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M5L22c.txt
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M5L22c.txt
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#
# File: content-mit-8-421-5x-subtitles/M5L22c.txt
#
# Captions for 8.421x module
#
# This file has 123 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
So how well can we measure the phase phi?
You should first assume the phase phi is perfectly
determined with extremely high accuracy
if you use a laser beam which has
a macroscopic electric field.
The phase phi is a classical variable,
and can be determined with arbitrary precision.
And therefore, the phase phi, which we have imprinted first
into the atomic wave function and then in the photon field,
is an exact number.
It comes from the laser beam.
Well, what I'm showing here is phase space plots
for the photon field.
I know we talk about photons two-dimensional phase space
distribution mainly in 8.422, but I
think the pictures speak for themselves.
So what you do here is you plot-- well, a lot of you
have seen the harmonic oscillator, and harmonic
oscillator-- if you start to prepare the system here,
this is position, this is momentum,
the system evolves in a circle-- a lot of you
have seen if we regard photon states as states of a harmonic
oscillator, which they are, that you have Fock states which
are just circles, or the vacuum state,
or the vacuum state is just a tiny circle at the center.
And if you have a coherent state,
the coherent state is maybe a little blob out there.
And for the coherent state, you can determine the phase
because the angle of this little block relative to the origin
is well-determined.
I think you all have seen a version of that.
So anyway, what is done here is I show you this phase space
plot for the photon field.
And what happens is if, initially, the excited
state was 0-- this is sort of just the ground
state of the harmonic oscillator-- it's a circle.
If the excited state was occupied
with unity probability, it's a Fock state with n equals 1.
And here you see the phase space plot of a Fock state with n
equals 1.
And of course, you realize if you've
exactly one photon or one atom in an excited state,
there is no phase information left,
because the phase is actually the relative phase
in the superposition between ground and excited state.
If you have an excited state, only
an excited state with a phase factor,
you know if a phase factor can simply
be factored out of the total wave function,
it's never measurable.
What is measurable are phases which
are relative phases between two amplitudes which are populated.
And of course, not surprisingly, if you now
vary the excited state fraction of the atom,
or that's a probability to have a photon in the photon field,
from 0 to 1, in between we sort of see that
this phase space distribution, it
points along the 45-degree axis.
And we can measure the phase.
And the most accurate phase measurement
can be done if the superposition between ground and excited
state, or it's 50-50, or talking about the photon field,
we've a 50-50 superposition state
between no photon and one photon.
But the phase here is undetermined,
and the phase here has quite a bit of, well, variance
because, well, if you've a single photon,
there's only so much accuracy for the phase.
I mean, sometimes-- it would require more discussion,
but sometimes you talk even about an uncertainty relation,
delta n delta phi equals 1.
So if you have one photon, you can only
measure the phase with the precision
on the order of unity.
If you have millions of photons, then you
can do a very accurate phase measurement.
So what we have is-- and let me just summarize a conclusion--
the phase phi is best defined in the atom--
And therefore also in the photon field-- when
we have an equal superposition of spin up and spin down,
of ground and excited state.
And you sort of also get that from the block vector picture.
If you have a block vector which is pointing like that,
it doesn't have a phase.
It's just pointing up.
If it's pointing down, there is no phase.
But if it's a 50% superposition state,
it points in the xy plane, and you have the best definition
of the relative phase of the amplitude between ground
and excited state.
So I mentioned this Heisenberg uncertainty relation.
So the fact is, just looking at these phase space plots,
you realize the angle at which-- the angle which
we can determine here for the photon distribution
will have quite a variance.
But now I want to discuss with you
how would we actually go about it,
how would we measure the phase of the photon field.
And this requires a homodyne experiment, a beat experiment
where we interfere the emitted photon with a local oscillator,
which is the laser beam which was used in first place
to excite the atom.
And what we will find out, and it's clear that we cannot
obtain a sharp value of the phase,
but these fluctuations in the phase do not come from any
partial trace, do not come from any fluctuations
in the Hamiltonian.
Just do a trace [INAUDIBLE] when we write down, determine
the Hamiltonian, the e.d term.
Yes, depending on the basis set, depending
how you define spin up, spin down,
and what phase factors you put into your basis set,
you may have a phase popping up in the Hamiltonian.
But this phase is purely definitional.
The phase I'm talking about is really
a relative phase between two amplitudes,
and it is independent of a phase which
may be your choice by choosing the basis set in which you
formulate the Hamiltonian.