Nonclassical Interferometry

Transcription

1 Nonclassical Interferometry Slide presentation accompanying the Lecture by Roman Schnabel Summer Semester 2004 Universität Hannover Institut für Atom- und Molekülphysik Zentrum für Gravitationsphysik Callinstr Hannover Germany Roman.

2 Nonclassical Interferometry I. Introduction, Basics and Tools 1. Classical Interferometry 2. The Standard Quantum Limit (SQL) 3. Noise Spectral Density 4. The Two-Photon-Formalism 5. Quadrature Input-Output Relations II. Schemes for Nonclassical (Quantum-Non-Demolition) Interferometers 6. Simple Michelson Interferometer with Squeezed Light Input 7. The Optical Spring Interferometer 8. Variational Output 9. A Squeezed Light Upgraded GEO600 Detector? 10. Interferometer Readout: Homodyne versus Heterodyne Detection 11. The Speed Meter Idea: Sloshing Cavity and Sagnac Interferometer 12. Optical Bars and Optical Levers 13. Kerr Effect Enhanced Gravitational Wave Interferometers

3 I. Introduction 1. Classical Interferometry Topics Waves Interferometer topologies Classical description of the interferometer The Michelson-Morley experiment Quantum model of the interferometer Shot noise Radiation pressure noise Standard Quantum Limit (SQL) Literature V. B. Braginski and F. Y. Khalili, Cambridge University Press (1995), Quantum measurement. P. R. Saulson, World Scientific (1994), Fundamentals of Interferometric Gravitational Wave Detectors, 91 US$ 3

4 I. 1. Classical Interferometry Waves Wave equation: 2 v E ( v r,t) 1 c 2 t 2 v E ( v r,t) = 0 Solution for the electric field vector: v E ω (r,t) = E ω 0 α( r v,t) e iωt + α *( r v,t) e iωt [ ] v p ( r v,t) With dimensionless complex amplitude: α( v r,t) 4

5 I. 1. Classical Interferometry Waves Dimensionless complex amplitude: α( r v, t) Monochromatic plane waves: α( r v, t) = α( z) = α0 e ikz Gaussian beam, TEM00-mode: α ikρ = z iz ( ) 0 2 z iz ρ ( r, t) exp, = x + y α v 5

6 I. 1. Classical Interferometry Waves Wave equation: 2 v 2 v 1 E( r, t) 2 c t v v E( r, t) = 0 Equivalent solution for the electric field vector: v E ω (r,t) = E ω [ 0 X 1 ( r v,t) cosωt + X 2 ( r v,t) sinωt] p v ( r v,t) Real and imaginary parts of the complex amplitude (quadrature amplitudes): X 1 ( r v,t) =α( r v,t) +α * ( r v,t) X 2 ( r v,t) = i α( r v,t) α * ( r v,t) [ ] 6

7 I. 1. Classical Interferometry Waves X 2 Phasor diagram: α l+m α m φ α l X 1 7

8 I. 1. Classical Interferometry The Michelson-Morley experiment 8

9 I. 1. Classical Interferometry The Michelson-Morley Experiment 9

10 I. Introduction 2. The Standard Quantum Limit Topics Quantum model of the interferometer Shot noise Radiation pressure noise Standard Quantum Limit (SQL) Literature C. M. Caves, Phys. Rev. D 23, 1693 (1981), Quantum-mechanical noise in an interferometer. C. M. Caves and B. L. Schumaker, Phys. Rev. A 31, 3068 (1985), New formalism for two-photon quantum optics. V. B. Braginski and F. Y. Khalili, Cambridge University Press (1995), Quantum measurement. 10

11 I. 2. The Standard Quantum Limit Quantum Noise Coherent state ˆ X 1 E aˆ =α + δaˆ ˆ X 2 t 11

12 Was ist Nichtklassische Interferometrie? Quantum Noise Coherent state ˆ X 1 E [ δˆ a,δˆ a ]=1 δx ˆ 1 = δˆ a + δˆ a ( a ) δx ˆ 2 = i δˆ a δˆ Commutator Quadrature noise operators ˆ X 2 aˆ =α + δaˆ δx ˆ 2 1 a ˆ a ˆ = n ˆ = α 2 ˆ n 2 = n ˆ 2 δx ˆ n ˆ 2 = α 2 Heisenberg Uncertainty Relation Photon number expectation value Photon number variance t 12

13 Was ist Nichtklassische Interferometrie? Quantum Noise in Interferometry b 1 a 1 b 2 c 1 c 2 a 2 13

14 I. 2. The Standard Quantum Limit Shot Noise Expectation value and variance at detector 1 n 1 n = 2 1 c 1 c = 1 cos ( φ / ) 2, ( ) c c = α cos ( φ / ) 1 2 = α 1 2 2, Expectation value and variance at detector 2 Expectation value and variance for a differential detector 1-2 n 2 n n 12 n = 2 2 = 2 12 c 2 = c c 1 = 2 sin ( φ / ) 2, ( ) c c = α sin ( φ / ) c 1 2 cos ( ) 2 2 c c c c = α. 1 2 = α 2 c 1 2 c = α 2, ( φ), 14

15 I. 2. The Standard Quantum Limit Shot Noise Shot noise for a single detector at dark port SNR = 1= φ = 0 2 SNR = 1= π φ = 2 dn n dn n = sin dφ sn sin = 2 ( φ / 2) cos( φ / 2) α dφsn 1 dφsn = α sin( φ / 2) α cos( φ / 2) dφ 1 = = α Shot noise for a differential detector at two half fringe ports sn ( φ) 2 α dφ α 1 = = α n sn 1 total dφ n 1 total sn = α 1 sin ( φ) 15

16 I. 2. The Standard Quantum Limit Radiation Pressure Noise Phase signal / radiation pressure noise - ratio for coherent states and a single detector at dark port SNR =1= dz rp dv mirror τ = dz rp m dp mirror τ, τ = time interval of measurement dp mirror = 2hω c dn = 2hω total c n total, dz = dφ ω c, dφ rp = 2hω2 mc 2 τ n total n total 16

17 I. 2. The Standard Quantum Limit SQL φ SQL = 4hω2 τ mc 2, Standard Quantum limit for a measurement of Phase z SQL = hτ m, Position h SQL = 2 L z SQL = 2 L hτ m. Strain induced by gravitational waves 17

18 I. Introduction 3. Noise Spectral Density Topics Noise spectral densities of - Shot noise - Radiation pressure noise - GEO The Standard Quantum Limit (SQL) Literature H. A. Haus, Electromagnetic Noise and Quantum Optical Measurement, Springer, Berlin (2000) C. Kittel, Elementary Statistical Physics, Wiley, New York (1958), F. Reif, Fundamentals of Statistical Physics, McGraw-Hill, New York (1965), H-A. Bachor and T. Ralph, Guide to Experiments in Quantum Optics, Springer, Berlin (2003), P. R. Saulson, World Scientific (1994), Fundamentals of Interferometric Gravitational Wave Detectors. 18

19 I. 2. The Standard Quantum Limit Standard-Quantum-Limit (SQL) Standard Quantum limit for a measurement of position (Heisenberg-Microscope approach) z mea p pert h 2 Perturbation on momentum due to position measurement on mass m z add = p pert τ /m hτ 2m z mea κ z mea Additional noise on position due to backaction after τ 2 z mea + κ 2 2 z mea min z mea = κ = hτ 2m = z add Minimum of sum assuming no correlations z SQL = 2 z mea + h 2 1 p pert 2 = hτ m SQL 19

20 I. 2. The Standard Quantum Limit SQL φ SQL = 4hω2 mc 2 τ, Standard Quantum limit for a measurement of Phase z SQL = h m τ, Position h SQL = 2 L z SQL = 2 L h m τ. Strain induced by gravitational waves 20

21 I. 2. The Standard Quantum Limit Shot Noise R-P Noise φ sn = hω Pτ φ rp = 4h 3 3 ω τ 2 4 m c P z sn = 2 c h 4ωPτ z rp = 3 hωτ P 2 2 m c h sn = 2 c h 2 L ωpτ h rp = 3 4hωτ P L m c 21

22 I. 3. Noise Spectral Density Noise Spectral Density Single-sided noise spectral density: S h ( f )df = h 2 0 QuickTime and a YUV420 codec decompressor are needed to see this picture. 22

23 I. 3. Noise Spectral Density Noise Spectral Density Single-sided noise spectral density: S h ( f ) = h 2 f,τ ( ) 1 f 1 Hz, τ 1 f S h,sn ( f ) = hc 2 ωpl 2 S h,rp ( f ) = 4hωP L 2 m 2 c 2 ( 2πf ) 4 (linear spectral density = square root of spectral density) 23

24 I. 3. Noise Spectral Density Effective length: L=1200 m 600 m Mirror masses: m=5.6 kg Laser 1064 nm Power- Recycling mirror 10 kw 600 m Photo diode 24

25 I. 3. Noise Spectral Density Quantum Noise of a Conventional MI Shot noise 25

26 I. 3. Noise Spectral Density Quantum Noise of a Conventional MI Shot noise Quantum noise in phase quadrature Radiation pressure noise Quantum noise with increased laser power (x100) Standard quantum limit (SQL) 26

27 I. Introduction 4. The Two-Photon-Formalism Topics One-photon quantum optics Two-photon quantum optics Quadrature noise operators Literature C. M. Caves, Phys. Rev. Lett. 31, 3068 (1985), New Formalism for two-photon quantum optics. H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin, Phys. Rev. D 65, (2001), Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics. 27

28 I. 1. Classical Interferometry Waves Wave equation: 2 v E ( v r,t) 1 c 2 t 2 v E ( v r,t) = 0 Solution for the electric field vector: v E ω (r,t) = E ω0 α ω ( r v,t) e iωt + α * ω ( r v,t) e iωt [ ] v p ( r v,t) With dimensionless complex amplitude: α ω ( v r,t) 28

29 I. 4. The Two-Photon-Formalism One-Photon-Formalism Time-dependent electric field operator: ˆ E (t) = 2π Ac 0 hω [ a ˆ ω e iωt + a ˆ ω e iωt ] dω 2π Commutation relations of time-independent creation and annihilation operators: [ a ˆ ω, a ˆ ω ]= 0 [ a ˆ ω, a ˆ ω ]= 0 [ a ˆ ω, a ˆ ω ]= 2πδ(ω ω ) 29

30 I. 4. The Two-Photon-Formalism Two-Photon Devices For independently excited modes: δˆ a (ω) δˆ a ( ω ) = 0 δˆ a = ˆ a ˆ a For Two-Photon devices: δˆ a (ω +Ω) δˆ a (ω Ω) 0 Optical Parametric Amplifier 30

31 I. 4. The Two-Photon-Formalism The Two-Photon-Formalism should - naturally describe squeezing - explicitly distinguish between shot noise and radiation pressure noise Carlton M. Caves, The Department of Physics and Astronomy at the University of New Mexico. -should clarify what observable is actually measured in an interferometer spectrum 31

32 I. 4. The Two-Photon-Formalism One-Photon-Formalism Time-dependent electric field operator: ˆ E (t) = 2π Ac 0 hω [ a ˆ ω e iωt + a ˆ ω e iωt ] dω 2π Commutation relations of time-independent creation and annihilation operators: [ a ˆ ω, a ˆ ω ]= 0 [ a ˆ ω, a ˆ ω ]= 0 [ a ˆ ω, a ˆ ω ]= δ(ω ω ) 32

33 I. 4. The Two-Photon-Formalism Two-Photon-Formalism ˆ E (t) = Time-dependent electric field operator: + 2π Ac 2π Ac M M hω 0 hω 0 [ X ˆ (Ω)e iωt + X ˆ 1 1 (Ω)e iωt ] dω 2π cosω t 0 [ X ˆ (Ω)e iωt + X ˆ 2 2 (Ω)e iωt ] dω 2π sinω 0t Commutation relations of quadrature operators: [ X ˆ 1 (Ω), X ˆ 2 ( Ω )]= 2iδ(Ω Ω ) [ X ˆ 1 (Ω), X ˆ 1 ( Ω )]= 2δ(Ω Ω ) Ω ω 0 33

34 I. 4. The Two-Photon-Formalism Phasors in frequency space X 1 E 0 ω Ω 0 X 2 ω+ω 0 Ω 34

35 Nonclassical Interferometry I. Introduction, Basics and Tools 1. Classical Interferometry 2. The Standard Quantum Limit (SQL) 3. Noise Spectral Density 4. The Two-Photon-Formalism 5. Quadrature Input-Output Relations II. Schemes for Nonclassical (Quantum-Non-Demolition) Interferometers 6. Simple Michelson Interferometer with Squeezed Light Input 7. Variational Output 8. Interferometer Readout: Homodyne versus Heterodyne Detection 9. The Optical Spring Interferometer 10. A Squeezed Light Upgraded GEO600 Detector? 11. Michelson Interferometers with Resonant Sideband Extraction (RSE) 12. The Speed Meter Idea 13. Sloshing Cavity and Sagnac Interferometer 14. Optical Bars and Optical Levers 15. Kerr Effect Enhanced Gravitational Wave Detectors

36 I. Introduction 5. Quadrature Input-Output Relations Topics Input-output-relations for optical devices: - free propagation - beam splitters - arm cavities - interferometers Quantum noise spectrum of the LIGO I detectors Literature M. J. Collet and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984), Jan Harms, Diploma Thesis, Hannover University 2002, Quantum Noise in the Laser-Interferometer Gravitational-Wave Detector GEO

37 I. 4. The Two-Photon-Formalism Two-Photon-Formalism Time-dependent electric field operator: E ˆ (t) = ( D+ E ˆ 1 (t))cosω 0 t + E ˆ 2 (t)sinω 0 t E D 1+ ˆ 1 (t) E cos ω 0 t ˆ 2 (t) D D, D >> ˆ E i ˆ E (t) = 2πhω 0 Ac P 0 + q ˆ 1 (t) cosω 0 t + q ˆ 2 (t)sinω 0 t hω 0 37

38 I. 4. The Two-Photon-Formalism Two-Photon-Formalism Normalized quadrature operators in time and frequency space: ˆ q 1 = ˆ q 2 = [ X ˆ (Ω)e iωt + X ˆ 1 1 (Ω)e iωt ] dω 2π, M [ X ˆ (Ω)e iωt + X ˆ 2 2 (Ω)e iωt ] dω 2π, M Quadrature amplitude vector v X (Ω) = X ˆ 1 (Ω) X ˆ 2 (Ω) = X ˆ 1 (Ω) X ˆ 2 (Ω) (hermitian) 38

39 I. 3. Noise Spectral Density Noise Spectral Density Single-sided noise spectral density: S h ( f ) = h 2 f,τ ( ) 1 f 1 Hz, τ 1 f S h,sn ( f ) = hc 2 ωpl 2 S h,rp ( f ) = 4hωP L 2 m 2 c 2 2πf ( ) 4 S h ( f ) = S h,rp ( f ) X ˆ S h,sn ( f ) X ˆ 2 2, ˆ X i 2 = ˆ X i 2 =1 39

40 I. 3. Noise Spectral Density Quantum Noise of a Conventional MI Shot noise Quantum noise in phase quadrature Radiation pressure noise Quantum noise with increased laser power (x100) Standard quantum limit (SQL) 40

41 I. 5. Quadrature Input-Output Relations Laser Power- Recycling mirror v a = X ˆ a 1 (Ω) X ˆ a 2 (Ω) v b = X ˆ b 1 (Ω) X ˆ b 2 (Ω) Photo diode 41

42 I. 5. Quadrature Input-Output Relations Laser Power- Recycling mirror v a = X ˆ a 1 (Ω) X ˆ a 2 (Ω) v b = X ˆ b 1 (Ω) X ˆ b 2 (Ω) Photo diode 42

43 I. 5. Quadrature Input-Output Relations LIGO Hanford LIGO Livingston 43

44 I. 5. Quadrature Input-Output Relations VIRGO Pisa 44

45 I. 5. Quadrature Input-Output Relations Laser Power- Recycling mirror v a = X ˆ a 1 (Ω) X ˆ a 2 (Ω) v b = X ˆ b 1 (Ω) X ˆ b 2 (Ω) Photo diode 45

46 I. 4. The Two-Photon-Formalism Two-Photon-Formalism ˆ E (t) = Time-dependent electric field operator: + 2π Ac 2π Ac M M hω 0 hω 0 [ X ˆ (Ω)e iωt + X ˆ 1 1 (Ω)e iωt ] dω 2π cosω t 0 [ X ˆ (Ω)e iωt + X ˆ 2 2 (Ω)e iωt ] dω 2π sinω 0t Commutation relations of quadrature operators: [ X ˆ 1 (Ω), X ˆ 2 ( Ω )]= 2iδ(Ω Ω ) [ X ˆ 1 (Ω), X ˆ 1 ( Ω )]= 2δ(Ω Ω ) Ω ω 0 46

47 I. 5. Quadrature Input-Output Relations Effective length: L=4000 m Power- Recycling mirror 4000 m Mirror masses: m=11 kg 4000 m Laser 1064 nm v a = 300 W X ˆ a 1 (Ω) X ˆ a 2 (Ω) v b = 30 kw X ˆ b 1 (Ω) X ˆ b 2 (Ω) Photo diode 47

48 I. 5. Quadrature Input-Output Relations 48

49 II. Schemes for Nonclassical Interferometers 6. Simple MI with Squeezed Light Input Topics A quantum noise MATLAB code for a simple Michelson Interferometer Squeezed states Optimized squeezed input Beating the Standard Quantum Limit Literature H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin, Phys. Rev. D 65, (2001), Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics 49

50 II. 6. Simple MI with Squeezed Light Input Laser Power- Recycling mirror v a = Squeezed vacuum state X ˆ a 1 (Ω) X ˆ a 2 (Ω) v b = X ˆ b 1 (Ω) X ˆ b 2 (Ω) Faraday rotator Photo diode 50

51 II. 6. Simple MI with Squeezed Light Input Coherent state δ ˆ E 1 E E 0 δ ˆ E 2 t 51

52 II. 6. Simple MI with Squeezed Light Input Phase squeezed state δ ˆ E 1 E E 0 δ ˆ E 2 t 52

53 II. 6. Simple MI with Squeezed Light Input Amplitude squeezed state δ ˆ E 1 E E 0 δ ˆ E 2 t 53

54 II. 6. Simple MI with Squeezed Light Input Phasors in frequency space X 1 E 0 ω Ω 0 X 2 ω+ω 0 Ω 54

55 II. 6. Simple MI with Squeezed Light Input 55

56 II. Schemes for Nonclassical Interferometers 7. The Optical Spring Interferometer Topics What is signal recycling? A quantum noise MATLAB code for GEO600, a signal recycled optical spring interferometer Beating the SQL by optical spring effect Beating the SQL by additional squeezed light input Literature J. Harms, Y. Chen, S. Chelkowski, A. Franzen, H. Vahlbruch, K. Danzmann, and R. Schnabel, Phys. Rev. D 68, (2003), Squeezed-input, optical-spring, signal-recycled gravitational-wave detectors. A. Buonanno and Y. Chen, Class. Quantum Gravity 18, L95 (2001), Optical noise correlations and beating the standard quantum limit in advanced gravitational-wave detectors. 56

57 II. 7. The Optical Spring Interferometer GEO600 in Ruthe near Hannover Hannover uni 57

58 North arm 600 m East arm 600 m 600 m vacuum tube

59 II. 7. The Optical Spring Interferometer 59

60 II. 7. The Optical Spring Interferometer Seismic Isolation Threefold pendulum 60

61 II. 7. The Optical Spring Interferometer Effective length: L=1200 m 600 m Mirror masses: m=5.6 kg Laser 1064 nm Power- Recycling mirror 10 kw 600 m Signal-recycling mirror SR Power reflectivity: r=0.99 Detuning: Φ= rad Photo diode 61

62 II. 7. The Optical Spring Interferometer 62

63 II. Schemes for Nonclassical Interferometers 8. Variational Output Topics Variational output (frequency dependent detection) Proposal for an experimental realization Ponderomotively squeezed light Variational output versus variational input (frequency dependent squeezing) Literature H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin, Phys. Rev. D 65, (2001), Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics J. Harms, Y. Chen, S. Chelkowski, A. Franzen, H. Vahlbruch, K. Danzmann, and R. Schnabel, Phys. Rev. D 68, (2003), Squeezed-input, optical-spring, signal-recycled gravitational-wave detectors. 63

64 I. 5. Quadrature Input-Output Relations Laser Power- Recycling mirror v a = X ˆ a 1 (Ω) X ˆ a 2 (Ω) v b = X ˆ b 1 (Ω) X ˆ b 2 (Ω) Photo diode 64

65 II. 8. Variational Output The Balanced Homodyne Detector Interfering 50/50 beam splitter Phase shift θ Intense local oscillator (Squeezed) signal beam Electrical current P( Ω) 2 I diff ( Ω) α 2 LO 2 Xˆ ( θ) 65

66 II. 8. Variational Output Variational Homodyne Detection Quantum noise for GEO 600 parameters without signal-recycling 1 16 π 3 16 π 5 16 π 8 16 π SQL 66

67 II. 8. Variational Output Variational Output Interferometer Filter cavities Photo diode Variational output interferometer. Two filter cavities are needed at the output [Kimble et al., Phys. Rev. D 65, (2001)]. 67

68 II. Schemes for Nonclassical Interferometers 9. A Squeezed Light Upgraded GEO600 Detector? Topics Variational output (frequency dependent detection) versus variational input (frequency dependent squeezing) Variational techniques and signal recycling (GEO600 scheme) Other noise sources than quantum in GEO600 Proposal for a squeezed light upgraded GEO600 detector Literature J. Harms, Y. Chen, S. Chelkowski, A. Franzen, H. Vahlbruch, K. Danzmann, and R. Schnabel, Phys. Rev. D 68, (2003), Squeezed-input, optical-spring, signal-recycled gravitational-wave detectors. R. Schnabel, J. Harms, K. Strain, and K. Danzmann, Class. Quantum Grav. 21, S1045 (2003), Squeezed light for the interferometric detection of high-frequency gravitational waves. 68

69 II. 9. A Squeezed Light upgraded GEO600 Frequency Dependent Squeezed Light Input Noise reduction by squeezed light, (- 6 db in variance) 69

70 II. 9. A Squeezed Light upgraded GEO600 Frequency Dependent Squeezed Light Input δê 2 δ ˆ E 1 OPA Faraday rotator Photo diode Ω SHG Filter cavities Reflecting the squeezed light at two subsequent filter cavities provide the desired frequency dependence in the generic case. [Kimble et al., Phys. Rev. D 65, (2001)]. 70

71 II. 9. A Squeezed Light upgraded GEO600 Squeezed-Variational Output Interferometer OPA Faraday rotator SHG Filter cavities Photo diode The optimum achievable is the squeezed variational output interferometer. Two filter cavities are needed at the output [Kimble et al., Phys. Rev. D 65, (2001)]. 71

72 II. 9. A Squeezed Light upgraded GEO600 Frequency Dependent Squeezed Light Input Variational output Noise reduction by squeezed light, (- 6 db in variance) 72

73 II. 9. A Squeezed Light upgraded GEO600 Effective length: L=1200 m 600 m Mirror masses: m=5.6 kg Laser Power- Recycling mirror 10 kw 600 m 1064 Signal-recycling nm mirror Squeezed vacuum state Faraday rotator Photo diode 73

74 II. 9. A Squeezed Light upgraded GEO600 Optical Spring SR Interferometers Quantum noise of GEO 600 in phase and amplitude Standard Quantum Limit Phase -6 db Amplitude [J. Harms et al., PRD (2003)] 74

75 II. 9. A Squeezed Light upgraded GEO600 Optical Spring SR Interferometers 75

76 II. 9. A Squeezed Light upgraded GEO600 Optical Spring SR Interferometers Var.Output Thermal noise (internal substrate + coating) 76

77 II. 9. A Squeezed Light upgraded GEO600 Squeezed Quantum Noise (-6 db) Thermal noise SQL 77

78 II. 9. A Squeezed Light upgraded GEO600 Squeezed Quantum Noise (-6 db) Phase squeezing Squeezing at 45 Squeezing at -45 Amplitude squeezing Frequency dependent squeezing 78

79 II. 9. A Squeezed Light upgraded GEO600 Variational Output? Amplitude Detection quadrature of amplitude quadrature ϕ = 0 ϕ = 0 ϕ = 0 ϕ = 45 ϕ = 90 79

80 II. 9. A Squeezed Light upgraded GEO600 A Squeezed Light upgraded GEO600? 80

81 II. 9. A Squeezed Light upgraded GEO600 81

82 II. 9. A Squeezed Light upgraded GEO600 Table-top Squeezing Experiment 82

83 II. 7. The Optical Spring Interferometer Pendulum 83

84 II. 7. The Optical Spring Interferometer Pendulum 84

85 II. Schemes for Nonclassical Interferometers 10. Homodyne versus Heterodyne Detection Topics Balanced homodyne detection of GW signal in phase quadrature Balanced homodyne detection of GW signal in amplitude quadrature Modulation/demodulation technique: heterodyne detection Balanced heterodyne detection Unbalanced heterodyne detection Quantum noise in heterodyne detection Literature A. Buonanno, Y. Chen, and N. Mavalvala, Phys. Rev. D 67, (2003), Quantum noise in laser-interferometer gravitational-wave detectors with a heterodyne readout scheme. 85

86 II. 10. Homodyne / Heterodyne Detection Conventional interferometer with balanced heterodyne or phase quadrature homodyne readout 86

87 II. 10. Homodyne / Heterodyne Detection Signal-recycled interferometer with unbalanced heterodyne or frequency dependent homodyne readout 87

88 II. Schemes for Nonclassical Interferometers 11. Speed Meter Idea and Sagnac Interferometer Topics Measurement backaction Quantum-Non-Demolition (QND) variables Speed meter QND-Sagnac-gravitational wave interferometer Literature V. B. Braginsky and F. Ya. Khalili, Rev. Mod. Phys. 68, 1 (1996), Quantum nondemolition measurements: the route from toys to tools. Y. Chen, Phys. Rev. D 67, (2003), Sagnac interferometer as a speed-meter-type, quantum-nondemolition gravitational-wave detector. P. Purdue and Y. Chen, Phys. Rev. D 66, (2002), Practical speed meter designs for quantum nondemolition gravitational-wave interferometers. 88

89 II. 11. Speed Meter/Sagnac Interferometer As in a Michelson interferometer bright and dark port are decoupled and can be recycled independently. Laser noise does not couple to the dark port. 89

90 II. 11. Speed Meter/Sagnac Interferometer 90

91 II. Schemes for Nonclassical Interferometers 12. Optical Bars and Optical Levers Topics Optical bar topology of a gravitational wave interferometer Weakly coupled oscillators Three-mirror-cavity QND and speed meter property of optical bar / lever How does optical bar relate to the signal-recycling topology? Literature V. B. Braginsky, M. L. Gorodetsky and F. Ya. Khalili, Phys. Lett. A 232, 340 (1997) Optical bars in gravitational wave antennas. P. Purdue, Phys. Rev. D 66, (2002), Analysis of a quantum nondemolition speed-meter interferometer. V. B. Braginsky and F. Ya. Khalili, Rev. Mod. Phys. 68, 1 (1996), Quantum nondemolition measurements: the route from toys to tools. 91

92 II. 12. Optical Bars and Optical Levers Weakly Coupled Oscillators ω g l mgl ml D J R 0 = = = ω 2 1 = ω D D F R ~ torsional moment due to spring ~ torsional moment due to gravity 92

93 II. 12. Optical Bars and Optical Levers Weakly Coupled Oscillators ω 0 ω 1 ω pend = ω 1 + ω 0 2 ω beat = ω 1 ω

94 II. 12. Optical Bars and Optical Levers Three-Mirror Cavity a t L 2 L 1 a i 0 M 3 M 2 M 1 a r a t 0 = t t t t t M 3 L 2 M 2 L 1 M 1 a i a r 94

95 II. 12. Optical Bars and Optical Levers Three-Mirror Cavity a t L 2 L 1 a i 0 t M 3 t L 2 t M 2 t L 1 t M 1 = M 3 M 2 M 1 i 1 ρ 3 e ikl 2 0 τ 1 τ 2 τ 3 ρ e ikl 2 1 ρ 2 e ikl 1 0 ρ e ikl 1 a r 1 ρ 1 ρ

96 II. 12. Optical Bars and Optical Levers Three-Mirror Cavity a t L 2 L 1 a i 0 M 3 M 2 M 1 a r ( ) τ 3M ( φ 1,φ 2 )= a t φ 1,φ 2 = ia i τ 1 τ 2 τ 3 ρ 3 e iφ ( 2 ρ 2 e iφ1 ρ 1 e iφ 1) e iφ 2 e iφ 1 ρ 1ρ 2 e iφ 1 ( ) φ 1 = kl 1 = ωl 1 c, φ = kl = ωl c 96

97 II. 12. Optical Bars and Optical Levers Three-Mirror Cavity Power Detuning Cavity 2 Detuning Cavity 1 A. Thüring 97

98 II. 12. Optical Bars and Optical Levers Three-Mirror Cavity Power in first cavity Power in second cavity A. Thüring 98

99 GW Detection - The Beginning t=t t=3t/4 t=t/2 t=t/4 t=0 Resonant bar antenna, Joe Weber,

100 Resonant Bar Detectors 1991 ALLEGRO Baton Rouge, (USA) EXPLORER Geneva, 1990CERN, INFN NAUTILUS Frascati, INFN 1995 (Italy) 1997 AURIGA Legnaro, INFN (Italy) M ~ 2000 kg, L ~ 3 m f ~ 900 Hz, f ~ 1 Hz h ~ NIOBE 1993 Perth, UWA (Australia) 100

101 AURIGA Legnaro, INFN (Italy) Electro mechanical transducer aluminum bar: length 3 m, diameter 60 cm, mass 2.3 tons, the mechanical quality factor is about 4x10 6 at 100 mk 101

102 II. 12. Optical Bars and Optical Levers 102

103 II. Schemes for Nonclassical Interferometers 13. Kerr Effect Enhanced GW Interferometers Topics Nonlinear dielectric polarization χ (2) and χ (3) nonlinearities Kerr effect Kerr squeezing Optical bistability Literature A.F. Pace, M.J. Collet and D.F. Walls, Phys. Rev. A 47,3173 (1993), Quantum Limits in interfreometric detection of gravitational radiation. Hyatt M. Gibbs: Optical bistability: controlling light with light, Acad. Pr., (1985). Henning Rehbein, Diploma Thesis, Hanover University (2004) Optische Bistabilität und gequetschtes Licht in einem Kerr-Interferometer. 103

104 II. 13. Kerr Effect Enhanced GW Int. χ (2) - Nonlinearity OPA/SHG layout MgO LiNO hemilithic crystal 7.5mm x 5mm x 2.5 mm Radius of curvature: 10 mm HR=99.97% at 1064 nm Flat surface AR at 1064 nm and 532 nm Output coupler: R=94.5% at 1064 nm Finesse ~ 100 Waist ~ 32 µm FSR ~ 3 GHz γ = 30 MHz A. Franzen 104

105 II. 13. Kerr Effect Enhanced GW Int. χ (2) - Nonlinearity (OPA) Squeezed Squeezed beam beam Coherent beam 105

106 II. 13. Kerr Effect Enhanced GW Int. χ (3) : Kerr Squeezing H. Rehbein 106

107 II. 13. Kerr Effect Enhanced GW Int. Optical Bistability in Kerr Cavities Phase shift on reflection H. Rehbein Detuning from linear cavity resonance 107

108 II. 13. Kerr Effect Enhanced GW Int. Optical Bistability in Kerr Cavities Reflectivity H. Rehbein Detuning from linear cavity resonance 108

109 II. 13. Kerr Effect Enhanced GW Int. Optical Bistability in Kerr Cavities Phase shift on reflection H. Rehbein Detuning from linear cavity resonance 109

110 II. 13. Kerr Effect Enhanced GW Int. Kerr Enhanced GW Interferometer + Linear noise spectral density H. Rehbein GW detection frequency 110