M1L7t.txt

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Why is squeezing important?
Well, squeezing caught the attention of the physics
community when it was suggested in connection
with the detection of gravitation waves.
As you know, the laser interferometer,
the most advanced one is LIGO, has a monumental task
in detecting a very small signal.
And pretty much everything which position metrology can provide
is being implemented for that purpose.
So you can say, this is like in position measurement
like maybe the trip to the moon was for aviation
several decades ago.
So everything is really-- a lot of things pushing
the frontier of position measurement
is motivated by the precision needed for gravitation wave
detection.
And what I want to show you here is
a diagram for what is called Advanced LIGO.
LIGO is currently operating, but there is an upgrade to LIGO
called Advanced LIGO.
And what you recognize here is we
have a laser which goes into a Michelson interferometer.
And this is how you want to detect gravitation waves.
But now you realize that the addition here
is a squeeze source.
And what you are squeezing is not--
but it should be clear to you now.
We're not squeezing the laser beam.
This would be much, much harder, because many, many photons are
involved.
But it is sufficient to squeeze the vacuum
and couple in squeezed vacuum into your gravitation wave
detector.
If you wonder, it's a little bit more
complicated, because people want to recycle light and have put
in other bells and whistles.
But in essence, a squeezed vacuum source
is a major addition to Advanced LIGO.
Yes.
Where is the squeeze vacuum coming into the system?
I see where it's drawn, but where is it actually
entering the interferometer?
At that first beamsplitter?
OK.
We have to now-- there are more things added here.
Ideally, you would think you have a beamsplitter here,
the laser comes in here.
And you're simply going to enter at the squeezed vacuum here.
And this is how we have explained it.
We have one beamsplitter in our interferometer.
There is an input part and an open part.
But what is important here is also the measurement.
Here you have a detector for reading out the interferometer.
And what is important is that the phase is balanced close
to the point where no light is coming out,
so you are measuring the zero crossing of a fringe.
But that would mean most of the light
would then exit the interferometer
at the other part.
But high power lasers are very important for keeping
the classical shot noise down.
So you want to work with the highest power possible.
And therefore, you can't allow the light to exit.
You want to recycle it.
You want to use enhancement cavities.
And what I can tell you is that this setup here integrates,
I think, the signal recycling, the measurement
at the zero fringe, and you see that it's kind
of those different parts are covered in a way which I didn't
prepare to explain it to you.
All I wanted to you to do is pretty much
recognize that a squeeze light generator is important.
And this enters the interferometer
as a squeezed vacuum.
What I find very interesting, and this
is what I want to discuss now, is
that when you have an interferometer like LIGO,
gravitational wave interferometer, and now you
want to squeeze.
Does it really help to squeeze?
Does it always help to squeeze?
Or what is the situation?
And this is what I want to discuss with you.
Let's forget about signal recycling, and enhancement
cavities, and things like this.
Let's just discuss the basic gravitation wave detector where
we have an input, we have the two arms of the Michelson
interferometer.
And to have more sensitivity the light bounces back and forth
in enhancement cavity.
You can say if the light bounces back and forth 100 times,
it is as if you had an arm length which
is 100 times larger.
And now we put in squeezed vacuum
at the open part of the interferometer.
And here we have our photodiodes to perform the measurement.
So the goal is to measure a small length scale.
If a gravitation wave comes by, gravitation waves
have quadrupolar character.
So the metric will be such that there
is a quadrupolar perturbation in the metric of space.
And that means that, in essence, one of the mirrors
is slightly moving out.
The other mirror is slightly moving in.
So therefore, the interferometer needs a very, very high
sensitivity to displacement of one of the mirrors
by an amount delta z.
And if you normalize delta z to the arm
lengths, or the arm length times the number of bounces,
the task is to measure the sensitivity in the length
displacement of 10 to the minus 21.
That's one of the smallest numbers
which have ever been measured.
And therefore, it is clear that this interferometer
should operate as close as possible to the quantum
limit of measurement.
So what you want to measure here is, with the highest accuracy
possible, the displacement of an object delta z.
However, your object fulfills an uncertainty relationship
that if you want to measure the position very accurately,
you also have to consider that it has a momentum uncertainty.
And this fulfills Heisenberg's uncertainty relation.
You would say, well, why should I
care about the momentum uncertainty
if all I want to measure is the position?
Well, you should care, because momentum uncertainty,
after a time tau, turns into position uncertainty.
Because position uncertainty is uncertainty in velocity.
And if I multiply it by the time tau
it takes you to perform the measurement,
you have now an uncertainty in position
which comes from the original uncertainty in momentum.
So if I use the expression for Heisenberg's uncertainty
relation, I find this.

And now what we have to minimize to get the highest precision
is a total uncertainty in position,
which is the original uncertainty
plus the uncertainty due to the motion of the mirror
during the measurement process.
So what we have here is we have delta z.
We have a contribution which goes as 1 over delta z.
And by just finding out what is the optimum choice of delta z,
you find the result above.
Or if you want to say you want that this delta z tau is
comparable to delta z, just set this equal to delta z,
solve for delta z, and you find the quantum limit
for the interferometer which is given up there.
So this has nothing to do with squeezing.
And you cannot improve on this quantum limit by squeezing.
This is what you've got.
It only depends on the duration of the measurement,
and it depends on the mass of the mirror.
There is a very influential and seminal paper
by Caves-- the reference is given here--
which was really laying out the concepts
and the theory for quantum limited measurements with such
an interferometer and the use of squeezed light.
Let me just summarize the most important findings.
So this paper explains that you have two contributions
to the noise, which depend on the laser power you
use for your measurement.
The first one is the photon counting noise.
If you use more and more laser power,
you have a better and better signal,
and your shot noise is reduced.
So therefore, you have a better read out of the interferometer.
And this is given here.
Alpha is the eigenvalue of the coherent state.
But there is a second aspect, which you may not
have thought about it.
And this is the following.
If you split a laser beam into two parts,
you have fluctuations.
The number of photons left and right are not identical.
And you have a coherent beam and you split it
into two coherent beams, and then
you have Poissonian fluctuations on either side.
But if you have no Poissonian fluctuations in the photon
number, if those photons are reflected off a mirror,
they transfer photon recoil to the mirror.
And the mirror is pushed by the radiation pressure.
And it's pushed, and there is a differential motion of the two
mirrors relative to each other due to the fluctuations
in the photon number in the two arms of the interferometer.
So therefore, what happens is you
have a delta z, and a deviation or variance
in the measurement of the mirror which
comes from radiation pressure.
It's a differential radiation pressure between the two arms.
And what Caves showed in this paper
is that the two effects which contribute
to the precision of the measurement
come from two different quadrature components.
For the photon counting, we always
want to squeeze the light in such a way
that we have the narrow part of the ellipse
in the quadrature component of our coherent beam.
We've discussed it several times.
So therefore, you want to squeeze it by e to the minus r.
However, when what has a good effect for the photon counting
has a bad effect for the fluctuations
due to radiation pressure.

So therefore, what happens is-- let's
forget squeezing for a moment.
If you have two contributions, one
goes with-- to the noise-- one goes with alpha, square root
of the number of photons.
One goes inverse with the square root of the number of photons.
You will find out that even in your interferometer
without squeezing, there is an optimum laser
power which you want to use.
Because if you use to low a power,
you lose in photon counting.
If you use too high a power, you lose in the fluctuations
of the radiation pressure.
So even without squeezing, there is an optimum laser power.
And what was shown in this paper is
whenever you choose the optimum power which
keeps a balance between photon counting and radiation
pressure, then you reach the standard quantum
limit of your interferometer.
But it turns out that for typical parameters,
this optimum power is 8,000 watt.
So that's why people at LIGO work
harder and harder to develop more and more powerful
lasers, because more laser power brings them closer and closer
to the optimum power.
But once they had the optimum power,
additional squeezing will not help them,
because they're already at the fundamental quantum limit.
So the one thing which squeezing does for you, it
changes the optimum power in your input beam
by a squeezing factor.
So therefore, if you have lasers which
have maybe 100 watt and not 8,000 watt,
then squeezing helps you to reach the fundamental quantum
limit of your interferometer.
So that's pretty much all I wanted
to say about precision measurements.
I hope the last example-- it's too complex
to go through the whole analysis,
but it gives you at least a feel that you
have to keep your eye on both quadrature components.
You can squeeze, you can get an improvement in one
physical effect, but you have to be
careful to consider what happens in the other quadrature
component.