Quantum Mechanical Noises in Gravitational Wave Detectors

Transcription

1 Quantum Mechanical Noises in Gravitational Wave Detectors Max Planck Institute for Gravitational Physics (Albert Einstein Institute) Germany

2 Introduction Test masses in GW interferometers are Macroscopic Objects, but measurement precision is extremely high. M=40kg, X=10-20 m Quantum Fluctuations of test masses becomes important This can be solved by designing the optical configuration carefully - applying techniques of non-linear optics Measurement back-action due to state reduction an important issue 2

3 Laser Light Quantum mechanics in GW interferometers pendulum Fabry-Perot Cavity pendulum phase shift (modulation) Shot Noise: Phase fluctuation, inversely proportional to: photon number # of round trips Local Mirror L km X=Lh Far Mirror phasor diagram This is just one arm, interferometer brings two arms together GW interferometers use light to measure relative motions of mirrors Need to measure position repeatedly in order to detect h(t), but [ˆxH (t 1 ), ˆx H (t 2 ) ] = i h t 2 t 1 M 0 Quantum mechanics does not allow us to do so!... at least not perfectly 3

4 The Standard Quantum Limit A Standard Quantum Limit was formulated by Braginsky in the 1960s Heisenberg Uncertainty Relation [ x(t 1 )] [ x(t 2 )] (t 2 t 1 ) M t 1 t 2 wavefunction widths of test mass t 1 : right after 1st measurement t 2 : right before 2nd measurement Standard Quantum Limit 4

5 The Standard Quantum Limit A Standard Quantum Limit was formulated by Braginsky in the 1960s Heisenberg Uncertainty Relation [ x(t 1 )] [ x(t 2 )] (t 2 t 1 ) M t 1 t 2 Laser Light Fabry-Perot Cavity wavefunction widths of test mass t 1 : right after 1st measurement t 2 : right before 2nd measurement Photon number fluctuation also causing noisy force Radiation-Pressure Noise X=Lh Standard Quantum Limit Increasing Photon Number... Lowers Shot Noise Raises Rad. Pres. Noise 5

6 Michelson Interferometer Common mode: provide strong carrier light in the arms Differential mode: GW-induced motion drives differential phaseshift 6

7 Shot & Radiation-Pressure Noises Rad. Pres. Noise Grows Shot Noise Drops Standard Quantum Limit Conventional Interferometer : - Shot & Rad. Pres. Noises uncorrelated. [Add powers] - Rad. Pres. Noise dominates at lower freq. s; Shot Noise at higher freq. s - Total Noise never surpasses the Standard Quantum Limit L = 4 km, M = 40 kg and γ = 100 Hz γ : optical bandwidth of arm cavity 7

8 Standard Quantum Limit The Standard Quantum Limit is indeed right beyond Advanced LIGO (LIGO-II), with 40kg test mass. If we want to improve by another factor of 10 in LIGO-III, or EGO, either - use 4000kg mirrors - or surpass the SQL - or some combination The Standard Quantum Limit can be surpassed, and ways of doing so will be of great interest for 3rdgeneration detectors! Figure from Cutler & Thorne 8

9 Surpassing the SQL Braginsky (1970s): Measure an observable that commutes at different times??? x t 1 +0 p t 2-0 Quantum Demolition Position eigenstates do not stay eigenstates during evolution --- being demolished continuously Quantum Non-Demolition Measure Quantities whose eigenstates stay eigenstates, e.g, momentum of free mass But the measurement here is only postulated in reality, they are executed by photons, which are quantized Quantum Electrodynamics Required 9

10 Classical Electrodynamics: Quadratures At a certain position, optical fields near central frequency! 0 (1015 Hz) E(t) = (t) cos! 0 t + (t) sin! 0 t,2 (t) slowly changing (~100Hz), called quadrature fields Large monochromatic field E cos! t (Local Oscillator) superimposed: 0 0 E tot (t) = [E 0 + (t)] cos! 0 t + (t) sin! 0 t E 0 [1 + (t)/e 0 ] cos [! 0 t - (t)/e 0 ] (t) amplitude modulation; (t) phase modulation (t)! (t) For a different Local Oscillator: E 0 cos (! 0 t - î) E 0 (t)cos î + (t) sin î amplitude modulation - (t) sin î + (t) cosî phase modulation Homodyne Detection: detecting the î quadrature by superimposing the above Local Oscillator and photodetection! E 0 (t) (t) 10

11 Quantum Mechanics of the Light: QED Commutation Relations: [ E1 (t), (t ) ] = [ (t), (t ) ] = 0, individual quadrature, QND also applies to E φ [ E1 (t), (t ) ] = i hδ(t t ) orthogonal quadratures not simultaneously measurable Homodyne Detection is QND measurement Individual Quadrature can be measured without further noise Heisenberg Uncertainty Principle for cross-quadrature measurement S E1 (Ω)S E2 (Ω) S 2 (Ω) h 2 Minimum Uncertainty States: - Vacuum state - Squeezed state - amplitude squeezed state - phase squeezed state Vacuum State Amplitude-Squeezed Phase-Squeezed State State 11

12 Quantum Noise in GW Detectors I 1/2 X coherent state amplitude fluctuation I 1/2 [I 1/2 /(M! 2 )] Mirror: Light: phase fluctuation Mẍ = IE in 1 (t) x = E1 out (t) = E1 in (t) E2 out (t) = E2 in (t) + Ix(t) x free [ x 0 + p ] 0t M + G(t) t = E2 in (t) + I t dt M 0 0 From Optical Fields + dt E in 1 (t) I M t 0 + [ I t dt 0 dt E in 1 (t ) x 0 + p ] 0t M + G(t) From Free Test Mass 12

13 Quantum Noise in GW Detectors I 1/2 X coherent state amplitude fluctuation I 1/2 [I 1/2 /(M! 2 )] Mirror: Light: phase fluctuation Mẍ = IE in 1 (t) x = E1 out (t) = E1 in (t) E2 out (t) = E2 in (t) + Ix(t) x free [ x 0 + p ] 0t M + G(t) t = E2 in (t) + I t dt M 0 0 From Optical Fields + dt E in 1 (t) I M t 0 + [ I t dt 0 dt E in 1 (t ) x 0 + p ] 0t M + G(t) From Free Test Mass 13

14 Quantum Noise in GW Detectors I 1/2 X coherent state amplitude fluctuation I 1/2 [I 1/2 /(M! 2 )] Mirror: Light: phase fluctuation Mẍ = IE in 1 (t) x = E1 out (t) = E1 in (t) E2 out (t) = E2 in (t) + Ix(t) x free [ x 0 + p ] 0t M + G(t) t = E2 in (t) + I t dt M 0 0 From Optical Fields + dt E in 1 (t) I M t 0 + [ I t dt 0 dt E in 1 (t ) x 0 + p ] 0t M + G(t) From Free Test Mass Optical-Field Commutator cancels Test-Mass Commutator Optical Fields and Test-Mass Position together form a QND observable! This includes all noises -- no extra noise dictated by Quantum Mechanics. 14

15 SQL derived from Quantum Measurement Theory The output fields E1 out (t) = E1 in (t) E2 out (t) = E2 in (t) + Ix(t) t = E2 in (t) + I t dt M 0 0 From Optical Fields In Frequency domain, if we measure out dt E in 1 (t) + [ I DC {}}{ x 0 + p 0t M +G(t) ] From Free Test Mass Noise Spectrum Uncertainty Principle S x = 1 I S Shot + I M 2 Ω 4 S Rad. Press. + 2 MΩ 2 S Correlation S E1 (Ω)S E2 (Ω) S 2 (Ω) h 2 15

16 SQL derived from Quantum Measurement Theory The output fields E1 out (t) = E1 in (t) E2 out (t) = E2 in (t) + Ix(t) t = E2 in (t) + I t dt M 0 0 From Optical Fields In Frequency domain, if we measure out dt E in 1 (t) + [ I DC {}}{ x 0 + p 0t M +G(t) ] From Free Test Mass Noise Spectrum Uncertainty Principle In Absence of Correlations... (e.g., vacuum input state) S x = 1 I S Shot + I M 2 Ω 4 S Rad. Press. + 2 MΩ 2 S Correlation S E1 (Ω)S E2 (Ω) S 2 (Ω) h 2 ( ) ( ) 1 I S x 2 I S M 2 Ω 4 S 2 h MΩ 2 SSQL x 16

17 Surpassing the SQL in a Michelson interferometer The Standard Quantum Limit only exists for specific readout scheme and input state SQL can be circumvented when either of the above are modified E1 out = E1 in E out 2 = E in 2 I MΩ 2 Ein 1 + IG - modification of input state: frequency dependent squeezing - modification of readout scheme: frequency dependent detection amplitude fluctuation phase fluctuation I 1/2 X x free I 1/2 [I 1/2 /(M! 2 )] 17

18 Surpassing the SQL in a Michelson interferometer The Standard Quantum Limit only exists for specific readout scheme and input state SQL can be circumvented when either of the above are modified E1 out = E1 in E out 2 = E in 2 I MΩ 2 Ein 1 + IG - modification of input state: frequency dependent squeezing - modification of readout scheme: frequency dependent detection amplitude fluctuation phase fluctuation I 1/2 X x free I 1/2 [I 1/2 /(M! 2 )] 18

19 Surpassing the SQL in a Michelson interferometer The Standard Quantum Limit only exists for specific readout scheme and input state SQL can be circumvented when either of the above are modified E1 out = E1 in E out 2 = E in 2 I MΩ 2 Ein 1 + IG - modification of input state: frequency dependent squeezing - modification of readout scheme: frequency dependent detection Both require frequency dependent rotation of quadratures, which can be realized by detuned Fabry-Perot Cavities. [Kimble et al., 2001; Appendix of Purdue & Chen, 2002] Bandwidth of typical filter cavities ~ 100Hz; loss has to be lower than squeeze factor. [Kimble et al., 2001] amplitude fluctuation phase fluctuation I 1/2 X x free I 1/2 [I 1/2 /(M! 2 )] 19

20 Surpassing the SQL in a Michelson interferometer Filter Cavity II Arm Cavity Filter Cavity II Arm Cavity Arm Cavity Arm Cavity Laser Laser Squeezed Vacuum Circulator Filter Cavity I Squeezed Vacuum Circulator Filter Cavity I Photodetector Homodyne detector frequency dependent input squeezed state frequency dependent homodyne detection (variational measurement) [Kimble et al., 2001] 20

21 Surpassing the SQL in a Michelson interferometer db Variational: Lossless f (Hz) Í = 2éÂ100 Hz Ic = 800 kw 10dB squeezing 20 ppm loss/round trip, 2 filters, each 4 km total loss ~ 1% 21

22 Other interferometer configurations: Speed Meters Cavity 1 beam L beam R Laser Laser East Arm Extraction Mirror Cavity 2 BS Sloshing Mirror Photodetection φ R φ L Photodetection extra mirror Michelson Speed Meter Sagnac Interferometer Braginsky & Khalili, 90s; Purdue & Chen, 02 Chen, 02; Danilishin, 03 They are equivalent, with optical responses characterized by Á and Í 22

23 Classical Speed Meters Speed Meter [for f<á ]: transfer function ~ f, i.e., suppressed at low frequencies Sensitivity traces the SQL: Equal amount of Shot Noise and Radiation-Pressure Noise Michelson Rad Pres. Noise Speed Meter Both Noises Michelson Shot Noise Speed Meters: Á = 2éÂ130 Hz Í = 2éÂ100 Hz Conventional: Í = 2éÂ100 Hz Ic = 800 kw 23

24 E1 out = E1 in E out 2 = E in 2 + Quantum Noise of Speed Meters I V BA {}}{ [ I ] M Ein 1 +V G V total Speed-Meter input-output relation for f<á Low frequencies: ordinary homodyne detection & squeezed state with fixed squeeze angle will be optimal. [But this will not be the usual phase quadrature.] High frequencies: optimal detection quadrature will be phase quadrature again. Á = 2éÂ173 Hz Í = 2éÂ200 Hz Ic = 800 kw 10dB squeezing 20 ppm loss/round trip 2 filters, each 4 km total loss ~ 1% Much less affected by losses!! 24

25 Vela Spindown Upper Limit BH/BH inspiral 100Mpc; NS/NS Inspiral 20Mpc "=10-9 "=10-7 Crab Spindown Upper Limit A: speed meter, wideband B: position meter with filters C: speed meter, broadband 1/2-1/2 ~ h(f) = S, Hz h BH/BH Inspiral 400Mpc; NS/NS Inspiral 80Mpc NS/NS Inspiral 300Mpc; BH/BH Inspiral, z=0.4 NB LIGO-II LIGO-I Sco X-1 Known Pulsar!=10-6, r=10kpc NS/NS SNR improvement A B C "=10-11 WB LIGO-II LMXBs!=10-7 r=10kpc NS/NS Range (z) Vol. Increase 0.33 [104] 0.41 [181] 0.49 [280] BH/BH (10+10) Range (z) Vol. Increase 1.4 [15] 2.0 [26] 2.4 [33] frequency, Hz Ic = 800 kw; 10dB squeezing; 20 ppm loss/round trip; 2 filters: each 4 km; total loss ~ 1% 25

26 Summary Test masses in GW interferometers are Macroscopic Objects, but measurement precision is extremely high. M=40kg, X=10-20 m The Standard Quantum Limit Quantum Fluctuations of test masses becomes important; Measurement back-action due to state reduction an important issue: The Standard Quantum Limit In GW Interferometers, measurement back-action/state reduction is implemented by Radiation-Pressure Noise This can be solved by designing the optical configuration carefully - applying techniques of non-linear optics 26

27 Quantization of Gravity? Gravitational waves we look for have very high number of gravitons Measurement back action (grav. l radiation reaction of mirrors) negligible. - GW-induced motion generates secondary wave: measurement back action. But this is weak!! - We estimate (quadrupole radiation) incident GW GW-induced motion secondary GW Proposal of Roger Penrose, Gravity-Induced Wavefunction Reduction (see his books) 27