`- [Tutor] And all right.
So I think it's good that
we spent a lot of time
on questions and you know,
the point here is not to
necessarily finish every single
slide and rapid fire at you
just like in the first lecture.
The point is that you take
something home from this
and you learn at least the concept,
you know, you might.
For the specialist in
the audience of course
the equations might be more interesting
but I also want to convey
some of the physical
and conceptual ideas of
how things generally work
and function and jiggle about and,
you know, how fun the quantum world is.
Alright, so let's get back
to the main part of this
which is what we do
with the Transmon qubit.
We have now broken up the
Transmon very neatly into
a linear piece and a nonlinear piece.
Let me expand the nonlinear piece
and let's try to look at what happens
assuming that this fluctuations phi ZPF
are very small, right?
Quantum fluctuations are usually small.
So it makes sense we
should say they're small.
And if you remember what that
corresponds to is the width
of the lowest energy state
which will come here.
If we make those fluctuations very large,
then that ground state, right?
If we make those zero
point quantum fluctuations
very very large,
then that ground state will
look really funny in here.
And you notice that some
weird stuff will happen
that it will start to
leak into nearby regions.
So we don't necessarily want to do that.
So let's look at this fourth order term.
Okay, so if I just look
at the nonlinearity,
it looks like negative EJ.
And then we have, zoom in a bit.
We have negative EJ phi
ZPF to fourth power.
This is really just some constant.
So I'm not even gonna worry about it.
I'm just going to leave it aside
and focus on a plus a
dagger to the fourth power.
And here I'll drop the hats
for notational simplicity
for the moment so I can
write a little quicker.
So if I expand this part of,
and I said I'll drop the
hats so let's drop them.
If I expand this term, you
notice of course that we have
just four copies of the exact same
ladder creation and animation operators.
If you multiply these out,
you can see that a lot
of terms will pop up
such as for instance, let's
start with the first one.
So we have aaaa, then
we can have aaaa dagger,
then we can have plus aaa
dagger a plus et cetera, right?
So there's quite a few terms.
There's a rather large multiplicity.
Now we're going to do a very
important and convenient trick
which is that every time
we see a term such as
let's take this term for instance,
let's take the term which is called
which is aa dagger aa.
A term that has a lowering
operator on the left side
of a raising operator.
We're going to move
through the raising
operator over to the left.
And the reason for that
is that if you remember
a applied to a Fock state n
produces a very simple result
and if we can write everything
in terms of products of
a dagger to some power m,
a to some power n called normal order.
We can very quickly and
easily evaluate the expression
applied onto a state such as this.
So let me give you an example here.
Suppose we have exactly
this term which goes like
aa dagger aa,
we can use the commutation relationships
between a and a dagger,
which commute to exactly one.
They're scaled precisely
to have this very nice
and easy relationship.
Oops, this is the commutator.
We can use repetitively the fact that
aa dagger is equal to one plus a dagger a.
And so substituting this
expression back into here.
I'm gonna apologize for the
extra long red line there.
Let's do that.
Substituting that back in there
and just focusing for
the moment on this term,
we can replace aa dagger
with a dagger a plus one
times aa.
Now you notice what has happened
is that a four body term,
aa dagger aa.
This is, if you want,
we like to call that four-wave mixing.
This four-wave mixing term
has now produced two terms.
It has produced a term
that goes like a dagger aaa
or in other words, a dagger a cubed
plus another term which is
just aa which is a squared.
And so we've broken up
a four body interaction
into two pieces.
A piece that is again
a four body interaction
that it has ignored the
commutation relationships.
And you can get purely classically
and a term that would not
exist in classical physics
when classical considerations that came
purely from the noncommutivity
of these operators.
So this is a non-classical term, right?
Whereas this is just the four-wave mixing.
Now you might be wondering,
why do I call these wave and wave mixings?
Well recall from the first lecture
or from the beginning of this lecture that
in phase space,
If I write the operator phi and Q here
and I look at alpha.
The trajectory of the oscillator, oops,
simply rotates in a circle like this.
And the rotation as a
function of time of alpha.
Let's maybe write it up here.
This evolution is governed
by alpha of t is equal to
alpha of some initial condition E
to the negative i omega not
t where omega not is the
frequency of the linear oscillator.
In the quantum case,
we similarly had that a hat of T
which is the Heisenberg
picture evolution of the
Masonic operator is equal to a hat
in the Schrodinger picture
or a hat at time zero,
if you want, times e to
the minus i omega not T.
Again this kind of semi
classical correspondence.
Now we can use this,
we can use this type of
picture analysis to think
of these operators up here
as waves that rotate, right?
They have some amplitude a
and they rotate at a particular frequency
E to the i omega not.
You'll also notice that if we
dagger one of these operators
then we dagger the initial operator here
and we complex conjugate the exponential.
In other words,
a rotates to the left and a
dagger rotates to the right.
So it represents motion
in the opposite direction.
If I substitute this dynamic
picture back into the
equations above for the terms,
you notice the terms that
as that look like say
a dagger a cubed in this picture
we'll combine to say a dagger
e to the plus i omega not t
times a cubed e to the
negative three i omega not t
which we can then say is
equal to a dagger a cubed
and then we can combine
the two oscillations into
e to the negative two i omega not t.
So you'll notice that this
term remains facilitatory
at the frequency two i omega not t
or the frequency is two omega not.
Now consider a term that looks different.
A term that looks like this,
a dagger squared, a squared.
Again, these are all the
possible terms that can come
from the expansion of the
fourth power of a plus a dagger
which seems simple enough,
but you'll notice that it creates
every possible combination
between a and a dagger and
with some multiplicity, right?
So let me just focus now on another term
in order to isolate the
different kinds of behaviors
you can have.
Well, this term, if you
think of it in this picture,
looks like a hat, let's
redraw that it looks like
a dagger e to the plus
i omega not t squared
times a e to the minus
i omega not t squared.
And so this looks like
the mixing of two waves,
a dagger squared a squared
times e to the a plus two I omega not t
e to the minus two i omega not t.
And you'll notice that
there's an interference here,
an interference which
precisely cancels out
the rotation of the
dagger with the a dagger.
And so this term is stationary.
This term doesn't
oscillate in this picture.
And so i can write this out
as a dagger squared a squared.
And so we can classify these
terms into two categories,
terms the do oscillate and
terms that don't oscillate.
And so I'll highlight
this in a green color
to indicate that this term
contrary to say the one that doesn't have
an equal number of a daggers and as,
this is a nonrotating term
whereas the one above is a rotating term.
Now you might be starting to wonder
why am I trying to identify
rotating and nonrotating terms.
And that's going to be along the lines of
pushing us into a very
important approximation
called the rotating wave approximation,
something you covered in lab yesterday.
And that approximation will say that well,
if I have a Hamiltonian,
which has terms that are stationary
or maybe I should just
write this as some H not,
which does not depend on time,
plus Hamiltonian that is rotating
at some interval t to
a first approximation.
Again, this is just an approximation
that works well If I assume that
this oscillation here is very fast and
the amplitude of H rotating is very small
to a first approximation.
I can say that this
Hamiltonian is approximately
equal to just H not.
I can essentially drop
all the rotating terms.
And I think Nick might have covered
should have covered this in more detail
in yesterday's lecture.
Now, I know we've kind of jumped into
a very few specific terms.
So let me go back to the
lecture and back out and look at
how does this term expand and
what are all the expansions.
So if you sit down and perform
exactly what we just did,
which is to take all of
the possible combinations
of the expansion of a plus a
dagger to the fourth power,
you get a zoo of terms.
And the zoo of term terms
comes into first two flavors.
The first are for body mixing terms,
which I've highlighted
here in purple, right?
And these are the terms that
you would obtain classically.
And also you get all the terms that result
from the commutation of a and a dagger.
And these are terms that appear strictly
because of the noncommutivity
of these two amplitudes.
Because of the noncompetitive
of these two amplitudes,
you also get some more zero
point fluctuation terms,
but this is a constant
stationary energy terms.
So we can always just drop that.
Now, notice that we can further subdivide
these terms into two pieces.
Pieces that oscillate at
frequencies, say two omega not
and pieces that oscillated frequencies
negative to omega not
and pieces that don't oscillate,
these that are stationary.
That similarly holds here, you know,
this a dagger to the fourth term in
the Heisenberg picture will
oscillate at a frequency of
say four omega not minus four mega not.
This term will oscillate at
negative three omega not, sorry,
two omega not.
This term here will
oscillate at no frequency,
in other words, it doesn't
oscillate, it's stationary term.
And of course this term will
oscillate at two omega not.
And this term will oscillate
at just four omega not.
The frequency omega is just given by the
difference in the number of
creation versus number of
annihilation operators.
And so what we can now say is
to very much in the same
way you did in your lab,
we can approximate, let's run this.
We can approximate all
the terms that, sorry,
do the rotating wave approximation
which is equivalent to four
story to probation theory
and to remove all the oscillating terms.
Again using the idea
that the ladder operator
rotates at this very particular frequency.
So if we simplify this Hamiltonian now.
Dropping all those terms,
it reduces to something
beautifully simple.
We get two contributions.
We get a contribution that we recognize
from the harmonic oscillator part.
That's just harmonic approximation.
We get a term that looks like
the harmonic oscillator energy
that's quadratic in energy, it's bilinear,
it's a dagger a that
comes exactly strictly
from the nonlinearity and
a term that is four body
or four-wave mixing term that
is new, that is nonlinear.
We haven't seen this term yet.
This will be crucial.
The first,
and so if we move down to this line,
the second term is due
to the non-linearity.
The first one is due to this piece here.
This Delta Q is due to the
commutation relationships
that arise from the nonlinearity.
Now what are these alpha and Q,
I've simply taken the
coefficients from the line above
and defined H bar Delta Q to
be actually the same value is
H bar alpha here.
And that equals one half
the Joseph Junction energy
times the zero point quantum fluctuations
of the reduced magnetic
flux to the fourth power.
Okay, and this is a crucial relationship
because on the left what we
have is the renormalization
of the linear frequency
of the oscillator due to
the zero point quantum
fluctuations, right?
The way we can interpret the
term over here on the left
is that this piece
would only exist because
a and a dagger don't commute,
which represents the quantum
zero point fluctuations.
And the amplitude of this
term is exactly dictated
by the size of the nonlinearity times
the amount of fluctuations of the mode.
Now, because it's a nonlinear
mode when it fluctuates
the fluctuation activates
the nonlinearity.
So as the ground state is fluctuating.
And some of the times it's sort of,
if you want climbs up here that
climbing up to a higher
potential allows it
to explore the nonlinearity
at a higher level.
And so that in effect then
addresses the frequency
of the qubit because
the particle is experiencing
a softer opening.
Provide this energy
landscape here opens up
the cosine opens up
relative to the quadratic
and so it lowers the frequency.
The second term here is the caton,
and that's going to be very crucial
as we'll see in the next slide.
So I can rewrite the equation
from the previous slide
using the notation N is a dagger a,
where N is the Fulton number operators.
So it says how many Fock states of
the classical harmonic oscillator
do I have in two pieces.
The first piece is using omega qubit and
which I'll define as the
dress frequency of the qubits.
So it's omega not,
that's the frequency of the
linear part of the qubit
minus Delta Q
without the Q is the lamp shift
or the dressing of the
frequency by the nonlinearity
and H not linear is the
non linear piece of this.
Now, the beautiful thing is
that we know the eigenstates
of the linear piece from
the harmonic oscillator,
our favorite bread and butter ingredient.
This nonlinear piece
has a very interesting
effect in that the
nonlinear piece goes like
N times N minus one.
And to first order,
it's dispersivity doesn't
change the eigenstates.
So maybe for the slightly
more familiar with this
with quantum physics,
if you apply first order
perturbation theory,
the energy change to
H lin due to H nonlin,
the nonlinear piece is
calculated by the overlap
between the eigenstate of H
lin or H zero in this case.
And that will be a correction
given by this term,
which we can also identify
easily because we can write
N times N minus one applied on
the Fock state N is equal to
N the number of excitations
times N minus one
times the Fock state N, right?
And that's because N hat
is equal to a dagger a.
Now, if you're wondering,
how did I jump in the previous
slide from a term that is
in the form of a dagger squared
a squared to this term here,
that's by again, using the
commutation relationships
but in the reverse order, right?
So we can write this term out as follows
using the commutation
relationships, I can again,
reverse the term and swap it out
so as to allow us to
get the aa dagger there.
Now, this more important piece
is that the eigenstate of the
harmonic oscillator, which
I'll label here's K zero,
this is the expression for
the perturbation theory
for the change in the
eigenstate, say K is zero,
the simple harmonic oscillator,
which are the eigenstate of H lin,
how those are affected
by the nonlinearity.
This is the first order correction
is given by how H nonlinear
links one Fock level,
maybe N to another Fock level,
maybe M and you notice that
because the composition of H nonlin is
only a dagger a terms, a dagger a, oops,
doesn't link a dagger a is only nonzero
in the case where M and
N, the two states of links
are the same.
So to first order perturbation theory,
the eigenstates don't change.
And that you can also visually
see in saying that, well,
you know,
to first order this purple
curve and this brown curve,
or basically the same,
it's only when you go to higher
excitation or higher order,
that the two curves really
look different, right?
So we can rewrite the energy within
the rotating wave
approximation of the transmon
to fourth order in the power
expansion for Fock States
like this.
And so what does this
physically begin to look like?
Well, we can see that,
let's look at some
experimental parameters.
So here's a device that I designed
and measured some time ago,
and it has the following real parameters.
The capacitance between
this pad and this pad,
which maybe we can write
like this is very small.
It's 65 femtofarads.
It's a pretty small number
but very typical for
these kinds of devices.
And then the inductance associated with
this nonlinear inductor,
the Josephson Junction here.
This inductance has a
value of 14 nanohenrys,
again a very small inductance
in the grand scheme of things.
Now these correspond to a
Josephson Junction energy
of about 12 gigahertz
and maybe for the expert in the audience,
this is a so-called charging energy,
the energy associated with the charge,
the momentum variable
is about 0.3 gigahertz.
So if we now look at how the
eigenstate of the oscillator
looked like to first order,
we can plot the oscillator eigenstates
familiar to us from the first lecture
here where the wave
function of the ground state
is a function of magnetic
flux phi is depicted here.
Again it has a mean position
of zero and it has a width,
a standard deviation of phi ZPF.
This amazing, very often
appearing parameter.
Of course, then this is
the one excited state,
two excited states and so forth.
Now, our approximation breaks down
as you get to very high states,
you notice that the harmonic
oscillator eigenfunction
for the fourth Fock state here
is bounded within the parabola
and not within the cosine.
So these eigenstates are much
more heavily renormalize,
but again, our interest is in a qubit.
So we're going to really,
almost exclusively focus on
what's happening down here at
the bottom between these two levels.
And you notice that the energy difference,
the splitting between the
ground and first excited level,
and for this qubit is
exactly about five gigahertz.
This was by design.
However, the energy difference
between the next two levels
and alpha here is the change, sorry.
So I guess maybe the way that
I should really write this is that
the difference between the energy of
the second level and the
energy of the first level
is equal to omega qubit minus alpha.
Whereas the difference between
the energy of the first level
and the energy of the ground
level is equal to omega qubit.
And this is precisely what we
set out at the very beginning
of our talk yesterday
that we wanted anharmonic level diagram,
a level diagram where the
levels are not equally spaced.
And so we have finally
shown from the beginning,
from almost first principles,
how you obtain this anharmonicity,
this non equal spacing
between the first two levels
and the second two levels and
the extra non-harmonicity here
is about 0.3 gigahertz
relative to five gigahertz.
So it's about a 10 ish percent correction.
Yesterday someone asked, well,
what is the impedance value
of these oscillators typically
or this transmon?
So for this transmon you notice
that it's about 450 amps.
So it's relatively large
cause this is an impedance
measured at 5.3 gigahertz.
And it's very difficult
to get above a kiloamp.
To first order, of course the
eigenstates haven't changed.
Now, if we look at the
zero point fluctuations,
which I alluded to earlier,
the zero point quantum
fluctuations of the charge
the corresponds to these real parameters
very much for a device like
the one you might measure in
in your just get experiments
are about one Cooper
pair or two electrons.
So in the ground state,
there is about exactly one
Coop pair worth of charge,
kind of fluctuating if
you want back and forth
from one island to the other island.
And then the magnetic flux is oscillating
at the level of about half
a magnetic flux quantum.
So very much in the same way that
two electrons is a
fundamental scale for charge,
the magnetic flux Quantum,
if you remember phi not was
Planck's constant reduced
divided by 2E.
This is the scale for flux.
And that one electron is a very
small number, because again,
there are about 10 to
the 12 mobile electrons
in each side of this transmon qubit.
So you can see why we have to work
at very very cold temperatures, right?
This explains in part why the environment
must be very pristine,
why we hope that things are
very isolated from noise,
because think about it.
Our fluctuations of the ground
state are one Cooper pair,
the fluctuations of the
excited state are roughly
two Cooper pairs.
And to distinguish that we need to,
we need to have the noise, not swamp,
one out of 10 to the 12.
That's a very minute number
but amazingly this microscopic
effect is highly robust.
And we've been able to
engineer these systems
within the community
to be able to work with
these systems almost
as if they're isolated,
ideal, real atoms.
What does the energy
level diagram look like
a bit more explicitly
of this transmon qubit
and what are the experimental
signatures you might see?
So again, let's look at the Fock levels
or the energy eigenstates.
So this is the ground
state let's say here,
then this might be the excited state one.
Then I'll exaggerate here
and draw the second excited
state two like this.
And I see that we have,
we can do that, right?
And then the energy of the ground state,
let's just label it as zero.
The energy of the first
excited state we'll label as
H bar omega Q.
The energy of the second excited state is
H bar omega Q minus alpha.
And so when you drive a qubit,
when you control a qubit,
you tend to drive transitions
at this frequency omega Q.
And you notice that that
frequency is very very different
from the frequency required to
transition to the second level
either from the ground or the first,
which is omega Q minus alpha.
However, it doesn't mean that
if you start in the ground state
and you now excite yourself
to a first excited state here,
it doesn't mean that
you can't suddenly apply
another frequency toner
at this lower frequency
which can allow you to then
hop from this ladder level
to the next,
or to use two photons
to excite from the bottom
level of the ground state
using two quanta of energy that combine
and take you from zero directly to two.
Now that process is very difficult
in the sense of it's not
very likely to occur.
You need to drive with a ton of photons
and put in a lot of power
cause photons don't
like to combine in a way
but it's possible to occur.
And so what you would experimentally
measure in an experiment and
resolve spectroscopically
is to say that as a
function of the frequency
that I probate I might see a
transition line in the spectrum
that appears at a frequency here.
And fortunately it seems
to have erased my diagram
so we can recreate these
energy levels very quickly,
that appears at a frequency omega Q
that corresponds to the transition between
the ground state and
the first excited state.
So this would be omega zero one,
which we'll call omega qubit
and this I'll called omega zero one.
This will call omega zero two,
or it's omega qubit minus alpha.
And then you might see another peak
which appears slightly lower infrequency,
space by about a difference of alpha
the anharmonicity of the transmon.
And this is the difference
between, excuse me,
between one and two.
And you can also potentially
see a peak that looks like
the difference between the
second and third and so forth.
So these are experimental
signatures you can observe
and in your demo lab at home.
Oh, and I think we have
covered that yesterday.
So that's a reminder that
the annihilation, creation
and hopping operators
have these relationships.
And the reason that I'd like
to know remind us of that
is that we have just
explained that using frequency
we can now isolate the qubit
in this frequency range.
What that means is that if we
only apply control or pulses
or frequencies of our drives,
that tried to steer the
atom by force in this range,
we can really isolate
the zero one transition,
and we can isolate the
qubit as zero and one.
Now, if you remember what the creation
and annihilation operators look like,
we can therefore restrict
ourselves to operate within
a manifold of the lowest
two energy states.
And so the photon number
operator a dagger a
which looks like this and
my notation here means that
you know, let's take the term,
let's take term two.
This term of the matrix
corresponds to an operator
that takes the state one
and map it to the state one
with amplitude two.
That's what my notation here represents.
Let's say matrix element here
would take the state three,
excuse me,
it would take the state three
and map it to the state one.
So this is where it comes in.
This is where it goes
out with amplitude zero.
So it's zero.
I'd like to just indicate
the basis of that
this matrix is all written
and you can always,
of course swap the basis,
but it's important to keep
track of what you use.
And because of the anharmonicity alpha
which is roughly 10% of the frequency
is about an order of magnitude.
We can restrict ourselves to
just the lowest two states,
zero and one.
And therefore restrict all
of our operators of the
Hamiltonian Fulton creation,
the annihilation operator
to work just in the qubit subspace.
And that's what allows
us to then identify that
the first part of the Hamiltonian here,
the Fock creation operator
minus some arbitrary shift overall shift
restricts ourself exactly to
the qubit Pauli sigma Z operator.
And of course our
favorite ladder operator a
becomes the fermionic
annihilation or excuse me,
shouldn't call it from me
like the two state restricted
annihilation operator
or the Pauli lowering
operator, sigma minus,
which is a combination
of the qubit Pauli X
minus i qubit Pauli Y.
So these are of course
the matrix representation.
So we've just taken a leap
from an oscillator to a spin.
And I hope that reminds
you of the picture of
slide and the first part of talk
were I think I mentioned
there are only two types of physicists
and maybe you can see which one I tend to
be most of the time.
