1. Resonant Mass Gravitational Wave Detectors David Blair University of Western Australia Historical Introduction Intrinsic Noise in Resonant Mass Antennas Transducers Transducer-Antenna interaction effects Suspension and Isolation Data Analysis

2. Sources and Materials These notes are about principles and not projects. Details of the existing resonant bar network may be found on the International Gravitational Events Collaboration web page. References and some of the content can be found in Ju, Blair and Zhou Rep Prog Phys 63,1317,2000. Online at www.iop.org/Journals/rp Draft of these notes available www.gravity.uwa.edu.au

3. Sao Paulo Leiden Frascati Sphere developments Existing Resonant Bar Detectors and sphere developments

4. AURIGA

5. EXPLORER

6. Weber’s Pioneering Work Joseph Weber Phys Rev 117, 306,1960 Mechanical Mass Quadrupole Harmonic Oscillator: Bar, Sphere or Plate Designs to date: Bar Sphere Torsional Quadrupole Oscillator Weber’s suggestions: Earth: GW at 10 -3 Hz. Piezo crystals: 10 7 Hz Al bars: 10 3 Hz Detectable flux spec density: 10 -7 Jm -2 s -1 Hz -1 ( h~ 10 -22 for 10 -3 s pulse)

7. Gravity Wave Burst Sources and Detection Energy Flux of a gravitational wave: Short Bursts of duration  g Assume J m -2 s -1 Total pulse energy density E G = S.  g J m -2 s -1 Jm -2

8. Flux Spectral Density Bandwidth of short pulse:  ~ 1/  g Reasonable to assume flat spectrum: F(  ) ~ E/  g ie: J.m -2.Hz -1 For short bursts: F(  ) ~ 20 x 10 34 h 2 Gravitational wave bursts with  g ~10 -3 s were the original candidate signals for resonant mass detectors. However stochastic backgrounds and monochromatic signals are all detectable with resonant masses.

9. Black Hole Sources and Short Bursts Start with Einstein’s quadrupole formula for gravitational wave luminosity L G : where the quadrupole moment D jk is defined as: Notice: for a pair of point masses D=ML 2, for a spherical mass distribution D=0 for a binary star system in circular orbit D varies as sin2  t

10. Burst Sources Continued Notice also that represents non-spherical kinetic energy ie the kinetic energy of non-spherically symmetric motions. For binary stars (simplest non sperically symmetric source), projected length (optimal orientation) varies sinusoidally, D~ML 2 sin 2 2  t, The numerical factor comes from the time average of the third time derivative of sin2  t. Now assume isotropic radiation but also use Note that KE= 1 / 2 M v 2 = 1 / 8 M L 2  2

11. To order of magnitude and Maximal source: E ns =Mc 2 ……merger of two black holes In general for black hole births Here  is conversion efficiency to gravitational waves

12. Weber used arguments such as the above to show that gravitational waves created by black hole events near the galactic centre could create gravitational wave bursts of amplitude as high as 10 -16. He created large Al bar detectors able to detect such signals. He identified many physics issues in design of resonant mass detectors. His results indicated that 10 3 solar masses per year were being turned into gravitational waves. These results were in serious conflict with knowledge of star formation and supernovae in our galaxy. His data analysis was flawed. Improved readout techniques gave lower noise and null results. Weber’s Research

13. Energy deposited in a resonant mass Energy deposited in a resonant mass E G  is the frequency dependent cross section F is the spectral flux density Treat F as white over the instrument bandwidth Then Paik and Wagoner showed for fundamental quadrupole mode of bar:

14. x y z Energy deposited in an initially stationary bar U s U s =F(  a ).sin 4  sin 2 2  Incoming wave Energy and Antenna Pattern for Bar Sphere is like a set of orthogonal bars giving omnidirectional sensitivity and higher cross section

15. Detection Conditions Detectable signal U s Noise energy U n Transducer: 2-port device: computer Amplifier, gain G, has effective current noise spectral density S i and voltage noise spectral density S e Mechanical input impedance Z 11 Forward transductance Z 21 (volts m -1 s -1 ) Reverse transductance Z 12 (kg-amp -1 ) Electrical output impedance Z 22

16. X 1 =Asin  Resonant mass transducer Vsin  a t ~ X G  X 2 =Acos  Reference oscillator multiply 0 o 90 o Bar, Transducer and Phase Space Coordinates  determines time for transducer to reach equilibrium X 1 and X 2 are symmetrical phase space coordinates Antenna undergoes random walk in phase space Rapid change of state measured by length of vector (P 1,P 2 ) High Q resonator varies its state slowly Asin(  a t+ 

17. Two Transducer Concepts Parametric Direct Signal detected as modulation of pump frequency Critical requirements: low pump noise low noise amplifier at modulation frequency Signal at antenna frequency Critical requirements: low noise SQUID amplifier low mechanical loss circuitry

18. Mechanical Impedance Matching High bandwidth requires good impedance matching between acoustic output impedance of mechanical system and transducer input impedance Massive resonators offer high impedance All electromagnetic fields offer low impedance (limited by energy density in electromagnetic fields) Hence mechanical impedance trasformation is essential Generally one can match to masses less than 1kg at ~1kHz

19. Mechanical model of transducer with intermediate mass resonant transformer Resonant transformer creates two mode system Two normal modes split by

20. Bending flap secondary resonator Microwave cavity

21.     Data Acquisition Mixers Phase shifters Filter Electronically adjustable phase shifter & attenuator   SO Filter Phase servo Frequency servo  W-amplifier Primary  W-amplifier Spare  W-amplifier Microstrip antennae Microwave interferometer Cryogenic components Bar Bending flap Transducer RF 9.049GHz451MHz 9.501GHz Composite Oscillator Microwave Readout System of NIOBÉ (upgrade)

22. Direct Mushroom Transducer A superconducting persistent current is modulated by the motion of the mushroom resonator and amplified by a DC SQUID.

23. Niobium Diaphragm Direct Transducer (Stanford)

24. Three Mode Niobium Transducer (LSU) Two secondary resonators Three normal modes Easier broadband matching Mechanically more complex

25. Three general classes of noise Brownian Motion Noise kT noise energy Series Noise Back Action Noise Low loss angle  compresses thermal noise into narrow bandwidth at resonance. Decreases for high bandwidth.(small  i ) Broadband Amplifier noise, pump phase noise or other additive noise contributions. Series noise is usually reduced if transductance Z 21 is high. Always increases with bandwidth Amplifier noise acting back on antenna. Unavoidable since reverse transductance can never be zero. A fluctuating force indistingushable from Brownian motion.

26. Noise Contributions Total noise referred to input: Reduces as  i /  a because of predictability of high Q oscillator Reduces as  i /M because fluctuations take time to build up and have less effect on massive bar Increases as M/  i reduces due to increased bandwidth of noise contribution, and represents increased noise energy as referred to input

27. Quantum Limits Noise equation shows any system has minimum noise level and optimum integration time set by the competing action of series noise and back action noise. Since a linear amplifier has a minimum noise level called the standard quantum limit this translates to a standard quantum limit for a resonant mass. Noise equation may be rewritten where A is Noise Number: equivalent number of quanta. The sum A B +A S cannot reduce below~1: the Standard Quantum Limit Burst strain limit~10 -22 (100t sphere) corres to h(  )~3.10 -24

28. Thermal Noise Limit Thermal noise only becomes negligible for Q/T>10 10 (100Hz bandwidth) (Q=  a /  Thermal noise makes it difficult to exceed h SQL

29. Ideal Parametric Transducer Noise temperature characterises noise energy of any system. Since photon energy is frequency dependent, noise number is more useful. Amplifier effective noise temperature must be referred to antenna frequency For example  a = 2  x 700Hz  pump = 2  x 9.2 GHz T n = 10K: Hence and T eff = 8  10 -7 K Cryogenic microwave amplifiers greatly exceed the performance of any existing SQUID and have robust performance Oscillator noise and thermal noise degrade system noise

30.  BPF LOOP OSCILLATOR Microwave Interferometer LO RF LNA Circulator Phase error detector mixer Loop filter Sapphire loaded cavity resonator Q e ~3  10 7  varactor DC Bias  W- amplifier Filtered output + + Non-filtered output Pump Oscillators for Parametric Transducer A low noise oscillator is an essential component of a parametric transducer A stabilised NdYAG laser provides a similar low noise optical oscillator for optical parametric transducers and for laser interferometers which are similar parametric devices.

32. Two Mode Transducer Model

33. Coupling and Transducer Scattering Picture aa pp +=p+a+=p+a -=p-a-=p-a ? transducer Pump photons Signal phonons Output sidebands Treat transducer as a photon scatterer Because transducer has negligible loss use energy conservation to understand signal power flow- Manley-Rowe relations. Note that power flow may be altered by varying  as  per previous slide Formal solution but results are intuitively obvious

34. Upper mode Lower mode Cold damping of bar modes by parametric transducer Bar mode frequency tuning by pump tuning Parametric transducer damping and elastic stiffness

35. Electromechanical Coupling of Transducer to Antenna signal energy in transducer signal energy in bar  In direct transducer  = ( 1 / 2 CV 2 )/M  2 x 2 In parametric transducer  =(  p /  a )( 1 / 2 CV 2 )/M  2 x 2 Total sideband energy is sum of AM and PM sideband energy, depends on pump frequency offset

36. Offset Tuning Varies Coupling to Upper and Lower Sidebands

37. Manley-Rowe Solutions If  p >>  a, P p ~ -(P + +P - ). If P + /  + < P - /  -,then P a < 0…..negative power flow…instability If P + /  + > P - /  -,then P a > 0…..positive power flow…cold damping By manipulating  using offset tuning can cold-damp the resonator…very convenient and no noise cost. Enhance upper sideband by operating with pump frequency below resonance.

38. Offset tuning to vary Q and  in high Q limit If transducer cavity has a Q e >  p /  a, then b is maximised near the cavity resonance or at the sideband frequencies. Strong cold damping is achieved for  p =  cavity -  a.

39. Thermal noise contributions from bar and secondary resonator Thermal noise components for a bar Q=2 x10 8 (antiresonance at mid band) and secondary resonator Q=5 x 10 7 m 2 Hz -1 Frequency Hz bar Secondary resonator

40. Low  high series noise, low back action noise Spectral Strain sensitivity SNR/Hz/mK Transducer Optimisation This and the following curves from M Tobar Thesis UWA 1993

41. Reduced Am noise Spectral Strain sensitivity SNR/Hz/mK

42. Higher secondary mass Q- factor Spectral Strain sensitivity SNR/Hz/mK

43. Reduced back action noise from pump AM noise Spectral Strain sensitivity SNR/Hz/mK

44. High Q e, high coupling Spectral Strain sensitivity SNR/Hz/mK

45. Allegro Noise Theory and Experiment

46. Relations between Sensitivity and Bandwidth Minimum detectable energy is defined by the ratio of wideband noise to narrow band noise Express minimum detectable energy as an effective temperature Optimum spectral sensitivity depends on ratio Independent of readout noise Bandwidth and minimum detectable burst depends on transducer and amplifier

47. Burst detection: maximum total bandwidth important Search for pulsar signals (CW) in spectral minima. More bandwidth=more sources at same sensitivity Stochastic background: use two detectors with coinciding spectral minima

48. Improving Bar Sensitivity with Improved Transducers High , low noise,3 mode Two mode, low , high series noise

49. Optimal filter Signal to noise ratio is optimised by a filter which has a transfer function proportional to the complex conjugate of the signal Fourier transform divided by the total noise spectral density Fourier tfm of impulse response of displacement sensed by transducer for force input to bar Fourier tfm of input signal force Double sided spectral density of noise refered to the transducer displacement

50. Monochromatic and Stochastic Backgrounds Both methods allow the limits to bursts to be easily exceeded. Monochromatic (or slowly varying) : (eg Pulsar signals):Long term coherent integration or FFT Very narrow bandwidth detection outside the thermal noise bandwidth. Stochastic Background: Cross correlate between independent detectors. Thermal noise is independent and uncorrelated between detectors.

51. Allegro Pulsar Search

52. Niobe Noise Temperature

53. Excess Noise and Coincidence Analysis Log number of samples Energy Excess noise Detector noise All detectors show non- thermal noise. Source of excess noise is not understood Similar behaviour (not identical) in all detectors. All excess noise can be elliminated by coincidence analysis between sufficient detectors. (>4) Measure noise performance by noise temperature. Typically h~(few x 10-17).T n 1/2

54. Coincidence Statistics Probability of event above threshhold: ( Event rate R, resolving time  r ) Prob of accidental coincidence in coincidence window  c If all antennas have same background Hence in time t tot the number of accidental coincidences is

56. Improvements through coincidence analysis

57. Suspension Systems General rule:Mode control. Acoustic resonance=short circuit. Low acoustic loss suspension: many systems. Low vibration coupling to cryogenics: Cable couplings: Taber isolators or non-contact readout Multistage isolation in cryogenic environment Room Temperature isolation stages Dead bug cables Nodal point Important tool: Finite element modelling Suspension choices

58. Niobe: 1.5 tonne Niobium Antenna with Parametric Transducer

59. Niobe Cryogenic System

60. Niobe Cryogenic Vibration Isolation

62. Nodal suspension Integrated secondary and tertiary resonators for reasonable bandwidth non-superconducting for efficient cooldown mass up to 100 tonnes Sphere

63. Current limits set by bars Bursts: 7 x 10 -2 solar masses converted to gravity waves at galactic centre (IGEC) Spectral strain sensitivity: h(f)= 6 x 10 -23 /Rt Hz (Nautilus) Pulsar signals in narrow band (95 days): h~ 3 x 10 -24 (Explorer) Stochastic background: h~10 -22 (Nautilus-Explorer)

64. Summary Bars are well understood Major sensitivity improvements underway SQUIDs for direct transducers now making progress (see Frossati’s talk) All significant astrophysical limits have been set by bars. At high frequency bars achieve spectral sensitivity in narrow bands that is likely to exceed interferometer sensitivity for the forseeable future.