Finally in this video we're going to discuss a very interesting class of
quantum phenomenon that appear in the presence of slow, or so called adiabatic
time-dependent perturbations. And such adiabatic perturbations give
rise to a very elegant mathematical structure.
And, in particular, this so called geometric, Berry Phase that we're going
to derive. so the this Barry Phase was discovered
theoretically by Sir Michael Barry. And this paper, published in 1984 in the
Proceedings of the Royal Society of London.
And this is a really very well written and elegant paper, and actually even
though it's a research-level paper which in all likelihood may lead to a Nobel
Prize for Sir Michael Berry you are well-equipped to actually understand
everything in this paper. So now, both the derivation and the final
results. because it requires nothing but the basic
knowledge of single particle quantum mechanics and Schrodinger equation.
So we're going to reproduce part of the derivation in today's lecture, in this
part but of course if you want to know more details I would like to refer you to
the original paper which again is very well written.
So the problem that Sir Michael Berry considered was well, in retrospect a very
simple problem. So basically it was the canonical
Schrödinger equation with some time-dependent Hamiltonian.
And he assumed two things. he looked into adiabatic perturbation.
That is to say that So the time-dependent in h of t is, is assumed to be slow.
what it means precisely, we're going to define in the next few slides, but at
this stage that's just leave it at that. So we have some very slowly changing
Hamiltonian. And also let's assume that the
Hamiltonian returns to itself after a certain period let me call it capital T.
So basically, we have a periodic in time slow perturbation.
And the question that he asked is what happens when the wave function As we get
to this moment of time, capital T. So what is the wave function at the final
moment of time? So and, it turned out that, as we will
see, so that this wave function has a very interesting contribution sort of
topilogical, or geometric contributions. There are various sort of descriptions of
it that appear in the literature. that was completely counterintuitive, and
that takes some time to digest. even after a derivation.
So now let me formulate the problem that he solved more precisely and proceed to
the actual solution of the problem. And to the formulation of these Theory of
the Berry phase. So, let's imagine that we have a
Hamiltonian that depends on a parameter lambda, okay?
So this could be a, anything. For example, it can be magnetic field or
it can be some parameters of a harmonic oscillator.
let's say, the frequency or something else, whatever you want to think about.
So let's have with some we have some set of parameters lambda And we combine those
parameter in a vector so in principle it could be more than one parameter.
And we just construct a, sort of a vector in the parameter space which does not
have to be three-dimensional but let me just for sake of simplicity, let me
imagine that they have a three-dimensional parameter space.
We will say this is going to be lambda x or lambda one, lambda two, lambda three.
Okay. And let me consider the station where
this parameter lambda is changing in, in time.
So, for example, again, it can be a magnetic field that is changing.
And also, let me consider the situation where the after a certain period of time
capital T, so the parameter. Returns to the original point.
So if this is my parameter space. So, basically what I'm talking about.
I'm talking about a slow evolution of this lambda such that it starts at a
certain point and returns to the same point after, time capital T.
So, and as I advertised in the previous slide, so what I want to know Is what
happens with the wave function? So, I have some initial condition for the
wave function. And,
So, the Hamiltonian, after these period. Capital T returns to itself.
So, but the wave function, does not necessarily have to return to itself.
So the question that we're interest in is what happens?
With this wave function. As a first step in this derivation, let
me define precisely what I mean by slow, or, adiabatic perturbation.
So what does it actually mean, that the Hamiltonian is changing slowly with time?
And to provide this definition let me formulate an eigenvalue problem.
Instantaneous eigenvalue problem for the Hamiltonian.
So the Hamiltonian itself is changing with time so there is no reason for us to
write the stationary Schrodinger equation.
But never the less mathematically it is well defined.
So, we can consider at any moment of time t, we can consider an eigenvalue problem
for this h of t. And we can find a set of eigenfunctions.
And that the responding eigen values which I denote as e sub n of t.
So, for the sake of simplicity let me actually assume that the
Spectrum of the Hamiltonian is discrete. So I have just a set of discrete levels.
So n equals 0, n equals 1, et cetera. Also, let me assume that the initial
condition for the wave function, psi at t equals 0, is one of the eigenstates of my
Hamiltonian at t equals 0. So, for the sake of concreteness, let me
actually assume that this is the ground state.
So n equals 0 at t equals 0. So by the way, it's not necessarily
really for the main conclusions of the derivation, but for the sake of
completeness, let me just be specific. Okay, now let us recall what we discussed
actually in the previous lecture, in the previous part of this lecture where we
dealt with the opposite case of fast or sudden perturbations.
So there, we know that if we change the Hamiltonian very fast, the state, the
initial state, which in this case is the ground state, would be sort of
redistributed all over the place. so the particle after the quench, or
after a fast sudden change, would exist in all possible eigenstates of the new
Hamiltonian. And sometimes it's natural to define a
small perterbation as the opposite to the sudden perturbation.
That is to say that, we will define a small perterbation aside that it does not
induce transitions. So, basically what happens here is, let's
say this is the parameter lambda. It's equal to 0.
And we have our initial state, the ground state, with n equals 0.
And, as I change my lambda, so, with time.
So, these energy levels start to move. So, the energy levels really are this
Solutions to this, eigenvalue problem. But the Hamiltonian was changing.
And so the, energy levels are moving. And the principal.
Also, the explicit, wave functions are changing.
So, and what I can imagine happening is that.
The wave, the particle, which initially, was a.
sort of confined to the ground state canon principal propagate to all other
states. And this indeed will, will happen if the
change in Hamiltonian is fast enough. But if this is slow, and this will give
the definition of slow, the the state of the system would remain in the
instantaneous ground state of my hamiltonian.
And also in this case, if I start form this point in the parameter space and
return to the same point in the parameter space so that at the end of this periodic
evolution basically I require then that the system, that the system would remain
in the old ground state. And this will be my definition of a slow
perturbation. So to mathematically define a precise
condition for this to happen requires a little bit of work and that provide this
attenuation in the supplementary material in the notes.
And if you go through the attenuation, you will see that the condition for this
to happen for the condition for having no transitions Is that the derivative of the
Hamiltonian, which is an operator, and the matrix element of this d H d t
between two states, let's say this ground state and for example the first excited
state, this must be much smaller than the level spacing, the distance.
between these energy levels divided by the typical time well, the concentric
time at which the change in the Hamiltonian occurs.
So this is the definition if you want, if you want or, the con, con, constraint
that we have to satisfy in order for the perturbation to be slow.
And one thing that we that we notice here is that, of course, if e n minus e of 0
vanishes. That is, if the levels cross, or if they
come very close to one another. So this condition cannot be satisfied
because when necessary we'll induce transitions between these states.
But if the gaps don't close. If the level spacing remains large enough
and if the change in the connectonium is slow enough so we can satisfy this
condition. An do good approximation, we can assume
that the system remains in the ground state during the revolution.
So, now we are at the position to actually derive this geometric berry
phase. And what it entails is essentially
solving the Schrodinger equation, the time dependent Schrodinger equation under
the assumption of this slow perturbation that does not induce the transitions.
So, this is my Schrodinger equation. I'm just writing it down again.
And here I'm basically requiring that psi of t remains proportional to the ground
state. And the only thing I can have here is a
phase. Right?
So, because, well, the total number of particles s conserved, so therefore I
cannot have here any, coefficient whose absolute value is larger than 1 because
it would imply that the probability to find a the particle in this instantaneous
ground is larger than 1, and it doesn't make any sense.
So, and I also require that well, within my approximation, there are no
transitions, so the particle remains in the ground state.
So there's basically only, only one possibility for this to happen.
if this coefficient relating the wave function at time t, and the instantaneous
ground state is a pure phase, either for i phi of t, or if i is a real function.
So under these assumptions, what remains is is to find this one phase.
Everything else we know so this is really the remaining unknown in our [INAUDIBLE]
time-dependent problem. And if we plug in this form of the
solution back into the Schrodinger equation and recall by definition that
this 0, t is the instantaneous eigenstate of the Hamiltonian.
So simply get, so basically combining these 2 things will lead to the
Schrodinger equation in this form. So it will be I H D over D T of phi of T,
in this form is equal to the energy of teh ground state which is time dependant.
And this is because energy is moving around.
si of t. Okay?
And well so this is no longer an operator actually.
This is just a function and naively you may sort of assume there exists a simple
solution to this problem. where if I would be equal to sort of a
generalization of usual quantum phase that appears even in the time-independent
problems. So we could naively just write it as
minus 1 over h bar, an integral from zero to t, E naught of t prime, d t prime.
So again, if the Hamiltonian were not to depend, on time, that is, in this picture
would mean that basically Lambda's a point which stays there.
So I would just get the quantum phase being e to the power i minus i over h
bar, energy times time, which is the usual quantum phase that appears in
stationary quantum mechanics. So here, instead, I may just go ahead and
integrate from 0 to t. Well it turns out however that this is
not entirely correct. And there exists an important
contribution which appears in well, to this quantum fix, and let me write this
contribution as gamma. And this addtional phase gamma that we
will derive in a second is exactly what is called the Berry phase.
So this is my, our main quantity of interest.
While the first part is, is sort of the usual dynamical phase.
It's actually called dynamical phase, let me, you note it as Theta with the
subscript of D, and this is called dynamical phase.
If we plug in now, back, you know, basically, this expression for the phase
back into the Schrodinger equation, we're going to get in the left hand side, we're
going to get the following term. So there's going to be derivative pf e to
the power i theta, e to the power i gamma.
And there will be the instantaneous ground state, 0 t, and the right hand
side will remain, will remain the same. So in here, we have the identity which is
going to act on. Three terms so they are going to, they
are going to be three contributions in the left hand side.
So the first contribution essentially, by construction, is going to cancel out the
right hand side. So if we differentiate this exponential
so let me just write it down. It's going to be i, h bar times i theta d
dot. Times psi.
And this theta d dot is this the derivative of this integral, which will
just pull out the value of the energy. So this whole thing is going to be equal
to the energy. as a function of time, which will be
cancelled by the same term in the right hand side.
So, there's going to be also another term here.
This is going to be i h bar i gamma dot times psi.
And finally, there's going to be a derivative of the wave function itself.
So the eigenstates are changing. So the Hamiltonian is changing and so are
the the eigenstate, so therefore there's going to be another term so let me write
a plus i h bar e to the power i theta plus gamma.
Here we're going to have the derivative of this of this guy and the right-hand
side is e naught times psi. Which as I already mentioned, is going to
be cancelled precisely by the first term in the left hand side.
That's why we introduced this dynamical phase in the first place.
And now, what remains is simply to balance out these 2 terms.
And as you can see, a lot of things cancel out.
So the Planc constant cancels out, so the phase factors.
In this term are going to be canceled by the corresponding phase factors in this
wave function. So this goes away.
And I times I is equal to minus one, and so what you can write is the following.
So let's say on the left hand side we're going to have a gamma dot the wave
function, the ground state wave function is equal, basically, to this guy, i d
over dt 0t. So finally, what we can do here, we can
take advantage of the fact that we have normalized states.
So the number of particles is conserved, and therefore, and hold up, and our
particles exist in this ground state, therefore, this bracket product of this
instantaneous ground state with itself is equal to one.
And so if we multiply this equation from the left by a broad vector of zero t, so
we're going to have simply one in the left hand side.
And we're going to have you know, zero t g over d t zero t in the right hand side.
So therefore, what we can do, we can write the, the Berry phase, well, this
Berry phase gamma at the time T as an integral from 0 to capital T.
So this is the time that it takes the particle, well, the, the system, to
return to the original Hamiltonian. And under this integral we're going to
have zero T, D over D T, zero T, D T. So it's an integral over time.
So we get this interesting term which is the Berry phase.
And and as we shall discuss in the last slide, so this term has a very
interesting, geometric rotation, that's why it's called a geometric phase.
To see this let us recall the definition of these eigenstates, zero T, so those
are the instantaneous eigenstates of the Hamiltonian.
But, per our assumption, so the Hamiltonian depends on time only through
the parameter lambda, this extra parameter lambda.
So instead of writing this guise as being parametrized by t, we can as well
parametrize them by this parameter lambda, in this parameter space here.
And if we do so, we can transform this integral over time into an integral over
lambda in this parameter space. And this would require a very curious, as
I said, geometric interpretation. So almost the last step here is to
perform a change of variables and to go from an integral over time, to an
integral over Landa. And this can be done by simply writing.
So here, this dd, d over dt can be written, so this is the identity.
It can be written as d lambda over dt d over d lambda.
Okay? And on the other hand, when this guy is
going to multiply. This differential, so we can simply
replace this product by d lambda. And so another way to write this this
Barry phase, so let me present it here, so gamma can be written as an integral
and now this integral will go actually uhhhm around this loop in this bramoshir
space. So it's going to be closed integral so
lets save this contour, we'll call it C, so it's going to be basically encircling
the C, and here were going to have zero d over d lambda 0 lambda d lambda.
Okay. So this is another way to write
essentially the same quantity, but the time is completely gone from our
description even though we started with time-dependent problem so here time is
not essential anymore. So, another thing we can do, we can
introduce a new notation, let me call it A a differentiation A is the function of
Lambda, which is by definition is going to be this guy.
And so in this notations we can, we can see that the,
A vary phase is nothing but a circulation of this potential if you want, A of
lambda, around this contour. Okay, so another thing that we can do
here. We can take advantage of certain
identities from the vector analysis. Hopefully familiar to some of you who
have studied the theory of electromagnetism, namely we can define
the curl of this vector A, so let's define the curl A Curl with this respect
to this lambda. And let me call it b of lambda.
And if we do so this berry phase gamma can be rewritten as a flux of this I will
call it, fictitious magnetic field, through the area enclosed by this loop
that corresponds to this quantum evolution, okay.
So essentially well it's very complicated but, so those of you who again who are
familiar with the theory of electromagnetism.
Now remember, that to describe magnetic field in real space is often times
convenient to introduce a vector potential such that the curl of this
vector potential is the magnetic field. And the, so here, instead of the real
space have this sort of fictitious parameter space, which does not have to
be a real space. It can be something else.
Can be just be a set of parameters in our problem.
And what we found is that the quantum phase that will be required by the wave
function as a result of this evolution. It's essentially a flux of some
fictitious magnetic field that exists in this model.
Okay? So, by the way, notice that even though
we started with a time-dependent problem, this b of lambda, and the, corresponding
vector potential a can be calculated by themselves.
So if we just have a, Hamiltonian, which depends on Lambda, whether or not it
depends on time is a separate matter, we can just go ahead and calculate these
properties. And so, what we may think about is that.
When we have a quantum system, which is parameterized by some parameter lambda,
so there exists an internal and in some sense fictitious magnetic field in this
parameter space. And so it exists whether or not we
perform a quantum evolution, but if we do perform quantum evolution and if we have
this sort of periodic Time depend on perturbation.
So the quantum phase that would be acquired by the particle is going to be
basically the flux of this magnetic field if you want, through this area enclosed
by this loop. And that's why it's called the geometric
phase, sometimes it's also called the pological phase.
It's a very non-trivial concept, so I mean, first of all it was not discovered
in the early days of quantum mechanics. It was discovered only in the 80s.
And furthermore, only now, and this is basically the subject of current
research, we're starting to understand the true importance of this Berry phase.
And, one of the recent discoveries that you might have heard about Is so called
topological insulators. Something I didn't have time to talk
about these quite amazing discoveries, but I just wanted to mention them in
passing to sort of emphasize that this is not just a pure mathematical construct,
which is sort of elegant by itself, but also does have Important physical
applications. And, this is the subject of current
research. So, the final thing I'm going to mention
here sort of concluding this technical discussion, is that unlike the magnetic
field in real space, which does not have sources or sinks.
So, there are no monopoles as we know. So, the divergence of the magnetic field
is 0. So this fictitious magnetic field
actually it can have sources. And this sources are, are, well some sort
of monopoles in this parameter space correspond to the degeneracies in this
spectrum of the problem. So what it actually means is and this is
one of the results eh, understood by Berry in his original paper.
Is that if we have some, value, some special values of parameter lambda, let's
say, equal lambda star, such that, there exist, two, there may exist two wave
functions with the same energy. So basically, the energy levels cross So
then these points, let's say this point, serve as a source of the monopoles of
this magnetic field. So in this parameter is best.
This is also a very interesting result. Well, it's a very interesting subject.
I would definitely talk more about it. It's a very exciting field.
But I feel that I probably should stop here.
Because I've been talking for twenty-five minutes now on this last part of the last
lecture. And I just would like to thank you very
much for your attention. I hope that this lecture and in general
the course were interesting and useful, at least some parts of it.
And I wish you all the best, thank you very much.