M1L4e.txt

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Any questions about that?
Yes, please?
So what determines the time constants,
like for dephasing time?

Well, that's a long discussion, and we
spend a long time in 8.422 discussing various processes.
But just to give you an example, if you have a gas of atoms
and there are slightly inhomogeneous magnetic field,
that would mean that each atom, if you look at it as precession
motion, precesses at slightly different rates,
and the atoms will lose-- will decohere.
They all will eventually wind up with a different phase,
that if you look at the average of coherence, it will go to 0.
So any form of inhomogeneity, which
is not quenching a quantum state, which is not creating
any form of deactivation of the excited state,
can actually decohere the phase.
And these are contributions to T2.
So often, contributions to T2 come
from inhomogeneous environments, but they are not changing
the population of states.
Whereas what contributes to T1 are often
collisions, collisions which, when an atom in an excited
state collides with a buffer gas atom,
it undergoes a transition from the excited to the ground
state.
So these are two distinctly different processes.
One is really a collision and energy transfer.
Each atom has to change its quantum state,
whereas decoherence can simply happen
that there is a small perturbation of the energy
levels due to external fields.
And then the system, as an ensemble, loses its phase.
In the simplest way, you can say--
you can assume inhomogeneous broadening.
But you can also assume if the whole ensemble
is subject to fluctuating fields, then
since you don't know how the fields exactly fluctuate,
after characteristic time you will no longer have
a phase-coherent system, where the phase at a later time
is deterministically related to the phase at which you prepared
it.
And that would mean the system has dephased,
and this dephasing time is called the T2 time.
Nancy?
I think I have two things.
First, you said that it's generally true
that T2 is less than T1.
Is it ever true that it's not the case?
Oh.

There is one exception, OK, and that's the following.
Let me put it this way.
Every process which contributes to T1
will also contribute to T2.
But there are lots of processes which only contribute to T2.
So therefore, in general, T2 is much faster
because many more processes can contribute to it.
However, now if you asked me, is that always true?
Well, there is one glitch, and this is the following.
T1 is the time to damp populations,
and that's a damping of psi square.
T2 is due to the damping of the phase.
And this is actually more a damping time of the wave
function itself.
And if you have a wave function psi which
is damped with a damping time tau,
psi square is damped with twice the damping time.
So if the only process you have is, for instance,
spontaneous emission, then you'll
find out that the damping rate for population is gamma.
This is the definition of the spontaneous emission rate.
But the damping rate 1 over T2 is 1/2 gamma.
But because simply the way how we have defined it,
one involves the square of the wave function.
The other one involves simply the wave function.
So there is this factor of 2, which
can make, by just a factor of 2, T1 faster than T2.
But apart from this factor of 2, if T2
would be defined in a way which would incorporate
the factor of 2, then T2 would always be faster than T1.
That makes sense physically for-- I can't imagine,
if the system has a smaller T1, then it still
has any coherence left in it.

So maybe to be absolutely correct I should say this.
T1 is much larger than-- it is larger or equal than T2 over 2.
In general, we have even the situation
that TI is much, much larger than T2.
But with this factor of 2, I've incorporated this subtlety
of the definition.
Other questions?
Yes.
I have a question about real motivation and using
plot divisions and [INAUDIBLE].
You mentioned-- I understand that.
I used to teach it for mixed states.
But you mentioned before that you
can't describe spontaneous emission with a Hamiltonian
formulation.
Yes.
But couldn't you use-- doesn't the Wigner-Weisskopf theory
when you just quantize your electromagnetic field
and look at the interaction between the electromagnetic
field and your atom.
Would you still get spontaneous emission out of coupling
to the continuum, with emission into the different modes
of the-- you don't necessarily need [INAUDIBLE].

Yes, but let me remind you of this.
If you're interested in the quantum state,
and it decays to a level, but you're not
really interested what this level is,
and we're not keeping track of the population here,
then we can describe the time evolution
of the excited state with an Hamiltionian,
because of the imaginary part.
The Hamiltonian is longer imaginary.
And this is what Wigner-Weisskopf theory does.
It looks at a system in the excited state
and looks at the time evolution of the excited state.
But if you want to include in this description what
happens in the ground state, you're
not taking this situation.
You have this situation.
And what eventually will happen is--
you can look at a pure state which decays.
And this is what is done in Wigner-Weisskopf theory.
But if you want to know now what happens in the ground state--
well, I'm speaking loosely, but that's
what really happens-- every spontaneous emission adds
something to the ground state but in incoherent way.
So what is being built up in the ground state
is not a wave function.
It's just population, which has to be described with a density
matrix.
Or in other words, if you have a coherent superposition between
excited and ground state, you cannot just say spontaneous
emission is now increasing the amplitude to be in the ground
state.
It really does something fundamentally different.
It puts population into the ground state
with-- I'm loosely speaking now-- but with a random phase.
And this can only be described probabilistically
by using the density matrix.
But what you're talking about is actually
for the Wigner-Weisskopf theory is pretty much
this part of the diagram.
We prepare an excited state, and we
study with all its glorious details
with the many modes of the electromagnetic field
how the excited state decays.

With the Hamiltonian formulation,
wouldn't you have to do the sum over every single mode you're
coupling to, which would be the exact same thing you
do when you do a partial trace over the environment?
Wouldn't the end result in sort of the same thing
when you have to do some very infinite sum and integral
for all the modes that you're coupling to?

You need a sum, but--
That's where the decoherence comes from.
Yes.
But if you're interested in only the decay of an excited state,
it can decay in many, many modes,
but all these different modes provide a contribution
to the decay rate gamma.
So at the end of the day, you have a Hamiltonian evolution
with a damping time gamma, and this damping time gamma
is the sum over all the other states.
So in other words, the loss of population from the excited
state is just incorporated by making a damping time
to the Schrodinger equation, because you're not
keeping track of the other modes where the population goes.
You're not keeping track.
You just say, excited state is lost.
You're not interested whether the atoms are now in the ground
state or some other state.
All you are describing the loss rate from the excited state.
And this is possible by simply doing-- by adding damping terms
to the Schrodinger equation.

In other words-- and what I'm saying
is actually fairly simple-- if you have a coherent state
and you lose it, it just lose amplitude.
What is left is coherent.
What is gone, it's gone.
You know?
You have a smaller amplitude, smaller probability.
And that's simple to describe.
What is harder to describe is if you accumulate population
in the ground state, and the population
arrives in incoherent pieces.
How to treat that, this is more complicated.
But simply the decay of a pure state,
it's just-- you have e to the i omega t, which
is a coherent evolution, and then you add an imaginary part,
and this is a damping time.
So what I'm saying-- it's sort of subtle,
but it's also very trivial.
I don't know if this addresses your question.

And in the end, in general you need the density matrix.
I just wanted to emphasize that there
is a little bit of decoherence where you can still get away
with a wave function description.
And actually Wigner-Weisskopf theory
is the wonderful example.