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M1L5v.txt
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M1L5v.txt
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#
# File: content-mit-8422-1x-captions/M1L5v.txt
#
# Captions for 8.422x module
#
# This file has 181 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
OK.
So that's almost trivial, but we need this definition.
We need this knowledge to take it to the next step.
Remember, we want to do something more interesting
than rotating two-level systems.
We want to get two qubits, combine them, entangle them
and all that.
So what we need for that is we have
to go-- we have to allow-- we have
to expand the Hilbert space.
I introduced the Mach-Zehnder interferometer as the device
to do it.
But now we need a way how a second photon
from another qubit can interact with the photon
of our first qubit.
And we want to do that by introducing
first the nonlinear Mach-Zehnder interferometer.
I want to throw in a nonlinear element.
And then we are ready to allow the second qubit,
through the nonlinear element, to manipulate the first qubit.
And then we have interactions between qubits.
We have [? two-qubit ?] gates.
Let me maybe just deviate from my notes.
I think I really want to show you what I'm doing
and that you have it clearly in mind where I'm aiming at.
So what we want to accomplish is,
instead of having a phase shift with an externally controlled
phase, I want to put in some nonlinear crystal which
has an index of refraction which has a phase shift.
But now the phase with the nonlinear crystal
has the property that its index of refraction
changes when I take another laser beam and send it through.
So the laser beam through-- because the index of refraction
is intensity-dependent, the green laser beam
changes the index of refraction of the nonlinear crystal,
also called Kerr medium.
And that means now, because the blue beam
goes through the same crystal, that the green beam controls
now the phase shift of the blue beam.
And now we have interacting photons.
One photon interacts with the second photon.
OK.
Let's work it out.
So our goal is to develop a description
for nonlinear Mach-Zehnder interferometer.
What we need is a nonlinear medium.
So let me just introduce, for two or three minutes, nonlinear
medium, how we describe it, and then we
put it into our interferometer.
So in linear optics, just to remind you,
we have a polarization, or an electric dipole moment,
which is proportional to the electric field of light.
And xi is the polarizability.
In the most complicated case, it can be polarizability tensor.
Now, this is linear.
If we want to go nonlinear, then we have the linear relationship
simply, as you can see, the first term in a Taylor
expansion.
So we expand the response of the medium in powers
of the electric field.
And we have the susceptibilities chi 1, chi 2, chi 3, and so on.
We have already encountered a nonlinear element which
had a chi 2 susceptibility.
And these were the crystals we put into our OPOs,
into our Optical Parametric Oscillators.
Remember, the optical parametric oscillator involved, how many
modes?
Three.
We had one photon, which was broken down into two photons.
But that also means that the reverse process
is you drive the beam-- you drive the OPO with omega.
But the E squared term creates an electric polarization
as 2 omega.
And if you have an oscillating polarization as 2 omega,
you create [INAUDIBLE] omega.
So I just want to remind you that the chi squared
term is what we have already encountered.
That's not what we need for the nonlinear Mach-Zehnder
interferometer.
What we need now is a chi 3 interaction,
which is the Kerr medium, because we want to realize
the following Hamiltonian.
So the Hamiltonian, the Kerr medium,
is also called cross, that's the x, cross-phase modulation.
One beam can affect the phase of another beam,
and this is called cross-phase modulation.
There's also self-phase modulation,
that one laser beam changes the index of refraction for itself.
But here we want the cross-phase modulation.
So we know that if we have a mode and we shift its phase,
we're not [? exchanging ?] any photons.
So if you're in an eigenstate state of mode a with one or two
or three photons and we simply shift its phase,
we are diagonal in a dagger a, and the phase shift
is proportional to pre-factor here.
So this Hamiltonian is simply a phase shift
of photons in mode a.
But if we now multiply that with b dagger b,
the phase shift is now proportional to the number
of photons in mode b.
So now we have a situation that the phase shift from mode a
is proportional to the intensity in b and b and vice versa.
So I think it's self-evident that this Hamiltonian is
the quantum description of the process we
want to introduce now.
So this is the cross-phase modulation Hamiltonian.
Isn't it much more elegant to describe nonlinear optics
with Hamiltonians than with parameterized susceptibilities
and all that?
I mean, it's just amazing.
You write in a few letters, and it
has all the power of the classical description
plus extra because it includes quantum
fluctuations and everything.
I mean, I hope you appreciate the power of the language
we are developing.
So this is the cross-phase modulation Hamiltonian.
And now we want to use this Hamiltonian
in a crystal of length L. Well, you
know if you have a time propagation operator,
you put the Hamiltonian in the exponent and multiply with T.
But for propagating laser beam, T and L are related.
So this is now the unitary transformation
if you propagate through that crystal.
So we are now defining that by saying we have this nonlinear
susceptibility.
Things will be proportional to the length
L. And then we have the operator a dagger a, b dagger b.
And let me choose for the following discussion
that the length is chosen such that when multiplied
with the nonlinear susceptibility,
we get a phase shift of pi.
OK.
So now we are ready to describe what
happens when we have a Kerr medium
and we have two input modes.
So let's write down the table of possible combinations
and operations.
Well, when we act on the vacuum, we get nothing.
What happens when we have one photon in one mode?
What happens to that photon?
Well, the extra phase shift coming
from cross-phase modulation is 0 because we have
no photon in one of the modes.
And a dagger a or b dagger b acting on this mode gives 0.
So therefore, as long as we have only one photon,
we have no phase shift.
The exponential factor is 1, and we simply
reproduce the original state.
So the only non-trivial situation, by construction,
is when we have one photon in each mode.
And then we get the matrix element e to the ikL.
And we adjusted the lengths of the crystal
that this is just minus 1.
So in other words, in this Hilbert space,
we get a phase shift of pi.
We get a minus sign when we have one photon in each mode.
That's all what this medium does.
And if you ever thought about it,
how to deal with the situation that when
I said there is one photon which provides a phase
shift with the other photon, but the other photon
is also providing a phase shift to the first photon.
So you may wonder when you have a combined state of the two,
how do you deal that one photon shifts the other photon
and vice versa?
But when you apply the operator, you don't even
have to think about it.
You see that the operator for the combined Hilbert
space of the two photon gives you a phase shift of pi.
It's not pi for one photon times pi for the other one.
It's pi for the whole quantum state.
Questions?
OK.
We have a Kerr medium, which means