Download Gravitational Wave Detection #2: Overview of detectors

Gravitational Wave Detection #3:
Precision of interferometry
Peter Saulson
Syracuse University
8 June 2004
Summer School on Gravitational
Wave Astronomy
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Outline
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Measurement noise vs. displacement noise
Review of interferometer response
Shot noise
Shot noise in an interferometer
Radiation pressure noise and
the quantum limit
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Summer School on Gravitational
Wave Astronomy
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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
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Three test masses
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Interferometer
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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.
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Interferometer output vs.
arm length difference
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0.9
0.8
Output power
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-1.5
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-0.5
0
0.5
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1.5
Optical Path Difference, modulo n wavelengths (cm) -5
x 10
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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
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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.
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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
2c
(photons per second.)
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Wave Astronomy
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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.
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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.
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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.
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Wave Astronomy
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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
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-1
-0.5
0
0.5
1
1.5
Optical Path Difference, modulo n wavelengths (cm) -5
x 10
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Wave Astronomy
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Working out the numbers
200 W of light @   1 m carries
1019 photons per second.
1 / 1019  3 1010 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.)
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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!
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Wave Astronomy
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“The Heisenberg microscope”:
a gedanken experiment
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Bohr derives least xp
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.]
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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.
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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 / m0  N
Too much laser power is as bad as too little!
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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.
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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?
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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.
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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
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