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Gravitational Wave Detection #3: Precision of interferometry Peter Saulson Syracuse University 8 June 2004 Summer School on Gravitational Wave Astronomy 1 Outline • • • • • Measurement noise vs. displacement noise Review of interferometer response Shot noise Shot noise in an interferometer Radiation pressure noise and the quantum limit 8 June 2004 Summer School on Gravitational Wave Astronomy 2 Two classes of noise Later, you will hear about noise that moves LIGO’s mirrors: seismic noise thermal noise We also have noise that only affects our ability to see where the mirrors are. Generically, one could call this readout noise. Our main source of readout noise is shot noise. 8 June 2004 Summer School on Gravitational Wave Astronomy 3 Three test masses 8 June 2004 Summer School on Gravitational Wave Astronomy 4 Interferometer 8 June 2004 Summer School on Gravitational Wave Astronomy 5 A length-difference-to-brightness transducer Wave from x arm. Light exiting from beam splitter. Wave from y arm. As relative arm lengths change, interference causes change in brightness at output. 8 June 2004 Summer School on Gravitational Wave Astronomy 6 Interferometer output vs. arm length difference 1 0.9 0.8 Output power 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -1.5 8 June 2004 -1 -0.5 0 0.5 1 1.5 Optical Path Difference, modulo n wavelengths (cm) -5 x 10 Summer School on Gravitational Wave Astronomy 7 Sensitivity of interferometer with “on/off” readout If we only distinguish between bright and dark output, interferometer wouldn’t be very sensitive. x 1 m L 4 km 25 x / L 10 8 June 2004 11 Summer School on Gravitational Wave Astronomy 8 The “fringe-splitting” solution We require ~10 more orders of magnitude in sensitivity, if we hope to see grav waves. If so, then we need to know much more than whether we are on the bright or dark point of a fringe. We need to know, to 1 part in 1010, where we are in the fringe. Is this possible? Yes. 8 June 2004 Summer School on Gravitational Wave Astronomy 9 Light and photons A beam of light consists of a stream of photons. A beam with power P (in Watts) is a stream with a mean flux of n P 2c (photons per second.) 8 June 2004 Summer School on Gravitational Wave Astronomy 10 Shot noise Nothing guarantees that n photons arrive each second. Sometimes more will arrive, sometimes fewer. The statistics of sets of independent events have been well studied. The behavior is called Poisson statistics or shot noise. Key result: The size of the fluctuation depends simply on the expected value. 8 June 2004 Summer School on Gravitational Wave Astronomy 11 N When you expect to find N independent events on average, you should also expect that the standard deviation in a set of counts will be N . The noise grows, although not as fast as N. What is the fractional precision for finding N? It is N/N = N /N = 1/ N . This shows that you really do better if you can count more. 8 June 2004 Summer School on Gravitational Wave Astronomy 12 Shot noise in an interferometer We measure the arm length diff by measuring the power out of the interferometer. If noise makes the power a bit high, we think the length diff is different from its true value. 8 June 2004 Summer School on Gravitational Wave Astronomy 13 Confusion of noise with signal 1 0.9 0.8 Output power 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -1.5 8 June 2004 -1 -0.5 0 0.5 1 1.5 Optical Path Difference, modulo n wavelengths (cm) -5 x 10 Summer School on Gravitational Wave Astronomy 14 Working out the numbers 200 W of light @ 1 m carries 1019 photons per second. 1 / 1019 3 1010 of (1 m / 2 ) We can reach h = 10-21 sensitivity by shining 200 W into the LIGO interferometer. This is what we do, using a 6 W laser (and a trick called “power recycling”. More on that in a later lecture.) 8 June 2004 Summer School on Gravitational Wave Astronomy 15 SNR vs. length Signal grows with length, shot noise doesn’t. This is why LIGO is long! Subtlety: Signal grows with the optical path length, which can be even longer if we make the light take many round trips. Shot noise is independent of the number of bounces off of mirrors. But displacement noise grows. Can’t use this trick too much! 8 June 2004 Summer School on Gravitational Wave Astronomy 16 “The Heisenberg microscope”: a gedanken experiment 8 June 2004 Summer School on Gravitational Wave Astronomy 17 Bohr derives least xp the Heisenberg microscope In using an optical instrument for determinations of position, it is necessary to remember that the formation of the image always requires a convergent beam of light. Denoting by the wave-length of the radiation used, and by e the so-called numerical aperture, that is, the sine of half the angle of convergence, the resolving power of a microscope is given by the well-known expression /2e. Even if the object is illuminated by parallel light, so that the momentum h/ of the incident light is known both as to regards magnitude and direction, the finite value of the aperture will prevent an exact knowledge of the recoil accompanying the scattering. Also, even if the momentum of the particle were accurately known before the scattering process, our knowledge of the component of momentum parallel to the focal plane after the observation would be affected by an uncertainty amounting to 2eh/. The product of the least inaccuracies with which the positional co-ordinate and the component of momentum in a definite direction can be ascertained is just given by [the uncertainty relation.] 8 June 2004 Summer School on Gravitational Wave Astronomy 18 The Quantum Limit A powerful microscope is used to see where an atom is located. Photons show where the atom is, but they also kick the atom by an unknown amount. A wide lens gives better position resolution. But a wide lens admits photons from greater angles. Position resolution vs. momentum uncertainty. 8 June 2004 Summer School on Gravitational Wave Astronomy 19 Quantum limit for LIGO Shot noise resolution x N / N 1 / N Shot momentum perturbation p Frad N Position error from momentum perturbation xrad p / m0 N Too much laser power is as bad as too little! 8 June 2004 Summer School on Gravitational Wave Astronomy 20 How big is the fluctuating force? 200 W of light @ 1 m carries 1019 photons per second. So the typical fluctuation in photon number in a second is 1019 3 109 . The radiation pressure force associated with the fluctuating number of photons is N1/2h/ = 2 10-18 newtons. Multiply this by the number of times the beam encounters the mirror (say 25 in LIGO I.) This causes fluctuating motion of a 10 kg mass of about 2.5 10-19 m on a 1 sec time scale. This is OK, but watch out for higher power in future interferometers. 8 June 2004 Summer School on Gravitational Wave Astronomy 21 Why is there radiation pressure noise in quantum mechanics? Understanding of the origin of this noise is subtle. We used semi-classical argument (discreteness of photons, assumption of independent behavior of photons at beamsplitter.) Orthodox QM can sound like this argument fails. Dirac wrote, “We must now describe the photon as going partly into each of the two components into which the beam is split.” If so, no imbalanced force in two arms, thus no noise in a grav wave interferometer. True? 8 June 2004 Summer School on Gravitational Wave Astronomy 22 Radiation pressure noise is real Some famous physicists thought this was true. But Dirac also writes about an interferometer where you measure the energy of recoil of a mirror. Then, “wavefunction collapses”, and photon will have to “choose” which arm to enter. Get recoil noise. Subtle. 8 June 2004 Summer School on Gravitational Wave Astronomy 23 The idea of squeezing Caves (1980) tried to find a way to get this problem right. His analysis of the coupled equations of mirrors/masses and EM field highlighted the role of vacuum fluctuations that enter the “output” port of the interferometer. Sounds mysterious, but has a practical consequence: If you can modify the properties of the fields entering that port, can modify the trade between shot noise and radiation pressure noise. This modification can be done – it is called squeezing. Consideration of squeezing is beyond the level of this course. But stay tuned for progress in the next decade. 8 June 2004 Summer School on Gravitational Wave Astronomy 24