M1L4b.txt

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I would like to take the topic one step further and discuss
the density matrix formalism.
So we have so far discussed purely
Hamiltonian unitary evolution, namely
the Schrodinger equation.
And of course, a unitary evolution
leaves the system, which is in a pure state, in a pure state.
It's just that the quantum state evolves.
However, that means we cannot describe processes like
decoherence or some losses away from the two levels we are
focusing on.
And so now we want to use the density operator, the density
operator formalism, to have a description of two-level system
which goes beyond that.
So Schrodinger equation deals only with pure states,
cannot describe loss of particles, loss of photons,
and decoherence.
Well, there is one exception.
If the incoherent, the decoherence process
is merely a state-dependent loss of atoms to a third state,
then you can still use the wave function formalism.
So this is the exception.
If you have two states and all what happens
is that you have some loss to some other levels
and their rate coefficients, then one
can still use a Hamiltonian description,
but you have to replace the eigenvalues by complex numbers.
In other words, you have to add an imaginary part to the energy
levels.
And that means the time evolution
is exponentially damped.
So that's as much as you can incorporate
decoherence and losses into a wave function formalism.
However, many other processes require the formalism
of the density matrix.
And the simplest process where a wave function formalism
is absolutely inadequate is the process
of spontaneous emission.
When you have a loss in the excited state,
you could still describe the excited state
with a complex energy eigenvalue.
But the fact that whatever is lost from the excited state
is added to the ground state, there
is no wave function formalism which can describe that.
So for those processes and for decoherence in general,
we require the use of the density operator.
So I know that most of you have seen
the density operator in statistical mechanics
or some advanced course in quantum mechanics.
So therefore, I'll only spend about five minutes on it.
So I want to kind of just remind you or give you
a very short introduction.
So for those of you who have never heard about it,
I probably say enough that you understand
the following discussion.
And for those of you who know already everything about it,
five minute of recapitulation is hopefully not too boring.
So my way of introducing the density operator
is first introduce it formally, write down
a few equations for a pure state, but then, in a moment,
add something to it.
So if we have a time-dependent wave function which
we expand into eigenfunctions, then we can, in this basis,
define arbitrary operators by matrices.
And we want to describe our system by finding out
measurable observables, expectation
values of operators, which, of course, depend on time.
And this is, of course, nothing else
than the expectation value taken with a time-dependent wave
function.
But now we can expand it into the bases m, n,
and we can then rewrite it as a matrix, which is a density
matrix, or simply as the trace of the density
matrix with the operator.
And what I introduced here as the density matrix
can be written as |psi(t)〉〈psi(t)|,
and the matrix element given by this combination of amplitudes
when we expand the wave function psi into its basis.
So this density matrix has diagonal
and off-diagonal matrix elements.
The diagonal matrix elements are called the populations,
the populations in state n.
And the off-diagonal matrix elements are called coherences.
OK.
So this is just rewriting Schrodinger's equation
expectation value in a matrix formalism.
But the reason why I wrote is that we want to now add
some probability to it.
We do not know for sure that the system is in a pure state.
We have probabilities Pi that the system
is in a quantum state psi i.
So we add another index to it.
And when we perform the expectation value--
there's also one star too many-- when we perform the expectation
value, we sort of do it for each quantum state
with a probability Pi.
So we are actually-- and this is what I wanted to point out,
this was the purpose of this whole discussion--
that we are now actually performing two averages.
One can be regarded as the normal quantum mechanical
average, when you find the average value
or the expectation value for a quantum state.
So this is sort of the statistics or the averaging
which is inherent in quantum physics.
But then in addition, there may simply
be another probabilistic average because you have not
prepared the system in a pure state,
or the system may undergo some stochastic process
and wind up in different states.
So there are two kinds of averages which are performed.
And the advantage of the density matrix formalism
is that both kinds of averages can be done simultaneously
in a very compact formalism.
So therefore, I put this probabilistic averaging now
into the definition of the density matrix,
or I can write the density matrix in this way.
And with this extended definition of the density
matrix, both kinds of averages are
done when I determine the expectation
value of an operator by performing the trace
with the density matrix.

A lot of properties of the density matrix.
I think you're familiar with many of those.
For instance, that Schrodinger's equation for the density matrix
becomes this following equation.
You can derive that if you take the Schrodinger equation
and you apply the Schrodinger equation to each state psi i,
and then you do the averaging with the probability pi.
You find that the Schrodinger equation for each state
psi i turns into this equation for the density matrix.
And we just write down that the purpose now
is we have two averages here.
One is the quantum mechanic average,
and one is sort of an ensemble average with the probabilities
Pi.
I want to discuss in a few moments
the density matrix for two-level system.
So I have to remind you of two properties,
that the density matrix is normalized to unity.
So that's probability of unity to find the system
in one of the states.
When we look at the square of the density
matrix, the trace of rho square, this
is simply the sum of the probability squared,
and this is smaller than 1.
And the only way that it is 1 is for pure state.
So pure state is characterized by the fact
that we can find a basis where only one of the probabilities
pi is non-vanishing.
And then, of course, almost trivially, the trace rho is 1
and the trace rho square is 1.
So so far, I've presented you the density matrix
just as an elegant way of integrating the two
averages into one formalism.
And in essence, this is what it is.
But you can now also use the density matrix
if the whole system undergoes a time evolution which
is no longer unitary, no longer described by a Hamilton
operator because you're interested in the time
evolution of a small system, which
is part of a bigger system.
The bigger system is always described by unitary time
evolution, but the smaller system is usually not described
by unitary time evolution.
And that's when the density matrix becomes crucial.
Of course, you can say this is just you describe the smaller
system and you do some probabilistic average, what
the other part of the system does,
and therefore it's just another version of doing two averages.
But this is sort of why we want to use the density
matrix in general.
So we want to use the density matrix for non-unitary time
evolution.
And the key word here is that this is often
the situation for open systems, where
we are interested in a small system,
but it is open to a bigger system.
Like we are interested that you describe an atom,
but the atom can spontaneously emit photons
into other parts of Hilbert space,
and we're not interested in those other parts
of Hilbert space.
So an open system for this purpose
is where we limit our description to a small part
of a larger system.
Again, an atom interacting with all the modes
of the electromagnetic field, but we simply
want to describe the atom.
And then we cannot use a wave function anymore, we have to use the density matrix.