M1L3g.txt

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The Hamiltonian is the spin coupled to a magnetic field.
And if you express that by angular momentum operators,
the gyromagnetic ratio, it involves the operator
for the spin in the z direction.
So, the two-level system has a splitting of h bar omega,
so it's plus 1/2 minus 1/2 h bar omega,
and that means the diagonal part, the non-driven part,
is simply given by the Pauli spin matrix sigma z
and omega 0 is the energy splitting.
Which is proportional to the applied magnetic field.
And up-down, or excited and ground
are the eigenstates of this Hamiltonian with energies plus
minus H bar omega 0 over 2.
OK, so this is the Zeeman Hamiltonian,
but now we add a real rotating B field, B1.
So the drive Hamiltonian, H1, is the same magnetic moment,
but now coupled to a time-dependent rotating field.
The amplitude of the rotating field
is the Rabi frequency divided by gamma.
And we assume that the field is rotating in the xy plane.
So, it's ex ey cosine omega t sine omega t.
And now, I put in two minus signs here for convenience.
If you want, I just shifted-- it's just a definition,
I have change the definition of the amplitude by a minus sign.
So this is the Rabi frequency and the magnetic moment
divided by gamma is nothing than the spin.
Magnetic moment is gamma times this spin.
So therefore, we are back to the spin operators,
and the spin operators, if I factor out 1/2 h bar,
are now the Pauli spin matrices sigma x and sigma y.

And if you look at those spin matrices,
then you realize that we go complex in our Hamiltonian.
Not because we have approximated a real field cosine omega
t by some e to the i omega t, but because when
you have rotating field and we write down it in Pauli spin
matrices, we get imaginary units from the sigma y spin matrix.
So, that means we have now, for this system,
rewritten the coupling term due to the rotating field H1 as 0 0
e to the plus i omega t e to the minus i omega t.
And therefore.
the Hamiltonian is the famous two-level Hamiltonian
with omega 0 in the Rabi frequency,
which I wrote down at the beginning.
So, you need that here, but we will use it even more in 8.421.
This is the famous dressed atom Hamiltonian.
It is the starting point, you calculate
eigenstates and eigenvalues, not just in perturbation theory.
You can go to oscillating fields at arbitrary strengths,
so you can solve exactly in the dressed atom
picture using this Hamiltonian.
The problem of a two-level system
plus one mode of the electromagnetic field,
no matter what the drive term, what
the strengths of the electromagnetic field
in this one mode is.
So, this describes the two-level system,
plus one mode of the electromagnetic field,
with arbitrary strengths.

And as I said, we talk about some things
here, but others I explored in 8.422.
Questions ?
Yes, Will.
We can refer to the eigenstates and eigenenergies
of this Hamiltonian as dressed states the same way
as we refer to dressed states in a fully quantized picture?
Do we refer both cases as dressed states?

Yes.
OK.
It's a good question.
The question is now, what are the dressed states, and Will,
I think you're referring that there
are two ways to talk about the coupling of a two-level system
to one mode of the electromagnetic field.
It is this semi-classical picture
where we introduce-- and let me say an analog amplitude
of the electromagnetic field, which drives it.
And then, there is a fully quantized picture,
where you first quantize the electromagnetic field,
and you couple to photon number states.
It's actually the beauty of it, that the two solutions
are exactly the same.
So, in other words, if you couple an atom
to one mode of the electromagnetic field,
we have two ways how we can solve it.
One is we introduce a coherent electromagnetic field,
and there is an exact unitary transformation
which tells us if you have the quantized field
in a coherent state.
We can do a unitary transformation, and what we get
is exactly this Hamiltonian.
So therefore, this is also-- you may not recognize it--
this is actually the quantum description
of the electromagnetic field when it's in a coherent state.
The other option is we use the dressed atom picture maybe
following some work of Claude Cohen-Tannoudji and others.
Where we assume this single mode of the electromagnetic field
has N photons, and then we solve it
for this photon number state.
So, in other words, these are the two ways
how we can relatively easily treat the problem.
Either we assume the quantum field is in a coherent state
or it's in a Fock state.
But, since the dressed atom picture in the standard way
assumes that the photon number, N, is large,
there is a correspondence that in the limit of N
being large the Fock state description
and the coherent state description fully agree.

And you pick what you want.
If you introduce the electromagnetic field
explicitly with it's quantum state,
you get the dressed atom picture as a solution
of time independent problem.
Whereas here, with a coherent state description,
the coherent state oscillates, cosine omega t,
with a time dependent problem.
And actually, I should say whenever
I get confused in one picture, I look in the other picture
and it becomes clear.
I generally prefer where we have N photons,
it's because we can discuss everything
in a time independent way.
But for certain intuitive aspects, this is also valuable.
So, in the end you have to learn both.
And, in your homework, you really
actually write down the general solution
for this Hamiltonian as an exercise.
Nancy.
I think I'm confused a little bit.
So, in the Fock state picture, the dressed states
can be exactly part of an independent matter of coupling
between a lesser photon number in an excited state.
So, we can write eN plus 1 and g N, or something like that.
Yeah, you couple the photon field
with N photons of energy N h bar omega
to N minus 1 h bar omega.
But in this one, is there a direct photon number thing,
because we haven't quantized the field yet?
What do the dressed states mean at this point?

Well, the fact is if you start out with a coherent state,
your photon field is not N photons, it's a laser beam.
The laser beam with a coherent state,
is in a quantized description, a superposition of many Fock
states.
So, therefore, the number photons in a coherent state
fluctuates, or has a large Poissonian statistics,
and if you take one photon out or not,
it doesn't make a big difference.
For instance, for those of you who
know how the coherent state is defined,
the coherent state is defined as when
you act on the coherent state with an annihilation operator,
you get the eigenvalue times a coherent state.
So, that tells you have a fully-quantized description
of your laser, in terms of a coherent state.
You take one photon out, and what do you get?
The same state back.
And this may immediately justify that what we write down here
is simply the coherent state with it's amplitude,
and the amplitude of the coherent state
would be B1, the amplitude of the drive field.
And we don't really need other states,
because to coherent state has the property.
You take a photon out and you still have the same state.
So therefore, we don't have to keep
track of the coherent state.
It's there all the time.
But, what I'm saying can be formulated more exactly when we
use the appropriate formalism.
But this is sort of the bridge.
That's why we do not have to keep track of the photon state.
It's because the coherent state has those wonderful properties.