Superconductivity is an example of an intrinsically many bodied quantum method,
which doesn't really have a single particle counterpart.
So it cannot be really understood based on just a single particle Schrodinger
equation. So in the context of superconductors, of
course, these many particles are electrons.
So that play together in some sense to create this amazing quantum coherent
state. Now, to understand the very complicated
array of many, Body quantum phenomena. The first step is to understand the
concepts of quantum statistics. Which is an increasing quantum
interaction between identical particles. It doesn't really have a classical
counterpart. So unlike let's say the electrostatic
let's say [UNKNOWN] which depends on distance between the two charges.
So this statistical interaction between identical particles is quite different.
It appears to never have identical particles and they feel each others
presence. independently of the distance separating
them, so this is very interesting. And so we're going to learn about it and
also we are going to learn about the two fundamental classes of particles, with
respect to their quantum statistics, that are called bosons and fermions.
And those guys they lose energy space, [INAUDIBLE] was very much drastically
different. And we will see that it's actually a
transition in some sense from fermion like behavior, to boson like behavior.
That is exactly what explains transition from a regular metal into a quantum
superconductor. The first step in defining quantum
statistics. Let us consider two identical quantum
particles. This blue represented by these blue dots
here. And let's assume that all intrinsic
properties of these particles are exactly the same.
And so the question I'm going to be asking is what happens if we braid two
particles? So, in other words, if these two
particles exchange places. So the first particle goes to the place
of the second one and vice-versa. So what it means is that the wave
functions. In this case, the wave function
describing this two, this two particle system depends well, on the coordinates
of, of these two particles, r1 and r2, in principle, some other quantum numbers
that I'm not writing. And so the question we're asking is what
happens with this wave function if we replace r1 with r2, and otherwise errors.
So it turns out that upon this braiding the wave function requires quantum
mechanical phase. So this is statistical phase phi.
And well we can sort of get an idea of why it might happen.
Because the wave function really is a tool for us to understand the observable
phenomenon, most observable answer. Through the let's say the absolute value
of the wave function or the various metric elements wave function phase is
not something we worry about. And so we may imagine that even though
from our perspective nothing has changed since two particles are identical which
one is at which point doesn't really matter.
So we still main principle get a quantum mechanical phase here, and indeed this
happens, okay? So and so the question that we can ask,
and that was asked a long time ago what are the possible values of this quantum
statistical phase? And the answer was provided in this
seminal paper entitled The Connection Between Spin and Statistics, by Wolfgang
Pauly, who proved there exists a one-to-one correspondence between the
internal angular momentum of a particle, so called Spin that you're going to learn
about a little later in the course and it's quantum statistics so I want to give
you a lecture on spin because there will be a separate one on that but that will
just mention that their are just two types of particles allowed in nature
which have either integer or half integer spin.
And those differ by how their wave functions transform under rotation.
So if we rotate our [INAUDIBLE] coordinate system by a certain angle of
this rotation [INAUDIBLE] that we would encounter would look differently
depending on what kind of spin of the particle has.
And, interestingly enough let's say spin one half particles an electron is an
example of such a particle. is going to be described by simple
representation of this rotation. Which upon repeating it by 360 degrees,
actually picks up a minus sign. As opposed to integer spin particles, or
particles with no spins that don't really change.
Whose wave function doesn't really change upon this rotation by 360 degrees.
Now I'm just sort of giving you a flavor, but the outcome of this so-called spin
statistics theory can be formulated in a very simple way.
So if a particle with integer a internal angle and momentum including zero the
phase here is really zero, so there is no change in the wave function.
If we replace r one bl-, by r2, nothing changes.
Well, for particles with a half integer spin this statistical [INAUDIBLE], phi is
equal to pi. Or, in other words, this psi of r1 and r2
is equal to minus of psi of r2 and r1. And these guys with half integer spins
are half integer spin are called fermions.
So, we'll just make one comment that it turns out that this classification is
specific to a three dimensional relativistic quantum theory to the world
that which we actually leave but if we consider two dimensional systems which
appear as a matter of fact in, let's say in condensed measure context.
So we have two dimensional metals and two dimensional super conductors.
So there it turns out that the possibilities for phi are much more
interesting. So apart from zero and pi we may have an
extra arbitrary phase of pi, which are called in the particles like that, so the
fictitious immersion particles like that are called anions this pi can be a matrix
in which case they are called [UNKNOWN] in news but this is the only thing I'm
going to mention about it. But it just, I want to want you to know
that the parts of these well known possibilities for fundamental particles.
So there are other situations which arise in a [UNKNOWN] manner of physics.
But now let me focus on this case. So well, so we for irregular particles we
have I, either a zero or pi. So let's assume we're dealing with
fermions, let's say we're dealing with electrons.
So what are the consequences of this minus sign that we pick up in front of
the wave function. Does it really matter at all, so should
we worry about it? So let us think about it.
So what we're really saying again is that psi of r 1 and r 2 is equal to minus of
psi of r 2 and r 1. And it immediately leads to a conclusion
that for an important case r 1 equals r 2.
So psi of r 1 and r 1 is equal to minus of psi of r 1 and r 1.
Which means that the size is equal to 0. So what does it mean?
It means that the probability of find two identical fermions in the same point is
0, because the size is 0. So, well and It means that no two
particles can be like no two fermions. Identical fermions cannot occupy the same
point. And more generally we can actually
generalize this for an arbitrary representation of the way function.
We can prove and what this is about exclusion principle is that no two
identical fermions cannot occupy the same quantum state.
They cannot have exactly the same quantum numbers.
But there is no such constraint for bosons okay.
So, this has very important consequences so this statement again is called the
Pauli exclusion principle and for bosons it doesn't apply.
So, and that the consequences are that if we have certain quantum state, let's see
an [UNKNOWN] quantum state i and we want to populate this state with fermions,
there are only two possibilities for the occupation number of this state.
So the occupation number can be either 0 or 1.
Well for bosons, it can be anything, it can be any 0, 1, 2, can put any, any
number of bosons we want in this quantum [INAUDIBLE] .
Now what does it mean in the context of many body systems, when we have actually
more than one and two particles, when we have billions and billions of identical
particles as we usually do in quantum systems such as, let's say, metals and
superconductors? So let us discuss this question first on
for many bosons system. And what we're actually asking is what
happens let's say if we have many energy levels that are initially empty and we
want to know what happens if we sort of pour identical bosons into this
Prescribe, landscape of quantum states, with some states I with some energy E,
energy E sub I. And more specifically what we're asking
is, what we're interested in is what is the lowest energy state, the so called
ground state, that the system would want to, Form at low temperatures.
So at low temperatures the excitations are going to sort of calm down and the
system would want to form the lowest energy state.
Now since any number of bosons can occupy any state, so clearly the lowest energy
state would correspond to essentially a state in which all bosons,
non-interacting bosons that is, are sitting just at the lowest possible
level. Now for a free particle with dispersion,
just having the kinetic energy dispersion p squared over 2 m.
Some sort of parabolic dispersion here, I'm just plotting it in sort of an
example of a two-dimensional system px, py and this is the energy basically all
bosons would want to drop into the zero momentum state.
Okay, and this kind of state is called Bose-Einstein condensate.
The state in which a single quantum mechanical level in this case the state
was zero momentum, zero velocity is microscopically occupied.
So this kind of phenomenon has been known for a long time, since Einstein.
But it was sort of, very observed in the clear way, only very recently in 1995
first. Now it's observed routinely in various
laboratories including our own joint quantum institute.
But back in 1995, it was new. And, this discovery was actually awarded
2001 Nobel Prize in physics. So here, I'm showing the [INAUDIBLE] .
So, basically, what the experimentalists did.
They trapped atoms, bosonic atoms in a confined geometry in this certain
trapping potential. And cooled them down to ultra cool
temperatures. And then let them remove the trap.
And let them expand. And they just sort of took photographs of
these atoms as time went by. And they saw this pic.
So this is basically the so called time of flight extension.
And this pic is shows the population. The density of particles in real space.
Large number of particles sort of don't, move too much, as time goes by.
And this speaks sort of [COUGH] in real space is [UNKNOWN] of condensation and
[UNKNOWN] space. So this corresponds to particles with
zero velocity so the just sort of stay still.
So now I would like to mention a very important circumstance namely that.
This Bose-Einstein condensates may host a very interesting phenomenon of
superfluidity where a condensate of interacting bosons form a superfluid
which has exact 0 viscocity and which can flow therefore without resistance.
So, here for example I'm showing a photo of superfluid.
Helium in a cup, the helium which can actually escape the cup climbing up the
walls. So let me mention that superfluidity, the
phenomenon of superfluidity has actually been known for a very very long time.
Since 1937 it was discovered then in Helium by Pyotr Kapitsa.
And, and Kapitsa got his Nobel prize for this discovery of 1978.
Well, it would be too late, 40 years after the initial discovery.
So the mathematical theory of superfluidty was put together by Lev
Landau a famous Russian theorist who got his Nobel prize for this theory in 1962.
So actually before Kapitsa and hey, let me just mention that the You know this
noble prize was awarded to him after Lev Landau got into a very serious accident.
Everybody thought he wouldn't survive but by some miracle he lived for a few more
years. And this allowed the Nobel community to
award him long overdue Nobel prize. So, in any case, I'm not going to
mention. I'm not going to talk too much about the
theory of superfluidity now because it actually requires understanding of the
interactions in Bose-Einstein condensates, and this is a non-trivial
business interacting many-body states. But I will just that, superfluidity sort
of rings a bell in the context of superconductivity because.
If we want to explain the main property of a superconductor, namely the zero
resistance state of the superconductor, we would want essentially a superfluid of
electrons, which would move with zero viscosity and without resistance.
This is exactly what the doctor had as prescribed, but there is one little
problem with this explanation of super, superconductivity, namely well electrons
in the metal are not [INAUDIBLE], they have half of half of the [INAUDIBLE], so
they spin one half, so they're [INAUDIBLE], so they do not form a
superfluid. So, what do they do?
And, to answer, to answer this question, I need to consider the ground state of
many-fermion system now. We talked about the ground state of
bosons. Now let me talk about what happens with
fermions if I pour them into a given energy landscape.
Let's say I have, Has some available levels and they have a many fermion
system that would want to occupy those levels and the question is, How will I
minimize energy? So due to the power of exclusion
principle I cannot allow my fermions to be sitting on the same lowest energy
state. So I can only at minimized energy, can
only sort of stack them up one by one to the lowest energy states as so until they
reach the the certain level called the fermion level or fermion energy up here
and basically this is the picture of fermion ground state, which is quite
different from bosonic ground state. Now in the context of a metal and three
dimensional, three dimensional metal, so what we have Is well our energy landscape
is given by the usual disbursement relation.
The energy versus momentum is equal to p squared over 2m.
So what I need to do in order to sort of satisfy the same picture with this energy
is that I will occupy all momentum stays. So here I have k x, k y,k z ,or I could
have written p x, p y, and p z. So below certain momentum is called
fermion momentum I will have all states occupied just like I have here all these
levels occupied and while these levels are all empty.
So and the threshold momentum it is just ignore this red and white dots for the
moment. So below this threshold momentum.
I have all states occupied above this threshold momentum I have all states
empty and this is exactly the picture which describes electrons in, in metals.
Actually it works amazingly well for metals.
And it's also known that the typical Fermi temperature in metal, the Fermi
temperature being the threshold momentum here The p, pF square divided by 2m this
is the Fermi energy divided by the Bols, Bolsman constants or constant which
converts energy into the temperature. So if we do this calculation the typical
Fermi temperature in the metal would be around 10000k, which is huge so even if
the room temperature left alone at 4 kelvin so this energy scale is normal as
compared to all other energy. [UNKNOWN] So and what the temperature,
well Fermi temperature usually does is it excited electrons from below the thermo
surface. For instance I can not excite electron
from here to here. I can not move, so let me just draw this
paralysis and then I will say it's not allowed.
So have, let's say, an electron here, I can move it to this point, it's not a
lot. Because both the stays are occupied.
So this stay is already occupied. So power exclusion wouldn't allow me to
do this thing. So, but what I can do if I have
temperature, I can, I have some energy to spend, so I can excite an electron let's
say from here to here. And so the y point here is an empty space
that the electron leaves behind. It's called a hole and the red one is the
actual electronic excitation. And this is exactly the picture of a
metal and so this guys, basically this red guys and this white guys.
these guys form basically they are the linked leaders in a typical metal.
But they do not condense, they don't form a super fluid.
They just move around and experience scattering you know from one point to a
different point and this result in a finite resist.
So how do you reconcile this picture of a metal with the presence of super
conductor is that is observed at low temperatures.
So this was again a major mystery for many many years, was not at all obvious
how to do this. But of course there is a hint about it
due to the presence of superfood so sort of in retrospect it's pretty natural to
ask whether or not it's possible to somehow convert to electronic liquid
[UNKNOWN] liquid Into a bosonic superfluid.
So this guys are fermions, so they're not bosons.
So can we make a bosonic superfluid out of a fermionic gas such as here?
Well, it's difficult unless, so unless I allow my electrons to form composite
objects. Of two electrons states whose total spin
is either one or a zero. Remember the speed of a single electron
is one half but if I combine the two of them together, so it's going to be an
integer. So they are going to be boson.
And so if I somehow find a way to say force objects well I could have actually
drawn it like this but well knowing the results, I'm drawing these objects this
week. But this station doesn't matter.
So if I allow the formation of to electron states, this, these 2 electrons
states are going to be bosons. And these bosons would be able to
condense. And these guys would be able to form a
super-fluid now, a charge super-fluid. And this is great because this state will
superconduct. And this is exactly what happens and this
is bound states of two electrons called Cooper burst.
And from this perspective superconductor is a Bose-Einstein condensate of cooper
burst. But the main question remains is why
would this Cooper burst would form? Why would such states appear?
And this is the question that we're actually going to answer in the remaining
two segments. And we're going to solve the cooper
pairing problem. And this was also established connection
between this lecture and the previous lecture.
When we discussed, [INAUDIBLE] states and weak potentials.