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PROFESSOR: Good afternoon.

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So it has been a little
bit more than a week

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when we met the last time.

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Before I continue our
discussion of metrology

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and interferometry.

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I just want to share
something I saw on my visit

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to the Netherlands.

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When I visited the University of
Delft and the Kavli Institute,

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they had just accomplished the
entanglement of two NV centers.

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And we had just talked in
class about the entanglement

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of two ions.

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So I'm sort of
excited to show you

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that people have just
taken the next step.

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And what often the
next step means

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is that in atomic physics, we
use pristine systems-- ions,

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00:01:16,780 --> 00:01:19,960
neutral atoms in
[INAUDIBLE] vacuum chamber.

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And we create new
forms of entanglement.

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Or with quantum gases, new
forms of quantum matter.

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But ultimately, we hope that
those concepts, those methods,

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and this knowledge
translates to some room

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00:01:35,507 --> 00:01:41,930
temperature materials or
solid state materials which

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can be handled more
easily, and are therefore

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much closer to applications.

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So NV centers are kind of
nature's natural ion trap,

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00:01:53,720 --> 00:01:55,850
or nature's natural
neutral atom trap.

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00:01:55,850 --> 00:01:59,870
Let's not discuss whether
this is neutral or ionized.

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00:01:59,870 --> 00:02:03,540
It's a vacancy in nitrogen.

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00:02:03,540 --> 00:02:06,990
And it has a spectrum
which looks like an atom.

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00:02:06,990 --> 00:02:10,880
So you can say once you have
such a defect in nitrogen,

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00:02:10,880 --> 00:02:13,410
you have an atom, a single
atom in an atom trap

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00:02:13,410 --> 00:02:14,920
or an ion in an ion trap.

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00:02:14,920 --> 00:02:16,770
And you don't have
to create the vacuum.

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00:02:16,770 --> 00:02:17,660
It's there.

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Every time you look
at it, it's there.

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And you can excite
it with the laser.

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It has a spectrum
similar to atoms.

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It has spin
structure So you have

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00:02:28,770 --> 00:02:30,350
these vacancies in diamond.

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00:02:33,820 --> 00:02:35,980
So these are little
quantum dots.

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00:02:35,980 --> 00:02:37,960
But now you have two problems.

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One is you want to collect
the light emitted by them.

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And what is best is to mill a
lens right into the material.

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00:02:49,930 --> 00:02:51,525
So this is the diamond material.

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00:02:51,525 --> 00:02:52,880
A lens is milled.

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00:02:52,880 --> 00:02:55,160
And that already gives
some [? culmination ?]

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00:02:55,160 --> 00:02:57,360
of the light emitted by it.

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But the bit problem
until recently

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has been when you create
those some people call it

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artificial atoms.

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Every atom is a
little bit different,

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because it experience a
slightly different environment.

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A crystal has strain, so if you
have seemingly two identical

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defects in a diamond
crystal, the two defects

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will have a
resonance line, which

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is a few gigahertz difference.

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You would say, well, maybe
it's just [? apart ?] in 10

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to the 5, but it means the
photons are distinguishable.

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So if you want to do
entanglement by having two such

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artificial atoms emitting a
photon onto a beam splitter,

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and then by performing
a measurement we project

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the atoms into Bell state-- I
hope you all remember what we

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discussed for the trapped ions--
then you have to make sure that

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fundamentally, those photons
cannot be distinguished.

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And the trick here is that
they put on some electrodes.

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And by adding an
electric field, they

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00:04:05,550 --> 00:04:07,790
can change the
relative frequency.

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00:04:07,790 --> 00:04:11,300
And therefore, within
the frequency uncertainty

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given by Heisenberg's
Uncertainty Relation,

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they can make the two
photons identical.

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And when, let's say, experiment.

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Well, you have two NV
centers, defects in diamond.

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You can manipulate with
microwaves coherently the spin.

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You need lasers for
initialization and readout.

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But then closer to what
we want to discuss,

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you need laser beams which
excite the NV center.

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And then the two NV
centers emit photons.

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00:04:54,510 --> 00:04:58,620
And what you see now is exactly
what we discussed schematically

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00:04:58,620 --> 00:05:03,500
and in context of the ions,
that the two NV centers now

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emit photons.

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00:05:06,100 --> 00:05:10,120
And by using polarization
tricks and the beam splitter,

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00:05:10,120 --> 00:05:14,120
you do a measurement
after the beam splitter.

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00:05:14,120 --> 00:05:16,420
And based on the outcome
of the measurement,

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you have successfully
projected the two NV centers

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into a [INAUDIBLE] state.

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OK, good.

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All right.

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00:06:04,450 --> 00:06:11,240
Let me just summarize what our
current discussion is about.

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00:06:11,240 --> 00:06:14,660
This section is called
Quantum Metrology.

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00:06:14,660 --> 00:06:18,690
It is a section where we want
to apply the concepts we have

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learned to precision
measurements.

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It's actually a chapter
which I find nice.

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We're not really
introducing new concepts.

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00:06:28,640 --> 00:06:31,270
We're using concepts
previously introduced.

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And now you see how
powerful those concepts

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are, what they can be used for.

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So we want to
discuss the precision

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we can obtain in
quantum measurement.

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We apply it here to an
atom in the interferometer,

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00:06:48,490 --> 00:06:50,900
to an optical interferometer.

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00:06:50,900 --> 00:06:53,190
We could also
discuss the precision

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00:06:53,190 --> 00:06:54,920
of spectroscopic measurements.

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00:06:54,920 --> 00:06:59,280
A lot of precision measurements
have many things in common.

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So what we discuss here
is as a generic example

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00:07:01,960 --> 00:07:04,710
for precision measurement
that we send light

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00:07:04,710 --> 00:07:06,580
from our center interferometer.

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00:07:06,580 --> 00:07:07,760
Here is a phase shift.

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And the question is,
how accurately can we

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measure the phase shift when
we use n photons as a resource?

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And of course, you
are all familiar

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with the fundamental limit of
standard measurements, which

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is short noise.

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And sort of as a warm up in
our last class Wednesday,

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week ago, I showed you that when
we use coherent states of light

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at the input of
the interferometer,

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we obtain the short noise.

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00:07:45,080 --> 00:07:48,130
Well, it may not be surprising,
because coherent light

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00:07:48,130 --> 00:07:53,900
is as close as possible, has
similar to classical light.

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But then we discussed single
mode, single photon input.

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And by using the formalism
of the Mach-Zehnder

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00:08:06,240 --> 00:08:10,190
interferometer, we've found that
the phase uncertainty is again

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00:08:10,190 --> 00:08:13,920
1 over square root n.

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So then the question is, how can
we go to the Heisenberg limit,

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00:08:19,080 --> 00:08:24,120
where we have an uncertainty
in the phase of 1 over n?

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00:08:24,120 --> 00:08:27,910
And just as a reminder,
I find it very helpful.

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I told you that you can always
envision if you have n photons

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and you multiply it, put
the n photons together

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00:08:39,840 --> 00:08:43,669
by multiplying the
frequency by n,

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then you have one photon
which [INAUDIBLE] with n times

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the frequency.

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And it's clear if you
do a measurement at n

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00:08:49,550 --> 00:08:52,100
times the frequency,
your precision in phase

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00:08:52,100 --> 00:08:54,700
is n times better.

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So what we have to do is
we have to sort of put

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the n photons together.

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00:09:00,130 --> 00:09:03,150
And then we can get a precision
of the measurement, which

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00:09:03,150 --> 00:09:06,680
is not square root n,
but n times better.

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00:09:06,680 --> 00:09:08,460
So this is what we
want to continue today.

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00:09:12,840 --> 00:09:16,330
This is the outline I gave
you in the last class.

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00:09:16,330 --> 00:09:19,620
If you have this
optical interferometer,

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00:09:19,620 --> 00:09:21,520
this Mach-Zehnder
interferometer.

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And if you use coherent
sets or single photons

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as the input state, we
obtain the short noise.

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Now we have to change something.

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00:09:29,150 --> 00:09:31,570
And we can change
the input state,

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00:09:31,570 --> 00:09:35,410
we can change the beam splitter,
or we can change the readout.

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So we have to change something
where we entangle the n

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photons, make sure in some sense
they act as one giant photons,

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with either n times
the frequency,

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but definitely with n times
the sensitivity to phase shift.

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Any questions?

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Good.

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The first example, which
we pretty much covered

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in the last class, was an
entangled state interferometer.

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00:10:08,740 --> 00:10:12,100
So instead of having the
Mach-Zehnder interferometer

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as we had before, we have
sort of a Bell state creation

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device.

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Then we provide the phase shift.

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And then we have a
Bell analysis device.

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00:10:23,610 --> 00:10:30,590
And I want to use here the
formalism and the symbols

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00:10:30,590 --> 00:10:32,770
we had introduced earlier.

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00:10:32,770 --> 00:10:40,090
So just as a reminder, what
we need is I need two gates.

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One is the Hadamard gate,
which in matrix representation

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has this matrix form.

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And that means if you have
a qubit which is either up

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or down and you apply
the matrix to it,

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00:10:55,840 --> 00:10:58,890
you put it in a superposition
state of up and down.

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00:10:58,890 --> 00:11:01,050
It's a single qubit rotation.

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00:11:01,050 --> 00:11:03,720
You can say for the bloch
on the Bloch sphere,

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00:11:03,720 --> 00:11:04,850
it's a 90 degree rotation.

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00:11:07,820 --> 00:11:11,290
The second gate we need
is the controlled NOT.

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00:11:14,250 --> 00:11:18,700
And the controlled NOT
we discussed previously

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00:11:18,700 --> 00:11:23,850
can be implemented by
having an interferometer

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00:11:23,850 --> 00:11:31,160
and using a non-linear Kerr
medium coupling another photon

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00:11:31,160 --> 00:11:33,510
to the interferometer
in such a way

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that if you have a
photon in mode C,

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it creates a phase shift
with the interferometer.

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00:11:38,280 --> 00:11:41,190
If there is no
photon in mode C, it

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00:11:41,190 --> 00:11:44,350
does not create an
additional phase shift.

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00:11:44,350 --> 00:11:46,940
And as we have
discussed, this can

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implement the controlled NOT.

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00:11:51,700 --> 00:11:54,220
So these are the ingredients.

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And at the end of
the last class,

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00:11:57,460 --> 00:12:03,450
I just showed you what those
quantum gates can do for us.

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00:12:03,450 --> 00:12:08,390
If you have two qubits at
the input, we apply the--

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00:12:08,390 --> 00:12:11,980
and I will assume they
are both in logical 0--

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00:12:11,980 --> 00:12:16,390
the Hadamard gate makes
the coherent superposition.

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00:12:16,390 --> 00:12:18,490
And then, we have
a controlled NOT

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00:12:18,490 --> 00:12:20,900
where this is the target gate.

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00:12:20,900 --> 00:12:25,960
Well, if this is 0, if
the control beat is 0,

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the target beat stays 0.

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00:12:27,990 --> 00:12:29,960
So we get 0, 0.

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00:12:29,960 --> 00:12:33,550
If the control beat is 1, the
target beat is flipped to 1.

200
00:12:33,550 --> 00:12:35,600
So we get 1, 1.

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00:12:35,600 --> 00:12:37,250
So the result is
that we have now

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00:12:37,250 --> 00:12:41,060
created a state 0, 0 plus 1, 1.

203
00:12:41,060 --> 00:12:44,760
And if we apply a phase shift
to all the photons coming out

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00:12:44,760 --> 00:12:50,050
on the right hand side, we get
a phase shift which is too fine.

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00:12:50,050 --> 00:12:51,720
So we already get the idea.

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00:12:51,720 --> 00:12:54,850
If we take advantage
of a state 1, 1,

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00:12:54,850 --> 00:12:58,880
it has twice the [? face ?]
sensitivity as a single photon.

208
00:12:58,880 --> 00:13:01,360
And we may therefore
get the full benefit

209
00:13:01,360 --> 00:13:04,580
of the factor of 2, and
not just square root 2.

210
00:13:04,580 --> 00:13:06,330
And this is what this
discussion is about.

211
00:13:08,947 --> 00:13:09,446
Questions?

212
00:13:13,200 --> 00:13:14,595
OK, so that's where we ended.

213
00:13:17,200 --> 00:13:26,650
We can now use
another controlled NOT

214
00:13:26,650 --> 00:13:28,255
and bring in the third qubit.

215
00:13:39,050 --> 00:13:44,240
So what we create here.

216
00:13:46,750 --> 00:13:50,200
Just make a reference, this
is where we start today.

217
00:13:53,280 --> 00:13:59,900
So we create here the state,
which is either all beats

218
00:13:59,900 --> 00:14:13,260
are 0, all beats are 1, and
then the phase shift gives us

219
00:14:13,260 --> 00:14:17,280
three times the phase shift phi.

220
00:14:17,280 --> 00:14:23,010
And therefore, by bringing
in more and more qubits,

221
00:14:23,010 --> 00:14:25,770
I've shown you n equals
1, n equals 2, n equals 3.

222
00:14:25,770 --> 00:14:27,650
So now you can go to n.

223
00:14:27,650 --> 00:14:34,740
We obtain states which have n
times the phase sensitivity.

224
00:14:43,690 --> 00:14:50,800
Let me just mention in
passing that for n equals 3,

225
00:14:50,800 --> 00:14:58,000
the superposition of 0, 0, 0
and 1, 1, 1 goes by the name

226
00:14:58,000 --> 00:15:01,710
not gigahertz, GHZ.

227
00:15:01,710 --> 00:15:02,210
Greenberger.

228
00:15:05,950 --> 00:15:07,180
This third one is Zeilinger.

229
00:15:07,180 --> 00:15:08,220
The second one is--

230
00:15:08,220 --> 00:15:08,887
AUDIENCE: Horne.

231
00:15:08,887 --> 00:15:09,594
PROFESSOR: Horne.

232
00:15:09,594 --> 00:15:10,260
Thank you.

233
00:15:10,260 --> 00:15:13,460
Greenberger-Horne-Zeilinger
state.

234
00:15:13,460 --> 00:15:15,600
And those states play
an important role

235
00:15:15,600 --> 00:15:18,120
in test of Bell's inequality.

236
00:15:18,120 --> 00:15:30,400
Or more generally, states which
are macroscopically distinct

237
00:15:30,400 --> 00:15:34,270
are also regarded cat states
or Schrodinger cat states.

238
00:15:42,620 --> 00:15:48,170
If we apply a phase
shift and go now

239
00:15:48,170 --> 00:15:50,920
through the
entangler in reverse,

240
00:15:50,920 --> 00:15:55,770
just a reverse sequence of
CNOT gates and Hadamard gates,

241
00:15:55,770 --> 00:16:03,010
then we have all
the other n minus,

242
00:16:03,010 --> 00:16:04,960
have the n minus 1 qubits.

243
00:16:04,960 --> 00:16:09,460
All the qubits except
for the first reset to 0.

244
00:16:09,460 --> 00:16:12,680
But the first qubit is now
in a superposition state

245
00:16:12,680 --> 00:16:15,850
where we have the
phase into the power n.

246
00:16:19,970 --> 00:16:22,460
And now, we can
make a measurement.

247
00:16:22,460 --> 00:16:27,110
And P is now the probability
to find a single photon

248
00:16:27,110 --> 00:16:29,150
in the first qubit.

249
00:16:29,150 --> 00:16:30,790
So in other words,
our measurement

250
00:16:30,790 --> 00:16:33,895
is exactly the same
as we had before where

251
00:16:33,895 --> 00:16:36,230
we had the normal
Mach-Zehnder interferometer.

252
00:16:36,230 --> 00:16:38,080
We put in one photon at a time.

253
00:16:38,080 --> 00:16:39,940
And we determined
the probability.

254
00:16:39,940 --> 00:16:46,150
What is the photon at one of the
outputs of the interferometer?

255
00:16:46,150 --> 00:16:49,620
But the only
difference is that we

256
00:16:49,620 --> 00:16:54,820
have now a factor of n in the
exponent for the phase shift.

257
00:16:54,820 --> 00:17:00,340
And I want to show you what
is caused by this factor of n.

258
00:17:04,380 --> 00:17:07,810
Let me first remind
you how we analyzed

259
00:17:07,810 --> 00:17:11,950
the sensitivity of
an interferometer

260
00:17:11,950 --> 00:17:14,349
for single photon input before.

261
00:17:14,349 --> 00:17:19,630
So for a moment,
set n equals 1 now.

262
00:17:19,630 --> 00:17:23,010
Then you get what we
discussed two weeks ago.

263
00:17:23,010 --> 00:17:26,349
The probability is [INAUDIBLE]
as an interferometer.

264
00:17:26,349 --> 00:17:31,320
Sine also, cosine also
infringes with cosine phi.

265
00:17:31,320 --> 00:17:34,140
If you do measurements
with a probability P,

266
00:17:34,140 --> 00:17:37,115
that's a binomial distribution.

267
00:17:37,115 --> 00:17:38,700
Then the deviation
or the square root

268
00:17:38,700 --> 00:17:41,890
of the variance in the binomial
distribution is P times 1

269
00:17:41,890 --> 00:17:49,300
minus P. And by inserting p
here, we get sine phi over 2.

270
00:17:49,300 --> 00:17:54,250
The derivative of P, which
is our signal, with a phase

271
00:17:54,250 --> 00:17:56,300
is given by that.

272
00:17:56,300 --> 00:17:58,590
And then just using
error propagation,

273
00:17:58,590 --> 00:18:01,200
the uncertainty in the
phase is the uncertainty

274
00:18:01,200 --> 00:18:03,670
in P divided by dp dt.

275
00:18:07,310 --> 00:18:11,280
So then we repeat the
whole experiment n times,

276
00:18:11,280 --> 00:18:15,350
and if your binomial
distribution and we

277
00:18:15,350 --> 00:18:19,520
do m [INAUDIBLE] of our
[INAUDIBLE] with probability P,

278
00:18:19,520 --> 00:18:21,630
we get an m under
the square root.

279
00:18:21,630 --> 00:18:24,040
And then we get 1
over square root m.

280
00:18:24,040 --> 00:18:26,354
And this was what I
showed you two weeks ago,

281
00:18:26,354 --> 00:18:27,645
the standard short noise limit.

282
00:18:30,670 --> 00:18:35,590
But now we have this
additional factor of n

283
00:18:35,590 --> 00:18:46,270
here, which means that
our probability has

284
00:18:46,270 --> 00:18:51,600
a cosine which goes with n phi.

285
00:18:51,600 --> 00:18:56,920
When we take the derivative,
we get a factor of n here.

286
00:18:56,920 --> 00:19:02,990
And then dp d phi
has the derivative.

287
00:19:02,990 --> 00:19:05,570
But then we have to
use a chain rule, which

288
00:19:05,570 --> 00:19:08,360
has another factor of n.

289
00:19:08,360 --> 00:19:13,220
And therefore, we obtain now
that the phase sensitivity

290
00:19:13,220 --> 00:19:14,805
goes as 1 over n.

291
00:19:18,610 --> 00:19:21,320
And therefore, we reach
the Heisenberg limit.

292
00:19:21,320 --> 00:19:26,400
So if you have now your n
photon entangler, the one I

293
00:19:26,400 --> 00:19:31,930
just-- the entanglement
operation with these n cubits

294
00:19:31,930 --> 00:19:36,760
is special beam spreader
replaced by the entangler,

295
00:19:36,760 --> 00:19:40,681
we have now a sensitivity,
which goes 1 over n.

296
00:19:40,681 --> 00:19:42,180
And then of course,
if you want, you

297
00:19:42,180 --> 00:19:44,100
can repeat the
experiment n times.

298
00:19:44,100 --> 00:19:45,950
And whenever you
repeat an experiment,

299
00:19:45,950 --> 00:19:49,254
you gain an addition with
the square root of m.

300
00:20:01,980 --> 00:20:03,400
Just give me one second.

301
00:20:03,400 --> 00:20:04,185
We have this.

302
00:20:07,420 --> 00:20:17,470
OK, so this is an example where
we had n qubits entangled,

303
00:20:17,470 --> 00:20:18,570
like here.

304
00:20:18,570 --> 00:20:22,720
And the sensitivity of
this interferometer scales

305
00:20:22,720 --> 00:20:26,070
now as the Heisenberg
limit with 1 over n.

306
00:20:31,810 --> 00:20:36,669
OK, I want to give you,
because they're all nice

307
00:20:36,669 --> 00:20:38,460
and they're all special,
I want to give you

308
00:20:38,460 --> 00:20:41,840
three more example how you can
reach the Heisenberg limit.

309
00:20:46,330 --> 00:20:52,300
The next one goes by the name
super beam splitter method.

310
00:20:52,300 --> 00:20:56,230
It's actually a very
fancy beam splitter

311
00:20:56,230 --> 00:20:59,680
where you have two prods
of your beam splitter.

312
00:20:59,680 --> 00:21:03,160
One has 0 and one has n photons.

313
00:21:03,160 --> 00:21:07,190
And behind the beam splitter,
you have two options.

314
00:21:07,190 --> 00:21:09,950
Either all the n photons
are in one state,

315
00:21:09,950 --> 00:21:12,970
or the n photons are
in the other state.

316
00:21:12,970 --> 00:21:16,900
You never have
any other mixture.

317
00:21:16,900 --> 00:21:18,990
Of course, you know your
standard half silvered

318
00:21:18,990 --> 00:21:24,440
mirror will not do that for
you, because it will split the n

319
00:21:24,440 --> 00:21:27,650
photon state with some
Poissonian statistics

320
00:21:27,650 --> 00:21:30,290
or whatnot into the
two arms and such.

321
00:21:30,290 --> 00:21:32,410
And you can calculate
exactly what a normal beam

322
00:21:32,410 --> 00:21:33,450
splitter does to that.

323
00:21:33,450 --> 00:21:35,040
It's very, very different.

324
00:21:35,040 --> 00:21:38,152
This here is a very, very
special beam splitter.

325
00:21:42,580 --> 00:21:47,070
Just to demystify this
beam splitter a little bit,

326
00:21:47,070 --> 00:21:56,660
I want to show you that, at
least conceptually, there

327
00:21:56,660 --> 00:22:00,550
is an easy implementation
with atoms.

328
00:22:00,550 --> 00:22:03,500
I really like in this section
to grab an example from atoms

329
00:22:03,500 --> 00:22:06,720
and to grab an example for
photons, because it really

330
00:22:06,720 --> 00:22:10,090
brings out that the
language may develop equally

331
00:22:10,090 --> 00:22:12,360
applies to atoms and photon.

332
00:22:12,360 --> 00:22:14,980
So if I take now
an easy example,

333
00:22:14,980 --> 00:22:17,790
if I take an example of the
Bose-Einstein condensate with n

334
00:22:17,790 --> 00:22:22,740
atoms, let's assume
we have a double well.

335
00:22:22,740 --> 00:22:24,950
But now we have
attractive interactions.

336
00:22:29,220 --> 00:22:31,050
This is not your standard BEC.

337
00:22:31,050 --> 00:22:33,130
Your standard
Bose-Einstein condensate

338
00:22:33,130 --> 00:22:36,840
has repulsive interaction,
with attractive interaction.

339
00:22:36,840 --> 00:22:40,740
If you go beyond a certain
size, certain a atom number,

340
00:22:40,740 --> 00:22:42,350
the condensate will collapse.

341
00:22:42,350 --> 00:22:45,190
So let's assume we have a
Bose-Einstein condensate with n

342
00:22:45,190 --> 00:22:45,900
atoms.

343
00:22:45,900 --> 00:22:47,940
There are some
attractive interactions,

344
00:22:47,940 --> 00:22:51,090
but we stay within
the stability diagram

345
00:22:51,090 --> 00:22:57,115
that those atoms do not-- that
those atoms do not collapse.

346
00:23:00,640 --> 00:23:03,900
So now, if the atoms
are attractive,

347
00:23:03,900 --> 00:23:05,280
they all want to be together.

348
00:23:05,280 --> 00:23:08,700
Because then they lower
the energy of each other.

349
00:23:08,700 --> 00:23:10,940
So if you have something
with attractive interaction,

350
00:23:10,940 --> 00:23:13,210
you want to have
n atoms together.

351
00:23:15,780 --> 00:23:21,390
But if your double
well potential

352
00:23:21,390 --> 00:23:28,990
is absolutely symmetric,
there is no symmetry breaking,

353
00:23:28,990 --> 00:23:32,480
and you have an equal
amplitude for all the atoms

354
00:23:32,480 --> 00:23:33,703
to be in the other well.

355
00:23:37,760 --> 00:23:46,020
So therefore, under the
very idealized situation

356
00:23:46,020 --> 00:23:48,410
I described here,
you will actually

357
00:23:48,410 --> 00:23:51,990
create a superposition
state of n atoms

358
00:23:51,990 --> 00:23:55,770
in one well with n
atoms in the other well.

359
00:23:55,770 --> 00:23:57,790
And this is exactly
what we tried

360
00:23:57,790 --> 00:23:59,690
to accomplish with
this beam splitter.

361
00:24:11,430 --> 00:24:17,690
OK, so if we have this
special beam splitter, which

362
00:24:17,690 --> 00:24:21,070
creates that state,
then when we add a phase

363
00:24:21,070 --> 00:24:25,680
shift to one arm of
the interferometer,

364
00:24:25,680 --> 00:24:31,860
we obtain-- which looks very
promising-- a phase shift

365
00:24:31,860 --> 00:24:33,800
phi multiplied by n.

366
00:24:38,120 --> 00:24:57,050
And if we read it
out, we have now

367
00:24:57,050 --> 00:25:04,850
if you pass it through the
other super beam splitter,

368
00:25:04,850 --> 00:25:10,460
we create again a
superposition of n0 and 0n.

369
00:25:14,390 --> 00:25:18,520
But now, because
of the phase shift

370
00:25:18,520 --> 00:25:23,280
with cosine and sine factors,
which involve n times

371
00:25:23,280 --> 00:25:24,150
the phase shift.

372
00:25:30,020 --> 00:25:33,080
For obvious reasons,
those states

373
00:25:33,080 --> 00:25:37,420
also go by the name noon states.

374
00:25:37,420 --> 00:25:39,940
That's the name which everybody
uses for those states,

375
00:25:39,940 --> 00:25:44,410
because if you just read
n00n, it gives noon.

376
00:25:44,410 --> 00:25:48,440
So this is the
famous noon state.

377
00:25:48,440 --> 00:25:51,880
So it seems it smells
already, right,

378
00:25:51,880 --> 00:25:55,040
because it has a
phase shift of n phi.

379
00:25:55,040 --> 00:26:00,720
Let me just convince you
that this is indeed the case.

380
00:26:00,720 --> 00:26:11,680
So the probability
of reading out

381
00:26:11,680 --> 00:26:15,730
0 photons in one arm
of the interferometer

382
00:26:15,730 --> 00:26:19,440
scales now with
the cosine square.

383
00:26:22,340 --> 00:26:27,810
If you do one
single measurement,

384
00:26:27,810 --> 00:26:40,860
the binomial
distribution is the same

385
00:26:40,860 --> 00:26:43,170
as before, but
with a factor of n.

386
00:26:47,310 --> 00:26:54,520
The derivative has
those factors of n.

387
00:26:54,520 --> 00:27:06,970
And that means that the variance
in the phase measurement

388
00:27:06,970 --> 00:27:08,080
is now 1 over n.

389
00:27:14,180 --> 00:27:17,950
If you want a reference,
I put it down here.

390
00:27:17,950 --> 00:27:20,260
As far as I know,
the experiment has

391
00:27:20,260 --> 00:27:22,680
been done with three
photons, but not

392
00:27:22,680 --> 00:27:23,840
with larger photon number.

393
00:27:32,306 --> 00:27:33,302
Questions?

394
00:27:33,302 --> 00:27:34,298
Collin.

395
00:27:34,298 --> 00:27:36,788
AUDIENCE: This picture
with the double well,

396
00:27:36,788 --> 00:27:41,072
isn't there going to be-- all
right, some people would say

397
00:27:41,072 --> 00:27:44,447
you spontaneously break the
symmetry and [INAUDIBLE]

398
00:27:44,447 --> 00:27:48,848
either ends up in one
side or the other.

399
00:27:48,848 --> 00:27:50,315
Like isn't this [INAUDIBLE]?

400
00:27:53,250 --> 00:27:55,940
PROFESSOR: Well,
Bose-Einstein condensation

401
00:27:55,940 --> 00:27:59,270
with attractive interactions
was observed in 1995.

402
00:27:59,270 --> 00:28:03,300
And now, 18 years later, nobody
has done this seemingly simple

403
00:28:03,300 --> 00:28:04,570
experiment.

404
00:28:04,570 --> 00:28:08,710
And what happens
is really that you

405
00:28:08,710 --> 00:28:10,900
have to be very,
very careful against

406
00:28:10,900 --> 00:28:13,210
any experimental imperfection.

407
00:28:13,210 --> 00:28:15,900
If the two double well
potentials are not

408
00:28:15,900 --> 00:28:18,740
exactly identical,
the bosons always

409
00:28:18,740 --> 00:28:21,300
want to go to the
lowest quantum state.

410
00:28:21,300 --> 00:28:23,020
Well, that's their
job, so to speak.

411
00:28:23,020 --> 00:28:24,370
That's their job description.

412
00:28:24,370 --> 00:28:26,540
And if you have a
minuscule difference

413
00:28:26,540 --> 00:28:30,590
between the two, the
dates of the two wells,

414
00:28:30,590 --> 00:28:32,250
you will not get a
superposition state.

415
00:28:32,250 --> 00:28:34,480
You will simply
populate one state.

416
00:28:34,480 --> 00:28:36,739
And how to make it
experimentally [INAUDIBLE],

417
00:28:36,739 --> 00:28:38,030
this is a really big challenge.

418
00:28:40,626 --> 00:28:41,125
[? Tino ?].

419
00:28:41,125 --> 00:28:42,740
AUDIENCE: I have a question.

420
00:28:42,740 --> 00:28:45,190
Let's say we're somehow
able to make the double well

421
00:28:45,190 --> 00:28:46,660
potential perfect.

422
00:28:46,660 --> 00:28:49,110
But if we didn't have
attractive interactions,

423
00:28:49,110 --> 00:28:51,720
then wouldn't he just
get a big product state

424
00:28:51,720 --> 00:28:54,310
of each atom being
in either well?

425
00:28:57,540 --> 00:29:00,414
PROFESSOR: OK, if you're
a non-interacting system,

426
00:29:00,414 --> 00:29:02,205
the ground state of a
double well potential

427
00:29:02,205 --> 00:29:06,990
is just your metric state 1
plus 2 over square root 2.

428
00:29:06,990 --> 00:29:09,920
And for non-interactive
BEC, you figure out

429
00:29:09,920 --> 00:29:11,900
what is the ground
state for one particle

430
00:29:11,900 --> 00:29:13,770
and then take it to the power n.

431
00:29:13,770 --> 00:29:15,106
This is the non-interactive BEC.

432
00:29:15,106 --> 00:29:17,605
STUDENT: So then entanglement
is because of the interaction,

433
00:29:17,605 --> 00:29:18,490
right?

434
00:29:18,490 --> 00:29:21,590
PROFESSOR: If you have
strong repulsive interaction,

435
00:29:21,590 --> 00:29:23,620
you have something
which should remind you

436
00:29:23,620 --> 00:29:25,900
of the [INAUDIBLE] insulator.

437
00:29:25,900 --> 00:29:27,110
You have n atoms.

438
00:29:27,110 --> 00:29:30,310
And n over 2 atoms go
to one well. n over 2 go

439
00:29:30,310 --> 00:29:31,840
to the other well.

440
00:29:31,840 --> 00:29:35,850
Because any form of number
fluctuations would be costly.

441
00:29:35,850 --> 00:29:38,400
It will cost you additional
repulsive interaction.

442
00:29:38,400 --> 00:29:40,460
So therefore, the
condensate wants

443
00:29:40,460 --> 00:29:43,640
to break up into
two equal parts.

444
00:29:43,640 --> 00:29:45,260
So that's actually
a way how you would

445
00:29:45,260 --> 00:29:48,650
create another non-classical
state, the dual flux

446
00:29:48,650 --> 00:29:51,690
state of n over 2,
n over 2 particle.

447
00:29:51,690 --> 00:29:58,140
And for attractive
interactions, well you

448
00:29:58,140 --> 00:29:59,540
should create the known state.

449
00:30:05,510 --> 00:30:09,340
OK, so this was now
a state n0 and 0n.

450
00:30:12,610 --> 00:30:16,410
There is another
state, which you

451
00:30:16,410 --> 00:30:19,530
have encountered
in your homework.

452
00:30:19,530 --> 00:30:25,350
And this is a superposition
not of n0 and 0n and n0.

453
00:30:25,350 --> 00:30:30,670
It's a superposition of n
minus 1n and n n minus 1.

454
00:30:30,670 --> 00:30:34,660
This state goes by the person
I think who invented it,

455
00:30:34,660 --> 00:30:36,560
the [? Yerkey ?] state.

456
00:30:36,560 --> 00:30:41,190
And you showed in your homework
that with that, you also

457
00:30:41,190 --> 00:30:45,260
reach Heisenberg Limited
Interferometry, where

458
00:30:45,260 --> 00:30:49,150
the phase scales is 1 over n.

459
00:30:49,150 --> 00:30:53,820
What I want to add
here to it is how

460
00:30:53,820 --> 00:30:56,560
one can create such
a [? Yerkey ?] state.

461
00:30:56,560 --> 00:30:59,960
And again, I want
to use the example

462
00:30:59,960 --> 00:31:02,800
with an atomic
Bose-Einstein condensate.

463
00:31:02,800 --> 00:31:06,135
And here's the reference where
this was very nicely discussed.

464
00:31:11,660 --> 00:31:16,420
So let's assume we can create
two Bose-Einstein condensates.

465
00:31:16,420 --> 00:31:19,300
And they have exactly n atoms.

466
00:31:19,300 --> 00:31:22,330
And I would actually refer
to [? Tino's ?] questions how

467
00:31:22,330 --> 00:31:30,050
to make them have
two n atoms in a trap

468
00:31:30,050 --> 00:31:32,310
and then make a
double well potential,

469
00:31:32,310 --> 00:31:35,250
deform the harmonic oscillator
potential to a double well

470
00:31:35,250 --> 00:31:36,260
potential.

471
00:31:36,260 --> 00:31:40,060
Then for strong
repulsive interactions,

472
00:31:40,060 --> 00:31:41,760
the condensate
will symmetrically

473
00:31:41,760 --> 00:31:47,580
split into two flux states,
each of which has n atoms.

474
00:31:47,580 --> 00:31:51,830
So now how can we create the
[? Yerkey ?] state from that?

475
00:31:51,830 --> 00:31:56,910
Well, we simply have--
we leak out atoms.

476
00:31:56,910 --> 00:31:59,980
We leak atoms out of the trap.

477
00:31:59,980 --> 00:32:03,640
My group demonstrated an RF
beam splitter, how you can just

478
00:32:03,640 --> 00:32:08,070
split in very
controlled way start

479
00:32:08,070 --> 00:32:11,810
rotating the cloud
on non-trapped state.

480
00:32:11,810 --> 00:32:14,970
And then atoms slowly leak out.

481
00:32:14,970 --> 00:32:17,240
Well, when you can
measure, of course,

482
00:32:17,240 --> 00:32:19,470
you can take an atom
detector and measure

483
00:32:19,470 --> 00:32:21,640
that an atom has
been out-coupled,

484
00:32:21,640 --> 00:32:26,065
that an atom has
leaked out of the trap.

485
00:32:26,065 --> 00:32:28,220
If you don't like
RF rotation, you

486
00:32:28,220 --> 00:32:31,810
can also think that there
is a tunneling barrier,

487
00:32:31,810 --> 00:32:35,240
and atoms slowly leak out
by whatever mechanism.

488
00:32:35,240 --> 00:32:38,290
And the moment you
detect that an atom,

489
00:32:38,290 --> 00:32:42,275
you then project the state in
the trap to n minus 1 atom,

490
00:32:42,275 --> 00:32:45,030
because you've measured
that one atom has come out.

491
00:32:45,030 --> 00:32:47,415
But now you use a beam splitter.

492
00:32:47,415 --> 00:32:51,610
And a beam splitter could
simply be a focused laser beam.

493
00:32:51,610 --> 00:32:54,030
And the atoms have
a 50% probability

494
00:32:54,030 --> 00:32:57,860
of being reflected or
end up tunneling through.

495
00:32:57,860 --> 00:33:01,250
So therefore, if you
have now a detector

496
00:33:01,250 --> 00:33:03,430
which measures the
atoms on one side

497
00:33:03,430 --> 00:33:08,470
and the atoms on the
other side, then you

498
00:33:08,470 --> 00:33:14,020
don't know anymore when the
detector makes click form which

499
00:33:14,020 --> 00:33:16,710
atom trap the atom came.

500
00:33:16,710 --> 00:33:20,260
Or more formally,
a beam splitter

501
00:33:20,260 --> 00:33:25,340
transforms the two input modes
ab into a plus b and a minus b

502
00:33:25,340 --> 00:33:27,790
normalized by square root 2.

503
00:33:27,790 --> 00:33:32,840
So therefore, if
this detector clicks,

504
00:33:32,840 --> 00:33:40,120
when you project the remaining
atoms into the symmetric state,

505
00:33:40,120 --> 00:33:44,840
here you detect one atom
in the mode a plus b.

506
00:33:44,840 --> 00:33:49,220
And that means that
the remaining atoms

507
00:33:49,220 --> 00:33:51,930
have been projected into
the [? Yerkey ?] state.

508
00:33:51,930 --> 00:33:54,070
If the lower atom
detector would click,

509
00:33:54,070 --> 00:33:56,600
well, you get a minus sign here.

510
00:33:56,600 --> 00:34:00,540
So that's one way how at
least in a conceptually simple

511
00:34:00,540 --> 00:34:05,120
situation, you can prepare
this highly non-classical state

512
00:34:05,120 --> 00:34:07,910
by starting with
a dual flux state

513
00:34:07,910 --> 00:34:11,480
of Bose-Einstein
condensates, and then using--

514
00:34:11,480 --> 00:34:16,050
and this is an ongoing theme
here-- by using a measurement,

515
00:34:16,050 --> 00:34:19,060
and then the
post-measurement state is

516
00:34:19,060 --> 00:34:21,469
the non-classical state
you wanted to prepare.

517
00:34:34,090 --> 00:34:34,830
Question?

518
00:34:34,830 --> 00:34:35,330
Yes.

519
00:34:35,330 --> 00:34:37,274
AUDIENCE: How can you
ensure that only one

520
00:34:37,274 --> 00:34:42,620
atom leaves the [INAUDIBLE]?

521
00:34:42,620 --> 00:34:44,650
PROFESSOR: The idea
is that we leak atoms

522
00:34:44,650 --> 00:34:46,870
out very, very slowly.

523
00:34:46,870 --> 00:34:48,989
And then we have
a detector, which

524
00:34:48,989 --> 00:34:52,090
we assume has 100%
quantum efficiency.

525
00:34:52,090 --> 00:34:55,820
So therefore, we simply wait
until the detector tells us

526
00:34:55,820 --> 00:34:58,060
that one atom has leaked out.

527
00:34:58,060 --> 00:35:02,050
And in an idealized experiment,
we know the atoms either

528
00:35:02,050 --> 00:35:03,791
have been measured
by the detector,

529
00:35:03,791 --> 00:35:05,040
or they are still in the trap.

530
00:35:07,781 --> 00:35:09,405
AUDIENCE: And also,
is there a property

531
00:35:09,405 --> 00:35:14,740
that holds the [INAUDIBLE]
you detect atoms [INAUDIBLE]?

532
00:35:18,150 --> 00:35:20,360
PROFESSOR: In principle
yes, but the idea here

533
00:35:20,360 --> 00:35:24,490
is if you have a very
slow leakage process,

534
00:35:24,490 --> 00:35:27,160
the probability that you detect
two atoms at the same time

535
00:35:27,160 --> 00:35:28,250
is really zero.

536
00:35:28,250 --> 00:35:31,270
You leak them continuously
and slowly, but then

537
00:35:31,270 --> 00:35:34,130
quantum mechanically, that
means for most of the time,

538
00:35:34,130 --> 00:35:34,970
you measure nothing.

539
00:35:34,970 --> 00:35:37,530
That means the leakage
hasn't taken place.

540
00:35:37,530 --> 00:35:40,430
The quantum mechanical
system has not developed yet.

541
00:35:40,430 --> 00:35:42,940
But the moment you
perform a measurement,

542
00:35:42,940 --> 00:35:45,040
you project-- it's
really the same

543
00:35:45,040 --> 00:35:47,680
if you say you have
n radioactive atoms.

544
00:35:47,680 --> 00:35:48,870
You have a detector.

545
00:35:48,870 --> 00:35:50,610
And when the
detector makes click,

546
00:35:50,610 --> 00:35:54,540
you know you have n minus
1 radioactive atoms left.

547
00:35:54,540 --> 00:35:57,900
It's just applied here
to two atom types.

548
00:35:57,900 --> 00:35:58,852
Other questions?

549
00:35:58,852 --> 00:35:59,768
AUDIENCE: [INAUDIBLE]?

550
00:36:02,522 --> 00:36:03,900
PROFESSOR: No.

551
00:36:03,900 --> 00:36:06,000
And maybe I'll tell
you now why not.

552
00:36:10,825 --> 00:36:14,080
We have discussed
the noon state.

553
00:36:14,080 --> 00:36:17,620
And we have discussed here
a highly non-classical

554
00:36:17,620 --> 00:36:19,970
superposition state.

555
00:36:19,970 --> 00:36:23,230
Let's just go back
to the noon state.

556
00:36:23,230 --> 00:36:28,060
We have n atoms here, zero here,
or n atoms, or the reverse.

557
00:36:28,060 --> 00:36:32,970
But now, assume a single atom
is lost, is lost from your trap

558
00:36:32,970 --> 00:36:35,230
by some background
gas collisions.

559
00:36:35,230 --> 00:36:39,880
And you have surrounded
the trap with a detector.

560
00:36:39,880 --> 00:36:43,540
So if you have the noon state,
the symmetric superposition

561
00:36:43,540 --> 00:36:46,030
state, all atoms here
and all atoms there.

562
00:36:46,030 --> 00:36:49,135
But by your background process,
by an inelastic collision,

563
00:36:49,135 --> 00:36:50,680
you lose one atom.

564
00:36:50,680 --> 00:36:51,950
And you detect it.

565
00:36:51,950 --> 00:36:53,705
You could set up your
detectors that you

566
00:36:53,705 --> 00:36:57,520
know from which
that you figure out

567
00:36:57,520 --> 00:36:59,900
from which trap was
the particle lost.

568
00:36:59,900 --> 00:37:03,000
So therefore, a
single particle lost

569
00:37:03,000 --> 00:37:07,080
if you localize from which
trap the particle is lost

570
00:37:07,080 --> 00:37:11,140
would immediately project
the noon state into a state

571
00:37:11,140 --> 00:37:15,670
where you know I have n or
n minus 1 atoms in one well,

572
00:37:15,670 --> 00:37:19,720
and 0 in the other well.

573
00:37:19,720 --> 00:37:23,510
So I've already told
you, with the attenuator,

574
00:37:23,510 --> 00:37:26,300
you can never assume
an attenuator is just

575
00:37:26,300 --> 00:37:27,345
attenuating a beam.

576
00:37:27,345 --> 00:37:31,050
An attenuator can always
regard it as a beam splitter.

577
00:37:31,050 --> 00:37:34,420
And you can do measurements at
both arms of the beam splitter.

578
00:37:34,420 --> 00:37:36,360
Or you have to consider
the vacuum noise which

579
00:37:36,360 --> 00:37:38,620
enters through the other
part of the beam splitter.

580
00:37:38,620 --> 00:37:44,580
And if you are now add that
those atoms, n atoms, in a trap

581
00:37:44,580 --> 00:37:47,050
have some natural loss
by inelastic collision

582
00:37:47,050 --> 00:37:49,670
or background gas
scattering, the loss

583
00:37:49,670 --> 00:37:53,110
is actually like a beam
splitter that particles

584
00:37:53,110 --> 00:37:56,280
don't stay in the trap, but
go out through the other part.

585
00:37:56,280 --> 00:37:58,070
And then you can measure it.

586
00:37:58,070 --> 00:38:01,400
So in other words,
every loss process

587
00:38:01,400 --> 00:38:03,836
should be regarded as
a possible measurement.

588
00:38:03,836 --> 00:38:05,210
And it doesn't
matter whether you

589
00:38:05,210 --> 00:38:08,240
perform the measurement or not.

590
00:38:08,240 --> 00:38:11,470
And I think it's just obvious
that the noon state, the moment

591
00:38:11,470 --> 00:38:15,920
one particle is lost and you
reduce this particle to figure

592
00:38:15,920 --> 00:38:19,410
out if n particles are here
or n particles are there,

593
00:38:19,410 --> 00:38:22,320
the whole superposition
state is lost if you just

594
00:38:22,320 --> 00:38:24,210
lose one single atom out of n.

595
00:38:28,970 --> 00:38:34,060
So the lifetime of a noon state
is then not your usual trap

596
00:38:34,060 --> 00:38:37,120
lifetime, where you
lose half of the atoms,

597
00:38:37,120 --> 00:38:38,660
1 over e of the atoms.

598
00:38:38,660 --> 00:38:42,280
It is n times faster because
it is the first atom lost,

599
00:38:42,280 --> 00:38:44,370
which is already
completely removing

600
00:38:44,370 --> 00:38:46,034
the entanglement of your state.

601
00:38:56,190 --> 00:39:00,450
So more quantitatively, to
say it more specifically,

602
00:39:00,450 --> 00:39:04,180
the limitation is loss.

603
00:39:04,180 --> 00:39:09,970
When you have a fully entangled
state, maximally entangled

604
00:39:09,970 --> 00:39:14,230
state, usually a
loss of one particle

605
00:39:14,230 --> 00:39:17,850
immediately removes
the entanglement.

606
00:39:17,850 --> 00:39:22,260
We had the situa-- no,
that's not a good example.

607
00:39:22,260 --> 00:39:26,060
But for the most
entangled state,

608
00:39:26,060 --> 00:39:27,840
usually-- and for
the noon state,

609
00:39:27,840 --> 00:39:31,820
it's trivial to see-- a
single particle lost allows

610
00:39:31,820 --> 00:39:35,560
you to measure on which side
of the potential barrier

611
00:39:35,560 --> 00:39:37,700
all the atoms are,
and all the beauty

612
00:39:37,700 --> 00:39:39,207
of the non-classical
state is lost.

613
00:39:42,870 --> 00:39:44,910
So if you assume that
in a time window,

614
00:39:44,910 --> 00:39:47,360
you have an infinitesimal
loss, a loss

615
00:39:47,360 --> 00:39:51,010
of epsilon, what
usually happens is

616
00:39:51,010 --> 00:39:53,730
if you have an entanglement
of n particles,

617
00:39:53,730 --> 00:39:56,330
then you lose your entanglement.

618
00:39:56,330 --> 00:39:58,820
If you expand 1
minus epsilon to n,

619
00:39:58,820 --> 00:40:02,470
it becomes 1 minus
n times epsilon.

620
00:40:02,470 --> 00:40:06,340
So that's one reason why
people have not scaled up

621
00:40:06,340 --> 00:40:09,160
those schemes to a
large number of photons,

622
00:40:09,160 --> 00:40:11,010
or a large number of atoms.

623
00:40:11,010 --> 00:40:14,060
Because the larger n
is, the more sensitive

624
00:40:14,060 --> 00:40:16,930
you are to even very,
very small losses.

625
00:40:26,630 --> 00:40:29,540
Questions?

626
00:40:29,540 --> 00:40:31,650
AUDIENCE: [INAUDIBLE]

627
00:40:31,650 --> 00:40:32,275
PROFESSOR: Yes.

628
00:40:32,275 --> 00:40:33,608
AUDIENCE: Or is that physically?

629
00:40:36,410 --> 00:40:38,910
PROFESSOR: The
super beam splitter

630
00:40:38,910 --> 00:40:42,580
would create the noon state.

631
00:40:42,580 --> 00:40:44,700
I've not explained to
you what it physically

632
00:40:44,700 --> 00:40:46,810
would be for photons,
but I gave you

633
00:40:46,810 --> 00:40:50,870
the example for atoms
to BECs in a double well

634
00:40:50,870 --> 00:40:53,540
potential with
attractive interaction.

635
00:40:53,540 --> 00:40:55,740
So to start with
the condensate in--

636
00:40:59,090 --> 00:41:00,840
AUDIENCE: Start with
a bigger [INAUDIBLE].

637
00:41:00,840 --> 00:41:03,200
PROFESSOR: If you start with
a double well potential,

638
00:41:03,200 --> 00:41:05,780
and you put in n
atoms initially in,

639
00:41:05,780 --> 00:41:10,030
but then you switch
on tunnel coupling,

640
00:41:10,030 --> 00:41:12,970
then you would
create the noon state

641
00:41:12,970 --> 00:41:15,040
if the interactions
are attractive,

642
00:41:15,040 --> 00:41:18,220
and if everything is idealized,
that you have a completely

643
00:41:18,220 --> 00:41:19,650
symmetric double well potential.

644
00:41:24,770 --> 00:41:31,336
OK, the last example is the
squeeze light Interferometer.

645
00:41:31,336 --> 00:41:33,250
I just want to
mention it briefly,

646
00:41:33,250 --> 00:41:35,880
because we've talked so
much about squeezed light.

647
00:41:35,880 --> 00:41:39,200
So now I want to show you
that squeezed light can also

648
00:41:39,200 --> 00:41:46,400
be used to realize Heisenberg
Limited Interferometery.

649
00:41:49,560 --> 00:41:57,520
So the idea is, when we plot
the electric field versus time,

650
00:41:57,520 --> 00:42:01,400
and if you do squeezing
in one quadrature,

651
00:42:01,400 --> 00:42:07,670
then for certain times,
the electric field

652
00:42:07,670 --> 00:42:09,770
has lower noise
and higher noise.

653
00:42:09,770 --> 00:42:11,130
We discussed that.

654
00:42:11,130 --> 00:42:15,090
And the idea is that if there is
no noise at the zero crossing,

655
00:42:15,090 --> 00:42:19,280
that this means we can determine
the zero crossing of the light,

656
00:42:19,280 --> 00:42:22,095
and therefore the phase
shift with higher accuracy.

657
00:42:27,130 --> 00:42:36,120
You may also argue if you
have this quasi-probabilities,

658
00:42:36,120 --> 00:42:45,260
and with squeezing, we have
squeezed the coherent light

659
00:42:45,260 --> 00:42:48,010
into an ellipse.

660
00:42:48,010 --> 00:43:01,070
And things propagate
with e to the i omega t.

661
00:43:01,070 --> 00:43:07,000
Then you can determine
a phase shift,

662
00:43:07,000 --> 00:43:10,370
which is sort of an angular
sector in this diagram,

663
00:43:10,370 --> 00:43:13,040
with higher precision
if you have squeezed

664
00:43:13,040 --> 00:43:16,890
this circle into an
ellipse like that.

665
00:43:16,890 --> 00:43:17,950
So that's the idea.

666
00:43:17,950 --> 00:43:24,780
And well, it's fairly clear that
squeezing, if done correctly,

667
00:43:24,780 --> 00:43:27,860
can provide a better
phase measurement.

668
00:43:27,860 --> 00:43:31,280
And what I want to show
you here in a few minutes

669
00:43:31,280 --> 00:43:36,040
is how you can think about it.

670
00:43:36,040 --> 00:43:40,500
So we have discussed at length
the optical Interferometer,

671
00:43:40,500 --> 00:43:44,440
where we have just a
coherence state at the input.

672
00:43:44,440 --> 00:43:47,360
This is your standard
laser Interferometer.

673
00:43:47,360 --> 00:43:51,880
But of course, very importantly,
the second input port

674
00:43:51,880 --> 00:43:53,120
has the vacuum state.

675
00:43:53,120 --> 00:43:55,540
And we discussed the
importance of that.

676
00:43:55,540 --> 00:43:57,920
So the one difference
we want to do now

677
00:43:57,920 --> 00:44:02,990
is that we replace the
vacuum at the second input

678
00:44:02,990 --> 00:44:06,540
by the squeezed
vacuum where r is

679
00:44:06,540 --> 00:44:09,200
the parameter of
the squeezed vacuum.

680
00:44:12,560 --> 00:44:14,660
OK, so that's pretty
much what we do.

681
00:44:14,660 --> 00:44:17,610
We take this state, we
plug it into our equation,

682
00:44:17,610 --> 00:44:19,340
we use exactly
the same formalism

683
00:44:19,340 --> 00:44:21,010
we have used for
coherence states.

684
00:44:21,010 --> 00:44:22,980
And the question,
what is the result?

685
00:44:22,980 --> 00:44:29,150
Well, the result will be that
the squeezing factor appears.

686
00:44:29,150 --> 00:44:38,560
Just as a reminder,
for our Interferometer,

687
00:44:38,560 --> 00:44:43,200
we derived the sensitivity
of the Interferometer.

688
00:44:43,200 --> 00:44:45,500
We had the quantities x and y.

689
00:44:45,500 --> 00:44:51,430
And the noise is delta in this
y operator squared divided by x.

690
00:44:51,430 --> 00:44:55,240
This was the result when we
operate the Interferometer

691
00:44:55,240 --> 00:44:58,050
at the phase shift
of 90 degrees.

692
00:44:58,050 --> 00:44:58,850
Just a reminder.

693
00:44:58,850 --> 00:45:00,010
That's what we have done.

694
00:45:00,010 --> 00:45:01,720
That's how we
analyze the situation

695
00:45:01,720 --> 00:45:02,795
with a coherent state.

696
00:45:05,770 --> 00:45:17,270
The signal x is
now-- the signal x

697
00:45:17,270 --> 00:45:22,475
is the number
measurement a dagger a.

698
00:45:26,480 --> 00:45:33,325
And we have now the input
of the coherent state.

699
00:45:37,120 --> 00:45:38,962
And b dagger b.

700
00:45:38,962 --> 00:45:41,340
We have an input
mode a and a mode b.

701
00:45:41,340 --> 00:45:42,480
They get split.

702
00:45:42,480 --> 00:45:44,370
And then we measure
it at the output.

703
00:45:44,370 --> 00:45:47,550
And we can now at the
output have photons--

704
00:45:47,550 --> 00:45:50,340
a dagger a, which come
from the coherent state,

705
00:45:50,340 --> 00:45:54,900
and b dagger b, which come
from the squeeze vacuum.

706
00:45:58,800 --> 00:46:03,410
So this is now using the
beam splitter formalism

707
00:46:03,410 --> 00:46:05,590
applied to the Interferometer.

708
00:46:05,590 --> 00:46:08,640
So this is now the
result we obtain.

709
00:46:08,640 --> 00:46:12,410
And in the strong local
oscillator approximation,

710
00:46:12,410 --> 00:46:16,790
it is only the first
part which contributes.

711
00:46:16,790 --> 00:46:20,550
And this is simply the number
or photons in the coherent beam.

712
00:46:23,810 --> 00:46:28,990
The expectation value
of y is 0, because it

713
00:46:28,990 --> 00:46:32,680
involves a b and
b dagger operator.

714
00:46:32,680 --> 00:46:40,310
And the squeezed
vacuum has only,

715
00:46:40,310 --> 00:46:43,400
if you write it down, in the
n basis, in the flux basis,

716
00:46:43,400 --> 00:46:44,920
has only even n.

717
00:46:44,920 --> 00:46:47,000
So if you change
n by one, you lose

718
00:46:47,000 --> 00:46:48,910
overlap with the
squeezed vacuum.

719
00:46:48,910 --> 00:46:52,480
So therefore, this
expectation value is 0.

720
00:46:52,480 --> 00:47:05,840
For the operator y squared,
you take this and square it.

721
00:47:05,840 --> 00:47:09,660
And you get many, many terms,
which I don't want to discuss.

722
00:47:09,660 --> 00:47:11,740
I use the strong
local oscillator.

723
00:47:11,740 --> 00:47:14,970
Limited a dagger and
a can be replaced

724
00:47:14,970 --> 00:47:18,660
by the eigenvalue alpha
of the coherent state.

725
00:47:18,660 --> 00:47:22,910
So therefore, I factor
out alpha squared

726
00:47:22,910 --> 00:47:29,570
in the strong local
oscillator limit.

727
00:47:29,570 --> 00:47:35,515
And then what is left is
b plus b dagger squared.

728
00:47:39,460 --> 00:47:44,900
And since we have
squeezed the vacuum,

729
00:47:44,900 --> 00:47:49,280
this gives us a factor
into the minus r.

730
00:47:49,280 --> 00:47:58,200
So if we put all those
results together,

731
00:47:58,200 --> 00:48:05,060
we find that the phase
uncertainty is now

732
00:48:05,060 --> 00:48:07,820
what we obtained when
we had a coherent state

733
00:48:07,820 --> 00:48:10,030
with the ordinary vacuum.

734
00:48:10,030 --> 00:48:12,460
And in the strong
local oscillator limit,

735
00:48:12,460 --> 00:48:15,050
the only difference
to the ordinary vacuum

736
00:48:15,050 --> 00:48:21,600
is that in this term, we've
got the exponential factor e

737
00:48:21,600 --> 00:48:25,780
to the minus r.

738
00:48:25,780 --> 00:48:27,770
And since we have
taken a square root,

739
00:48:27,770 --> 00:48:31,340
it's e to the minus r over 2.

740
00:48:31,340 --> 00:48:38,750
So that result would
actually suggest

741
00:48:38,750 --> 00:48:47,730
that the more we squeeze,
that delta phi should go to 0.

742
00:48:47,730 --> 00:48:50,430
So it seems even better
than the Heisenberg limit.

743
00:48:53,660 --> 00:48:59,460
However-- well, this is
too good to be true--

744
00:48:59,460 --> 00:49:02,310
what I've neglected here
is just the following.

745
00:49:02,310 --> 00:49:05,660
When you squeeze more
and more, the more you

746
00:49:05,660 --> 00:49:08,920
squeeze the vacuum,
the more photons

747
00:49:08,920 --> 00:49:12,250
are in the squeezed vacuum,
because this ellipse stitches

748
00:49:12,250 --> 00:49:15,880
further and further out and
has overlap with flux states

749
00:49:15,880 --> 00:49:18,230
at higher and higher
photon number.

750
00:49:18,230 --> 00:49:24,670
So therefore, when you go to
the limit of infinite squeezing,

751
00:49:24,670 --> 00:49:26,530
you squeeze out
of the limit where

752
00:49:26,530 --> 00:49:29,480
you can regard the local
oscillator as strong,

753
00:49:29,480 --> 00:49:31,530
because the squeezed
vacuum has more photons

754
00:49:31,530 --> 00:49:32,980
than your local oscillator.

755
00:49:32,980 --> 00:49:36,820
And then you have to
consider additional terms.

756
00:49:36,820 --> 00:49:39,070
So let me just write that down.

757
00:49:39,070 --> 00:49:52,620
However, the squeezed vacuum has
non-zero average photon number.

758
00:49:57,990 --> 00:50:00,670
And the photon number
of the squeezed vacuum

759
00:50:00,670 --> 00:50:06,880
is, of course, apply b dagger
b to the squeezed vacuum.

760
00:50:06,880 --> 00:50:11,730
This gives us a sinc function.

761
00:50:11,730 --> 00:50:14,480
And we can call this
the number of photons

762
00:50:14,480 --> 00:50:15,515
in the squeezed vacuum.

763
00:50:20,410 --> 00:50:26,880
So we have to consider now
this contribution to y square.

764
00:50:33,130 --> 00:50:36,770
So we have to
consider the quadrator

765
00:50:36,770 --> 00:50:39,620
of the ellipse, the long
part of the ellipse,

766
00:50:39,620 --> 00:50:42,390
the non-squeezed
quadrator component.

767
00:50:50,890 --> 00:50:52,910
And we have to
consider that when

768
00:50:52,910 --> 00:50:57,010
we calculate the expectation
value of y square.

769
00:50:57,010 --> 00:51:00,270
And then we find
additional terms,

770
00:51:00,270 --> 00:51:02,800
which I don't want
to derive here.

771
00:51:02,800 --> 00:51:06,886
And the question is then, if
you squeeze too much, you lose.

772
00:51:06,886 --> 00:51:08,635
So there's an optimal
amount of squeezing.

773
00:51:13,220 --> 00:51:19,290
And for this optimal
amount of squeezing,

774
00:51:19,290 --> 00:51:25,670
the phase uncertainty
becomes approximately one

775
00:51:25,670 --> 00:51:28,610
over the number of photons
in the coherent state,

776
00:51:28,610 --> 00:51:31,880
plus the number of photons
in your squeeze vacuum.

777
00:51:31,880 --> 00:51:35,345
So this is, again, very close
to the Heisenberg limit.

778
00:51:43,810 --> 00:51:46,560
So the situation
with squeezed light

779
00:51:46,560 --> 00:51:50,830
is less elegant, because
if you squeeze too much,

780
00:51:50,830 --> 00:51:52,510
you have to consider
additional terms.

781
00:51:52,510 --> 00:51:55,190
This is why I gave you the
example of the squeezed light

782
00:51:55,190 --> 00:51:56,690
and the squeezed
vacuum as the last.

783
00:51:56,690 --> 00:52:00,080
But again, the Heisenberg
limit is very fundamental

784
00:52:00,080 --> 00:52:01,190
as we discussed.

785
00:52:01,190 --> 00:52:03,710
And for an optimum
arrangement of the squeezing,

786
00:52:03,710 --> 00:52:06,830
you can also use a
squeezed vacuum input

787
00:52:06,830 --> 00:52:09,890
to the Interferometer to
realize the Heisenberg limit.

788
00:52:13,420 --> 00:52:14,823
Any questions?

789
00:52:22,320 --> 00:52:24,710
Why is squeezing important?

790
00:52:24,710 --> 00:52:29,350
Well, squeezing caught the
attention of the physics

791
00:52:29,350 --> 00:52:32,840
community when it was
suggested in connection

792
00:52:32,840 --> 00:52:35,080
with gravitational--
with the detection

793
00:52:35,080 --> 00:52:36,810
of gravitational waves.

794
00:52:36,810 --> 00:52:40,620
As you know, the
laser Interferometer,

795
00:52:40,620 --> 00:52:44,630
the most advanced one is
LIGO, has a monumental task

796
00:52:44,630 --> 00:52:47,290
in detecting a
very small signal.

797
00:52:47,290 --> 00:52:54,230
And pretty much everything which
precision metrology can provide

798
00:52:54,230 --> 00:52:56,870
is being implemented
for that purpose.

799
00:52:56,870 --> 00:53:00,230
So you can see, this is
like precision measurement.

800
00:53:00,230 --> 00:53:02,120
Like maybe the trip
to the moon was

801
00:53:02,120 --> 00:53:10,290
for aviation in
several decades ago.

802
00:53:10,290 --> 00:53:13,690
So everything is really--
a lot of things pushing

803
00:53:13,690 --> 00:53:15,430
the frontier of
precision measurement

804
00:53:15,430 --> 00:53:19,290
is motivated by the precision
needed for gravitational wave

805
00:53:19,290 --> 00:53:20,070
detection.

806
00:53:20,070 --> 00:53:27,110
And what I want to show
you here is a diagram

807
00:53:27,110 --> 00:53:29,240
for what is called
advanced LIGO.

808
00:53:29,240 --> 00:53:31,470
LIGO is currently
operating, but there

809
00:53:31,470 --> 00:53:34,500
is an upgrade to LIGO
called advanced LIGO.

810
00:53:34,500 --> 00:53:36,860
And what you
recognize here is we

811
00:53:36,860 --> 00:53:44,010
have a laser which goes into
a Michelson Interferometer.

812
00:53:44,010 --> 00:53:46,640
And this is how you want to
detect gravitational waves.

813
00:53:46,640 --> 00:53:50,030
But now you realize
that the addition here

814
00:53:50,030 --> 00:53:52,430
is a squeezed source.

815
00:53:52,430 --> 00:53:55,191
And what you are
squeezing is not,

816
00:53:55,191 --> 00:53:56,690
while it should be
clear to you now,

817
00:53:56,690 --> 00:53:58,680
we're not squeezing
the laser beam.

818
00:53:58,680 --> 00:54:01,470
This would be much, much harder,
because many, many photons are

819
00:54:01,470 --> 00:54:02,700
involved.

820
00:54:02,700 --> 00:54:10,360
But it is sufficient to
squeeze the vacuum and couple

821
00:54:10,360 --> 00:54:14,420
in squeezed vacuum into your
cavitational wave detector.

822
00:54:25,890 --> 00:54:29,300
If you wonder, it's a
little bit more complicated

823
00:54:29,300 --> 00:54:32,820
because people want to
recycle light and have

824
00:54:32,820 --> 00:54:35,650
put in other bells and whistles.

825
00:54:35,650 --> 00:54:38,590
But in essence, a
squeezed vacuum source

826
00:54:38,590 --> 00:54:41,950
is a major addition
to advanced LIGO.

827
00:54:41,950 --> 00:54:42,758
Yes.

828
00:54:42,758 --> 00:54:44,382
AUDIENCE: Where is
the squeeze actually

829
00:54:44,382 --> 00:54:45,830
coming into the system?

830
00:54:45,830 --> 00:54:48,370
I see where it's drawn,
but where is it actually

831
00:54:48,370 --> 00:54:50,064
entering the interferometer?

832
00:54:50,064 --> 00:54:54,420
At that first beam splitter?

833
00:54:54,420 --> 00:54:56,670
PROFESSOR: OK.

834
00:54:56,670 --> 00:54:59,739
We have to now-- there are
more things added here.

835
00:54:59,739 --> 00:55:02,030
Ideally, you would think you
have a beam splitter here,

836
00:55:02,030 --> 00:55:04,380
the laser comes in
here, and you simply

837
00:55:04,380 --> 00:55:06,550
want to enter the
squeezed vacuum here.

838
00:55:06,550 --> 00:55:08,270
And this is how we
have explained it.

839
00:55:08,270 --> 00:55:10,660
We have one beam splitter
in our Interferometer.

840
00:55:10,660 --> 00:55:13,630
There is an input
port and an open port.

841
00:55:13,630 --> 00:55:16,480
But what is important
here is also

842
00:55:16,480 --> 00:55:22,900
that the measurement--
here you have

843
00:55:22,900 --> 00:55:27,040
a detector for reading
out the Interferometer.

844
00:55:27,040 --> 00:55:31,260
And what is important
is that the phase

845
00:55:31,260 --> 00:55:35,750
is balanced close to the point
where no light is coming out.

846
00:55:35,750 --> 00:55:39,090
So you're measuring the
zero crossing of a fringe.

847
00:55:39,090 --> 00:55:41,180
But that would mean
most of the light

848
00:55:41,180 --> 00:55:43,150
would then exit
the Interferometer

849
00:55:43,150 --> 00:55:44,920
at the other port.

850
00:55:44,920 --> 00:55:49,650
But high power lasers are
very important for keeping

851
00:55:49,650 --> 00:55:51,995
the classical short noise down.

852
00:55:51,995 --> 00:55:54,740
So you want to work with
the highest power possible.

853
00:55:54,740 --> 00:55:57,760
And therefore, you can't
allow the light to exit.

854
00:55:57,760 --> 00:55:59,040
You want to recycle it.

855
00:55:59,040 --> 00:56:01,450
You want to use
enhancement cavities.

856
00:56:01,450 --> 00:56:05,020
And what I can tell you is that
this set up here integrates,

857
00:56:05,020 --> 00:56:08,440
I think, the signal
recycling, the measurement

858
00:56:08,440 --> 00:56:09,790
at the zero fringe.

859
00:56:09,790 --> 00:56:12,000
And you see that kind
of those different parts

860
00:56:12,000 --> 00:56:14,320
are copied in a
way which I didn't

861
00:56:14,320 --> 00:56:17,090
prepare to explain it to you.

862
00:56:17,090 --> 00:56:19,660
All I wanted you to
do is pretty much

863
00:56:19,660 --> 00:56:23,410
recognize that a squeeze
light generator is important.

864
00:56:23,410 --> 00:56:25,660
And this enters
the interferometer

865
00:56:25,660 --> 00:56:27,130
as a squeezed vacuum.

866
00:56:34,230 --> 00:56:35,860
What I find very
interesting, and this

867
00:56:35,860 --> 00:56:38,640
is what I want to
discuss now is,

868
00:56:38,640 --> 00:56:42,120
that when you have an
interferometer like LIGO,

869
00:56:42,120 --> 00:56:45,080
cavitational wave
interferometer,

870
00:56:45,080 --> 00:56:48,380
and now you want to squeeze.

871
00:56:48,380 --> 00:56:50,150
Does it really help to squeeze?

872
00:56:50,150 --> 00:56:51,960
Does it always help to squeeze?

873
00:56:51,960 --> 00:56:54,780
Or what is the situation?

874
00:56:54,780 --> 00:56:56,750
And this is what I want
to discuss with you.

875
00:57:03,350 --> 00:57:11,440
So let's forget about signal
recycling and enhancement

876
00:57:11,440 --> 00:57:13,160
cavities and things like this.

877
00:57:13,160 --> 00:57:17,690
Let's just discuss the basic
cavitation wave detector, where

878
00:57:17,690 --> 00:57:20,440
we have an input,
we have the two arms

879
00:57:20,440 --> 00:57:22,480
of the Michelson interferometer.

880
00:57:22,480 --> 00:57:27,160
And to have more
sensitivity, the light

881
00:57:27,160 --> 00:57:30,310
bounces back and forth
in an enhancement cavity.

882
00:57:30,310 --> 00:57:33,100
You can say if the light bounces
back and forth 100 times,

883
00:57:33,100 --> 00:57:35,390
it is as if you had
an arm length which

884
00:57:35,390 --> 00:57:37,830
is 100 times larger.

885
00:57:37,830 --> 00:57:41,010
And now we put in
squeezed vacuum

886
00:57:41,010 --> 00:57:43,540
at the open part of
the interferometer.

887
00:57:43,540 --> 00:57:45,945
And here we have
our photo-diodes

888
00:57:45,945 --> 00:57:47,070
to perform the measurement.

889
00:57:53,190 --> 00:58:00,710
So the goal is to measure
a small length scale.

890
00:58:00,710 --> 00:58:03,680
If a cavitation wave comes
by, cavitational waves

891
00:58:03,680 --> 00:58:05,410
have quadrupolar character.

892
00:58:05,410 --> 00:58:07,970
So the metric will
be such that there's

893
00:58:07,970 --> 00:58:11,070
a quadrupolar perturbation
in the metric of space.

894
00:58:11,070 --> 00:58:16,090
And that means that, in
essence, one of the mirrors

895
00:58:16,090 --> 00:58:17,560
is slightly moving out.

896
00:58:17,560 --> 00:58:19,910
The other mirror is
slightly moving in.

897
00:58:19,910 --> 00:58:22,680
So therefore, the interferometer
needs a very, very high

898
00:58:22,680 --> 00:58:26,080
sensitivity to displacement
of one of the mirrors

899
00:58:26,080 --> 00:58:28,380
by an amount delta z.

900
00:58:28,380 --> 00:58:33,650
And if you normalize
delta z to the arm length,

901
00:58:33,650 --> 00:58:36,790
or the arm length times
the number of bounces,

902
00:58:36,790 --> 00:58:41,020
the task is to
measure sensitivity

903
00:58:41,020 --> 00:58:43,315
in a length displacement
of 10 to the minus 21.

904
00:58:46,350 --> 00:58:48,320
That's one of the
smallest numbers

905
00:58:48,320 --> 00:58:50,220
which have ever been measured.

906
00:58:50,220 --> 00:58:54,680
And therefore, it is clear
that this interferometer

907
00:58:54,680 --> 00:58:57,886
should operate as close
as possible to the quantum

908
00:58:57,886 --> 00:58:58,760
limit of measurement.

909
00:59:03,340 --> 00:59:06,880
So what you want to measure here
is, with the highest accuracy

910
00:59:06,880 --> 00:59:12,420
possible, the displacement
of an object delta z.

911
00:59:15,090 --> 00:59:21,050
However, your object fulfills
and uncertainty relationship,

912
00:59:21,050 --> 00:59:24,580
that if you want to measure
the position very accurately,

913
00:59:24,580 --> 00:59:31,020
you also have to consider that
it has a momentum uncertainty.

914
00:59:31,020 --> 00:59:34,020
And this fulfills Heisenberg's
uncertainty relation.

915
00:59:34,020 --> 00:59:37,460
You will say, well, why should
I care about the momentum

916
00:59:37,460 --> 00:59:41,740
uncertainty if all I want
to measure is the position.

917
00:59:41,740 --> 00:59:46,850
Well, you should care
because momentum uncertainty

918
00:59:46,850 --> 00:59:50,630
after a time tau turns
into position uncertainty,

919
00:59:50,630 --> 00:59:53,950
because position uncertainty
is uncertainty in velocity.

920
00:59:53,950 --> 00:59:56,120
And if I multiply
it by the time tau

921
00:59:56,120 --> 00:59:58,420
it takes you to perform
the measurement,

922
00:59:58,420 --> 01:00:00,890
you have now an
uncertainty in position,

923
01:00:00,890 --> 01:00:03,215
which comes from the original
uncertainty in momentum.

924
01:00:08,090 --> 01:00:13,040
So if I use the expression
for Heisenberg's Uncertainty

925
01:00:13,040 --> 01:00:17,130
Relation, I find this.

926
01:00:19,690 --> 01:00:24,170
And now, what we
have to minimize

927
01:00:24,170 --> 01:00:28,350
to get the highest precision
is the total uncertainty

928
01:00:28,350 --> 01:00:34,850
in position, which is
the original uncertainty,

929
01:00:34,850 --> 01:00:37,694
plus the uncertainty due
to the motion of the mirror

930
01:00:37,694 --> 01:00:38,985
during the measurement process.

931
01:00:43,040 --> 01:00:45,270
So what we have here
is we have delta z.

932
01:00:45,270 --> 01:00:48,320
We have a contribution which
scores as 1 over delta z.

933
01:00:48,320 --> 01:00:53,850
And by just finding out what is
the optimum choice of delta z,

934
01:00:53,850 --> 01:00:56,940
you find the result above.

935
01:00:56,940 --> 01:01:01,050
Or if you want to say you
want that this delta z tau is

936
01:01:01,050 --> 01:01:05,460
comparable to delta z, just
set this equal to delta z,

937
01:01:05,460 --> 01:01:12,510
solve for delta z, and
you find the quantum limit

938
01:01:12,510 --> 01:01:19,560
for the interferometer,
which is given up there.

939
01:01:19,560 --> 01:01:21,640
So this has nothing
to do with squeezing.

940
01:01:21,640 --> 01:01:25,340
And you cannot improve on this
quantum limit by squeezing.

941
01:01:25,340 --> 01:01:27,200
This is what you got.

942
01:01:27,200 --> 01:01:32,610
It only depends on the
duration of the measurement.

943
01:01:32,610 --> 01:01:35,696
And it depends on the
mass of the mirror.

944
01:01:41,410 --> 01:01:53,860
Now-- just get my
notes ready-- there

945
01:01:53,860 --> 01:01:57,880
is a very influential
and seminal paper

946
01:01:57,880 --> 01:02:00,400
by Caves-- the reference
is given here--

947
01:02:00,400 --> 01:02:03,000
which was really
laying out the concepts

948
01:02:03,000 --> 01:02:06,910
and the theory for quantum
limited measurements with such

949
01:02:06,910 --> 01:02:10,220
an interferometer, and
the use of squeezed light.

950
01:02:10,220 --> 01:02:15,420
Let me just summarize the
most important findings.

951
01:02:15,420 --> 01:02:20,150
So this paper explains that
you have two contributions

952
01:02:20,150 --> 01:02:27,760
to the noise, which
depend on the laser

953
01:02:27,760 --> 01:02:30,030
power you use for
your measurement.

954
01:02:30,030 --> 01:02:33,390
The first one is the
photon counting noise.

955
01:02:33,390 --> 01:02:36,870
If you use more and
more laser power,

956
01:02:36,870 --> 01:02:38,760
you have a better
and better signal,

957
01:02:38,760 --> 01:02:41,420
and your short noise is reduced.

958
01:02:41,420 --> 01:02:45,170
So therefore, you have a better
read out of the interferometer.

959
01:02:45,170 --> 01:02:46,970
And this is given here.

960
01:02:46,970 --> 01:02:51,240
Alpha is the eigenvalue
of the coherent state.

961
01:02:51,240 --> 01:02:54,810
But there is a second
aspect which you may not

962
01:02:54,810 --> 01:02:57,560
have thought about it,
and this is the following.

963
01:02:57,560 --> 01:03:02,140
If you split a laser
beam into two parts,

964
01:03:02,140 --> 01:03:04,060
you have fluctuations.

965
01:03:04,060 --> 01:03:09,350
The number of photons left
and right are not identical.

966
01:03:09,350 --> 01:03:11,420
You have a coherent
bean and you split it

967
01:03:11,420 --> 01:03:13,450
into two coherent
beams, and then you

968
01:03:13,450 --> 01:03:16,060
have Poissonian
fluctuations on either side.

969
01:03:16,060 --> 01:03:19,820
But if you have now Poissonian
fluctuations in the photon

970
01:03:19,820 --> 01:03:23,400
number, if those photons
are reflected off a mirror,

971
01:03:23,400 --> 01:03:26,510
they transfer photon
recoil to the mirror.

972
01:03:26,510 --> 01:03:32,860
And the mirror is pushed
by the radiation pressure.

973
01:03:32,860 --> 01:03:35,008
And it's pushed,
and it has-- there

974
01:03:35,008 --> 01:03:37,560
is a differential motion
of the two mirrors

975
01:03:37,560 --> 01:03:40,545
relative to each other due to
the fluctuations in the photon

976
01:03:40,545 --> 01:03:43,740
number in the two arms
of the interferometer.

977
01:03:43,740 --> 01:03:50,530
So therefore, what happens is
you have a delta z deviation

978
01:03:50,530 --> 01:03:55,580
or variance in the measurement
of the mirror, which

979
01:03:55,580 --> 01:03:57,930
comes from radiation pressure.

980
01:03:57,930 --> 01:04:01,970
It's a differential radiation
pressure between the two arms.

981
01:04:01,970 --> 01:04:06,860
And what Caves
showed in this paper

982
01:04:06,860 --> 01:04:10,520
is that the two effects
which contribute

983
01:04:10,520 --> 01:04:12,580
to the precision
of the measurement

984
01:04:12,580 --> 01:04:15,670
come from two different
quadrature component.

985
01:04:15,670 --> 01:04:18,770
For the photon
counting, we always

986
01:04:18,770 --> 01:04:21,900
want to squeeze the
light in such a way

987
01:04:21,900 --> 01:04:25,870
that we have the narrow
part of the ellipse

988
01:04:25,870 --> 01:04:31,880
in the quadrature component
of our coherent beam.

989
01:04:31,880 --> 01:04:33,960
We've discussed
it several times.

990
01:04:33,960 --> 01:04:39,340
So therefore, you want to
squeeze it by e to the minus r.

991
01:04:39,340 --> 01:04:45,190
However, when what has a good
effect for the photon counting

992
01:04:45,190 --> 01:04:47,092
has a bad effect
for the fluctuations

993
01:04:47,092 --> 01:04:48,175
due to radiation pressure.

994
01:04:59,850 --> 01:05:03,350
So therefore, what
happens is-- let's

995
01:05:03,350 --> 01:05:05,240
forget squeezing for a moment.

996
01:05:05,240 --> 01:05:10,440
If you have two contributions,
one goes to the noise,

997
01:05:10,440 --> 01:05:13,890
one goes with alpha squared
of the number of photons.

998
01:05:13,890 --> 01:05:16,820
One goes inverse with the square
root of the number of photons.

999
01:05:16,820 --> 01:05:20,390
You will find out that
even in the interferometer

1000
01:05:20,390 --> 01:05:23,490
without squeezing, there
is an optimum laser

1001
01:05:23,490 --> 01:05:25,630
power, which you want to use.

1002
01:05:25,630 --> 01:05:28,100
Because if you use
two lower power,

1003
01:05:28,100 --> 01:05:29,470
you lose in photon counting.

1004
01:05:29,470 --> 01:05:31,570
If you use two higher
power, you lose

1005
01:05:31,570 --> 01:05:35,600
in the fluctuations of
the radiation pressure.

1006
01:05:35,600 --> 01:05:43,610
So even without squeezing,
there is an optimum laser power.

1007
01:05:43,610 --> 01:05:48,900
And for typical parameters,
so there is an optimum power.

1008
01:05:48,900 --> 01:05:52,020
And what we're
shown in this paper

1009
01:05:52,020 --> 01:05:56,620
is whenever you choose
the optimum power which

1010
01:05:56,620 --> 01:05:59,810
keeps a balance between
photon counting and radiation

1011
01:05:59,810 --> 01:06:02,462
pressure, then you reach
the standard quantum

1012
01:06:02,462 --> 01:06:03,670
limit of your interferometer.

1013
01:06:06,760 --> 01:06:10,240
But it turns out that
for typical parameters,

1014
01:06:10,240 --> 01:06:13,580
this optimum is 8,000 watt.

1015
01:06:13,580 --> 01:06:15,960
So that's why
people at LIGO work

1016
01:06:15,960 --> 01:06:18,840
harder and harder to develop
more and more powerful

1017
01:06:18,840 --> 01:06:21,510
lasers, because more
laser power brings

1018
01:06:21,510 --> 01:06:24,370
them closer and closer
to the optimum power.

1019
01:06:24,370 --> 01:06:27,360
But once they had
the optimum power,

1020
01:06:27,360 --> 01:06:29,730
additional squeezing
will not help them

1021
01:06:29,730 --> 01:06:33,430
because they are already at
the fundamental quantum limit.

1022
01:06:33,430 --> 01:06:38,090
So the one thing which
squeezing does for you, it

1023
01:06:38,090 --> 01:06:42,420
changes the optimum
power in your input beam

1024
01:06:42,420 --> 01:06:44,810
by a squeezing factor.

1025
01:06:44,810 --> 01:06:47,320
So therefore, if you
have lasers which

1026
01:06:47,320 --> 01:06:51,220
have maybe 100 watt
and not 8,000 watt,

1027
01:06:51,220 --> 01:06:55,792
then squeezing helps you to
reach the fundamental quantum

1028
01:06:55,792 --> 01:06:57,000
limit of your interferometer.

1029
01:07:08,130 --> 01:07:10,680
So that's pretty
much all I wanted

1030
01:07:10,680 --> 01:07:13,320
to say about precision
measurements.

1031
01:07:13,320 --> 01:07:16,520
I hope the last example--
it's too complex to go

1032
01:07:16,520 --> 01:07:19,170
through the whole analysis--
but it gives you at least a

1033
01:07:19,170 --> 01:07:22,740
feel that you have
to keep your eye

1034
01:07:22,740 --> 01:07:24,310
on both quadrature components.

1035
01:07:24,310 --> 01:07:27,200
You can squeeze, you
can get an improvement,

1036
01:07:27,200 --> 01:07:30,440
in one physical
effect, but you have

1037
01:07:30,440 --> 01:07:34,020
to be careful to consider what
happens in the other quadrature

1038
01:07:34,020 --> 01:07:34,724
component.

1039
01:07:34,724 --> 01:07:37,140
And in the end, you have to
keep the two of them balanced.

1040
01:07:42,430 --> 01:07:43,210
Any questions?

1041
01:07:50,970 --> 01:07:51,470
Oops.

1042
01:08:07,600 --> 01:08:09,100
OK.

1043
01:08:09,100 --> 01:08:17,520
Well, we can get started with
a very short chapter, which

1044
01:08:17,520 --> 01:08:27,180
is about g2.

1045
01:08:27,180 --> 01:08:31,410
The g2 measurement
for light and atoms.

1046
01:08:44,260 --> 01:08:51,750
I don't think you will
find the discussion I

1047
01:08:51,750 --> 01:08:53,910
want to present to
you in any textbook.

1048
01:08:53,910 --> 01:09:00,729
It is about whether g2 is 1 or
2, whether we have fluctuations

1049
01:09:00,729 --> 01:09:01,899
or not.

1050
01:09:01,899 --> 01:09:05,819
And the discussion will be
whether g2 of 2 and g2 of 1

1051
01:09:05,819 --> 01:09:08,899
are quantum effect
or classical effects.

1052
01:09:08,899 --> 01:09:16,810
So I want to give you
here in this discussion

1053
01:09:16,810 --> 01:09:21,920
four different derivations of
whether g2 is 1 or g2 is 2.

1054
01:09:21,920 --> 01:09:23,649
And they look very,
very different.

1055
01:09:23,649 --> 01:09:25,370
Some are based on
classical physics.

1056
01:09:25,370 --> 01:09:28,420
Some are based on the
concept of interference.

1057
01:09:28,420 --> 01:09:31,270
And some are based on the
quantum indistinguishability

1058
01:09:31,270 --> 01:09:33,270
of particles.

1059
01:09:33,270 --> 01:09:36,869
And once you see you all those
four different explanations,

1060
01:09:36,869 --> 01:09:38,410
I think you'll see
the whole picture.

1061
01:09:38,410 --> 01:09:41,680
And I hope you
understand something.

1062
01:09:41,680 --> 01:09:45,430
So again, it's a long
story about factors of 2.

1063
01:09:45,430 --> 01:09:49,020
But there are some factors of 2
which are purely calculational,

1064
01:09:49,020 --> 01:09:51,060
and there are other
factors of 2 which

1065
01:09:51,060 --> 01:09:53,779
involve a hell of
a lot of physics.

1066
01:09:53,779 --> 01:09:56,740
I mean, there are people
who say the g2 factor of 2

1067
01:09:56,740 --> 01:10:00,210
is really the difference between
ordinary light and laser light.

1068
01:10:00,210 --> 01:10:03,390
For light from a
light bulb, g2 is 2.

1069
01:10:03,390 --> 01:10:05,550
For light from a laser, g2 is 1.

1070
01:10:05,550 --> 01:10:07,590
And this is the only
fundamental difference

1071
01:10:07,590 --> 01:10:09,410
between laser light
and ordinary light.

1072
01:10:09,410 --> 01:10:12,460
So this factor of
2 is important.

1073
01:10:12,460 --> 01:10:19,890
And I want to therefore have
this additional discussion

1074
01:10:19,890 --> 01:10:20,745
of the g2 function.

1075
01:10:23,380 --> 01:10:32,390
So let me remind
you that g2 of 0

1076
01:10:32,390 --> 01:10:45,900
is the normalized probability
to detect two photons or two

1077
01:10:45,900 --> 01:10:52,770
particles simultaneously.

1078
01:11:01,300 --> 01:11:16,140
And so far, we have
discussed it for light.

1079
01:11:16,140 --> 01:11:21,205
And the result we obtained by
using our quantum formulation

1080
01:11:21,205 --> 01:11:26,400
of light with creation
annihilation operator,

1081
01:11:26,400 --> 01:11:31,480
we found that g2 is 0.

1082
01:11:31,480 --> 01:11:36,810
In the situation that we had
black body radiation, which

1083
01:11:36,810 --> 01:11:41,950
we can call thermal
light, it's sometimes

1084
01:11:41,950 --> 01:11:45,020
goes by the name chaotic light.

1085
01:11:45,020 --> 01:11:47,190
Sometimes it's called
classical light,

1086
01:11:47,190 --> 01:11:50,350
but that may be a misnomer,
because I regard the laser

1087
01:11:50,350 --> 01:11:52,930
beam as a very
classical form of light.

1088
01:11:56,580 --> 01:11:59,340
This is sometimes
called bunching

1089
01:11:59,340 --> 01:12:01,270
because 2 is larger than 1.

1090
01:12:01,270 --> 01:12:03,780
So pairs of photons
appear bunched up.

1091
01:12:03,780 --> 01:12:07,310
You have a higher probability
than you would naively

1092
01:12:07,310 --> 01:12:12,270
expect of detecting two
photons simultaneously.

1093
01:12:12,270 --> 01:12:19,670
And then we had the situation of
laser light and coherent light

1094
01:12:19,670 --> 01:12:24,240
where the g2 function was 1.

1095
01:12:24,240 --> 01:12:28,660
And I want to shed some
light on those two cases.

1096
01:12:28,660 --> 01:12:35,640
We have discussed the extreme
case of a single photon

1097
01:12:35,640 --> 01:12:38,430
where the probability
of detecting two photons

1098
01:12:38,430 --> 01:12:40,040
is 0 for trivial reasons.

1099
01:12:40,040 --> 01:12:42,662
So you have a g2 function of 0.

1100
01:12:45,760 --> 01:12:47,930
But this is not what I
want to discuss here.

1101
01:12:47,930 --> 01:12:51,570
I want to shed some
light on when do we

1102
01:12:51,570 --> 01:12:53,720
get a g2 function of 1.

1103
01:12:53,720 --> 01:12:56,800
When do we get a
g2 function of 2.

1104
01:12:56,800 --> 01:13:04,010
And one question
we want to address,

1105
01:13:04,010 --> 01:13:09,560
when we have a g2
function of 2, is

1106
01:13:09,560 --> 01:13:19,790
this a classical
or quantum effect?

1107
01:13:28,320 --> 01:13:31,010
Do you have an opinion?

1108
01:13:31,010 --> 01:13:32,133
Who thinks-- question.

1109
01:13:32,133 --> 01:13:34,133
AUDIENCE: When we did the
homework problem where

1110
01:13:34,133 --> 01:13:35,930
we had the linear's
position, like alpha

1111
01:13:35,930 --> 01:13:38,870
minus alpha
[INAUDIBLE] plus alpha,

1112
01:13:38,870 --> 01:13:41,810
we found that one can
have a g2 greater than 1,

1113
01:13:41,810 --> 01:13:43,770
and one can have
a g2 less than 1.

1114
01:13:43,770 --> 01:13:46,710
So maybe g2 isn't a
great discriminator,

1115
01:13:46,710 --> 01:13:49,160
whether it's very
quantum or very classic.

1116
01:13:53,017 --> 01:13:54,725
PROFESSOR: Let's hold
this thought, yeah.

1117
01:13:54,725 --> 01:13:55,830
You may be right.

1118
01:14:00,920 --> 01:14:03,430
Let's come back to that.

1119
01:14:03,430 --> 01:14:06,580
I think that's one opinion.

1120
01:14:06,580 --> 01:14:09,700
The g2 function may
not be a discriminator,

1121
01:14:09,700 --> 01:14:14,070
because we can have g2 of 1
and g2 of 2 purely classical.

1122
01:14:14,070 --> 01:14:18,350
But why classical light
behaves classically,

1123
01:14:18,350 --> 01:14:21,200
maybe that's what we
can understand there.

1124
01:14:21,200 --> 01:14:22,670
And maybe what I
want to tell you

1125
01:14:22,670 --> 01:14:27,490
is that a lot of classical
properties of classical light

1126
01:14:27,490 --> 01:14:31,110
can be traced down to the
indistinguishability of bosons,

1127
01:14:31,110 --> 01:14:33,020
which are photons.

1128
01:14:33,020 --> 01:14:35,610
So in other words, we
shouldn't be surprised

1129
01:14:35,610 --> 01:14:38,340
that something which
seems purely classical

1130
01:14:38,340 --> 01:14:41,090
is deeply rooted
in quantum physics.

1131
01:14:41,090 --> 01:14:44,290
But I'm ahead of my agenda.

1132
01:14:44,290 --> 01:14:46,390
So let me start now.

1133
01:14:46,390 --> 01:14:48,510
I want to offer you
four different views.

1134
01:14:59,440 --> 01:15:05,034
And the first one is that
we have random intensity

1135
01:15:05,034 --> 01:15:05,575
fluctuations.

1136
01:15:08,230 --> 01:15:09,960
Think of a classical
light source.

1137
01:15:15,790 --> 01:15:20,450
And we assume that if
things are really random,

1138
01:15:20,450 --> 01:15:22,590
they are described by a
Gaussian distribution.

1139
01:15:25,650 --> 01:15:28,930
And if you switch on
a light bulb, what

1140
01:15:28,930 --> 01:15:31,920
you get if you measure the
intensity when you measure what

1141
01:15:31,920 --> 01:15:36,290
is the probability that the
momentary intensity is I,

1142
01:15:36,290 --> 01:15:40,240
well, you have to normalize
it to the average intensity.

1143
01:15:40,240 --> 01:15:47,890
But what you get is pretty much
an exponential distribution.

1144
01:15:47,890 --> 01:15:49,610
And this exponential
distribution

1145
01:15:49,610 --> 01:15:51,980
has a maximum at I equals 0.

1146
01:15:51,980 --> 01:15:55,790
So the most probable
intensity of all intensities

1147
01:15:55,790 --> 01:15:57,420
when you switch
on a light bulb is

1148
01:15:57,420 --> 01:16:00,210
that you have 0 intensity
at a given moment.

1149
01:16:00,210 --> 01:16:06,140
But the average
intensity is I average.

1150
01:16:06,140 --> 01:16:12,550
So you can easily, for
such a distribution,

1151
01:16:12,550 --> 01:16:14,350
for such an exponential
distribution,

1152
01:16:14,350 --> 01:16:18,710
figure out what is the
average of I to the power n.

1153
01:16:18,710 --> 01:16:22,000
It is related to I
average to the power n,

1154
01:16:22,000 --> 01:16:25,060
but it has an n factorial.

1155
01:16:25,060 --> 01:16:28,730
And what is important is the
case for our discussion of n

1156
01:16:28,730 --> 01:16:32,770
equals 2, where the
square of the intensity

1157
01:16:32,770 --> 01:16:36,850
averaged is two times the
average intensity squared.

1158
01:16:39,700 --> 01:16:44,120
And classically, g2
is the probability

1159
01:16:44,120 --> 01:16:46,900
of detecting two
photons simultaneously,

1160
01:16:46,900 --> 01:16:49,990
which is proportional
to I square.

1161
01:16:49,990 --> 01:16:51,230
We have to normalize it.

1162
01:16:51,230 --> 01:16:54,220
And we normalize it
by I average square.

1163
01:16:54,220 --> 01:16:56,940
And this gives 2.

1164
01:16:56,940 --> 01:17:01,910
So simply light with
Gaussian fluctuations

1165
01:17:01,910 --> 01:17:06,630
would give rise to
a g2 function of 2.

1166
01:17:06,630 --> 01:17:10,800
Since it's random fluctuations,
it's also called chaotic light.

1167
01:17:10,800 --> 01:17:13,780
And the physical pictures
is the following.

1168
01:17:13,780 --> 01:17:19,350
If you detect a photon,
the light is fluctuating.

1169
01:17:19,350 --> 01:17:22,440
But whenever you detect
the first photon,

1170
01:17:22,440 --> 01:17:25,440
it is more probable that you
detect take the first photon

1171
01:17:25,440 --> 01:17:28,490
when the intensity
happened to be high.

1172
01:17:28,490 --> 01:17:30,760
And then since the
intensity is high,

1173
01:17:30,760 --> 01:17:32,720
the probability
for the next photon

1174
01:17:32,720 --> 01:17:34,390
is higher than the
average probability.

1175
01:17:38,240 --> 01:17:43,340
So therefore, you get
necessarily a g2 function of 2.

1176
01:17:43,340 --> 01:17:44,560
So this is the physics of it.

1177
01:17:47,140 --> 01:17:50,040
So let me just write that down.

1178
01:17:50,040 --> 01:18:07,800
The first photon is more
likely to be detected

1179
01:18:07,800 --> 01:18:19,400
when the intensity
fluctuation-- when

1180
01:18:19,400 --> 01:18:21,720
intensity fluctuations
give high intensity.

1181
01:18:26,820 --> 01:18:29,330
And then we get this result.

1182
01:18:40,950 --> 01:18:42,780
Yes.

1183
01:18:42,780 --> 01:18:48,350
Let us discuss a second
classical view, which

1184
01:18:48,350 --> 01:18:50,690
I can call wave interference.

1185
01:18:58,500 --> 01:18:59,610
This is really important.

1186
01:18:59,610 --> 01:19:01,890
A lot of people get
confused about it.

1187
01:19:01,890 --> 01:19:07,120
If you have light
in only one mode,

1188
01:19:07,120 --> 01:19:10,550
this would be the
laser of, for atoms,

1189
01:19:10,550 --> 01:19:11,935
the Bose-Einstein condensate.

1190
01:19:15,430 --> 01:19:19,310
One mode means a single wave.

1191
01:19:19,310 --> 01:19:24,470
So if we have plain waves, we
can describe all the photons

1192
01:19:24,470 --> 01:19:26,350
or all the atoms by
this wave function.

1193
01:19:29,330 --> 01:19:31,910
So what is the g2 function
for an object like this?

1194
01:19:35,350 --> 01:19:35,850
AUDIENCE: 1.

1195
01:19:35,850 --> 01:19:39,920
PROFESSOR: Trivially
1, because if something

1196
01:19:39,920 --> 01:19:48,835
is a clean wave, a single
wave, all correlation functions

1197
01:19:48,835 --> 01:19:49,335
factorize.

1198
01:19:54,600 --> 01:20:00,140
You have the situation that
I to the power n average

1199
01:20:00,140 --> 01:20:03,230
is I average to the power n.

1200
01:20:03,230 --> 01:20:08,570
And that means that
gn is 1 for all n.

1201
01:20:13,670 --> 01:20:19,850
OK, but let's now
assume that we have two.

1202
01:20:19,850 --> 01:20:23,000
We can also use more, but
I want to restrict to two.

1203
01:20:23,000 --> 01:20:25,840
That we have only two modes.

1204
01:20:25,840 --> 01:20:29,170
And two modes can interfere.

1205
01:20:29,170 --> 01:20:35,550
So let me apply to those
two modes a simple model.

1206
01:20:35,550 --> 01:20:39,650
And whether it's simple
or not is relative.

1207
01:20:39,650 --> 01:20:42,070
So it goes like as follows.

1208
01:20:42,070 --> 01:20:49,990
If you have two modes,
both of Unity intensity,

1209
01:20:49,990 --> 01:20:53,430
then the average intensity is 2.

1210
01:20:57,920 --> 01:21:09,430
But if you have interference,
then the normalized intensity

1211
01:21:09,430 --> 01:21:13,450
will vary between 0 and 4.

1212
01:21:13,450 --> 01:21:16,040
Constructive interference
means you get twice as much

1213
01:21:16,040 --> 01:21:16,650
as average.

1214
01:21:16,650 --> 01:21:19,640
Destructive means
you get nothing.

1215
01:21:19,640 --> 01:21:22,380
So therefore, the
[? nt ?] squared

1216
01:21:22,380 --> 01:21:30,300
will vary between 0 and 16.

1217
01:21:30,300 --> 01:21:33,940
So if I just use the two
extremes, it works out well.

1218
01:21:33,940 --> 01:21:36,830
You have an I
square, which is 8,

1219
01:21:36,830 --> 01:21:39,110
the average of
constructive interference

1220
01:21:39,110 --> 01:21:41,370
and destructive interference.

1221
01:21:41,370 --> 01:21:44,900
And this is two times
the average intensity,

1222
01:21:44,900 --> 01:21:48,220
which is 2 squared.

1223
01:21:48,220 --> 01:21:52,400
So therefore, if you
simply allow fluctuations

1224
01:21:52,400 --> 01:21:54,860
due to the interference
of two modes,

1225
01:21:54,860 --> 01:21:57,960
we find that the
g2 function is 2.

1226
01:22:00,550 --> 01:22:13,280
So this demonstrates
that g2 of 2

1227
01:22:13,280 --> 01:22:18,875
has its deep origin
in wave interference.

1228
01:22:27,890 --> 01:22:34,230
And indeed, if you take a
light bulb which emits photons,

1229
01:22:34,230 --> 01:22:37,760
you have many, many atoms
in your tungsten filament

1230
01:22:37,760 --> 01:22:40,620
which can emit, re-emit waves.

1231
01:22:40,620 --> 01:22:42,760
And since they have
different positions,

1232
01:22:42,760 --> 01:22:46,790
the waves arrive at your
detector with random phases.

1233
01:22:46,790 --> 01:22:49,290
And if you really write
that down in a model--

1234
01:22:49,290 --> 01:22:51,540
this is nicely done in
the book by Loudon--

1235
01:22:51,540 --> 01:22:54,390
you realize that
random interference

1236
01:22:54,390 --> 01:22:58,590
between waves results in
an exponential distribution

1237
01:22:58,590 --> 01:23:00,600
of intensity.

1238
01:23:00,600 --> 01:23:02,700
So most people wouldn't
make the connection.

1239
01:23:02,700 --> 01:23:06,050
But there is a deep and
fundamental relationship

1240
01:23:06,050 --> 01:23:11,270
between random interference and
the most random distribution,

1241
01:23:11,270 --> 01:23:13,780
the exponential
distribution, intensity

1242
01:23:13,780 --> 01:23:18,700
which characterises thermal
light or chaotic light.

1243
01:23:22,330 --> 01:23:25,380
So let me just write that down.

1244
01:23:25,380 --> 01:23:34,680
So the Gaussian intensity
distribution-- actually

1245
01:23:34,680 --> 01:23:39,350
it's an exponential
intensity distribution.

1246
01:23:39,350 --> 01:23:41,290
But if you write the
intensity distribution

1247
01:23:41,290 --> 01:23:43,280
as a distribution in
the electric field,

1248
01:23:43,280 --> 01:23:44,910
intensity becomes e square.

1249
01:23:44,910 --> 01:23:46,850
Then it becomes a
Gaussian [INAUDIBLE].

1250
01:23:46,850 --> 01:23:50,650
So Gaussian or exponential
intensity distribution

1251
01:23:50,650 --> 01:23:57,130
in view number one is the
result, is indeed the result,

1252
01:23:57,130 --> 01:23:59,055
of interference.

1253
01:24:02,890 --> 01:24:05,110
Any questions?

1254
01:24:05,110 --> 01:24:06,656
So these are the
two classical views.

1255
01:24:06,656 --> 01:24:07,780
I think we should stop now.

1256
01:24:07,780 --> 01:24:13,590
But on Wednesday, I will present
you alternative derivations,

1257
01:24:13,590 --> 01:24:18,690
which are completely
focused on quantum operators

1258
01:24:18,690 --> 01:24:21,960
and quantum counting statistics.

1259
01:24:21,960 --> 01:24:25,430
Just a reminder, we have
class today and Friday.

1260
01:24:25,430 --> 01:24:28,390
And we have class
this week on Friday.