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M1L1b.txt
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M1L1b.txt
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#
# File: content-mit-8-421-1x-subtitles/M1L1b.txt
#
# Captions for 8.421x module
#
# This file has 103 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
Suppose for the laser-- I first explain the laser to you,
and then we go back to the pure quantum system, which has
actually certain subtleties.
But for the laser, it is obvious--
at least if I tell you how I want to measure it--
because I can take the laser and create a beat note
with another very stable laser.
And there we caught the speed note on a photodiode.
I can realize by making this other laser,
the local oscillator, stronger and stronger,
I can create a beat note, which is larger and larger,
and corresponds to a macroscopic electric current, which
can be measured with very high precision.
So you can realize an arbitrarily high
signal-to-noise ratio by using a strong local oscillator.
And then you can actually say the photocurrent, which
comes out of the photodiode is actually-- you can regard it
with macroscopic current as purely classical,
and then of course, the answer to the first question applies.
So that takes care of question 3 by mapping it actually
on question 2.
But now by saying that the laser also
has a quantum mechanical limit and I'm not changing anything,
we realize that probably the answer to question 2
should also be yes.
So let's therefore ask ourself, what
is the situation when the Heisenberg uncertainty
relation applies?
One is we have to be really careful.
It predicts the outcome of a single measurement
on a single quantum system.
Or if the Heisenberg uncertainty relation
sets a limit how well we prepare a quantum system,
it's about a single quantum system,
and then we perform a single measurement.
So in a sense, if we would say all
you have is a single photon, which is a very special quantum
system.
You have a single photon, and you measure the frequency
of the photon only once.
Then you will find the limit, which is the Heisenberg limit.
You cannot with a single measurement on a single photon
determine the accuracy of the frequency better than this.
And of course, you can get higher accuracy
by doing repeated measurements, or by using many photons.
We talk about it more in 8.422.
But I just want to remind you if you
have n uncorrelated photons-- in other words we perform
n measurements on n different objects-- then
the signal-to-noise ratio is, just
by Poisson distribution, square root n.
And therefore the resolution for the frequency of the photons
is better than the Heisenberg or the Fourier limit
by 1 over the square root n.
Some of you-- and actually in Professor Vuletic's group,
there is research on it-- that if you have correlated-- well
in his case, correlated atoms-- but if you
had correlated photons, then you can even do better.
You can reach what is sometimes called the Heisenberg
limit, where you are better than the limitation
given by Fourier's theorem or by the Heisenberg uncertainty
relation by a factor of 1/n.
OK.
So as far as the question 2, where the quantum harmonic
oscillator is concerned, we would
say the answer is yes, if you have
a single photon at frequency omega 0,
which interacts with the quantum harmonic oscillator
at frequency omega 0.
However, the answer would even in that case
be no, if you have harmonic oscillator levels,
and you take a photon, and by a non-linear process,
excite the n-th level.
So you have a single photon now.
You can resolve the energy, delta E, of this level.
A single photon, a single quantum object.
You can define the energy.
That's Heisenberg's uncertainty relation.
The energy is determined to that precision.
But your frequency, omega, of the optical pulse
is n times omega 0, using non-linear process.
And then you can determine the frequency of the harmonic
oscillator, even for a single quantum
system and a single photon with a precision
which is 1/n times better.
So you have to be also careful-- but I
don't want to beat it to death now--
to distinguish between the accuracy
at which Heisenberg's uncertainty relation maybe
limits the measurement of an energy level,
and how this is related to the frequency
of the harmonic oscillator.
And by going immediately to the n-th level,
you can, of course, measure the distance between two levels
more accurately, because you have increased your lever arm.