M1L4d.txt

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So now we are just one tiny step away from introducing the Bloch
equations.
We will fully feature the optical Bloch equations
in 8.422.
But since we have discussed two-level systems to quite some
extent, I cannot resist to show you now--
in three or four minutes-- what the Bloch equations are,
and then when you take the second part of the course you
will already be familiar with it.
So let me just now tell you what has
to be added to do this step from the previous formulation
to the Bloch equations and how we-- and this is the one
step you have to do-- we have to include relaxation
processes into the description.
So my less-than-five minute way to now
derive the Bloch equations for you,
goes as follows.
I first remind you that everything has
to come to thermal equilibrium.
In other words, if you have an atomic system--
if you have a quantum computer, whatever system you have,
and you're prepare it in a pure state,
you know if you will wait forever,
this system will be described by a density matrix, which
is the density matrix of thermal equilibrium, which has
only diagonal matrix elements.
The populations follow the Boltzmann factor.
And everything is normalized by the partition function.
So we know that this will happen after long, long times.
So no matter with what density matrix we start out.
If we start with the density matrix rho pure state,
for instance, they will be inevitably
some relaxation process which will restore rho to rho t
to the thermal equilibrium.

Now how this happens can be formulated in a microscopic way
and we will go through a beautiful derivation
of a master equation and really provide some insight
what causes relaxation.
But here for the purpose of this course,
I want to say that there is relaxation
and I want to introduce now in the phenomenological way
damping and damping times.
My phenomenological way-- way to introduce damping,
goes as follows.
Our equation of motion for the density matrix
was that this is a unitary evolution described
by the Schrodinger equation, was that the density matrix evolves
according to the commutator with the Hamiltonian
But now-- and I have to put quotation marks around it
because this is not a mathematically exact way
of writing it-- but now we're going
to introduce some term which will damp the density
matrix to the thermal equilibrium density matrix,
with some equilibration time Te.

In other words-- I mean this is what you can always
do if you know the system is damped,
you have some coherent evolution but eventually you
add a damping term and you make the damping term-- you
formulate in such a way that asymptotically the system will
be damped to the thermal equilibrium.
In other words, the damping term will
have no effect on the dynamics once you've
reached equilibrium.
So it does all the right things.
Of course, we have to be a little bit careful
because everything is either an operator or matrix
and I was just adding the damping term
as you would probably do it to a one-dimensional equation.
So therefore it may be a little bit more specific.
In many cases, you will find that there
are two distinctly different relaxation times.
In other words, the system will have usually at least two
physically distinct relaxation times.
They're traditionally called T1 and T2.
T1 is the damping time for population differences.

So this is the damping time to shuffle population
from some inverted state or some other state
into equilibrium state.
That usually involves the removal
of energy out of the system.
So it's an energy decay time.
And if you would inspect our parameterization of the Bloch
vector or population or population differences
are described by the Z component, the third component
of the Bloch vector.
Well we have other components of the Bloch vector
which correspond to coherences, the off diagonal matrix
element of the density matrix, and they are only non-zero
if you have two states populated with a well-defined relative
phase. So when you lose-- the system, quantum
mechanical system loses its memory of the phase,
the r1 and r2 component of the Bloch vector go to 0.
So therefore the time T2, the time T2,
is a time which describes the loss of coherences,
the de-phasing times, and in most situations-- well,
if you lose energy, you've also lost-- if you lose energy,
because you quench a quantum state,
you've also lost the phase, so therefore, in general,
T2 is smaller than T1, often by a lot.
So with those remarks about the two damping times,
I can now go back to the equation at the top, which
was sort of written with quotation marks,
and write it in a more accurate way as a matrix equation
for the damping of the components of the density
matrix expressed by the Bloch vector.
In other words, the equation of motion
for the Z component of the Bloch vector,
which is describing the population
has a coherent part, which is this generalized precession,
and then it has a damping part which
damps the populations to the equilibrium
value with a damping time T1.
And then we have the corresponding equations
for the x and y, or the 1 and 2 component of the optical Bloch
vector.
We'll just replace the Z index by x and y
from the equation above, but then we
divide by a different relaxation time, T2.
So what we have found here, these are the famous Bloch
equations, which were introduced by Bloch in 1946, introduced
first for magnetic resonance, but they are also
valid in the optical domain.
For magnetic resonance, you have a two-level system,
spin up and spin down.
In the optical domain, you have a ground and excited state.
In the latter case, they are often referred to as the Optical
Bloch equations.