ θ θ θ 2i 1 2 t SA1 r bs rm rm tm tm 2k + bs π ( a ) sin 1 = e . nefrmtrtopology Interferometer 4.1 t ftesadn aeas pro- also wave standing the of and ( + + + + + − bs 2 + 2n 1/ ikl r k k 2k 2k t cos m m ( ( x √ + Θ l l b ( ( e r a a − l l 2 ( 2 SA2 iθ bs 1 b a 4k∆x + + hssmvnse if vanishes sum this , ) ) ) y t MI2 r . n , bs l l b b = , e . ) ) iθ )) ∈ π , SA2 r + , , N kl . b . (without r resulting , nn the oning hthick- th (4.17) (4.16) (4.18) (4.15) (4.19) (4.14) (4.20) (4.13) (4.21) aves, ctly 31 of e-

Chapter 4 4 Michelson-Sagnac interferometer in a node, thereby minimising the optical absorption. In [84], it was demonstrated that the resulting optical absorption can be minimized to below 5.4 ppm for a membrane of thickness d = 60 nm at a laser wavelength of 1064 nm if exploiting a node position.

4.2 Interferometric signals and noise

The interferometer output port can be monitored by a single photo detector. It generates an electric current proportional to the incident light power. If the membrane moves (oscillates), a change (oscillation) of the output power will be detected. The amplitude of the detected signal depends on the interferometer tuning, since the derivative ∂Pout/∂x is depending on the tuning. The maximal signal is obtained if the interferometer is operated on mid-fringe, whereas no signal can be detected on dark- and bright fringe.

4.2.1 The shot noise limit The photons detected in the interferometer output are considered to be uncorrelated in time. The one-sided, linear spectral density of the optical shot noise [89] can then be written as [90] (compare Eq. 2.13)

4πhcP out Gshot = 2hωP out = . (4.22) r λ p The shot noise is proportional to the square root of the interferometer output power, whereas an interferometric signal is proportional to the optical power incident on the interferometer. Both result in a ﬂuc- tuation of detected output power, rendering both indistinguishable. Optical shot noise results in an apparent length noise. Division of the shot noise by the signal for some tuning (arm length diﬀerence) ∆x of the interferometer yields

G hcλ 1 + cos(4kx) shot = . (4.23) | | 2 | ( )| ∂Pout/∂x s8πrmPin p sin 4kx The last equality was achieved by inserting Eq. (4.12) for the interferometer output power Pout. The minimal value of Eq. (4.23) is achieved π π if x = . This is the dark-fringe condition, Pout( ) = 0. The ± 4k 4k single-sided linear spectral density of the relative error caused by shot noise on dark-fringe is

32 oawie igesddfreseta est f[86] cor of density and spectral (2.13) force Eq. single-sided of white, out a follows to then density spectral force pnsto sponds tefa ahpoo eeto sascae ihafre a force, a with associated is measur position reflection membrane the the photon to each due is as noise itself quantum of source second A noise back-action Quantum 4.2.2 wa gravitational large-scale 95]. [94, and tabl in 91–93] demonstrated [22, been erferometers has st as vacuum port squeezed output a interferometer of injection the is possibility Another codn oE.(.0,ti oc assa siltrdis by oscillator given an density causes spectral force a this with (3.10), Eq. to According eu a elwrdb nraigteicdn ae power laser incident the increasing by lowered be can setup ity o ie ae wavelength laser given a For ulse n[6.A soitdeeti odrmtv squ ponderomotive is effect associated An n [96]. back-action quantum in of published observation oscilla experimental or first mass, interferometer. The test Michelson-Sagnac the the of of rela motion case phase i differential splitter anti-correlated a beam are cause the hence they of such, because As arms origi interferometer [16]. t the port through explain splitter interferometer to beam the able entering equall was fluctuations, Caves fluctuations vacuum laser arms. (quantum) beam interferometer balanced translate two distur perfectly not a would would why on noise unclear ter shot an rad was quantum terminated it as not Caves as to measurement, or Carlton referred whether 1980, on also In discussion is noise. noise shot shot pressure back-action photon This r m 2 h htnieipeiino h oiinmaueeti t in measurement position the of imprecision noise shot the , P/ ( hω

F ph ) = htn e ieui,sc that such unit, time per photons G rpn x 2 hk

G G rpn F = shot x e htn enotclpower optical mean A photon. per | G = = λ ( ω r . nefrmti inl n noise and signals Interferometric 4.2 n ebaewt oe reﬂectiv- power with membrane a and s ) 16π | r 16πr 16π hr hcλ

cλ m 2 m 2 hr

P cλ P in in m 2 P in . . . F ( = edetectors ve 2/c t nothe into ate placement, esecond he h two the n o nthe in tor in and tions -o int- e-top responds eigof eezing P tn on cting nthe in y iewas oise ) P in P with n ement (4.25) (4.24) (4.26) corre- iation The . the b going split- only. his 33

Chapter 4 4 Michelson-Sagnac interferometer

Thermal noise Q = 106, T = 300 K 10 -13 ) Radiation pressure noise

1/2 Shot noise 10 -14

10 -15

10 -16

10 -17

10 -18

10 -19 Displacement linear spectral density (m/Hz linear Displacement

10 -20 1 k 10 k 100 k 1 M Frequency (Hz)

Figure 4.3: Radiation pressure, thermal noise and shot noise for an input power 6 of Pin = 1 W, and an oscillator with Rm = 0.3, Q = 10 , T = 300 K and mass m = 125 ng, with a resonance frequency of fres = 75 kHz, considered for a laser wavelength λ = 1064 nm. the light ﬁeld reﬂected of the membrane, which has been demonstrated in [97, 98]. For experimental parameters, Fig. 4.3 depicts the linear spectral densities of thermal and quantum noise, for an interferometric operation on dark-fringe and a laser input power of Pin = 1W. As can be seen, the radiation pressure noise is about 3 orders of magnitude smaller than the other contributions. For a direct observation, the shot noise level has to be decreased, by either applying 1 kW of optical input power at the beam splitter (which will only be possible by a power recycling mirror), or alternatively, the equivalent gain by signal-recycling. In either case, the thermal noise of the oscillator has to be reduced drastically, which makes it necessary to operate the experiment with cryogenic temperatures. The noise contributions are considered in [86], and depicted in Fig. 4.4 For a temperature T = 1 K, the thermal noise is reduced by a factor of √300, (compare Eq. (3.25)). The mechanical quality factor is assumed to be increased by one order of magnitude at low temperatures, as it has been observed by [77] previously. In this case, the spectrum is dominated by radiation pressure noise at all frequencies.

34 n q 42)adrgrigtersm hc sasmdt be to assumed is which sum, their regarding and (4.26) uncorrelated. Eq. and iue4.4: Figure and K 1 xssa pia ae power laser sta freq optimal precis each the an for readout to exists interferometer, interferometric Michelson-Sagnac rise the the give of for case noise (SQL) back-action limit and quantum noise shot Optical limit quantum Standard 4.2.3 frequen a of because response. arises dependenc mechanical spect displacement pendent frequency power the The as of well independent. measurement as t frequency force and Both, are discrete sities a are statistics. photons noise Poisson that a pressure fact obey radiation the both, from originate that noise mentioning worth is It measur the case, this noise. In shot cm. pressure 12 radiation be by to limited assumed is cavity recycling Displacement linear spectral density (m/Hz 1/2 ) 10 10 10 10 10 10 10 10 -20 -19 -18 -17 -16 -15 -14 -13 1 k R sr h xeietlniebde iha siltro temperat of oscillator an with budget noise experimental The = G 99.6 total x Thermal noiseQ=10 oe eetvt inlrccigmro.Telnt fth of length The mirror. signal-recycling reﬂectivity power % 2 Radiation pressurenoise ( P = ) = 10 k 10 G 16πr Shot noise 7 shot x , T=1K hcλ

. nefrmti inl n noise and signals Interferometric 4.2 P m 2 2 sql Frequency (Hz) ( P P hscnb enb q (4.23) Eq. by seen be can This . in + ) + G | G rpn x ( ω 2 ) ( | P 2 100 k 100 ) 16π hr

cλ m 2 P mnswudbe would ements in ec there uency a they hat nthe in e a den- ral o.In ion. dshot nd ure yde- cy ndard (4.27) T 1 M 35 = e

Chapter 4 4 Michelson-Sagnac interferometer

2 The sum in Eq. 4.27 is minimized if ∂/∂PGtotal(P) = 0, which is the case if the input power P = Psql, where

cλ = Psql 2 . (4.28) 16π|G(ω)|rm

Equation (4.28) shows that Psql is a function of frequency. The minimal total noise is given by

x Gtot,sql = 2h |G(ω)| . (4.29) p The maximal value of the SQL is reached at the mechanical resonance frequency,

2hQ x = Gtot,sql 2 . (4.30) smωm

It is a common misassumption [99, 100] that achieving a total noise below this peak value at arbitrary frequencies is of any importance for a displacement (or force) measurement or would beat the standard quantum limit. It is rather obvious from equation (4.29) that such a situation always occurs if the mechanical quality Q is high enough. The comparison between measurement precision at one frequency, and the standard-quantum limit at another frequency is invalid. A measurement can only claim to beat the SQL at some frequency if the observed total noise is smaller than the SQL at this very frequency [83]. The ’correct’ way to beat the SQL is based on another idea of Caves. He suggested to replace the vacuum state entering the interferometer by a squeezed vacuum state of light. [20]. Squeezed states of light exhibit unequal variances in both quadratures, with a variance smaller than the variance of a vacuum state in one particular quadrature. Thereby, it is possible to reduce the noise in one quadrature at the expense of the other, orthogonal quadrature. The ideal squeezing quadrature depends on the frequency, as do radiation pressure and optical shot noise. Kimble proposed optical ﬁlter cavities [25], which induce a frequency dependent phase rotation, to accomplish a frequency dependent squeezing angle. It is thereby possible to beat the SQL on a broad frequency range.

36 power eotdi [96]. in reported scnb enfo h rtsto qain,teobservation the equations, likel of most is set temperature ﬁrst factors given quality mechanical the at noise from pressure seen radiation be can As is,teseta este ttemcaia eoac f resonance mechanical the t at for Michel densities given ( a spectral is for the comparison noise this First, displacement Here, thermal interferometer. spectra and Sagnac displacement quantum individual the both, compare of to useful is It contribution noise individual of Comparison ua rvsos.Tedslcmn pcrldniisare densities spectral displacement ( The mechanism damping viscous). the or on tural depends noise thermal the quency, G o frequencies For ω G G G therm x G G shot x therm x, therm x, G = rpn x shot x st vi rpn x ω P m ( ( ( ( in = = = f f f f ,wihi sue as assumed is which ), = ) = ) = ) = ) n bevto ttemcaia eoac rqec,as frequency, resonance mechanical the at observation and 2.01 4.72 3.55 f 3.08 4.72 3.55 2.01 · · · ≪ 10 10 10 − − − f · · · · m 14 17 11 10 10 10 10 √ √ √ mle hntemcaia eoac fre- resonance mechanical the than smaller − − − − m m m 17 17 17 20 Q Hz Hz Hz o siltrmass oscillator low , √ √ √ √ m m m m Hz Hz Hz Hz . nefrmti inl n noise and signals Interferometric 4.2 10 W 1 ng 125 P Q ω in m 6 m W 1 ng 125 ng 125 W 1 04nm 1064 P P = in in m m 10 W 1 2π Q λ P 0.3 nm 1064 r 6 in m 2 · 0 K 300 0 K 300 K 300 5kzyield kHz 75 04nm 1064 0.3 λ r T T T m 2 0.3 r m 2 04nm 1064 λ m 10 0.3 r Q 1/2 ihotclinput optical high , 1/2 m 2 1/2 λ 6 2 ng 125 kHz 1 1/2 m f , , 2 ng 125 , ie by given m o high for y densities l ocases. wo , requency 1/2 1/2 struc- 1/2 son- . , 37 of .

Chapter 4 4 Michelson-Sagnac interferometer

0.3 C = 1.0 0.25 C = 0.8 C = 0.5 0.2 in

/P 0.15 out

Figure 4.5: Ratio of Michelson- P 0.1 Sagnac interferometer output power Pout/Pin in case of a balanced beam 0.05 splitter for three values of interferometric contrast C. Here, a mem- 0 2 λ λ brane power reﬂectivity rm = 0.3 was 0 /8 /4 assumed. Membrane displacement

4.3 Inﬂuence and suppression of laser noise

This section describes the eﬀect classical laser amplitude and frequency noise. The considerations focus on technical laser amplitude and frequency noise. Their impact on the interferometric measurement of the membrane position is described with regards to a realistic experimental situation: a small, but non-vanishing arm length diﬀer- ence of the interferometer, and an imperfect interference contrast.

4.3.1 Interferometer contrast Although the signal to shot noise ratio is maximized mathematically on dark-fringe, measurements with a single photo diode will become insensitive at this point of operation, such that an infinitesimal offset from dark-fringe is required. Here, it is shown that this offset can not be arbitrarily small in the experiment. So far, Eq. (4.11) was taken as complete description of the interferometer output power, but silently assumed a perfect interferometer contrast C, which is defined as follows

P − P C = max min . (4.31) Pmax + Pmin Here Pmax and Pmin are the maximal and minimal interferometer output power. In the perfect case, and for a balanced beam splitter, 2 Pmin = 0 and Pmax = rmPin, such that it gives C = 1 for Pmin = 0. In reality, C is a measure of the interferometer quality, and usually C < 1. This is caused by a variety of reasons, such as diﬀerential losses in the interferometer arms, imperfect alignment or non-ideal mode matching.

38 oe eetvt a assumed The was values. reflectivity contrast power interferometer different two with Michelson-Sagnac same eter the Figur of sensitivity offset. displacement dark-fringe the interferometer val some maximal for an i achieved exhibits associated (and then ratio sensitivity) noise displacement to ter microscopi signal The membrane however. the zero, for is derivative its dark-fringe, On nefrmtrwrigpitdpnso h nefrnec interference the th on case, any depends In point sa detector. working the photo interferometer cases, the r in both be gain could In electronic noise higher this 4.3.4. although assumed, Sec. is dark-noise in tronic described is which noise, akfig snnvnsig h nefrmtrotu p (4.12)) output Eq. interferometer (compare The writes non-vanishing. is dark-fringe contrast g A electronic higher by reduced is it experiment, the detector. In ments. n right and h nefrmtrtnn fdr-rne et:interferom : Left dark-fringe. of tuning interferometer the of iue4.6: Figure n ihritreec otatalw o oeprecise more dark-fringe. a the for to allows closer contrast ment, interference higher a and Displacement spectral density (m/Hz 1/2 ) P 10 10 10 10 in -17 -16 -15 -14 0.001 = C 0 W h ltas nldstcncllsramplitude laser technical includes also plot The mW. 100 htnie ehia ae os n eetrdr os as noise dark detector and noise laser technical noise, Shot = 1 < C 99.999 Locking position(Rad) 0.01 hsipista h nefrmtrotu oe on power output interferometer the that implies thus .Eetoi akniei sue qa nbt measure- both in equal assumed is noise dark Electronic %. P out = . nuneadsprsino ae noise laser of suppression and Influence 4.3 P in 0.1 r 2 m 2 ( r 1 m 2 + = C 0.3 cos 10 10 10 10 n notcliptpower input optical an and , ( -16 -15 -14 -17 4kx 0.001 )) Laser intensity noise Detector darknoise Locking position (Rad) trcontrast eter . 0.01 Shot noise i ftephoto the of ain e hc is which ue, dcdb a by educed nterferome- . shows 4.6 e interferom- optimum e membrane ontrast wrthen ower position c ucinof function measure- eelec- me C (4.32) = 0.1 99 39 C % ,

Chapter 4 4 Michelson-Sagnac interferometer

4.3.2 Interferometer arm length differences Non-ideal destructive interference of the two Michelson modes at the beam splitter and hence, C < 1 can be caused by a macroscopic interferometer arm length difference of the interferometer. This can be modelled by two Gaussian modes with same waist size but a lateral displacement ∆zR of the two waist positions. The overlap of such two optical modes is given by Eq. (4.33) and depicted in Fig. 4.7. For C < 1, the interferometer will transmit a fraction of the incident laser carrier and thus couple technical amplitude fluctuations into the read out:

2√z √z C = R1 R2 2 2 (z1 − z2) + (zR1 + zR2) 2z 1 = p R = . (4.33) 2 2 1/2 ∆z (∆z + 4zR) 1 + ( )2 2zR q 2 Here zR denotes the Rayleigh range zR = πω0/λ. A laser wavelength λ = 1064 nm and a waist size of ω0 = 500 µm were chosen. Despite suppression of the tech- 1 nical amplitude laser noise in the read out,the interferometer con- 0.9 1 trast becomes important when a high-reflective recycling mirror is 0.8 to be installed in the interferom- 0.995 eter output port, because the in- 0.7 terference contrast should exceed 0.6 the recycling mirror reflectivity 0.99 0 0.05 in order to provide an under- Contrast C 0.5 coupled cavity condition. The higher the desired contrast 0.4 is, the smaller the arm-length difference has to be. In the ex- 0.3 periment, a piezo stepped stage 0.2 is placed underneath the mem- Interference contrast brane holder to allow the mem- 0.1 brane to be positioned such 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Shift of beam waists (m) that the arm-length difference is smaller than 1mm. It is an unanswered question whether Figure 4.7: Achievable interferometer contrast C as function of arm length differences.

40 tcnb prxmtl rjce s[0,106] [102, as projected approximately th be for t relevant can only is It noise signifi Here, frequency a laser resonator. with the laser of H sources the contribution laser of frequencies. other linewidth the high cavity using for using larger when achieved when important only be factor is can limit limiting this a as not laser, is this Fortunately, an of noise quency where needn n ie by given and independent δ yeo eghnie ftelsrehbt rqec noise frequency exhibits laser the If noise. length of type Where h arsoi r eghdifference length arm macroscopic The noise frequency Laser 4.3.3 t by eliminate dark-fringe would to this interferometer frontal as the 0, stabilize is to sibility difference arm-length best the and fteitreoee r eghieult sue obe to assumed inequality length arm interferometer the If (Hz density spectral Shawlow-Townes frequency called the so p the that to the states rise perturbing It gives fluctuations, This fundamental quantum carrier. A from the o power. results fluctuations pump noise by or of caused length te be crystal Both can resonator 105]. laser noise 104, frequency [102, in and example amplitude for times, numerous sured stecs o n ae) hslast naprn eghno length apparent an to leads this laser), by any given for case the is and ν = δ P H o the for kHz 1 ν ˜ = ν ( δ ouainshm [101]. scheme modulation 0 kHz 100 ν ,E.(.5 yields (4.35) Eq. W, 1 stema ae rqec.Tetpcltcncllsrfre laser technical typical The frequency. laser mean the is stelsrrsntrlnwdh hc sseie 13 to [103] specified is which linewidth, resonator laser the is = ) Innolight δ s Hz 1 ST Innolight ν ˜ ( f ( f = ) . nuneadsprsino ae noise laser of suppression and Influence 4.3 s = ) / ST √ 0kHz 10 z hslast naprn eghnoise length apparent an to leads this Hz, ( δ 3.05 f ehsoNR ae.With laser. NPRO Mephisto s ˜ = ) 13 ehsoNR ae a mea- was Laser NPRO Mephisto [103] = · δ δ ν 10 ν ν ˜ · r √ L, ∆L − 1 Hz 7 hπc

Pλ Hz ∆L · / Hz 1 √ sas ujc oanother to subject also is f . Hz / . . √ Hz ) experiment. e h so-called the λ sfrequency is ∆L eclassical he = δ wvr it owever, ν Innolight ˜ s [102], ise ii [7]. limit = 04nm 1064 epos- he aeof hase chnical (which source cantly (4.34) (4.35) (4.36) mm 1 the f 41 -

Chapter 4 4 Michelson-Sagnac interferometer

) arm length di!erence of... 1 cm 1/2 1 mm 0.1 mm 10 -16

10 -17

10 -18

Displacement spectral density (m/Hz Displacement 10 -19 1 k 10 k 100 k 1 M Frequency (Hz)

Figure 4.8: Laser frequency noise projection into apparent length noise for the Mephisto NPRO laser for three absolute arm length diﬀerences of the Michelson interferometer. Note that the 1/f projection is an approximation, which is less accurate for frequencies above 100 kHz.

of δs˜ = 3.546 10−18 m/√Hz. Thus, the arm length diﬀerence has to · be kept smaller than 1 mm or the laser has to have an active laser frequency stabilisation to suppress the frequency noise at 100 kHz. Frequency noise becomes important again as soon as the experiment uses a signal recycling mirror (see Sec. 5).

42 r omni ohitreoee rs uhta suppress a that fluc such laser arms, the tor interferometer fluctuations, both vacuum in common the are to contrast In ations. nt.Ti seuvln to sh equivalent ratio is phot their compared, This of are unity. densities noise spectral laser excitation technical power the and the noise which noise, in quantum situation, by a nated achieve To introduced. with siltr uta ncs fqatmso os.I assa causes It noise. shot a of quantum density force, of spectral ’classical’ case ment a in as cause te just intensity First, oscillator, laser two-fold. the is of noise intensity tuations laser of influence The yia I ausfrhg ult,cmeca ae sourc laser commercial quality, high for 2 values RIN Typical doe but dependent, frequency power are detected average noise the the on fre for Fourier depend a the sources by on the divided depends typically re as fluctuations RIN their power The by is power. detected characterised which (RIN), times noise often tensity o are fluctuation Lasers technical is intensity. noise of source important Another noise intensity Laser 4.3.4 osesand lossless qain(.8 sbs nesodwe acltn t lef its calculating when For understood best is (4.38) Equation ae powers laser if filled ipaeetnie hte rntteeflcutosaene are mean fluctuations a these fluctuati into offset, not Thus, translate or dark-fringe directly Whether detector. and power noise. photo out output displacement the read interferometer by the DC measured of of be case conce will In is power fluctuations intensity out. laser read of influence second The · 10 ∆ λ − | bs S = 7 I / ( < P 04n,i rewrites it nm, 1064 = √ ω zfrfeunisblw1Mz fteba pitris splitter beam the If MHz. 1 below frequencies for Hz ) | t | r bs 2 P = 0 W nodrt civ h aertoa higher at ratio same the achieve to order In mW. 100 | bs 2 S h rdc of product the , G − I P shot = ( ω · | r S I en h pcrldniyo h ae fluctu- laser the of density spectral the being RIN bs 2 0.505 6 ) x | | ( · ersnigteba pitrublnigis unbalancing splitter beam the representing , 1 > ω 10 ) . nuneadsprsino ae noise laser of suppression and Influence 4.3 | − ⇒ = ⇔ 10 r 2 c ∆ r | bs S Hz W ∆ 4π I ( = bs λ ω hc

∆ > · 10 ) I a ob reduced. be to has RIN | r ∆ > − m 2 bs 2 ∆ P q 43)wl nyb ful- be only will (4.38) Eq. , · bs . bs RIN | G · RIN ( ω · √ ) | · P √ , . P tn nthe on cting hia ﬂuc- chnical adside. hand t udexceed ould h laser the f nn the rning aiein- lative apparent displace- tuations sdomi- is quency, nshot on o fac- ion gligible verage sare es (4.37) (4.38) not s ons 43

Chapter 4 4 Michelson-Sagnac interferometer or dominant in the output spectrum depends on the laser quality and the amount of detected light in the interferometer output. The relative shot noise is

G 4πhc shot = . (4.39) P r λP If an optical power of Pout = 1 mW is to be detected, the technical laser noise at a RIN level of 10−7 is

−10 |SI(ω)| = 10 W/√Hz . The shot noise level at this power is (compare Eq. 2.12)

−11 Gshot = 1.93 10 W/√Hz . · If no other noise sources are present, the measured spectrum will thus not be dominated by shot noise at low frequencies but technical laser noise. Hence, performing precision measurements at these frequencies thus requires a stabilised laser. Laser stabilisation can either be done in active or passive schemes, as outlined in the following. Passive stabilisation Passive stabilisation schemes exploit the amplitude transfer function of an optical cavity (see section 5.1) , which can be modelled as the equivalent to an optical low pass filter. The laser is incident to an impedance matched cavity, which is kept resonant. Amplitude fluctuations are transmitted only within the cavity linewidth (see section 5.1). Passive stabilisation requires the cavity optical linewidth to be much smaller than the frequency at which the noise is to be suppressed, which turns out to be challenging in table-top experiments when noise at frequencies around 100 kHz is to be suppressed. At low frequencies, passive stabilisation schemes are inferior to active stabilisations. Active stabilisation Active stabilisation schemes consist of an electronic feedback loop and some power actuating device. The advantage of active stabilisation is that gain and resulting noise suppression is variable. The limitations for an active stabilisation are given by the speed of the electronics and the actuator in use. Also, at 10 MHz, each meter of electric connection already causes a phase loss of about 18 deg [107], which is why unity-gain frequencies above 1 MHz get more and more difficult

44 pcrlrneo 5Mz h nsei 5.Tegendse l dashed of green suppression The A 150. filter. is lowpass which finesse order m, first The 10 a is MHz. by length 15 approximation cavity of The range kHz. 100 spectral of linewidth optical an iue4.9: Figure hnfdbc oteEA.Ti ceehsaraybe sdin used been already has scheme pro This electronic EOAM. after the which [106]. to signal, back error fed an an then as signal used the ultr det is both an is between erence to difference light compared The laser is the reference. and current of arrangement, fraction this small of A transmission actuator. 4.1 power splitte beam a Fig. polarising as a in with depicted combination in is which (EOAM), used scheme amplitude fast a The a uses kHz, required. 100 was at beca isation suppression kHz noise sufficient some a to For actuator. limited piezo unity-g the be The of will resonances phase. stabilisation mechanical relative this the actuate of piez to A quency used port. be output can interferometer li mirror of other fraction the variable towards A mitted port. constructiv output interferometer to i the intensity close in laser operating the interferometer, stabilise auxiliary actively to possibility One th is stabilisation active stabilisations. frequencies, passive high At achieve. to b just or can finesse higher frequency means this which at linewidth, optical suppression narrow Additional more kHz. 100 at Power transmission (dB) -50 -40 -30 -20 -10 0 1 k 1 λ/2 rnmte oe hog nipdnemthdotclcavi optical matched impedance an through power Transmitted aepaei rn fa lcr-pi mltd modulator amplitude electro-optic an of front in plate wave Lowpass 1storder Cavity exact 10 k 10 . nuneadsprsino ae noise laser of suppression and Influence 4.3 100 k Frequency (Hz) 1 M aiylength. cavity 10 M 10 − 3 interference e yuigan using by s civdb a by achieved e eut nafree a in results sifro to inferior us Bi achieved is dB h strans- is ght n hw the shows ine esn is cessing h ref- the d ,served r, driven o ce in ected a-stable i fre- ain stabil- ywith ty s of use .It 1. 100 M 100 45

Chapter 4 4 Michelson-Sagnac interferometer

Mach-Zehnder stabilisation EOAM stabilisation

Piezo actuated mirror

λ 2

EOAM

PBS

Figure 4.10: Left: power stabilisation via a Mach-Zehnder interferometer. Right: high-speed setup with an EOAM. In both cases, one port of the beam splitter (PBS) is used to dump a part of the power, and a small pick-up in the output port is required.

4.3.5 Laser light preparation

In the experiment, an Innolight Mephisto non-planar ring oscillator, operated at a laser wavelength of λ = 1064 nm with an optical output power of 2 W was used. This type of laser source was characterized in different works before [106, 108], and was found to exhibit a low RIN level of about 2 10−7 √Hz at frequencies f< 1 MHz. The laser · relaxation oscillation at 1 MHz was suppressed by an integrated noise- eater function. The laser light preparation is shown in Fig. 4.11. To eliminate higher order optical modes, and hence, inject only a TEM00 mode into the interferometer, the laser light was spatially filtered by means of a mode cleaner cavity [109]. The cavity was stabilised on the laser frequency by a Pound-Drever-Hall scheme [110]: a phase modulation of 12.14 MHz was imprinted on the laser beam by an electro-optic modulator (EOM). The light in reflection of the mode cleaner cavity was detected with a photo detector, resonant at 12.14 MHz, and demodulated with the modulation frequency. The result error-signal was low pass filtered and used for actuation of the cavity length by means of a piezo actuated end-mirror. A Farraday-isolator was used to prevent a direct reflection of the interferometer (not shown in Fig 4.11) through the mode cleaner and to the laser. In case of perfect alignment, this beam would be transmitted through the mode cleaner in reverse di-

46 iue4.12: Figure 4.11: Figure h ih a vial o neto noteinterferomet the into thr injection transmission for After available clarity. was of light sake the the for shown not are ehia apiue ae os ssprse by suppressed is noise laser (amplitude) technical anfeunyi rud1Mz Below MHz. 1 around is frequency gain eitdi i.41.A cieniesprsinof suppression noise active stabili intensity An laser the 4.12. of Fig. performance rig stabili The the in on active depicted shown 4.10. the and by section Fig. previous stabilised of the was in power described scheme beam The rection. detector. photo the in ﬁlter high-pass electronic civda frequencies at achieved Noise power spectral density (dBm/Hz) 2WNPRO MephistoInnolight -110 -105 -100 -80 -75 -70 -65 -95 -90 -85 10k Free runningNPRO1064nm xmlr efrac ftelsritniystabilisation intensity laser the of performance Exemplary λ 4 ceai ftelsrlgtpeaain essadsteeri and Lenses preparation. light laser the of Schematic λ 2 Photo detector darknoise isolator Farraday 21 H ie o-asServo Low-pass Mixer 12.14 MHz 12.5 dB . nuneadsprsino ae noise laser of suppression and Inﬂuence 4.3 f EOM Stabilised 6 0 kHz. 100 100k Fast servo λ 2 EOAM Frequency (Hz) f = 6kz esniei eetddean due detected is noise less kHz, 16 PBS RIN PD 12.5 Pick-up 1 % Ba 0 H.Teunity- The kHz. 100 at dB er. HV uhtemd cleaner, mode the ough 1M λ 2 Locking PDLocking 12.5 h NPRO The . Mode cleanerMode gmirrors ng ainis sation Bwas dB tside ht sation 5M 47

Chapter 4 4 Michelson-Sagnac interferometer

4.4 Realisation of the Michelson-Sagnac interferometer

The first version of the experiment was set up in [76, 81] and was used in the beginning of this thesis. Some disadvantages could be identified and were taken into account during the re-design the interferometer. In this section a short summary of identified and solved problems is given and the improved interferometer design is presented.

Initial setup and related problems The Michelson-Sagnac interferometer was build up on a 60 cm diameter, 2 cm thick baseplate made of stainless steel. The baseplate itself was rigidly ﬁxed to the vacuum tank by three posts of stainless steel, sized 30 cm with a 3 cm radius. The main problems identiﬁed with this setup were

Loss of interferometer alignment in vacuum. The rigid connec- • tion exposed the interferometer to deformation and bending of the vacuum tank during evacuating the system. The misalignment could only be compensated partially by remote control of central optics.

The rigid connection also allowed undesired introduction of acous- • tic coupling. All pumps had to be shut down and a quiet laboratory was mandatory to obtain useful output spectra.

Disadvantageous mechanical resonances of the interferometer • baseplate. The output spectrum contained beat signals around the membrane resonance frequency, separated by 400 kHz and 1 kHz as depicted in Fig. 4.13. These frequencies coincided with mechanical resonances of the baseplate and the posts underneath.

48 iue4.14: Figure 4.13: Figure sdpce srdln,teitreoee uptba sde is beam output interferometer the line, line. red as depicted is ftosmti ek ihaba rqec f40H n H.Righ vacu kHz. evacuated 1 and and closed Hz with 400 speaker of frequency audio beat an acc a by with is excitation peaks resonance symetric mechanical two fundamental of membrane The tion. Noise power (dBm) -120 -110 -100 -90 -80 -70 -60 -50 -40 -30 to TMP andScroll 100 Satellites nta nefrmtrisd h aumtn.Teincident The tank. vacuum the inside interferometer Initial no excitation by speaker . elsto fteMcesnSga interferometer Michelson-Sagnac the of Realisation 4.4 102 Membrane resonanceMembrane et yia uptpwrsetu ro ointerferometer to prior spectrum power output Typical Left: Frequency (kHz) Optical viewport 104

60 cm Beam splitterBeam 106 400 Hz 1 kHz 108 Folding mirror 110 Steering mirror -120 -110 -100 -90 -80 -70 -60 -50 -40 -30 Electric feedthroughElectric 100 400 Hz 1 kHz audio speakerexcitation 102 SiN membrane Frequency (kHz) itda e,dashed red, as picted Steering mirror 104 mchamber. um maidb set a by ompanied 106 :external t: ae beam laser 108 isola- 110 49

Chapter 4 4 Michelson-Sagnac interferometer

Membrane holder 1“ Steering 0.5“ optic mounts Mirror Newport Agilis MNv050

1“ Recycling mirror mount

1“ Beam splitter

Interferometer baseplate

Isolation Lower baseplate stages

Figure 4.15: CAD Design for the Michelson-Sagnac interferometer. It consists of three double-stage isolation footers. On top of these, a heavy plate ensures suﬃcient weight to push in the viton material. The baseplate itself is then mounted on a 6 cm thick baseplate made out of stainless steel.

4.4.1 Interferometer design The new interferometer design had to fulﬁl several requirements. First, the interferometer should have a reduced size. An interferometer arm length of less than 10 cm was aimed for. A compact design was expected to exhibit a higher tolerance against small misalignments of the central optics due to reduced leverage. Furthermore, in case of signal-recycling (compare chapter 5) a smaller interferometer results in a larger free spectral range, and thereby bigger optical linewidth with same recycling gain. The interferometer baseplate was required to be as ﬁrm as possible. Moreover, it should not to be rigidly connected to the vacuum tank. Remote control for the central optics (except beam splitter) was required to maximize the interferometer contrast under vacuum conditions. During a supervised masters thesis [111], the interferometer was redesigned according to these requirements. A schematic of the revised interferometer is depicted in Fig. 4.15

50 etdvacompressed via nected ntpo he apn otr tidioainsae,whi stage), isolation thin (third on footers placed were damping selves three of top on h ev aso h bv base above eigenfrequency was the resulting The of mm. 5 plates. mass heavy by of compressed the height were cylinders a These and diameter, h otr,respectively. footers, in cylinders the viton and plates) base the sltn ffc sepce tfrfre- for at quencies expected is effect isolating es epciey h ytmehne eoatexcitati resonant frequencies enhances at system excitations The suppresses respectively. ness, aeil on’ ouu,wihwsdtrie to determined was which modulus, Young’s materials a ie by plate given three base was lower the and footers between damping connection The du compressed, by wi stages with isolatio orubber separate construction connecting The the by suspending but plates. by dulums, achieved base not individual was two case is vertical of four-level consisting a on tem, built was interferometer new The isolation Vibration 4.4.2 uee with buffered was stage their individual the screws, to connection the a by avoid connection To rigid screws. ap- auxiliary was in by force plied and additional plates footers, base the two the the on between pressure sufficient For stage. isolation lar [111]. ihrsnnefrequency resonance with as and mass, ie lc fauiimadwspae ntpo eodbase second a of was top mass on placed combined was Their and aluminium plate. of block sized .) oedtie ecito bu h slto a be 12 a can of isolation consisted plate the base about interferometer The description detailed [111]. in more A 3.1). ω 0 A / ( Viton > f 2π and D . elsto fteMcesnSga interferometer Michelson-Sagnac the of Realisation 4.4 ) Viton h siltrsrn constant spring oscillator the 4 zfrti particu- this for Hz 140 ≈ Viton ahcneto hsatda pigms damper, spring-mass as acted thus connection Each . L 0 z uhta an that Such Hz. 100 r h compressed the are yidr f7mm 7 of cylinders ig i aeof case (in rings Viton Viton ω res 2 ig.Telwrbspaewsstacked was baseplate lower The rings. Viton = ic lws,fut slto stage). isolation fourth (lowest, discs D/m m ω Viton emc.Teioainhsbeen thesis masters has [111] a isolation during The developed and seimics. acoustics against isolation sive 4.16: Figure = exc where , 12 D > . ufc raadthick- and area surface = g ohwr con- were Both kg. 4 √ EA/L 2ω m ceai ftepas- the of schematic A Interferometer baseplateInterferometer res stesuspended the is Smart Smart Table ST-series E Here, . cmaeFig. (compare ≈ × Laboratory oorLaboratory First baseplate lto sys- olation 5 17 . N 8 hthem- ch nthis in n n,but ons, tl flu- ctile × E epen- re / cm 3 found sthe is mm 51 2 3

Chapter 4 4 Michelson-Sagnac interferometer

-40 Bottom of vacuum tank -50 Interferometer Dark noise -60 -70 -80 -90 -100 Amplitude (dBV) -110 -120 -130 50 100 150 200 250 300 350 400 450 Frequency (Hz)

Figure 4.17: Seismometer measurement on top of the interferometer baseplate. The measurement shows that above 140 Hz the detected signal is signiﬁcantly reduced.

To measure the performance of the isolation, seismometer measurements were carried out on the optical table and compared to measurements on top of the interferometer baseplate. The device used was a Model L-4 Seismometer by Mark Products with a measurement range of < 400 Hz. The result for the isolation in z (vertical) direction is depicted in Fig. 4.17. The measured signal is reduced for frequencies above 140 Hz by more than 30 dB, the electronic dark noise of the measurement is depicted as black trace. For low frequencies, no suppression was expected.

52 ne n ue yidr,wihwr le ntodffrn s two different two of on consisted glued It were 25 which 4.18. cylinders, outer Fig. [ and in in inner separati depicted suggested is holdered, as spatially, holder was membrane membrane d membrane the the the and to actuator Thus, piezo perpendicular piezo the plane extension. the to r its in applied membrane shortened of was is the piezo high-voltage of a a modification cause m when induced w between frequency stress this connection to nance rigid to direct, led previous a piezo piezo that ring and out a turned to It th glued [81]. along was membrane membrane the the of path, position microscopic the actuate To holdering Membrane int 4.4.3 fixed was ring outer hold The figure. membrane the assembled of ring. side The right inner later. the positioner to controlable glued is membrane 12 oiinr uhta h iz oe nyteinrcy inner the only thereby. moved membrane controllable the piezo remote and the extending a that when within such fixed positioner, piez was [112] the cylinder both outer through The driven was cylinder. which outer the cylinder, inner the of n n ue ig hc r le obt ie fa25 a of sides both to glued are which ring, outer one and 4.18: Figure Outer ring Complete membrane holder SiN membrane . mrn iz.Temmrn a le oahlo extension hollow a to glued was membrane The piezo. ring mm 4 25.4 mmPiezo . elsto fteMcesnSga interferometer Michelson-Sagnac the of Realisation 4.4 ceai ftemmrn odr h odrcnit fone of consists holder The holder. membrane the of Schematic Extension Outer ring Newport MN100nv6Newport Newport LS 25v6Newport . mrn iz.The piezo. ring mm 4 ri eitdo the on depicted is er remote a o 1.The 81]. embrane Newport dso a of ides irection beam e gthe ng drilled and o linder be- , inner eso- ork 53

Chapter 4 4 Michelson-Sagnac interferometer

a) b) c)

Figure 4.19: Examplary situations during the Sagnac interferometer alignment. a) Ideal situation: dark output, incident and reflected beam are folded. b) If both steering mirrors a rotated in the same direction, this will result in two parallel beams inside and outside the interferometer. c) The steering mirrors are rotated by the same angle as in b), but with different orientations. This can result in a dark output, but incident and reflected beam are pointing in different directions.

4.4.4 Interferometer alignment

This section addresses the Michelson-Sagnac interferometer alignment procedure, as it is crucial for a potential cavity arrangement. The setup requires two iterative alignment steps: ﬁrst, the internal alignment of the Sagnac interferometer is completed (Step 1). Second, the external (lateral) mode matching can be done.

Step 1: Sagnac alignment The laser light is spatially filtered by a mode cleaner cavity and is incident on a balanced beam splitter under 45 ◦. After confirming that the beam splitter does not al- ter the vertical beam direction, the steering mirrors are installed, giving rise to two counter propagating beams and the Sagnac interferometer topology. A CCD camera in the interferometer output can be used to identify misalignment of these mirrors. Fig. 4.19 depicts exemplary situations during the initial setup of the Sagnac interferometer. It illustrates how both steering mirrors are used to overlap both beams. Step 1 is completed if destructive interference in the output port is established, and both, incident and reflected beam are overlapping at every point along their path.

Step 2: Lateral mode matching Step 1 guarantees that internal degrees of freedom are set correctly. In the second step, the lateral beam waist must coincide with the later membrane position. This can be checked by verifying that the interferometer folds

54 b) a) oiin as ieadba ipaeet )Witsz as size Waist b) displacement. par beam steeri beam z and Sagnac all size for the waist reference of position, alignment steering as served coarse cavity for cleaner used were apertures ehtb h emwe en lcdisd h interferomete the inside placed being when membran beam the the ﬁxed, by are hit alignment be of degrees internal the Once lines. black dashed, as indicated iue4.20: Figure

rmteiiilwito h oecenrcavity cleaner mode the of waist initial the from Waist size (mm) 0.5 1.5 Mode cleanercavityMode w =370µm,z0 emi efcl oddbc.Teiiilpstoso h r the of soft the positions with initial calculated JamMT the be The when approximately maximal, can back. matching lenses be folded quired will mode It perfectly the at beam. is shifting the resonant by beam of kept maximised path ligh the is be along reflected (which can lenses the cavity It of cleaner time). fraction mode all a incide the 1), the enter Step to will in matched clean already mode described is initial (as beam the beam reflected by the provided if is coarse check cavity: a auxiliary c just An interferometer is the interference) tor. destructive Since of shar size. (degree beam and trast reflected position and waist incident same that the such beam, incident the 0 1 2 0 Iris 1Iris . elsto fteMcesnSga interferometer Michelson-Sagnac the of Realisation 4.4 )AinetstpfrteMcesnSga interferomete Michelson-Sagnac the for setup Alignment a) 0.25 [113]. f =-167mm R =-75mm z =86.4cm 0.5 Forward beam PD Alignment check Mode matchingMode lenses 0.75 R =100mm f =222mm z =99.2cm 1 z (m) 1.25 ω 1.5 = 370 Iris 2Iris 1.75 PD contrast check µ mtr uha waist as such ameters .Ln oiin are positions Lens m. gmros mode A mirrors. ng ucino distance of function Waist size 2 CCD Camera .Toiris Two r. z =220.5cm w =251µm 2.25 .Since r. a to has e indica- ware on- 2.5 55 nt er e- e t

Chapter 4 4 Michelson-Sagnac interferometer

6 Measurement by Ring-down 10 Theoretical model

10 5 Quality factor Quality 10 4

Pirani gauge Penning gauge 10 3 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 Pressure (mbar)

Figure 4.21: Measurement of the mechanical quality factor Q of a membranes fundamental oscillation mode for different vacuum qualities, as shown in [76]. Only for pressures P 6 4 · 10−6 mbar the Q-factor is not limited by residual gas damping. no membrane positioning in the plane perpendicular to the beam is possible, the incident beam has to be translated by an external set of steering mirrors. This is possible since the Sagnac interferometer is indifferent to pure displacement of the incident beam (as long as no beam clipping occurs). The additional degrees of freedom of the membrane (pitch, yaw) can cause misalignment of the Michelson mode with respect to the Sagnac mode, such that a fine alignment of the membrane is required. The effect of a misalignment of the membrane is depicted on the right side of Fig. 6.14.

4.4.5 Vacuum requirements

The mechanical quality factor Q of the high-stress silicon nitride membrane oscillation modes is around 106, but can be dramatically reduced if the oscillation is damped. Major damping is caused by interaction with gas molecules. To not limit the Q-factor, a vacuum environment for the system is necessary. Fig. 4.21 illustrates the dependence of mechanical quality factor and environmental pressure used in the experiment. An analytical relation between residual gas pressure and mechanical quality factor of the oscillator is given in [114, 115].

56 where rn thickness, brane constant, empt ssona e ie h nefrmtrotu dir output interferometer the line, red as shown is path beam AG-LS25V6 ilis eeb wthdo.Tepesr nietevcu ytmdi system th vacuum between the valve inside pressure a 1 The exceed data, off. pum mechanical switched take both be and To closed were was TMP the 20’. and chamber plus vacuum ’VacIon pump getter emitie t0 at maintained be finally, and nitrogen) case this (in gas damping the of os u oaniyotu fahg-otg mlfir Figur linea amplifier. additional high-voltage An a p of interferometer. additional assembled output no introducing the noisy depicts not a with to were steppers, due they piezo noise Thus, are positioners input. The voltage closed. moun chamber were holder AG-MN50NV6 membrane positioners ilis controllable the the remote and opening compatible, mirrors vacuum thereby into all (and system Hence, the mis again). of the corrected disassembly be a to possible without interferom not the was of it interferometer. alignment components, tical the the over within control exper occur remote the without to of found version was initial misalignment the in significant contrast a interferometer caused chamber interferometer the Evacuating alignment Remote 4.4.6 h rsuei h aumtn a ob eo 4 below be to had tank vacuum the in pressure The pressure. ufcs,tepesr a tblsda hsvleb a by value this at stabilised boiled was was pressure hydrogen residual the (when surfaces), pumping of hours several i-resrl uppeeautdtesse o2 to [81] system thesis master the a pre-evacuated during pump up set scroll system, oil-free c pump vacuum two-stage resona the evacuating a fundamental by by achieved factor was this quality experiment, mechanical the the limit not hl unn,ti uppoie rsueo 3 of pressure a provided pump this running, While Varian ρ P50 ub oeua upwsue ntescn stage. second the in used was pump molecular turbo TPH520M . steseicms ftemmrn , membrane the of mass specific the is 4 T · . elsto fteMcesnSga interferometer Michelson-Sagnac the of Realisation 4.4 10 steaslt temperature, absolute the is − 6 a ntle nenahtemmrn odr The holder. membrane the underneath installed was br au elblwterqieet n could and requirements the below well value a mbar, f 0 . 7 Q stersnnefrequency, resonance the is 12 oalwfrrainetwt h vacuum the with realignment for allow to [112] · 10 = − 7 π 2 bratrlne upn periods. pumping longer after mbar 2 3 ρHf 0 s M RT mol M p 1 mol R H , stemlrweight molar the is steuieslgas universal the is · eoe h mem- the denotes 10 · − 10 · 7 10 − epr Ag- Newport p br After mbar. Varian mn.The iment. − 2 stage r yreduced ly stegas the is 6 alignment br A mbar. trsop- eter’s brto mbar cinis ection c.In nce. hamber Hence, osition ffthe off 4.23 e (4.40) not d high- An . tank ion- Ag- ted 57 ps e

Chapter 4 4 Michelson-Sagnac interferometer

Ion-getter pump

Penning Pirani gauge gauge 1 mbar Valve 3

Valve 1 Valve 2 -6 10 mbar

Electrical Feedthrough -2 3 10 mbar 10 mbar Optical Feedthrough Turbo molecular pump Scroll pump

Figure 4.22: Vacuum setup, [81]. A turbo-molecular pump (TMP) provides a −6 vacuum with Pvac < 10 mbar. The high-pressure side of the TMP is pumped via a scroll pump. For measurements, the valves 1 and 2 were closed, and all pumps except for the ion-getter pump were switched oﬀ. shown as red, dashed line. Full remote access allowed to restore the high initial interference contrast of C = 99.7 % in the evacuated vacuum tank. Thereby, the technical laser noise in the readout could be suppressed signiﬁcantly. A measured fringe of the interferometer is presented in Fig. 4.24.

4.5 Interferometer calibration

This section addresses the calibration of the interferometer. The output is analysed in form of a power spectrum, which has to be converted to an equivalent displacement spectrum. Diﬀerent calibration methods can be used to achieve this.

4.5.1 Calibration via the interferometer fringe In case of DC readout with one single photo detector in transmission of the interferometer, the measured output power spectra can be calibrated via the interferometer fringe. Equation (4.11) relates the interferometer output power (and output power derivative) to the differential microscopic membrane position. To calibrate a spectrum, a saw tooth voltage U(t) is applied to the membrane piezo high-voltage

58 hc sgudwt ml rpof drop small a with glued is which airto a oacutfrteeetoi rnfrfunc anal transfer spectrum electronic a the into for fed account is detector to me photo has the the Calibration of of density output spectral AC displacement the linear the spectrum obtain To displacement membrane the Obtaining 4.5.2 measurement). each eoee uig ievra h ipaeetprdetecte per displacement the nm versa, in Vice tuning. ferometer h ecldtertclcrei eie ihrsett th to V respect of ratio with a derived yielding is position, curve brane theoretical occur rescaled power t The of output independent maximum are ratio. that and splitter fact positions the membrane equidistant of for use makes one Here glsAG-LS25V6 Agilis AG-M50NV6 seta opnnsaelbld h ebaei one i mounted is membrane The labeled. are components Essential 4.23: Figure ilepeso o h displacement the wi (4.11) for expression expression analytical from mial non-linear the infered ﬁtting the be by to can voltage displacement plied Due membrane time. actual interferomete the in thereby piezo, periodically and ’scanned’ position is ing membrane The ampliﬁer. / skonfrec ebaepsto wihi eoe for denoted is (which position membrane each for known is V htgaho h eeindMcesnSga interfero Michelson-Sagnac redesigned the of photograph A on,wihi uni tahdo o falna iz stepp piezo linear a of top on attached is turn in which mount, h ne hw lsrve oa0 a to view closer a shows inset The . VacSeal / mdslcmn o ahinter- each for displacement nm x nec corner. each in . nefrmtrcalibration Interferometer 4.5 ( U ( t )) otemaue fringe. measured the to . 5 × t a nto 0 . cm 5 epr Agilis Newport hapolyno- a th 2 ino the of tion t fthe of ity membrane, voltage d ebeam he mbrane, mem- e h ap- the rstage er tun- r meter. yzer. 59 s

Chapter 4 4 Michelson-Sagnac interferometer

14 Fit 0.15 12 0.1 10 0.05 8 0 6 Volt/nm -0.05 4 Output voltage (V) Output -0.1 2 -0.15 0 λ /8 λ /4 3/8 λ λ /2 5/8 λ Membrane displacement

Figure 4.24: Interferometer output signal detected with a single photo diode (grey dots). The theoretical output fringe (black solid line) was rescaled to account for piezo non-linearities. The dashed green line depicts the derivative of the rescaled theoretical output fringe, relating the membrane displacement to change of detected output voltage for each membrane position detector. In case of the spectra presented here, the AC path featured two ﬁrst-order high pass ﬁlters with corner frequencies f0,i and a proportional electronic gain, such that the ratio G between DC and AC output voltage for each frequency f was given by

f f U G(f)= AC = Gain f0,1 f0,2 , (4.41) UDC · 1 + ( f )2 1 + ( f )2 f0,1 f0,2 q q where Gain = 23.4, f0,1 = 16.67 kHz, f0,2 = 1.52 kHz which results in G = 23.04 at f = 100 kHz. The spectrum analyser was set to measure dBm. In this case, the input power level is compared to 1 mW with an input impedance of R = 50 Ω of the spectrum analyser.

L P U2 p = 10 log , P = (4.42) dBm 1 mW R U = (R 1 mW)1/2 10 Lp /20dBm ⇔ · ·

Here, Lp is the noise level in dBm/Hz. For another measurement resolution bandwidth (RBW), Lp is to be rescaled Lp = Lp −10 log RBW.

60 iiyaon h udmna ehnclrsnnefreque displacemen n resonance achieved mechanical laser The fundamental 3 technical the by line). around (dashed limited tivity noise is shot sensitivity optical The bilisation. on ob tbea 0 interferomete at the stable of be mid-fringes to ca and found was dark- displacement multiple peak for The for marker method. calibration the calibr fringe were of which the spectra displacement d displacement untouched The from determined were was piezo measurements. and following amplifier all voltage high amplitude, xiaino h siltrwsgnrtdb pligasin a applying of by of generated quency was a oscillator Therefore, calibrated the introduced. of a was excitation fringe, peak interferometer marker the displacement scan fined calibrati to a need provide no to and with procedure calibration the ease peak To marker a using Calibration 4.5.3 iue42 hw airtditreoee uptspect output contrast interferometer interferometer calibrated refined a shows 4.25 Figure ato eoac,btlsradeetoi aknieaen at are marker dark-noise calibrated electronic A and line, ble. laser dashed but grey, resonance, as off plotted nant is level noise shot calculated o ihotcliptpwr,a h ro eu a limited powers was input setup for prior limite noise the noise as laser shot powers, nical a input achieve to optical was high thesis for this sensi the reduced sho of with be goal but powers, first will input measurement optical ( displacement low power for the optical limited Thus, detected the (4.39)). on technical depends Eq. of noise ratio shot the optical chapter, to previous the in detector explained photo As single a with Readout 4.6.1 detection. homodyne d balanced photo a single with a as with well analysed as was output interferometer The sensitivity displacement Achieved 4.6 offset. voltage piezo a by caused displacement ouaint h ebaepeoatao.Mroe,teob the Moreover, actuator. piezo membrane the to modulation estvt iie to limited sensitivity . 6 · 10 − 16 m / f √ = zfra pia nu oe of power input optical an for Hz 2 H otemmrn iz cutr Frequency, actuator. piezo membrane the to kHz 128 3 · . 10 9 · − 10 f 15 = − . civddslcmn sensitivity displacement Achieved 4.6 m 12 2 H a nrdcdb pliga applying by introduced was kHz 128 C / √ ,wt oprevbevrainof variation perceivable no with m, ≈ P Hz. in 99.7 > 50 n ae mltd sta- amplitude laser and % µ ,wt displacement a with W, P in = 0 W The mW. 100 n sdomi- is and sia fre- usoidal iiy The tivity. nmethod on ,adwas and r, ae noise laser edout read d tnegligi- ot u with rum tdwith ated i de- a via ieand oise compare c was ncy ytech- by librated sensi- t defined etector noise t tained uring peak 61

Chapter 4 4 Michelson-Sagnac interferometer

1/2 -11 Thermal noise 10 Shot noise calculated Unstabilized 10 -12 Stabilized Dark noise Sum 10 -13

-14 10 Calibrated marker peak

10 -15

10 -16 Displacement spectral density (m/Hz ) Displacement 115 120 125 130 135 140 Frequency (kHz)

Figure 4.25: Membrane displacement spectrum around the fundamental mode. Off resonance, the spectrum is limited by laser intensity noise. Shot noise as well as detector dark noise are below. At the very resonance, the spectrum is dominated by the thermal noise (dashed red line) which here was calculated for Q = 5.8 · 106 and a temperature of T = 300 K. Off resonance the highest displacement sensitivity is 3.6 · 10−16m/√Hz. This measurement was taken before the final displacement of the marker was fixed, such that the marker peak here corresponds to 1.25 · 10−12 m/√Hz. spectra did not contain satellite peaks. This was attributed to the performance of the vibration isolation described in Sec. 4.4.2. Despite its simplicity, the DC readout has three drawbacks. First, it only allows for measuring the amplitude quadrature of the output field. Second, when obtaining the output power spectrum, it is not clear whether or not the spectrum is limited by shot noise. This can be checked by measuring the output spectrum at high frequencies f > 10 MHz and comparing the relative power noise levels. This measurement relies on a flat (or precisely known) transfer function of the photo detector, however. The shot noise equivalent displacement noise can also be calculated, according to Eq. (4.24), but this method requires knowledge about the optical loss during detection. Third, the measurements have to be taken off dark-fringe to provide some local oscillator power. Due to the finite contrast C (compare Fig. 4.6) the offset from interferometer dark-fringe must exceed a certain value. Thereby technical laser amplitude noise couples into the measurement. Moreover, this reduces the interferometer amplitude reflectivity rifo, which is wanted to be as high as possible for the

62 emwsajsal yapeoatae irr eaieph relative A mirror. and actuated oscillator piezo a local of m by difference between following adjustable the phase ba was in a relative used beam readout, The and DC up a set surements. was of detector drawbacks homodyne mentioned anced above the resolve To readout homodyne Balanced 4.6.2 5. Chap. see techniques, recycling of adaption whereas o inlwsue oatvl tbls h eaiephase relative the T stabilise actively signal. to error used appropriate s was an resulting signal provided The ror and signal. filtered pass modulation signal low the the mixing of wit by ’copy’ the achieved demodulated electronic is was quadrature, subtr This detectors amplitude the frequency. photo and the modulation homodyne MHz, to 10 both with lock of modulated To signal i phase was quadrature 4.26. oscillator amplitude cal the Fig. trans measuring zero in homo DC for the picted the scheme of at re output locking measured the DC is The quadrature, the quadrature this by phase as provided the detector, directly For is stabilized. signal error be to phase had relative signal the o quadrature field, particular put a observe To quadrature. amplitude inl rvdn nerrsga o eaiepaeof phase relative a w for analyse demodulated signal was spectrum error currents a an filte photo into providing AC signal, fed both pha The was of A beam. detector subtraction oscillator direct homodyne splitter. local the the beam of on output balanced imprinted an a was beam) MHz on (signal 10 overlapped field are output oscillator interferometer The quadrature. iue4.26: Figure 4 -10mW Local oscillator 10 MHz modulation10 MHz 30 kHz Low pass!lter 2nd order θ = okn ceefrterltv phase relative the for scheme Locking θ π/2 = π orsod oamaueeti h orthogonal, the in measurement a to corresponds EOM eest esrmn ntepaequadrature, phase the in measurement a to refers θ . civddslcmn sensitivity displacement Achieved 4.6 DC ewe h oa siltradthe and oscillator local the between Phase shifter θ - θ Balanced homodyne detector θ AC coe oteamplitude the to (close) = θ π/2 Spectrum analyzerSpectrum Signal beam<0.5 mW t h modulation the ith . togrlocal stronger a d emdlto of modulation se e,differential red, ,weesthe whereas r, ga was ignal h out- the f θ ihan with ient. ythe by i er- his quired signal acted the h de- s dyne ase ea- lo- 63 l-

Chapter 4 4 Michelson-Sagnac interferometer

UDC 1

π 3π θ 2 π 2 2π −1

Figure 4.27: Homodyne detector DC signal (green) and demodulated error signal (black dashed line) in arbitrary units. The amplitude quadrature refers to a relative phase of θ = π/2, whereas the phase quadrature is measured at θ = 0 (mod π). piezo actuated mirror. The modulation/ demodulation technique is described in section 5.2.1. A calculated error signal is shown in Fig. 4.27. To avoid beam pointing and associated interference reduction because of lever, the actuated mirror was positioned close to the homodyne beam splitter. The overlap of both optical fields on the homodyne beam splitter exhibits an interference contrast Chomo, just as in case of the Michelson- Sagnac interferometer. Measuring the maximum and minimum power in each of the beam splitter output ports Pmax, Pmin, yields the same expression for C as given in Eq. (4.31). For equal optical input powers, the contrast can be limited by mode mismatch of the two beams. For the detection efficiency, a non-ideal interference contrast has the same effect as optical loss in the path [67]

2 Power Loss = Chomo . (4.43) In the experiment, the mode matching achieved a value of 90 %, causing 19 % optical loss. In the abscence of squeezing, the loss was found tolerable. Together with a homodyne detector eﬃciency of about 80 % for the photo diodes in use, an overall read out loss of 35 % was assumed. The loss results in an increased relative shot noise level. With- out the loss associated with the detection, the displacement sensitivity would have been better by a factor of 1/√0.65 1.24. ≈ The homodyne detection requires a small coherent amplitude of the signal beam (compare Eq. (2.42)). Scanning the interferometer is thus not possible, as was done in case of DC readout and fringe calibration. The output spectra were thus calibrated by the calibrated marker peak. The resulting output spectra are depicted in Fig. 4.28. All measurements were limited by optical shot noise oﬀ

64 eoac o h ebaeseictos htnielimi noise shot 1 A of sensitivity specifications. placement membrane calcu the theoretical the for matched resonance noise thermal The resonance. dependence slight a where traces to the Due power, Hz). 100 input grey. (about res optical dark frequency thermal to plotte in off frequency is plotted limited resonance measurement noise is each shot noise for is (gr dark measurement level powers Each noise input shot lines. optical The different for traces). spectra the calibrate eeto.Amre at marker A detection. iue4.28: Figure for iie ipaeetmaueetfrfrequencies for sho measurement a demonstrated displacement span limited frequency broad a over Measurements n or epciey h htniepwrlvlwsobser was level power noise shot noise, The by shot drop fa by respectively. a si limitation by four, consecutively the the and attenuated verify blocking was To power by opt oscillator dump. measured to local beam normalized was a were which with measurements level, path The power 4.29. noise shot Fig. in seen be yavraino h ooyeraotangle readout homodyne the of variation a By hw 8]ta h siltrtemlniei rsn only present t A is field. noise output interferometer thermal the oscillator of quadrature the The amplitude that field. optical [85] the shown of quadratures arbitrary investigate Displacement spectral density (m/Hz1/2 ) P 10 10 10 10 10 10 10 in -16 -15 -14 -13 -12 -11 -10 120 = 3 0 Wwsachieved. was mW 200 B sepce hnhligtelcloclao power. oscillator local the halving when expected as dB, airtditreoee uptsetu ihblne h balanced with spectrum output interferometer Calibrated Dark noise Shot noise Calibrated marker 200 mW 20mW 50mW 125 f @ 128kHz = . 9 2 H fwl-nw ipaeetwsue to used was displacement well-known of kHz 128 · 10 . civddslcmn sensitivity displacement Achieved 4.6 TN +SN − Frequency (kHz) 16 Thermal noise m 130 / 200 mW √ zfra pia nu power input optical an for Hz SQL T =300K fundamental mode Thermally excited θ > f e,prl n blue and purple een, ti osbeto possible is it , 135 nne Electronic onance. nge (dashed) grey in t lgtysitdin shifted slightly 0kz scan as kHz, 60 fmechanical of eyi was it reby tro two of ctor ain on lations riangular omodyne e dis- ted noise t e to ved nthe in gnal ical the 140 65

Chapter 4 4 Michelson-Sagnac interferometer

50 Homodyne spectrum 200 mW Shot noise 40 Detector Dark noise 30 excess noise 20

Noise power (dBm) 0 ... -10 membrane mechanical modes

-20 10 k 100 k 1 M 10 M 20 M Frequency (Hz)

Figure 4.29: Balanced homodyne output power spectrum, normalised to the mean optical shot noise level. The spectra are shot noise limited with Pin = 200 mW for frequencies f > 60 kHz. The electronic dark noise clearance is about 15 dB in the relevant frequency range. voltage was applied to the piezo on the phase shifter mirror and the a spectrum analyser was set to measure with a minimum frequency span at the mechanical resonance frequency. The result is shown in Fig. 4.30. At t = 2.08 s, the detected noise power (shown as green trace) is maximum. This corresponds to a measurement in amplitude quadratute. At t 1s and t 1.5 s, the measured noise ≈ ≈ power drops to the shot noise level, which was measured by blocking the interferometer output port, and is shown as grey trace. This corresponds to a measurement in phase quadrature. The electronic dark noise of the measurement is shown as black trace. The ability to investigate arbitrary quadratures is necessary to perform opto-mechanical entanglement analysis [116] or to detect non- classical properties of the interferometer output light, such as ponderomotive squeezing [117].

4.7 Chapter summary

In this chapter, the Michelson-Sagnac interferometer topology was presented, and a detailed explanation about interferometer input and output optical ﬁelds was provided. The achievable interferometer sensitivity was investigated under consideration of quantum and classi-

66 ihlo-ancitreoee nhg aum( vacuum high in interferometer Michelson-Sagnac a o engetd nitreec otatof contrast interference An noi neglected. frequency be and intensity not laser can namely sources, noise cal thi for RBW orth The the trace. in black mea kHz. as noise 10 shown a thermal is large, noise to dark and theoretica corresponds Electronic quadrature the This phase shows in line noise magenta detected. The is quadrature. power amplitude noise maximum the oe rp ont h neednl esrdso os le noise shot t measured quadrature, independently at phase the In happened to time. This of down t function drops a to as power applied shown voltage is triangular thus, a and by modulated periodically os iie ipaeetsniiiyo 1 of provide and sensitivity out displacement read the i limited in w the noise modes dar method suppressed transversal on order detection calibration higher homodyne operated peak of balanced marker was The a interferometer purpose, troduced. the d this and homodyne For high balanced up, realised fringe. a set a Next, was of scheme means stabilisation. by Tec intensity decreased developed. laser further be could was procedure noise standard laser a ﬁ a which requires for tank both, vacuum ment, nois combining the when in laser interferometer contrast Michelson technical high and of this achieve inﬂuence To the readout. reducing achieved, was sepce hnsiligtedtcinfo ta ih,a light, stray from detection the freq low shielding the when of reduction expected further A is light. in of the state sche of output reconstruction detection ter tomographic homodyne a balanced perform to The necessary kHz. 60 above quencies out read homodyne The trace). (green membrane SiN the of quency 4.30: Figure Noise power (dBm) -100 -30 -20 -10 -90 -80 -70 -60 -50 -40 (phase quadrature) No thermal noise 0.6 esrdniepwra h udmna ehnclresonan mechanical fundamental the at power noise Measured 0.8 t = and s 1 1 t 1.2 = 2 (amplitude quadrature) Maximal thermalnoise . 8s napiuequadrature amplitude In s. 08 1.4 Time (s) 1.6 . 9 · . hpe summary Chapter 4.7 10 1.8 p − epaesitrpiezo shifter phase he 16 C = oe o vacuum for model l esrmn was measurement s gnlquadrature. ogonal m = 1 emaue noise measured he 2 . / e ge trace). (grey vel 0 99.7 √ · ec noise uency zfrfre- for Hz 10 ea these as se terferome- ueetin surement ( 2.2 dditional t fthe of % phase − ealign- ne nthe in e shot a d etection ≈ 6 Sagnac -speed mpact hnical sin- as mbar) eis me efre- ce 1 . s 5 θ 2.4 67 k- is ) ,

Chapter 4 4 Michelson-Sagnac interferometer improvement can be expected with reduced optical path lengths, as beam pointing on the detector and the interferometer beam splitter can thus be reduced, due to a reduced lever. The fact that the obtained spectra were limited by thermal and optical shot noise only implies that the interferometer sensitivity can also be increased by injecting a squeezed vacuum state in the interferometer output port.

68 rnmsiiyof transmissivity e iuto r ie.Tefloigcluain aeus make calculations following The f requirements given. amplitude and are cavity situation optical e the ter resonantly of (signal-recy also cavity description to the short inside a and generated mirror bands initial side signal the transmission paragraph. complete on next a field incident the for power optical in allows m input shown arrangement two the be the of result, will resonance, consists as to cavity enhanced, brought optical resonantly is linear cavity a the of If case simplest The cavity Optical 5.1 53]. [52, coupling mechanis dissipative shap This as the o position. to cavity, modifications micro-oscillator direct the linear by to a finesse due ity resultin complex, to more the contrast is for In spring model optical a spring. presents optical and plex mirror circumstanc recycling certain the whic under by interactions mirror opto-mechanical only the recycling port issues a also output chapter as the addressed, in briefly signal The is the gain. cavity recycling the signal in as mirror loss such discussed benefits additional is possible an to port gards by output interferometer interferometer the Michelson-Sagnac ca of optical an extension of concept the basic the provides first chapter This 5 a interferometer Michelson-Sagnac Signal-recycled in hpe 5 Chapter nieto irroewt mltd eetvt and reflectivity amplitude with one mirror on incident r 1 t , 1 h eodmro ssprtdb noptical an by separated is mirror second The . r enabled are h ln) Here, cling). sreferred is m iy Next, vity. rtelat- the or ihre- with , foptical of matof impact h cav- the f enhances fthe of e fthe of e s The es. ,com- g, nhance a be can nthe in irrors. sa As 69

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer

Mirror1b1 Mirror2

ain a1 a2 aout

v aref a4 a3

(r1, t1)d (r2, t2)

Figure 5.1: Schematic illustration of a two mirror cavity [118]. The optical axis is depicted as the black arrow, optical fields are depicted as green arrows. Light is assumed to be incident from the left side only. The mirrors exhibit amplitude re- flexion and transmission coefficients r1, t1 and r2, t2 respectively. v 6 1 corresponds to loss within the cavity distance d with respect to the first mirror, and exhibits amplitude reflectivity and transmissivity r2, t2. In a lossless scenario, the coeffi- 2 2 cients fulfill ri + ti = 1 due to energy conservation. The fields inside the cavity can be calculated following [118, 119]. A phase flip of π/2 is assumed for every transmission. Loss inside the cavity can be modelled with an additional factor (v 6 1). The resulting set of equations is

a1 = it1ain + r1a4 , −ikd a2 = ve (a1 + b1) ,

a3 = r2a2 , −ikd a4 = va3e ,

aref = r1ain + it1a4 , (5.1)

aout = it2a2 . (5.2)

Here, k is the optical wave vector k = 2π/λ with λ being the optical wavelength. An input signal b1 can be used to compute the cavity transfer function [119], for signals generated inside the cavity. It will be set to zero here b1 = 0. Then, by solving the above set of equations for a4 and a2 results in

iv2r t e−ik2d ivt e−ikd = 2 1 = 1 a4 ain 2 −ik2d , a2 ain 2 −ik2d . (5.3) 1 − v r1r2e 1 − v r1r2e

70 d with aiylength cavity erwitnas rewritten be called h aiyehbt utpersnn lengths resonant multiple exhibits cavity The fE.56i ufild n o qa reflectivities equal for and fulfilled, is 5.6 Eq. If aiyrsnnecniinis condition resonance cavity h eetdadtasitdfil fulfil field transmitted and reflected The netn hs eut akit q 51,52 n assum and ( (5.1),(5.2) case Eq. into back results these Inserting ihti ento,tecvt finesse cavity the definition, this With pia length optical uhta osctv eoacsaesprtdb h free- the by separated are (FSR), resonances range consecutive that such h eodcaatrsi uniyo notclcvt si is cavity optical an of quantity characteristic second The hc sdfie ntrso h ulwdha afmxmm(F cavity. half-maximum at the full-width of the transmission of power terms in defined is which r With . N v impedance = en nitgr ti fe ie sflt eaaethe separate to useful times often is It integer. an being 1 k yields ) = a a d out ∆f d 2πf/c ref FWHM noamcocpclength macroscopic a into o ntereflectivities the on not , r ace.Teeutos(.)te yield then (5.5) equations The matched. = = = e n h ento FSR definition the and − S.TeFRol eed ntecavity’s the on depends only FSR The FSR. a a ik2d = a in in ref 2 F r FSR Nπ e = r 1 π − = = 1 − − ik2d ,a 0, − 1 t = FWHM 1 r 1 arcsin 1 r ⇔ FSR − t out 2 FSR r 2 πf 2 ( = e r kd r e 1 − r 1 2 = − . 1 r ikd r + ik2d 2 F a e = 1 t in | , 2 − r sdfie as defined is a . 1 2 L i √ − π, Nπ ik2d out ) e ftemirrors. the of r n irsoi length microscopic a and r . = − 1 | 1 2 d r ik2d r c/ 2 r + 2 . r , 1 ( | . pia cavity Optical 5.1 a 2d = refl ) . r q 58 can (5.8) Eq. . | 2 2 , h aiyis cavity the , = n lossless a ing sfinesse ts 1 ∀ spectral WHM) d (5.11) (5.10) The . (5.8) (5.7) (5.6) (5.5) (5.4) (5.9) 71 F ,

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer

10 0

10 -1

-2

in 10 /P

trans -3 P 10

10 -4

Ptrans/P in 10 -5 -0.5 0 0.5 1 1.5 Detuning (x/FSR)

Figure 5.2: Cavity resonance peaks for an impedance-matched cavity r1 = r2. On cavity resonance, all incident power is transmitted through the cavity. The distance between two resonances is given by the cavity free spectral range (FSR).

Cavity oﬀset from a resonance For small frequencies (f FSR), the cavity acts as an optical low-pass ≪ ﬁlter [54] with corner frequency fc. This can be seen from the Taylor expansion of the exponential function in denominator of equation (5.5)

1 1/(1 − r r ) FWHM 1 2 = −ik2d , fc . (5.12) 1 − r1r2e ≈ 1 + if/fc 2 Inserting Eq. (5.10) into Eq. (5.12) yields

c 1 − r1r2 1 − r2 fc = arcsin , (5.13) 2πd √r r ≈ 2πd 1 2 with the last approximation being valid for small arguments x, when arcsin(x) x. This is however ensured by the fact that r ,r 1. ≈ 1 2 ≈ A comparison of the exact cavity solution and the low-pass approximation is depicted in Fig. 4.9. The DC enhancement (f 0) is thus → given by 1/(1 − r1r2).

72 mltd eeto n rnmsincecet a ewri be can coefficients transmission and reflection amplitude hieo emslte hssa nSc . sue.Te,am a Then, used. is 4.1 Sec. in as position phases brane splitter beam of choice eoee mltd eeto sgvnb q 41) o si assumed, For th is for (4.13). splitter expression Eq. beam general by balanced the given a is whereas reflection (4.11), amplitude ferometer Eq. in inter Michelson-Sagnac given the is of amplitude output general The power output cavity Effective 5.1.1 enpromdfrablne emslte n eyln m recycling a and splitter beam balanced of a reflectivity for pos performed mirror been recycling and membrane both A of function as power) irr hs mltd eeto offiin eedo h m the express for on above the (5.5) depend Inserting coefficient position. microscopic reflection brane’s amplitude whose a as mirror, considered be can interferometer Michelson-Sagnac The eitdi i.55 cno h inlrccigmro i mirror signal-recycling the outco of the scan mirror, A recycling memb the 5.5. the scanning of Fig. When tuning in 5.4. fixed depicted Fig. a inte of an for axis to the position corresponds of this one modified, to is parallel parameter one only If Finesse Laser input P in t 1 iuain[2]o h eaiecvt uptpwrhas power output cavity relative the of [120] simulation and r 2 2 x ( Compound mirror x a a ( r Signal-recycling mirror = a a ifo r r out ifo refl ifo = sr 1 in in ( R x t , ) lost aclt h aiyotu ed(and field output cavity the calculate to allows sr t , 0 sr = = ifo ) = qasabih-rneadteinterferometer the and bright-fringe a equals ( r Membrane t x ) ifo ifo 95 ) .Tersl sdpce nFg 5.4. Fig. in depicted is result The %. = = t r m m cos + ftemembrane. the position of microscopic on depends iue5.3: Figure mltd reflectivity variable amplitude of mirror latter compound as The acts interferometer. Sagnac Michelson- the signal- and mirror high-reflective recycling a of consists r r ( m BS 2 2kx i sin = ) ( t 2kx BS 2 , h ffciecavity effective The ) = . pia cavity Optical 5.1 0.5 . r ifo n h same the and osit Eq. into ions ( x ) effective n which , ferometer mplicity, tnas tten rsection inter- e nstead ition. (5.14) (5.15) eis me irror x rane emem- 73

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer

1 1 0.9 0.8 0.8 0.7 0.6 0.5 Pout/P in 0.6 0.4 0.3 0.2 0.4 0.1 0 0.2 5/16

0 0 1/8 1/4 λ 1/4 SR Detuning ( ) Membrane Detuning (λ) 3/8 1/2 3/16

Figure 5.4: Normalised optical output power as function of membrane and recycling mirror detuning. White lines depicts levels of constant height. Interferometer and recycling mirror tuning are correlated, such that the tuning of the recycling mirror 2 has to be adjusted for each membrane position. Here, rm = 0.17 was assumed. For 2 a pure Michelson interferometer (rm = 1), both tunings are independent, and all interferometer dark-fringes line up at the same tuning of the SRM. Spikes on top are caused by the finite resolution of the simulation. produces a cavity resonance profile as described in Fig. 5.2. A maximal of ratio of Pout/Pin = 0.9 was transmittable through the cavity. The difference to a complete transmission of input power was due to optical loss, which caused an impedance mismatch of the cavity.

5.1.2 Signal-Recycling The additional mirror in the interferometer output port can be used to increase the interferometer sensitivity [54, 121–125], as signals that are generated inside the cavity (interferometric signals) are resonantly enhanced. The amplitude gain can be calculated from the ratio a0/b1 (compare Fig. 5.1). Solving the set of equations yields

a t eikd 0 = 2 2ikd . (5.16) b1 1 − r1r2e In the case of the signal-recycled Michelson-Sagnac interferometer, r1 = rifo and r2 = rsr are the interferometer and signal recycling

74 hsmdfid[6 to [86] modified thus f < f q 51)smlfisto simplifies cavi (5.16) the On Eq. respectively. reflectivities, amplitude mirror undercoupled. is cavity the pitrrtoo 05 n ebaepwrrflciiyof reflectivity power membrane a perfe and a 50/50 have of to ratio assumed splitter is interferometer Michelson-Sagnac iue5.5: Figure ace h inlrccigreflectivity recycling signal the matches aumflcutosaeapie with amplified are fluctuations Vacuum euigo h nefrmtri hsnsc htteinter the that such chosen is interferometer the of detuning hc smxmzdfor maximized is which lc uv.Tega radpcsastaini hc the which in situation a depicts area o gray interferometer The The curve. transmitted. is black light incident all and odto.Tnn h nefrmtrsc that such interferometer the Tuning condition. neculdcvt a enfo h nu ot.Tepower The port). input the from G seen (as cavity undercoupled

sr Pout /P in G sgvnby given is 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 c sr 0 1 h xrsinfrterltv htnielvl(q (4.23 (Eq. level noise shot relative the for expression The . 0 = re:Nraie rnmte oe hog h ﬀcieca eﬀective the through power transmitted Normalised Green:

a b 1 0

λ /16 2 G = shot x ( 1 = − λ /8 t s r sr 2 r ifo sr a b 16πr ) 1 0 = 2 = Interferometer Detuning = 1 Node position hcλ

m 2 hsi h nefrmtrdark-fringe interferometer the is This . 1 G ( ( 1 1 − r sr − sr − − P r t h aiybcmsipdnematched impedance becomes cavity the in ifo Signal recycling output R Signal recycling output r r sr λ /4 sr sr s r )( )( sr 1 1 1 √ + + − , G r r Cavity undercoupled Impedance matchedImpedance sr Anti-node position sr sr f f c ) ) r apiuegi)for gain) (amplitude ifo eoee reﬂectivity ferometer tu oe ssonas shown is power utput = . pia cavity Optical 5.1 2 3/8 r > 1 1 tcnrs,abeam a contrast, ct r λ ifo r + − , R m 2 sr bs bs r > r r =0.5 yresonance, ty =0.5 = sr sr eut nan in results sr 17 . uhthat such , .I the If %. iy The vity. (5.19) (5.18) (5.17) gain )is )) r λ 75 /2 ifo

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer

1 0.8 P /P out in 0.6 0.4 sr 0.2 R ifo < R sr R ifo = R 0 sr R ifo > R (node) 1 R ifo max. 0.015 2 0.01 Mem. position. 0.005 3 0 4 -0.005 -0.01 Time (ms) -0.015

1 0.8 P /P out in 0.6 0.4 0.2 0 1 2 0.005 SRM position 3 0 4 -0.005 5 -0.01 Time (ms) -0.015

Figure 5.6: Scans of the signal recycling mirror position for different membrane positions (upper plot). The dashed line (membrane position 0) shows the applied piezo voltage in each plot. Bottom plot: Scans of the microscopic membrane position for different (fixed) signal recycling mirror positions. The recycling mirror 2 reflectivity was rsr = 99 %. whereas the quantum back-action noise (Eq. (4.26)) with recycling mirror is

2 x 16πhrmGsrPin 1 Grpn = |G(ω)| . (5.20) cλ 2 r 1 + (f/fc) q Fig. 5.7 illustrates the interferometer sensitivity gain achieved by 2 signal-recycling for various recycling mirror power reﬂectivities rsr and allows for a comparison without signal-recycling. 2 The calculation assumes rm = 0.17, d = Lsr + Lifo = 2(1.2 cm + 7.5 cm) = 17.4 cm and an optical input power of Pin = 1W with no loss in the cavity and a laser wavelength λ = 1064 nm. The above considerations result in a constant optical gain-bandwidth

76 pia an h aiyhst esot o inlrecycli signal a For short. be to has cavity the gain, optical ieit.Tu,t ananasffiin pia ieit ( linewidth optical frequency sufficient resonance a mechanical narrower a maintain the of to than expense Thus, the at comes linewidth. gain optical higher Hence, iue5.7: Figure eod safnto ftecvt euig scmaio,t comparison, correspo As (this shown detuning. is cavity mirror the recycling of without function sensitivity a as second, notcliptpwro a sue,adammrn power membrane a and assumed, was W 1 of r power input optical an product is,sona ucino inlrccigmro oe r power mirror signal-recycling of function a as shown First, m 2 Displacement sensitivity (m/Hz1/2 ) Displacement sensitivity (m/Hz1/2 ) = 10 10 10 10 10 10 10 10 0.17 -15 -18 -17 -16 -15 -18 -17 -16 10k 10k . ipaeetsniiiyehneetdet signal-recycl a to due enhancement sensitivity Displacement no recycling R = 99.00 % R =99.00 % R =95.00 % R =90.00 R =99%100MHzdet. R =99%10MHzdet. 100k 100k G sr R =99%tuned · f c = c 1M 1M ( 1 Frequency (Hz) Frequency (Hz) 2πL + r sr ) ≈ 10M 10M πL f c res ≈ . . pia cavity Optical 5.1 d to nds 0 H)frhigh for kHz) 100 eflectivity 100M 100M einterferometer he R sr eetvt of reflectivity = n mirror. ing 0 ggain ng .Here, ). R bigger cavity (5.21) sr and 77 1G 1G

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer

√Gsr = 30 and a linewidth of 200 kHz, the cavity length (including the interferometer arm length) has to be smaller than 10 cm [76]. This length was already considered when designing the experiment. The interferometer arm length was chosen to about 7.5 cm, and the distance between beam splitter and signal-recycling mirror is 1.2 cm.

5.1.3 Design considerations for the effective cavity When designing the effective cavity of length L, the cavity stability criterion has to be fulfilled

|g1g2 − 1| < 1 . (5.22)

Here, gi = 1 − L/Ri with Ri being the radius of curvature of mirror i and L is the optical length. As the membrane is ﬂat, (Rm = ), Eq. (5.22) results in 0

z 2 w(z)= w0 1 + , (5.23) s z0 2 where z0 = πω0/λ is the Rayleigh range, w0 is the minimum beam size (z = 0), and λ is the optical wavelength. The wavefront curvature of the beam is a function of propagated distance from waist position along the beam path

z 2 R(z)= z 1 + 0 . (5.24) z The radii of curvature of the wave front and of the mirror surfaces have to match. For the plane membrane this corresponds to a waist position at the membrane position, such that the radius of curvature R(z) of the recycling mirror is determined by Eq. (5.24). For the waist size on the membrane another restriction is given by the membrane ﬁnite size. The waist size has to be small compared to the size of the membrane, otherwise beam clipping occurs, resulting in optical loss within the cavity. Assuming the membrane as circular aperture (radius a = Lx/2), the loss for a spot diameter ω = 2ω0 can be approximated to [126]

78 eoac n nefrmtro akfig)cnb writte be can function dark-fringe) transfer on the interferometer equations, and of resonance set general the Solving h aiytase ucin osi h aiyi modelled is cavity the in Loss deriv be de-amplification can function. amplitude cavity transfer the within cavity interferometer. loss the the acceptable of of loss estimate optical An intrinsic the to rable n ed ob osdrd hsi fpriua importance transmissivity particular power of h whose is effect, mirrors, This this recycling considered. by reflective be compensated ’mode-heali to be needs so-called cannot and loss the optical by The compensated [54]. im partially The interferometer. is int the first maximall non-ideal within loss as the optical such limit and reflectivity, contrast reasons mirror of recycling variety cavi signal a the within however, still reality, are In frequencies relevant the as shot long relative as low arbitrary thereby and factor enhancement da n hr em orao o oincrease to not a reason were no optics seems all there and and alignment interferometer ideal ca mirror the recycling far, not. signal or Thus poss - a sensitivity was w of interferometer loss usage length the optical the arm the reality, on In small depend cavity. arbitrarily would gain an signal-recycling (if the cavit limiting the if even not [76]: was by mentioned already is losses issue Another optical of impact The squar 5.1.4 a is area membrane the because disc. smaller, a even is loss the than For If o hscieinrslsi h aiu cetbeloss acceptable maximum the in results criterion this for h aiyof cavity the | a / ω 5/2 > a 0 40 /b Loss 1 p.Ti sacnevtv aclto,hwvr nrealit In however. calculation, conservative a is This ppm. | = = 1 h inlrcciggi suiy ovn q (5.26) Eq. Solving unity. is gain signal-recycling the , h oscue yba lpigtu eoe smaller becomes thus clipping beam by caused loss the , πω 2 2

a b 1 0 Z 0

a = a b 2πte ! 1 0 1 v = ⇒ ihntecvt cmaeFg 5.1). Fig. (compare cavity the within − 1 2t 1 − − 2 − /ω ivt v v 2 2 2 sr r d sr ≈ t t = sr 2 . e 4. /4 − 2a 2 /ω . pia cavity Optical 5.1 r sr 2 o maximal a for . T 1 ylinewidth. ty erferometric sr − linewidth y ato the of pact os level, noise frcavity (for as n increase n scompa- is g eﬀect ng’ v ti the ithin usable y 2 o high for dfrom ed owever, ssumed within yits by (5.27) (5.25) (5.26) ,not e, ible), 79 y,

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer

10 3

10 2

10 1 1/2 SR G 10 0

-1 No loss 10 v = 1-5e-5 v = 1-5e-4 v = 1-5e-3 10 -2 10 -3 10 -2 10 -1 tsr

Figure 5.8: Signal-recycling gain as function of the signal-recycling mirror amplitude transmissivity tsr for diﬀerent loss parameters v in the cavity. For zero loss (v = 1), the signal-recycling gain can reach arbitrarily high values. In all other cases, it is limited and a maximum recycling gain exists.

2 2 For a signal-recycling mirror with rsr = 0.99, tsr = 0.01, recycling is only beneﬁcial for a loss signiﬁcantly smaller than 0.25 %. For the model discussed above, it is also possible to calculate an optimum value topt for the signal-recycling mirror transmissivity. Deriving Eq. (5.26) yields

∂ −ivt sr =! = − 4 2 0 topt 1 v . (5.28) ∂t 1 − v rsr ⇔ p The resulting signal-recycling gain is depicted in Fig. 5.8. The optimum signal-recycling gain is a function of loss, and is achieved for the optimum amplitude transmissivity topt = √Tsr, such that tsr = 0.1 corresponds to a power reﬂectivity of Rsr = 99 %. The maximum signal-recycling gain is given by

1/2 v Gsr,max = . (5.29) √1 − v4

80 i membrane SiN d losses overall the properti to optical cantly. their contribution of individual means their by inter ever, imperfect the be Michelson-S of to the component assumed Each within is sens loss presented. displacement be the will for reduced terferometer inter estimation a the brief cause in a losses can Here, the cavity) section, effective previous (and the in discussed As cavity the in Loss 5.1.5 teigmirrors Steering ieetls otiuin r eitdi i.59 cons A 5.9. Fig. in splitt assumes depicted beam scenario the are tive by contributions dominated loss thus is different loss optical overall The R emsplitter Beam l pia osi oiatadteotclls exceeds loss optical the and dominant is loss optical all AR = frfato fslcnntie( ind nitride an silicon the for that in of shown wave refraction have standing we of the [84], of In node interferometer. the possibl Sagnac to in is membrane it Due the membrane, the position membrane. of thickness the sub-wavelength through the transmission during sorption 100 of wavelength laser u oecnevtv siainis estimation conservative more a but h ult fterH otn.Hg ult,inba sput- beam ion achieve quality, can High coatings coating. multilayer HR tered their of quality the a esalrthan smaller be can e fa nh uhta ahtasiso under transmission each that such inch, an of ter orsod oarudti osof loss trip round a to corresponds inof tion n ntebc ieo h emslte.Mnfcue speci are Manufacturer coatings splitter. AR beam quality the high of for side fications back the on ing 0.3 usrt,cuigls yasrto.Hg ult ue s th fused through quality transmitted High as be such absorption. to materials, by has loss power causing laser substrate, the First, loss. 0.01 p os h eodsuc fls steat-eetv coa anti-reflective the is loss of source second The loss. ppm − .Ete a,teba pitrcnrbto oteover- the to contribution splitter beam the way, Either %. 500 0.3 h nefrmtrba pitri ulsuc of source dual a is splitter beam interferometer The p,cuigals of loss a causing ppm, ppm h oso h ebaei asdb h nt ab- finite the by caused is membrane the of loss The h oscue ytesern irr eed on depends mirrors steering the by caused loss The / m yia emslte hcns saquar- a is thickness splitter beam typical A cm. R ursl3001 Suprasil AR λ 57 = = ppm. 04n 3] h ebaeotclloss optical membrane the [37], nm 1064 0.05 ,amr piitcetmt is estimate optimistic more a %, n

R 2 SiN · 17 rvd pia absorp- optical provide [127] 2 ( 100 · = 10 R R 2.2 − ppm. AR HR 500 R + . pia cavity Optical 5.1 ≈ AR = i ppm · 0.01 99.999 6 1.5 200 45 ) 0.01 ffr signifi- iffers · . − 10 ◦ ferometer ferometer ppm. ga in- agnac 0.05 which % s How- es. − assa causes [128], % r The er. 4 itivity. ta at ) erva- % ilica to e 81 ex = t- e -

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer

10 3 10 3

10 2 10 2

10 1 10 1 Loss (ppm)Loss

10 0 10 0 BS absorp. clipping membrane BS absorp. clipping membrane membrane absorption steering mirror loss membrane absorp. steering mirror loss BS AR coating BS AR coating 10 -1 10 -1 conservative scenario optimistic scenario

Figure 5.9: Loss channels in the eﬀective cavity. The beam splitter anti-reﬂective coating is the largest source of loss. Left: conservative estimate of optical loss, beam splitter RAR = 0.05 %. Right: more optimistic scenario, beam splitter RAR = 0.01 % and thinner membrane, with slightly reduced spotsize on the membrane.

5.2 Realisation of the recycled interferometer

When installing a high-reﬂective signal-recycling mirror into the interferometer output port, this mirror must match the optical mode deﬁned by the interferometer optical mode, which is not guaranteed a priori. The mirror radius of curvature has to match the curvature of the optical wavefront, and the interferometer mode might be translated in the horizontal or vertical direction with respect to the central axis of the recycling mirror. This problem is similar to the situation in which the membrane might not be hit by the Sagnac mode, because the Sagnac interferometer is invariant to pure beam translations, with one additional complication: when the cavity is not resonant, no light is transmitted. The amount of light transmitted on cavity resonance furthermore depends on the interferometer tuning, which is unsta- ble while working in the vacuum tank. During the installation, the signal-recycling mirror position was continuously scanned, until optical modes were visible in transmission of the cavity. Fine positioning was then accomplished by the remote controllable mounts. A scan of the signal-recycling cavity length is depicted in Fig. 5.11. The two

82 ouaino n ftebas nteeprmn,ti phase this experiment, the In defined beams. a the requires of It one on technique. was appropria modulation demodulation signal an / error this light, modulation experiment, laser a the In the for required. is resonant signal cavity the keep To scheme stabilisation length Cavity 5.2.1 attribu interferometer. was the and within eliminated misalignment be resonanc small not smaller a could a to which and mode, mode TEM10 TEM00 the a to correspond resonances output the of density o spectral splitter sign power their beam the of two provides difference and The The sp analyzer detectors. beam frequency. photo two homodyne optical by part balanced same detected main second, the with The a have overlapped finally cavity. on fields and recycling oscillator) shifter the local phase lock (the a to to used transmitted is is off split beam iue5.10: Figure nefrmtrotu oei rnmte hog a through transmitted is mode output interferometer @ 1064nm Laser input (P <10mW) Local oscillator p =1.010mbar Vacuum tank Phase modulation eu ftesga-eyldMcesnSga interferome Michelson-Sagnac signal-recycled the of Setup 10 MHz -6 EOM Phase shifter . elsto ftercce interferometer recycled the of Realisation 5.2 R =50% splitterBeam R =10% Pick-up Balanced homodyne detector SR Mirror R=99.97%SR Mirror - R =17% SiN membrane Locking PDLocking Spectrum analyzerSpectrum 10 emslte.The splitter. beam % li e oaspectrum a to fed is al togrlgtfield light stronger a field. itr ohlight Both litter. 1 MHz tu ot are ports utput rvddby provided fteoutput the of 128 kHz LP 300kHz eerror te e.The ter. phase mod- of e ted 83

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer

0.8 Output power Piezo voltage 0.6

0.4 (V) TEM 00 resonance OUT

P TEM 01 resonance 0.2

-15 -10 -5 0 5 10 Time (ms)

Figure 5.11: To scan the cavity, a ramp voltage has been applied to the SRM piezo, thus periodically changing the cavity length. The interferometer tuning was chosen as to maximise the power in the TEM00 mode (corresponds to 3.0 % fraction of ≈ input power). The smaller peak is a resonance of TEM10 mode. ulation was imprinted on the light field exiting the recycling cavity by applying a sinusoidal frequency to the signal-recycling mirror piezo actuator. The cavity output power Ptrans(x) (for a fixed interferometer tuning) is a typical cavity resonance profile (Airy peak) and was to be maximized. Therefore, the microscopic position x of the recycling mirror was modulated with a known frequency ωmod, such that x(t)= x0 + a cos(ωmodt). Taylor expansion of Pout(x) at x0 now yields

′ Ptrans(x) = Ptrans(x0)+ Ptrans(x0)(a cos(ωmodt)) 1 + P ′′ (x )(a cos(ω t))2 + .... (5.30) 2 trans 0 mod

A demodulation at frequency ωdemod = ωmod (multiplication with cos(ωmodt)) thus provides a DC signal proportional to the derivative ′ Ptrans(x0) around x0. A fraction of the output power was split off the interferometer output mode and detected by a photo detector. The AC-filtered output was mixed electronically with a copy of the modulation signal. The mixer output was low-pass filtered (fc = 300 kHz) and fed back to a control loop for the SR mirror position. In the experiment, the modulation frequency fmod = ωmod/(2π) ≈

84 eas ofrhrcntanso h function the on constraints however. further dark-fringe, no to Because arbitrari tuned becomes dark-f is signal interferometer to error this the close of operated ratio interferometer noise to the as signal for signals 5.12 error reson acceptable Fig. provide piezo in to mechanical ex found narrow an was very scheme generate a This to exploiting possible by still piezo zer was the typically it are but high transducer frequencies, the ceramic high both, piezo of the and responses typic amplifier The is (which modulat resonances). interest the mechanical of of the range harmonics frequency se the order outside higher exhibits low-pa well second, after frequency and bandwidth high signal control-loop relatively the high a This First, benefits. chosen. was MHz re 1 been have detecto traces photo (both a signal on error voltage tra the above detected is black scan green the a cavity, as is side, the right Black depicted the is On depicted. power is green. in output plotted is fractional derivative The profile. iue5.12: Figure inlrcciggi.Frarno oiino h signal- the of position depen gain random lock i the a interferometer mirror, the For the lock complicat for to gain. is signal used signal-recycling situation error be The the also that can dark-fringe. o fact it local to the principle, close between In ferometer phase relative beam. the signal s lock locking and to this applied value, extreme be some also exhibits actually it than aiu n iiu value minimum and maximum a Ptrans /P in G 0.6 0.8 0.2 0.4 0 1 = -2 g g max min aiyotu oe n eiaie et hoeia cavity theoretical Left: derivative. and power output Cavity Detuning (x/FWHM) -1 = G 1 1 0 o h nefrmtrcnrllo aisbetween varies loop control interferometer the for + − . elsto ftercce interferometer recycled the of Realisation 5.2 r r sr sr 1 2 r , 2 sr g max = Voltage V (V) √ Out and 0.95 0.5 1.5 2.5 0 1 2 -0.127 g ⇒ min P G trans with ≈ scaled). 00. 6000 h aiyresonance cavity the Time (s) ntasiso of transmission in r eegvnother given were -0.126 e t rtorder first its ce, ysalwhen small ly lyaround ally ig.The ringe. sfiltering ss hm can cheme iainof citation db the by ed so the on ds tsuch at o recycling depicted resonance ance. scillator -voltage o are ion (5.31) veral nter- -0.125 85

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer

Laser source Amplitude Mode cleaner E!ective cavity modulator

EOAM

PBS

RF Out Input A a) b) Fast detector Network analyzer AC output

Figure 5.13: Schematic of the setup for measuring the eﬀective cavity linewidth. An EOAM modulates the input beam power, which is transmitted through the mode cleaner. This acts as an optical lowpass. The transfer function was measured a) directly in transmission of the mode cleaner and b) for additional transmission through the eﬀective cavity.

For a derivation, the reader is referred to the thesis of G. Heinzel [119]. Eq. (5.31) explains why the signal-recycling mirror has to be close to resonance before the interferometer can be locked. The gain for the signal-recycling mirror on the other hand depends on the interferometer tuning. Thus, to lock both, interferometer and cavity, the membrane position already has to be correct before an appropriate error signal for its lock can exist. The coupling of individual error signals is of fundamental importance for any complex interferometric setup and for gravitational wave detectors in particular [54].

5.2.2 Effective cavity optical linewidth To measure the optical linewidth of the effective cavity, its frequency response function was probed. Therefore, the laser light amplitude was modulated by an electro-optic amplitude modulator in front of the mode cleaner cavity, and the transfer function of the recycling cavity was measured with a network analyser. This method has been described in [129]. In the experiment, the setup used is depicted in Fig. 5.13. A reference output signal was applied to the EOAM in front of the first mode cleaner cavity. The AC filtered output of a single photo detector in transmission of the mode cleaner was used to obtain the transfer function of the subsystem : EOAM, mode cleaner and photo detector.

86 o asi hw sbak ahdln.Tecre rqec ( frequency corner The line. line dashed finite black, the as to shown due trace), is (grey pass is cavity low behavior effective pass the low to additional sion An trace). (green todetector flectivity a sue,adamd acigof matching mode a and assumed, was mlpsil ieit.Telnwdh(WM a determin th was achieve (HWHM) to linewidth as The the such linewidth. measurement, opera possible this was imal in interferometer dark-fringe The the function, linewidth. transfer to cavity overall finite the the in to behaviour shows cavity, pass effective low the tional of transmission in measurement A MHz. 1 about of frequencies for reached ia oso about of loss tical 1 was splitter beam and mirror al rcino h pia nu oe.Ti rcinmea fraction This power. trans input maximal optical the be the measuring of by fraction T determined table cavity. was the loss into of loss amount incorporate to allows simulation The match. parameters experimental iue5.14: Figure eylditreoee a acltdnmrclywt a with numerically calculated fo was density interferometer spectral recycled displacement equivalent noise shot The sensitivity Displacement 5.2.3 measurement. the from inferred be could MHz 1 of efrmtri e odr-rneadi sue oexhibit to signal-recycl assumed a is Furthermore, r and contrast. dark-fringe 8. interference Appendix to fect see set [118], is interferometer terferometer the of script ulation r eghwsstt 7 to set was length arm

sr 2 Mag. (dB) 3 = -3 -9 -6 − 0 cmaeFg .5 o h E0 oe hc ilsa op- an yields which mode, TEM00 the for 5.15) Fig. (compare % 10 k 10 3 99.97 Blvl h euti eitdi i.51.Acvt linewi cavity A 5.14. Fig. in depicted is result The level. dB r m 2 omlsdtase ucino h ouao,md cleane mode modulator, the of function transfer Normalised After e!ective cavity oe eetnewsasmdadteinterferometer the and assumed was reﬂectance power % After modecleaner modelled lowpass = 17 1 .A qiaetotcliptpwro 0 mW 200 of power input optical equivalent An %. nteitreoee.Cmae otelossless the to Compared interferometer. the in % . elsto ftercce interferometer recycled the of Realisation 5.2 . m h itnebtensignal-recycling between distance The cm. 5 100 k 100 Frequency (Hz) . m n h ebaepwrre- power membrane the and cm, 2 80 .Tu,smlto and simulation Thus, %. bevdi transmis- in observed it.Temodelled The width. 1 M − n irrof mirror ing 3 Finesse h signal- the r B)lvlis level dBm) .Tein- The 1. attributed ecorrect he e close ted n pho- and r ue to sured naddi- an min- e per- a das ed sim- mit- dth 3 M 3 87

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer

10 -15 1 0.05

0.8 0.04 1/2

10 -16 0.6 0.03 in /P out P 0.4 0.02 10 -17

0.2 0.01 Displacement sensitivity (m/Hz ) Displacement no loss loss 10 -18 0 0 10k 100k 1M 10M 100M -1/32 0 1/32 Frequency (Hz) Membrane detuning (λ)

Figure 5.15: Left: Simulated displacement equivalent shot noise level of the signal- 2 recycled Michelson-Sagnac interferometer on dark-fringe with rsr = 99.97 %. Here, the tuned case is considered for no loss (grey) and loss within the cavity (green). Right: Transmittable fraction of optical input power for no loss (green) and loss (grey). In case of 1 % optical loss, a maximal fraction of 0.03 of optical input power is transmittable through the effective cavity. case, the displacement sensitivity is significantly reduced. A comparison of the shot noise equivalent displacement in both cases is depicted in Fig. 5.15. The recycling gain is approximately unity, such that no increase in displacement sensitivity was expected. It is necessary to keep in mind which assumptions were made to sim- ulate the sensitivity and to critically analyse to which degree they are justified. Especially the condition for dark-fringe operation of the interferometer is not fulfilled in reality, as this eliminates the error signal of both the cavity and the homodyne phase lock. The interferometer is tuned slightly off dark-fringe, causing an increased amplitude transmissivity towards the laser port and will thereby increase the relative shot noise level even further, as will optical loss in the detection path. The above considerations may be summarized as follows: In case of signal-recycling, and to achieve an enhanced displacement sensitivity, it is mandatory to operate the recycling cavity undercoupled. Side bands generated inside the cavity will then leave towards the detection port. For given optical loss, the signal-recycling gain exhibits a maximum value for one particular reflectivity of this mirror. A further increase will result in the same resonant enhancement of signals, but a smaller fraction is transmitted to the detection side.

88 nu irrapiuereflectivity amplitude mirror input h olwn eto osdr nua frequencies angular considers section following the t o sasmdt xii nt mltd reflectance amplitude unity oscill exhibit The to fix. assumed assumed h is is as mirror ror serving second mirror c the one cavity while with The oscillator 5.16), 130]. Fig. 49, (compare 36, mirrors 29, t two [28, topology opto-mechanics used of frequently field a the is cavity Fabry-Perot linear, A cavity Fabry-Perot signal-re on cooling resonance. and additio i instabilities exhibits It conditional interaction cavity. as full the such the of that linewidth dependence mo the thereby this and It frequency (5.10). com resonance Eq. the the in modifies mirror but effective cavity, the re effective of d flectivity the the of position of length case membrane the In modify the frequency. oscillat interferometer, resonance a the Michelson-Sagnac cavity frequency of the resonance thereby associated position (and and sp microscopic length beam cavity a the central the cavity, from fies a linear different on a is modes In cavity optical Michelson-S (or two linear Michelson linea overlaps a a A terferometer mode, that optical single mind nature. a by contains in interferometer kept Michelson-type be two-path to the has mirror position. It effective membrane this microscopic of the opera reflectivity on when The mirror effective dark-fringe. an to Michel as effective, acts the which by replaced interferometer, is mirrors cavity two In the as of system features. these the exhibits imagine also to which intuitive cavity, Fabry-Perot is parametric it spring, cooling, optical optical the and as such effects, interaction understand To interferometer. Michelson-Sagnac w recycled interactions opto-mechanical the describes section This interaction Opto-mechanical 5.3 w Section. This next the end-mirrors. the high-reflective in Yet, very install again. reduced to is reasons gain signal-recycling the Hence, 1 2 ( = 1 − r 1 2 ) ncs fzr oswti h aiy o convenience, For cavity. the within loss zero of case in , . pomcaia interaction Opto-mechanical 5.3 r 1 1 < n transmissivity and ω r = 2 yln cavity cycling ti signal- a ithin hscs,one case, this a features, nal = l eissued be ill 2πf instabilities oeo the of some eaegood are re neffective an son-Sagnac ie both difies hroughout depends n tn mir- ating ga)in- agnac) 1 dFSR), nd nit of onsists rmodi- or e close ted u to due s fthe If . armonic cavity r t n the and e not oes lxre- plex in cycled litter. with 89

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer

a) Input mirror oscillator (r1, t1) (r = 1) Frp

Figure 5.16: a) A high-reﬂective oscillator (r = 1) and an input mirror (r1, t1) build a linear cavity. L The laser ﬁeld is incident from the b) P left and indicated in green color. cav dFrp b) Power build-up in the cavity as dx function of oscillator position. On cavity resonance (dashed, grey line, the power is maximal. The deriva- oscillator pos. tive dFrp/dx vanishes on resonance. macroscopic cavity length is L, then the cavity is resonant for frequencies nc ωn = π , n N , (5.32) L ∈ such that FSR = πc/L. The cavity linewidth (HWHM) γcav then is (compare Eq. (5.10))

1 FSR 1 − r1r2 γcav = FWHM = arcsin 2 π 2√r r 1 2 c 1 − r c 1 − r = arcsin 1 1 L 2√r ≈ L 2√r 1 1 c |t |2 (1 − r ) c 1 . (5.33) ≈ 2L 1 ≈ 4L Here, three approximations were made. First, arcsin(x) x, which is ≈ valid for x 1. Second, √r 1, and ﬁnally ≪ 1 ≈ |t |2 = (1 − r2) = (1 + r )(1 − r ) 2(1 − r ) . (5.34) 1 1 1 1 ≈ 1

All of these approximations are valid for r1 1. On resonance, the ≈ 2 power build-up in the cavity is maximal, as can be seen from |a2| in Eq. 5.3. In this situation, a constant optical force Frp is caused by the power within the cavity, which has to be stabilised by some kind of feedback. The oscillator now experiences a position dependent optical force. The sign of the force depends on the detuning: Suppose an initial cavity length L0, which is larger than the cavity resonance

90 t ognrt eaieotcldamping to optical Due negative (5.38). generate Eq. to of ity out follows consequence interesting An instability Parametric li 5.3.1 associated small, (and finesse cavity high γ a Thus, factor. atof part ehnclrsnnefeuny h siltreetv d effective by oscillator given an The then damping mechanical frequency. the resonance both mechanical modify expressions above The Here, constant h oie pigcntn lorslsi neetv mech effective an in results also frequency resonance constant spring modified The tcnb enfo q 53)ta h tegho h optical the of strength the power that input available (5.37) the Eq. on from depends seen be can It length irr oe(es oe sbidu ihntecvt,cau cavity, the le initial within the If build-up force. is optical repulsive power (smaller) larger (less) more mirror, httettl(ope)otclsrn sgvnb [132] by given is spring optical (complex) total the that ya diinlotclforce optical motio of additional ch equation suc an the oscillators by The instantaneous, between delay force. length optical time of the dependent length change frequency in a a [131] r to exists spring decay there react cavity optical not finite an can the cavity of as because considered position However, The be approximation. attractive. can but force repulsive optical not is force optical cav sbeneficial. is ) Γ P L κ cav ω − = hnteoclao oe oad aa rm h input the from) (away towards moves oscillator the When . ( K eff 2 ω K n pia damping optical and stelsrpwrisd h aiy h pia spring optical The cavity. the inside power laser the is ) = 2ω with , = 1 ω ℜ κ ℑ 0 2 ( [ [ κ κ ω + ( ( γ = ) ω ω ω ω eff ]= )] ]=− = )] OM 2 eff 2 = 4ω γ ω , cL c + P 0 2 cav 4ω γ F = 4ω rp OM Γ cL c ( ( / ω , k/m γ eutoto h el n imaginary and real- the of out result P ω cL c γ , . pomcaia interaction Opto-mechanical 5.3 cav cav P − = ) cav P − | δ in δ OM 2 | ( iω δ δ δ n h aiyenhancement cavity the and ( + κ 2 OM 2 2 γ ( = ) ( + ω OM + 2 γ . Γ = ) + cav γ γ x 2δγ cav 2 ( cav / . K/m 0 < δ ω 2 − cav ) − − iω tcnb shown be can It . ti osbeto possible is it , . ngth ω iω ) 2 2 ) ) | smodified is n 2 2 L | 2 dependent 0 migis amping , h abil- the L < newidth t,the ate, neof ange . that h iga sing spring (5.37) (5.35) anical (5.38) (5.39) (5.36) linear the d the , 91

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer

generate an anti-damped oscillator γeff < 0, with an effective quality 2 factor Qeff = mωeff/γeff < 0. In this case, the faintest excitation at this frequency gets amplified exponentially, as can be seen from Eq. (3.33). Note that in the case of a linear cavity, this is possible only for one sign of the cavity detuning, namely where the optical spring acts anti-damping and restoring. In fact, this behaviour can be observed in the signal-recycled Michelson-Sagnac interferometer as well. When a signal-recycling mirror with a power reflectivity of RSR = 99.0 % was installed and the interferometer input power was increased above 1 mW, a parametric instability was observed for one particular detuning of the cavity. To demonstrate the effect, the cavity length was scanned by applying a saw-tooth high voltage to the signal- recycling mirror piezo actuator, which thereby modified the length of the effective cavity. The membrane position was chosen near to the interferometer dark-fringe for these measurements. The cavity output power time series is depicted in Fig. 5.17. A parametric instability is observed only for one particular sign of cavity detuning δ. The frequency of oscillations were found very close to the fundamental resonance frequency of the oscillator, such that within the measurement precision, no noticeable change of frequency was observed. This is explainable by the fact that K and Γ can still be small to produce the instability: In case of a high-frequency, high-Q mechanical oscillator, the mechanical damping γ is small and it only requires slight optical anti-damping to overcompensate it. Thus, the (absolute) modification to the mechanical resonance frequency will also be small. Parametric instabilities are the mechanical analog to the process of optical lasing. They are a well known feature of optical springs, and have been observed in other opto-mechanical setups [44, 133–136].

5.3.2 Optical cooling For the opposite detuning of the cavity, the optical spring causes optical damping Γ, which suppresses oscillator position fluctuations. The thermal excitation spectrum of the oscillator is unchanged by optical forces [44], such that to maintain the form of Eq. (3.23) (and neglect- ing technical laser noise), an effective temperature Teff is introduced, which is given by γ Teff = T (5.40) γeff The bottom plot in Fig. 5.17 depicts the measured cavity resonance

92 n sfeunydpnet oemcaia oe ih ec be. be not might might modes others mechanical while Some optical, dependent. frequency is ing pia oln sntietclt hsclcoig nthe In cooling. temperature physical sample to a identical the cooling not optical is to corresponds cooling anti-dampi this Optical and (5.40), damping Eq. optical to of cording regions two the and profile n are and up, build not can oscillations plot). the exponentiall high, is is oscillation velocity this f of membrane amplitude the The precisely frequency. with oscillations excited cavity the h oto neuldces nteqaiyfco.As,the Also, factor. quality Thi the in in oscillator. dependence decrease the frequency equal of implicit an temperature of effective cost the the th Therefore reduce to temperature. zero is reservoir at effectively temperature is additional which an field, to coupled is cillator efre iha with performed iue5.17: Figure PD Out (V) PD Out (V) PD Out (V) -0.2 -0.2 -0.2 0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6 0.8 0 0 1 0 -30 -30 -3 cn ftesga-eyln aiylnt.Temeasurement The length. cavity signal-recycling the of Scans R SR -20 -20 -2 = 99.0 exp. amplication pia etn opticalcooling optical heating irr ftesa eoiyi lw n euigof detuning one slow, is velocity scan the If mirror. % T -10 -10 srdcd hl ntefis ae h os- the case, first the in while reduced, is -1 Time (ms) Time (ms) Time (ms) Γ . pomcaia interaction Opto-mechanical 5.3 , 0 0 0 Γ = Γ ( ω ) 10 10 1 uhta h damp- the that such , Slow scan 2 Slow scan 1 Slow scan nraig ftescan the If increasing. y naetlresonance undamental Fast scan tosre (bottom observed ot 20 20 2 h optical the , dheating. nd atrcase, latter e effect net e comes s ei an is re g Ac- ng. ooled 3 30 30 was 93

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer

Dispersive and dissipative coupling In a linear cavity, the oscillator position modifies the cavity length L. Thereby, the cavity resonance frequency ωc is modified, resulting in an altered resonance condition for the optical field within the cavity (dispersive coupling). The cavity linewidth γ is also affected by the oscillator position, as can be seen from Eq. (5.33). The oscillator thereby causes a modified cavity decay rate, such that this mechanism is referred to as dissipative coupling. Their individual strength is given by

∂ω ∂γ g = c , g = cav . (5.41) ω ∂x κ ∂x Their ratio is gω/gκ = ωc/γcav = F , with F being the cavity finesse. Since typical cavity finesse values fulfil F 1, the coupling is highly dispersive. Dispersive coupling is ≫ the leading order interaction in linear cavities. In the effective cavity of the signal-recycled Michelson-Sagnac interferometer, the microscopic position of the membrane modifies the compound Michelson-Sagnac mirror reflectivity rifo, whereas the total optical length of the cavity remains constant (one arm extends while the other is shortened). Thus, a direct modification of the cavity finesse F is possible (compare Eq. (5.10)).

Dissipative coupling and its implications have been theoretically studied in [52, 137]. In case of this coupling mechanism, the unsymetrised back-action force spectral density features a Fano line shape (instead of a Lorentzian resonance proﬁle). This is a direct result of the interference of a resonant and a non resonant process. As a consequence, the coupling allows for strong optical cooling in the unresolved side band regime. This is impossible in linear cavity opto-mechanical setups. An experimental demonstration of dissipative coupling is given in [138]. In [53] it was shown that the signal-recycled Michelson-Sagnac interferometer can allow for a strong and purely dissipative opto- mechanical coupling, under the assumption of a perfect mirror in the interferometer output port. In the previous chapter, the consequence of such a choice of signal-recycling mirror to achievable displacement sensitivity was discussed.

94 nta,teepeso o ml nefrmtrostfr offset interferometer [140] this small by of given a is scope cas for fringe the Michelson-Sagnac expression beyond the the and In fo Instead, cumbersome [141] quiet approach fields. is matrix optical calculation transfer output a and using Mich input membrane fiel a the the in calculating on spring by ing derived optical be complex can the interferometer Sagnac for equation general The interferomet Michelson-Sagnac the in spring Optical 5.3.3 analysis Michelson- signal-recycled theoretical the the to a terferometer. applied be by also descr provided can accurate is which more a interaction moment, choic the the of of at method detectors the GW is scale pic operation) large cavity fringe covered linear (off-dark deduced not readout spring are reflectivit DC optical which mirror the effects, compound for back-action expression modified dark-f anomalous a A in in also results only: only tuning r not interferometer operation set of dark-fringe choice However, particular a fringe. mirro dark recycling on reflective highly operation a with scenario Fabry-Perot interferometer the an that to states [139] law scaling The law Scaling Laser input P in L arm l sr ( ( Signal-recycling mirror splitterBeam r r l bs sr a t , t , sr L bs arm ) ) ( Membrane r m t , l b m ) . pomcaia interaction Opto-mechanical 5.3 iue5.18: Figure nefrmtr h effective The length Michelson-Sagnac interferometer. signal-recycled L = l sr + L L fterccigcvt is cavity recycling the of arm + ceai fthe of Schematic l . ,i aeof case in r, ue Since ture. simping- ds ancin- Sagnac s applies lso mdark- om ig off- ringe epresents n[140], in o all for e nthe in ,but y, ,the e, iption elson- work. the r 95 er

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer

4ω r2 P 1 κ(ω)= 0 m in (5.42) 2 2 cL γtot + ∆ · δ [γ2 + ∆2 − 4(γ γ + ∆δ )] sr tot tot m m + 2 2 ∆ + (γtot − iω) 2 2i(γsrδm + γmδsr)ω + δmω 2 2 . ∆ + (γtot − iω)

Here, the notation is as follows. The total length of the signal- recycling cavity is L, which is composed of

L = lsr + Larm + l , (5.43) where Larm is the distance between beam splitter and steering mirrors, lsr depicts the distance between signal-recycling mirror and beam splitter. l is exactly half the distance between the steering mirrors, which corresponds to a dark-fringe here (this can be achieved by an appropriate choice of beam splitter phases). Then, the arm lengths due to membrane displacement δl/2 = ǫ from dark-fringe are la = l−δl/2 and lb = l + δl/2, such that lb − la = δl and lb + la = 2l. The total cavity linewidth γ is a composition of two terms now [140], given by

γtot = γsr + γm , (5.44) where γsr and γm are the contribution from both the ﬁnite recycling mirror and interferometer transmissivity, respectively, and are written as

ct2 ct2 cr2 (k ǫ)2 γ = sr , γ = ifo = m 0 . (5.45) sr 4L m 4L 4L

Here, the expression for γm is derived from Eq. (4.12) for the interferometer close to dark-fringe and with 2x = ǫ. Equation (5.44) provides an intuitive understanding of why the perfect mirror results 2 in the strongest dissipative coupling: For tsr = 0, the contribution γsr = 0, and γtot is given by the γm only. The cavity will be over- coupled for any membrane position. The total cavity detuning ∆ is deﬁned by

(k ǫ)2 ∆ = δ + δ , δ = ω − ω , δ = cr t 0 , (5.46) sr m sr 0 c m m m 4L

96 mgnr ato h pia pigi q 54) hssitu This (5.42). Eq. in spring optical the of part imaginary ac.Ti sms biu ytels contribution last the by obvious most signal-recycling is on This oscillator the nance. of modification optical eyln irri zero, is mirror recycling u fE.(.2.Ee ftedetuning the if Even (5.42). Eq. of out noie yteculn otecvt ed hrfr,the Therefore, field. For cavity holds. the [86] to in budget coupling p the oscillator resonan by the on unmodified cavity dark-fringe, signal-recycling on the precisely for interferometer Thus, 5.19. Fig. nefrmtroeae ihapprox. with operated interferometer ar-eo aiy tte xssol n eocosn o zero-crossing f one ones only the exists resembles Γ then expression It the that cavity. Fabry-Perot such (5.42), Eq. the h etfiue h ebaepsto scoe ocicd w coincide to chosen is position membrane the position, figure, left the spring iue5.19: Figure -5 -4 -3 -2 -1 uigteitreoee odr-rne( dark-fringe to interferometer the Tuning t detuning ity h pia spring optical The respectively. interferometer where eed nbt,itreoee euigfo dark-fringe from detuning interferometer cavity Fabry-Perot both, operation. a on dark-fringe of depends close-to case of in approximation that the than with shape complex more a 0 1 2 3 4 5 , Γ ( ǫ -4 H K ω κ 6= δ ( = ) ω sr γ Detuning 0 m ) nte etr fteculn eoe vdn directly evident becomes coupling the of feature another , and -2 el( Real nteMcesnSga nefrmtr acltdfor calculated interferometer, Michelson-Sagnac the in = 0 δ 0 sr ⇔ h pia iiiyaddmigvnse for vanishes damping and rigidity optical The . δ oeseilcsssalb osdrdhr briefly. here considered be shall cases special Some . m K 0 ( lc ie n mgnr at(,genln)o h optic the of line) green (H, part imaginary and - line) black , ∆/γ δ r h euigo h inlrccigcvt and cavity signal-recycling the of detuning the are sr κ o h ihlo-ancitreoee exhibits interferometer Michelson-Sagnac the for = tot 2 ) 0 hsstaini eitdo h etof left the on depicted is situation This . δ sr 4 = 0 /0λ 1/30 o l em iapa,implying disappear, terms all not , . pomcaia interaction Opto-mechanical 5.3 akfig offset. dark-fringe -4 -3 -2 -1 0 1 2 3 ǫ -4 δ = sr 0 otiuino signal- of contribution Detuning → -2 δ m 0 ( t dark-fringe a ith = rmtr are arameters ∆/γ δ ∆ 0 tespecially It m aiyreso- cavity ω simplifies ) eadthe and ce ǫ ω = damping f tot 2 = n cav- and 2 0 o the rom to is ation ) Right: . ω nthe in even , m noise 4 In . 97 al

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer

Dark-fringe operation small o!set from dark-fringe 10 9 10 9 10 8 10 8 10 7 10 7 10 6 10 6

e! 5 e! 5 Q 10 Q 10 10 4 10 4 conditional 10 3 10 3 instability 10 2 10 2 10 1 10 1 -1 0 1 2 3 4 5 -1 0 1 2 3 4 5

Detuning (∆/γ tot ) Detuning (∆/γ tot )

Figure 5.20: Numerically calculated modification of the membrane mechanical quality factor due to the optical spring. Here, an optical input power Pin = 50 mW was assumed. Left: Operation on dark-fringe. The quality factor is reduced for positive detunings ∆. Right: dark-fringe offset of ksi = 0.03λ. The initial quality factor (shown as dashed line) can be decreased by several orders of magnitude on resonance, e.g. ∆ = 0. The initial quality factor is depicted as black, dashed line. A region of anomalous (conditional) instability is expected on for 2γtot <∆<4γtot. depicted on the right side in Fig. 5.19. These exemplary situations highlight the difference to the linear, Fabry- Perot cavity setup, in which the optical spring vanishes on cavity resonance.

5.3.4 Conditional instability

For a sufficiently large dark-fringe offset, a second region of anomalous instability is predicted by theory [140, 142] for the (otherwise) damping detuning of the signal-recycling cavity. A comparison of the modified mechanical quality factor for operation on and off dark-fringe is depicted in Fig. 5.20. The anomalous instability occurs in the right figure, in a region of 2γtot <∆<4γtot. It is referred to as conditional instability in the following context, as it emerges only if the optical input power and the dark-fringe offset of the interferometer exceed a certain threshold. According to Eq. (5.42), the threshold detuning of the signal-recycling cavity δsr for this instability depends on the optical input power and the interferometer detuning from dark-fringe. Indeed, this behaviour is observable in the experiment. For a demonstration, the interferom-

98 tblt sitrutdol yteogigsito h sig the around of shift at ongoing cond mirror the The contra by only detuning. in interrupted is opposite stability observed, the is for instability ampliﬁcation parametric ’condit exponential this no During sel bility, a amplitude. into increased switches with behaviour instabi oscillation oscillator parametric the The and instantly, detuning). (zero resonance reached rdcdi ueydsesv opig nwihcs h d the case which in coupling, dispersive purely a in produced nnefo h etsd.At side. left appro the resonance), from from detuning onance (cavity time of function as ihtemmrn eoac frequency resonance o membrane increase the exponential with an causing instability, parametric h nefrmtrotu oe,nraie otemxmmt maximum the to sca P normalised fast power, a output with interferometer capture the (as peak resonance cavity undisturbed iedpcsteata esrdfato fotu power output Th of 5.11). fraction Fig. measured in proﬁ actual shown the (as resonance velocity depicts cavity scan line unperturbed high a the with shows captured trace 5.21. green Fig. Fig. in The depicted (compare is detector outcome photo time measurement locking power The output the The with actuator. triangu monitored piezo a was SRM with the again to once applied signal- scanned age, the was and position dark-fringe, mirror from cavity detuned manually was eter opposi the for instability conditional was power of input region second a as iue5.21: Figure max Pout /P max h xetdisaiiyo h etsd ftersnneis resonance the of side left the on instability expected The . 0.2 0.4 0.6 0.8 0 1 0 ntblt nbt ie ftecvt eoac.Tegenl green The resonance. cavity the of sides both on Instability P in = 2 t 0mW. 50 = 10 instability parametric . .Tecniinlisaiiycnntbe not can instability conditional The s. 5 4 t = ,tecvt ece h eieof regime the reached cavity the s, 2 . pomcaia interaction Opto-mechanical 5.3 Time (s) 6 conditional instability ω 0 At . 8 Cavity resonance t edtnn.Hr,the Here, detuning. te = ) rysosteis the shows Grey n). P asitbepower ransmittable 5 out . ,tecavity the s, 5 bevda well as observed /P nal-recycling oscillations f 10 oa’insta- ional’ max f-sustaining cigres- aching P toa in- itional tt the to st stops lity recycling out a volt- lar n sthe is ine grey e /P 5.10). series amp- eas le max 12 99

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer

ing for this sign of the signal-recycling cavity detuning δsr is always positive, never negative. During the parametric instability, the time series of the interferometer output power contained a dominant modulation at the frequency of the fundamental mechanical resonance of ωm/(2π) = 133 kHz. Dur- ing the self-sustaining conditional instability, the frequency spectrum of the output power time series was dominated by one a multiple of the fundamental mechanical resonance frequency, such that the time series contained a dominant modulation at ω/(2π)= n 133 kHz, with · n being an integer. For multiple measurements, n was mostly found within a small interval, n {1, 2, 3}. At this time, the exact dynamics ∈ causing this behaviour are not fully understood and need to be investigated in more detail, both experimentally and theoretically. It is possible that the effect is in analogy with the so-called limit-cycle behaviour [135, 143], known from linear cavity opto-mechanical setups. Here, the threshold of the conditional instability was investigated closer. Therefore, the membrane microscopic position was first chosen to maximize the transmittable output power Ptrans of the cavity while scanning the microscopic position of the recycling-mirror. The cavity resonance profile in transmission of the cavity was monitored continuously with the locking photo detector again. The scanning of the recycling mirror was stopped, and the SRM position was manually shifted towards cavity resonance by applying a high-voltage to the piezo ceramic transducer of the SRM mirror. Thus, both ∆ and δsr were reduced from an initial positive detuning (ap- proaching the cavity from the conditionally stable side of the resonance), and the relative height of the cavity resonance profile was denoted, allowing to calculate the detuning δSR. This threshold tuning was determined for different optical input powers and interferometer positions. The result is depicted in Fig. 5.22. Here, different lines correspond to measurements at constant interferometer detuning from dark-fringe (constant ksi). The dark-fringe offsets were chosen such as to (first) maximise the transmittable laser power through the system. Then, the dark-fringe offset was reduced to transmit a half (quarter) of the maximum transmittable power. For each input power and dark-fringe offset, the green areas indicate an area of conditional instability. On the right side of the figure, the relative height was converted to corresponding total detuning ∆ in units of the total cavity linewidth γtot. The existence of the conditional instability confirms the theoretical prediction qualitatively. However,

100 iue5.22: Figure odtoa ntblt.Lnsaeue oidct measur indicate (constant to dark-fringe used from are offset interferometer Lines instability. conditional odtoa aoaos ntblt.Rgt ovrino Conversion Right: detuning instability. (anomalous) conditional h okwt o nu oes( me powers This input low feed-back al electronic resonance. with is the leave lock in cavity to the gain the adaptable need an since no require but would is cavity, there ins detuned that (tuned) such the resonant power, for input a the occur th In increase with would to stabilisation. resonant possible the cavity on be switch the should to make and non-v power to with input powers be optical input might optical offset high dark-fringe for e solution mixer possible the A error-signal. saturate the to spoofing thereby found am and was huge oscillation The increasing reached. is the resonance encounte will before state instability detuned metric far Ap opposite, an r resonant. from will is resonance combi cavity interferometer the the the if before of re instability, possible offset conditional to not dark-fringe length is and cavity state power the detuned put Tuning far a 5.12). from manually Fig. with (see only fo sign) acquisi linewidth error-signal right ity lock the The (exhibits that meaningful non-trivial. stated becomes but lock be possible can is cavity it the ongoing section, of this subject summarise the To is thre values the calculated for numerically data the experimental of comparison quantitative a erdcdwe nraigteotclgi yahge input higher a by gain optical the increasing when reduced be Ptrans /P max 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1 ∆ ie nuiso oa linewidth total of units in given , esrdtrsodo h inlrccigcvt euigf detuning cavity signal-recycling the of threshold Measured 50 P P P P out out out in 100 (mW) =0.25max =0.50max =1.00max 150 200 P in . pomcaia interaction Opto-mechanical 5.3 ∆ < (γtot ) γ tot 0m) hc hnnesto needs then which mW), 10 0.5 1.5 2.5 k 0 1 2 3 . si .Genaesdpc ein of regions depict areas Green ). 50 h etdt nototal into data left the f mnsa particular a at ements P in 100 (mW) rahn the proaching h cavity the r h para- the r ntecav- the in research. ssae it state, is oinitiate to hl and shold ltd of plitude 150 sl na in esult tabilities lectronic anishing sonance e in- ned inof tion power. ready rthe or thod 101 low 200

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer

10 mW 50 mW 200 mW 80 80 80

70 70 70

60 60 60

50 50 50

40 40 40

30 Q = 21300 30 Q = 5112 30 Q = 1730

20 20 20

Relative noise power (dB) noise Relative 10 10 10

0 0 0

-10 -10 -10 130 135 140 145 130 135 140 145 130 135 140 145 Frequency (kHz) Frequency (kHz) Frequency (kHz)

Figure 5.23: Interferometer output power spectra (green lines) around the mechanical resonance frequency for different optical input powers Pin. The noise level were normalised to the individual, measured shot noise level (plotted in grey). The effective quality factor Qeff was inferred from the FWHM of the mechanical resonance.

The realisation of such an electronic feed-back is feasible and should then allow to operate the experiment precisely on resonance. For the interferometer on dark-fringe and operation precisely on cavity resonance, the optical spring is ’switched oﬀ’ and the quantum noise calculations in [86] hold.

5.3.5 Observation of optical damping

When increasing the optical input power Pin, the shape of the mechanical resonance in the interferometer output spectrum was broadened, indicating optical damping as expected from the complex optical spring Eq. (5.42). Moreover, the signal to shot noise ratio seems reduced, even though the optical input power was increased. This was attributed to the effect of optical cooling to an effective temperature Teff < Tini = 300 K, as is anticipated from Eq. (5.40). Interferometer output spectra for three different optical input powers are are shown in Fig. 5.23. The FWHM of the resonances was inferred from the measured output

102 opgi,nrb nicesdcnrllo adit.Tesi kHz 160 The about bandwidth. by loop detuned control thus co increased was higher cavity an a by by recycling reduced nor be not gain, could loop and lock during noticed was iiu rqec eouino zo h pcrmanalyse spectrum the of of factor Hz quality 1 a erro of absolute resolution the of th frequency factor, because minimum quality damping initial small high for of reliable Because less (3.17). Eq. mechanica by the tor determination direct a for allowing spectra, al 5.1: Table w spring, optical the to related was effect this that possible ooti h pcr,tesga-eyln aiywslock was cavity signal-recycling resonance, the spectra, the obtain mHz. To 224 is FWHM the inlwt orc ffe a hw nFg .2.I okdw locked of If error non-vanishing 5.12). a Fig. signal, in error shown pk-pk (as mV approp offset 100 an correct despite resonance, with on signal precisely cavity the lock to P of

in Q ± e! 10 10 10 10 iha r eghiblneof imbalance length arm an with 2 zcue iiu n aiu ffcieqaiyfactor quality effective maximum and minimum a causes Hz 3 4 5 6 -0.4 0 W8 z18413 671774 1687 1730 138.4 Hz 80 mW 200 0m 27 mW 50 0m 6 mW 10 eemndmcaia ult atr o ieetoptical different for factors quality mechanical Determined -0.2 δ E!ective quality factor sr Detuning P in 0 6 0.1γ 0.2 FWHM 5.8 200 mW . . 10 mW 50 mW sr ( z10651 765513 4766 5112 140.6 Hz 5 z18523712530666 16235 21307 138.5 Hz 5 0.4 δ · ncs of case In . sr 10 Q /γ ini 5 0.6 tot n eoac rqec of frequency resonance a and f ) res 0.8 k si ( kHz = 1 P . pomcaia interaction Opto-mechanical 5.3 .15λ 0.0125 in ) = | atro approximatley of factor noacut For account. into notclls of loss optical An 0 Wwt ninterferometer of imbalance an length with arm mW 200 inlrccigdetuning signal-recycling o he ieetotclinput optical different calculated powers three is damping for optical The factor quality 5.24: Figure expected. nt fttlcvt linewidth cavity total of units δ sr 0 W twsntpossible not was it mW, 200 | naslt esrmn error measurement absolute An . Q 6 eff . γ 0.1 P in = Q tot eff min neetv quality effective an , acltdeffective Calculated 0m,5 Wand mW 50 mW, 10 Q eff V ≈ P lock Q 1 sfnto of function as f in Q ult fac- quality l 0.1γ do cavity on ed smto is method is a taken was % eff max u othe to due r = nu powers input eff max it error riate = = ihpre- hich 3 kHz, 130 Q , sr 0 mW, 200 .15λ 0.0125 2300 0mV 10 tis It . eff min δ .For r. ntrol- sr gnal- t a ith γ 103 tot in is . .

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer vents operation precisely on cavity resonance. This seems conceivable because this detuning bears resemblance to the membrane mechanical resonance frequency.

The setup for the measurement is shown in Fig. 5.10. The output power spectra were analysed with the balanced homodyne detector in transmission of the cavity. Thereby, the full laser power was avoided on the detector, which would be the case for a detection in reflection of the cavity. All spectra were limited by optical shot noise. According to Eq. (5.40), a reduced quality factor is associated with a reduced, effective temperature Teff. When laser noise is neglected, this allows to calibrate the spectrum to m/√Hz, as the displacement noise close to the mechanical resonance has to match the predicted thermal noise. Then, the relative shot noise level is determined as thereby. It was found at 8.0 10−16 m/√Hz, and therefore higher than the value that · was estimated from the model presented in Sec. 5.2.3. This was attributed to the dark-fringe offset of the interferometer, which results in an over coupled signal-recycling cavity and thereby reduced the signal-recycling gain significantly. However, an independent calibration is required in a future setup. Here, the calibration follows out of Teff 1 K and Eq. (5.40). ≈ An accordingly calibrated interferometer output spectrum for an input power of Pin = 200 mW is depicted in Fig. 5.25. It shows the interferometer output spectrum around the fundamental resonance of the membrane, which was increased by about 5 kHz (3.8 % of the initial frequency). The orange line depicts thermal noise of the membrane with Teff 1.0K and Qeff 1730. The measured mechanical ≈ ≈ quality factors are in reasonable agreement with the theoretical prediction, if optical loss is considered and given the uncertainty of the experimental parameters. Also, the interferometer was not locked during these measurements. Instead, a constant dark-fringe offset during the measurements was ensured by scanning the position of the signal-recycling mirror before and after the measurment of the spectrum. Subsequently, the cavity resonance output power profiles were compared measurements were. In the case of same mean membrane position, the maximum transmitted power through the cavity is the same. The calculated optical damping for this particular optical input power is shown in Fig 5.24. The resulting shot noise level is higher than the value derived in Sec. 5.1.4. This was attributed to the fact that the measurements were taken with a small dark-fringe offset and

104 8 hnclrsnne h u fso os ge rc)adth and effect trace) The (grey line. noise dashed orange, shot as of Q depicted long-te sum is ordinary oscillator The the the to due resonance. increased chanical is frequency resonance inlrccigmro gentae.I h rtcs,the case, first the In trace). resonance (green mirror signal-recycling iue5.25: Figure etro ashdcagd-ohrie hseeti unexpla is effect this otherwise, - changed the of had excitation mass the reduced of was that center marker indicating the powers, of input level spectra optical noise I shot the manner. signal-to calibrate unpredictable the an to lar, in used varied displacement not its was cause method marker-peak The peak marker the prev about Comment all in observed was membrane mec SiN the a on f of influence measurements. about factor perceivable of quality conditi no fraction laboratory ical However, by a example temperature. to for as inve only influenced, the is stable lon of and long-term by frequency percent but is resonance spring, The membrane optical SiN resonance. the the t by of caused without drifts not frequency is resonance mirror measured freq higher cycling the towards to kHz 3 difference fundamen about The the by of shifted optic was frequency internal mode resonance to cillation the due measurement, cavity the the of In mismatch impedance the to

. 1/2 0 eﬀ Displacement spectral density (m/ Hz ) · 10 10 10 10 10 10 ≈ 10 -16 -15 -14 -13 -12 -11 1730 − 15 δ m orsodn to corresponding , sr / 130 airtdotu oe pcrmwt bu rc)adwith and trace) (blue with spectrum power output Calibrated √ 0.1γ < Hz. tot h ehnclrsnnei lal raee.The broadened. clearly is resonance mechanical The . T eﬀ ≈ 135 1 . .Terltv htnielvlte yields then level noise shot relative The K. 0 Frequency (kHz) . pomcaia interaction Opto-mechanical 5.3 Shot noisewithout SRM 200 mWwithout SRM Output spectrum 140 Thermal noise v ult atrwas factor quality ive Shot noise mdito h me- the of drift rm aiywscoeto close was cavity ra os of noise ermal membrane particu- n o higher for n such ons stigated uencies. lloss. al g-term inable. a os- tal ere- he han- ious 105 be- 145 out ew

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer

This was interpreted as follows: The excitation for the calibrated marker was not directly given to the membrane fundamental mode (as this is not possible), but instead was applied to the piezo indirectly connected to the silicon frame. In presence of a strong optical spring with perceivable suppression of membrane motion (optical damping), the frame might still perform the same movement but the membrane center does not. Since only the membrane displacement is read out, the marker peak will not be a reliable calibration method. If this explanation is correct, a possible solution is to actuate not the membrane (as it is subject to the optical spring), but introduce a signal by actuating one of the steering mirrors instead. The eﬀect of the optical spring to this massive object is expected small, such that it could provide a more reliable and independent calibration method.

5.3.6 Reduction of interferometer loss

If the optical losses within the interferometer can be reduced by one order of magnitude to about 0.1%, an even stronger optical damping (and cooling) is expected. Such loss should be feasible given the considerations in Sec. 5.1.5. In the case of this loss value, the effect of the optical spring on the effective mechanical quality factor and the mechanical resonance frequency is shown in Fig. 5.26. Here, different values for ksi are considered to illustrate the emergence of the conditional instability. The right subplot depicts the induced frequency shift of the mechanical resonance, which is found approximately 1 kHz around zero detuning of the signal-recycling cavity δsr.

A minimum value for Qeff 20 is achieved for a detuned signal- ≈ recycling cavity in the case of operation on dark-fringe. For a dark- fringe offset of (ksi = 0.005 λ), the effective quality factor on signal- recycling cavity resonance is reduced to about 60, a reduction of about 4 orders of magnitude. The achievable temperature in this case would be estimated according to Eq. (5.40) to Teff = 30 mK, which, in turn results in a mean occupation number oscillator occupation number of (compare Eq. (6.3)) of n = k Teff/(ωmh ) 4718, in the case of B ≈ an initial oscillator temperature of T = 300 K (room temperature). Conducting the experiment at cryogenic temperatures can decrease this number significantly. Subsequently, the application of cryogenic cooling is presented in the next chapter.

106 diinlotcldmigdet h ope pia sprin optical complex the to expe due as damping powers, input optical i optical additional the high in for broadened spectra t was output on resonance ometer as mechanical dark-fringe The from power. instabilit offset laser interferometer This the on c resonance. pendent cavity (otherwise) the signal-recycling on the instability of qualitativel conditional and a tested was observing spring by Th optical setups the spring. for both optical model individual cal Thus, their to regard with experiments. line compared opto-mechanical from known type features Perot multiple Miche exhibits signal-recycled interferometer the laser the in for interaction resonant cavity opto-mechanical the keep implicati to and used esti its setup scheme cavity locking to and realised experimentally discussed. model introduced, the was presents A was also sensitivity func interferometer loss input. (recycled) transfer optical the (signal) cavity of internal the impact as to the external regard for with both interfero investigated Michelson-Sagnac b was the established put in cavity, definitions mirror effective summarised high-reflective An and a cavity arrangement. t this linear in two-mirror scribe opto-mechanics a of of principle ple the described chapter This summary Chapter 5.4 nu the of method the to due is figure left the in traces the of end P iue5.26: Figure ihnteitreoee o ieetvle of values different for interferometer the within in Qe! 10 10 10 10 10 10 10 10 = 1 2 3 4 5 6 7 8 0 mW, 200 -1 Detuning E!ective quality factor 0 oicto ftemcaia ult atrwt . optic % 0.1 with factor quality mechanical the of Modification Q ini 1 = 0.0150 0.0125 0.0100 0.0075 0.0050 0.0025 ( k δ 5.8 si sr =0 2 /γ · tot 10 ) 5 3 and 4 ω m / ( 2π = ) fres (kHz) k 3 H r sue.Teabrupt The assumed. are kHz 133 130 132 134 136 si -0.4 nuiso wavelength of units in . hpe summary Chapter 5.4 Detuning -0.2 Frequency shift eia calculation. merical 0 sbt de- both is y ( h chapter The δ xlisthe explains lson-Sagnac confirmed y sr ee out- meter oigside ooling r Fabry- ar, theoreti- e /γ ed The field. 0.2 tdfrom cted placing y eexam- he einput he .From g. tot nterfer- λ n to ons ) ode- to Here, . tions, lloss al mate were 0.4 107

Chapter 5 5 Signal-recycled Michelson-Sagnac interferometer the FWHM of the mechanical resonance, a minimal effective quality factor of Qeff = 1730 could be inferred, which is in good agreement calc with the theoretical prediction of Qeff = 2300. In contrast to previous experiments [111], a balanced homodyne detector was used for read out, and it could be confirmed that the spectra were limited by optical shot noise.

108 n xeietlprmtr,terrtoi ie by given is ratio their parameters, experimental and o elsi cnro( scenario realistic a For be h ai ftemlniet aito rsuenoise pressure yielding (4.26), radiation Eq. to and noise (3.25) thermal Eq. form of b derived will ratio effects The pressure excited radiation able. thermally before by long membrane caused the fluctuation measurem of the phase as the signal-recycling, by by limited increased measur be the to when is interest disp sion particular applicatio the of of the t is enhancement This 4.29), the significant sensitivity. a by cause Fig. limited can (compare is cooling oscillator genic interferometer the the of which noise in situation a temperatures In low for need The 6.1 overv an and solutions. K possible 10 and below problems m identified temperatures of about presentation at the spectra with closes displacement chapter te The cryogenic a presented. at principle are operation function for prototype its interferometer system, exp tom cryogenic associated The and Michelson-S challenges. temperatures a cryogenic operating at of effort terferometer the addresses chapter This 6 interferometer the of cooling Cryogenic s hpe 6 Chapter G G therm vis rpn = s r m 2 4π = hr

mk .,ω 0.3, m 2 B G Tω sr P 0 0 / 0 /Q ( / 2π ( cλ = ) ) ∝ 5kHz 75 s Q T m , . mn preci- ement mperatures da cus- an nd n ilbe will ent = fcryo- of n erimental ga in- agnac observ- e lacement embrane 2 ng) 125 motion hermal a be can (6.1) 109 iew

Chapter 6 6 Cryogenic cooling of the interferometer

Gvis 1W λ 106 T 1 therm = 442.5 . (6.2) G P 1064 nm Q 300 K G s rpn · r r s r r sr For the reasons discussed in Section 5, the signal-recycling gain will be limited and probably √Gsr < 30. Without signiﬁcant enhancement of the input power (power-recycling), and for typical laser wavelengths 532 nm <λ< 1550 nm, achieving a unity ratio is rendered impossible at room temperature.

6.2 Concepts for cryogenic cooling

Cryogenic cooling can be provided in several ways, and there are various cryogenic topologies for manifold experimental situations [144]. The choice of which cryogenic system to use depends on several aspects. Obviously, the minimal achievable temperature is important, but the complexity of the cryogenic system itself and the effort to operate it also impact the decision. Operation and maintenance costs are to be considered as well. Usage of ultra cold gases, such as 4He or its rare, much more expensive isotope 3He allow for temperatures below 4 K and < 300 mK, respectively. Modern dilution refrigerators also allow for a ’dry operation’, without the need of external supply of liquid gases, which simplifies their application as infrastructural requirements are eased, e.g. no helium gas recovery lines and collecting is necessary. However, pumps and compressors are still required. These components typically contain moving parts and produce noise and vibrations, which are to be kept to a minimal when performing low frequency (f 6 10 kHz), high precision measurements nearby. Thus, application of cryogenic cooling in gravitational wave detectors is still uncommon. Future gravitational wave detectors are expected to be limited by mirror thermal noise in their most sensitive frequency region however (compare the anticipated noise budget of the advanced LIGO detector in Fig. 1.1). For this reason, the usage of cryogenic temperatures in a GW detector will be realised in the Japanese KA- GRA [145] detector for the first time. Beyond this, temperaturess of around 10 K are also anticipated for the Einstein telescope [146] to overcome sensitivity limitations caused by thermal noise. In opto-mechanical experiments, cryogenic cooling is well-established. This is mainly due to two reasons. First, the majority of these exper-

110 siltro eoac frequency resonance of oscillator AI.Tesse sacso ae lsdcycle closed In made, Albert-Einstein custom the a at is up system set The been inter (AEI). has Michelson-Sagnac temperatures a low of at [ operation thesis for masters supervised system a cryogenic within and thesis, this During cryocooler Gifford-McMahon The 6.3 occup state ground close-to unr to the lead in also cooling [53]. can numbers Optical regime 50]. band opt [49, side resolved techniques assisted stat cryogenic cooling ground of ult the band combination the Instead, a to cryostat. by coupling the i reached pure by by as supplied achievable bath (such not heat frequencies is state resonance ground lower the with r oscillators is For reader [147 the order in methods [148]. cryogenic two to reported about temperature and overview base extended mK 20 an the than about higher art of the magnitude state-of temperatures than base temperatures larger magnitude for refrigerators of state order ground one its perature in is system nnefeunyof sta frequency ground mechanical onance its to brought micro-oscillator first mech number a occupation of phonon state small ground a opto-mechani mechanical e.g. in the oscillator, osc achieve cooling micro-mechanical to cryogenic the is for periments cool motivation system only cryogenic major to the One designed co of or very volume working times small the often kept that are such setups measureme size, required in the the second, degrade And likely come. less disturbances mechanical innme sgvnby given is number tion e ento,temcaia rudsaeo oini reac is motion of state ground mechanical the definition, Per 1 < n frequencies high very at operate typically iments n xaso fa da a hr,hlu) aetemperat base a helium), (here, gas ideal com controlled an By of 156]. expansion 155, and [144, cycle thermodynamic 154] Research Jannis sflle.CnieigE.(.)i sntsrrsn that surprising not is it (6.3) Eq. Considering fulfilled. is 11.I ssatosae iodMMhn[152– Gifford-McMahon two-staged a uses It [151]. E = ω n m hω

/ ( 2π m = = ) . h iodMMhncryocooler Giﬀord-McMahon The 6.3 k B ω H 5] o hsfeuny the frequency, this For [51]. GHz 6 T 0 ⇔ n temperature and n = hω

k B T m < T > f n 4 . ecycoe by cryocooler He o harmonic a For . 8 K tem- a mK, 280 0 H,where kHz, 100 T h occupa- the , ehdares- a had te 5] ﬁrst a 150], ferometer e when hed clside- ical [149]), n dilution- a be can e t out- nts pression esolved acold ra eferred a ex- cal illator. a be can stitute mpact .For ]. r of ure anical ation (6.3) of s 111 the

Chapter 6 6 Cryogenic cooling of the interferometer

6 K is achieved. A review about the state of art cryocoolers is given in [157]. Their principle of operation can be understood when recalling the ideal gas law

pV = nRT , (6.4) where p is the pressure, V is the volume, n is the amount of substance, T is the temperature and R the universal gas constant. Equation (6.4), which can be used to understand individual steps of the cryocooler’s thermodynamic cycle.

6.3.1 Thermodynamic cycle This section outlines the working principle of the Gifford-McMahon cryocooler and its thermodynamic cycle. The system is composed of a closed volume (cold head) and a motorized displacer unit, which sep- arates the volume into two parts (which will be referred to as cool and warm side of the volume). The displacer is tight, such that gas can only be exchanged via the so-called regenerator in the displacer. More- over, the warm side can be selectively connected to a high pressure reservoir of gaseous helium at 200 bar (P2) or atmospheric pressure (P1). A schematic of the cold head is depicted in Fig. 6.1 a). GM cryocoolers are simple, cost efficient devices with a high reliability, as there are few, slow moving parts at room temperature. Generally they are cheaper than pulse type cryocoolers, but exhibit a higher vibration level due to the moving mechanical displacer unit. Vibrations caused by the expansion and compression of the gas inside are present in both types, however. The system undergoes a thermodynamic cycle which can be split into four consecutive steps c) - f) , which will be described briefly in the following and are depicted in Fig 6.1. c) The displacer is on the left side of the cold head, the high pressure valve is opened. Gas streams into the volume and is stored on the right (warm) side of the cold head. The pressure is increased from P1 to P2 at constant volume, such that according to Eq. (6.4) the temperature of the gas must increase. The water cooling of the compressor is used to cool the gas. d) With the high pressure valve opened, the displacer is moved to the right side. The working volume (on the left, cool side) is maximised, the gas streams through the regenerator. The pressure

112 c) iue6.1: Figure a) ouepaeadGffr-cao yl.c )rpeetdiffer represent v e) head - ri cold c) the cycle. the cycle. the to inside Gifford-McMahon volume pressure and left-sided plane atmospheric volume the or from high flow for allow can gas which through )Nw h ihpesr av scoe n h o pressure low the and closed is valve pressure high the Now, e) )Wt h ae o-rsuevleoee,tedslcri displacer the opened, valve low-pressure same, the With f) f) odsd Warm side Cold side Volume He ieto n xrcssm eteeg u fteregenerat cooled. the rev slightly of in out is regenerator energy the regenerator heat through some flows extracts pres gas furth and the the is direction of because temperature part cooled, its A is that creased. such side volume, left constant the at on drops gas The opened. is tr vraanfo tt ) h nieccerpaswith repeats cycle entire and state The initial the c). Hz. in 1 state is of from frequency cycle h again The some over latter. taking start the regenerator, of the M out through volume. energy flowing working is the minimizing gas side, exhaust left the to back moved hog h eeeao,apr fisha nryi trdi stored passe is energy gas heat the its When of part regenerator. a step. regenerator, this the during through high constantly remains 4 Regenerator Displacer )Tecl ed ooie ipae otisteregener the contains displacer motorised A head. cold The a) Motor Gas !owGas P P P Valves (closed) 1 1 1 . h iodMMhncryocooler Gifford-McMahon The 6.3 1 Hz P P P 2 2 2 e) d) b) P P P f) 2 1 )e) c) V 1 d) f) V lm.b pressure- b) olume. 2 h.Tovalves Two ght. n iutosin situations ent V P P r the or, now s 1 1 rde- er the n valve sure ator, erse 113 can ore eat P P 2 2 a s

Chapter 6 6 Cryogenic cooling of the interferometer

It can be shown [158, 159], that the total refrigeration Q of the cycle is

Q = V dP = P dV = V(P2 − P1) , (6.5) I I where P2 and P1 are high and low pressures and V is the working volume inside the cold head.

6.3.2 Cryocooler components This section brieﬂy outlines all components required to operate the cryocooler. A schematic of the setup is shown in Fig. 6.2.

Cold head As described, the cold head consists of a motorized displacer unit with integrated regenerator. An inlet and an outlet valve connect the room temperature side of the cold head with the compressor unit.

Vibration isolation stage Cold head and vacuum stage are connected via a passive isolation stage of rubber bellows as depicted in Fig. 6.2. Internal heat links are flexible, such that no rigid connection is required to establish a thermal contact between the cold stages and the sample stage. During operation, the cold head is fixed rigidly to a massive superstructure of Item profiles with no connection to the optical table. To open the vacuum chamber, the cold head and vacuum stage are fixed with steel spacers, and the connection to the superstructure is released. Thus lifting the cryocooler by means of a crane is possible. A similar vibration isolation has been developed for CLIO, a precursor of KAGRA [160].

Exchange gas Puriﬁed helium 4.6 (purity > 99.996 %) was used as exchange gas. To avoid contamination of the system, the cold head had to be evacuated to 10−2 mbar before inlet of the helium.

Regenerator The regenerator consists of a porous or granular material with small ﬂow resistance and ideally no thermal conductivity in ﬂow direction. Lead was formerly used, but modern re- generators use lanthanide materials like Er3Ni or HoCu2, which are well suited due to their high heat capacity [161].

Compressor unit A compressor unit was used to recycle the exhaust, low-pressure gas from the cold head. The compressor type was a

114 aumsystem Vacuum shield Heat eprtr controller Temperature . nefrmtrpooyefrcryogenic for prototype Interferometer 6.4 nyt oltemmrn,btteetr nefrmtr in interferometer, entire the decide but was membrane, It the desired. cool was to configuration only ex stable initial and the with compact experience at A the operation on for based interferometer was prototype temperatures the of design The rvddpevcu f10 of vacuum pre provided h apehle n a edslydo tblsdt de- 33 a Model to a is stabilised unit or controller t by displayed manufactured The attached be temperature. diode can working silicon and sired calibrated holder col a the sample of with min- the temperature the measured The until is head, reached. finger cold is the temperature of base side imal temperature low the down rvdn rsueo 10 of pressure a a was providing unit This pump. molecular ftrepmswsue.A used. was pumps three of t optics. on interferometer condensate the will of therm gas instead in residual shield, brought finger heat is cold interferometer the stage the cold with first before contact cen the that the to such reach connected to K, is co light 60 shield The It laser the chamber. walls. allow the outside of to ’hot’ holes small the two of tains radiation heat from chamber co water required itself overheating. compressor avoid The to oil. residual inate C41from HC-4E1 eprtr,i eesr,adteeyatae h temper the r actuates to thereby and thermally necessary, finger if cold temperature, the contacts loop control feedback ueisd ftevcu akwsmntrdwt combined from a 90 with PTR monitored pres- was gauge The tank Pirani vacuum Penning/ vibrations. the the avoid with of level inside to vacuum sure off the keep chamber switched to vacuum pumps used the mechanical was to pump getter conncection ion available An only the of tion environment . nefrmtrpooyefrcygncenvironment cryogenic for prototype Interferometer 6.4 ealcsil rtcsteinrpr ftevacuum the of part inner the protects shield metallic A opoieterqie aumlvl combination a level, vacuum required the provide To Sumitomo aeSoeCygnc Inc Cryogenics Shore Lake hnsatd h roolrbgn ocool to begins cryocooler the started, When tas lee h eimgst elim- to gas helium the filtered also It . − − 5 2 brdet h iie rs sec- cross limited the to due mbar Variant brt prt eod turbo second, a operate to mbar fie Vacuum Pfeiffer elknLeybold Oerlikon rSrl 0 colpump scroll 300 TriScroll nitgae PID integrated An . iae300, HiPace periment. cryogenic . cluding ature. not d oling oom 115 tre he at n- 5, al d o .

Chapter 6 6 Cryogenic cooling of the interferometer

Compressor unit

Motor Scroll pump

Rubber isolation Turbo molecular pump 10 -3 mbar

Pirani / Penning High vacuum cold nger gauge valve

He 4 Vacuum chamber 200 bar 10 -6 mbar Ion getter pump

Figure 6.2: Schematic of the cryogenic system. A compressor is used to recycle the exhaust helium from the cold head. The motor and cold head are separated against the lower (vacuum stage) by a rubber isolation and fixed to an external superstructure of item profiles. The heat shield is depicted as dashed black line. The cold plate is depicted gold. A combination of scroll, turbo pump and ion getter pump provides the necessary vacuum level, which was measured with a combined Pirani/Penning gauge. all optics and the interferometer baseplate in order to eliminate heat radiation from close-by components to the membrane. Such effect has shown to limit the achievable base temperature in other experiments, for example in [130]. The heat transfer rate per unit area Q/A˙ caused by heat radiation of a nearby component of area A1 can be calculated with the modified Stefan-Boltzmann law [144] to be

˙ Q 4 4 = F12ǫ12σ(T1 − T2 ) , (6.6) A1 where T1 and T2 are the temperatures the two surfaces and F12 is a configuration factor relating the two surfaces. ǫ1 is the emissivity of surface 1 (ǫ = 1 is a perfectly reflective surface, ǫ = 0 corresponds to a perfect black-body radiator) and σ is the Stefan-Boltzmann constant. If geometrical aspects shall be left out, it becomes clear from equation (6.6) that, if possible, all components should be cooled, such that no or little net heat flow is present. Due to the cryocooler layout, the interferometer had to be attached

116 iersepr cnigsaeAP11 oainsaeAN stage rotation A ANPx101. stage scanning stepper/ linear o prto tlwtmeaue,i a eie oueasta a use to decided was it temperatures, no low three interfe are an at transducers ’scan’ ceramic operation po to piezo for be path standard a Since had beam the Sagnac it fringe. along ter overlap third, membrane to and, the order interferometer shift to in the membrane of must the holder mode membrane Michelson of the alignment Moreover, fine repla be for experiments). to had all membrane the in ess First, (as Three met. crucial. be to is were membrane quirements the aligning and Positioning positioning and alignment Membrane 6.4.1 c conductivity. was thermal good Copper its cryocooler. to due the material of baseplate cold-plate as the to down upside euemut.Tecnrlba pitri iil one i 2 mounted of rigidly diameter is a in splitter has mounted beam and are central mirrors The steering mounts. The flexure plate thick diameter. cm cm 5 12 a with is baseplate interferometer The prototype. iue6.3: Figure forBlock-out recycling !exure mounts Top adjustable SiN membrane Attocube A eino h o-eprtr ihlo-ancinterf Michelson-Sagnac low-temperature the of design CAD . nefrmtrpooyefrcygncenvironment cryogenic for prototype Interferometer 6.4 oiinr.Telws oiinri hssaki a is stack this in positioner lowest The positioners. . 4cm. 54 Stainless steel retainer 0.5“ Steering mirror 1“ Beam splitter1“ Beam Siskyou fd-xdzdcopper, de-oxidized of t utmholder custom a nto 12 cmcopper baseplate o adjustable top specified t nilre- ential erometer ceable rome- hosen ssible allow R101 kof ck 117 nd

Chapter 6 6 Cryogenic cooling of the interferometer

a) b) ANR v51 rotation stage Copper holder

Copper wires

ANR 101 rotation stage

ANPx101 linear stepper / scanner

Figure 6.4: Left: Attocube positioner stack. Right: The membrane is glued on a copper holder, which is connected to the cryocooler cold plate by copper wires. Two remote controllable Attocube positioners allow for adjusting the orientation of the membrane in pitch and yaw. A third linear stage below allows to scan and step the membrane along the direction of the beam.

is mounted on top, allowing for rotation in the plane of the interferometer baseplate (yaw) and, ﬁnally, another rotation stage ANRv51 allowed for pitch alignment. The two rotation stages are depicted in Fig. 6.4. The positioner speciﬁcations are given in Table 6.1.

The use of piezo actuators between the interferometer baseplate and the membrane was expected to provide poor thermal conductivity and reduced thermal contact between the interferometer baseplate and the membrane. The best cooling was expected for a direct connection from the membrane to the cold plate. Thus, the membrane was ﬁnally mounted on a small holder made of copper, which was connected to the cold stage directly by several copper wires. The mount contained a cone-shaped drilling on the back side to avoid beam clipping.

The Attocube positioners were found to introduce a significant noise level because of the noise in the applied high-voltage. Without electronic low-pass filtering, the output spectra were spoiled by the positioner noise. The controller unit ANC300 provides two integrated low-pass filters with corner frequencies of 16 Hz and 160 Hz respectively, which provided a sufficient noise suppression and thus had to be enabled during measurements of the interferometer output spectrum.

118 n tcygnctmeaue [162] temperatures cryogenic at ing vn rqec ( frequency evant prt h xeieta hsotclwvlnt.Telase The a wavelength. was decid optical choice was this it of at minimised, be experiment to the was 15 membrane operate of the wavelength mini to laser the entry a heat that at As achieved found was was membrane the It to absorpti [163]. entry in optical wavelengths their laser for ferent investigated were membranes SiN setup optical The 6.4.2 6.1: Table h I ee fti ae a esrdt be th to masters measured supervised was a laser Within this of stage. level amplifier RIN W the 5 a for laser tonics h aeitniysaiiainshm a e p providi up, set was scheme lasers. NPRO stabilisation typical intensity than specification same worse manufacturer magnitude the the of order with one agreement about good in found ih a ptal lee yarn oecenrcvt and cavity cleaner mode interferometer. ring the a into by injection 4.3.4 for filtered available Section spatially (see modul stage was i for stabilisation rate light EOM power extinction an active through higher the transmission even and After an provide direction. int to ward w an table isolator with Faraday optical Th collimator additional the An fibre source. combined isolator. laser a optical Faraday the from noi in emitted internal included was an light by already suppressed was re be which laser could The function, kHz 6.5). 330 Fig. at (compare oscillation kHz 100 at suppression noise cnigrne@30K 300 @ range scanning cnigrne@4K 4 @ range scanning rvlrne5mm 5 range travel i.se ie@4K1 m0 nm 10 K 4 @ size step min. i.se ie@30K5 m1m 1 nm 50 K 300 @ size step min. odcpct 0 0 0g 30 g 100 g 100 capacity load eouinsb nm sub- resolution tcnitdo 0m belsr hc a sda seed as used was which laser, fibre mW 50 a of consisted It . Attocube . nefrmtrpooyefrcygncenvironment cryogenic for prototype Interferometer 6.4 oea osi E-15 Boostik Koheras eoepstoe aafrmmrn lgmn n position and alignment membrane for data positioner remote < f H) opr i.65 10.Ti was This [150]. 6.5. Fig. compare MHz), 1 6 ierrtto rotation rotation ANRv51 linear ANR101 ANPx101 6 5 0 . µ 8 m µ auatrdby manufactured m 6 360 µ 6 . ◦ m 5 70 14 2.5 ◦ ◦ m m ◦ · ◦ ◦ 10 − 6 0 m 1 360 6 µ 6 . ◦ m 5 ,telaser the ), o h rel- the for K Pho- NKT 40 6 sue on used as ◦ na dif- at on ◦ m ng a heat mal system r laxation m eeater se ◦ back- n ◦ 0nm. 50 laser e ,and s, ◦ 15 Here, ation ernal dto ed esis, now 119 dB -

Chapter 6 6 Cryogenic cooling of the interferometer

) noise eater OFF free running noise eater ON stabilised 1/2 10 -5 10 -5

-6 -6 10 10 x 0.17 Relative intensity noise (1/Hz intensity Relative

100 k 1 M 50 k 100 k 150 k Frequency (Hz) Frequency (Hz)

Figure 6.5: Left: Relative intensity noise measurement of the 1550 nm laser system. The RIN level at 100 kHz was found to be 2.0 · 10−61/√Hz. For 10 A ampliﬁer ≈ current, the laser relaxation oscillation was found at 330 kHz and could be suppressed by an internal noise eater. Right: performance of the laser stabilisation. The RIN level is suppressed by a factor of about 5.6.

6.4.3 Interferometer installation procedure

Because of the cryocooler layout, in which the interferometer is hanging upside down in the closed vacuum chamber, the interferometer could not be aligned in the cryocooler. The alignment of both, Sagnac and Michelson mode is already difficult when IR-Cameras provide vi- sion and optical sensor cards can be used to follow the various beam paths within the interferometer. Aligning the combined interferometer as ’black-box’ is not possible. Thus, another installation procedure was chosen. First, the distance between the last steering mirror (or any other point of reference along the beam path) was determined by the CAD construction of the interferometer and the lengths specifications of the cryo chamber and an auxiliary optical path of the same length was established outside of the tank. A flip mirror was used to switch between a position for the interferometer in- or outside of the cryocooler. The interferometer was then assembled and aligned on the optical table following the procedure given in Section 4.4.4, using the auxiliary beam. The outside alignment of the interferometer was complete when an interference contrast of C = 97 % was established, and the beam reflected by the interferometer was transmitted through the initial mode cleaner cavity. In this situation, the interferometer could be treated as a ’black-box’, which is an effective mirror for the ’correct’ optical input mode. Thus,

120 efrmtrbspae hncpe ie eecapdt t to clamped were wires copper Thin drilli through baseplate. guided Cabl were terferometer positioners cold-finger. cou piezo the cryocooler’s interferometer of the the nection to and down opened s upside was a installed chamber by viewports. cryo corrected optical the be the Next, could close o which mirrors thus mode, steering can two input contrast of wrong ins interference the interferometer reduced by the the caused a of that chamber, case assuming cryo In the cryocooler, stable. the into remains installed alignment be could it instal the rotated simplicity. encased be for shield here to heat shown o has The not set interferometer is a viewports. but the by the figure, interferometer for this the account in to aligned that l m is Note is w interferometer) beam the copper and input between insulated plate thin, optical cold contact by (including thermal head established cryo A is the plate cold plate. the suffic do cold and upside bot with cryocooler installed When splitter the is beam to interferometer aligned. the t entire and on the on installed overlapped contrast, aligned is are is membrane mode interferometer the Michelson Sagnac step, the second First, a plate. cold iue6.6: Figure Complete Sagnac alignment Step 1 ntlainpoeueo h nefrmtrit h cryo the into interferometer the of procedure Installation . nefrmtrpooyefrcygncenvironment cryogenic for prototype Interferometer 6.4 Align Michelson-Sagnac membrane +positionersInsert Step 2 Close system heat shieldInstall Attach IFOto cold plate Step 3

Copper wires nadatce directly attached and wn wrd(lsd n the and (closed) owered Cryo chamberCryo rs hnattached, When ires. Cold !nger height 75mm Viewports eotcltbe In table. optical he Cold plate e interferometer, led xenlmirrors. external f y10dge to degree 180 by etinterference ient mrn holder embrane g ntein- the in ngs ,Sga and Sagnac h, hme and chamber sfrcon- for es emem- he aldto talled internal l be nly dbe ld 121 et

Chapter 6 6 Cryogenic cooling of the interferometer

a) b) Item superstructure 2x ANC300 controller unit Temp. controller unit Crane connection Motor unit

Connection to the ITEM Frame

Rubber isolation

High-vacuum valve

Vacuum chamber

Copper base plate d) c)

Figure 6.7: Photos of the realised cryocooler setup. a) Crycooler, Item superstructure, temperature and Attocube controller units (left). b) Cold head and connection to the Item structure. Helium inlet and evacuation hose for the cold head are attached. c) Lower part of the cryocooler: Vaccum chamber with view ports, rubber Viton isolation and vacuum hoses for the high vacuum chamber. d) Interferometer during external alignment process (without membrane holder installed). brane holder in order to improve heat exchange between membrane and cold plate. After isolating the copper wires with thin Teflon tape, the cryocooler radiation shield was attached and the cryo chamber was closed by lowering the cryocooler. The Teflon was used to bundle the copper wires, avoiding contact to other surfaces (and thereby reducing the heat transfer) and also allowing to prevent the wires from hanging in the beam path. The work flow of the installation process is depicted in Fig. 6.6.

122 wthdo eoetemaueet n h ihvcu valv vacuum high the and measurement, pu a getter the ion well before The as on monitored. temperature switched were comp cold-finger system the the the within and temperature, pressure off base switched the was cham i reached unit cryo the cryocooler the except the to valves pumps After the all and cool-down, opened. running, the were kept During were b pump interferometer cooled. getter massive be the d that to cool fact had in the difference to The sh attributed operation. K was of 8 Accord hours wor of 1.5 temperature slowly. starts after base reached drops immediately a just specifications, compressor temperature manufacturer the the the depi Although on, is w switched measurement pressure 6.8. when The the and Fig. measured. temperature in were finger system mounted cold cryogenic interferometer the the the plate, with sequence cold down the cool first a In performance Cooling K. d 10 i obtained below the presents operation and on interferometer membrane temperatures an the low and of sensitivity effect ometer the describes section This Results 6.5 example, minutes. for 360 after of, measured surfaces is syst cold K the the 10 below on within calibrate temperature out a pressure freeze via The molecules finger gas cold resistance. cryocooler electrical the pendent at measured is ature iue6.8: Figure Cold !nger temperature (K) 100 300 10 30 5 1 oldw ftesse ihitreoee ntle.Tet The installed. interferometer with system the of Cool-down Cold !ngertemperature 10 Pressure Time (min) 100 mi eraigfor decreasing is em ,tmeauede- temperature d, h etsil.A shield. heat the . Results 6.5 1000 w time own s plate ase pwas mp udbe ould nterfer- t for ata n to ing emper- ressor below 1 2 3 4 5 6 7 8 the s ithin king cted to e 123 ber on Pressure (10 -6 mbar)

Chapter 6 6 Cryogenic cooling of the interferometer

30 30 Cold !nger temperature Pressure

20 mbar) Compressor switched on -6

10 10 Pressure (10 Cold !nger temperature (K) Cold !nger

5 0 5 10 15 20 Time (min)

Figure 6.9: Switching off the compressor of the cryocooler results in a slow increase of cold-finger temperature. The pressure inside the vacuum chamber shortly in- creases, but then stabilises on a sufficiently low level to operate the experiment. the TMP and scroll pump was close, only the evacuation valve to the sample chamber was remained open. After 14 minutes, the compressor was started again. The temperature immediately started to drop, reaching 7 K after about 5 minutes. The measurement indicates that the temperature increase is rather fast without the compressor running. Averaging the output spectra was thus not possible at minimal temperature, because the mechanical resonance frequencies shifted by more than the mechanical linewidth during two consecutive measurements of the output spectrum (compare Sec. 6.5.2).

6.5.1 Cryogenic interferometer performance

During the cool process, the membrane position was continuously scanned to observe the interferometer contrast, piezo travel range and optical output mode. All positioners were found to be operational at 8 K. The interferometer fringe with the maximal amplitude of the applied high voltage (U = 150 V) contained two dark-fringes, corresponding to a maximal scanning range of 400 nm, which is within the manufacturer speciﬁcations for the ANPx101 under load and at low temperatures. The interferometer contrast was found to be severely reduced from 97 % to about 70 % within the ﬁrst 20 − 40 K below room temperature, and was found to stabilise at C 20 % at a temperatures of ≈

124 h eu sdi eitdi i.6.10. Fig. in depicted su is not was used thus setup the and The interferometer, The the noise. inside subtract laser generated be correlated was could operated the outputs AC were cancelled two therefore The both electroni and and power. detectors beam match, The mean the same the light. to p laser adjusted and were input interferometer functions the the detect of photo of fraction auxiliary front a an in with positioned readout was to the detector extended in was por noise exci setup output laser the thermally Therefore, the the the membrane. in resolve the to of photodetector nances unable single was a interferometer output with dominated the noise readout laser hi a a the in Thus, and resulted contrast noise interferometeric laser reduced nical the Both, K. 20 el subtracted is detectors analyser. photo spectrum two a the sp into of install beam output is light polarising which filtered interferometer, laser a AC the the of onto of incident reflection is fraction in laser A detector cavity. photo cleaner auxiliary mode a by filtered iue6.10: Figure λ 2

EOM EOAM PBS ceai ftecygnceprmn.Telsrlgti spa is light laser The experiment. cryogenic the of Schematic Mode cleanercavityMode 5 W Laser@1550nm λ 2 PBS environment highvacuumCryogenic,

PD 1 Laser noise reference AC Substraction Beam splitterBeam R =10% - PD 2 dit h roolr The cryocooler. the into ed Spectrum analyzerSpectrum Beam splitterBeam crnclyadfed and ectronically itr h residual The litter. spce po an on up picked is Oscilloscope . Results 6.5 SiN membrane mlsignal rmal spectrum. transfer c btracted. e reso- ted ce up icked r This or. htech- gh CCD Camera cancel with tially of t 125 ed

Chapter 6 6 Cryogenic cooling of the interferometer

155 245 (1,1) Mode (1,2) Mode 150 240

145 235

140 230

135 225 (1,1) resonance frequency (kHz) resonance (1,1) (1,2) resonance frequency (kHz) resonance (1,2) 130 220 100 150 200 250 300 Temperature (K)

Figure 6.11: Resonance frequencies of the (1,1) and the (1,2) oscillation mode during cool down. Both modes experience a frequency decrease of about 10 % of their initial resonance frequency.

6.5.2 Influence of low temperatures on the membrane Cryogenic temperatures also affected the SiN membrane mechanical resonance frequencies. Figure 6.11 depicts the measured resonance frequencies for the (1,1) and (1,2) mode of a low-stress membrane during the cool down process. The resonances frequencies were found to decrease to about 90 % of their initial values, indicating a reduction of internal stress (compare Sec. 3.4.2). A direct measurement of the membrane temperature was impossible as no temperature sensor could be attached to the fragile membrane. It could be assumed that the membrane resonance frequencies are stable in case of thermal equilibrium with the membrane environment. The membrane mechanical quality factor was determined for both, cryogenic and close-to room temperature (T = 260 K) environment by ring-down measurements. As already reported in [77], the quality 5 factor increased by about one order of magnitude, Q300 K = 5.2 10 6 · and Q8 K = 6.8 10 for temperatures below 20 K. Our observation · is different from [77], where the Q factor increased rapidly for temperatures below 8 K, such that it might be attributed to a different mechanism, which might be related to the method of fixing the membrane frame. Figure 6.12 shows a comparison of the exponential decay of the oscillation amplitude after sudden stop of the excitation. Calibrated interferometer output spectra were obtained at temperatures T = 260 K, 20 K and at 8 K, respectively, and are shown in Fig. 6.13. All three spectra all three are dominated by laser noise due to

126 iue6.12: Figure eetaserdt h xeietvatevcu connectio outer vacuum the the on via fixed experiment were compress the these the of to vibrations high transferred to were due was runn This without obtained unit. were pressor temperatures low for spectra The frequency. resonance mechanical rqec ositb oete h ehncllnwdhper linewidth mechanical the then res the more caused by also shift This to off. frequency switched was increa compressor to the co started when immediately spectra temperature the the mec factor), as narrow averaged, the quality of mechanical Because (high indirectly. linewidth motor the of motion the rn t20K(rycrls n at and circles) (grey K 260 at brane eue otata o eprtrsa icse ntene the of in sensitivity discussed displacement as a temperatures with low tion, at contrast reduced th a for used not were seconds last the that such that th (such zero), that level same dark-noise Note the relative different graph. slightly semi-logarithmic a had lower, the on depicted Normalised amplitude |x/x 0| ln(|x/x 0|) 0.2 0.4 0.6 0.8 -2 -1 0 0 1 -2 -2 igdw esrmnso o-tes lme iio ni silicon clamped low-stress, a of measurements Ring-down 0 0 2 2 4 4 6 6 8 8 10 10 item T = 12 12 gencrls.A xoeta tis fit exponential An circles). (green K 8 Time (s) Time (s) rfie,adteeytransferred thereby and profiles, 14 14 5 16 16 · 10 18 18 − 15 20 20 re osntapproach not does green m / w measurements two e 22 22 √ fit. e 260 K!t zaon the around Hz 24 24 8 K!t 260 K 260 . Results 6.5 8 K l o be not uld esteadily se 26 26 rd mem- tride r which or, n com- ing second. hanical onance tsec- xt s(as ns 28 28 127 30 30

Chapter 6 6 Cryogenic cooling of the interferometer

a) ) -11 Output spectrum 1/2 10 Dark noise Thermal noise 260 K -12 Laser noise 10 Sum

10 -13

10 -14

10 -15 Displacement spectral density (m/Hz 10 -16 147.6 147.8 148.0 148.2 148.4 Frequency (kHz) b) c) ) )

1/2 10 -11 1/2 10 -11

10 -12 10 -12

10 -13 10 -13

10 -14 10 -14

10 -15 10 -15 20 K 8 K

Displacement spectral density (m/Hz spectral density Displacement 10 -16 (m/Hz spectral density Displacement 10 -16 141.6 141.7 141.8 141.9 142.0 141.6 141.7 141.8 141.9 142.0 142.1 Frequency (kHz) Frequency (kHz)

Figure 6.13: Displacement noise linear spectral densities for the interferometer operated at a) T = 260 K. The temperature was inferred from the calibrated output spectrum. The orange trace depicts the sum of technical, thermal (red) and electronic dark noise. b) Calibrated output spectrum for T = 20 K. The quality 6 factor was measured to Q20 K = 6.5 · 10 . The spectrum is limited by technical laser noise, which is at a higher level due to the reduced contrast of C 20 %. ≈ c) Spectrum for T = 8 K. The thermal noise is plotted as dark red line. For a comparison, the expected thermal noise at 260 K is depicted as red dashed trace. The measurement agrees with the anticipated thermal noise, given the oscillator parameters.

128 h otatcntnl erddfrom temperature. degraded room constantly below contrast K The misalig 30 The first the however. within reduced, curred is ratio intentionally signal-to-noise (here, individua present The respective still the modes, are in modes Michelson power Michelson output the The th the by from in seen compensated field be electric be can result non-vanishing the cannot The and mode camera 6.14. (Sagnac) CCD a Fig. inter with in The analysed depicted was expected. mode were output side correspo membrane eter spots, each two on only the reflection and alignment, the vanishing, ideal and be splitter should beam m and amplitude balanced Michelson by a the verified For separate was to modes. th This membrane of the port. misaligning output interference ally interferometer destructive the m at non-ideal steering mode the in of resulted misalignment which caused process cool-down The problems Identified 6.5.3 Sagnac and Michelson- separate to interferomet membrane Sagnac the the of of alignment ce misalignment bright a A from colors). originating (inverted membrane the of misalignment 6.14: Figure about lgmn a on ob reesbeb etn ptesyst the up heating the by Moreover, irreversible again. temperature be system. room to the found into r of was be coupled disassembly alignment r not noise complete could the laser a interferometer at technical without Sagnac spectrum The result, power a output frequencies. evant the As dominated and K. 20 measurement as low as Michelson 1Michelson a) 70 t20K n a eue oabout to reduced was and K, 260 at % )CDpcueo h nefrmtrotu oewt intent with mode output interferometer the of picture CCD a) Sagnac mode Bright Michelson 2Michelson b) 97 tro eprtr to temperature room at % 20 r )Itninlmis- Intentional b) er. ttemperatures at % ta pti observed, is spot ntral modes. misaligned). . Results 6.5 mn oc- nment Sagnac e central e dn to nding modes. l ealigned Sagnac Sagnac ferom- irrors, mto em anu- mis- ional 129 the el- as is

Chapter 6 6 Cryogenic cooling of the interferometer

6.5.4 Next steps

To increase the interferometer sensitivity, a list of possible improvements could be derived. First, the reduction technical laser noise in the read out is the major priority. The cryogenic experiment highlighted the necessity for a remote control of the interferometer key components, namely the steering mirrors. Currently, the steering mirror mounts are exchanged for additional Attocube positioners. Full remote control is expected to provide an interferometer contrast of C > 99 % at low temperatures, as it has been demonstrated for the Michelson-Sagnac with signal-recycling already. Moreover, a new seed laser model with significantly reduced RIN level of 10−7/√Hz was or- dered and is currently being installed. Second, the isolation stage between cold head and experimental stage may not be ’shortened’ with the flexible hoses for the vacuum system. Those were fixed at the item superstructure, which also supported the cold head with a rigid connection. This is a clear weakness, which can be eliminated by a separating the cold head and hoses support. A better isolation should allow for running the compressor while taking (and averaging) output spectra, providing a clearer measurement of the oscillator displacement spectrum. Beside these technical improvements, the experiment is planned to operate with injection of a squeezed vacuum state into the interferometer output. Within a masters thesis, a squeezed light source is currently being set up and is expected to provide a reduced shot noise level of 6 − 10 dB. Thereby, the relative shot noise level can be decreased. At the same time, injection of a squeezed vacuum state is compatible with signal-recycling [94], such that both can be used to enhance the displacement sensitivity.

6.6 Chapter summary

The last chapter contains provides an overview about the installation and operation of a Michelson-Sagnac at cryogenic temperatures. For this purpose, the ﬁrst cryogenic system at the Albert-Einstein Institute was set up from scratch. The utilised Giﬀord-McMahon cryocooler was presented and its working principle was explained. An interferometer prototype for operation at cryogenic temperatures was designed and tested. An oscillator temperature of T = 8 K was inferred from the calibrated interferometer output spectra. The re-

130 sdfraineto h ebae h iuto ste sim of contrast then condit vacuum interferometer is under the situation and situation, temperature The this room For at membrane. alignment en the were the be of to which easily alignment res positioners, can for The piezo alignment used speciﬁed the contrast. to low-temperature interferometer access by high steerin remote the a as of maintain promising, control to remote order for in need the highlighted sults esntae ihnti thesis. this within demsonstrated . hpe summary Chapter 6.6 99 > a already was % mirrors g already l is ult abled ions. 131 ilar

Chapter 6

iha 97%wsahee nhigh-vacuum in achieved was % interferenc 99.7 mount an as Thereby, were high components positioners. these controllable interf remote Moreover, the of misalignment components. angular optical of influence the minished nefrmtr ml nefrmtramlnt fabout of length Michelso arm combined interferometer a small to A rise interferometer. giving a interferometer, end which Sagnac common a membrane, and nitride oscillator micro-mechanical silicon interfehigh-quality, Michelso translucent The signal-recycled a investigated. a corporated and thesis, realised this was of interferometer framework the In nteeprmnswe eyln irrwt oe reflec power a with situatio mirror recycling This a when falls reduced. experiments the gain is loss in sensitivity signal-recycling optical displacement the intrinsic the case, the and latter below, tr the or mirror In to, signal-recycling cavity. close for are optic int importance that internal recycled of ities cavity the is of This for presence model given. the cavi analytical in effective An sensitivity an ometer in port. i mirror output mirror compound ometer signal-recycling high-reflective a the a as by allowed operated established particular be in to contrast ometer interferometer high achieved.The was MHz 25 to up kHz 60 from aacdhmdn edu ceewsetbihd htn shot A 1 of established. sensitivity was displacement scheme limited readout homodyne stabilisatio balanced eliminat intensity To laser a high-speed suppres port. a were noise, readout laser interferometer light ing the laser in the degree of high fluctuations power technical 7 Summary hpe 7 Chapter . 9 · 10 − 16 ( p m 6 / √ 10 zfrfrequencies for Hz − a e pand up set was n 6 br.Hence, mbar). h interfer- the n otatas contrast e eo unity, below oee in- rometer occurred n llse is losses al ihnthe within ansmissiv- mro in -mirror 7 remain- e n-Sagnac n-Sagnac erometer . e oa to sed mdi- cm 5 interfer- iiyof tivity tdas cted erfer- din ed oise 133 ty,

Summary 7 Summary

Rsr = 99.97% was installed. The high power reflectivity is required in order to achieve a strong dissipative opto-mechanical coupling, as described in [53]. A model for the resulting complex optical spring in the case of close-to dark-fringe operation of the interferometer has been developed in close collaboration with the theoretical quantum optics group [140], and its implications are discussed. In particular, the optical spring depends on both the signal-recycling cavity detuning and the interferometer offset from the dark-fringe. This dependence gives rise to anomalous, opto-mechanical back-action effects, such as a conditional instability and cooling on signal-recycling cavity resonance. The former was observed in the experiment, confirming the theoretical model. In addition, the predicted optical damping was verified qualitatively in the experiment by inferring a reduced effective quality factor of the oscillator for different optical input powers. For an optical input power of 200 mW, the mechanical quality factor was reduced by more than two orders of magnitude, yielding Qeff = 1730, limited by a 1% optical loss within the interferometer.

In a second experiment, a full Michelson-Sagnac interferometer was cooled to cryogenic temperatures. Therefore, a Giﬀord-McMahon type cryocooler was set up, and an interferometer prototype for operation at temperatures close to absolute zero was realised. The experiment was carried out at a laser wavelength of 1550 nm to minimise optical absorption. The analysis of calibrated interferometer output spectra demonstrated a reduction of the oscillator temperature from T = 300 K down to 8 K, reducing the associated displacement thermal noise around the fundamental mechanical resonance by more than one order of magnitude.

In conclusion, a signal-recycled Michelson-Sagnac interferometer was established as a novel opto-mechanical topology. It allows the study of dissipative opto-mechanical coupling and associated optical cooling in the unresolved side band regime. The theoretical model for the complex optical spring was conﬁrmed in the experiment by observing a predicted conditional instability, as well as the anticipated optical damping. Moreover, the achievement of cryogenic oscillator temperatures resolves the restriction to conduct the experiments at room temperature, and hence marks a major step towards the regime of an all-quantum noise dominated experiments.

134 s l uos04 5ne1dm dump dump nMem1 n5 45 0 $uloss 0 ul1 bs1 n4 n5 n3 n2 n2 0.0375 n1 airgaparm1 45 s 0 0.5 0.5 msbs bs 0 uloss #const 0.002 uloss const Interferometer fi Michelson-Sagnac a ======via # modulation and input laser the code, defined. the are of EOM part n1 this nin In 0 pm 2 0.17 $fMi nin eom1 0 mod 0.05 l1 l 12.14M demod const 100k fMi const modulation Laser [118]. ======thi to about # refered information kindly more is For reader the used. software, source was script Fin [120] a interferometer, 0.998) noi Michelson-Sagnac shot the the and of field sensitivity output interferometer the calculate To File Simulation Finesse 8.1 8 pedxA Appendix hpe 8 Chapter se(version esse elimited se open s 135 rst

Appendix 8 Appendix A m mMembran 0.17 0.83 135 nMem1 nMem2 fsig sig1 mMembran 1 0 bs1 ul2 0 $uloss 0 45 nMem2 n9 dump dump s airgaparm2 0.0375 n9 n3

The simulation is either done with or without loss being considered. Then the central beam splitter is deﬁned, as well as the interferometer arm lengths. The steering mirrors to fold both arms are deﬁned as beam splitters here, for mirrors only having two optical ports in Finesse and are usually considered under normal incidence.

# ======Signal recycling mirror const delta 0.99970 s airgapsr 0.012 n4 n13 m2 mSR $delta 0 102.175 n13 n14 # Rsr = 99.97 % Tuned s airgapst2 0.01 n14 n15

Straight forward, the distance from the central beam splitter to the recycling mirror as well as the signal recycling mirror are deﬁned. Some tuning is necesseray to calculate the resonant case (or a detuned one). The detuning depends on the mirror reﬂectivity.

# ======Propagation loss to detector bs2 verlustbs 0.64 0 0 45 n15 ndump n16 ndump

A beam splitter is placed between the homodyne detector to pick up light. In the experiment, this pickup provided the error signal to keep the signal recycling cavity resonant. Changing the achieveable displacement sensitivity due to its power reflectivity, it has to be considered in the simulation file. The same argument also applies for the quantum efficiency of the detector in use, which was set to 0.7 for the simulations.

136 h atprgahcnan ltigotosue yGnuplo by used options plotting contains paragraph last The notitle ($1):($2) END u "misasr2012.out" plot ’P_{out}/P_{in}’ (rad)’ ylabel detuning set ’Membrane xlabel set [0.9:1] yrange set noplot GNUPLOT $x1 f2 3000 qnoise $x1 1000M put f2 10 pdmisa log put f sig1 xaxis abs lin #yaxis $tr1p 1064.0E-9* / tr2p 2*3.141 noplot = tr2p abs func pdmisa n16 tr1p max set 1 pdmisa max noplot $fMi pdmisa 1064E-9) pd2 / qn2p sqrt($qn1p*6.6262E-34*299792458.0 noplot = qn2p re n16 func qnoise max qn1p 1 set max qnoise $fMi noplot 2 qnoise qshot n16 out (Homodyne) pd 2 out Read ======# uncons left was it wa that It such well. simulation. however, as the here experiment, to added the be in can used mirror recycling power A npr2 npr1 0 0 $delta n1 mPR npr2 #m2 0.01 airgappr #s 0.99 delta #const mirror recycling Power ======# . ies iuainFile Simulation Finesse 8.1 t. idered not s 137

Appendix 8 Appendix A

8.2 Matlab Files

A Matlab ﬁle allows to calculate the optical spring in the signal- recycled Michelson-Sagnac interferometer [164]. Note that the script assumes the interferometer to be tuned close to dark-fringe, also, the optical spring is not calculated as function of frequency ω, but as function of detuning. clear; clc; c = 299792458; hbar = 1.054e-34;

L_sr0 = 0.087; tau_sr0 = L_sr0/c; lambda_initial = 1.064*10^(-6); N = round(2*L_sr0/lambda_initial); k0 = pi*N/L_sr0; lambda0 = 2*pi/k0; omega0 = c*k0; P_in = 200*10^(-3);

R_m_power = 17; R_m = sqrt(R_m_power/100); T_m = sqrt(1 - R_m^2); R_sr_power = 99.97; R_sr = sqrt(R_sr_power/100); T_sr = sqrt(1 - R_sr^2); delta_bs_percent = 0.1; delta_bs = delta_bs_percent/100; R_bs = sqrt(0.5 - delta_bs/2); T_bs = sqrt(0.5 + delta_bs/2); m = 80*10^(-12); f_m = 130*10^3; omega_m = 2*pi*f_m; Q_m = 5.8*10^5; gamma_mem = omega_m/(2*Q_m); delta_l_DP = (1/k0)*acos((T_m/R_m)*(-delta_bs/sqrt(1-delta_bs^2))); ksi = 0.0125*lambda0;

138 pit(i,’E% E% En,ifoarray); %E\n’, %E %E fclose(fid); %E ’%E ’w’); fprintf(fid, fopen(filename1, = fid ’springdata.dat’; = filename1 [x;real(K);imag(K);Q;Q_intr]; = ifoarray []; = ifoarray on; grid \gamma}’); / xlabel(’{\Delta legend(’In semilogy(-x,Q_intr,-x,Q,’Linewidth’,1.5); figure; (delta-delta_m)/gamma; = x_sr (delta)/gamma; = x -10*gamma:0.001*gamma:10*gamma; = delta end -10*gamma:0.001*gamma:10*gamma = delta for 0; = y -c*R_m*T_m*(k0*ksi/2)^2/L_sr0; = delta_m gamma_m; + gamma_s = gamma c*R_m^2*(k0*ksi/2)^2/L_sr0; = gamma_m c*(T_sr^2+loss)/(4*L_sr0); = gamma_s 0.01; = loss delta_l/lambda0; omega_m; ksi; = + Omega delta_l_DP = delta_l _nry Q_m; = Q_intr(y) omega_m/(2*gamma_full(y)); gamma_OM(y); = + Q(y) gamma_mem = gamma_full(y) -(1/(2*m*Omega))*imag(K(y)); = gamma_OM(y) (4*k0*R_m^2*P_in/L_sr0)*(1/((gamma-i*Omega)^2+ = K(y) delta_m; - delta 1; = + delta_s y = y /(delta^2+gamma^2); 2*i*(gamma_s*delta_m+gamma_m*delta_s)*Omega+delta_ + *(delta_s*(gamma^2+delta^2-4*(gamma*gamma_m+delta*d . albFiles Matlab 8.2 itial’,’Final’) et^) ... delta^2)) lam)... elta_m)) 139 m*Omega^2)...

Appendix 8 Appendix A

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[lsc6] J. Abadie... H. Kaufer ...J. Zweizig, Search for gravitational waves from intermediate mass binary black holes, Phys. Rev. D, 85, 102004, (2012).

[lsc7] J. Abadie... H. Kaufer ...J. Zweizig, Upper limits on a stochastic gravitational-wave background using LIGO and Virgo interferometers at 600-1000 Hz, Phys. Rev. D, 85, 122001, (2012).

[lsc8] Abadie... H. Kaufer ...J. Zweizig, First low-latency LIGO+Virgo search for binary inspirals and their electromagnetic counterparts, As- tronomy & Astrophysics, 541, (2012).

[lsc9] J. Abadie... H. Kaufer ...J. Zweizig, Search for gravitational waves from low mass compact binary coalescence in LIGO’s sixth science run and Virgo’s science runs 2 and 3, Phys. Rev. D, 85, 82002, (2012).

[lsc10] J. Abadie... H. Kaufer ... J. Zweizig, All-sky search for periodic gravitational waves in the full S5 LIGO data, Phys. Rev. D, 85, 22001, (2012).

[lsc11] J. Abadie... H. Kaufer ...W. Zheng, Implementation and testing of the ﬁrst prompt search for gravitational wave transients with electro- magnetic counterparts, Astronomy & Astrophysics, 539, (2012).

156 obaigm o-ogo oddrn h tesu period stressful the al during for can mood and past. encouragement, I not-so-good and my whom love your forbearing on for Steffen, you thank brother Marina, my and you. Thank made, financia Jü rely. I both always and life, decision Angelika entire parents every my my during in you me family: of supported my all who for to Kaufer, fortunate thanks very typos, of am number I enormous people my of lot eliminate Täsch’. A to ’auf still and we where problems who which experimental tha electronics, solving Mehmet, to in Moritz help like to Dr. also willing pre and always would Vahlbruch and Henning I past group, Dr. atmosphere. Schnabel Postdocs working the pleasant of the members all for thank a Friedrich to Daniel want I Dr. group: grating Britzger. Mein the Michael with Melanie times and good mad Kleybolte had you Lisa Mensa-cr although the Audley, of institute, Heather rest the the Pal-Singh, forget of the to miss Not outside to and skyrocket. want Nerd-Level in not would both mat I had, bureau we and Gerberding. companions Oliver PhD and my thank Ast I Stefan crueltie laboratories. had, we the the fun all in the all electronics ignore For the to to enough occur wit kind might excel was not our who particularly without Weidner, workshop, possible dreas been mechanical have and not would electronics work my of SawadskMost a Andreas to accompany, Nia. to willing Moghadas to allowed Ramon always like was was would I wor I who students to signal-recycling. master Heinzel, great about Gerhard questions was tha thank some It swer I to Tarabrin. like Sergey you. would with Dr. I and ch support, Hammerer the theoretical Klemens had and excellent experiment. it exciting their of really For part est a being to on enjoyed managed work You I la to mem group, thesis. a of working my as great supervising field me for a accepting the and for Schnabel group joined Roman his I in Prof. reasons to grateful the am I of about only one enthusiasm not was His interferometry. Danzmann, and does. Karsten infectious he way is Prof. the thank institute to this leading like would I First, Acknowledgements ees kind so were w Amrit ew: oprovide to hn the thank otAn- hout kProf. nk nthe in s physics l and lly es I ders. kour nk dDr. nd that s ablish your l and y . my e time ance rgen sent lent 157 ber ser for es, n- re k

Acknowledgements

ono oebr1t,18 nLht,Niedersachen Lehrte, in 1983 18th, November on Born [email protected] Uhldingen-Mühlhofen88690 1 Buchenweg Kaufer Henning vitae Curriculum 020 /09Pyisstudies, Physics 3/2009 - 10/2003 /90-719 rnshl Ahlten Grundschule Lehrte-Süd Orientierungsstufe Abitur Abschluss 7/1994 Lehrte, - Gymnasium 8/1990 7/1996 - 8/1994 7/2003 - 8/1996 thesis, Diploma 3/2009 - 03/2008 studies Doctoral 10/2013 - 10/2009 assistant Scientific 10/2013 - engineer 10/2009 System present - 10/2013 ebi Universität Hannover Leibniz Quantenpunkte Split-Gate lateraler isierung Education Universität Hannover Leibniz Universität Hannover Leibniz www.spacetech-i.com GmbH, Spacetech studies University eseln n Charakter- und Herstellung 159

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