Optical Interferometry for Gravitational Wave Detection

Transcription

1 Fundamentals of Optical Interferometry for Gravitational Wave Detection Yanbei Chen California Institute of Technology

2 Gravitational Waves 2 accelerating matter oscillation in space-time curvature Propagation A perturbation of ~Minkowski space-time

3 Linearized Einstein s Equations 3 Near flat spacetime, metric η is corrected by h (relative correction in time 2 or length 2 ) ds 2 = g µν dx µ dx ν g µν = η µν + h µν, trace-reversed perturbation satisfies wave eqn, sourced by energy and momentum h µν = 16πT µν T00: energy density, T 01,02,03 : momentum density, T11,12,...33: stress Leading multipole radiation is mass quadrupole (analogous to Electric Quadrupole) Magnitude is very very small analogous to EM: A µ = 4π J µ Q h ~ d ~ ML2 Ω 2 d ~ Mv2 d 1 m away from the most powerful H-bombs tested ( J): h ~ Total mass of 5M colliding at v~0.3c, at VIRGO cluster: h ~

4 Evidence of Gravitational Waves 4 Hulse-Taylor binary pulsar discovered in 1974 Two 1.4 M neutron stars orbiting around each other with period 7.75h, one emitting radio pulses Energy carried away by GW causing orbital period to shrink Current GW frequency (twice orbital freq): 71µHz Orbital decay will cause merger in 300M years (GW frequency will reach 10Hz - khz during merger, the final several minutes) [Weisberg 2004] estimated merger rate /Myr in Milky Way [e.g., Kalogera et al 2004]

5 Sources of Gravitational Waves Strain in Space Compact Binaries Black Hole Binary Coalescence 10 6 M Black Hole Formation 10 6 M Black Hole Binary 10 5 M Compact Binary Coalescence Stellar White Dwarf Binaries BH-BH 10 3 M Collapse Frequency [Hz] Space Based Detectors merging supermassive black holes smaller BHs falling into supermassive BHs binaries with larger separations stochastic background Ground Based Detectors merging neutron stars/black holes rotating aspherical neutron stars collapsing stars stochastic background

6 Ground-Based Detectors 6 How do we detect gravitational waves on the earth? - Effect of GW in a small region (compared with wavelength) - Optical Interferometry with short arms - Quantum enhancement on the ground - Limitations of GW detection on the ground

7 Plane Gravitational Wave 7 Coordinates can be chosen such that a plane wave along z direction can be written as g µν = η µν + h µν, h ij TT (t, x, y,z) = h + (t z) h (t z) 0 h (t z) h + (t z) , i, j = x, y,z This is called the TT gauge because h is transverse, and traceless. h+, are the two polarizations of the plane GW This coordinate system is convenient in describing wave propagation, but not for describing relative motions of nearby objects

8 Influence of GW on Light and Matter... in a region with spatial size much less than GW wavelength we go to the Local Lorenz Frame low-velocity objects feel tidal gravity force: M x j = 1 2 M h TT jk x k + F j h TT array of free masses will be distorted with strain ~ h h + h 0 = jk h h squeezing stretching squeezing stretching + polarization polarization Light propagation is unaffected by gravitational wave Problem reduced to the measurement of a (very weak) classical force field

9 A Global Network 9 LIGO-H GEO 600 Hannover, Germany KAGRA Kamioka, Japan LIGO- L VIRGO Pisa, Italy TAMA 300 Tokyo, Japan LIGO-India???

10 Laser Interferometer Gravitational-wave Observatory (LIGO) 10

11 Laser Interferometer Gravitational-wave Observatory (LIGO) 10 LIGO Hanford, WA site LIGO Livingston, LA site

12 Ground-Based Laser Interferometer GW Detector 11 4 km 4 km ~ free horizontally Arm Cavity signal light ~ h Local Oscillator Schematic drawing of LIGO Detectors

13 Sensitivity achieved in first-generation LIGO 12 achieving /rthz at ~ 200 Hz Spectral Density: Noise Power Per Frequency Band h rms ~ f S h

14 Michelson Interferometer: Sensitivity Estimate 13 Lh/2 test-mass mirror input light Y arm X-arm α Y-arm Laser Light I0, ω0 beamsplitter Lh/2 α = 2π λ Lh X arm test-mass mirror + polarizad plane GW along z axis phasor diagram Resolvable phase: ~1/(Number of Photons) 1/2 Photon Number: Power Duration/(Energy of Photon) assuming λ=1μm δh = λ 2π L ω 0 I 0 τ S h = λ 2π L ω 0 = km I 0 L 5W I 0 1/2 Hz 1/2 rms error shot noise spectral density (noise power/frequency band) initial LIGO factor of away!

15 Resonant Enhancement of Sensitivity 14 Laser Light I0, ω0 Lh/2 arm cavity power recycling mirror Above bandwidth: test-mass mirror beamsplitter arm cavity Ω GW > γ τ GW < τ storage Lh/2 test-mass mirror within light storage time, GW already changes sign. Resonant enhancement deteriorates! 2 / T Cavity Gain = 1+ (Ω / γ ) 2 LIGO I: Power-recycling gain ~ [noise ~ 1/(Mich. input power) 1/2 ] # of bounces in arm cavity ~40 Total factor of improvement ~300 Improvement is only below the bandwidth of the cavity. no cavity # of bounces with cavity S h shot = λ 2π L ω 0 1+ (Ω / γ ) 2 I 0 2 / T

16 Radiation Pressure Noise 15 Lh/2 test-mass mirror with cavity without cavity ~Ω -2 Y arm Laser Light I0, ω0 beamsplitter Lh/2 X arm test-mass mirror + polarizad plane GW along z axis δp = δ N 2 ω 0 c without cavity... = N 2 ω 0 c rms momentum of mirror given by photon # fluctuation = I 0 τ 2 ω 0 ω 0 c S F = 4 ω 0 I 0 c 2 S x = 1 4 ω 0 I 0 mω 2 c 2 S h rad pres = with cavity gain 1 mω 2 L 4 ω 0 I 0 c 2 δf = δp /τ = 4 ω 0 I 0 c 2 τ S h rad pres = 1 mω 2 L 4 ω 0 I 0 c 2 2 / T 1+ (Ω / γ ) 2

17 If we place the two types of noise together Standard Quantum Limit 16 S h shot = λ 2π L ω 0 I 0 = 1 L c 2 1+ (Ω / γ ) 2 I 0 ω 0 2 / T S h rad pres = 1 mω 2 L 4 I 0 ω 0 c 2 2 / T 1+ (Ω / γ ) = 2 2 mω 2 L I 0 ω 0 c 2 2 / T 1+ (Ω / γ ) 2 Their dependences on power and cavity gain are opposite 1/10 power 1 power 10 power Total Noise Never Surpasses the Standard Quantum Limit S h SQL = 8 mω 2 L 2

18 Quantum Optical Noise in LIGO-I 17 square root of noise spectral density noise from quantum optical fluctuations noise from thermal fluctuations

19 Generations of GW Detectors 18 initial LIGO (2007) no detections Standard Quantum Limit 10kg 10 x Advanced LIGO Being installed 2015: first science run (hopefully) first detections Standard Quantum Limit 40kg 10 x LIGO-III Einstein Telescope (after detections) precise knowledge of waveforms

20 Where does quantum noise come from? 19 Lh/2 test-mass mirror Laser Light I0, ω0 arm cavity beamsplitter arm cavity Lh/2 power recycling mirror fluctuations entering from dark port test-mass mirror

21 Quadratures, Homodyne Detection and Squeezing 20 Optical field close to ω0 can ben written in the quadrature representation E(t) = E 1 (t)cosω 0 t + E 2 (t)sinω 0 t E 1,2 (t) : slowly varying Act as modulations when superimposed with single-frequency carrier at ω0 E2 E1 phasor diagram

22 Quadratures, Homodyne Detection and Squeezing 20 Optical field close to ω0 can ben written in the quadrature representation E(t) = E 1 (t)cosω 0 t + E 2 (t)sinω 0 t E 1,2 (t) : slowly varying Act as modulations when superimposed with single-frequency carrier at ω0 E2 carrier A cos ω0t E1 phasor diagram

23 Quadratures, Homodyne Detection and Squeezing 20 Optical field close to ω0 can ben written in the quadrature representation E(t) = E 1 (t)cosω 0 t + E 2 (t)sinω 0 t E 1,2 (t) : slowly varying Act as modulations when superimposed with single-frequency carrier at ω0 carrier A cos ω0t phase quadrature E2 amplitude quadrature E1 phasor diagram

24 Quadratures, Homodyne Detection and Squeezing 20 Optical field close to ω0 can ben written in the quadrature representation E(t) = E 1 (t)cosω 0 t + E 2 (t)sinω 0 t E 1,2 (t) : slowly varying Act as modulations when superimposed with single-frequency carrier at ω0 phase quadrature E2 amplitude ϕ carrier A cos ω0t quadrature E1 A cos (ω0t+ϕ) another carrier phasor diagram

25 Quadratures, Homodyne Detection and Squeezing 20 Optical field close to ω0 can ben written in the quadrature representation E(t) = E 1 (t)cosω 0 t + E 2 (t)sinω 0 t E 1,2 (t) : slowly varying Act as modulations when superimposed with single-frequency carrier at ω0 E2 E1 phase quadrature E2 amplitude ϕ carrier A cos ω0t quadrature E1 A cos (ω0t+ϕ) another carrier phasor diagram

26 Quadratures, Homodyne Detection and Squeezing 20 Optical field close to ω0 can ben written in the quadrature representation E(t) = E 1 (t)cosω 0 t + E 2 (t)sinω 0 t E 1,2 (t) : slowly varying Act as modulations when superimposed with single-frequency carrier at ω0 E2 - E1 sin (ϕ) + E2 cos (ϕ) E1 cos (ϕ) + E2 sin (ϕ) E1 phase quadrature E2 amplitude ϕ carrier A cos ω0t quadrature E1 A cos (ω0t+ϕ) another carrier phasor diagram

27 Quadratures, Homodyne Detection and Squeezing 20 Optical field close to ω0 can ben written in the quadrature representation E(t) = E 1 (t)cosω 0 t + E 2 (t)sinω 0 t E 1,2 (t) : slowly varying Act as modulations when superimposed with single-frequency carrier at ω0 E2 - E1 sin (ϕ) + E2 cos (ϕ) E1 cos (ϕ) + E2 sin (ϕ) E1 phase quadrature E2 amplitude ϕ carrier A cos ω0t quadrature E1 A cos (ω0t+ϕ) another carrier phasor diagram Homodyne Detection

28 Quadratures, Homodyne Detection and Squeezing 21 Heisenberg Uncertainty In the Frequency Domain S a1 a 1 S a2 a 2 S a1 a Minimum Uncertainty Gaussian States are: vacuum state coherent states squeezed vacuua squeezed states

29 Quadratures, Homodyne Detection and Squeezing 21 Heisenberg Uncertainty In the Frequency Domain S a1 a 1 S a2 a 2 S a1 a Minimum Uncertainty Gaussian States are: vacuum state coherent states squeezed vacuua squeezed states E2 E1 vacuum

30 Quadratures, Homodyne Detection and Squeezing 21 Heisenberg Uncertainty In the Frequency Domain S a1 a 1 S a2 a 2 S a1 a Minimum Uncertainty Gaussian States are: vacuum state coherent states squeezed vacuua squeezed states E2 coherent state E1 vacuum

31 Quadratures, Homodyne Detection and Squeezing 21 Heisenberg Uncertainty In the Frequency Domain S a1 a 1 S a2 a 2 S a1 a Minimum Uncertainty Gaussian States are: vacuum state coherent states squeezed vacuua squeezed states E2 coherent state E1 vacuum

32 Quadratures, Homodyne Detection and Squeezing 21 Heisenberg Uncertainty In the Frequency Domain S a1 a 1 S a2 a 2 S a1 a Minimum Uncertainty Gaussian States are: vacuum state coherent states squeezed vacuua squeezed states E2 coherent state E1 vacuum

33 Quadratures, Homodyne Detection and Squeezing 21 Heisenberg Uncertainty In the Frequency Domain S a1 a 1 S a2 a 2 S a1 a Minimum Uncertainty Gaussian States are: vacuum state coherent states squeezed vacuua squeezed states E2 coherent state E1 vacuum

34 Quadratures, Homodyne Detection and Squeezing 21 Heisenberg Uncertainty In the Frequency Domain S a1 a 1 S a2 a 2 S a1 a Minimum Uncertainty Gaussian States are: vacuum state coherent states squeezed vacuua squeezed states E2 coherent state E1 vacuum phase squeezed

35 Quadratures, Homodyne Detection and Squeezing 21 Heisenberg Uncertainty In the Frequency Domain S a1 a 1 S a2 a 2 S a1 a Minimum Uncertainty Gaussian States are: vacuum state coherent states squeezed vacuua squeezed states E2 coherent state E1 amplitude squeezed vacuum phase squeezed

36 Quadratures, Homodyne Detection and Squeezing 21 Heisenberg Uncertainty In the Frequency Domain S a1 a 1 S a2 a 2 S a1 a Minimum Uncertainty Gaussian States are: vacuum state coherent states squeezed vacuua squeezed states amplitude squeezed E2 coherent state E1 amplitude squeezed vacuum phase squeezed

37 Quadratures, Homodyne Detection and Squeezing 21 Heisenberg Uncertainty In the Frequency Domain S a1 a 1 S a2 a 2 S a1 a Minimum Uncertainty Gaussian States are: vacuum state coherent states squeezed vacuua squeezed states phase squeezed amplitude squeezed E2 coherent state E1 amplitude squeezed vacuum phase squeezed

38 Amplitude Noise & Phase Noise 22 E 2 amplitude fluctuation I 1/2 X I 1/2 [I 1/2 E 1 /(MΩ 2 )] E 2 E 1 E 1 incoming vacuum phase fluctuation x free out-going field squeezing phase noise will lower shot noise, but increase radiation-pressure noise (good for first-generation detectors, but doesn t help beating the SQL) squeezing a combination of input amplitude and phase will help, but only narrow band squeezing a frequency-dependent combination will help beat the SQL broadband. detecting a combination of output amplitude and phase may even completely remove back-action noise

39 Surpassing the SQL in a Michelson interferometer 23 Filter Cavity II Arm Cavity Filter Cavity II Arm Cavity Laser Arm Cavity Laser Arm Cavity Squeezed Vacuum Circulator Filter Cavity I Squeezed Vacuum Circulator Filter Cavity I Photodetector Homodyne detector frequency dependent input squeezed state frequency dependent homodyne detection [Kimble et al., 2001]

40 Frequency Dependent Squeezing & Detection Sh H f L H1ê Hz L Standard Quantum Limit f HHzL

41 Frequency Dependent Squeezing & Detection Sh H f L H1ê Hz L Standard Quantum Limit 10 db input squeezing lossless, with filters f HHzL

42 Frequency Dependent Squeezing & Detection Sh H f L H1ê Hz L % total loss Standard Quantum Limit 10 db input squeezing lossless, with filters f HHzL

43 Frequency Dependent Squeezing & Detection 24 Sh H f L H1ê Hz L % total loss Standard Quantum Limit ( loss squeeze factor) 1/4 10dB input squeezing and 1% loss Loss Limit Loss Limit = 1/5 SQL 10 db input squeezing lossless, with filters f HHzL

44 Frequency Dependent Squeezing & Detection 24 Sh H f L H1ê Hz L % total loss Standard Quantum Limit ( loss squeeze factor) 1/4 10dB input squeezing and 1% loss Loss Limit Loss Limit = 1/5 SQL 10 db input squeezing lossless, with filters f HHzL Quantum Enhancement of Sensitivity Requires Low Loss!

45 Generation of Squeezed Vacuum 25 Nonlinear Optics H I = χe 3 quantize H =...+ a 2ω0 a ω0 +Ω a ω 0 Ω + a 2ω 0 a ω 0 +Ω a ω 0 Ω H =...+ when non-linear medium pumped with 2ω0 and phase-matching condition satisfied dω 2π A * 2ω 0 a a + A ω0 +Ω ω 0 Ω 2ω 0 a ω 0 +Ω a ω 0 Ω this term becomes effective and generates squeezing T=0 cavity resonant with both ω0 and 2ω0 nonlinear medium T>0 pumped with 2ω0 squeezed vac at ω0

46 Squeezing for GW Detectors Power Relative to the SNL [db] (a) -2-4 (b) -6-8 (c) Frequency[Hz] Observatory noise, calibrated to GW-strain (Hz 1/2 ) k 2k 3k 4k 5k Frequency (Hz) Observatory noise, calibrated to test mass displacement (m Hz 1/2 ) Low-Frequency Squeezing K. McKenzie et al., 2004 [GEO Squeezing result, LIGO Scientific Collaboration, 2011] First demonstration of squeezing in the GW band (sub khz) [K. McKenzie et al., 2004] Squeezing injection at the Caltech 40 m prototype lab [K. Goda et al., 2008] 3.5 db Squeezing at GEO 600 detector [LSC, 2011; H. Vahlbruch, 2010] 2+dB squeezing of LIGO Hanford, achieving best-ever sensitivity to GWs at 200+Hz.

47 Squeezing Status & Prospects 27 initial LIGO Advanced LIGO aim of future LIGO total loss 55-60% 20% <2% detected 2+dB 6dB 10-15dB [slide & numbers from Sheila Dwyer, GW Adv Detector Workshop, 2012]

48 Low-frequency barrier on the earth? 28 Suspension Thermal Noise - a pendulum s thermal noise seems a strong limitation - other methods are being considered - magnetic levitation - juggling mirrors? - atom interferometers - TOBA? 2 T 2 L c 2 T L c T 2 L c T L c T 0 0 x 1 x 2 L atom cloud A atom cloud B [Dimopoulos, Graham, et al.] L c access down to 0.1 Hz experimental prototypes built in Kyoto & Tokyo [Ando et al., 2010]

49 Gravity Gradient Noise 29 Seismic motion driving fluctuations in newtonian gravity field gravitational attraction propagation of surface wave on the surface of the earth [figure from Pitkin et al. 2011] Can be suppressed by monitoring ground motion and subtracting the predicted effect. For LIGO (between 10 Hz and 20 Hz) - 5x suppression required to not affect Advanced LIGO - 30x suppression required to not affect 3rd generation designs [J. Driggers, 2012] Moving detector underground may suppress level and allow better subtraction.

50 Space-Based GW Detection 30 Space-based GW - interferometers with long arms (compared with GW wavelength) - Laser Interferometer Space Antenna (LISA) - quantum enhancements of a LISA-like mission? - Other space missions

51 Plane Gravitational Wave 31 Coordinates can be chosen such that a plane wave along z direction can be written as g µν = η µν + h µν, h ij TT (t, x, y,z) = h + (t z) h (t z) 0 h (t z) h + (t z) , i, j = x, y,z This is called the TT gauge because h is transverse, and traceless. h+, are the two polarizations of the plane GW

52 Influence of GW on Light and Matter 32 Propagation of Light in the Transverse Traceless gauge the scalar wave equation g ab a b Φ = 0 µ ( gg µν ν Φ) = 0 Φ = Aexp(ik µ x µ + iδφ) = Aexp( iωt + ik x + iδφ ) flat-space solution plus additional phase due to GW k u = (ω,k) = ω(1, ˆk) 4-wavevector ang freq 3-wavevector propagation direction δϕ slowly varying δφ(t 0 + L,x 0 + ˆkL) = ω 2c k µ µ δφ = htt µν k µ k ν 2 additional phase accumulates along rays as wave propagates L 0 ˆk i ˆk j h ij (t 0 + ξ,x 0 + ˆkξ)dξ Light propagation is modified by GW t ray of flat space propagation direction k µ δϕ accumulates no longer a ray! z

53 Response of Laser Interferometers: TT Gauge 33 For larger separation (~ reduced wavelength): oscillatory nature matters h p = H p e iω(t z), p = +,... plane wave with propagation direction N δφ(t 0 + L,x 0 + ˆkL) = ω 0 c LH p e p ij ˆk i ˆk j 2 e iω(t z) e iωl(1 ˆkz ) 1 iωl(1 ˆk z ) GW along z k z same as before this favors k orthogonal to N (transverse wave) proportional to L additional phase factor due to propagation effect this favors k along N x θ + polarized y dfêhlh+l π/2 π/4 π/ WLêH2pL suppression of phase shift from simple Lh

54 Response of Masses and Building an Interferometer In TT gauge: low-speed motion of test masses not affected by GW! But test masses won t stay at fixed locations; they will be moving under noisy forces! Simplest interferometer - A, B, and C freely fall + noisy motion - A sends light to B and C - B and C reflect light back to A - A compares phase between light from B and light from C. This gives signal of (δϕ1 + δϕ2) - (δϕ3+δϕ4) Plus local displacement noises (driven by force noise) & shot noise time 34 C B δϕ4 δϕ2 A δϕ3 δϕ1 space C A B

55 Arm Length? 35 What if frequency is f = 10mHz. Reduced wavelength is λ/(2π) ~ m ~ km This is the most optimal arm length to reduce effect of local force noise D~12.5 km D~0.5 m A L = m B λ=1μm very small amount of light (~0.2%) is received by B to collect most of the light, the mirror diameter has to be > (λl) 1/2 ~ 71 m or, reduce to L < D 2 /λ ~ 250 km

56 Laser Interferometer Space Antenna km 1 AU Earth 20º Relative Orbit of Spacecraft Venus Sun Mercury 60º L = km= m Equilateral Triangle, tilted at 60 degrees

57 LISA s Time-Delay Interferometry (TDI) LIGO-like interferometry does not work for LISA, because - light is too weak - arm lengths are not equal enough Armstrong, Estabrook & Tinto s Time-Delay Interferometry - light not bounced back by mirrors, but detected - interferometry signal synthesized, with length difference accounted for time t12 37 t space naive view: test masses compare each other s clock by sending & receiving light pulses 6 links between the 3 spacecraft, each with 1 clock 6 channels - 3 clock noises = 3 noise-free channels t21(l12)+ t12(2l12): cancels noise of 1 t23(l23)+ t32(2l23): cancels noise of 3 subtracting the two doesn t cancel clock noise of 2!! but we can complete the loop!!

58 The Real Time-Delay Interferometry 38 Two Lasers & Two Test masses on board each spacecraft - 6 additional links - 3 additional channels of laser noise - 3 additional test-mass degrees of freedom - these are arranged to also cancel Tinto, Estabrook & Armstrong (2002)

59 LISA Noise Spectrum 39 acceleration noise averaged over all directions longer arm shorter arm [LISA Science Requirement Document]

60 Squeezing? 40 D~12.5 km D~0.5 m A L = m modulated by GW B λ=1μm Signal mode: very wide Gaussian cut by B s aperture, flat-top mode Local oscillator at B must match this mode (mixing in any other mode will only lose) Can we squeeze this mode (or approximately this mode)? - let s propagate it backwards... - it s not possible to squeeze this mode, unless we have larger apertures!! Being limited by Aperture Size & Acceleration Noise, LISA cannot be improved quantum mechanically...

61 Beyond LISA: BBO & DECIGO 41 amplitude spectral density (strain Hz 1/2 ) Initial LIGO Advanced LIGO Einstein Telescope AURIGA/ALLEGRO/NAUTILUS LISA DECIGO BBO Pulsar Timing Array (Current) Pulsar Timing Array (SKA) Frequency (Hz)

62 DECIGO 42 Photo detector FP cavity Mirror L = 1000 km Laser Photo detector Beam splitter Drag-free spacecraft

63 Summary 43 Laser Interferometry can be used to detect gravitational waves. Squeezing already improves sensitivity of ground-based interferometry. Space-based GW detection goes after low-frequency sources