Hello and welcome, let us continue our discussions
on the superconductivity and towards the end
we will discuss the topological states of
matter that is something I alluded to in the
beginning. These are extremely new topics
of almost at the level of research, but one
should have some idea of what it is and I
will also give references from which you can
pick up the basics of it.
So, we were continuing with Ginzburg - Landau
theory of superconductor, in particular we
were interested in finding a problem the solution
of a problem which was undertaken by Abrikosov
in former USSR using Ginzburg - Landau theory.
The problem is this that if you have strong
magnetic field and if you have a superconductor
typically these are called type - 2 superconductors.
And in these superconductors is it possible
that you have a state where the magnetic flux
penetrates the superconducting system and
by the by definition at the core of those
fluxes the magnetic order the superconducting
order parameter goes to 0 so, it becomes normal.
Now if one wants to work that out one starts
from this phenomenological theory called Ginzburg-Landau
theory which we outlined how does it work.
And it works by a purely phenomenological
idea that close to the critical point the
transition the free energy can be expanded
around the normal state value. That means,
the state in which there is no order the free
energy of that plus a polynomial and some
gradients and suitable terms that account
for all the energies that are there in the
system due to the formation of the order.
And if the ordered state has lower free energy
then the transition takes place and that is
how it is set up and that is what gave us
this equation in the absence of fields.
And from which we actually found out another
length scale which is Ginzburg - Landau coherence
length. And then of course, we already had
the effective London penetration depth which
in a disordered system for example, gets modified
considerably from the original lambda sub
L that we got from London equation London
penetration depth.
So, this effective lambda and the ratio of
this Ginzburg-Landau coherence length decides
which is the way the superconductor will behave
whether it will be type - I or type - II.
So, then these are the length scale in 2 different
cases in type - I the c is large and in type
- II c is small compared to lambda.
And this is how the B versus H curve will
behave in type-II superconductor for example,
and in type-I type-I of course, you have only
1 H c at which superconductivity is destroyed
and type-II has two H c values H c 2 and H
c 1.
Next figure will show it better here it is.
So, the M is proportional to H is inside a
superconductor minus M is proportional to
H and that is what is plotted. So, it increases
and then at H c it drops to 0 and the super
conductivity vanishes so that is what type
- 1 it does. And in type- 2 it linearly goes
up of course, but then there is a flux penetration
field penetrates the superconductor. So, there
is a slow variation and then finally, at H
c 2 the superconductivity vanishes finally,
H c somewhere between H c 1 and H c 2.
And the vortex solution we outlined as to
how Abrikosov got it is very much like the
principles the mathematics is almost the algebra
is more or less similar to what one did in
quantum hall effect integer quantum Hall effect
and this is the Landau famous Landau problem
that you put a good electrons in a perpendicular
magnetic field strong magnetic field.
And then of course, you find the solution
and the idea is that you from that you find
out what is the first value of magnetic field
at which superconducting solution becomes
non zero. So, it is called nucleation of superconducting
conductivity and it starts at a particular
value of H which we called H c 2 if you remember
that picture this is where the superconductivity
starts.
So, psi becomes nonzero here ok. So, that
is how one finds out the coexistence coexisting
state, but that just tells you that there
is a coexistence of both H and B and psi mod
psi square in this formulation inside the
superconductor ok.
That is that was the theory and then therefore,
of course, you can estimate the H c 2 in terms
of H c and find out it find that one finds
that it is root 2 kappa times H c. So, kappa
equal to 1 by root 2 as we showed earlier
is the point at which the two kinds of superconductors
are distinguished. So, that is the distinction
point between two superconductors type -I
and type - II. So, all these came out of this
calculation of Abrikosov, but he went further.
And then he of course, found out something
remarkable that he showed that these superconduct
this the flux that goes inside the superconductor
goes in quantized vortices and these quantized
vortices are like flux tubes and they are
not random inside the superconductor, they
have a lattice structure; that means, they
are like crystalline lattice.
So, if you look at from the top of the superconductor
where the flux tubes are coming out you will
see that they have a regular pattern and what
is what was found that these pattern is a
triangular lattice and that is what he actually
showed. So, this calculation is a bit involved
I am not getting into the details he did fantastically
intuitive calculation.
And found out that of course, he is there
was slight numerical mistake he first found
a square lattice structure, but later on it
was corrected to a triangular lattice. These
this is a remarkable calculation and if you
want you can look up in the literature , but
that the facts is that the flux does not penetrate
randomly it penetrates in tubes the magnetic
field and in the flux goes through tubes in
quantized vortices and. So, the flux value
is quantized and the structure of these tubes
is not random it is a triangular lattice ok.
So, flux quantization is a bit I mean this
is almost straightforward to see why it happens
and if you remember your London equation you
can you see that it is proportional to A and
you can write down these wave function the
superconducting wave function for the pair
for pair wave function as root over n r e
to the power i theta r.
And then you can that is the. So, n r is kind
of rho square and then you can write psi r
is rho e to the power i theta and then of
course, just use London equation and assume
that the. So, amplitude of the superconductive
wave function is more or less well formed
and it does not vary strongly. So, grad write
psi as a grad of psi as h cross grad theta
so, that is how the p operator will be.
So, p minus e A e star A e star is the charge
of the pair by c is h cross grad theta my
the canonical momentum in presence of a magnetic
field will just become this h cross grad theta
minus e A by c. This we have also seen in
the current expression from the Ginzburg - Landau
equation and of course, London equation says
that this must be 0 alright.
And that is exactly what happens and then
you can just integrate over a curve inside
the body of superconducting ring and then
you will get this expression A dot dl which
is B curl A dot dS is e star by c into B dot
Ds. Therefore, grad theta dot dl is equal
to e star by h cross c into phi the flux.
Now, theta being the phase of the wave function
it has to be single valued over a closed loop.
So, it has to come back to the to either change
will be either 0 or multiple of 2 pi and that
is what is used here and that immediately
gives you the quantization.
So, these fluxes are quantized and the quantum
is n times this is the number of that number
n 2 pi n into hc by e see hc by e is we know
that quantum of flux of course, here it is
e star which will turn out to be 2 e that
we know. So, this is the value.
Then we did the Josephson Effect which is
basically coupled super conductors with a
thin insulating layer.
And we found out that in even without a potential
between the two junctions across the junction
you can get a current and this is remarkable
when you bring two superconductors in proximity.
And so, that just comes out of the fact that
these two the super conductivity conductor
has a uniform has one wave function uniform
wave function throughout and the two theta
has tried to become the same at the across
the junction and that means, they will exchange
cooper pairs and that is exactly what this
calculation shows.
And it shows interestingly that if V equal
to 0 the voltage across this junction is 0
then you have a non zero current whereas,
if the voltage is finite then of course, the
current fluctuates so fast that you will effectively
get average current 0 ok.
So, then you can extend it to AC fields and
you can choose a particular frequency at which
you will get currents. So, there is called
AC Josephson effect.
There is a an instrument that we I mentioned
briefly where you have this the flux due to
a magnetic field can be measured of course,
flux can be measured in many other ways even
in MSc labs you measure magnetic field and
flux and so on, but the precision of this
instrument is extraordinarily high and that
is why this is a an instrument of choice for
anybody doing research in basic sciences or
material sciences or even in cases where you
want to find out extremely low magnetic fields
somewhere.
So, the accuracy the resolution is almost
like 10 to the power minus 14 Tesla and this
is just remarkable and how does it work. It
works very simply just as we said that the
Josephson effect it depends on the difference
of the two phases right. And when there is
a magnetic field of course, there is an additional
phase that comes in which is just 1 by phi
not A dot dl right, A is the vector potential.
So or 2 pi by phi not the way you write typically
is sorry. So, this is the magnetic additional
theta change in theta that creeps in because
of the magnetic field.
And the instrument can sense this at this
change in phase because the current depends
on the of the phase and that is now, that
is converted into a voltage here in this instrument
called a superconducting quantum interference
device or SQUID. This device is shown here,
there is two junctions it is a two junction
SQUID. So, these are two Josephson junctions
and the current gets divided into 2 and then
you can just work out.
And you will see that the enclosed flux if
you have a magnetic field B that is perpendicular
to this plane or I mean the if it is not perpendicular
you just take the perpendicular component
to calculate the flux times the area and then
you will get this gives you the value of the
phase difference between the two leads. Suppose
initially they were there were no phase differences
and once you put this magnetic field due to
this vector potential the flux threading through
the ring you will get a change in this and
change in the phase is just 2 pi phi by phi
naught which is what is I have written here
2. So, this is the change.
So, that is what one basically measures and
then you just see that there is a kind of
an interference pattern because of the because;
obviously, the once the flux goes through
this integer multiples of 1 2 phi by phi naught
is 1 2 3 4 at these points there are minima,
you can look at this expression this is the
I max is twice I c cos of pi phi by phi naught.
So, as a function of phi which means the magnetic
field times the area you will get a get this
oscillatory pattern and from that you can
figure out what phi is.
It is actually done this way the device you
converts this into a voltage. So, it is the
magnetic field to voltage transit transducer
in some sense. So, flux to voltage transducer.
And you measure the voltage across this inductively
inductance and this basically between these
two points. And then it induces another voltage
here you can multiply amplify it and this
output voltage is what is measured and from
that one can read of the magnetic field or
change in field.
So, that is the device and this device is
so important and the theory is fairly simple
that this is one of the most used magnetic
measurement technique in all of all over the
world and you can go to any lab in a so, we
seen more or less a research institute any
research institute or universities or and
then or IITs or somewhere and you will find
this instrument and see how it works ah.
It has actually revolutionized the way we
do magnetism in these days it of course, got
a Nobel Prize also Brian Josephson got Nobel
Prize with two others.
Now, the very recently you must have heard
this quantum supremacy and all that there
is a lot of noise there is a paper in nature.
Then of course, question is what is quantum
computing I will not get into that is not
a subject which we are studying here, but
I will just outline what it is because it
uses this one uses solid state techniques
to generate this so, called qubits. So, and
the technique one uses in one of the techniques
one uses is using Josephson junctions.
So, you see the use of Josephson junction
it can it is basically a quantum interference
device and so it can be used to generate qubits
to two different states which between which
you can you have a superposition. So, I will
not get into the details the quantum computers
is you must have heard it so many times, it
is a buzz word and people are pursuing it
and they use basic quant like atoms, photons,
spins to generate these states quantum mechanical
states.
And then unlike in the classical case where
0 and 1 are the only two states you need in
a in a bit classical qubit a classical bit
you here you can actually have a superposition
between the these two. So, it is a some if
alpha 0 plus beta 1 kind of divided by root
over alpha square plus beta square divided
by these so, you normalize it by these alpha
square plus beta square.
So, that kind of so, here for example, the
1 the 2 coefficients are 1 and 1. So, the
normalization is 1 by root 2 this is simple
superposition of 2 states 0 and 1. So; that
means, you have now if you have n such qubits
then you can have 2 to the power n such states
to play with and that is those states have
to be generated they have to be stabilized,
they have they should not become decoherent.
So, that is where the trouble is and of course,
then you have to manipulate them also read
and write and so on.
So, those I will not get into, but the only
reason we are discussing this is that one
of the methods of generating this is using
superconducting qubits micrometer or less
size Josephson junctions as qubits. So, the
it is just as he says if there are equal number
of qubits and regular bits which means classical
bits then the qubits will hold twice the information.
That is n qubits in a superconductor in a
superconductor will have 2 to the power n
different states.
So, experimentally it can hold much more information
compared to a regular digital qubit that we
nowadays use and so, the speed of the system
increases exponentially because it is a 2
to the power to the power something. So, it
is like an exponent exponential increase.
So, the these the ones which are suggested
long back not very long, but in the late 1990s
and early 2000 was this from and this is came
out in science and this is people are trying
to use it. And so, quantum supremacy claims
that they have this they refer to this paper
as source as how they generate the qubit the.
So, I mean as a possible route to generate
the qubit.
So, this is simply three junction three Josephson
superconducting three and four junction qubits.
So, in as you can see here you have two directions
of current and you can linearly super superpose
them. So, this superposition is at the heart
of quantum computing and the elementary unit
is a two state quantum system called a qubit
which I am mentioning so far so, long.
Computations are performed by the creation
of quantum superposition states of these qubits
and by controlled entanglement of the information
on the qubits. So, that control is a bit non
trivial and the whole system has to be kept
in very low temperatures so, few milli Kelvin's
apparently. And so, this is how the basic
architecture of the qubit works in a quantum
computer of course, the algorithm and other
things are very different ball game and that
we will not discuss here right.
This is a now a separate course to discuss
quantum computing and it is algorithms and
so on the mathematics of it. But at the core
of it we are these Josephson junctions as
a possible qubit. This is a 4 junction qubit
on the right and. So, you can create more
states and entangle them.
So, of course, you have to prevent decoherence
as I said. So, you they typically work at
extremely low temperatures like milli Kelvin.
So, it has to be completely isolated from
the from fields and other things from outside.
So, these are things that you have to go you
have to take care if you are doing a quantum
computation at the level of devices, because
uncertainty principle will become important
at that level.
And so, you cannot simultaneously measure
conjugate properties besides to try to measure
you know that there is collapse that can happens
so, the system can go to a state collapse
to a state. So you have to have a measure
properties without disturbing the system much
and so on ok.
So, those are details I will not get into,
these are applications where the latest one
has a is a on 22 of October 2019.
Google has announced in a nature paper they
have announced also in a press conference
there is a machine called quantum supremacy
which is a programmable superconducting processor
which uses programmable superconducting processor.
So, they have been able to entangle states,
retrieve states and entangle them in a way
they want I mean. So, that is remarkable achievement
if it is a really achieved and of course,
the at the heart maybe these superconducting
Josephson junctions as the qubits. They have
used 53 of them to do the computation so,
2 to the power 53 is the number of states
there.
They claim that the sycamore processor which
is which they called sycamore processor was
able to perform a calculation in 200 seconds
that would have taken the world's most powerful
supercomputer 10000 years. This was immediately
disputed within a few days and IBM claimed
that it is the they can do the computation
even plus even with a usual regular computer
they are much much lesser time 2.5 days or
so. So, no matter what it is it is still an
achievement and it is a first step towards
doing many other companies and research institutes
are engaged in it.
And this is the picture of the quantum supremacy
basic architecture these 53 qubits. So, it
has a dimension of 2 to the power 53 state
space. So, that is where the those are the
states that are manipulated, that is about
10 to the power 16 which is enormous I mean
this is just really very very big. Of course,
to have a real quantum computer one needs
to go to much much larger state space typically
10000 qubits are a possibility that people
are talking about and that is where the real
difference will come up will show up.
So, these are these couple Josephson junctions
as you can see these are problem these are
the you needs. The processor is fabricated
using aluminum for metallization and Josephson
junctions and indium for bump bonds between
two silicon wafers. The chip is wire bonded
to a superconducting circuit board and cooled
to below 20 milli Kelvin in a dilution refrigerator.
The processor is connected to filters and
attenuators to room temperature electronics
which synthesizes the control signals. So,
that is how the states are manipulated the
read and write and information. So, the qubits
can be read simultaneously the state of all
the qubits can be read simultaneously by using
some multiplexing techniques which is a technical
issue, but this is at the heart is are these
Josephson junctions that we have just studied
and that is the reason they are so important
is because they are extraordinarily sensitive
and they are quantum interference devices.
So, they are natural choice for qubits. So,
this is all superconductivity that we will
do there are lot more this is a fascinating
subject, there is this new high DC superconductors
which have dramatically changed our perception
of super conductivity and they are the possibility
of possibilities are endless in fact, every
other day new materials are coming up and
their order parameter is an interesting issue,
because the psi in those new superconductors
may have may not be isotropic that it is not
a constant gap is not a constant.
And therefore, there are nodes in the gap,
there are 0s in the gap, which are fascinating
physics connected to them. And those who are
interested can look up some of the literature
in there are there are n number of literatures
in large number of literatures available on
high temperature superconductor and that is
you are welcome to delve into it.
